Package 'energy' reference manual

Title:	E-Statistics: Multivariate Inference via the Energy of Data
Description:	E-statistics (energy) tests and statistics for multivariate and univariate inference, including distance correlation, one-sample, two-sample, and multi-sample tests for comparing multivariate distributions, are implemented. Measuring and testing multivariate independence based on distance correlation, partial distance correlation, multivariate goodness-of-fit tests, k-groups and hierarchical clustering based on energy distance, testing for multivariate normality, distance components (disco) for non-parametric analysis of structured data, and other energy statistics/methods are implemented.
Authors:	Maria Rizzo [aut, cre], Gabor Szekely [aut]
Maintainer:	Maria Rizzo <[email protected]>
License:	GPL (>= 2)
Version:	1.7-12
Built:	2024-10-26 05:11:08 UTC
Source:	https://github.com/mariarizzo/energy

E-statistics: Multivariate Inference via the Energy of Data

Description

Description: E-statistics (energy) tests and statistics for multivariate and univariate inference, including distance correlation, one-sample, two-sample, and multi-sample tests for comparing multivariate distributions, are implemented. Measuring and testing multivariate independence based on distance correlation, partial distance correlation, multivariate goodness-of-fit tests, clustering based on energy distance, testing for multivariate normality, distance components (disco) for non-parametric analysis of structured data, and other energy statistics/methods are implemented.

Author(s)

Maria L. Rizzo and Gabor J. Szekely

References

G. J. Szekely and M. L. Rizzo (2013). Energy statistics: A class of statistics based on distances, Journal of Statistical Planning and Inference.

M. L. Rizzo and G. J. Szekely (2016). Energy Distance, WIRES Computational Statistics, Wiley, Volume 8 Issue 1, 27-38. Available online Dec., 2015, doi:10.1002/wics.1375.

G. J. Szekely and M. L. Rizzo (2017). The Energy of Data. The Annual Review of Statistics and Its Application 4:447-79.

G. J. Szekely and M. L. Rizzo (2023). The Energy of Data and Distance Correlation. Chapman & Hall/CRC Monographs on Statistics and Applied Probability. ISBN 9781482242744. https://www.routledge.com/The-Energy-of-Data-and-Distance-Correlation/Szekely-Rizzo/p/book/9781482242744.

Double centering and U-centering

Description

Stand-alone double centering and U-centering functions that are applied in unbiased distance covariance, bias corrected distance correlation, and partial distance correlation.

Usage

Dcenter(x)
Ucenter(x)
U_center(Dx)
D_center(Dx)
Dcenter(x)
Ucenter(x)
U_center(Dx)
D_center(Dx)

Arguments

`x`	dist object or data matrix
`Dx`	distance or dissimilarity matrix

Details

In Dcenter and Ucenter, x must be a dist object or a data matrix. Both functions return a doubly centered distance matrix.

Note that pdcor, etc. functions include the centering operations (in C), so that these stand alone versions of centering functions are not needed except in case one wants to compute just a double-centered or U-centered matrix.

U_center is the Rcpp export of the cpp function. D_center is the Rcpp export of the cpp function.

Value

All functions return a square symmetric matrix.

Dcenter returns a matrix

$A_{ij}=a_{ij} - \bar a_{i.} - \bar a_{.j} + \bar a_{..}$

as in classical multidimensional scaling. Ucenter returns a matrix

$\tilde A_{ij}=a_{ij} - \frac{a_{i.}}{n-2} - \frac{a_{.j}}{n-2} + \frac{a_{..}}{(n-1)(n-2)},\quad i \neq j,$

with zero diagonal, and this is the double centering applied in pdcov and pdcor as well as the unbiased dCov and bias corrected dCor statistics.

Note

The c++ versions D_center and U_center should typically be faster. R versions are retained for historical reasons.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2014), Partial Distance Correlation with Methods for Dissimilarities, Annals of Statistics, Vol. 42, No. 6, pp. 2382-2412.

Examples

 x <- iris[1:10, 1:4]
 dx <- dist(x)
 Dx <- as.matrix(dx)
 M <- U_center(Dx)

 all.equal(M, U_center(M))     #idempotence
 all.equal(M, D_center(M))     #invariance
x <- iris[1:10, 1:4]
 dx <- dist(x)
 Dx <- as.matrix(dx)
 M <- U_center(Dx)

 all.equal(M, U_center(M))     #idempotence
 all.equal(M, D_center(M))     #invariance

Distance Correlation t-test for High Dimensions

Description

Defunct: use dcorT.test and dcorT.

Usage

dcor.t(x, y, distance = FALSE)
dcor.ttest(x, y, distance = FALSE)
dcor.t(x, y, distance = FALSE)
dcor.ttest(x, y, distance = FALSE)

Arguments

`x`	data or distances of first sample
`y`	data or distances of second sample
`distance`	TRUE if x and y are distances, otherwise FALSE

Details

See dcorT.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

Distance Correlation t-Test

Description

Distance correlation t-test of multivariate independence for high dimension.

Usage

dcorT.test(x, y)
dcorT(x, y)
dcorT.test(x, y)
dcorT(x, y)

Arguments

`x`	data or distances of first sample
`y`	data or distances of second sample

Details

dcorT.test performs a nonparametric t-test of multivariate independence in high dimension (dimension is close to or larger than sample size). As dimension goes to infinity, the asymptotic distribution of the test statistic is approximately Student t with $n(n-3)/2-1$ degrees of freedom and for $n \geq 10$ the statistic is approximately distributed as standard normal.

The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values.

The t statistic (dcorT) is a transformation of a bias corrected version of distance correlation (see SR 2013 for details).

Large values (upper tail) of the dcorT statistic are significant.

