Package {DistributionalDDA}

Type:	Package
Title:	Distributional Data Analysis in Bayes Spaces
Version:	0.1.1
Maintainer:	K. Gerald van den Boogaart <support@boogaart.de>
Description:	Statistical analysis of data sets whose observations are whole probability distributions, using Bayes space calculus. Distributions (provided as samples, histograms or density functions) are treated as elements of a Hilbert space, so that operations such as mean, variance, distance, principal component analysis and clustering carry over to densities. Reference measures defining the geometry of the space can be chosen and combined. The methodology was first published with peer review by van den Boogaart, Egozcue and Pawlowsky-Glahn (2014) <doi:10.1111/anzs.12074>.
License:	LGPL-3
Encoding:	UTF-8
LazyData:	true
Depends:	R (≥ 4.1), compositions, splines
Imports:	fda, methods, graphics, stats, utils, rlang, digest, MASS, Matrix, grDevices
Suggests:	knitr, igraph, ggraph, rmarkdown, testthat (≥ 3.0.0)
Collate:	DistributionalDDA-package.R topics.R helpers.R integration.R ListData.R classes.R converters.R Bayes_spaces.R Cartesian.R operations.R summaries.R plots.R simulations.R datasets.R linear_models.R PCA.R bulding_blocks.R statistics.R
VignetteBuilder:	knitr, rmarkdown
Config/testthat/edition:	3
Config/roxygen2/version:	8.1.0
RoxygenNote:	7.3.3
NeedsCompilation:	no
Packaged:	2026-10-08 14:59:46 UTC; tolosa53
Author:	Ivana Pavlu [aut], Raimon Tolosana-Delgado [aut], Matthias Templ [aut], K. Gerald van den Boogaart [aut, cre]
Repository:	CRAN
Date/Publication:	2026-10-20 15:10:02 UTC

DistributionalDDA: Distributional Density Data Analysis

Description

Tools for the statistical analysis of *datasets of distributions* — where each observation is a whole probability distribution rather than a single number — using **Bayes space** calculus. A Bayes space is a Hilbert space whose elements are distributions, so the usual operations (mean, variance, distance, principal component analysis, clustering) carry over to densities once a reference measure and a finite-dimensional basis are fixed.

Details

## Getting started — the typical workflow

1. Choose a **reference measure** that defines the geometry of the space and matches the support of your data: [ReferenceMeasureUniform], [ReferenceMeasureTransformed] (e.g. a Gaussian support via its CDF), or [ReferenceMeasureDiscrete].

2. Wrap it in a **Bayes space**: [BayesSpace] (predefined: 'B2Unif01', 'B2N01').

3. Pick a finite-dimensional **basis**: [BayesSpaceBasis] (predefined: 'B2Unif01Basis5', 'B2Unif01Basis7', ...). The default basis uses zero-integral B-splines (ZB-splines); see [getMetricTensor] for the resulting metric.

4. Put your data into a **BData** container that matches its form (see below).

5. Analyse: [mean][mean.BDataSplines], [mvar][mvar.BData], [msd][mvar.BData], [dist][dist.BData], [scale][scale.BData], [princomp][princomp.BData] (SFPCA), and the plotting methods.

## BData representations (which container to use)

[BDataSamples]

raw samples drawn from each distribution.

[BDataFrequencyHistograms] / [BDataDensityHistograms]

binned data.

[BDataPDFs] / [BDataLogPDFs]

densities w.r.t. the Lebesgue measure.

[BDataDensities] / [BDataLogDensities]

densities w.r.t. the reference measure.

[BDataSplines]

spline coefficients — the compact representation used for computation (PCA, distances, inner products).

Convert between them with the 'as.BData*' functions (e.g. [as.BDataDensities], [as.BDataSplines]); the usual path is Samples/Histograms -> Densities -> Splines.

## Key concepts

Bayes space algebra

In a Bayes space, addition of two distributions is the (closed) product of their densities, subtraction the ratio, and scalar multiplication a power transform. See [Add], [Sub], [Mul], [Div].

clr / closure

[clr] is the centred log-ratio transform into the L2 space; [clo] closes a density to integrate to one.

Gram matrix / metric tensor

The basis carries a Gram matrix 'S' with ‘S[i,j] = <psi_i, psi_j>'; inner products in coefficient space are 'c1’ S c2'. For ZB-splines 'S' is not the identity. See [getMetricTensor].

## Datasets

[age] (age distributions of Upper Austrian communes) and [DTD] (a simulated particle dataset) are included for examples.

Author(s)

Maintainer: K. Gerald van den Boogaart support@boogaart.de (ORCID)

Authors:

• Ivana Pavlu ivana.pavlu@upol.cz (ORCID)

• Raimon Tolosana-Delgado (ORCID)

• Matthias Templ (ORCID)

See Also

The package vignettes for worked examples: vignette("DistributionalDDA") and vignette("PCA_distributional_data").

Operator to act on a list

Description

The operator acts on all elements of a list and returns a list of the results. The right hand side can be a name in which case the '$' operator is used to e

Usage

X %.% expr

X %..% expr

Arguments

Value

a list, with results of the operations on the individual list elements.

Functions

• X %..% expr: same as '

Examples

data(DTD)
DTD$data %.% shape
DTD$data %.% summary(.$shape)
DTD$data %..% summary(.$shape)

Operations for BData objects

Description

BData objects are considered a elements of a BayesSpace and operated accordingly to Bayes space rules, using the functions [Add], [Sub], [Mul], [Div]. For operating with a scalar, please use the BData object first.

Usage

## S3 method for class 'BData'
x + y

## S3 method for class 'BData'
x - y

## S3 method for class 'BData'
x * y

## S3 method for class 'BData'
x / y

Arguments

Value

A BData object of the same class as 'x', resulting from the corresponding Bayes-space operation (addition, subtraction, scalar multiplication, or scalar division).

Functions

• - : Bayes space subtraction divides densities

• * : Bayes space multiplication works only with scalars

• / : Bayes space division only works with scalars and leads to do the y-th root

Examples

X <- BDataSplines(t(1:5),B2Unif01Basis5)
X+X

Compute the density relativ to a measure from histogram weights.

Description

The function does not normalize the density.

Usage

.DensityFromHist(
  weights,
  breaks,
  background = 0,
  dmu = function(x) rep(1, numberOfCases(x))
)

Arguments

Value

a function that can be evaluated on the support of the underlying measure (given by dmu) and provides the a density at every point.

See Also

Other .DensitFromHist: .LogDensityFromHist()

Examples

f = .DensityFromHist(1:5,0:5)
x = seq(-1,6,by=0.01)
plot(x,f(x),type="b")

Compute the log density relativ to a measure from histogram weights.

Description

The function does not normalize the density.

Usage

.LogDensityFromHist(
  weights,
  breaks,
  background = mean(weights)/(length(weights) * 10),
  dmu = function(x) rep(1, numberOfCases(x))
)

Arguments

Value

a function that can be evaluated on the support of the underlying measure (given by dmu) and provides the a density at every point.

See Also

Other .DensitFromHist: .DensityFromHist()

Examples

f = .LogDensityFromHist(1:5,0:5)
x = seq(-1,6,by=0.01)
plot(x,f(x),type="b")

Adds functional objects

Description

This additional is also able to add functions and BayesSpace objects. The sum of two functions is a function that results in the sum of the two results. The sum of two BayesSpace objects (in BData), is given by the BayesSpace operations, which is given by the product of the densities with respect to the reference measure.

Usage

Add(x, y)

## S3 method for class ''function''
Add(x, y)

## S3 method for class 'list'
Add(x, y)

## S3 method for class 'numeric'
Add(x, y)

## S3 method for class ''function''
Add.function(x, y)

## S3 method for class 'list'
Add.function(x, y)

## S3 method for class 'numeric'
Add.function(x, y)

## S3 method for class ''function''
Add.numeric(x, y)

## S3 method for class 'numeric'
Add.numeric(x, y)

## S3 method for class 'numeric'
Add.list(x, y)

## S3 method for class 'list'
Add.numeric(x, y)

## S3 method for class 'list'
Add.list(x, y)

## S3 method for class ''function''
Add.list(x, y)

## S3 method for class 'BDataDensities'
Add(x, y)

## S3 method for class 'BDataLogDensities'
Add(x, y)

## S3 method for class 'BDataSplines'
Add(x, y)

## S3 method for class 'BDataHistograms'
Add(x, y)

## S3 method for class 'BDataDensities'
Add.BDataDensities(x, y)

## S3 method for class 'BDataLogDensities'
Add.BDataLogDensities(x, y)

## Default S3 method:
Add.BDataLogDensities(x, y)

## Default S3 method:
Add.BDataDensities(x, y)

## S3 method for class 'BDataSplines'
Add.BDataSplines(x, y)

## Default S3 method:
Add.BDataSplines(x, y)

## S3 method for class 'BDataHistograms'
Add.BDataHistograms(x, y)

## S3 method for class 'BDataDensities'
Add(x, y)

## S3 method for class 'BDataSplines'
Add(x, y)

## S3 method for class 'BDataLogDensities'
Add(x, y)

## S3 method for class 'BDataPDFs'
Add(x, y)

## S3 method for class 'BDataLogPDFs'
Add(x, y)

Arguments

Value

A BData object of the same subclass as the first argument (the Bayes-space sum / difference / scalar multiple / quotient).

Methods (by class)

• Add(`function`): method for functions

• Add(list): method for lists, iterating over them

• Add(numeric): method for numerics, considering them as constant functions

• Add(BDataDensities): method for BDataDensities (effectively multiplying the densities as addition in the BayesSpace)

• Add(BDataLogDensities): method for BDataLogDensities

• Add(BDataSplines): method for BDataSplines

• Add(BDataHistograms): method for BDataHistograms

• Add(BDataDensities): BayesSpaceAddition for BDataDensities

• Add(BDataSplines): BayesSpaceAddition for BDataSplines

• Add(BDataLogDensities): BayesSpaceAddition for BDataLogDensities

• Add(BDataPDFs): BayesSpaceAddition for BDataPDFs

• Add(BDataLogPDFs): BayesSpaceAddition for BDataLogPDFs

Functions

• Add.function(`function`): method for adding two functions, doing it by creating a function adding their values.

• Add.function(list): method adding a function to a list, creating a list with entries for each list

• Add.function(numeric): method adding a constant to a function

• Add.numeric(`function`): method for adding a constant to a function

• Add.numeric(numeric): method for two numerics, simply adding the values

• Add.list(numeric): method for adding a numeric to every value of the list

• Add.numeric(list): method for adding a numeric to every value of the list

• Add.list(list): method for vectorized addition of list elements

• Add.list(`function`): method for adding a function to every element of a list

• Add.BDataDensities(BDataDensities): method for two BDataDensities

• Add.BDataLogDensities(BDataLogDensities): method for adding two BDataLogDensities

• Add.BDataLogDensities(default): method converting the second argument to a BDataLogDensity and then adding

• Add.BDataDensities(default): method converting the second argument to a BDataDensity object and then adding.

• Add.BDataSplines(BDataSplines): method for two BDataSplines

• Add.BDataSplines(default): method for BDataSplines default

• Add.BDataHistograms(BDataHistograms): method for BDataHistograms

See Also

Other Add: Div(), Mul(), Sub()

Examples

Add(sin,cos)
Add(list(sin,cos),sin)
Add(cos,list(sin,cos))
Add(list(sin,cos),list(sin,cos))
Add(list(sin,cos),10)

A distribution of distributions based a distribution family with random hyperparameters.

Description

A distribution of distributions based a distribution family with random hyperparameters.

Usage

B2DistHyper(
  density,
  basis,
  paramSimulator,
  ...,
  simulator = NULL,
  .hyperparameters = list(...)
)

Arguments

Value

A B2DistHyper object encoding the distribution.

Examples

D = B2DistHyper(dbeta,B2Unif01Basis5,function() rexp(2),simulator=rbeta)
simulate(D,4,seed=47)

A Gaussian Process distribution for probabilities

Description

A Gaussian Process distribution for probabilities

Simulate a Gaussian Distribution on B2

Usage

B2GaussianDistribution(
  mean = BDataLogDensities(list(function(x) rep(0, length(x))), basis = B2Unif01Basis7),
  var = varKernel(function(x, y) exp(-abs(outer(x, y, "-"))), basis =
    getBayesSpaceBasis(mean)),
  nEval = 513
)

## S3 method for class 'B2GaussianDistribution'
simulate(object, nsim = 1, seed = NULL, ..., nEval = object$nEval)

Arguments

Value

B2GaussianDistribution: a B2GaussianDistribution object which can be simulated with, simulate: BDataLogDensities-Object with nsim cases with the same BayesSpace and BayesSpaceBasis as the mean.

Methods (by generic)

• simulate(B2GaussianDistribution): Simulaton method

See Also

[

[stats::simulate]

Examples

m = BDataPDFs(list(dunif),basis=B2Unif01Basis7)
G1 = B2GaussianDistribution(m,function(x,y) exp(-(4*abs(x-y)^1.7)))
plot(simulate(G1,10,seed=47),col=2:11)

GD = B2GaussianDistribution(mean=BDataPDFs(\(x)x,basis=B2Unif01Basis5),
                            var=\(x,y) exp(-abs(x-y)))
plot(simulate(GD,5))
GD = B2GaussianDistribution(mean=BDataPDFs(\(x) dnorm(x),basis=B2N01Basis5),
                            var=\(x,y) exp(-abs(x-y)))
plot(as.BDataPDFs(simulate(GD,5)))

The standard B2(N(0,1)) Bayes space.

Description

B2N01 is the Bayes space B2(N(0,1)), and B2N01BasisN its B-Spline basis with N elements of order 2

Usage

B2N01

B2N01Basis5

B2N01Basis7

B2N01Basis20

Format

An object of class BayesSpace of length 1.

An object of class BayesSpaceBasis of length 5.

An object of class BayesSpaceBasis of length 5.

An object of class BayesSpaceBasis of length 5.

Functions

• B2N01Basis5: a basis for B2N01 with 5 elements

• B2N01Basis7: a basis for B2N01 with 7 elements

• B2N01Basis20: a basis for B2N01 with 20 elements

See Also

[BayesSpaceBasis]

Other ReferenceMeasures: ReferenceMeasureDiscrete(), ReferenceMeasureTransformed(), ReferenceMeasureUniform(), StdReferenceMeasureUnif01, expectation(), getSample(), isInDomain()

Examples

getReferenceMeasure(B2N01)
getMetricTensor(B2N01Basis5)

Distributional data containers (BData)

Description

'BData' is the base class for the data containers in DistributionalDDA. A 'BData' object holds a data set of distributions together with the [BayesSpaceBasis] in which they are represented. Several subclasses store the same information in different forms; pick the one that matches how your data arrive, then convert to splines for computation.

Details

# Which representation should I use?

Raw samples

[BDataSamples] – one numeric sample per distribution.

Binned data

[BDataFrequencyHistograms] (counts) or [BDataDensityHistograms] (densities).

Density functions w.r.t. the Lebesgue measure

[BDataPDFs] / [BDataLogPDFs].

Density functions w.r.t. the reference measure

[BDataDensities] / [BDataLogDensities].

Spline coefficients

[BDataSplines] – the compact representation used for computation (PCA, distances, inner products).

# Conversion pipeline The 'as.BData*' converters move between representations. The usual flow is

  Samples / Histograms  ->  Densities  ->  Splines  ->  PDFs 

Converting towards densities/splines requires a basis and is an approximation (the data are projected onto the finite-dimensional spline space); extracting samples or re-expressing a representation as itself is lossless. For most analyses, convert to [BDataSplines] with [as.BDataSplines].

See Also

[BayesSpace], [BayesSpaceBasis], [ReferenceMeasure], [as.BDataSplines], [as.BDataDensities]

Constructor of BDataDensities objects representing a Dataset of Densities in a BayesSpace

Description

The dataset is represented by a list of functions, associated with a Bayes space. The basis is used for functions relying on projections, e.g. when the data is converted to a BDataSpline object.

Usage

BDataDensities(f, basis, ...)

BDataPDFs(f, basis, ...)

## S3 method for class 'list'
BDataLogDensities(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'list'
BDataLogPDFs(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class ''function''
BDataDensities(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class ''function''
BDataPDFs(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'list'
BDataDensities(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'list'
BDataPDFs(f, basis, ..., close = FALSE, closed = FALSE, centered = FALSE)

Arguments

Details

BDataDensities represent densities (aka as Radon-Nikodym derivatives) with repect to the reference measure of the BayesSpace.

BDataPDFs for (Probablity Density function) represents densities with respect to the Lebesgue measure (Haar measure) of the support. I.e. the measure that is invariant under shift or permutation of the space. BDataPDFs are only reasonable if the reference measure is absolutely continuous, i.e. itself has a finite Lebesgue density.

For the special case of uniform reference measures BDataDensities and BDataPDFs of the same function, represent the same measure.

Value

An object of class 'c("BDataDensities","BData")' representing the densities with respect to the reference measure.

Methods (by class)

• BDataDensities(`function`): Method for a single function

• BDataDensities(list): Method for a list of functions

Functions

• BDataPDFs(): Object representing Lebesgue densities

• BDataLogDensities(list): Method for list of functions interpreted as log densities.

• BDataLogPDFs(list): Method for list of functions interpreted as log probability density functions (PDFs).

• BDataPDFs(`function`): Method for a single function

• BDataPDFs(list): Method for a list of functions

See Also

[BayesSpaceBasis]

Other BData: BDataDensityHistograms(), BDataLogDensities()

Examples

BDataDensities(list(function(x) x, function(x) 1-x),B2Unif01Basis5)

Data container for distributions as histograms

Description

Simple S3 constructor of a data container for distributional data where each distribution is represented as a density histogram.

Usage

BDataDensityHistograms(matrixDensities, breaks, basis)

Arguments

Value

An object of class 'c("BDataDensityHistograms","BDataHistograms","BData")', with the matrix densities as main value and 'breaks' and 'basis' as attributes.

See Also

Other BData: BDataDensities(), BDataLogDensities()

Examples

set.seed(47)
breaks = seq(-3,3,by=1)
inRange = function(x) -3<x & x<3
X = t(replicate(10,hist(Filter(inRange,rnorm(1000)),breaks=breaks,plot=FALSE)$density))
hists = BDataDensityHistograms(X,breaks=breaks,basis=B2N01Basis5)
plot(hists)

Data container for distributions as histograms

Description

Simple S3 constructor of a data container for distributional data where each distribution is represented as a (absolute) frequency histogram.

Usage

BDataFrequencyHistograms(matrixCounts, breaks, basis)

Arguments

Value

An object of class 'c("BDataFrequencyHistograms","BDataHistograms","BData")', with the matrix counts as main value and 'breaks' and 'basis' as attributres.

See Also

Other BDataHistograms: as.BDataDensityHistograms(), as.BDataDensityHistograms.list(), as.BDataFrequencyHistograms.list(), getBreaks(), plot.BDataHistograms()

Examples

set.seed(47)
breaks = seq(-3,3,by=1)
inRange = function(x) -3<x & x<3
X = t(replicate(10,hist(Filter(inRange,rnorm(1000)),breaks=breaks)$counts))
hists = BDataFrequencyHistograms(X,breaks=breaks,basis=B2N01Basis5)
plot(hists)

BayesSpace data in different formats

Description

A BDataGeneral object is essentially a list of different representations of BayesSpace elements of the same BayesSpace. They are treated individually as long as possible.

Usage

BDataGeneral(
  data,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(data),
  centered = isCentered(data)
)

Arguments

Value

An object of class 'c("BDataGeneral","BData")': a list bundling the several distributional representations for a common Bayes space.

Examples

X = BDataGeneral(list(sin,hist(runif(100)),BDataSamples(list(runif(100)),basis=B2Unif01Basis7)),
             basis=B2Unif01Basis5)
Xs= as.BDataSplines(X)

Constructor of BDataLogDensities objects representing a Dataset of log Densities in a BayesSpace

Description

The dataset is represented by a list of functions, associated with a Bayes space. The basis is used for functions relying on projections, e.g. when the data is converted to a BDataSpline object

Usage

BDataLogDensities(f, basis, ...)

BDataLogPDFs(f, basis, ...)

## S3 method for class 'BDataLogDensities'
BDataDensities(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'BDataLogDensities'
BDataDensities(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'BDataDensities'
BDataLogDensities(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'BDataLogPDFs'
BDataPDFs(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'BDataPDFs'
BDataLogPDFs(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

Arguments

Value

An object of class 'c("BDataLogDensities","BData")' representing the log-densities with respect to the reference measure.

Methods (by class)

• BDataLogDensities(BDataDensities): conversion method

Methods (by generic)

• BDataDensities(BDataLogDensities): Method for a list of functions

• BDataDensities(BDataLogDensities): conversion method

Functions

• BDataLogPDFs(): Object representing Lebesgue densities

• BDataPDFs(BDataLogPDFs): conversion method

• BDataLogPDFs(BDataPDFs): conversion method

See Also

[BayesSpaceBasis]

Other BData: BDataDensities(), BDataDensityHistograms()

Examples

BDataDensities(list(function(x) x, function(x) 1-x),B2Unif01Basis5)

BData container with distributions as samples

Description

Constructs a BData object where each distribution is represented as a sample. Such a sample can be a vector of values (for univariate cases) or some data structure convertible to a "data.frame" (for multivariate cases).

