parDist: Parallel Distance Matrix Computation using multiple Threads

Description

Calculates distance matrices in parallel using multiple threads. Supports 41 predefined distance measures and user-defined distance functions.

Usage

parDist(x, method = "euclidean", diag = FALSE, upper = FALSE, threads = NULL, ...)
parallelDist(x, method = "euclidean", diag = FALSE, upper = FALSE, threads = NULL, ...)

Arguments

a numeric matrix (each row is one series) or list of numeric matrices for multidimensional series (each matrix is one series, a row is a dimension of a series)

method

the distance measure to be used. A list of all available distance methods can be found in the details section below.

diag

logical value indicating whether the diagonal of the distance matrix should be printed by print.dist.

upper

logical value indicating whether the upper triangle of the distance matrix should be printed by print.dist

threads

number of cpu threads for calculating a distance matrix. Default is the maximum amount of cpu threads available on the system.

...

additional parameters which will be passed to the distance methods. See details section below.

Value

parDist returns an object of class "dist".

The lower triangle of the distance matrix stored by columns in a vector, say do. If n is the number of observations, i.e., n <- attr(do, "Size"), then for $i < j \le n$, the dissimilarity between (row) i and j is do[n*(i-1) - i*(i-1)/2 + j-i]. The length of the vector is $n*(n-1)/2$, i.e., of order $n^2$.

The object has the following attributes (besides "class" equal to "dist"):

Size

integer, the number of observations in the dataset.

Labels

optionally, contains the labels, if any, of the observations of the dataset.

Diag, Upper

logicals corresponding to the arguments diag and upper above, specifying how the object should be printed.

call

optionally, the call used to create the object.

method

optionally, the distance method used; resulting from parDist(), the (match.arg()ed) method argument.

Details

User-defined distance functions

custom

Defining and compiling a user-defined C++ distance function, as well as creating an external pointer to the function can easily be achieved with the cppXPtr function of the RcppXPtrUtils package. The resulting Xptr external pointer object needs to be passed to parDist using the func parameter.

Parameters:

func (Xptr)
External pointer to a user-defined distance function with the following signature: double customDist(const arma::mat &A, const arma::mat &B) Note that the return value must be a double and the two parameters must be of type const arma::mat &param. More information about the Armadillo library can be found at http://arma.sourceforge.net/docs.html or as part of the documentation of the RcppArmadillo package.

An exemplary definition and usage of an user-defined euclidean distance function can be found in the examples section below.

Available predefined distance measures (written for two vectors $x$ and $y$)

Distance methods for continuous input variables

bhjattacharyya: The Bhjattacharyya distance. Type: continuous Formula: $sqrt(sum_i (sqrt(x_i) - sqrt(y_i))^2))$. Details: See pr_DB$get_entry("bhjattacharyya") in proxy.
bray: The Bray/Curtis dissimilarity. Type: continuous Formula: $sum_i |x_i - y_i| / sum_i (x_i + y_i)$. Details: See pr_DB$get_entry("bray") in proxy.
canberra: The Canberra distance (with compensation for excluded components). Terms with zero numerator and denominator are omitted from the sum and treated as if the values were missing. Type: continuous Formula: $sum_i |x_i - y_i| / |x_i + y_i|$. Details: See pr_DB$get_entry("canberra") in proxy.
chord: The Chord distance. Type: continuous Formula: $sqrt(2 * (1 - xy / sqrt(xx * yy)))$. Details: See pr_DB$get_entry("chord") in proxy.
divergence: The Divergence distance. Type: continuous Formula: $sum_i (x_i - y_i)^2 / (x_i + y_i)^2$. Details: See pr_DB$get_entry("divergence") in proxy.

dtw

Implementation of a multi-dimensional Dynamic Time Warping algorithm. Type: continuous Formula: Euclidean distance $sqrt(sum_i (x_i - y_i)^2)$. Parameters:

window.size (integer, optional)
Size of the window of the Sakoe-Chiba band. If the absolute length difference of two series x and y is larger than the window.size, the window.size is set to the length difference.
norm.method (character, optional)
Normalization method for DTW distances.
- path.length Normalization with the length of the warping path.
- n Normalization with n. n is the length of series x.
- n+m Normalization with n + m. n is the length of series x, m is the length of series y.
step.pattern (character or stepPattern of dtw package, default: symmetric1)
The following step patterns of the dtw package are supported:
- asymmetric (Normalization hint: n)
- asymmetricP0 (Normalization hint: n)
- asymmetricP05 (Normalization hint: n)
- asymmetricP1 (Normalization hint: n)
- asymmetricP2 (Normalization hint: n)
- symmetric1 (Normalization hint: path.length)
- symmetric2 or symmetricP0 (Normalization hint: n+m)
- symmetricP05 (Normalization hint: n+m)
- symmetricP1 (Normalization hint: n+m)
- symmetricP2 (Normalization hint: n+m)
For a detailed description see stepPattern of the dtw package.

euclidean

The Euclidean distance/L_2-norm (with compensation for excluded components). Type: continuous Formula: $sqrt(sum_i (x_i - y_i)^2))$. Details: See pr_DB$get_entry("euclidean") in proxy.