Value

dcorT returns the dcor t statistic, and dcorT.test returns a list with class htest containing

`method`	description of test
`statistic`	observed value of the test statistic
`parameter`	degrees of freedom
`estimate`	(bias corrected) squared dCor(x,y)
`p.value`	p-value of the t-test
`data.name`	description of data

Note

dcor.t and dcor.ttest are deprecated.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2013). The distance correlation t-test of independence in high dimension. Journal of Multivariate Analysis, Volume 117, pp. 193-213.
doi:10.1016/j.jmva.2013.02.012

Szekely, G.J., Rizzo, M.L., and Bakirov, N.K. (2007), Measuring and Testing Dependence by Correlation of Distances, Annals of Statistics, Vol. 35 No. 6, pp. 2769-2794.
doi:10.1214/009053607000000505

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1236-1265.
doi:10.1214/09-AOAS312

Examples

 x <- matrix(rnorm(100), 10, 10)
 y <- matrix(runif(100), 10, 10)
 dcorT(x, y)
 dcorT.test(x, y)
x <- matrix(rnorm(100), 10, 10)
 y <- matrix(runif(100), 10, 10)
 dcorT(x, y)
 dcorT.test(x, y)

Distance Covariance Test and Distance Correlation test

Description

Distance covariance test and distance correlation test of multivariate independence. Distance covariance and distance correlation are multivariate measures of dependence.

Usage

dcov.test(x, y, index = 1.0, R = NULL)
dcor.test(x, y, index = 1.0, R)
dcov.test(x, y, index = 1.0, R = NULL)
dcor.test(x, y, index = 1.0, R)

Arguments

`x`	data or distances of first sample
`y`	data or distances of second sample
`R`	number of replicates
`index`	exponent on Euclidean distance, in (0,2]

Details

dcov.test and dcor.test are nonparametric tests of multivariate independence. The test decision is obtained via permutation bootstrap, with R replicates.

The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values.

The index is an optional exponent on Euclidean distance. Valid exponents for energy are in (0, 2) excluding 2.

Argument types supported are numeric data matrix, data.frame, or tibble, with observations in rows; numeric vector; ordered or unordered factors. In case of unordered factors a 0-1 distance matrix is computed.

Optionally pre-computed distances can be input as class "dist" objects or as distance matrices. For data types of arguments, distance matrices are computed internally.

The dcov test statistic is $n \mathcal V_n^2$ where $\mathcal V_n(x,y)$ = dcov(x,y), which is based on interpoint Euclidean distances $\|x_{i}-x_{j}\|$ . The index is an optional exponent on Euclidean distance.

Similarly, the dcor test statistic is based on the normalized coefficient, the distance correlation. (See the manual page for dcor.)

(1) $\mathcal R(X,Y)$ is defined for $X$ and $Y$ in arbitrary dimension.

(2) $\mathcal R(X,Y)=0$ characterizes independence of $X$ and $Y$ .

Characterization (2) also holds for powers of Euclidean distance $\|x_i-x_j\|^s$ , where $0<s<2$ , but (2) does not hold when $s=2$ .

Distance correlation satisfies $0 \le \mathcal R \le 1$ , and $\mathcal R = 0$ only if $X$ and $Y$ are independent. Distance covariance $\mathcal V$ provides a new approach to the problem of testing the joint independence of random vectors. The formal definitions of the population coefficients $\mathcal V$ and $\mathcal R$ are given in (SRB 2007). The definitions of the empirical coefficients are given in the energy dcov topic.

For all values of the index in (0,2), under independence the asymptotic distribution of $n\mathcal V_n^2$ is a quadratic form of centered Gaussian random variables, with coefficients that depend on the distributions of $X$ and $Y$ . For the general problem of testing independence when the distributions of $X$ and $Y$ are unknown, the test based on $n\mathcal V^2_n$ can be implemented as a permutation test. See (SRB 2007) for theoretical properties of the test, including statistical consistency.

Value

dcov.test or dcor.test returns a list with class htest containing

`method`	description of test
`statistic`	observed value of the test statistic
`estimate`	dCov(x,y) or dCor(x,y)
`estimates`	a vector: [dCov(x,y), dCor(x,y), dVar(x), dVar(y)]
`condition`	logical, permutation test applied
`replicates`	replicates of the test statistic
`p.value`	approximate p-value of the test
`n`	sample size
`data.name`	description of data

Note

For the dcov test of independence, the distance covariance test statistic is the V-statistic $\mathrm{n\, dCov^2} = n \mathcal{V}_n^2$ (not dCov).

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1236-1265.
doi:10.1214/09-AOAS312

Szekely, G.J. and Rizzo, M.L. (2009), Rejoinder: Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1303-1308.

Examples

 x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 set.seed(1)
 dcor.test(dist(x), dist(y), R=199)
 set.seed(1)
 dcov.test(x, y, R=199)
x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 set.seed(1)
 dcor.test(dist(x), dist(y), R=199)
 set.seed(1)
 dcov.test(x, y, R=199)

Fast dCor and dCov for bivariate data only

Description

For bivariate data only, these are fast O(n log n) implementations of distance correlation and distance covariance statistics. The U-statistic for dcov^2 is unbiased; the V-statistic is the original definition in SRB 2007. These algorithms do not store the distance matrices, so they are suitable for large samples.

Usage

dcor2d(x, y, type = c("V", "U"))
dcov2d(x, y, type = c("V", "U"), all.stats = FALSE)
dcor2d(x, y, type = c("V", "U"))
dcov2d(x, y, type = c("V", "U"), all.stats = FALSE)

Arguments

`x`	numeric vector
`y`	numeric vector
`type`	"V" or "U", for V- or U-statistics
`all.stats`	logical

Details

The unbiased (squared) dcov is documented in dcovU, for multivariate data in arbitrary, not necessarily equal dimensions. dcov2d and dcor2d provide a faster O(n log n) algorithm for bivariate (x, y) only (X and Y are real-valued random vectors). The O(n log n) algorithm was proposed by Huo and Szekely (2016). The algorithm is faster above a certain sample size n. It does not store the distance matrix so the sample size can be very large.

Value

By default, dcov2d returns the V-statistic $V_n = dCov_n^2(x, y)$ , and if type="U", it returns the U-statistic, unbiased for $dCov^2(X, Y)$ . The argument all.stats=TRUE is used internally when the function is called from dcor2d.

By default, dcor2d returns $dCor_n^2(x, y)$ , and if type="U", it returns a bias-corrected estimator of squared dcor equivalent to bcdcor.

These functions do not store the distance matrices so they are helpful when sample size is large and the data is bivariate.

Note

The U-statistic $U_n$ can be negative in the lower tail so the square root of the U-statistic is not applied. Similarly, dcor2d(x, y, "U") is bias-corrected and can be negative in the lower tail, so we do not take the square root. The original definitions of dCov and dCor (SRB2007, SR2009) were based on V-statistics, which are non-negative, and defined using the square root of V-statistics.