Usage

BDataSamples(datalist, basis)

## S3 method for class 'BDataUnivariateSamples'
summary(object, ...)

## S3 method for class 'BDataSamples'
summary(object, ...)

## S3 method for class 'BDataUnivariateSamples'
x[...]

## S3 method for class 'BDataSamples'
x[i, j]

Arguments

Value

In multivariate cases, the outcome is an object of class 'c("BDataSamples","BData")' encoding a list of data.frames of the same number of columns. In univariate cases, the outcome is an object of class 'c("BDataUnivariateSamples","BDataSamples","BData")' encoding a list of vectors of samples of each individual distribution.

Methods (by generic)

• summary(BDataSamples): summary method for multivariate cases

• [: subsetting method for multivariate samples

Functions

• summary(BDataUnivariateSamples): summary method for univariate cases

• [: subsetting method for univariate samples

Examples

x = lapply(rep(1000,4), runif) |> BDataSamples(basis=B2Unif01Basis5)
summary(x)

create a 'BDataSplines' object: a set of distributions represented as splines; possible representation of individual distribution through a vector of coefficients.

Description

create a 'BDataSplines' object: a set of distributions represented as splines; possible representation of individual distribution through a vector of coefficients.

Usage

BDataSplines(matrixCoefs, basis)

## S3 method for class 'BDataSplines'
x[i, j, ...]

Arguments

Value

object of class 'BDataSplines' describing a set of distributions represented as splines

Methods (by generic)

• [: subsetting method, keeps structure if only i is given.

See Also

BData

Other BDataSplines: plot.BDataSplines()

Examples

coefs=matrix(1:15,nrow=3)
bssplines=BDataSplines(coefs,B2Unif01Basis5)
bssplines

Creates the Bayes space for a reference measure

Description

A Bayes Space is a Hilbert space of distribution and likelihoods having a non zero density with respect to the reference measure (which needs) to be a distribution itself.

Usage

BayesSpace(referenceMeasure = ReferenceMeasureUniform(0, 1))

Arguments

Value

An object of class '"BayesSpace"': a list holding the reference measure.

See Also

[ReferenceMeasure]

Other BayesSpace: L2Space()

Examples

bs = BayesSpace(ReferenceMeasureUniform(0,10))
bs
X = BDataSamples(replicate(10,runif(100,0,10)),basis=BayesSpaceBasis(bs))

Create a basis for a Bayes space

Description

Creates the finite-dimensional basis in which distributional data are represented for computation. Besides the functional spline basis and its [BayesSpace], the returned object stores the transformation matrix 'A' and the Gram matrix 'S' (the inner products of the basis elements).

Usage

BayesSpaceBasis(basis, ...)

## S3 method for class 'basisfd'
BayesSpaceBasis(
  basis,
  bayesSpace = BayesSpace(ReferenceMeasureUniform(basis$rangeval[1], basis$rangeval[2])),
  A = computeL22BayesSpacesBasisMatrix(basis),
  S = A %*% fda::inprod(basis, basis) %*% t(A)/diff(basis$rangeval),
  orthonormalize = FALSE,
  ...
)

## S3 method for class 'fd'
BayesSpaceBasis(basis, ...)

## S3 method for class 'BayesSpace'
BayesSpaceBasis(basis, ...)

Arguments

Details

The default basis (constructed from a reference measure or an 'fda' B-spline basis) consists of **zero-integral B-splines (ZB-splines)**: spline functions that integrate to zero, which is required to represent centred-log-ratio (clr) transformed densities. The object stores

'basis'

a function evaluating the underlying (B-spline) basis,

'A'

the transformation matrix mapping those values to the ZB-spline basis, so the basis functions are 'zeta = A %*% B',

'S'

the Gram (metric) matrix 'S[i,j] = <zeta_i, zeta_j>',

'bayesSpace', 'centered'

the space and a centring flag.

ZB-splines are **not orthonormal**, so in general 'S' is not the identity; inner products, norms, variances and PCA all use 'S' (see [getMetricTensor], [princomp.BData]). Pass an explicit 'S' only if you have a better value than the internally computed one.

Value

An object of class '"BayesSpaceBasis"': a list with components 'basis', 'bayesSpace', 'A', 'S' and 'centered'.

Methods (by class)

• BayesSpaceBasis(basisfd): Construct from an basisfd object from functional data

• BayesSpaceBasis(fd): Construct from an fd object from functional data

• BayesSpaceBasis(BayesSpace): Calls BayesSpaceBasis on the Reference measure

See Also

[getMetricTensor], [BayesSpace], [as.BDataSplines], [princomp.BData]

[fda::basisfd]

[fda::fd]

Examples

basis <- BayesSpaceBasis(B2Unif01, nbasis = 7)
getMetricTensor(basis)        # Gram matrix S (not the identity for ZB-splines)

Create a BayesSpaceBasis for a Cartesian Product of Bayes Spaces

Description

Create a BayesSpaceBasis for a Cartesian Product of Bayes Spaces

Usage

## S3 method for class 'BayesSpaceCartesianProduct'
BayesSpaceBasis(basis, ..., useBases = lapply(basis, BayesSpaceBasis, ...))

## S3 method for class 'ReferenceMeasureCartesianProduct'
BayesSpaceBasis(
  basis,
  ...,
  useBases = lapply(bayesSpace, BayesSpaceBasis, ...),
  bayesSpace = BayesSpaceCartesianProduct(lapply(basis, BayesSpace))
)

Arguments

Value

A BayesSpaceBasisCartesianProduct of the bases in useBases, for the BayesSpace basis.

Functions

• BayesSpaceBasis(ReferenceMeasureCartesianProduct): creates the BayesSpaceCartesian Product for this reference measure and the corresponding product basis.

See Also

BayesSpaceBasis

Other Cartesian: BayesSpaceBasisCartesianProduct(), BayesSpaceCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), cartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

Examples

BayesSpaceBasis(BayesSpaceCartesianProduct(
    A=BayesSpace(StdReferenceMeasureUnif01),
    B=BayesSpace(ReferenceMeasureDiscrete(c("A","B","C"),c(0.2,0.3,0.5))),
    C=BayesSpace(StdReferenceMeasureN01)
    )
)

Coded Basis for ReferenceMeasureDiscrete

Description

Coded Basis for ReferenceMeasureDiscrete

Usage

## S3 method for class 'ReferenceMeasureDiscrete'
BayesSpaceBasis(
  basis,
  bayesSpace = BayesSpace(basis),
  orthonormalize = FALSE,
  ...
)

Arguments

Value

A BayesSpaceBasis for that discrete reference measure. If a nonstandard A is used it might be nonorthogonal.

Examples

X=c("A","B","C")
rmd = ReferenceMeasureDiscrete(X)
basis = BayesSpaceBasis(rmd)
as.function(basis)(X)
weightedStats(rmd,as.function(basis),function(x) rep(1,length(x)))

Generates a BayesSpaceBasis for transformed Bayes Spaces

Description

The basis is a transformation of the Basis that would be computed for the underlying untransformed BayesSpace.

Usage

## S3 method for class 'ReferenceMeasureTransformed'
BayesSpaceBasis(
  basis,
  bayesSpace = getBayesSpace(basis),
  orthonormalize = FALSE,
  ...
)

Arguments

Value

an object of class '"BayesSpaceBasis"', transforming the basis of the underlying untransformed BayesSpace.

Examples

rmt <- ReferenceMeasureTransformed("pnorm", mean=2, sd=2)
basis <- BayesSpaceBasis(rmt)
basis

create the 'BayesSpaceBasis' object for 'ReferenceMeasureUniform'

Description

create the 'BayesSpaceBasis' object for 'ReferenceMeasureUniform'

Usage

## S3 method for class 'ReferenceMeasureUniform'
BayesSpaceBasis(
  basis,
  bayesSpace = getBayesSpace(basis),
  orthonormalize = FALSE,
  nbasis = 7,
  order = 4,
  breaks = seq(basis$a, basis$b, length.out = (nbasis - order + 3)),
  ...
)

Arguments

Value

object of class 'BayesSpaceBasis'

Examples

basis <- BayesSpaceBasis(ReferenceMeasureUniform(0,1))
basis
#basis(seq(0,1,length.out=10))

Creates a BayesSpaceBasis from a function

Description

The function needs a vectorized function mapping the support of the Bayes space to real densities.

Usage

## S3 method for class ''function''
BayesSpaceBasis(
  basis,
  bayesSpace = B2Unif01,
  A = diag(rep(1, ncol(basis(getSample(bayesSpace))))),
  S = A %*% scalarProductOfLogDensities(basis, space = bayesSpace) %*% t(A),
  orthonormalize = FALSE,
  ...,
  centered = FALSE
)

Arguments

Details

A and S only need to be provided if there is a better result availabel than from the internal computation of the Bayes space

Value

A BayesSpaceBasis

See Also

BayesSpaceBasis

Examples

BayesSpaceBasis(function(x) outer(x,1:5,function(a,b) cos(a*b*pi)))
BayesSpaceBasis(function(x) outer(x,1:5,function(a,b) cos(a*b*pi)),orthonormalize=TRUE)

Computes the product basis of BayesSpaceBasis for the product BayesSpace

Description

Computes the product basis of BayesSpaceBasis for the product BayesSpace

Usage

BayesSpaceBasisCartesianProduct(
  ...,
  .LIST = .listCatcher(...),
  bayesSpace = BayesSpaceCartesianProduct(.LIST = lapply(.LIST, getBayesSpace))
)

Arguments

Value

A Bayes Space Basis for the product space as defined the DDA Book.

See Also

[BayesSpaceBasis]

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), cartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

The Cartesian Product of Bayes Spaces

Description

The Cartesian Product of Bayes Spaces

Usage

BayesSpaceCartesianProduct(...)

L2SpaceCartesianProduct(...)

Arguments

Value

the Product space of the BayesSpaces, a BayesSpace itself.

Functions

• L2SpaceCartesianProduct(): Product of L2 spaces

See Also

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceBasisCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), cartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

Computes a covariance with respect to a measure. The function can be considered internal implementation details of the package, but needs to be exported due to its generic nature.

Description

The implementation depends on the measure

Usage

CovarianceOperator(measure, X, Y, ...)

## S3 method for class 'ReferenceMeasure'
CovarianceOperator(
  measure,
  X,
  Y,
  ...,
  method = c("default", "sample", "MonteCarlo", "integration"),
  tol = 0.01,
  close = FALSE
)

## S3 method for class 'ReferenceMeasureDiscrete'
CovarianceOperator(measure, X, Y, ...)

Arguments

Value

the scalar products (covariances) of 'X' and 'Y' with respect to the reference measure. For the default ReferenceMeasure method these are obtained from [stats::cov()] applied to ‘X' and 'Y' evaluated on the measure’s sample. The result is a numeric scalar, vector, or matrix depending on the number of columns returned by 'X' and 'Y' (one entry per pair of output components); when 'X' or 'Y' is a list, the results are combined columnwise.

Methods (by class)

• CovarianceOperator(ReferenceMeasure): method for ReferenceMeasures using a MonteCarlo integration based on the sample provided by getSample(sample)

• CovarianceOperator(ReferenceMeasureDiscrete): method for ReferenceMeasuresDiscrete explicit computation

Examples

CovarianceOperator(StdReferenceMeasureUnif01,
                   function(x) cbind(x,-x),
                   function(x) cbind(x,-x,2*x+1)
)

An example dataset for distributional data analysis

Description

This is a simulated dataset, which can be used to try different tools for distributional data analysis. The idea is that there is a set of samples 'sample' from three regions, each containing many particles. From each particle we have observed a size parameter 'size' (volume equivalent diameter), a density 'density', and a shape parameter 'shape'

Usage

DTD

Format

A data frame with 20 rows (material samples) and 9 columns. Several columns hold one *function* per row (the particle PDFs / CDFs), and data holds one nested data frame per row:

sample

The id of the material sample

region

The region the sample comes from.

d50

A measured median particle size of the sample

shapePDF

Particle shape distribution in sample, provide as a probability density function. Shape is measured by a measure for sphericity.

shapeCDF

Particle shape distribution in sample, provided as Cummulative distribution.

sizePDF

Particle size distribution in the sample (diameter), provide as probability density function. Size is measured by equivalent diameter. The sieving diameter relevant for d50 is slightly different.

sizeCDF

Particle size distribution in the sample provided as cummulative distribution function (cdf).

rhoCDF

Particle mass density distribution provided as cummulative distribution function (CDF). Particle density does not have probability density, because liberated particles containing only one mineral all can only have very specific densities (e.g. Quarz 2.25 kg/l and Magnetit at 5.2 kg/l)

data

data.frames containing samples of the particles in each material sample. It has four columns: sample (the id of the sample), size (equivalent diameter of the particle), shape (sphericity), and density (mass density) of the particle.

Source

Simulated data, not based on real measurements. Generated by data-raw/DTD.R with set.seed(47) (N = 20 samples drawn from three regions A/B/C). Shipped as data/DTD.rda; see data-raw/DTD.R to regenerate.

Examples

data("DTD")
DTD
boxplot(DTD$d50)
# Looking at the size variable
plotFuncs(DTD$sizePDF)
plotFuncs(\(x)(dlnorm(x,meanlog=-0.2,sdlog=0.5)),add=TRUE,col="red",lwd=3)
RMForSize = ReferenceMeasureTransformed("plnorm",meanlog=-0.2,sdlog=0.5)
B2ForSize  = BayesSpace(RMForSize)
BasisForSize = BayesSpaceBasis(B2ForSize,nbasis=6)

sizePDFs = BDataPDFs(DTD$sizePDF,basis=BasisForSize)
plot(sizePDFs)
sizeDensities = as.BDataDensities(sizePDFs)
plot(sizeDensities)
sizeSplines   = as.BDataSplines(sizeDensities)
#plot(sizeSplines,add=TRUE,col="blue")
DTD$data[[1]]
summary(DTD$data[[1]])

Creates a function based on a regular discretization

Description

A simple fast tool for defining a discretized function defined continuously on [a,b] and discrete values in X, with equidistant data values y in between. The resulting function is vectorized and interpolates linearily with errors or NA outside the region. It handles infinite supports, but not infinite results on the continuous region.

Usage

DiscretizedFunction(y, ...)

## S3 method for class 'numeric'
DiscretizedFunction(y, ..., support = Support(...), transferCdf = NULL)

## S3 method for class ''function''
DiscretizedFunction(
  y,
  a = 0,
  b = a,
  n = 501,
  X = c(),
  transferCdf = NULL,
  transferQf = cdf2qf(transferCdf, ...),
  ...
)

## S3 method for class 'Discretizer'
DiscretizedFunction(y, Y, ...)

## S3 method for class 'ReferenceMeasureTransformed'
DiscretizedFunction(y, f, n = 513, support = getSupport(y), ...)

## S3 method for class 'ReferenceMeasureUniform'
DiscretizedFunction(y, f, n = 513, support = getSupport(y), ...)

Arguments

Value

a function interpolating linearly between the given evaluation points and working in a vectorized form.

Methods (by class)

• DiscretizedFunction(numeric): method for precomputed values

• DiscretizedFunction(`function`): method for functions that need to be computed first.

• DiscretizedFunction(Discretizer): method for Discretizer

• DiscretizedFunction(ReferenceMeasureTransformed): method for functions on Transformed Reference measures.

• DiscretizedFunction(ReferenceMeasureUniform): method for functions on Transformed Reference measures.

Examples

x=seq(0,1,length.out=100)
f = DiscretizedFunction(sin(x),0,1)
x=seq(0,1,length.out=1000)
max(abs(f(x)-sin(x)))

Division of functional objects

Description

This division is also able to divide functions and BayesSpace objects with constants. The quotient of two functions is a function that results in the quotient of the two results. The quotient of a BayesSpace objects (in BData) with a constant, is given by the BayesSpace operations of powering with the inverse value 1/y.

Usage

Div(x, y)

## S3 method for class ''function''
Div(x, y)

## S3 method for class 'list'
Div(x, y)

## S3 method for class 'numeric'
Div(x, y)

## S3 method for class 'BData'
Div(x, y)

## S3 method for class ''function''
Div.function(x, y)

## S3 method for class ''function''
Div.list(x, y)

## S3 method for class 'list'
Div.function(x, y)

## S3 method for class 'numeric'
Div.function(x, y)

## S3 method for class ''function''
Div.numeric(x, y)

## S3 method for class 'numeric'
Div.numeric(x, y)

## S3 method for class 'numeric'
Div.list(x, y)

## S3 method for class 'list'
Div.numeric(x, y)

## S3 method for class 'list'
Div.list(x, y)

Arguments

Value

A BData object of the same subclass as the first argument (the Bayes-space sum / difference / scalar multiple / quotient).

Methods (by class)

• Div(`function`): method for functions

• Div(list): method for vectorizing over lists

• Div(numeric): method for numerical values seen as constant functions

• Div(BData): method for operating any BData object (argument 'x') by a numeric value (argument 'y'); defers calculations to 'Mul()'

Functions

• Div.function(`function`): method for deviding two functions

• Div.list(`function`): method for multiplying a function with each element of a list

• Div.function(list): method for multiplying a function with each element of a list

• Div.function(numeric): method for dividing a function with a constant

• Div.numeric(`function`): method for deviding a constant with a function

• Div.numeric(numeric): method for deviding two numeric values

• Div.list(numeric): method for dividing each element of a list with a constant

• Div.numeric(list): method for deviding a constant with each element of a list

• Div.list(list): method for vectorized division of the elements of lists

See Also

Other Add: Add(), Mul(), Sub()

Examples

Div(sin,cos)
Div(list(sin,cos),sin)
Div(cos,list(sin,cos))
Div(list(sin,cos),list(sin,cos))
Div(list(sin,cos),10)

Creates the L2-space for a Reference Measure

Description

An L2-space is a Hilbert space of with respect to the reference measure square integrable functions. They are vector addition and multiplication with a scalar is done on the results. The scalar product is defined integrating the product of two functions over the reference measure.

Usage

L2Space(x = StdReferenceMeasureUnif01, ...)

## S3 method for class 'ReferenceMeasure'
L2Space(x, ...)

## S3 method for class 'BayesSpace'
L2Space(x, ...)

Arguments

Value

An object of class '"L2Space"': a list holding the reference measure.

Methods (by class)

• L2Space(ReferenceMeasure): Method for ReferenceMeasure

• L2Space(BayesSpace): Method for BayesSpaces.

See Also

[ReferenceMeasure]

Other BayesSpace: BayesSpace()

Examples

space = L2Space(ReferenceMeasureUniform(0,10))
space
bs = BayesSpace(ReferenceMeasureUniform(0,10))
space2 = L2Space(bs)
space2

Makes a list includable in a data.frame

Description

data.frame objects treat lists as variables and not as cases. This functions presents the list as data to a data.frame. For better printing the elements of the list will be represented as a name.

Usage

ListDataEnv

ListData(x)

ListDataRecover(hash)

storingDigester(x, env = environment(x))

## S3 method for class 'ListData'
format(x, ...)

## S3 method for class 'BData'
format(x, ...)

## S3 method for class 'ListData'
as.data.frame(x, row.names = NULL, optional = FALSE, ...)

## S3 method for class 'BData'
as.data.frame(x, row.names = NULL, optional = FALSE, ...)

Arguments

Format

An object of class environment of length 1.

Details

The first letter of the name is the first letter of the class, then comes a '_' and then a alphanumerical digest, of the objects. As long as the object is in reach, it can be inspected by this name.

Afterwards an NA of class "ForgottenObject" is returned.

Value

the object x with an additional "ListData" class attribute

Methods (by generic)

• format(ListData): formats as the digest

• as.data.frame(ListData): converts to a data.frame

Functions

• ListDataEnv: the environment in which the digests can be found

• ListDataRecover(): recovers object from the digest

• storingDigester(): returns a recoverable digest of the object

• format(BData): formats as the digest

• as.data.frame(BData): converts to a data.frame

Examples

FX = ListData(
   list(function(x) x,
        function(x) x^2,
        function(x) x^3))
FX
D = data.frame(X=FX,Y=FX)
D
N = format(FX)[[1]]
N
ListDataRecover(N)
attach(ListDataEnv)
get(N)
N2 = storingDigester(function(x) sin(x))
N2
ListDataRecover(N2)
detach(ListDataEnv)

Functional mapping a function to its exp version

Description

Exp maps f(x) to Exp(f(x)). Ln is its inverse operation.

