Examples

CosinusDistance

The cosine distance is a distance-measure based on the cosine similarity. Let $A$ be the data matrix and $A_i$ , $A_j$ some row vectors of $A$ . The cosine similarity is then defined as $\begin{equation} \text{s(i,j)} = \cos(\theta) = \frac{\mathbf{A_i} \cdot \mathbf{A_j}}{|\mathbf{A_i}| |\mathbf{A_j}|} \end{equation}$ , and the cosine distance as $d(i,j)=\max{s}-s(i,j)$ .

data(Hepta) 
distMatrix = CosinusDistance(Hepta$Data)

Dist2All

The Dist2All function calculates the distances of a given point $x$ , to all other points (rows) of a given data matrix $A$ . For the calculation of the distances, various distance-measures can be chosen, for e.g. Euclidean, Manhattan (City Block), Mahalanobis, Bhjattacharyya, for a complete list see parallelDist. The distance-measure can be specified with the method argument. The function returns an ordered vector of the distances from point $x$ to all points in $A$ in ascending order, as well as the indices of k-nearest-neighbors for the chosen distance measure.

data(Hepta)
V = Dist2All(Hepta$Data[1,],Hepta$Data, method = "euclidean", knn=3)
# Vector of distances from Hepta$Data[1,] to all other rows in Hepta$Data
print(V$distToAll)
#>   [1] 0.00000000 0.08058895 0.04781043 0.09214454 0.03886835 0.09699465
#>   [7] 0.09783282 0.06061589 0.10267484 0.08356635 0.12395455 0.13170909
#>  [13] 0.09367107 0.07311107 0.10489254 0.07201004 0.16944962 0.08932404
#>  [19] 0.13545775 0.03665085 0.12492320 0.13485587 0.05292353 0.03097211
#>  [25] 0.14614814 0.08677934 0.02394002 0.11479889 0.03230518 0.14903284
#>  [31] 0.13893677 0.14948497 2.52446917 3.01913398 3.21579110 3.17836461
#>  [37] 3.15769630 2.57796199 3.31600855 3.42089925 3.83891632 3.58352663
#>  [43] 3.82807942 3.74241501 3.66695458 2.56495911 3.48798635 2.51574735
#>  [49] 2.47210300 3.08088830 3.29044933 2.51797162 2.77927187 3.11074797
#>  [55] 3.09722767 3.55496535 3.33477595 3.09117752 3.14815058 2.98838821
#>  [61] 2.90743022 2.93734164 2.66675926 2.64259662 3.36701734 3.90809765
#>  [67] 3.30677234 2.66204951 2.20011186 3.05133415 3.51307757 2.65792595
#>  [73] 3.32292220 2.87368651 2.72029774 2.61406504 2.92254677 2.83595565
#>  [79] 3.55531532 2.53112234 2.76796248 2.90106297 3.11617310 3.60091602
#>  [85] 3.55476025 2.31093773 2.62329978 3.22935873 2.56992529 3.40588864
#>  [91] 3.90022679 2.81386462 3.02034146 3.05283900 2.13176098 3.07322426
#>  [97] 3.49643390 3.33818899 3.28321841 2.64631876 3.34413644 3.69818991
#> [103] 2.86346004 3.65485742 3.78012860 3.58415974 2.65018304 3.56255550
#> [109] 3.65163561 2.91422029 3.07258132 2.45181926 2.29991462 3.20917355
#> [115] 3.70924494 2.59280107 2.97424022 2.83887470 3.53219603 2.70771842
#> [121] 3.03205030 3.31160172 2.47996181 3.05245948 3.12721819 3.63906971
#> [127] 3.07121966 2.40720597 2.77981952 3.75378880 3.93878434 2.63787864
#> [133] 3.57013739 3.00944011 3.00081140 3.10025752 2.44570366 3.09900684
#> [139] 2.94566780 3.22610410 3.77257806 2.95948219 3.04835200 3.29707317
#> [145] 2.38829944 3.36077136 3.68833648 2.18316289 2.99890839 2.81540383
#> [151] 2.42404613 3.81733227 2.92926568 3.45549966 3.21561093 3.37903200
#> [157] 2.41146632 3.09742210 2.93177839 3.02379783 3.01282943 2.31164299
#> [163] 2.92613725 3.30081802 2.89988712 2.83634572 2.95293088 3.38450777
#> [169] 2.22953148 2.83342086 3.52553473 2.32071642 2.65455358 2.52694921
#> [175] 2.78506782 3.55896170 2.21862698 3.10491516 2.20840668 2.95602706
#> [181] 3.02296244 3.87704358 3.02381731 3.93379495 3.70924221 3.03949680
#> [187] 3.13826953 3.00121181 2.97098494 2.90194795 3.67516270 3.42685363
#> [193] 3.65196565 2.40343230 3.17742347 2.80353846 3.04065098 2.98600351
#> [199] 3.22565744 3.16701313 2.52899115 3.72693787 2.57746647 3.77579621
#> [205] 2.94798545 3.06495823 2.52541787 2.76796966 3.30391078 2.95077124
#> [211] 3.67311616 2.69897901
# Vector of the indices of the k-nearest-neighbors, according to the euclidean distance
print(V$KNN)
#> [1]  1 27 24

DistanceMatrix

For a given $[1:n, 1:d]$ data matrix $A$ , with $n$ cases and $d$ variables, the function calculates the symmetric $[1:n, 1:n]$ distance matrix, given a chosen distance-measure. The method argument specifies the distance-measure (euclidean by default).

data(Hepta)
Dmatrix = DistanceMatrix(Hepta$Data, method='euclidean')

Options for method include :

‘euclidean’, ‘sqEuclidean’, ‘binary’, ‘cityblock’, ‘maximum’, ‘canberra’, ‘cosine’, ‘chebychev’, ‘jaccard’, ‘mahalanobis’, ‘minkowski’ ,‘manhattan’ , ‘braycur’ ,‘cosine’.

For the method ‘minkowski’, the parameter dim, can be used to specify the value of p in $\left( \sum_{i=1}^{n} |A_{j i} - A_{l i}|^p \right)^{1/p}$

Dmatrix = DistanceMatrix(Hepta$Data, method='minkowski', dim=3)

Fractional Distances

The fractional distance function uses the formula of the Minkowski-metric to calculate the distances and allows the usage of fractional values $p \in [0,1]$ , which can be useful for high-dimensional data [Aggrawal et al., 2001].

data(Hepta)
distMatrix = FractionalDistance(Hepta$Data, p = 1/2)

Tfidf-distance

The term frequency-inverse document frequency (Tf-idf) is a statistical measure of relevance of a term $t$ to a document $d$ in a collection of documents $D$ . The Tfidf-distance for two documents $d_i$ , $d_j \in D$ is then the absolute difference between the Tfidf-values.

An exemplary usage for bioinformatic data is the calculation of distances between genes using the Tfidf-distance, based on GO-Terms (Gene-Ontology-terms). For this a matrix $A$ of $n$ genes as rows and $m$ GO-Terms as columns is used, where genes can be interpreted as documents and GO-terms as terms [Thrun, 2022].

data(Hearingloss_N109)
V = Tfidf_dist(Hearingloss_N109$FeatureMatrix_Gene2Term, tf_fun = mean)
# Get distances
dist = V$dist
# Get weights
TfidfWeights = V$TfidfWeights

For the calculation of the (augmented) term-frequency, per default the mean of the non-zero entries is used, but can be specified with the argument tf_fun.

BIDistances

Introduction to Bioinformatic Distances