It has been suggested that instead of taking the square root of the U-statistic, one could take the root of $|U_n|$ before applying the sign, but that introduces more bias than the original dCor, and should never be used.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Huo, X. and Szekely, G.J. (2016). Fast computing for distance covariance. Technometrics, 58(4), 435-447.

Szekely, G.J. and Rizzo, M.L. (2014), Partial Distance Correlation with Methods for Dissimilarities. Annals of Statistics, Vol. 42 No. 6, 2382-2412.

Examples

  
    ## these are equivalent, but 2d is faster for n > 50
    n <- 100
    x <- rnorm(100)
    y <- rnorm(100)
    all.equal(dcov(x, y)^2, dcov2d(x, y), check.attributes = FALSE)
    all.equal(bcdcor(x, y), dcor2d(x, y, "U"), check.attributes = FALSE)

    x <- rlnorm(400)
    y <- rexp(400)
    dcov.test(x, y, R=199)    #permutation test
    dcor.test(x, y, R=199)
      
## these are equivalent, but 2d is faster for n > 50
    n <- 100
    x <- rnorm(100)
    y <- rnorm(100)
    all.equal(dcov(x, y)^2, dcov2d(x, y), check.attributes = FALSE)
    all.equal(bcdcor(x, y), dcor2d(x, y, "U"), check.attributes = FALSE)

    x <- rlnorm(400)
    y <- rexp(400)
    dcov.test(x, y, R=199)    #permutation test
    dcor.test(x, y, R=199)

Unbiased distance covariance statistics

Description

This function computes unbiased estimators of squared distance covariance, distance variance, and a bias-corrected estimator of (squared) distance correlation.

Usage

dcovU_stats(Dx, Dy)
dcovU_stats(Dx, Dy)

Arguments

`Dx`	distance matrix of first sample
`Dy`	distance matrix of second sample

Details

The unbiased (squared) dcov is inner product definition of dCov, in the Hilbert space of U-centered distance matrices.

The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values. The arguments must be square symmetric matrices.

Value

dcovU_stats returns a vector of the components of bias-corrected dcor: [dCovU, bcdcor, dVarXU, dVarYU].

Note

Unbiased distance covariance (SR2014) corresponds to the biased (original) $\mathrm{dCov^2}$ . Since dcovU is an unbiased statistic, it is signed and we do not take the square root. For the original distance covariance test of independence (SRB2007, SR2009), the distance covariance test statistic is the V-statistic $\mathrm{n\, dCov^2} = n \mathcal{V}_n^2$ (not dCov). Similarly, bcdcor is bias-corrected, so we do not take the square root as with dCor.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2014), Partial Distance Correlation with Methods for Dissimilarities. Annals of Statistics, Vol. 42 No. 6, 2382-2412.

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1236-1265.
doi:10.1214/09-AOAS312

Examples

 x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 Dx <- as.matrix(dist(x))
 Dy <- as.matrix(dist(y))
 dcovU_stats(Dx, Dy)
 x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 Dx <- as.matrix(dist(x))
 Dy <- as.matrix(dist(y))
 dcovU_stats(Dx, Dy)

distance components (DISCO)

Description

E-statistics DIStance COmponents and tests, analogous to variance components and anova.

Usage

disco(x, factors, distance, index=1.0, R, method=c("disco","discoB","discoF"))
disco.between(x, factors, distance, index=1.0, R)
disco(x, factors, distance, index=1.0, R, method=c("disco","discoB","discoF"))
disco.between(x, factors, distance, index=1.0, R)

Arguments

`x`	data matrix or distance matrix or dist object
`factors`	matrix or data frame of factor labels or integers (not design matrix)
`distance`	logical, TRUE if x is distance matrix
`index`	exponent on Euclidean distance in (0,2]
`R`	number of replicates for a permutation test
`method`	test statistic

Details

disco calculates the distance components decomposition of total dispersion and if R > 0 tests for significance using the test statistic disco "F" ratio (default method="disco"), or using the between component statistic (method="discoB"), each implemented by permutation test.

If x is a dist object, argument distance is ignored. If x is a distance matrix, set distance=TRUE.

In the current release disco computes the decomposition for one-way models only.

Value

When method="discoF", disco returns a list similar to the return value from anova.lm, and the print.disco method is provided to format the output into a similar table. Details:

disco returns a class disco object, which is a list containing

`call`	call
`method`	method
`statistic`	vector of observed statistics
`p.value`	vector of p-values
`k`	number of factors
`N`	number of observations
`between`	between-sample distance components
`withins`	one-way within-sample distance components
`within`	within-sample distance component
`total`	total dispersion
`Df.trt`	degrees of freedom for treatments
`Df.e`	degrees of freedom for error
`index`	index (exponent on distance)
`factor.names`	factor names
`factor.levels`	factor levels
`sample.sizes`	sample sizes
`stats`	matrix containing decomposition

When method="discoB", disco passes the arguments to disco.between, which returns a class htest object.

disco.between returns a class htest object, where the test statistic is the between-sample statistic (proportional to the numerator of the F ratio of the disco test.

Note

The current version does all calculations via matrix arithmetic and boot function. Support for more general additive models and a formula interface is under development.

disco methods have been added to the cluster distance summary function edist, and energy tests for equality of distribution (see eqdist.etest).

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

M. L. Rizzo and G. J. Szekely (2010). DISCO Analysis: A Nonparametric Extension of Analysis of Variance, Annals of Applied Statistics, Vol. 4, No. 2, 1034-1055.
doi:10.1214/09-AOAS245

Examples

      ## warpbreaks one-way decompositions
      data(warpbreaks)
      attach(warpbreaks)
      disco(breaks, factors=wool, R=99)
      
      ## warpbreaks two-way wool+tension
      disco(breaks, factors=data.frame(wool, tension), R=0)

      ## warpbreaks two-way wool*tension
      disco(breaks, factors=data.frame(wool, tension, wool:tension), R=0)

      ## When index=2 for univariate data, we get ANOVA decomposition
      disco(breaks, factors=tension, index=2.0, R=99)
      aov(breaks ~ tension)