Usage

Ln(f)

## S3 method for class 'numeric'
Ln(f)

## S3 method for class ''function''
Ln(f)

## S3 method for class 'list'
Ln(f)

Exp(f)

## S3 method for class 'numeric'
Exp(f)

## S3 method for class ''function''
Exp(f)

## S3 method for class 'list'
Exp(f)

Arguments

Value

same type as f but exponentiated

Methods (by class)

• Exp(numeric): numeric method just evaluates exp

• Exp(`function`): Function method maps the function

• Exp(list): list method iterates in a vectorized manner over the list

Functions

• Ln(): Inverse of Exp

• Ln(numeric): method for numerics just applies the natural logarithm

• Ln(`function`): Inverse functional mapping

• Ln(list): list method operates in a vectorized way over the list

Examples

Exp(1)
f = function(x) x^2
expf = Exp(f)
expf
lnf = Ln(f)
Exp(lnf)
Ln(expf)

Multiplication of functional objects

Description

This multiplication is also able to multiply functions and BayesSpace objects with constants. The product of two functions is a function that results in the product of the two results. Product of a BayesSpace objects (in BData), is given by the BayesSpace operations, which is given powering the density with respect to the reference measure, with the given scalar.

Usage

Mul(x, y)

## S3 method for class ''function''
Mul(x, y)

## S3 method for class 'list'
Mul(x, y)

## S3 method for class 'numeric'
Mul(x, y)

## S3 method for class ''function''
Mul.function(x, y)

## S3 method for class ''function''
Mul.list(x, y)

## S3 method for class 'list'
Mul.function(x, y)

## S3 method for class 'numeric'
Mul.function(x, y)

## S3 method for class ''function''
Mul.numeric(x, y)

## S3 method for class 'numeric'
Mul.numeric(x, y)

## S3 method for class 'BData'
Mul.numeric(x, y)

## S3 method for class 'numeric'
Mul.list(x, y)

## S3 method for class 'list'
Mul.numeric(x, y)

## S3 method for class 'list'
Mul.list(x, y)

## S3 method for class 'BDataDensities'
Mul(x, y)

## S3 method for class 'BDataLogDensities'
Mul(x, y)

## S3 method for class 'BDataPDFs'
Mul(x, y)

## S3 method for class 'BDataLogPDFs'
Mul(x, y)

## S3 method for class 'BDataSplines'
Mul(x, y)

Arguments

Value

A BData object of the same subclass as the first argument (the Bayes-space sum / difference / scalar multiple / quotient).

Methods (by class)

• Mul(`function`): method for multiplying functions

• Mul(list): method for vectorized operation over lists

• Mul(numeric): method for numerical values treated as constant functions

• Mul(BDataDensities): BayesSpace scaling multiplication

• Mul(BDataLogDensities): BayesSpace scaling multiplication

• Mul(BDataPDFs): BayesSpace scaling multiplication

• Mul(BDataLogPDFs): BayesSpace scaling multiplication for log-PDFs

• Mul(BDataSplines): BayesSpace scaling multiplication

Functions

• Mul.function(`function`): method for operating two functions

• Mul.list(`function`): method for multiplying a function with each element of a list

• Mul.function(list): method for multiplying a function with each element of a list

• Mul.function(numeric): method for multipying a function with a numeric value

• Mul.numeric(`function`): method for multiplying numeric constant with a function

• Mul.numeric(numeric): method for multiplying two constant values

• Mul.numeric(BData): method for multiplying a numeric object by a BData object

• Mul.list(numeric): method for multiplying each element of a list with a numeric value

• Mul.numeric(list): method for multiplying a numeric value with each element of a list

• Mul.list(list): method for vectorized multiplication of the elements of a list.

See Also

Other Add: Add(), Div(), Sub()

Examples

Mul(sin,cos)
Mul(list(sin,cos),sin)
Mul(cos,list(sin,cos))
Mul(list(sin,cos),list(sin,cos))
Mul(list(sin,cos),10)

Reference measures: the geometry of a Bayes space

Description

A reference measure is the probability distribution that defines a [BayesSpace]: it plays the role of the origin, and all modelled distributions are expressed relative to it. Choosing it fixes the support and the geometry of the analysis.

Details

# Choosing a reference measure Pick a measure whose support matches your data, and ideally whose mass is distributed similarly:

[ReferenceMeasureUniform]

uniform on an interval '(a, b)'; the simplest choice for data on a bounded interval (predefined: 'B2Unif01').

[ReferenceMeasureTransformed]

a transformed support, e.g. a Gaussian reference via its CDF (predefined: 'B2N01'); useful for data on the real line or with a known reference shape.

[ReferenceMeasureDiscrete]

a finite set of categories with given probabilities.

A poor choice (e.g. a uniform reference for strongly skewed data) still works but gives a less informative geometry. Inspect a measure with [getReferenceMeasure] and the resulting metric with [getMetricTensor].

See Also

[BayesSpace], [BayesSpaceBasis], [BData]

Compute a cartesian Product of measures

Description

Compute a cartesian Product of measures

Usage

ReferenceMeasureCartesianProduct(..., nSamples = 1e+05)

Arguments

Value

A ReferenceMeasureCartesianProduct, which is essentially a list of the measures, in an object of that class, that acts like a ReferenceMeasure for the Cartesian Product of the supports of the individual measures.

The product measure corresponds gives a any product set the measure of the that corresponds to the product of the individual measures. In case of a probablity distribution, that would be the measure representing an independent distribution, with the initial measures as marginal distributions. This concept is used to establish a reference measure for multivariate BayesSpaces which can also contain non independent measures.

See Also

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceBasisCartesianProduct(), BayesSpaceCartesianProduct(), SupportCartesianProduct(), cartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

Examples

 RMCP3 = ReferenceMeasureCartesianProduct(A=StdReferenceMeasureN01,
                                          B=StdReferenceMeasureUnif01,
                                          C=ReferenceMeasureDiscrete(c("A","B","C"),c(0.2,0.3,0.5))
                                           )

Discrete Reference Measure for BayesSpaces

Description

A discrete reference measure is a measure on a discrete support.

Usage

ReferenceMeasureDiscrete(
  X,
  p = rep(1/length(X), length(X)),
  sampler = NULL,
  support = Support(X = X)
)

Arguments

Value

An object of type ReferenceMeasureDiscrete encapsulating this measure

See Also

Other ReferenceMeasures: B2N01, ReferenceMeasureTransformed(), ReferenceMeasureUniform(), StdReferenceMeasureUnif01, expectation(), getSample(), isInDomain()

Examples

X <- c("A","B","C")
( rmd <- ReferenceMeasureDiscrete(X) )

A general continuous reference measure on the real line

Description

The reference measure is created by transforming a ReferenceMeasure Uniform to a given distributions based using a distribution function and a quantile function.

Usage

ReferenceMeasureTransformed(
  cdf = c("pnorm", "pbeta", "pcauchy", "pchisq", "pexp", "pf", "pgamma", "plnorm",
    "plogis", "pt", "punif", "pweibull", "pwilcox"),
  qf = cdf2qf(cdf, ...),
  ...,
  pdf = cdf2density(cdf, ...),
  reference = StdReferenceMeasureUnif01,
  .args = list(...),
  .eps = 1e-07
)

## S3 method for class 'ReferenceMeasureTransformed'
getSample(x, ...)

Arguments

Value

a reference measure with representing the reference measure described by the distribution and quantile function.

Methods (by generic)

• getSample(ReferenceMeasureTransformed): returns transformed sample from the underlying reference

See Also

Other ReferenceMeasures: B2N01, ReferenceMeasureDiscrete(), ReferenceMeasureUniform(), StdReferenceMeasureUnif01, expectation(), getSample(), isInDomain()

Examples

rmt = ReferenceMeasureTransformed(pnorm,qnorm,mean=2,sd=2)
isInDomain(0:17,rmt)
expectation(rmt,\(x) x^2)
hist(getSample(rmt))

create the ReferenceMeasureUniform class

Description

create the ReferenceMeasureUniform class

Usage

ReferenceMeasureUniform(
  a = 0,
  b = 1,
  sample = makeUnifGrid(2^12, a, b),
  seed = NULL
)

Arguments

Value

a 'ReferenceMeasureUniform class'

See Also

[BData]

Other ReferenceMeasures: B2N01, ReferenceMeasureDiscrete(), ReferenceMeasureTransformed(), StdReferenceMeasureUnif01, expectation(), getSample(), isInDomain()

Examples

min <- -5
max <- 5
RMU <- ReferenceMeasureUniform(min,max)
RMU
## reproducible construction, RNG state left untouched:
RMU1 <- ReferenceMeasureUniform(0, 1, seed = 20260702)
RMU2 <- ReferenceMeasureUniform(0, 1, seed = 20260702)
identical(RMU1$sample, RMU2$sample)

The standard B2(Unif(0,1)) Bayes space.

Description

B2Unif01 is the Bayes space B2(Unif(0,1)), and B2Unif01BasisN its B-Spline basis with N elements of order 2

Usage

StdReferenceMeasureUnif01

StdReferenceMeasureN01

B2Unif01

B2Unif01Basis5

B2Unif01Basis7

B2Unif01Basis20

Format

An object of class ReferenceMeasureUniform (inherits from ReferenceMeasure) of length 3.

An object of class ReferenceMeasureTransformed (inherits from ReferenceMeasure) of length 5.

An object of class BayesSpace of length 1.

An object of class BayesSpaceBasis of length 5.

An object of class BayesSpaceBasis of length 5.

An object of class BayesSpaceBasis of length 5.

Functions

• StdReferenceMeasureUnif01: the underlying standard reference measure

• StdReferenceMeasureN01: the underlying standard reference measure

• B2Unif01Basis5: a basis for B2Unif01 with 5 elements

• B2Unif01Basis7: a basis for B2Unif01 with 7 elements

• B2Unif01Basis20: a basis for B2Unif01 with 20 elements

See Also

[BayesSpaceBasis]

Other ReferenceMeasures: B2N01, ReferenceMeasureDiscrete(), ReferenceMeasureTransformed(), ReferenceMeasureUniform(), expectation(), getSample(), isInDomain()

Examples

getReferenceMeasure(B2Unif01)
getMetricTensor(B2Unif01Basis5)

Substract functional objects

Description

This subtraction is also able to add functions and BayesSpace objects. The sum of two functions is a function that results in the difference of the two results. The difference of two BayesSpace objects (in BData), is given by the BayesSpace operations, which is given by the quotient of the densities with respect to the reference measure.

Usage

Sub(x, y)

## S3 method for class ''function''
Sub(x, y)

## S3 method for class 'list'
Sub(x, y)

## S3 method for class 'numeric'
Sub(x, y)

## S3 method for class ''function''
Sub.function(x, y)

## S3 method for class ''function''
Sub.list(x, y)

## S3 method for class 'list'
Sub.function(x, y)

## S3 method for class 'numeric'
Sub.function(x, y)

## S3 method for class ''function''
Sub.numeric(x, y)

## S3 method for class 'numeric'
Sub.numeric(x, y)

## S3 method for class 'numeric'
Sub.list(x, y)

## S3 method for class 'list'
Sub.numeric(x, y)

## S3 method for class 'list'
Sub.list(x, y)

## S3 method for class 'BDataHistograms'
Sub(x, y)

## S3 method for class 'BDataHistograms'
Sub.BDataHistograms(x, y)

## S3 method for class 'BDataDensities'
Sub(x, y)

## S3 method for class 'BDataSplines'
Sub(x, y)

## S3 method for class 'BDataLogDensities'
Sub(x, y)

## S3 method for class 'BDataPDFs'
Sub(x, y)

## S3 method for class 'BDataLogPDFs'
Sub(x, y)

Arguments

Value

A BData object of the same subclass as the first argument (the Bayes-space sum / difference / scalar multiple / quotient).

Methods (by class)

• Sub(`function`): method for functions

• Sub(list): method for lists, iterating over the list

• Sub(numeric): method for numerics, treating them as constant functions

• Sub(BDataHistograms): method for BDataHistograms (operating in BayesSpaces)

• Sub(BDataDensities): BayesSpace Difference for BDataDensities

• Sub(BDataSplines): BayesSpace Difference for BDataSplines

• Sub(BDataLogDensities): BayesSpace Difference for BDataLogDensities

• Sub(BDataPDFs): BayesSpace Difference for BDataPDFs

• Sub(BDataLogPDFs): BayesSpaceAddition for BDataLogPDFs

Functions

• Sub.function(`function`): method for two functions, returning a function giving the difference of the two as a result

• Sub.list(`function`): method for a function and list, iterating over the list

• Sub.function(list): method for a function and list, iterating over the list

• Sub.function(numeric): method substracting a constant value from a function

• Sub.numeric(`function`): method substracting a function from a constant

• Sub.numeric(numeric): method substracting two values

• Sub.list(numeric): method for substracting a value from each element of a list, returning a list.

• Sub.numeric(list): method each element of a list from a constant

• Sub.list(list): method for vectorized application of sub on the elements of lists

• Sub.BDataHistograms(BDataHistograms): method for two BDataHistograms

See Also

Other Add: Add(), Div(), Mul()

Examples

Sub(sin,cos)
Sub(list(sin,cos),sin)
Sub(cos,list(sin,cos))
Sub(list(sin,cos),list(sin,cos))
Sub(list(sin,cos),10)

The support of the measures

Description

For this package the support tells what is needed to know for plotting

Usage

Support(a, b, X = c(), lim)

getSupport1DPlottingLimits(x)

getSupportAtoms(x)

getSupport(x, ...)

## S3 method for class 'Support'
getSupport(x, ...)

## S3 method for class 'BayesSpace'
getSupport(x, ...)

## S3 method for class 'BayesSpaceBasis'
getSupport(x, ...)

## S3 method for class 'BData'
getSupport(x, ...)

## S3 method for class 'ReferenceMeasureUniform'
getSupport(x, ...)

## S3 method for class 'ReferenceMeasureDiscrete'
getSupport(x, ...)

## S3 method for class 'ReferenceMeasureTransformed'
getSupport(x, ..., alpha = 0.01)

## S3 method for class 'list'
getSupport(x, ...)

## S3 method for class ''function''
getSupport(
  x,
  ...,
  extrems = c(-1e+20, 1e+20),
  starts = c(exp(seq(-20, 20, by = 0.2)), -exp(seq(-20, 20, by = 0.2)))
)

## S3 method for class 'numeric'
getSupport(x, ...)

## S3 method for class 'ListData'
getSupport(x, ...)

## S3 method for class 'ReferenceMeasureCartesianProduct'
getSupport(x, ...)

## S3 method for class 'varKernel'
getSupport(x, ...)

## S3 method for class 'variationKernel'
getSupport(x, ...)

Arguments

Value

An object of class "Support" with the given parameters

Methods (by generic)

• getSupport(Support): method for Support returns itself

Functions

• getSupport1DPlottingLimits(): method for getting plotting limits of supports

• getSupportAtoms(): method for getting values with mass

• getSupport(): get the support associated to an object

• getSupport(BayesSpace): method for BayesSpaces

• getSupport(BayesSpaceBasis): method for BayesSpaces

• getSupport(BData): method for BData

• getSupport(ReferenceMeasureUniform): method for ReferenceMeasureUniform

• getSupport(ReferenceMeasureDiscrete): method for ReferenceMeasureDiscrete

• getSupport(ReferenceMeasureTransformed): method for ReferenceMeasureTransformed

• getSupport(list): combines the supports of all Elements in the list

• getSupport(`function`): tries to identify pdfs and cdfs and useds pdf2Support or cdf2Support, a simple 0,1 otherwise.

• getSupport(numeric): guesses the support through sample2Support

• getSupport(ListData): just calls NextMethod to hand over to the other classes

• getSupport(ReferenceMeasureCartesianProduct): returns the product support

• getSupport(varKernel): getter method for varKernel

• getSupport(variationKernel): getter method for variationKernel

See Also

[ReferenceMeasureCartesianProduct], [SupportCartesianProduct]

Other Support: cdf2Support(), pdf2Support(), plotFuncs(), sample2Support()

Examples

Support(0,1)
Support(X=1:4)
getSupport(StdReferenceMeasureUnif01)
getSupport(StdReferenceMeasureN01)

Creates a cartesian product of supports

Description

Creates a cartesian product of supports

Usage

SupportCartesianProduct(...)

Arguments

Value

A Support of type SupportCartesianProduct that provides the Cartesian product of the supports

Supports are essentially encoded sets or spaces of possible values. The cartesian product of such spaces is the space containing the tuples of with one entry from each space. In R it will be encoded by the columns of a data.frame.

See Also

[Support], [data.frame]

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceBasisCartesianProduct(), BayesSpaceCartesianProduct(), ReferenceMeasureCartesianProduct(), cartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

Examples

 s1 = Support(-Inf,Inf,lim=c(-2,2))
 s2 = Support(0,2)
 s3 = Support(X=c(1,2,3))
 s4 = Support(X=c("A","B","C","D"))
 s  = SupportCartesianProduct(s1,s2,s3,s4)

Subsetting BDataHistograms

Description

A subsetting function selecting only a part of the histograms

Usage

## S3 method for class 'BDataHistograms'
x[i, j, ...]

Arguments

Value

A BDataHistograms object with the subset of histograms described by the basis

Examples

set.seed(47)
breaks = seq(-3,3,by=1)
inRange = function(x) -3<x & x<3
X = t(replicate(4,hist(Filter(inRange,rnorm(1000)),breaks=breaks,plot=FALSE)$counts))
hists = BDataFrequencyHistograms(X,breaks=breaks,basis=B2N01Basis5)
hists[1:2, ]

Pool absolute frequencies

Description

Sum together the absolute frequencies of two representations with this information

Usage

addFrequency(x, y, ...)

## S3 method for class 'BDataFrequencyHistograms'
addFrequency.BDataFrequencyHistograms(x, y, ...)

## S3 method for class 'BDataUnivariateSamples'
addFrequency.BDataUnivariateSamples(x, y, ...)

## S3 method for class 'BDataUnivariateSamples'
addFrequency.BDataFrequencyHistograms(
  x,
  y,
  .baseSpaceBasis = getBayesSpaceBasis(x),
  ...
)

## S3 method for class 'BDataFrequencyHistograms'
addFrequency.BDataUnivariateSamples(
  x,
  y,
  .baseSpaceBasis = getBayesSpaceBasis(x),
  ...
)

Arguments

Value

An object of the least informative kind from the two pooled together, which contains the added absolute frequencies of both input objects. The Bayes Space basis is controlled by the first argument.

Functions

• addFrequency.BDataFrequencyHistograms(BDataFrequencyHistograms): method for two "BDataFrequencyHistograms" object. Returns a "BDataFrequencyHistograms" where the frequencies are summed (if breaks are equivalent), or where they are distributed uniformly along the interval bins and rebinned by summing with a break set resulting from the combination of both input bin sets.

• addFrequency.BDataUnivariateSamples(BDataUnivariateSamples): method for two "BDataUnivariateSamples" object. Returns a "BDataUnivariateSamples" where the observations are pooled together (in the order they are given), pair by pair. It can recycle a "BDataUnivariateSamples" of one single population with a multi-population "BDataUnivariateSamples", but will not work with any other combination of "BDataUnivariateSamples" of different lengths.

• addFrequency.BDataFrequencyHistograms(BDataUnivariateSamples): method for a "BDataFrequencyHistograms" and a "BDataUnivariateSamples" objects; classifies the data sets from "BDataUnivariateSamples" into the bins of "BDataFrequencyHistograms". As with other methods of this function, it only works for 'x' and 'y' of the same length, or else either 'x' or 'y' must have length 1.

• addFrequency.BDataUnivariateSamples(BDataFrequencyHistograms): method for a "BDataUnivariateSamples" and a "BDataFrequencyHistograms" objects; same as in the reverse order

Examples

xh = lapply(rep(1000,4), runif) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5)
yh = lapply(rep(1000,4), runif) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5)
addFrequency(xh,yh)
xh1 = list(runif(300)) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5)
addFrequency(xh,xh1)
xh1 = list(runif(300)) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5, breaks=c(0:4)/4)
addFrequency(xh,xh1)
xs = lapply(rep(200,4), runif) |> BDataSamples(B2Unif01Basis5)
ys = lapply(rep(10,4), runif) |> BDataSamples(B2Unif01Basis5)
addFrequency(xs,ys) |> summary()
x1 = BDataSamples(list(runif(100)), B2Unif01Basis5)
addFrequency(xs, x1)  |> summary()
xs = lapply(rep(200,4), runif) |> BDataSamples(B2Unif01Basis5)
yh = lapply(rep(1000,4), runif) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5)
addFrequency(xs,yh)

Age Distribution Data for Upper Austrian Communes

Description

Age distributions for 57 communes in Upper Austria (Oberoesterreich), separated by sex. Each observation represents the age distribution of either the male or female population in a commune, given as proportions across 19 five-year age groups.

Usage

age

Format

A list with three components:

data

A data frame with 114 rows (57 communes x 2 sexes) and 22 columns:

commune

Character. Four-digit commune code.

municipality

Character. Name of the municipality (e.g., "Linz", "Steyr"). German umlauts have been transliterated to ASCII (ue, oe, ae, ss).

sex

Factor with levels "male" and "female".

AGE_2, AGE_7, ..., AGE_92

Numeric. Proportions for each age group. Column names indicate the midpoint of the 5-year age interval (e.g., AGE_2 = ages 0-4, AGE_7 = ages 5-9, ..., AGE_92 = ages 90+).

ages

Numeric vector of length 19. The midpoint ages: 2, 7, 12, ..., 92.

distributions

A 114 x 19 matrix of age proportions. Row names are formatted as "municipality_sex" (e.g., "Linz_male"). Each row sums to 1.