fJaccard

The fuzzy Jaccard distance. Type: binary Formula: $sum_i (min{x_i, y_i}) / sum_i(max{x_i, y_i})$. Details: See pr_DB$get_entry("fJaccard") in proxy.

geodesic

The geoedesic distance, i.e. the angle between x and y. Type: continuous Formula: $arccos(xy / sqrt(xx * yy))$. Details: See pr_DB$get_entry("geodesic") in proxy.

hellinger

The Hellinger distance. Type: continuous Formula: $sqrt(sum_i (sqrt(x_i / sum_i x) - sqrt(y_i / sum_i y)) ^ 2)$. Details: See pr_DB$get_entry("hellinger") in proxy.

kullback

The Kullback-Leibler distance. Type: continuous Formula: $sum_i [x_i * log((x_i / sum_j x_j) / (y_i / sum_j y_j)) / sum_j x_j)]$. Details: See pr_DB$get_entry("kullback") in proxy.

mahalanobis

The Mahalanobis distance. The Variance-Covariance-Matrix is estimated from the input data if unspecified. Type: continuous Formula: $sqrt((x - y) Sigma^(-1) (x - y))$. Parameters:

cov (numeric matrix, optional)
The covariance matrix (p x p) of the distribution.
inverted (logical, optional)
If TRUE, cov is supposed to contain the inverse of the covariance matrix.

Details: See pr_DB$get_entry("mahalanobis") in proxy or mahalanobis in stats.

manhattan

The Manhattan/City-Block/Taxi/L_1-norm distance (with compensation for excluded components). Type: continuous Formula: $sum_i |x_i - y_i|$. Details: See pr_DB$get_entry("manhattan") in proxy.

maximum

The Maximum/Supremum/Chebyshev distance. Type: continuous Formula: $max_i |x_i - y_i|$. Details: See pr_DB$get_entry("maximum") in proxy.

minkowski

The Minkowski distance/p-norm (with compensation for excluded components). Type: continuous Formula: $(sum_i (x_i - y_i)^p)^(1/p)$. Parameters:

p (double, optional)
The $p$th root of the sum of the $p$th powers of the differences of the components.

Details: See pr_DB$get_entry("minkowski") in proxy.

podani

The Podany measure of discordance is defined on ranks with ties. In the formula, for two given objects x and y, n is the number of variables, a is is the number of pairs of variables ordered identically, b the number of pairs reversely ordered, c the number of pairs tied in both x and y (corresponding to either joint presence or absence), and d the number of all pairs of variables tied at least for one of the objects compared such that one, two, or thee scores are zero. Type: continuous Formula: $1 - 2 * (a - b + c - d) / (n * (n - 1))$. Details: See pr_DB$get_entry("podani") in proxy.

soergel

The Soergel distance. Type: continuous Formula: $sum_i |x_i - y_i| / sum_i max{x_i, y_i}$. Details: See pr_DB$get_entry("soergel") in proxy.

wave

The Wave/Hedges distance. Type: continuous Formula: $sum_i (1 - min(x_i, y_i) / max(x_i, y_i))$. Details: See pr_DB$get_entry("wave") in proxy.

whittaker

The Whittaker distance. Type: continuous Formula: $sum_i |x_i / sum_i x - y_i / sum_i y| / 2$. Details: See pr_DB$get_entry("whittaker") in proxy.

Distance methods for binary input variables

Notation:

a: number of (TRUE, TRUE) pairs
b: number of (FALSE, TRUE) pairs
c: number of (TRUE, FALSE) pairs
d: number of (FALSE, FALSE) pairs

Note: Similarities are converted to distances.