      ## Multivariate response
      ## Example on producing plastic film from Krzanowski (1998, p. 381)
      tear <- c(6.5, 6.2, 5.8, 6.5, 6.5, 6.9, 7.2, 6.9, 6.1, 6.3,
                6.7, 6.6, 7.2, 7.1, 6.8, 7.1, 7.0, 7.2, 7.5, 7.6)
      gloss <- c(9.5, 9.9, 9.6, 9.6, 9.2, 9.1, 10.0, 9.9, 9.5, 9.4,
                 9.1, 9.3, 8.3, 8.4, 8.5, 9.2, 8.8, 9.7, 10.1, 9.2)
      opacity <- c(4.4, 6.4, 3.0, 4.1, 0.8, 5.7, 2.0, 3.9, 1.9, 5.7,
                   2.8, 4.1, 3.8, 1.6, 3.4, 8.4, 5.2, 6.9, 2.7, 1.9)
      Y <- cbind(tear, gloss, opacity)
      rate <- factor(gl(2,10), labels=c("Low", "High"))

	    ## test for equal distributions by rate
      disco(Y, factors=rate, R=99)
	    disco(Y, factors=rate, R=99, method="discoB")

      ## Just extract the decomposition table
      disco(Y, factors=rate, R=0)$stats

	    ## Compare eqdist.e methods for rate
	    ## disco between stat is half of original when sample sizes equal
	    eqdist.e(Y, sizes=c(10, 10), method="original")
	    eqdist.e(Y, sizes=c(10, 10), method="discoB")

      ## The between-sample distance component
      disco.between(Y, factors=rate, R=0)
## warpbreaks one-way decompositions
      data(warpbreaks)
      attach(warpbreaks)
      disco(breaks, factors=wool, R=99)
      
      ## warpbreaks two-way wool+tension
      disco(breaks, factors=data.frame(wool, tension), R=0)

      ## warpbreaks two-way wool*tension
      disco(breaks, factors=data.frame(wool, tension, wool:tension), R=0)

      ## When index=2 for univariate data, we get ANOVA decomposition
      disco(breaks, factors=tension, index=2.0, R=99)
      aov(breaks ~ tension)

      ## Multivariate response
      ## Example on producing plastic film from Krzanowski (1998, p. 381)
      tear <- c(6.5, 6.2, 5.8, 6.5, 6.5, 6.9, 7.2, 6.9, 6.1, 6.3,
                6.7, 6.6, 7.2, 7.1, 6.8, 7.1, 7.0, 7.2, 7.5, 7.6)
      gloss <- c(9.5, 9.9, 9.6, 9.6, 9.2, 9.1, 10.0, 9.9, 9.5, 9.4,
                 9.1, 9.3, 8.3, 8.4, 8.5, 9.2, 8.8, 9.7, 10.1, 9.2)
      opacity <- c(4.4, 6.4, 3.0, 4.1, 0.8, 5.7, 2.0, 3.9, 1.9, 5.7,
                   2.8, 4.1, 3.8, 1.6, 3.4, 8.4, 5.2, 6.9, 2.7, 1.9)
      Y <- cbind(tear, gloss, opacity)
      rate <- factor(gl(2,10), labels=c("Low", "High"))

	    ## test for equal distributions by rate
      disco(Y, factors=rate, R=99)
	    disco(Y, factors=rate, R=99, method="discoB")

      ## Just extract the decomposition table
      disco(Y, factors=rate, R=0)$stats

	    ## Compare eqdist.e methods for rate
	    ## disco between stat is half of original when sample sizes equal
	    eqdist.e(Y, sizes=c(10, 10), method="original")
	    eqdist.e(Y, sizes=c(10, 10), method="discoB")

      ## The between-sample distance component
      disco.between(Y, factors=rate, R=0)

Distance Correlation and Covariance Statistics

Description

Computes distance covariance and distance correlation statistics, which are multivariate measures of dependence.

Usage

dcov(x, y, index = 1.0)
dcor(x, y, index = 1.0)
dcov(x, y, index = 1.0)
dcor(x, y, index = 1.0)

Arguments

`x`	data or distances of first sample
`y`	data or distances of second sample
`index`	exponent on Euclidean distance, in (0,2]

Details

dcov and dcor compute distance covariance and distance correlation statistics.

The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values.

The index is an optional exponent on Euclidean distance. Valid exponents for energy are in (0, 2) excluding 2.

Optionally pre-computed distances can be input as class "dist" objects or as distance matrices. For data types of arguments, distance matrices are computed internally.

Distance correlation is a new measure of dependence between random vectors introduced by Szekely, Rizzo, and Bakirov (2007). For all distributions with finite first moments, distance correlation $\mathcal R$ generalizes the idea of correlation in two fundamental ways: (1) $\mathcal R(X,Y)$ is defined for $X$ and $Y$ in arbitrary dimension. (2) $\mathcal R(X,Y)=0$ characterizes independence of $X$ and $Y$ .

The empirical distance covariance $\mathcal{V}_n(\mathbf{X,Y})$ with index 1 is the nonnegative number defined by

$\mathcal{V}^2_n (\mathbf{X,Y}) = \frac{1}{n^2} \sum_{k,\,l=1}^n A_{kl}B_{kl}$

where $A_{kl}$ and $B_{kl}$ are

$A_{kl} = a_{kl}-\bar a_{k.}- \bar a_{.l} + \bar a_{..}$

$B_{kl} = b_{kl}-\bar b_{k.}- \bar b_{.l} + \bar b_{..}.$

Here

$a_{kl} = \|X_k - X_l\|_p, \quad b_{kl} = \|Y_k - Y_l\|_q, \quad k,l=1,\dots,n,$

and the subscript . denotes that the mean is computed for the index that it replaces. Similarly, $\mathcal{V}_n(\mathbf{X})$ is the nonnegative number defined by

$\mathcal{V}^2_n (\mathbf{X}) = \mathcal{V}^2_n (\mathbf{X,X}) = \frac{1}{n^2} \sum_{k,\,l=1}^n A_{kl}^2.$

The empirical distance correlation $\mathcal{R}_n(\mathbf{X,Y})$ is the square root of

$\mathcal{R}^2_n(\mathbf{X,Y})= \frac {\mathcal{V}^2_n(\mathbf{X,Y})} {\sqrt{ \mathcal{V}^2_n (\mathbf{X}) \mathcal{V}^2_n(\mathbf{Y})}}.$

See dcov.test for a test of multivariate independence based on the distance covariance statistic.

Value

dcov returns the sample distance covariance and dcor returns the sample distance correlation.