Details

The data originates from the Austrian Register Census 2021 (Registerzaehlung 2021), which provides population counts by gender, age, and citizenship at the municipality level. The original counts have been converted to proportions (probability distributions) for distributional data analysis.

This dataset is useful for demonstrating PCA on distributional data, as it contains real-world probability distributions that vary in shape (e.g., rural vs urban communes have different age structures).

License

Creative Commons Attribution 4.0 International (CC BY 4.0). https://creativecommons.org/licenses/by/4.0/

Note

Municipality names have been transliterated from German to ASCII for cross-platform compatibility, following the convention ae / oe / ue / ss for the umlauts and sharp-s (so e.g. the German "Altmuenster" and "Schoenegg" are stored with ue / oe). For an unambiguous key independent of spelling, use the four-digit codes in the 'commune' column.

Source

Data from the Austrian Register Census 2021 (Registerzaehlung 2021), provided by the State of Upper Austria (Land Oberoesterreich) via the Open Data Portal Austria.

References

Land Oberoesterreich (2021). Bevoelkerung nach Geschlecht und Alter. Open Data Oesterreich. Old broken link: https://www.data.gv.at/katalog/dataset/land-ooe_bevolkerung-geschlecht-und-alter Current dataset: https://www.data.gv.at/katalog/datasets/cd33f19a-7bcc-4454-987c-9f2d76542872

Examples

data("age")

# Examine the structure
str(age)
head(age$data)

# The age midpoints
age$ages

# Plot a few age distributions
matplot(age$ages, t(age$distributions[1:6, ]), type = "l",
        xlab = "Age", ylab = "Proportion",
        main = "Age distributions for first 6 observations")

# Convert to BData for distributional analysis
# First, create a basis for age distributions (support 0-100)
rmAge <- ReferenceMeasureUniform(0, 100)
B2Age <- BayesSpace(rmAge)
basisAge <- BayesSpaceBasis(B2Age, nbasis = 7)

# Create histograms from the proportional data
breaks <- c(5*(0:18), 100)
ageHist <- BDataDensityHistograms(
    age$distributions, breaks = breaks, basis = basisAge
    )

# Convert to splines: histograms -> densities -> splines
ageDensities <- as.BDataDensities(ageHist)
ageSplines <- as.BDataSplines(ageDensities)
pcaAge <- princomp(ageSplines)
summary(pcaAge)

# Visualize PCA results
plot(pcaAge, type = "screeplot")
plot(pcaAge, type = "scores")

# Score plot with municipality labels
plot(pcaAge, type = "scores", labels = age$data$municipality, cex = 0.6)

# Color by sex
sexColors <- ifelse(age$data$sex == "male", "blue", "red")
plot(pcaAge, type = "scores", col = sexColors)
legend("topright", c("Male", "Female"), col = c("blue", "red"), pch = 19)

Converts to a BDataDensities object

Description

This represents BayesSpace data by a list of density functions, i.e. one density (with respect to the reference measure) per observation. Use this representation to inspect or plot the distributions as densities; for computation (PCA, distances) convert further to splines with [as.BDataSplines].

Usage

as.BDataDensities(x, ...)

## S3 method for class 'BDataDensities'
as.BDataDensities(x, ...)

## S3 method for class 'BDataSplines'
as.BDataDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogDensities'
as.BDataDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataPDFs'
as.BDataDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogPDFs'
as.BDataDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataHistograms'
as.BDataDensities(
  x,
  ...,
  background = 0,
  basis = getBayesSpaceBasis(x),
  dmu = getPDF(getReferenceMeasure(basis))
)

## S3 method for class 'BDataSamples'
as.BDataDensities(
  x,
  ...,
  method = c("Fisher", "kde"),
  basis = getBayesSpaceBasis(x)
)

Arguments

Value

A BDataDensities object (densities w.r.t. the reference measure).

Methods (by class)

• as.BDataDensities(BDataDensities): method for the class itself das nothing

• as.BDataDensities(BDataSplines): constructs the projected functions

• as.BDataDensities(BDataLogDensities): calls [BDataDensities.BDataLogDensities]

• as.BDataDensities(BDataPDFs): divides PDFs by PDF of reference measure

• as.BDataDensities(BDataLogPDFs): divides PDFs by PDF of reference measure

• as.BDataDensities(BDataHistograms): method for histograms (TODO: Uses a very crude heuristic to estimate the log density)

• as.BDataDensities(BDataSamples): constructs a density from the samples

See Also

[.DensityFromHist]

Other BayesSpaceConverters: as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

X = BDataLogDensities(list(sin,cos),basis=B2Unif01Basis5)
as.BDataDensities(X)

Convert to a BDataDensityHistograms dataset

Description

Density histograms do not require count information, but provide density information about the relative density in the bars.

Usage

as.BDataDensityHistograms(X, ...)

## S3 method for class 'BDataUnivariateSamples'
as.BDataDensityHistograms(
  X,
  ...,
  breaks = "Sturges",
  basis = getBayesSpaceBasis(X)
)

Arguments

Value

a BDataDensityHistograms object

Methods (by class)

• as.BDataDensityHistograms(BDataUnivariateSamples): method for samples computing the histogram

See Also

[as.BDataDensityHistograms.list]

Other BDataHistograms: BDataFrequencyHistograms(), as.BDataDensityHistograms.list(), as.BDataFrequencyHistograms.list(), getBreaks(), plot.BDataHistograms()

Other BayesSpaceConverters: as.BDataDensities(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

s = BDataSamples(list(rnorm(300), rnorm(300,1)), basis=B2N01Basis5)
as.BDataDensityHistograms(s)

Build a distributional dataset as density histograms from a list

Description

Constructor for a distributional data set of class "BDataDensityHistograms" out of a list of either precalculated histograms or of numeric vectors.

Usage

## S3 method for class 'list'
as.BDataDensityHistograms(X, breaks, basis, ...)

Arguments

Value

A distributional data container of class ['BDataDensityHistograms'].

See Also

Other BDataHistograms: BDataFrequencyHistograms(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms.list(), getBreaks(), plot.BDataHistograms()

Examples

x = lapply(rep(1000,4), runif)
(bddh = as.BDataDensityHistograms(x, breaks="Sturges", basis=B2Unif01Basis5) )
(bdfh = as.BDataFrequencyHistograms(x, breaks="Sturges", basis=B2Unif01Basis5) )
as.BDataDensityHistograms(bdfh)

Converts to a BDataFrequencyHistograms

Description

Only samples and BDataFrequencyHistograms themselfs can be effectively converted to histograms, because all other representations lack information on the certainty

Usage

as.BDataFrequencyHistograms(X, ...)

## S3 method for class 'BDataUnivariateSamples'
as.BDataFrequencyHistograms(
  X,
  breaks = "Sturges",
  ...,
  basis = getBayesSpaceBasis(X)
)

Arguments

Value

a BDataFrequencyHistograms object representing the same dataset

Methods (by class)

• as.BDataFrequencyHistograms(BDataUnivariateSamples): method for BDataUnivariateSamples

See Also

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

Xsamples = BDataSamples(lapply(1:10,function(i) runif(100)),basis=B2Unif01Basis5)
Xhist    = as.BDataFrequencyHistograms(Xsamples)

Build a distributional dataset as histograms from a list

Description

Constructor for a distributional data set of class "BDataFrequencyHistograms" out of a list of either precalculated histograms or of numeric vectors.

Usage

## S3 method for class 'list'
as.BDataFrequencyHistograms(X, breaks, basis, ...)

Arguments

Value

A distributional data container of class ['BDataFrequencyHistograms'].

See Also

Other BDataHistograms: BDataFrequencyHistograms(), as.BDataDensityHistograms(), as.BDataDensityHistograms.list(), getBreaks(), plot.BDataHistograms()

Examples

x = lapply(rep(1000,4), runif)
as.BDataFrequencyHistograms(x, breaks="Sturges",basis=B2Unif01Basis5)

Represent data as a BDataHistogram (BDataDensityHistogram or BDataFrequencyHistogram

Description

The function tries to guess whether its datasets, distributions, Histograms,...

Usage

as.BDataHistograms(X, ...)

## S3 method for class 'list'
as.BDataHistograms(
  X,
  ...,
  breaks = NULL,
  basis = NULL,
  nsim = NULL,
  seed = NULL
)

## S3 method for class 'BDataHistograms'
as.BDataHistograms(
  X,
  ...,
  breaks = NULL,
  basis = NULL,
  nsim = NULL,
  seed = NULL
)

## S3 method for class 'histogram'
as.BDataHistograms(X, ..., breaks = NULL, basis, nsim = NULL, seed = NULL)

## S3 method for class 'BDataFrequencyHistograms'
as.BDataHistograms(X, ...)

## S3 method for class 'BDataDensityHistograms'
as.BDataHistograms(X, ...)

Arguments

Value

either a BDataFrequencyHistograms (if counts are availalbe) , or a BDataDensityHistograms object if only densities are known

Methods (by class)

• as.BDataHistograms(list): method for lists of possibly heterogenious data

• as.BDataHistograms(BDataHistograms): checks breaks if given

• as.BDataHistograms(histogram): checks breaks if given

• as.BDataHistograms(BDataFrequencyHistograms): identity converter

• as.BDataHistograms(BDataDensityHistograms): identity converter

See Also

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

myBreaks = seq(0,1,by=0.2)
as.BDataHistograms(replicate(10,runif(100),simplify=FALSE),
                   breaks=myBreaks,basis=B2Unif01Basis5)

Converts to a BDataLogDensities object

Description

This represents BayesSpace data by a list of log-density functions (the logarithm of the density with respect to the reference measure). Log-densities are the natural representation for Bayes-space operations, where adding distributions corresponds to adding their log-densities.

Usage

as.BDataLogDensities(x, ...)

## S3 method for class 'BDataDensities'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogPDFs'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataPDFs'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSplines'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogDensities'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSamples'
as.BDataLogDensities(
  x,
  ...,
  method = c("Fisher", "kde"),
  basis = getBayesSpaceBasis(x)
)

## S3 method for class 'list'
as.BDataLogDensities(x, ..., basis)

## S3 method for class 'BDataHistograms'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

Arguments

Value

A BDataLogDensities object.

Methods (by class)

• as.BDataLogDensities(BDataDensities): method for BDataDensities

• as.BDataLogDensities(BDataLogPDFs): method for BDataLogPDFs

• as.BDataLogDensities(BDataPDFs): method for BDataPDFs

• as.BDataLogDensities(BDataSplines): constructs the projected functions

• as.BDataLogDensities(BDataLogDensities): calls [BDataLogDensities.BDataLogDensities]

• as.BDataLogDensities(BDataSamples): constructs a density from the samples

• as.BDataLogDensities(list): constructs a density from the samples

• as.BDataLogDensities(BDataHistograms): constructs a density from the samples (TODO: Uses a very crude heuristic to estimate the log density)

See Also

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

as.BDataLogDensities(list(sin,cos),basis=B2N01)

Converts to representations as Log Probability Density functions

Description

A probability density function or Lebesgue density of a measure can only be computed for Lebesgue continuous distributions. As a dataset of BayesSpace elements (BData) they can be encoded as BDataLogPDFs objects. This function converts other representations to this representation.

Usage

as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataDensities'
as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogDensities'
as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSplines'
as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataPDFs'
as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogPDFs'
as.BDataLogPDFs(x, ..., basis)

## S3 method for class 'BDataDensityHistograms'
as.BDataLogPDFs(x, ..., background = 0, basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataFrequencyHistograms'
as.BDataLogPDFs(x, ..., background = 0, basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSamples'
as.BDataLogPDFs(
  x,
  ...,
  method = c("Fisher", "kde"),
  basis = getBayesSpaceBasis(x)
)

Arguments

Value

a BDataLogPDFs object

Methods (by class)

• as.BDataLogPDFs(BDataDensities): method for BDataDensities multiplies with reference measure density

• as.BDataLogPDFs(BDataLogDensities): method for BDataLogDensities adds the log density of the reference measure

• as.BDataLogPDFs(BDataSplines): method for BDataSplines

• as.BDataLogPDFs(BDataPDFs): method for BDataLogPDFs exponentiates and multiplies with reference measure density

• as.BDataLogPDFs(BDataLogPDFs): method for BDataLogPDFs possibly changes the basis

• as.BDataLogPDFs(BDataDensityHistograms): method for BDataDensityHistograms interprets histogram as density.

• as.BDataLogPDFs(BDataFrequencyHistograms): method for BDataFrequencyHistograms

• as.BDataLogPDFs(BDataSamples): method for BDataSamples

See Also

[BDataLogPDFs], [as.BDataPDFs]

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

X <- BDataDensities(list(\(x)(1),\(x)(x)),basis=B2N01Basis5)
Xlpdfs = as.BDataLogPDFs(X)
Xlpdfs
X1 = as.BDataLogPDFs(as.BDataLogDensities(X))
X2 = as.BDataLogPDFs(as.BDataLogPDFs(X))
X3 = as.BDataLogPDFs(Xlpdfs)
set.seed(47)
Xs = BDataSamples(replicate(7,runif(200),simplify=FALSE),B2Unif01Basis5)
X4 = as.BDataLogPDFs(Xs) # TODO
X5 = as.BDataLogPDFs(as.BDataFrequencyHistograms(Xs,breaks=seq(0,1,by=0.1)))
X6 = as.BDataLogPDFs(as.BDataDensityHistograms(Xs,breaks=seq(0,1,by=0.1)^2))
# plot(Xlpdfs)

Converts to representations as Probability Density functions

Description

A probability density function or Lebesgue density of a measure can only be computed for Lebesgue continuous distributions. As a dataset of BayesSpace elements (BData) they can be encoded as BDataPDFs objects. This function converts other representations to this representation.

Usage

as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataDensities'
as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogDensities'
as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogPDFs'
as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataPDFs'
as.BDataPDFs(x, ..., basis)

## S3 method for class 'BDataDensityHistograms'
as.BDataPDFs(x, ..., background = 0, basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataFrequencyHistograms'
as.BDataPDFs(x, ..., background = 0, basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSamples'
as.BDataPDFs(
  x,
  ...,
  method = c("Fisher", "kde"),
  basis = getBayesSpaceBasis(x)
)

## S3 method for class 'BDataSplines'
as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

Arguments

Value

a BDataPDFs object

Methods (by class)

• as.BDataPDFs(BDataDensities): method for BDataDensities multiplies with reference measure density

• as.BDataPDFs(BDataLogDensities): method for BDataLogDensities exponentiates and multiplies with reference measure density

• as.BDataPDFs(BDataLogPDFs): method for BDataLogPDFs exponentiates and multiplies with reference measure density

• as.BDataPDFs(BDataPDFs): method for BDataPDFs possibly changes the basis

• as.BDataPDFs(BDataDensityHistograms): method for BDataDensityHistograms interprets histogram as density.

• as.BDataPDFs(BDataFrequencyHistograms): method for BDataFrequencyHistograms

• as.BDataPDFs(BDataSamples): method for BDataSamples

• as.BDataPDFs(BDataSplines): method for BDataSplines

See Also

[BDataPDFs]

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

X <- BDataDensities(list(\(x)(1),\(x)(x)),basis=B2N01Basis5)
Xpdfs = as.BDataPDFs(X)
Xpdfs
X1 = as.BDataPDFs(as.BDataLogDensities(X))
X2 = as.BDataPDFs(as.BDataLogPDFs(X))
X3 = as.BDataPDFs(Xpdfs)
set.seed(47)
Xs = BDataSamples(replicate(7,runif(200),simplify=FALSE),B2Unif01Basis5)
# X4 = as.BDataPDFs(Xs) # TODO
X5 = as.BDataPDFs(as.BDataFrequencyHistograms(Xs,breaks=seq(0,1,by=0.1)))
X6 = as.BDataPDFs(as.BDataDensityHistograms(Xs,breaks=seq(0,1,by=0.1)^2))
# plot(Xpdfs)

Converts to representations as BDataSamples

Description

Convert any representation to "BDataSamples", most commonly by simulating the data by means of [simulate.BData()].

Usage

as.BDataSamples(x, ...)

## S3 method for class 'BData'
as.BDataSamples(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSamples'
as.BDataSamples(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataUnivariateSamples'
as.BDataSamples(x, ..., basis = getBayesSpaceBasis(x))

Arguments

Value

a BDataSamples object

Methods (by class)

• as.BDataSamples(BData): for any "BData" object

• as.BDataSamples(BDataSamples): for "BDataSamples" object, as identity or to change the basis

• as.BDataSamples(BDataUnivariateSamples): for "BDataUnivariateSamples" object, as identity or to change the basis

See Also

[simulate.BData]

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSplines(), idt.BData(), toHistogram()

Convert distributional data to a spline representation

Description

Returns a dataset of densities represented by zero-integral B-spline (ZB-spline) coefficients in a [BDataSplines] object. The spline representation is the compact form used for computation (PCA, distances, inner products). The conversion relies on a [BayesSpaceBasis] that is either taken from the object or supplied via 'basis'.

Usage

as.BDataSplines(X, ...)

## S3 method for class 'BDataLogDensities'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataLogPDFs'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataPDFs'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataDensities'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataSplines'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataSamples'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataGeneral'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class ''function''
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'histogram'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BayesSpaceBasis'
as.BDataSplines(X, ...)

Arguments

Value

A distributional data container of class ['BDataSplines'].

Methods (by class)

• as.BDataSplines(BDataLogDensities): method for BDataLogDensities

• as.BDataSplines(BDataLogPDFs): method for BDataLogPDFs

• as.BDataSplines(BDataPDFs): method for BDataPDFs

• as.BDataSplines(BDataDensities): method for BDataDensities

• as.BDataSplines(BDataSplines): method for BDataSplines, eventually reprojects if basis given

• as.BDataSplines(BDataSamples): method for BDataSamples

• as.BDataSplines(BDataGeneral): method for BDataGeneral

• as.BDataSplines(`function`): method for function interpreted as density with respect to the reference measure.

• as.BDataSplines(histogram): method for function interpreted as density with respect to the reference measure. (TODO: Uses a very crude heuristic to estimate the log density)

• as.BDataSplines(BayesSpaceBasis): the distributions corresponding to the basis elements.

Note

When converting from densities, log-densities or PDFs, the spline coefficients are the L2 projection onto the (non-orthonormal) ZB-spline basis: they solve the normal equations 'S c = <f, zeta>' with the Gram matrix 'S = getMetricTensor(basis)' (see [getMetricTensor]). The inner products '<f, zeta>' are estimated by Monte-Carlo integration over the reference-measure sample, so the coefficients carry a small sampling error that decreases as the sample size grows.

See Also

[as.BDataDensities], [BayesSpaceBasis], [getMetricTensor]

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), idt.BData(), toHistogram()

Examples

X = BDataDensities(list(\(x) x, \(x) x^2),basis=B2Unif01Basis5)
as.BDataSplines(X)
as.BDataSplines(as.BDataSplines(X))
as.BDataSplines(as.BDataLogDensities(X))
as.BDataSplines(BDataSamples(replicate(10,runif(100),simplify=FALSE),basis=B2Unif01Basis5))

Finds a BayesSpaceBasis representation of the object

Description

Finds a BayesSpaceBasis representation of the object

Usage

as.BayesSpaceBasis(x, ...)

## S3 method for class 'BayesSpaceBasis'
as.BayesSpaceBasis(x, ...)

## S3 method for class 'BayesSpace'
as.BayesSpaceBasis(x, ...)

Arguments

Value

a [BayesSpaceBasis] object

Methods (by class)

• as.BayesSpaceBasis(BayesSpaceBasis): returns x

• as.BayesSpaceBasis(BayesSpace): Calls 'BayesSpaceBasis(x,...)' to create an appropriate basis

See Also

[BayesSpaceBasis]

Examples

as.BayesSpaceBasis(B2Unif01Basis5)
as.BayesSpaceBasis(B2Unif01)

Evaluates a BayesSpaceBasis

Description

The basis is represented by log-densities. The function is vectorized, i.e. multiple x values may be provided.

Usage

basisEval(basis, x, ...)

## S3 method for class 'BayesSpaceBasis'
basisEval(basis, x, ...)

## S3 method for class 'basisfd'
basisEval(basis, x, ...)

## S3 method for class 'BayesSpaceBasis'
as.function(x, ...)

Arguments

Value

a matrix with the evaluation for one each of the x in a row.

Methods (by class)

• basisEval(BayesSpaceBasis): the standard method for BayesSpaceBasis

• basisEval(basisfd): a method implementation for basisfd

Functions

• as.function(BayesSpaceBasis): the basis can also be converted to a function or called directly

Examples

x = seq(0,1,by=0.1)
basisEval(B2Unif01Basis5,x)

Biplot for PCA of Distributional Data

Description

Creates a biplot that simultaneously displays the principal component scores (observations) and loadings (variables/coefficients) in the same plot. This visualization helps understand relationships between observations and the underlying coefficient structure.