binary: The Jaccard Similarity for binary data. It is the proportion of (TRUE, TRUE) pairs, but not considering (FALSE, FALSE) pairs. Type: binary Formula: $a / (a + b + c)$. Details: See pr_DB$get_entry("binary") in proxy.
braun-blanquet: The Braun-Blanquet similarity. Type: binary Formula: $a / max{(a + b), (a + c)}$. Details: See pr_DB$get_entry("braun-blanquet") in proxy.
cosine: The cosine similarity. Type: continuous Formula: $(a * b) / (|a|*|b|)$. Details: See pr_DB$get_entry("cosine") in proxy.
dice: The Dice similarity. Type: binary Formula: $2a / (2a + b + c)$. Details: See pr_DB$get_entry("dice") in proxy.
fager: The Fager / McGowan distance. Type: binary Formula: $a / sqrt((a + b)(a + c)) - sqrt(a + c) / 2$. Details: See pr_DB$get_entry("fager") in proxy.
faith: The Faith similarity. Type: binary Formula: $(a + d/2) / n$. Details: See pr_DB$get_entry("faith") in proxy.
hamman: The Hamman Matching similarity for binary data. It is the proportion difference of the concordant and discordant pairs. Type: binary Formula: $([a + d] - [b + c]) / n$. Details: See pr_DB$get_entry("hamman") in proxy.
hamming: The hamming distance between two vectors A and B is the fraction of positions where there is a mismatch. Formula: $\textit{\# of }(A != B) / \textit{\# in A (or B)}$
kulczynski1: Kulczynski similarity for binary data. Relates the (TRUE, TRUE) pairs to discordant pairs. Type: binary Formula: $a / (b + c)$. Details: See pr_DB$get_entry("kulczynski1") in proxy.
kulczynski2: Kulczynski similarity for binary data. Relates the (TRUE, TRUE) pairs to the discordant pairs. Type: binary Formula: $[a / (a + b) + a / (a + c)] / 2$. Details: See pr_DB$get_entry("kulczynski2") in proxy.
michael: The Michael similarity. Type: binary Formula: $4(ad - bc) / [(a + d)^2 + (b + c)^2]$. Details: See pr_DB$get_entry("michael") in proxy.
mountford: The Mountford similarity for binary data. Type: binary Formula: $2a / (ab + ac + 2bc)$. Details: See pr_DB$get_entry("mountford") in proxy.
mozley: The Mozley/Margalef similarity. Type: binary Formula: $an / (a + b)(a + c)$. Details: See pr_DB$get_entry("mozley") in proxy.
ochiai: The Ochiai similarity. Type: binary Formula: $a / sqrt[(a + b)(a + c)]$. Details: See pr_DB$get_entry("ochiai") in proxy.
phi: The Phi similarity (= Product-Moment-Correlation for binary variables). Type: binary Formula: $(ad - bc) / sqrt[(a + b)(c + d)(a + c)(b + d)]$. Details: See pr_DB$get_entry("phi") in proxy.
russel: The Russel/Raosimilarity for binary data. It is just the proportion of (TRUE, TRUE) pairs. Type: binary Formula: $a / n$. Details: See pr_DB$get_entry("russel") in proxy.
simple matching: The Simple Matching similarity for binary data. It is the proportion of concordant pairs. Type: binary Formula: $(a + d) / n$. Details: See pr_DB$get_entry("simple matching") in proxy.
simpson: The Simpson similarity. Type: binary Formula: $a / min{(a + b), (a + c)}$. Details: See pr_DB$get_entry("simpson") in proxy.
stiles: The Stiles similarity. Identical to the logarithm of Krylov's distance. Type: binary Formula: $log(n(|ad-bc| - 0.5n)^2 / [(a + b)(c + d)(a + c)(b + d)])$. Details: See pr_DB$get_entry("stiles") in proxy.
tanimoto: The Rogers/Tanimoto similarity for binary data. Similar to the simple matching coefficient, but putting double weight on the discordant pairs. Type: binary Formula: $(a + d) / (a + 2b + 2c + d)$. Details: See pr_DB$get_entry("tanimoto") in proxy.
yule: The Yule similarity. Type: binary Formula: $(ad - bc) / (ad + bc)$. Details: See pr_DB$get_entry("yule") in proxy.
yule2: The Yule similarity. Type: binary Formula: $(sqrt(ad) - sqrt(bc)) / (sqrt(ad) + sqrt(bc))$. Details: See pr_DB$get_entry("yule2") in proxy.

Examples

Run this code

# NOT RUN {
## predefined distance functions
# defining a matrix, where each row corresponds to one series
sample.matrix <- matrix(c(1:100), ncol = 10)

# euclidean distance
parDist(x = sample.matrix, method = "euclidean")
# minkowski distance with parameter p=2
parDist(x = sample.matrix, method = "minkowski", p=2)
# dynamic time warping distance
parDist(x = sample.matrix, method = "dtw")
# dynamic time warping distance normalized with warping path length
parDist(x = sample.matrix, method = "dtw", norm.method="path.length")
# dynamic time warping with different step pattern
parDist(x = sample.matrix, method = "dtw", step.pattern="symmetric2")
# dynamic time warping with window size constraint
parDist(x = sample.matrix, method = "dtw", step.pattern="symmetric2", window.size=1)

## multi-dimensional distance functions using list of matrices
# defining a list of matrices, where each list entry row corresponds to a two dimensional series
tmp.mat <- matrix(c(1:40), ncol = 10)
sample.matrix.list <- list(tmp.mat[1:2,], tmp.mat[3:4,])

# multi-dimensional euclidean distance
parDist(x = sample.matrix.list, method = "euclidean")
# multi-dimensional dynamic time warping
parDist(x = sample.matrix.list, method = "dtw")

## user-defined distance function
library(RcppArmadillo)
# Use RcppXPtrUtils for simple usage of C++ external pointers
library(RcppXPtrUtils)

# compile user-defined function and return pointer (RcppArmadillo is used as dependency)
euclideanFuncPtr <- cppXPtr(
"double customDist(const arma::mat &A, const arma::mat &B) {
  return sqrt(arma::accu(arma::square(A - B)));
}", depends = c("RcppArmadillo"))

# distance matrix for user-defined euclidean distance function (note that method is set to "custom")
parDist(matrix(1:16, ncol=2), method="custom", func = euclideanFuncPtr)
# }

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