Note

Note that it is inefficient to compute dCor by:

square root of dcov(x,y)/sqrt(dcov(x,x)*dcov(y,y))

because the individual calls to dcov involve unnecessary repetition of calculations.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1236-1265.
doi:10.1214/09-AOAS312

Szekely, G.J. and Rizzo, M.L. (2009), Rejoinder: Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1303-1308.

Examples

 x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 dcov(x, y)
 dcov(dist(x), dist(y))  #same thing
x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 dcov(x, y)
 dcov(dist(x), dist(y))  #same thing

Distance Matrices

Description

Utilities for working with distance matrices. is.dmatrix is a utility that checks whether the argument is a distance or dissimilarity matrix; is it square symmetric, non-negative, with zero diagonal? calc_dist computes a distance matrix directly from a data matrix.

Usage

is.dmatrix(x, tol = 100 * .Machine$double.eps)
calc_dist(x)
is.dmatrix(x, tol = 100 * .Machine$double.eps)
calc_dist(x)

Arguments

`x`	numeric matrix
`tol`	tolerance for checking required conditions

Details

Energy functions work with the distance matrices of samples. The is.dmatrix function is used internally when converting arguments to distance matrices. The default tol is the same as default tolerance of isSymmetric.

calc_dist is an exported Rcpp function that returns a Euclidean distance matrix from the input data matrix.

Value

is.dmatrix returns TRUE if (within tolerance) x is a distance/dissimilarity matrix; otherwise FALSE. It will return FALSE if x is a class dist object.

calc_dist returns the Euclidean distance matrix for the data matrix x, which has observations in rows.

Note

In practice, if dist(x) is not yet computed, calc_dist(x) will be faster than as.matrix(dist(x)).

On working with non-Euclidean dissimilarities, see the references.

Author(s)

Maria L. Rizzo [email protected]

References

Szekely, G.J. and Rizzo, M.L. (2014), Partial Distance Correlation with Methods for Dissimilarities. Annals of Statistics, Vol. 42 No. 6, 2382-2412.

Examples

x <- matrix(rnorm(20), 10, 2)
D <- calc_dist(x)
is.dmatrix(D)
is.dmatrix(cov(x))
x <- matrix(rnorm(20), 10, 2)
D <- calc_dist(x)
is.dmatrix(D)
is.dmatrix(cov(x))

E-distance

Description

Returns the E-distances (energy statistics) between clusters.

Usage

 edist(x, sizes, distance = FALSE, ix = 1:sum(sizes), alpha = 1,
        method = c("cluster","discoB"))
edist(x, sizes, distance = FALSE, ix = 1:sum(sizes), alpha = 1,
        method = c("cluster","discoB"))

Arguments

`x`	data matrix of pooled sample or Euclidean distances
`sizes`	vector of sample sizes
`distance`	logical: if TRUE, x is a distance matrix
`ix`	a permutation of the row indices of x
`alpha`	distance exponent in (0,2]
`method`	how to weight the statistics

Details

A vector containing the pairwise two-sample multivariate $\mathcal{E}$ -statistics for comparing clusters or samples is returned. The e-distance between clusters is computed from the original pooled data, stacked in matrix x where each row is a multivariate observation, or from the distance matrix x of the original data, or distance object returned by dist. The first sizes[1] rows of the original data matrix are the first sample, the next sizes[2] rows are the second sample, etc. The permutation vector ix may be used to obtain e-distances corresponding to a clustering solution at a given level in the hierarchy.

The default method cluster summarizes the e-distances between clusters in a table. The e-distance between two clusters $C_i, C_j$ of size $n_i, n_j$ proposed by Szekely and Rizzo (2005) is the e-distance $e(C_i,C_j)$ , defined by

$e(C_i,C_j)=\frac{n_i n_j}{n_i+n_j}[2M_{ij}-M_{ii}-M_{jj}],$

where

$M_{ij}=\frac{1}{n_i n_j}\sum_{p=1}^{n_i} \sum_{q=1}^{n_j} \|X_{ip}-X_{jq}\|^\alpha,$

$\|\cdot\|$ denotes Euclidean norm, $\alpha=$ alpha, and $X_{ip}$ denotes the p-th observation in the i-th cluster. The exponent alpha should be in the interval (0,2].

The coefficient $\frac{n_i n_j}{n_i+n_j}$ is one-half of the harmonic mean of the sample sizes. The discoB method is related but with different ways of summarizing the pairwise differences between samples. The disco methods apply the coefficient $\frac{n_i n_j}{2N}$ where N is the total number of observations. This weights each (i,j) statistic by sample size relative to N. See the disco topic for more details.

Value

A object of class dist containing the lower triangle of the e-distance matrix of cluster distances corresponding to the permutation of indices ix is returned. The method attribute of the distance object is assigned a value of type, index.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G. J. and Rizzo, M. L. (2005) Hierarchical Clustering via Joint Between-Within Distances: Extending Ward's Minimum Variance Method, Journal of Classification 22(2) 151-183.
doi:10.1007/s00357-005-0012-9

M. L. Rizzo and G. J. Szekely (2010). DISCO Analysis: A Nonparametric Extension of Analysis of Variance, Annals of Applied Statistics, Vol. 4, No. 2, 1034-1055.
doi:10.1214/09-AOAS245

Szekely, G. J. and Rizzo, M. L. (2004) Testing for Equal Distributions in High Dimension, InterStat, November (5).

Szekely, G. J. (2000) Technical Report 03-05, $\mathcal{E}$ -statistics: Energy of Statistical Samples, Department of Mathematics and Statistics, Bowling Green State University.

Examples

     ## compute cluster e-distances for 3 samples of iris data
     data(iris)
     edist(iris[,1:4], c(50,50,50))
    
     ## pairwise disco statistics
     edist(iris[,1:4], c(50,50,50), method="discoB")  

     ## compute e-distances from a distance object
     data(iris)
     edist(dist(iris[,1:4]), c(50, 50, 50), distance=TRUE, alpha = 1)

     ## compute e-distances from a distance matrix
     data(iris)
     d <- as.matrix(dist(iris[,1:4]))
     edist(d, c(50, 50, 50), distance=TRUE, alpha = 1)

 ## compute cluster e-distances for 3 samples of iris data
     data(iris)
     edist(iris[,1:4], c(50,50,50))
    
     ## pairwise disco statistics
     edist(iris[,1:4], c(50,50,50), method="discoB")  

     ## compute e-distances from a distance object
     data(iris)
     edist(dist(iris[,1:4]), c(50, 50, 50), distance=TRUE, alpha = 1)

     ## compute e-distances from a distance matrix
     data(iris)
     d <- as.matrix(dist(iris[,1:4]))
     edist(d, c(50, 50, 50), distance=TRUE, alpha = 1)

Deprecated Functions

Description

These deprecated functions have been replaced by revised functions and will be removed in future releases of the energy package.