Usage

## S3 method for class 'princomp.BData'
biplot(
  x,
  choices = 1:2,
  scale = 0,
  pc.biplot = FALSE,
  main = NULL,
  col = c("black", "red"),
  cex = c(0.8, 0.8),
  xlabs = NULL,
  ylabs = NULL,
  expand = 1,
  arrow.len = 0.1,
  ...
)

Arguments

Details

## Biplot Interpretation

In a biplot:

• Observations (numbers/text) represent individual distributions. Observations close together have similar distribution shapes.

• Loading arrows (red arrows labeled C1, C2, etc.) represent spline basis coefficients. The direction and length indicate how each coefficient contributes to the principal components.

• Projecting an observation onto a loading arrow approximates that observation's value on that coefficient (relative to the mean).

• Arrow angles: Arrows pointing in similar directions indicate positively correlated coefficients; opposite directions indicate negative correlation; perpendicular arrows indicate uncorrelated coefficients.

## Note on Coefficient Interpretation

Unlike PCA on raw data where variables have direct meaning, the "variables" here are spline basis coefficients. These are abstract and their individual interpretation may not be intuitive. For understanding how distributions vary, the plot.princomp.BData function with 'type="eigenfunctions"' provides a more interpretable visualization showing how distribution shapes change along each PC direction.

## Why Biplot Works for ZB-Splines

The biplot is meaningful for the default ZB-spline basis because ZB-splines have **local support**: each basis function is non-zero only in a specific region of the domain. This means coefficient i primarily reflects the behavior of the distribution near knot region i. When a loading arrow for coefficient Ci points strongly in a PC direction, it indicates that variation in that region of the domain is associated with that principal component.

**Important:** If orthonormal splines were used instead, the biplot would lose this interpretability because orthonormal basis functions have global support (they are non-zero across the entire domain), so each coefficient would mix information from everywhere rather than representing a local region.

Value

Invisibly returns a list containing the scaled 'scores' and 'loadings' matrices as plotted.

Author(s)

Matthias Templ

References

Gabriel, K. R. (1971). The biplot graphic display of matrices with application to principal component analysis. Biometrika, 58(3), 453-467.

See Also

princomp.BData for performing the PCA.

plot.princomp.BData for other visualization options, especially 'type="loadings"' for functional interpretation.

biplot for the generic biplot function.

Other PCA: plot.princomp.BData(), plot.princomp.BData.grouped(), plot.princomp.BData.multivariate(), predict.princomp.BData(), princomp.BData(), screeplot.princomp.BData(), summary.princomp.BData()

Examples

## Create example PCA result
meanDist <- BDataPDFs(list(dunif), basis = B2Unif01Basis7)
gaussProc <- B2GaussianDistribution(meanDist, function(x,y) exp(-4*abs(x-y)^1.7))
pcaResult <- princomp(as.BDataSplines(simulate(gaussProc, 25, seed = 42)))

## Basic biplot
biplot(pcaResult)

## Biplot of PC2 vs PC3
biplot(pcaResult, choices = c(2, 3))

## Correlation biplot (scale=1)
biplot(pcaResult, scale = 1)

## Customized biplot
biplot(pcaResult,
       col = c("blue", "darkgreen"),
       cex = c(1, 0.7),
       expand = 1.5,
       main = "PCA Biplot of Distributional Data")

Functional Boxplot for BData objects

Description

Currently this function used the [fda::fbplot] function on the evaluated functions in compbination with [depthRank] as depth measure.

Usage

## S3 method for class 'BData'
boxplot(
  x,
  ...,
  depth,
  depthType = c("depthquantile", "spatialdepth", "meandistance", "median"),
  support = getSupport(x),
  SD = supportDiscretization(support, nEval),
  nEval = 513,
  xlim = getSupport1DPlottingLimits(support)
)

## S3 method for class 'BDataSplines'
boxplot(x, ...)

## S3 method for class 'BDataSamples'
boxplot(x, ...)

Arguments

Value

'NULL' invisibly, called for the side effect of producing a functional boxplot.

Functions

• boxplot(BDataSplines): method for BDataSplines uses classical boxplot for the coefficients.

• boxplot(BDataSamples): method for BDataSamples uses classical boxplot for the the datasets.

See Also

fda::fbplot

Examples

X = BDataSplines(matrix(rnorm(10*5),ncol=5),basis=B2Unif01Basis5)
boxplot(X)
boxplot(clr(X),depthType="meandistance")
boxplot(clr(X))

Compute the cartesian product

Description

Compute the cartesian product

Usage

cartesianProduct(...)

## S3 method for class 'Support'
cartesianProduct(...)

## S3 method for class 'ReferenceMeasure'
cartesianProduct(...)

## S3 method for class 'BayesSpace'
cartesianProduct(...)

## S3 method for class 'BayesSpaceBasis'
cartesianProduct(...)

Arguments

Value

the Cartesian product of the objects given. Check [SupportCartesianProduct()] for details about this structure.

The cartesian product of a space (like a vector space or support) is the space of tuples, where each tuple element comes from the space of the same position or name. The cartesian product of measures is the product measure, which gives a cartesian product of sets the product of the measures. The cartesian product of functions(like basis) is a function on the cartesian product of supports and gives the product of the function values for each of the components.

Methods (by class)

• cartesianProduct(Support): method for Supports

• cartesianProduct(ReferenceMeasure): method for ReferenceMeasure

• cartesianProduct(BayesSpace): method for BayesSpace

• cartesianProduct(BayesSpaceBasis): method for BayesSpaceBasis

See Also

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceBasisCartesianProduct(), BayesSpaceCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

Expected values of functions of distributions

Description

Ideally this should always be called on the original samples.

Usage

caseE(x, X, ...)

## S3 method for class 'BDataSamples'
caseE(x, X, ...)

## S3 method for class 'BData'
caseE(x, X, ...)

Arguments

Value

a vector, matrix or with the Expected values for each element of x. if the expected value is multivariate the last dimension iterates over the cases in x.

Methods (by class)

• caseE(BDataSamples): method for BDataSamples simply uses means

• caseE(BData): method for distributions

See Also

Other caseStatistics: caseLogRatios(), caseMeans(), caseQuantiles(), caseSds(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseE(X,X=function(x) x,probs=c(0.25,0.75))
caseE(X,X=function(x) x^2,probs=0.5)

Log-density ratios for each distribution (case)

Description

Density log ratios independent of the reference measure and any other values.

Usage

caseLogRatios(x, numerator, denominator, ...)

Arguments

Value

a dataset of the same number of cases as x of the corresponding log rations

See Also

Other caseStatistics: caseE(), caseMeans(), caseQuantiles(), caseSds(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X  = BDataPDFs(DTD$shapePDF,basis=B2Unif01Basis5)
caseLogRatios(X,0.8,0.9)

Expected values (means) of real-valued distributions

Description

Ideally this should always be called on the original samples.

Usage

caseMeans(x, ...)

Arguments

Value

a vector or a matrix (with different cases in columns) of the means of the distributions in the dataset. For multivariate (vector-valued) means, the last dimension iterates over the elements of x.

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseQuantiles(), caseSds(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseMeans(X)
X = BDataSamples(DTD$data,basis=B2Unif01Basis5)
caseMeans(X)

Quantiles of distributions in a data set of distributions

Description

Ideally this should always be called on the original samples.

Usage

caseQuantiles(X, ...)

## S3 method for class 'BDataSamples'
caseQuantiles(X, ...)

## S3 method for class 'BData'
caseQuantiles(X, ...)

Arguments

Value

a vector or a matrix (with different probabilities in colums) of quantiles of the distributions in the dataset.

Methods (by class)

• caseQuantiles(BDataSamples): method for BDataSamples, simply calles [stats::quantile]

• caseQuantiles(BData): method for BData not yet implemented

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseSds(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseQuantiles(X,probs=c(0.25,0.75))
caseQuantiles(X,probs=0.5)

Standard deviations of real-valued distributions

Description

Ideally this should always be called on the original samples.

Usage

caseSds(x, ...)

Arguments

Value

a vector the standard deviations of the distributions in the dataset.

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseQuantiles(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseSds(X)

Variances of real-valued distributions

Description

Ideally this should always be called on the original samples.

Usage

caseVars(x, ..., X = function(x) x)

## S3 method for class 'BDataSamples'
caseVars(x, ..., X = function(x) x)

## S3 method for class 'BData'
caseVars(x, ..., X = function(x) x)

Arguments

Value

A vector or tensor of variances. In case of multiple dimensional means, the last dimension iterates over the elements of x.

Methods (by class)

• caseVars(BDataSamples): method for samples

• caseVars(BData): method for distributions

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseQuantiles(), caseSds(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseVars(X)
X = BDataSamples(DTD$data,basis=B2Unif01Basis5)
caseVars(X)

Try to guess the support from cdf

Description

This function essentially tries to infere the support from a cdf.

Usage

cdf2Support(cdf, alpha = 0.01, eps = 1e-07, digits = 5)

Arguments

Value

a 'Support' object describing the support inferred from the 'cdf'. Its 'a' and 'b' elements give the (possibly infinite) lower and upper support boundaries, its 'lim' element the effective lower and upper boundaries (a numeric vector of length two, cut at probability 'alpha/2' and '1 - alpha/2' when the support is unbounded), and 'X' any detected points of positive probability mass.

See Also

Other Support: Support(), pdf2Support(), plotFuncs(), sample2Support()

Examples

cdf2Support(pnorm)
cdf2Support(punif)
cdf2Support(
  function(x){
  ifelse(x< -pi,0,((pmax(x,0)+pi)/pi)*0.5) +
      ifelse(x<0,0,0.25+(pmax(x,pi)/pi)*0.25)
  }
)

numerically compute a density from a cdf

Description

The function tries to keep the (open) support, by using a one sided derivative at limits of the support, while using a two sided in the middle of the support.

Usage

cdf2density(cdf, ..., .args = list(...), eps = 1e-07, cdf2 = NULL)

Arguments

Value

a vectorized function giving the density obtained by numerically differentiating 'cdf' (a one-sided difference at the boundaries of the support, a two-sided difference in the interior). If 'cdf2' is supplied the result is the density relative to that reference measure instead of the Lebesgue density.

Examples

dUnif = cdf2density(punif,min=-0.2,0.4)
x=seq(-1,2,by=0.01)
plot(x,dUnif(x))

Centers the representation

Description

This centers an representation in a way that a direct converstion to a log Density has integral 0.

Usage

cen(X, ..., center = TRUE)

## S3 method for class ''function''
cen(X, measure = getReferenceMeasure(X), ..., center)

## S3 method for class 'BDataDensities'
cen(X, ..., center)

## S3 method for class 'BDataPDFs'
cen(X, ..., center)

## S3 method for class 'BDataLogDensities'
cen(X, ..., center)

## S3 method for class 'BDataLogPDFs'
cen(X, ..., center)

## S3 method for class 'BayesSpaceBasis'
cen(X, ..., center)

Arguments

Details

Centring subtracts the integral mean (with respect to the reference measure) so that the result integrates to zero; equivalently it makes the clr representative of the distribution centred.

Value

and object of the same type as the incoming, but closed.

Methods (by class)

• cen(`function`): method for functions

• cen(BDataDensities): method for BDataDensities

• cen(BDataPDFs): method for BDataPDFs

• cen(BDataLogDensities): method for BDataLogDensities

• cen(BDataLogPDFs): method for BDataLogPDFs

• cen(BayesSpaceBasis): method for BayesSpaceBasis returns the centered logDensities

See Also

[clo]

Examples

cen(function(x) x^2,measure=ReferenceMeasureUniform(0,1))

Finds the centering constant(s)

Description

A centering constant is a real value, by which needs to be substracted from a function (of its log in case of densities) get an integral of 0.

Usage

centeringConstant(f, ...)

## S3 method for class ''function''
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BayesSpaceBasis'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataDensities'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataLogDensities'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataLogPDFs'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataPDFs'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataSplines'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

Arguments

Value

a (vector) of numeric centering constants. Might contain Inf in case of infinite measures and 0 in case of functions that are constant zeros

Methods (by class)

• centeringConstant(`function`): method for function

• centeringConstant(BayesSpaceBasis): method for BayesSpaceBasis

• centeringConstant(BDataDensities): method for BDataDensities

• centeringConstant(BDataLogDensities): method for BDataLogDensities

• centeringConstant(BDataLogPDFs): method for BDataLogPDFs

• centeringConstant(BDataPDFs): method for BDataPDFs

• centeringConstant(BDataSplines): method for BDataSplines

Examples

# centering constant of f(x) = x^2 on Uniform(0,1)
centeringConstant(function(x) x^2, measure = ReferenceMeasureUniform(0,1))

centralityQuantile

Description

Computes quantile of distanceToCenter of data in a dataset, 0 is the median, 1 is the farthest point

Usage

centralityQuantile(x, ...)

## Default S3 method:
centralityQuantile(
  x,
  ...,
  d = distanceToCenter(x, ..., distType = distType),
  distType = c("meandistance", "median", "spatialdepth"),
  ties.method = "first"
)

centralityRank(x, ...)

## Default S3 method:
centralityRank(
  x,
  ...,
  d = distanceToCenter(x, ..., distType = distType),
  distType = c("meandistance", "median", "spatialdepth"),
  ties.method = "first"
)

Arguments

Value

A numeric vector in [0,1], 0 for the most central point and 1 for the most peripheral.

Methods (by class)

• centralityQuantile(default): default method

Functions

• centralityRank(): returns the rank rather than the quantile: an integer vector of ranks, 1 for the most central point up to the number of observations for the most peripheral.

• centralityRank(default): default method returning ranks

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseQuantiles(), caseSds(), caseVars(), distanceToCenter(), median.BData(), spatialDepth()

Examples

x = rnorm(100)
plot(x,centralityQuantile(x))
plot(x,centralityRank(x))

Closes to a given total measure

Description

This is a generic version of clo overwriting the one in compositions

Usage

clo(X, ..., close = TRUE)

## Default S3 method:
clo(X, ..., total = 1, close)

## S3 method for class ''function''
clo(X, measure = getReferenceMeasure(X), ..., total = 1, close)

## S3 method for class 'BDataDensities'
clo(X, ..., total = 1, close)

## S3 method for class 'BDataPDFs'
clo(X, ..., total = 1, close)

## S3 method for class 'BDataLogDensities'
clo(X, ..., total = 1, close)

## S3 method for class 'BDataLogPDFs'
clo(X, ..., total = 1, close)

## S3 method for class 'BayesSpaceBasis'
clo(X, ..., total = 1)

Arguments

Details

Closure rescales a (positive) function so that it integrates to 'total' (one by default) with respect to the reference measure, i.e. it projects the function to a proper density and thereby picks the representative of its Bayes-space class.

Value

and object of the same type as the incoming, but closed.

Methods (by class)

• clo(default): the default is [compositions::clo]

• clo(`function`): method for functions

• clo(BDataDensities): method for BDataDensities

• clo(BDataPDFs): method for BDataPDFs

• clo(BDataLogDensities): method for BDataLogDensities

• clo(BDataLogPDFs): method for BDataLogPDFs

• clo(BayesSpaceBasis): method for BayesSpaceBasis returns the closed logDensities

See Also

[compositions::clo] for parameters of the default, [cen]

Examples

clo(function(x) x^2,measure=ReferenceMeasureUniform(0,1))

Closure methods for BData objects

Description

Closure functions to enforce a distributional representation to integrate to one, if needed.

Usage

## S3 method for class 'BDataHistograms'
clo(X, ...)

Arguments

Value

A fitting object with elements now integrating to one, depending on the specific method.

Examples

# close frequency histograms to density histograms integrating to one
counts = rbind(c(10,20,30,40), c(25,25,25,25))
fh = BDataFrequencyHistograms(counts, breaks = seq(0,1,by=0.25),
                              basis = B2Unif01Basis5)
clo(fh)

Finds the closing constant(s)

Description

A closing constant is a real value, by which the function (or its Exp in case of log functions) needs to be divided to get the representative with measure 1.

Usage

closingConstant(f, ...)

## S3 method for class ''function''
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BayesSpaceBasis'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataDensities'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataLogDensities'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataLogPDFs'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataPDFs'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataSplines'
closingConstant(f, measure = getReferenceMeasure(f), ...)

Arguments

Value

a (vector) of numeric closing constants. Might contain Inf in case of infinite measures and 0 in case of functions that are constant zeros

Methods (by class)

• closingConstant(`function`): method for function

• closingConstant(BayesSpaceBasis): method for BayesSpaceBasis

• closingConstant(BDataDensities): method for BDataDensities

• closingConstant(BDataLogDensities): method for BDataLogDensities

• closingConstant(BDataLogPDFs): method for BDataLogPDFs

• closingConstant(BDataPDFs): method for BDataPDFs

• closingConstant(BDataSplines): method for BDataSplines

Examples

# closing constant of f(x) = x^2 on Uniform(0,1)
closingConstant(function(x) x^2, measure = ReferenceMeasureUniform(0,1))

compute the clr transform

Description

This is a generic version of clr overwriting the one in compositions

Usage

clr(x, ...)

## Default S3 method:
clr(x, ...)

## S3 method for class 'BData'
clr(x, ...)

Arguments

Details

The centred log-ratio map sends a distribution to its log-density centred to integrate to zero, i.e. to its coordinates in the L2 space underlying the Bayes space. It is an isometry from the Bayes space onto the centred subspace of that L2 space, so Bayes-space distances and inner products become the ordinary L2 ones.

Value

the centred log-ratio (clr) representation of 'x'. For 'BData' objects this is the centred log-density representation (an object of class 'BDataLogDensities' whose log-density integrates to zero); the default method returns the clr transform computed by [compositions::clr].

Methods (by class)

• clr(default): the default is [compositions::clr]

• clr(BData): returns a centered representation as BDataLogDensities

Examples

X1 = BDataPDFs(basis=B2Unif01Basis5,
   list(
    \(x)(dbeta(x,1,2)),
    \(x)(dbeta(x,1,1)),
    \(x)(dbeta(x,1.5,0.7))
   ) )
plot(X1,main="Beta Densities on Unif(0,1)")
plot(clr(X1),main="CLRs of Beta Densities on Unif(0,1)")

Computes the matrix mapping an L2-space basis to a Bayes-Space-Basis.

Description

This computes a (n-1)x1 matrix mapping the basis results of the L2 basis to L^2_0 basis for a subspace of the Log-Space of a BayesSpace.

Usage

computeL22BayesSpacesBasisMatrix(x, ...)

Arguments

Value

the transformation matrix 'A' mapping the L2 (B-spline) basis to the ZB-spline basis of the Bayes space. For a B-spline basis with 'nbasis' functions this is an '(nbasis - 1) x nbasis' matrix, i.e. one row per ZB-spline basis function.

create the transform matrix A from a B-spline basis (fda::create.bspline.basis())

Description

create the transform matrix A from a B-spline basis (fda::create.bspline.basis())

Usage

## S3 method for class 'basisfd'
computeL22BayesSpacesBasisMatrix(x, ...)

Arguments

Value

matrix

Examples

computeL22BayesSpacesBasisMatrix(
         fda::create.bspline.basis(rangeval = c(0,1),
                                  nbasis=9,
                                  norder=4,
                                  breaks=seq(0,1,length.out=7)
                                  )
                                )
# a cubic B-spline basis with 8 functions yields a 7 x 8 matrix
A = computeL22BayesSpacesBasisMatrix(
        fda::create.bspline.basis(c(0,1), nbasis=8, norder=4)
    )
dim(A)

computes the limit of a function

Description

computes the limit of a function

Usage

computeLimit(f, x, d = 1)

Arguments

Value

the limit it will still return

Examples

computeLimit(\(x)(2*(x-pi)/(x-pi)),x=pi)
np = computeLimit(\(x)(pi*(x-pi)/(x-pi)),x=pi)
np-pi

Distance method for BData objects

Description

Computes inter-distances between all objects of 'x'

Usage

## S3 method for class 'BData'
dist(x, ...)

## S3 method for class 'BDataLogDensities'
dist(x, ...)

## S3 method for class 'BDataSplines'
dist(x, ...)

Arguments

Value

a distance object, as 'stats::dist()'. The implementations rely on transformation to BDataSplines; the spline coefficients are first mapped to orthonormal coordinates via the Gram matrix of the basis, so that the default 'method = "euclidean"' yields the exact Bayes-space (clr-L2) distance also for non-orthonormal bases such as ZB-splines. Other 'method's (Manhattan, Minkowski, ...) operate on these orthonormal coordinates and depend on the chosen basis.

Functions

• dist(BDataLogDensities): calculations for BDataLogDensities objects

• dist(BDataSplines): calculations for BDataSplines objects

Examples

lbdl = BDataLogDensities(basis=B2N01Basis5, closed=TRUE,
  list(
    \(x)(dnorm(x,2,1,log=TRUE)),
    \(x)(dnorm(x,1,1,log=TRUE)),
    \(x)(dnorm(x,-1,1,log=TRUE)),
    \(x)(dnorm(x,-2,1,log=TRUE))
    ))
dist(lbdl)

Distance to Center

Description

Computes the distance of a datapoint to the center of a dataset, for the definition of central regions.

Usage

distanceToCenter(x, na.rm = TRUE, ...)

## Default S3 method:
distanceToCenter(
  x,
  na.rm = TRUE,
  ...,
  distType = c("meandistance", "median", "spatialdepth")
)

Arguments

Value

A numeric vector of distances of each observation to the centre of the dataset.

Methods (by class)

• distanceToCenter(default): default method

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseQuantiles(), caseSds(), caseVars(), centralityQuantile(), median.BData(), spatialDepth()

Examples

x = rnorm(100)

plot(x,distanceToCenter(x,distType="median"),col="green")
points(x,distanceToCenter(x,distType="meandist"))
points(x,distanceToCenter(x,distType="spatialdepth"),col="red")

data(DTD)
Xpdf    = BDataPDFs(DTD$shapePDF,basis=B2Unif01Basis5)
distanceToCenter(Xpdf)

Returns the argument, but ensures that it has a certain class.

Description

If the x has not the given type it, the function prints information about the problem and stops with an error.

Usage

ensureType(x, type)

Arguments

Value

x

Examples

ensureType(3,"numeric")
ensureType(diag(2),"matrix")

Computes expectations from reference measures

Description

Computes expectations from reference measures

Usage

expectation(measure, f, ..., method)

## S3 method for class 'ReferenceMeasure'
expectation(
  measure,
  f,
  ...,
  method = c("default", "sample", "MonteCarlo", "integration"),
  tol = 0.01
)

## S3 method for class 'ReferenceMeasureDiscrete'
expectation(measure, f, ..., method = c("sample"))

Arguments

Value

the numerically computed expected value of f with respect to the measure. The method

Methods (by class)

• expectation(ReferenceMeasure): general method supporting only the "sample" method.

• expectation(ReferenceMeasureDiscrete): a computing the results explicitly

See Also

Other ReferenceMeasures: B2N01, ReferenceMeasureDiscrete(), ReferenceMeasureTransformed(), ReferenceMeasureUniform(), StdReferenceMeasureUnif01, getSample(), isInDomain()

Examples

m <- ReferenceMeasureUniform(0,10)
expectation(m, function(x) x^2)

integrate a function over the uniform reference measure

Description

integrate a function over the uniform reference measure

Usage

## S3 method for class 'ReferenceMeasureUniform'
expectation(measure, f, ..., method = c("sample", "integrate"))

Arguments

Value

the integral of f over c(x[1],x[2])

Examples

RMU <- ReferenceMeasureUniform(a=-5,b=5)
expectationRMU <- expectation(measure=RMU,f=cos,method="integrate")
expectationRMU

Finds jumps of a function of a certain minimum height

Description

This is a helper tool to infere supports from cummulative density functions.

Usage

findJumps(f, a, b, minJump = 1/100, eps = 1e-07, digits = 5, n = 100)

Arguments

Value

a vector of numerical values at which the function f jumps by at least minJump in an eps distance, possibly rounded to digits.

Examples

findJumps(function(x) pbinom(x,4,0.2),-pi,17)
findJumps(pnorm,-3,3)
p <- findJumps(function(x) ifelse(x<pi,x,x+pi),0,6,eps=0)
p==pi # TRUE

Shortcut for creating functions of x

Description

'fx(body)' is essentially equal to 'function(x) body'. This can be used to quickly create functions, with minimal syntactic burden.

Usage

fx(
  .body,
  template = function(x) {
 },
  .env = parent.frame()
)

Arguments

Value

a function with .body as body

Examples

f = fx(sin(x)+cos(x))
f
f(pi)

Generic function returning the Bayes space associated to the given object.

Description

Generic function returning the Bayes space associated to the given object.

Usage

getBayesSpace(X, ...)

## S3 method for class 'BayesSpace'
getBayesSpace(X, ...)

## S3 method for class 'BayesSpaceBasis'
getBayesSpace(X, ...)

## S3 method for class ''NULL''
getBayesSpace(X, ..., Default)

## S3 method for class 'BData'
getBayesSpace(X, ...)

## S3 method for class 'ReferenceMeasure'
getBayesSpace(X, ...)

## Default S3 method:
getBayesSpace(X, ..., Default = attr(X, "BayesSpace"))

## S3 method for class 'L2Space'
getBayesSpace(X, ...)

## S3 method for class 'L2Space'
getL2Space(X, ...)

Arguments

Value

The associated [BayesSpace] object (class '"BayesSpace"').

Methods (by class)

• getBayesSpace(BayesSpace): method for BayesSpace returning itself

• getBayesSpace(BayesSpaceBasis): method for BayesSpaceBasis

• getBayesSpace(`NULL`): method for NULL, returns default

• getBayesSpace(BData): method for BData

• getBayesSpace(ReferenceMeasure): creates the BayesSpace for the reference measure

• getBayesSpace(default): default method tries to get an attribute

• getBayesSpace(L2Space): returns BayesSpace with same reference measure

Functions

• getL2Space(L2Space): method for BayesSpace returning itself

See Also

Other getBayesSpace: getL2Space()

Examples

B2Unif01Basis5
unclass(B2Unif01Basis5)
getBayesSpace(B2Unif01Basis5)

Construct the product Bayes Space for a product reference measure

Description

Construct the product Bayes Space for a product reference measure

Usage

## S3 method for class 'ReferenceMeasureCartesianProduct'
getBayesSpace(X, ...)

Arguments

Value

A BayesSpaceCartesianProduct corresponding to the cartesian Product of Bayes measures, itself also a Bayes measure.

See Also

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceBasisCartesianProduct(), BayesSpaceCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), cartesianProduct(), getCartesianFactors(), subBasisIndices()

Generic function returning the BayesSpaceBasis associated to the given object.

Description

Generic function returning the BayesSpaceBasis associated to the given object.

Usage

getBayesSpaceBasis(X, ...)

## S3 method for class 'BayesSpaceBasis'
getBayesSpaceBasis(X, ...)

## S3 method for class 'BData'
getBayesSpaceBasis(X, ...)

## S3 method for class 'BayesSpace'
getBayesSpaceBasis(X, ...)

## S3 method for class 'varKernel'
getBayesSpaceBasis(X, ...)

## S3 method for class 'variationKernel'
getBayesSpaceBasis(X, ...)

Arguments

Value

The associated [BayesSpaceBasis] object (class '"BayesSpaceBasis"').

Methods (by class)

• getBayesSpaceBasis(BayesSpaceBasis): returns itself

• getBayesSpaceBasis(BData): method for Datasets

• getBayesSpaceBasis(BayesSpace): constructs a default vasis from the reference measure. Use ... as further parameters to generic constructor.

• getBayesSpaceBasis(varKernel): method for varKernel

• getBayesSpaceBasis(variationKernel): method for variationKernel

See Also

[BayesSpaceBasis], [getBayesSpace], [BayesSpace]

Examples

getBayesSpaceBasis(B2Unif01)
set.seed(47)
X = BDataGeneral(list(function(x) x, function(x) x^2),basis=B2Unif01Basis5)
getBayesSpaceBasis(X)
getBayesSpace(X)
getReferenceMeasure(X)
getBayesSpaceBasis(B2Unif01Basis5)

the dimension of the basis

Description

the dimension of the basis

Usage

getBayesSpaceBasisDimension(basis)

## S3 method for class 'BayesSpaceBasis'
getBayesSpaceBasisDimension(basis)

Arguments

Value

a integer providing the dimension of the basis

Methods (by class)

• getBayesSpaceBasisDimension(BayesSpaceBasis): the method for a standard BayesSpaceBasis

Examples

getBayesSpaceBasisDimension(B2Unif01Basis5)
getBayesSpaceBasisDimension(B2Unif01Basis7)

Common utilities for BDataHistograms objects

Description

Common utilities for BDataHistograms objects

Usage

getBreaks(x)

## S3 method for class 'BDataHistograms'
getBreaks(x)

Arguments

Value

depeding on the function, this can print the content, extract the breaks attribute in a convenient fashion or do subsetting.

Methods (by class)

• getBreaks(BDataHistograms): Method for BDataHistograms

See Also

Other BDataHistograms: BDataFrequencyHistograms(), as.BDataDensityHistograms(), as.BDataDensityHistograms.list(), as.BDataFrequencyHistograms.list(), plot.BDataHistograms()

Examples

set.seed(47)
H = hist(rnorm(30))
getBreaks(H)
X = as.BDataFrequencyHistograms(list(H,H),basis=B2N01Basis5)
getBreaks(X)
X
X[1]

Decomposes a cartesian product into a list of its factors.

Description

Decomposes a cartesian product into a list of its factors.

Usage

getCartesianFactors(...)

Arguments

Value

If x is not a cartesian product the function should return list(x). if x is a cartesian product it should provide its factors. Check [SupportCartesianProduct()] for details about this latter structure.

See Also

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceBasisCartesianProduct(), BayesSpaceCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), cartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), subBasisIndices()

Gets number of elements in a Basis

Description

Returns the dimension of an object.

Usage

getDimension(x, ...)

## S3 method for class 'BayesSpaceBasis'
getDimension(x, ...)

## S3 method for class 'BDataSplines'
getDimension(x, ...)

## S3 method for class 'matrix'
getDimension(x, ...)

## S3 method for class 'numeric'
getDimension(x, ...)

Arguments

Value

an integer giving the number of dimensions of the object

Methods (by class)

• getDimension(BayesSpaceBasis): returns number of elements in the basis

• getDimension(BDataSplines): returns number of elements in the basis

• getDimension(matrix): method for matrix returns number of columns

• getDimension(numeric): method for numeric returns number of elements

Examples

getDimension(1:3)
getDimension(matrix(rnorm(10),ncol=2))
getDimension(B2N01Basis5)

Computes Discretization points for nonequidistant discretization

Description

This is a helper function to create nonequidistant discretizations

Usage

getDiscretizer(x, nEval = 513, ...)

## S3 method for class 'BayesSpaceBasis'
getDiscretizer(x, nEval = 513, ...)

## S3 method for class 'ReferenceMeasureDiscrete'
getDiscretizer(x, nEval, ...)

## S3 method for class 'ReferenceMeasureUniform'
getDiscretizer(x, nEval = 513, ...)

## S3 method for class 'ReferenceMeasureTransformed'
getDiscretizer(x, nEval = 513, ...)

Arguments

Value

an object that can be use together with its evaluation on object$all to create a DiscreteFunction

Methods (by class)

• getDiscretizer(BayesSpaceBasis): method for BayesSpaceBasis

• getDiscretizer(ReferenceMeasureDiscrete): method for ReferenceMeasure discrete

• getDiscretizer(ReferenceMeasureUniform): method for ReferenceMeasureUniform

• getDiscretizer(ReferenceMeasureTransformed): method for ReferenceMeasureTransformed

Examples

getDiscretizer(B2Unif01Basis5)
getDiscretizer(B2N01Basis5)

Generic function returning the L2Space associated to the given object.

Description

Generic function returning the L2Space associated to the given object.

Usage

getL2Space(X, ...)

## S3 method for class 'BayesSpaceBasis'
getL2Space(X, ...)

## S3 method for class ''NULL''
getL2Space(X, ..., Default)

## S3 method for class 'BData'
getL2Space(X, ...)

## S3 method for class 'ReferenceMeasure'
getL2Space(X, ...)

## Default S3 method:
getL2Space(X, ..., Default = getL2Space(attr(X, "BayesSpace")))

## S3 method for class 'BayesSpace'
getL2Space(X, ...)

Arguments

Value

The associated [L2Space] object (class '"L2Space"').

Methods (by class)

• getL2Space(BayesSpaceBasis): method for BayesSpaceBasis

• getL2Space(`NULL`): method for NULL, returns default

• getL2Space(BData): method for BData

• getL2Space(ReferenceMeasure): creates the BayesSpace for the reference measure

• getL2Space(default): default method tries to get an attribute

• getL2Space(BayesSpace): returns L2Space with same reference measure

See Also

Other getBayesSpace: getBayesSpace()

Examples

getL2Space(B2Unif01Basis5)
set.seed(47)
X = BDataGeneral(list(function(x) x, function(x) x^2),basis=B2Unif01Basis5)
getL2Space(X)
getL2Space(StdReferenceMeasureUnif01)

Metric tensor of a space or basis

Description

The metric tensor is the Gram matrix ‘S', used in '|v|^2 = v’ S v' to compute the metric of the space when 'v' is the vector of basis coefficients. 'S' contains the scalar products of the basis elements, 'S[i,j] = <psi_i, psi_j>'.

Usage

getMetricTensor(x, ...)

Arguments

Details

For an orthonormal basis 'S' is the identity matrix. The default zero-integral B-spline (ZB-spline) basis is **not** orthonormal, so 'S' is a banded, non-identity matrix; inner products ([scalarProduct]), norms ([norm.BData]), metric variance ([mvar][mvar.BData]) and PCA ([princomp.BData]) all rely on it. You can check whether a basis is orthonormal by comparing the result to 'diag(nrow(S))'.

Value

A symmetric positive-definite matrix 'S' (the Gram matrix / metric tensor), with 'S[i,j]' the inner product of basis elements 'i' and 'j'.

See Also

[BayesSpaceBasis], [scalarProduct], [princomp.BData]

Examples

getMetricTensor(B2Unif01Basis5)   # not the identity: ZB-splines are not orthonormal

returns the probability density function

Description

The probability density function, also known as the Lebesgue density of the measure is returned. In case of datasets a dataset of such functions is returned. (To be encoded as BDataPDF)

Usage

getPDF(X, ...)

## S3 method for class 'ReferenceMeasureUniform'
getPDF(X, ...)

## S3 method for class 'ReferenceMeasureTransformed'
getPDF(X, ...)

## S3 method for class 'BData'
getPDF(X, ...)

Arguments

Value

a function (or a list of functions if a dataset of distributions is given) that can compute the probably density function.

Methods (by class)

• getPDF(ReferenceMeasureUniform): method for ReferenceMeasureUniform

• getPDF(ReferenceMeasureTransformed): method for ReferenceMeasureUniform

• getPDF(BData): method for BData objects, returns BDataPDFs (TODO)

Examples

x = seq(-1,2,by=0.1)
f1 = getPDF(StdReferenceMeasureUnif01)
plot(x,f1(x))
f2 = getPDF(StdReferenceMeasureN01)
plot(x,f2(x),type="l")
f3 = getPDF(BDataLogDensities(list(sin),basis=B2Unif01Basis5))
plot(x,f3(x),type="l")
getPDF(BDataLogDensities(list(sin,cos,\(x)(1)),basis=B2Unif01Basis5))

Getter for the ReferenceMeasure of an object

Description

BayesSpaces and L2Spaces are defined on underlying reference measures. This function returns this measure.

Usage

getReferenceMeasure(x)

## S3 method for class 'BayesSpace'
getReferenceMeasure(x)

## S3 method for class 'L2Space'
getReferenceMeasure(x)

## S3 method for class 'ReferenceMeasure'
getReferenceMeasure(x)

## Default S3 method:
getReferenceMeasure(x)

Arguments

Value

The reference-measure object (class '"ReferenceMeasure"') underlying 'x'.

Methods (by class)

• getReferenceMeasure(BayesSpace): method for [BayesSpace]

• getReferenceMeasure(L2Space): method for [L2Space]

• getReferenceMeasure(ReferenceMeasure): method for ReferenceMeasure, returns the measusre itself

• getReferenceMeasure(default): default method tries to get a BayesSpace and the corresponding reference measure from that.

See Also

[ReferenceMeasure], [BayesSpace], [L2Space]

Examples

StdReferenceMeasureUnif01
getReferenceMeasure(B2Unif01)
getReferenceMeasure(B2Unif01Basis5)
getReferenceMeasure(L2Space(StdReferenceMeasureUnif01))
getReferenceMeasure(BDataSamples(replicate(2,runif(10)),basis=B2Unif01Basis5))

Get a sample from the reference measure

Description

This provides a sample of data form the support of the reference measure, e.g. to test functions or to be used as integration points. The sample can optionally come as a weighted sample to ensure integration. In order to achieve good performance and reproducability the sample is typically fixed and not random, except in cases where no good algorithm for a deterministic grid could be implemented.

This function is here for fast and reproducable Monte Carlo integration, with respect to a measure. Unlike a [simulate] it is allowed to return the same sample in different calls and to provide default size rather than a requested one.

Usage

getSample(x, ...)

## S3 method for class 'ReferenceMeasure'
getSample(x, ...)

## S3 method for class 'ReferenceMeasureCartesianProduct'
getSample(x, ...)

## S3 method for class 'BayesSpace'
getSample(x, ...)

## S3 method for class 'L2Space'
getSample(x, ...)

getSample(x, ...)

## S3 method for class 'ReferenceMeasureDiscrete'
getSample(x, ...)

hasWeights(x)

## Default S3 method:
hasWeights(x, ...)

getWeights(x, ...)

Arguments

Value

a sample of the distribution in the form that would be expected from a dataset, like e.g. a vector or a data.matrix.

Methods (by class)

• getSample(ReferenceMeasure): method for referencemeasure, please overwrite

• getSample(ReferenceMeasureCartesianProduct): method for ReferenceMeasureCartesianProduct

• getSample(BayesSpace): method for BayesSpace

• getSample(L2Space): method for L2Space

• getSample(ReferenceMeasureDiscrete): method for discrete measures, returning a weighted sample

Functions

• hasWeights(): generic method checking whether its weighted sample

• hasWeights(default): default method checking for a weights attribute.

• getWeights(): generic method for retrieving weights of a sample

See Also

Other ReferenceMeasures: B2N01, ReferenceMeasureDiscrete(), ReferenceMeasureTransformed(), ReferenceMeasureUniform(), StdReferenceMeasureUnif01, expectation(), isInDomain()

Other ReferenceMeasures: B2N01, ReferenceMeasureDiscrete(), ReferenceMeasureTransformed(), ReferenceMeasureUniform(), StdReferenceMeasureUnif01, expectation(), isInDomain()

Examples

getSample(B2Unif01)

hist(getSample(StdReferenceMeasureUnif01))
mean(cos(getSample(StdReferenceMeasureN01)))

Coefficients of a single BayesSpace element.

Description

The coefficients are computed with respect to the given basis. Only a single density data element my be provide, but in many different froms including single case BData* objects, density functions, samples, frequency and density histograms. A density histogram is a histogram structure without counts. Samples need to provide at least minSampleSize elements, to avoid simply misinterpreting inadmissible object types as samples.

Usage

getSplineCoefs(x, ..., basis)

## S3 method for class 'BDataSplines'
getSplineCoefs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class ''function''
getSplineCoefs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'hist'
getSplineCoefs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BData'
getSplineCoefs(x, ..., basis = getBayesSpaceBasis(x))

## Default S3 method:
getSplineCoefs(x, ..., minSampleSize = 10, basis = getBayesSpaceBasis(x))

Arguments

Value

a numerical vector of coefficients in the given basis

Methods (by class)

• getSplineCoefs(BDataSplines): the method for BDataSpline objects.

• getSplineCoefs(`function`): method for density functions. Convert log densities to densities by applying 'Exp()'.

• getSplineCoefs(hist): method for histograms

• getSplineCoefs(BData): method for distributions represented as a single element in a BData structure.

• getSplineCoefs(default): interpreting it as a sample of data. It must have at least 10 observations.

Examples

getSplineCoefs(function(x) x^2,basis=B2Unif01Basis5)
getSplineCoefs(rbeta(1000,2,2),basis=B2Unif01Basis5)

Checks whether two objects have the same BayesSpace

Description

The function fails if no BayesSpace is associated to any of the objects.

Usage

hasSameBayesSpace(x, y)

Arguments

Value

a boolean

Examples

hasSameBayesSpace(B2Unif01Basis5,B2Unif01Basis7) # TRUE
hasSameBayesSpace(B2Unif01Basis5,B2N01Basis5) # FALSE

Checks whether the associated BayesSpaceBasis of to objects is the same

Description

Checks whether the associated BayesSpaceBasis of to objects is the same

Usage

hasSameBayesSpaceBasis(x, y)

Arguments

Value

a boolean

Examples

x = BDataSamples(list(runif(100)), basis=B2Unif01Basis5)
hasSameBayesSpaceBasis(B2Unif01Basis5,x) # TRUE
hasSameBayesSpaceBasis(B2Unif01Basis5,B2Unif01Basis7) # FALSE
hasSameBayesSpace(B2Unif01Basis5,B2N01Basis5) # FALSE

Isometric default representations for BData objects

Description

The isometric default transform for BData objects is return as BDataSplines..