Usage

DCOR(x, y, index=1.0)
DCOR(x, y, index=1.0)

Arguments

`x`	data or distances of first sample
`y`	data or distances of second sample
`index`	exponent on Euclidean distance in (0, 2)

Details

DCOR is an R version replaced by faster compiled code.

Hierarchical Clustering by Minimum (Energy) E-distance

Description

Performs hierarchical clustering by minimum (energy) E-distance method.

Usage

    energy.hclust(dst, alpha = 1)
energy.hclust(dst, alpha = 1)

Arguments

`dst`	`dist` object
`alpha`	distance exponent

Details

Dissimilarities are $d(x,y) = \|x-y\|^\alpha$ , where the exponent $\alpha$ is in the interval (0,2]. This function performs agglomerative hierarchical clustering. Initially, each of the n singletons is a cluster. At each of n-1 steps, the procedure merges the pair of clusters with minimum e-distance. The e-distance between two clusters $C_i, C_j$ of sizes $n_i, n_j$ is given by

$e(C_i, C_j)=\frac{n_i n_j}{n_i+n_j}[2M_{ij}-M_{ii}-M_{jj}],$

where

$M_{ij}=\frac{1}{n_i n_j}\sum_{p=1}^{n_i} \sum_{q=1}^{n_j} \|X_{ip}-X_{jq}\|^\alpha,$

$\|\cdot\|$ denotes Euclidean norm, and $X_{ip}$ denotes the p-th observation in the i-th cluster.

The return value is an object of class hclust, so hclust methods such as print or plot methods, plclust, and cutree are available. See the documentation for hclust.

The e-distance measures both the heterogeneity between clusters and the homogeneity within clusters. $\mathcal E$ -clustering ( $\alpha=1$ ) is particularly effective in high dimension, and is more effective than some standard hierarchical methods when clusters have equal means (see example below). For other advantages see the references.

edist computes the energy distances for the result (or any partition) and returns the cluster distances in a dist object. See the edist examples.

Value

An object of class hclust which describes the tree produced by the clustering process. The object is a list with components:

`merge:`	an n-1 by 2 matrix, where row i of `merge` describes the merging of clusters at step i of the clustering. If an element j in the row is negative, then observation -j was merged at this stage. If j is positive then the merge was with the cluster formed at the (earlier) stage j of the algorithm.
`height:`	the clustering height: a vector of n-1 non-decreasing real numbers (the e-distance between merging clusters)
`order:`	a vector giving a permutation of the indices of original observations suitable for plotting, in the sense that a cluster plot using this ordering and matrix `merge` will not have crossings of the branches.
`labels:`	labels for each of the objects being clustered.
`call:`	the call which produced the result.
`method:`	the cluster method that has been used (e-distance).
`dist.method:`	the distance that has been used to create `dst`.

Note

Currently stats::hclust implements Ward's method by method="ward.D2", which applies the squared distances. That method was previously "ward". Because both hclust and energy use the same type of Lance-Williams recursive formula to update cluster distances, now with the additional option method="ward.D" in hclust, the energy distance method is easily implemented by hclust. (Some "Ward" algorithms do not use Lance-Williams, however). Energy clustering (with alpha=1) and "ward.D" now return the same result, except that the cluster heights of energy hierarchical clustering with alpha=1 are two times the heights from hclust. In order to ensure compatibility with hclust methods, energy.hclust now passes arguments through to hclust after possibly applying the optional exponent to distance.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G. J. and Rizzo, M. L. (2004) Testing for Equal Distributions in High Dimension, InterStat, November (5).

Szekely, G. J. (2000) Technical Report 03-05: $\mathcal{E}$ -statistics: Energy of Statistical Samples, Department of Mathematics and Statistics, Bowling Green State University.

Examples

   ## Not run: 
   library(cluster)
   data(animals)
   plot(energy.hclust(dist(animals)))

   data(USArrests)
   ecl <- energy.hclust(dist(USArrests))
   print(ecl)
   plot(ecl)
   cutree(ecl, k=3)
   cutree(ecl, h=150)

   ## compare performance of e-clustering, Ward's method, group average method
   ## when sampled populations have equal means: n=200, d=5, two groups
   z <- rbind(matrix(rnorm(1000), nrow=200), matrix(rnorm(1000, 0, 5), nrow=200))
   g <- c(rep(1, 200), rep(2, 200))
   d <- dist(z)
   e <- energy.hclust(d)
   a <- hclust(d, method="average")
   w <- hclust(d^2, method="ward.D2")
   list("E" = table(cutree(e, k=2) == g), "Ward" = table(cutree(w, k=2) == g),
    "Avg" = table(cutree(a, k=2) == g))
  
## End(Not run)
 ## Not run: 
   library(cluster)
   data(animals)
   plot(energy.hclust(dist(animals)))

   data(USArrests)
   ecl <- energy.hclust(dist(USArrests))
   print(ecl)
   plot(ecl)
   cutree(ecl, k=3)
   cutree(ecl, h=150)

   ## compare performance of e-clustering, Ward's method, group average method
   ## when sampled populations have equal means: n=200, d=5, two groups
   z <- rbind(matrix(rnorm(1000), nrow=200), matrix(rnorm(1000, 0, 5), nrow=200))
   g <- c(rep(1, 200), rep(2, 200))
   d <- dist(z)
   e <- energy.hclust(d)
   a <- hclust(d, method="average")
   w <- hclust(d^2, method="ward.D2")
   list("E" = table(cutree(e, k=2) == g), "Ward" = table(cutree(w, k=2) == g),
    "Avg" = table(cutree(a, k=2) == g))
  
## End(Not run)

Multisample E-statistic (Energy) Test of Equal Distributions

Description

Performs the nonparametric multisample E-statistic (energy) test for equality of multivariate distributions.