Usage

## S3 method for class 'BData'
idt(x, ...)

## S3 method for class 'BData'
idtInv(x, orig = attr(x, "orig"), ...)

## S3 method for class 'BDataSplines'
idtInv(x, orig = attr(x, "orig"), ...)

Arguments

Value

a BSpline representation of the Distributional data

Functions

• idtInv(BData): invert representation of an object 'x' according to the representation of 'orig'

• idtInv(BDataSplines): returns directly to splines

See Also

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), toHistogram()

Examples

# BData densities
bdld = BDataLogDensities(list(\(x)(x^2-x^3), \(x)(sin(x*pi))), basis=B2Unif01Basis20)
plot(bdld, col=c(2,4))
coefs = as.BDataSplines(bdld)
mn = colMeans(coefs)
recasted = idtInv(mn, orig=bdld)
class(recasted)
f = as.function(recasted[[1]])
curve(f, add=TRUE)

# BSpline coefficients
coefs=matrix(1:15,nrow=3)
bssplines=BDataSplines(coefs,B2Unif01Basis5)
cf = idt(bssplines)
mn = colMeans(cf)
idtInv(mn, orig=bssplines)

Returns whether its know that the object is closed or centered

Description

Returns whether its know that the object is closed or centered

Usage

isClosed(x)

## Default S3 method:
isClosed(x)

## S3 method for class 'BData'
isClosed(x)

isCentered(x)

## Default S3 method:
isCentered(x)

## S3 method for class 'BData'
isCentered(x)

## S3 method for class 'BDataSplines'
isCentered(x)

## S3 method for class 'BayesSpaceBasis'
isCentered(x)

Arguments

Value

a boolean, if true the system believes that the object is already closed

Methods (by class)

• isClosed(default): default methods always returns FALSE

• isClosed(BData): BData returns true if it was closed or declared closed

Functions

• isCentered(): Checks for centering

• isCentered(default): default methods always returns FALSE

• isCentered(BData): BData returns true if it was closed or declared closed

• isCentered(BDataSplines): BDataSplines returns true if the basis is centered

• isCentered(BayesSpaceBasis): BayeSpaceBasis requests whether the basis is considered centered

Examples

isClosed(dnorm) # FALSE, because we have not way of knowing
isClosed(BDataDensities(list(dnorm),basis=B2N01Basis5,closed=TRUE)) # TRUE by assertation
isClosed(BDataDensities(list(Mul(2,dnorm)),basis=B2N01Basis5,close=TRUE))  # TRUE by making it
isCentered(dnorm) # FALSE, because we have not way of knowing
isCentered(BDataDensities(list(dnorm),basis=B2N01Basis5,closed=TRUE)) # FALSE
isCentered(cen(BDataDensities(list(Mul(2,dnorm)),basis=B2N01Basis5)))  # TRUE by making it

Checks whether values are in the domain of the ReferenceMeasure

Description

The function acts vectorized, i.e. in case of numbers or other atomic objects it expects a vector of the given type. In case of values that can not be stored in a vector it expects a list of the elements.

Usage

isInDomain(x, referenceMeasure)

## S3 method for class 'BayesSpace'
isInDomain(x, referenceMeasure)

## S3 method for class 'L2Space'
isInDomain(x, referenceMeasure)

## S3 method for class 'BayesSpaceBasis'
isInDomain(x, referenceMeasure)

## S3 method for class 'ReferenceMeasureTransformed'
isInDomain(x, referenceMeasure)

## S3 method for class 'ReferenceMeasureDiscrete'
isInDomain(x, referenceMeasure)

## S3 method for class 'ReferenceMeasureCartesianProduct'
isInDomain(x, referenceMeasure)

Arguments

Value

a logical vector of the same length as 'x', 'TRUE' where the corresponding value lies in the domain (support) of the reference measure and 'FALSE' otherwise.

Methods (by class)

• isInDomain(BayesSpace): method for BayesSpaces

• isInDomain(L2Space): method for L2Spaces

• isInDomain(BayesSpaceBasis): method for BayesSpaces

• isInDomain(ReferenceMeasureTransformed): method for ReferenceMeasureTransformed. The domain is always an open interval, where 0<cdf(x)<1.

• isInDomain(ReferenceMeasureDiscrete): method for discrete Reference Measures

• isInDomain(ReferenceMeasureCartesianProduct): method for product measures

See Also

isInDomain.ReferenceMeasureUniform

Other ReferenceMeasures: B2N01, ReferenceMeasureDiscrete(), ReferenceMeasureTransformed(), ReferenceMeasureUniform(), StdReferenceMeasureUnif01, expectation(), getSample()

Examples

isInDomain(1:10,B2Unif01)

assess whether a vector of points belong to the domain of the uniform reference measure

Description

unlike typical measures the domain is closed, i.e. the extreme values are considered part of the support.

Usage

## S3 method for class 'ReferenceMeasureUniform'
isInDomain(x, referenceMeasure)

Arguments

Value

a logical vector of TRUE/FALSE with a length length(x)

Examples

isInDomain(c(1:10),ReferenceMeasureUniform(0,1))

Checks that an object is a BayesSpaceBasis

Description

Checks that an object is a BayesSpaceBasis

Usage

isValidBayesSpaceBasis(X)

Arguments

Value

boolean

Examples

isValidBayesSpaceBasis(B2Unif01Basis7)

Checks whether an object is an element (or dataset of elements) of a BayesSpace

Description

May different R objects can be interpreted as elements of a BayesSpace including density functions, histograms, and BData objects.

Usage

isValidBayesSpaceElement(X, basis = NULL)

## S3 method for class ''function''
isValidBayesSpaceElement(X, basis = NULL)

## S3 method for class 'histogram'
isValidBayesSpaceElement(X, basis = NULL)

## S3 method for class 'BData'
isValidBayesSpaceElement(X, basis = NULL)

## Default S3 method:
isValidBayesSpaceElement(X, basis = NULL)

Arguments

Value

boolean, FALSE if it is definitely not an element of a BayesSpace

Methods (by class)

• isValidBayesSpaceElement(`function`): functions are valid BayesSpaceElements

• isValidBayesSpaceElement(histogram): histograms are valid BayesSpaceElements

• isValidBayesSpaceElement(BData): BData objects contain valid BayesSpaceElements if they are for the same BayesSpace

• isValidBayesSpaceElement(default): method for samples, tries to check whether the values are in the domain.

Examples

isValidBayesSpaceElement(sin)
isValidBayesSpaceElement(hist(runif(100), plot=FALSE))

Computes an evaluable kernel density estimate on a dataset

Description

The distance is measured in the metric d(x,y)= sqrt((x-y)^t M (x-y)) given by the 'Metric' tensor M. The kernel density is estimated as

Usage

kdeDensity(X, ..., bandwidth, Metric, kernel, K, Var, sdFactor)

## S3 method for class 'matrix'
kdeDensity(
  X,
  ...,
  bandwidth = NULL,
  Metric = if (is.null(bandwidth)) invertVariance(Var * sdFactor^2) else
    diag(1/bandwidth^2, nrow = ncol(X)),
  K = 1/sqrt(2 * pi)^(NCOL(X) - 1) * sqrt(det(Metric)),
  kernel = dnorm,
  Var = var(X),
  sdFactor = (0.2)^(1/NCOL(X))
)

## S3 method for class 'data.frame'
kdeDensity(
  X,
  ...,
  bandwidth = NULL,
  Metric = if (is.null(bandwidth)) invertVariance(Var * sdFactor^2) else
    diag(1/bandwidth^2, nrow = ncol(X)),
  K = 1/sqrt(2 * pi)^(NCOL(X) - 1) * sqrt(det(Metric)),
  kernel = dnorm,
  Var = var(X),
  sdFactor = (0.2)^(1/NCOL(X))
)

## S3 method for class 'list'
kdeDensity(
  X,
  ...,
  bandwidth = NULL,
  Metric = if (is.null(bandwidth)) 1/(Var * sdFactor^2) else 1/bandwidth^2,
  K = sqrt(Metric),
  kernel = dnorm,
  Var = var(X),
  sdFactor = 0.2,
  minus = "-"
)

## S3 method for class 'numeric'
kdeDensity(
  X,
  ...,
  support = Support(a = -Inf, b = Inf, lim = range(X)),
  bandwidth = NULL,
  Metric = if (is.null(bandwidth)) 1/(Var * sdFactor^2) else 1/bandwidth^2,
  K = sqrt(Metric),
  kernel = dnorm,
  Var = var(X),
  sdFactor = 0.2,
  minus = "-"
)

Arguments

Details

f(x) = 1/N sum_i=1^n scale*kernel(d(x,X_i))

The computation is done explicity in all calls to the resulting function.

Value

a function that can be evaluated with a dataset similar to X

Methods (by class)

• kdeDensity(matrix): method for multivariate datasets provided as matrix

• kdeDensity(data.frame): method for multivariate datasets provided as data.frame

• kdeDensity(list): method for datasets provided as list. The minus operator must be defined on the elements.

• kdeDensity(numeric): method for datasets provided as a numeric vector.

Examples

X = rnorm(10)
x=seq(-4,4,by=0.1)
f = kdeDensity(X,sdFactor=0.5)
plot(x,f(x),type="b")

# data generation and kde function construction
xx = cbind(rnorm(100, sd=2.5), rnorm(100, sd=0.5))
kdxy = kdeDensity(xx)
# check that the integral is approx 1:
x0 = seq(from=-2.5, to=2.5, length.out=25)
xys = expand.grid(x0*2.5, x0*0.5)
zs =  kdxy(xys)
(dx0 = x0[2]-x0[1])
(A = sum(zs)*dx0^2*2.5*0.5) # bad approx but within expectations given low precision
# compare with kde from stats::density for one marginal:
dim(zs) = c(25,25)
dx = density(xx[,1], kernel="gaussian")
dx |> plot()
lines(x0*2.5, apply(zs,1,sum)*dx0*0.5/A, col=2)
rug(xx[,1])
xx = rnorm(1000, sd=2.5) # simulate some data
# generate and plot the kde function
kdx = kdeDensity(xx)
plot(kdx, from=-5, to=5)
# ensure that the integral is approx 1
xs = seq(from=-5, to=5, by=0.01)
ys = kdx(xs)
sum(ys)*0.01             #  approx 1.0
# graphically compare with kde from stats::density
dx = density(xx, kernel="gaussian")
with(dx, lines(x, y, col=2))
plot(dx)
curve(kdx, from=-4, to=4, col=2, add=TRUE)

Accesses list elements with an expression

Description

This is the generic behind ' to modify the behaviour of '

Usage

listDataElementAccess(X, .body, .env)

Arguments

Value

a list from the results of body evaluated in .env, with . standing in for the elements of X, or if .body is a name with .$name evaluted.

Examples

L = list(list(name = "a", val = 1), list(name = "b", val = 2))
# a bare name applies the corresponding `$` accessor to each element
listDataElementAccess(L, quote(name), environment())
# a full expression uses `.` for each list element
listDataElementAccess(L, quote(.$val * 10), environment())

Makes a vectorized 2-ary function into one that returns a matrix

Description

A vectorized function of two arguments is made returning the result for each pair structured as a matrix, or if it already returns an n-ary array, as an n+1-ary array with the last dimension splitted

Usage

makeFunctionOuter(f)

Arguments

Value

a function like f but returning the result for each pair of input structured with one more dimension.

Examples

makeFunctionOuter(\(x,y) exp(-abs(x-y)))(1:4,1:4)

Makes a real function returning limits at undefined points

Description

The function identifies Punctures in the definition of a function, by getting a non finite result (Inf,-Inf,Nan) and than asks computeLimit to compute the corresponding directional limit (for finite values given by d). The value can still be infinite.

Usage

makeLimitStable(f, x0 = 0.01)

Arguments

Value

a numerically stabilised version of 'f': a function that evaluates 'f' but replaces any non-finite result (Inf, -Inf, NaN) by the corresponding directional limit obtained from [computeLimit] (which may itself still be infinite).

Gets coordinates on an axis for possibly non real data

Description

Gets coordinates on an axis for possibly non real data

Usage

mapToRealLineWithSupport(X, support, d = 1)

Arguments

Value

the value(s) of 'X' mapped onto the 'support', i.e. possibly the same values of 'X' (i.e. if the support is a subset of the real line) or the indices corresponding to the atoms in the support equal to the values of 'X' (if the support is a discrete set of points or of categories).

Compute Maximum and Minimum Functions

Description

Compute Maximum and Minimum Functions

Usage

maxFunc(..., na.rm = FALSE, bound = NULL, Funcs = c(...))

minFunc(..., na.rm = FALSE, bound = NULL, Funcs = c(...))

Arguments

Value

a function taking like the given functions, and returning the maximum (or minium) of the values not exceeding the boundary.

Functions

• minFunc(): computes the minimum

Examples

F = maxFunc(\(x) dnorm(x),\(x) dnorm(x,mean=2),bound=\(x) rep(0.3,length(x)))
x=seq(-2,5,by=0.01)
plot(x,F(x))
F = minFunc(\(x) dnorm(x),\(x) dnorm(x,mean=2),bound=\(x) rep(0.1,length(x)))
plot(x,F(x))

Computes the distribution mean for BDataSplines.

Description

This function simply averages the spline coefficients and returns a correponding spline.

Usage

## S3 method for class 'BDataSplines'
mean(x, ..., method = c("classical", "mcd", "mve"))

## S3 method for class 'BDataDensities'
mean(x, ..., method = c("functions", "classical", "mcd", "mve"))

## S3 method for class 'BDataPDFs'
mean(x, ..., method = c("functions", "classical", "mcd", "mve"))

## S3 method for class 'BDataLogPDFs'
mean(x, ..., method = c("functions", "classical", "mcd", "mve"))

## S3 method for class 'BDataLogDensities'
mean(x, ..., method = c("functions", "classical", "mcd", "mve"))

Arguments

Value

a BDataSplines with a single observation representing the mean

Functions

• mean(BDataDensities): method for full densities

• mean(BDataPDFs): method for full PDFs

• mean(BDataLogPDFs): method for full logPDFs

• mean(BDataLogDensities): method for full logDensities

Examples

data(DTD)
Xpdf    = BDataPDFs(DTD$shapePDF,basis=B2Unif01Basis5)
Xspline = as.BDataSplines(Xpdf)
plot(Xspline,col=2:12)
meanXspline = mean(Xspline)
plot(meanXspline,col=1,lwd=3,add=TRUE)

plot(Xspline,as="PDF")
plot(meanXspline,col="red",lwd=3,add=TRUE,as="PDF")

Xdensities = as.BDataDensities(Xpdf)

plot(Xdensities)
meanXdensities = clo(mean(Xdensities))
plot(meanXdensities,col="green",lwd=7,add=TRUE)
plot(meanXspline,col="red",lwd=3,add=TRUE,as="density")

plot(Xpdf)
meanXpdf = mean(Xpdf)
plot(clo(meanXpdf),col="green",lwd=7,add=TRUE)
plot(meanXspline,col="red",lwd=3,add=TRUE,as="PDF",close=TRUE)
plot(as.BDataPDFs(meanXdensities),col="blue",lwd=3,add=TRUE,close=TRUE)
plot(as.BDataPDFs(mean(as.BDataLogPDFs(Xpdf))),col="orange",lwd=3,add=TRUE,close=TRUE)
plot(as.BDataPDFs(mean(as.BDataLogDensities(Xpdf))),col="magenta",lwd=3,add=TRUE,close=TRUE)

A distributional median

Description

The distributional median is the distribution (from a dataset), that minimizes the sum over the Bayes Space distances in the dataset. The sample median selects the distribution from the dataset that minimizes this distance (or maximizes the spatial depth).

Usage

## S3 method for class 'BData'
median(x, na.rm = TRUE, ..., distType = c("meandistance", "spatialdepth"))

Arguments

Value

An object of the same type as x, but with only the median, defined as the value with the smallest distance to all others (distType "mean"), or the largest spatial depth (distType "spatialdepth").

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseQuantiles(), caseSds(), caseVars(), centralityQuantile(), distanceToCenter(), spatialDepth()

Examples

data(DTD)
Xpdf    = BDataPDFs(DTD$shapePDF,basis=B2Unif01Basis5)
median(Xpdf)

Compute the metric variance

Description

The metric variance is the expected squared distance from the mean in the metric of the enclosing space. By construction the metric variance computed from a spline representation should be smaller than metrix variances computed on functional data, because the spline captures only a part of the variance.

Usage

## S3 method for class 'BData'
mvar(x, ..., meanx = mean(x), method = "classical")

## S3 method for class 'BDataSplines'
mvar(x, ..., method = c("classical", "mcd", "mve"), meanx = mean(x, method))

## S3 method for class 'BData'
msd(x, ...)

Arguments

Value

a positive real number representing the metric variance

Functions

• mvar(BDataSplines): a specialized method for BDataSpline, which does not require integration.

• msd(BData): compute the metric standard deviation, proportional to the square root of the metric variance

Examples

data(DTD)
Xpdf    = BDataPDFs(DTD$shapePDF,basis=B2Unif01Basis5)
Xsplines= as.BDataSplines(Xpdf)
mvar(Xsplines)
 mvar(Xsplines,method="mcd")
mvar(as.BDataDensities(Xpdf))
mvar(as.BDataLogDensities(Xpdf))
mvar(as.BDataLogPDFs(Xpdf))
mvar(Xpdf)

Accessing cases regardless how a dataset is stored.

Description

Accessing cases regardless how a dataset is stored.

Usage

nCases(X)

## Default S3 method:
nCases(X)

selectCases(X, I)

## Default S3 method:
selectCases(X, I)

Arguments

Value

returns the subset of cases.

Methods (by class)

• nCases(default): default method working for most know types of datasets, by using the first array dimension (if applicable) or the length as number of cases.

Functions

• selectCases(): selects the cases given by the index I

• selectCases(default): selects the cases given by the index I

Examples

nCases(1:5)
nCases

Computes the Bayes-Space norm of BData objects

Description

Computes the Bayes-Space norm of BData objects

Usage

norm(x, ...)

## S3 method for class 'BData'
norm(x, ..., measure = getReferenceMeasure(x))

## S3 method for class 'BDataSplines'
norm(x, ...)

Arguments

Value

A numeric vector of the L2 (Bayes-space) norms, one per observation.

Functions

• norm(): generic functionality computing Euclidean norms of vectors (default method) and redirecting to 'base::norm()' for matrices.

• norm(BDataSplines): fast exact norm for BDataSplines via the Gram matrix of the basis: the squared norm of each observation is the quadratic form of its coefficient vector with the metric tensor S.

Examples

lbdl = BDataLogDensities(basis=B2N01Basis5, closed=TRUE,
  list(
    \(x)(dnorm(x,2,1,log=TRUE)),
    \(x)(dnorm(x,1,1,log=TRUE)),
    \(x)(dnorm(x,-1,1,log=TRUE)),
    \(x)(dnorm(x,-2,1,log=TRUE))
    ))
norm(lbdl)

A generic function computing the number of cases.

Description

Different representations of datasets provide its elements in a very different way. This function provides a uniform interface to figure get the number of cases in a dataset.

Usage

numberOfCases(x)

## S3 method for class 'data.frame'
numberOfCases(x)

## Default S3 method:
numberOfCases(x)

Arguments

Value

an integer

Methods (by class)

• numberOfCases(data.frame): method for dataframes

• numberOfCases(default): method for lists and numeric, BDataDensities,...

Examples

data(iris)
numberOfCases(iris$Sepal.Length)
numberOfCases(iris)
numberOfCases(lapply(1:numberOfCases(iris),function(i) iris[i,]))

Guesses the support from a pdf

Description

The function is designed to allow a quick start into distributional data analysis and plotting.

Usage

pdf2Support(
  pdf,
  starts = exp(seq(-20, 20, by = 0.2)),
  minDensityFactor = dnorm(qnorm(alpha/2))/dnorm(0),
  alpha = 0.01
)

Arguments

Value

a Support object with the guessed support

See Also

Other Support: Support(), cdf2Support(), plotFuncs(), sample2Support()

Examples

pdf2Support(dnorm)
pdf2Support(dunif)

Plot histograms as segments

Description

Plot a histograms dataset as a set of "stairs" using matplot.

Usage

## S3 method for class 'BDataHistograms'
plot(
  x,
  probability = is(x, "BDataDensityHistograms"),
  xlab = "x",
  ylab = ifelse(probability, "density", "frequency"),
  ...
)

Arguments

Value

nothing, called for its side effect to produce a plot. Each histogram is represented only as its outline.

See Also

Other BDataHistograms: BDataFrequencyHistograms(), as.BDataDensityHistograms(), as.BDataDensityHistograms.list(), as.BDataFrequencyHistograms.list(), getBreaks()

Examples

x = lapply(rep(1000,4), runif)
bdfhx = as.BDataFrequencyHistograms(x, breaks="Sturges",basis=B2Unif01Basis5)
plot(bdfhx)
bdfhx2 = as.BDataFrequencyHistograms(x, breaks=sqrt(0:10)/sqrt(10),basis=B2Unif01Basis5)
plot(bdfhx2, probability=TRUE)
# plot(bdfhx2) # this generates a warning

Plotting BDataSample Objects

Description

Plotting BDataSample Objects

Usage

## S3 method for class 'BDataSamples'
plot(x, ..., pch = 20, method = "stack", as = c("points"), add = FALSE)

Arguments

Value

'NULL' invisibly, called for the side effect of producing a plot.