Usage

eqdist.etest(x, sizes, distance = FALSE,
    method=c("original","discoB","discoF"), R)
eqdist.e(x, sizes, distance = FALSE,
    method=c("original","discoB","discoF"))
ksample.e(x, sizes, distance = FALSE,
    method=c("original","discoB","discoF"), ix = 1:sum(sizes))
eqdist.etest(x, sizes, distance = FALSE,
    method=c("original","discoB","discoF"), R)
eqdist.e(x, sizes, distance = FALSE,
    method=c("original","discoB","discoF"))
ksample.e(x, sizes, distance = FALSE,
    method=c("original","discoB","discoF"), ix = 1:sum(sizes))

Arguments

`x`	data matrix of pooled sample
`sizes`	vector of sample sizes
`distance`	logical: if TRUE, first argument is a distance matrix
`method`	use original (default) or distance components (discoB, discoF)
`R`	number of bootstrap replicates
`ix`	a permutation of the row indices of x

Details

The k-sample multivariate $\mathcal{E}$ -test of equal distributions is performed. The statistic is computed from the original pooled samples, stacked in matrix x where each row is a multivariate observation, or the corresponding distance matrix. The first sizes[1] rows of x are the first sample, the next sizes[2] rows of x are the second sample, etc.

The test is implemented by nonparametric bootstrap, an approximate permutation test with R replicates.

The function eqdist.e returns the test statistic only; it simply passes the arguments through to eqdist.etest with R = 0.

The k-sample multivariate $\mathcal{E}$ -statistic for testing equal distributions is returned. The statistic is computed from the original pooled samples, stacked in matrix x where each row is a multivariate observation, or from the distance matrix x of the original data. The first sizes[1] rows of x are the first sample, the next sizes[2] rows of x are the second sample, etc.

The two-sample $\mathcal{E}$ -statistic proposed by Szekely and Rizzo (2004) is the e-distance $e(S_i,S_j)$ , defined for two samples $S_i, S_j$ of size $n_i, n_j$ by

$e(S_i,S_j)=\frac{n_i n_j}{n_i+n_j}[2M_{ij}-M_{ii}-M_{jj}],$

where

$M_{ij}=\frac{1}{n_i n_j}\sum_{p=1}^{n_i} \sum_{q=1}^{n_j} \|X_{ip}-X_{jq}\|,$

$\|\cdot\|$ denotes Euclidean norm, and $X_{ip}$ denotes the p-th observation in the i-th sample.

The original (default method) k-sample $\mathcal{E}$ -statistic is defined by summing the pairwise e-distances over all $k(k-1)/2$ pairs of samples:

$\mathcal{E}=\sum_{1 \leq i < j \leq k} e(S_i,S_j).$

Large values of $\mathcal{E}$ are significant.

The discoB method computes the between-sample disco statistic. For a one-way analysis, it is related to the original statistic as follows. In the above equation, the weights $\frac{n_i n_j}{n_i+n_j}$ are replaced with

$\frac{n_i + n_j}{2N}\frac{n_i n_j}{n_i+n_j} = \frac{n_i n_j}{2N}$

where N is the total number of observations: $N=n_1+...+n_k$ .

The discoF method is based on the disco F ratio, while the discoB method is based on the between sample component.

Also see disco and disco.between functions.

Value

A list with class htest containing

`method`	description of test
`statistic`	observed value of the test statistic
`p.value`	approximate p-value of the test
`data.name`	description of data

eqdist.e returns test statistic only.

Note

The pairwise e-distances between samples can be conveniently computed by the edist function, which returns a dist object.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G. J. and Rizzo, M. L. (2004) Testing for Equal Distributions in High Dimension, InterStat, November (5).

M. L. Rizzo and G. J. Szekely (2010). DISCO Analysis: A Nonparametric Extension of Analysis of Variance, Annals of Applied Statistics, Vol. 4, No. 2, 1034-1055.
doi:10.1214/09-AOAS245

Szekely, G. J. (2000) Technical Report 03-05: $\mathcal{E}$ -statistics: Energy of Statistical Samples, Department of Mathematics and Statistics, Bowling Green State University.

Examples

 data(iris)

 ## test if the 3 varieties of iris data (d=4) have equal distributions
 eqdist.etest(iris[,1:4], c(50,50,50), R = 199)

 ## example that uses method="disco"
  x <- matrix(rnorm(100), nrow=20)
  y <- matrix(rnorm(100), nrow=20)
  X <- rbind(x, y)
  d <- dist(X)

  # should match edist default statistic
  set.seed(1234)
  eqdist.etest(d, sizes=c(20, 20), distance=TRUE, R = 199)

  # comparison with edist
  edist(d, sizes=c(20, 10), distance=TRUE)

  # for comparison
  g <- as.factor(rep(1:2, c(20, 20)))
  set.seed(1234)
  disco(d, factors=g, distance=TRUE, R=199)

  # should match statistic in edist method="discoB", above
  set.seed(1234)
  disco.between(d, factors=g, distance=TRUE, R=199)
data(iris)

 ## test if the 3 varieties of iris data (d=4) have equal distributions
 eqdist.etest(iris[,1:4], c(50,50,50), R = 199)

 ## example that uses method="disco"
  x <- matrix(rnorm(100), nrow=20)
  y <- matrix(rnorm(100), nrow=20)
  X <- rbind(x, y)
  d <- dist(X)

  # should match edist default statistic
  set.seed(1234)
  eqdist.etest(d, sizes=c(20, 20), distance=TRUE, R = 199)

  # comparison with edist
  edist(d, sizes=c(20, 10), distance=TRUE)

  # for comparison
  g <- as.factor(rep(1:2, c(20, 20)))
  set.seed(1234)
  disco(d, factors=g, distance=TRUE, R=199)

  # should match statistic in edist method="discoB", above
  set.seed(1234)
  disco.between(d, factors=g, distance=TRUE, R=199)

Eigenvalues for the energy Test of Univariate Normality

Description

Pre-computed eigenvalues corresponding to the asymptotic sampling distribution of the energy test statistic for univariate normality, under the null hypothesis. Four Cases are computed:

Simple hypothesis, known parameters.
Estimated mean, known variance.
Known mean, estimated variance.
Composite hypothesis, estimated parameters.

Case 4 eigenvalues are used in the test function normal.test when method=="limit".