Examples

s <- BDataSamples(list(rnorm(200), rnorm(200,1)), basis=B2N01Basis5)
plot(s)

Plot a set of spline functions (using visualisation tools from fda)

Description

Plot a set of spline functions (using visualisation tools from fda)

Usage

## S3 method for class 'BDataSplines'
plot(
  x,
  ...,
  as = c("coef", "PDF", "CDF", "density", "logPDF", "logDensity", "clr"),
  close = TRUE
)

Arguments

Value

nothing or an invisible version of the object actually plotted

See Also

Other BDataSplines: BDataSplines()

Examples

X = BDataSplines(matrix(rnorm(30),ncol=5),basis=B2Unif01Basis5)
plot(X, as = "coef")
plot(X, as = "PDF")
plot(X, as = "clr")  # centered log-ratio representation

plot the 'BayesSpaceBasis'

Description

plot the 'BayesSpaceBasis'

Usage

## S3 method for class 'BayesSpaceBasis'
plot(
  x,
  ...,
  as = c("is", "logPDF", "PDF", "CDF", "density", "logDensity", "clr", "coefs")
)

Arguments

Value

typically some representation of the data plotted

Examples

print(B2Unif01Basis5)

Plot Method for PCA of Distributional Data

Description

A versatile plotting function for visualizing principal component analysis results on distributional data. Provides multiple plot types for comprehensive exploration of the PCA results, including the unique ability to visualize loadings as functional perturbations of the mean distribution.

Usage

## S3 method for class 'princomp.BData'
plot(
  x,
  type = c("screeplot", "variance", "scores", "eigenfunctions", "loadings", "biplot",
    "covariance", "reconstruction", "boxplot"),
  choices = 1:2,
  scale = 1,
  pc.biplot = FALSE,
  main = NULL,
  col = NULL,
  lwd = 1,
  lty = 1,
  legend = TRUE,
  labels = FALSE,
  ...,
  npcs = min(10, length(x$sdev)),
  as = c("PDF", "density", "logPDF", "logDensity", "clr", "coef"),
  scale.sdev = 1
)

Arguments

Details

## Understanding the Eigenfunctions Plot

The 'type="eigenfunctions"' plot is the most important visualization for interpreting SFPCA results on distributional data. It answers the question: "How does the distribution shape change as we move along each principal component?"

The plot shows:

• Black thick line: The mean distribution (center of the data).

• Colored solid lines: Distributions at +'scale.sdev' standard deviations from the mean along each PC direction (positive perturbation).

• Colored dashed lines: Distributions at -'scale.sdev' standard deviations from the mean along each PC direction (negative perturbation).

For example, if PC1 captures variation in the "location" of distributions, the PC1+ line will be shifted right and PC1- will be shifted left compared to the mean. If PC1 captures variation in "spread", PC1+ will be wider and PC1- will be narrower.

## Choosing the Number of Components

Use the screeplot and variance plot to decide how many components are meaningful:

• Screeplot: Look for an "elbow" where the variance drops off. Components before the elbow are typically retained.

• Variance plot: Choose enough components to explain a target percentage (e.g., 80% or 90%) of total variance.

Value

Invisibly returns the data used for plotting (varies by plot type).

Author(s)

Matthias Templ

See Also

princomp.BData for performing the PCA.

biplot.princomp.BData for detailed control over biplots.

screeplot.princomp.BData for the standalone screeplot function.

Other PCA: biplot.princomp.BData(), plot.princomp.BData.grouped(), plot.princomp.BData.multivariate(), predict.princomp.BData(), princomp.BData(), screeplot.princomp.BData(), summary.princomp.BData()

Examples

## Setup: Create PCA result for examples
meanDist <- BDataPDFs(list(dunif), basis = B2Unif01Basis7)
gaussProc <- B2GaussianDistribution(meanDist, function(x,y) exp(-4*abs(x-y)^1.7))
pcaResult <- princomp(as.BDataSplines(simulate(gaussProc, 30, seed = 42)))

## Screeplot: Decide how many components to keep
plot(pcaResult, type = "screeplot")
# Interpretation: The bars show the percentage of total variance explained by
# each PC. Here, PC1 and PC2 dominate, suggesting that the main structure in
# the data can be captured by just two components. The "elbow" after PC2 or
# PC3 indicates where additional components add little explanatory power.

## Cumulative variance: What percentage is explained?
plot(pcaResult, type = "variance")
# Interpretation: This shows how total explained variance accumulates as more
# components are included. The dashed reference lines help identify how many
# components are needed to reach common thresholds (80%, 90%, 95%). Here,
# approximately 3 components explain over 90% of the variance.

## Score plot: Explore structure in the data
plot(pcaResult, type = "scores")
# Interpretation: Each point is one distribution projected onto PC1 and PC2.
# The spread along PC1 (x-axis) shows variation in the first mode; spread
# along PC2 (y-axis) shows the second mode. Clusters indicate groups of
# similar distributions; outliers are distributions with unusual shapes.

## Score plot with labels: Identify individual distributions
plot(pcaResult, type = "scores", labels = TRUE)
# Using labels=TRUE replaces points with observation numbers (1, 2, 3, ...),
# allowing you to identify which specific distribution each position
# represents. This is useful for tracking outliers or interesting cases
# back to the original data.

## Eigenfunctions plot: THE KEY INTERPRETIVE PLOT
plot(pcaResult, type = "eigenfunctions", npcs = 2)
# Interpretation: The black curve is the mean (average) distribution. For
# each PC, solid lines show how distributions look when they score HIGH on
# that component, dashed lines show LOW scores. Compare the colored curves
# to the black mean to see what "high PC1" or "low PC2" means in terms of
# actual distribution shape (e.g., shifted location, changed spread, altered
# skewness, or different modality).

## More extreme perturbations (2 standard deviations)
plot(pcaResult, type = "eigenfunctions", npcs = 2, scale.sdev = 2)
# Using scale.sdev=2 shows more extreme deviations from the mean, making
# the differences between PC directions more visible.

## Show loadings as log-PDFs (useful for seeing tail behavior)
plot(pcaResult, type = "loadings", npcs = 2, as = "logPDF")
# The log-PDF representation emphasizes differences in the tails of the
# distributions, which can be important for detecting outlier behavior.

## Show loadings in CLR (centered log-ratio) representation
plot(pcaResult, type = "loadings", npcs = 2, as = "clr")
# The CLR representation shows the additive structure in Bayes space.
# Values are centered around zero, making it easy to see which parts of the
# distribution are above or below the geometric mean. This is the natural
# representation for PCA in Bayes space.

## Covariance surface: See which parts of the domain co-vary
plot(pcaResult, type = "covariance")
# Interpretation: This heatmap shows functional covariance Cov(f(s), f(t))
# across the domain. High values (red) indicate that when one part of the
# distribution is above average, the other tends to be as well. The diagonal
# shows variance at each point. Off-diagonal patterns reveal correlation
# structure (e.g., if left and right tails move together or oppositely).

## Reconstruction: How well do k PCs approximate the data?
plot(pcaResult, type = "reconstruction", npcs = 2)
# Shows all distributions reconstructed using only the first 2 PCs.
# Compare to the original data to assess information loss.
# If reconstructions look similar to originals, 2 PCs are sufficient.

plot(pcaResult, type = "reconstruction", npcs = 1)
# Using only 1 PC shows more loss of detail - useful for understanding
# what information each additional PC captures.

## Boxplot: Distribution of scores across PCs
plot(pcaResult, type = "boxplot", npcs = 4)
# Shows the spread of observations along each PC direction.
# Useful for identifying outliers (points beyond whiskers) and comparing
# the amount of variation captured by each component. The percentage
# labels show variance explained by each PC.

Plot Method for Grouped PCA of Distributional Data

Description

Creates multi-panel comparison plots for grouped PCA results. Each group is shown in a separate panel, allowing visual comparison of PCA results across different subgroups of the data.

Usage

## S3 method for class 'princomp.BData.grouped'
plot(
  x,
  type = c("screeplot", "variance", "scores", "eigenfunctions", "loadings", "covariance",
    "reconstruction", "boxplot"),
  mfrow = NULL,
  main = NULL,
  ...
)

Arguments

Details

This function sets up a multi-panel plotting layout and calls plot.princomp.BData for each group. The layout is automatically determined based on the number of groups, but can be overridden using 'mfrow'.

Common use cases:

• Compare loadings across groups to see if different subgroups have different modes of variation.

• Compare screeplots to assess if dimensionality differs by group.

• Compare score plots to see internal structure within each group.

Value

Invisibly returns the grouped PCA object.

Author(s)

Matthias Templ

See Also

princomp.BData for performing grouped PCA. plot.princomp.BData for single-group plot options.

Other PCA: biplot.princomp.BData(), plot.princomp.BData(), plot.princomp.BData.multivariate(), predict.princomp.BData(), princomp.BData(), screeplot.princomp.BData(), summary.princomp.BData()

Examples

# Create grouped PCA using age distribution data
data("age")

# Set up basis
rmAge <- ReferenceMeasureUniform(0, 100)
B2Age <- BayesSpace(rmAge)
basisAge <- BayesSpaceBasis(B2Age, nbasis = 7)

# Create histograms and convert to splines
breaks <- c(0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 100)
ageHist <- BDataDensityHistograms(age$distributions, breaks = breaks, basis = basisAge)
ageSplines <- as.BDataSplines(as.BDataDensities(ageHist))

# Perform grouped PCA by sex
pcaGrouped <- princomp(ageSplines, group = age$data$sex)
pcaGrouped

# Compare screeplots by group
plot(pcaGrouped, type = "screeplot")

# Compare loadings by group
plot(pcaGrouped, type = "loadings", npcs = 2)

# Custom layout (2 rows)
plot(pcaGrouped, type = "scores", mfrow = c(2, 1))

Plot Method for Multivariate SFPCA of Distributional Data

Description

Creates visualization plots for multivariate SFPCA (2D-SFPCA) results. Eigenfunctions span multiple groups and are shown with one panel per group.

Usage

## S3 method for class 'princomp.BData.multivariate'
plot(
  x,
  type = c("screeplot", "variance", "scores", "eigenfunctions", "loadings"),
  npcs = 2,
  pcs = c(1, 2),
  as = "PDF",
  ...
)

Arguments

Details

For multivariate SFPCA, eigenfunctions are joint across all groups. The eigenfunctions plot shows how each group's component of the eigenfunction affects the distribution shape.

Value

Invisibly returns the multivariate PCA object.

Author(s)

Matthias Templ

See Also

princomp.BData for performing multivariate SFPCA.

Other PCA: biplot.princomp.BData(), plot.princomp.BData(), plot.princomp.BData.grouped(), predict.princomp.BData(), princomp.BData(), screeplot.princomp.BData(), summary.princomp.BData()

Examples

data("age")
basisAge <- BayesSpaceBasis(BayesSpace(ReferenceMeasureUniform(0, 100)), nbasis = 7)
breaks <- c(0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 100)
ageHist <- BDataDensityHistograms(age$distributions, breaks = breaks, basis = basisAge)
ageSplines <- as.BDataSplines(as.BDataDensities(ageHist))
pcaMulti <- princomp(ageSplines, group = age$data$sex, multivariate = TRUE)
plot(pcaMulti, type = "eigenfunctions", npcs = 2)

Plotting varKernel objects

Description

Plots a varKernel object as an image or contour plot.

Usage

## S3 method for class 'varKernel'
plot(
  x,
  ...,
  support = getSupport(x),
  SD = supportDiscretization(support, nEval),
  C = x(SD$all, SD$all),
  col = par("col"),
  pch = par("pch"),
  lty = par("lty"),
  lwd = par("lwd"),
  cex = par("lwd"),
  type = c("both", "image", "contour"),
  nEval = 513,
  add = FALSE,
  xlim = getSupport1DPlottingLimits(support),
  ylim = rev(xlim),
  zlim = range(breaks),
  breaks = pretty(c(C), ncol),
  ncol = 10,
  cols = grDevices::hcl.colors(length(breaks) - 1, "YlOrRd", rev = TRUE),
  dWidth = (xlim[2] - xlim[1])/200,
  xlab = "x",
  ylab = xlab,
  zlab = "c",
  notColour = "#FFFFFF"
)

## S3 method for class 'variationKernel'
plot(
  x,
  ...,
  support = getSupport(x),
  SD = supportDiscretization(support, nEval),
  V = x(SD$all),
  col = par("col"),
  pch = par("pch"),
  lty = par("lty"),
  lwd = par("lwd"),
  cex = par("lwd"),
  type = c("both", "image", "contour"),
  nEval = 513,
  add = FALSE,
  xlim = getSupport1DPlottingLimits(support),
  ylim = rev(xlim),
  zlim = range(breaks),
  breaks = pretty(V, ncol),
  ncol = 10,
  cols = grDevices::hcl.colors(length(breaks) - 1, "YlOrRd", rev = TRUE),
  dWidth = (xlim[2] - xlim[1])/200,
  xlab = "x",
  ylab = "x",
  zlab = "c",
  notColour = "#FFFFFF"
)

Arguments

Value

'NULL' invisibly, called for the side effect of producing an image and/or contour plot of the kernel.

Functions

• plot(variationKernel): method for variationKernel

Examples

C = varKernel(function(x,y) exp(-outer(x,y,"-")^2),basis=B2N01Basis5)
plot(C)
V = variationKernel(function(x) 1-exp(-outer(x,x,"-")^2),basis=B2N01Basis5)
plot(V)

Plot multiple coefficients in a parallel coordinate plot

Description

The function currently uses the same axes,

Usage

plotCoefs(
  x,
  type = "b",
  ...,
  commonAxis = TRUE,
  sequence = 1:ncol(x),
  col = par("col"),
  pch = par("pch"),
  lty = par("lty"),
  lwd = par("lwd"),
  cex = par("cex"),
  nEval = NULL,
  add = FALSE,
  xlim = c(0.5, length(sequence) + 0.5),
  ylim = range(c(x)),
  xlab = "j",
  axes = rep(list(c(0, 1)), ncol(x)),
  ylab = expression(c["." * j]),
  xaxt = "s"
)

Arguments

Value

The coefficient matrix 'x' (invisibly), called for the side effect of producing the parallel coordinate plot.

Examples

plotCoefs(matrix(rnorm(20),nrow=5))

Helper function to plot multiple functions

Description

This function uses a loop over lines and points to plot multiple functions with different parameters into one plot.

Usage

plotFuncs(
  x,
  ...,
  support = getSupport(x),
  col = par("col"),
  pch = par("pch"),
  lty = par("lty"),
  lwd = par("lwd"),
  nEval = 1500,
  add = FALSE,
  xlim = getSupport1DPlottingLimits(support),
  ylim = NULL,
  xs = xL[!xL %in% X],
  xL = if (xlim[1] != xlim[2]) seq(xlim[1], xlim[2], length.out = nEval),
  X = getSupportAtoms(support),
  xlab = "x",
  ylab = "f(x)",
  close = FALSE
)

## S3 method for class 'BDataDensities'
plot(
  x,
  ...,
  support = getSupport(x),
  col = par("col"),
  pch = par("pch"),
  lty = par("lty"),
  lwd = par("lwd"),
  nEval = 1500,
  add = FALSE,
  xlim = getSupport1DPlottingLimits(support),
  ylim = NULL,
  xs = xL[!xL %in% X],
  xL = if (xlim[1] != xlim[2]) seq(xlim[1], xlim[2], length.out = nEval),
  X = getSupportAtoms(support),
  xlab = "x",
  ylab = expression(d * X/d * P),
  close = FALSE
)

## S3 method for class 'BDataLogDensities'
plot(
  x,
  ...,
  support = getSupport(x),
  col = par("col"),
  pch = par("pch"),
  lty = par("lty"),
  lwd = par("lwd"),
  nEval = 1500,
  add = FALSE,
  xlim = getSupport1DPlottingLimits(support),
  ylim = NULL,
  xs = xL[!xL %in% X],
  xL = if (xlim[1] != xlim[2]) seq(xlim[1], xlim[2], length.out = nEval),
  X = getSupportAtoms(support),
  xlab = "x",
  ylab = expression(ln(dX/d * P)),
  close = FALSE
)

## S3 method for class 'BDataPDFs'
plot(
  x,
  ...,
  support = getSupport(x),
  col = par("col"),
  pch = par("pch"),
  lty = par("lty"),
  lwd = par("lwd"),
  nEval = 1500,
  add = FALSE,
  xlim = getSupport1DPlottingLimits(support),
  ylim = NULL,
  xs = xL[!xL %in% X],
  xL = if (xlim[1] != xlim[2]) seq(xlim[1], xlim[2], length.out = nEval),
  X = getSupportAtoms(support),
  xlab = "x",
  ylab = expression(pdf(x) == d * X/d * lambda),
  close = FALSE
)

## S3 method for class 'BDataLogPDFs'
plot(
  x,
  ...,
  support = getSupport(x),
  col = par("col"),
  pch = par("pch"),
  lty = par("lty"),
  lwd = par("lwd"),
  nEval = 1500,
  add = FALSE,
  xlim = getSupport1DPlottingLimits(support),
  ylim = NULL,
  xs = xL[!xL %in% X],
  xL = if (xlim[1] != xlim[2]) seq(xlim[1], xlim[2], length.out = nEval),
  X = getSupportAtoms(support),
  xlab = "x",
  ylab = "f(x)",
  close = FALSE
)

Arguments

Value

invisible the results of the computation

• x - the evaluation points for continuous evaluation

• y - the evaluated values for the continuous evaluation, each function gets a column

• X - the evaluation points for the discrete support

• Y - the evaluated values for the discrete support, each function gets a column

Functions

• plot(BDataDensities): plot method for BDataDensities

• plot(BDataLogDensities): plot method for BDataLogDensities

• plot(BDataPDFs): plot method for BDataPDFs

• plot(BDataLogPDFs): plot method for BDataLogPDFs

See Also

Other Support: Support(), cdf2Support(), pdf2Support(), sample2Support()

Examples

plotFuncs(list(fx(sin(10*x)),fx(cos(10*x))),support=Support(0,pi,0:3),pch=2:3,col=2:3)
X = BDataPDFs(list(fx(dnorm(x,0,1)),fx(dnorm(x,0.3,0.5)),fx(dnorm(x,0,1.5))),
              basis=B2N01Basis5)
plot(X,col=2:4)
plot(as.BDataLogPDFs(X),col=2:4)
plot(as.BDataDensities(X),col=2:4)
plot(as.BDataLogDensities(X),col=2:4)

Predict Scores for New Distributional Data

Description

Projects new distributional data onto the principal component space defined by a fitted PCA model. This allows new distributions to be positioned relative to the original dataset without re-running the entire PCA.

Usage

## S3 method for class 'princomp.BData'
predict(object, newdata, ...)

Arguments

Details

The prediction process:

1. Converts 'newdata' to spline coefficients using the same basis as the original PCA.

2. **For non-orthonormal bases:** Applies the same Gram matrix transformation that was used during fitting (see princomp.BData details).

3. Centers the (transformed) coefficients using the center from the original PCA.

4. Projects the centered coefficients onto the loading vectors to obtain scores.

The resulting scores can be compared directly to the scores from the original analysis. New observations with scores similar to original observations have similar distribution shapes.

**Note:** For non-orthonormal bases, the function automatically applies the same transformation that was used during fitting. This ensures consistency between training and prediction.

Value

A matrix of principal component scores with observations in rows and components in columns. Each row contains the coordinates of one new distribution in the PC space.

Author(s)

Matthias Templ

See Also

princomp.BData for fitting the PCA model.

plot.princomp.BData with 'type="scores"' for visualizing scores.

Other PCA: biplot.princomp.BData(), plot.princomp.BData(), plot.princomp.BData.grouped(), plot.princomp.BData.multivariate(), princomp.BData(), screeplot.princomp.BData(), summary.princomp.BData()

Examples

## Fit PCA on training data
meanDist <- BDataPDFs(list(dunif), basis = B2Unif01Basis7)
gaussProc <- B2GaussianDistribution(meanDist, function(x,y) exp(-4*abs(x-y)^1.7))
trainData <- as.BDataSplines(simulate(gaussProc, 30, seed = 42))
pcaModel <- princomp(trainData)

## Generate new data and predict scores
newData <- as.BDataSplines(simulate(gaussProc, 5, seed = 999))
newScores <- predict(pcaModel, newData)
print(newScores)

## Compare: new observations vs training data
trainScores <- pcaModel$scores
plot(trainScores[,1], trainScores[,2], pch = 19, col = "gray50",
     xlab = "PC1", ylab = "PC2", main = "Training (gray) vs New (red)")
points(newScores[,1], newScores[,2], pch = 17, col = "red", cex = 1.5)
legend("topright", c("Training", "New"), pch = c(19, 17),
       col = c("gray50", "red"))

Principal Component Analysis for Distributional Data

Description

Performs Simplicial Functional Principal Component Analysis (SFPCA) on distributional data where each observation is an entire probability distribution. This function extends the classical 'princomp' to work with BData objects, enabling dimensionality reduction and exploratory analysis of datasets of distributions.

Usage

## S3 method for class 'BData'
princomp(
  x,
  cor = FALSE,
  scores = TRUE,
  covmat = NULL,
  center = NULL,
  group = NULL,
  multivariate = FALSE,
  ...
)

## S3 method for class 'princomp.BData'
print(x, ...)

Arguments