Usage

data(EVnormal)data(EVnormal)

Format

Numeric matrix with 125 rows and 5 columns; column 1 is the index, and columns 2-5 are the eigenvalues of Cases 1-4.

Source

Computed

References

Szekely, G. J. and Rizzo, M. L. (2005) A New Test for Multivariate Normality, Journal of Multivariate Analysis, 93/1, 58-80, doi:10.1016/j.jmva.2003.12.002.

Energy-tests of Independence

Description

Computes a multivariate nonparametric test of independence. The default method implements the distance covariance test dcov.test.

Usage

indep.test(x, y, method = c("dcov","mvI"), index = 1, R)
indep.test(x, y, method = c("dcov","mvI"), index = 1, R)

Arguments

`x`	matrix: first sample, observations in rows
`y`	matrix: second sample, observations in rows
`method`	a character string giving the name of the test
`index`	exponent on Euclidean distances
`R`	number of replicates

Details

indep.test with the default method = "dcov" computes the distance covariance test of independence. index is an exponent on the Euclidean distances. Valid choices for index are in (0,2], with default value 1 (Euclidean distance). The arguments are passed to the dcov.test function. See the help topic dcov.test for the description and documentation and also see the references below.

indep.test with method = "mvI" computes the coefficient $\mathcal I_n$ and performs a nonparametric $\mathcal E$ -test of independence. The arguments are passed to mvI.test. The index argument is ignored (index = 1 is applied). See the help topic mvI.test and also see the reference (2006) below for details.

The test decision is obtained via bootstrap, with R replicates. The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values.

These energy tests of independence are based on related theoretical results, but different test statistics. The dcov method is faster than mvI method by approximately a factor of O(n).

Value

indep.test returns a list with class htest containing

`method`	description of test
`statistic`	observed value of the test statistic $n \mathcal V_n^2$ or $n \mathcal I_n^2$
`estimate`	$\mathcal V_n$ or $\mathcal I_n$
`estimates`	a vector [dCov(x,y), dCor(x,y), dVar(x), dVar(y)] (method dcov)
`replicates`	replicates of the test statistic
`p.value`	approximate p-value of the test
`data.name`	description of data

Note

As of energy-1.1-0, indep.etest is deprecated and replaced by indep.test, which has methods for two different energy tests of independence. indep.test applies the distance covariance test (see dcov.test) by default (method = "dcov"). The original indep.etest applied the independence coefficient $\mathcal I_n$ , which is now obtained by method = "mvI".

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3 No. 4, pp. 1236-1265. (Also see discussion and rejoinder.)
doi:10.1214/09-AOAS312

Bakirov, N.K., Rizzo, M.L., and Szekely, G.J. (2006), A Multivariate Nonparametric Test of Independence, Journal of Multivariate Analysis 93/1, 58-80,
doi:10.1016/j.jmva.2005.10.005

Examples


 ## independent multivariate data
 x <- matrix(rnorm(60), nrow=20, ncol=3)
 y <- matrix(rnorm(40), nrow=20, ncol=2)
 indep.test(x, y, method = "dcov", R = 99)
 indep.test(x, y, method = "mvI", R = 99)

 ## dependent multivariate data
 if (require(MASS)) {
   Sigma <- matrix(c(1, .1, 0, 0 , 1, 0, 0 ,.1, 1), 3, 3)
   x <- mvrnorm(30, c(0, 0, 0), diag(3))
   y <- mvrnorm(30, c(0, 0, 0), Sigma) * x
   indep.test(x, y, R = 99)    #dcov method
   indep.test(x, y, method = "mvI", R = 99)
    }
  
## independent multivariate data
 x <- matrix(rnorm(60), nrow=20, ncol=3)
 y <- matrix(rnorm(40), nrow=20, ncol=2)
 indep.test(x, y, method = "dcov", R = 99)
 indep.test(x, y, method = "mvI", R = 99)

 ## dependent multivariate data
 if (require(MASS)) {
   Sigma <- matrix(c(1, .1, 0, 0 , 1, 0, 0 ,.1, 1), 3, 3)
   x <- mvrnorm(30, c(0, 0, 0), diag(3))
   y <- mvrnorm(30, c(0, 0, 0), Sigma) * x
   indep.test(x, y, R = 99)    #dcov method
   indep.test(x, y, method = "mvI", R = 99)
    }

K-Groups Clustering

Description

Perform k-groups clustering by energy distance.

Usage

kgroups(x, k, iter.max = 10, nstart = 1, cluster = NULL)
kgroups(x, k, iter.max = 10, nstart = 1, cluster = NULL)

Arguments

`x`	Data frame or data matrix or distance object
`k`	number of clusters
`iter.max`	maximum number of iterations
`nstart`	number of restarts
`cluster`	initial clustering vector

Details

K-groups is based on the multisample energy distance for comparing distributions. Based on the disco decomposition of total dispersion (a Gini type mean distance) the objective function should either maximize the total between cluster energy distance, or equivalently, minimize the total within cluster energy distance. It is more computationally efficient to minimize within distances, and that makes it possible to use a modified version of the Hartigan-Wong algorithm (1979) to implement K-groups clustering.

The within cluster Gini mean distance is

$G(C_j) = \frac{1}{n_j^2} \sum_{i,m=1}^{n_j} |x_{i,j} - x_{m,j}|$

and the K-groups within cluster distance is

$W_j = \frac{n_j}{2}G(C_j) = \frac{1}{2 n_j} \sum_{i,m=1}^{n_j} |x_{i,j} - x_{m,j}.$

If z is the data matrix for cluster $C_j$ , then $W_j$ could be computed as sum(dist(z)) / nrow(z).

If cluster is not NULL, the clusters are initialized by this vector (can be a factor or integer vector). Otherwise clusters are initialized with random labels in k approximately equal size clusters.

If x is not a distance object (class(x) == "dist") then x is converted to a data matrix for analysis.

Run up to iter.max complete passes through the data set until a local min is reached. If nstart > 1, on second and later starts, clusters are initialized at random, and the best result is returned.

Value

An object of class kgroups containing the components

`call`	the function call
`cluster`	vector of cluster indices
`sizes`	cluster sizes
`within`	vector of Gini within cluster distances
`W`	sum of within cluster distances
`count`	number of moves
`iterations`	number of iterations
`k`	number of clusters

cluster is a vector containing the group labels, 1 to k. print.kgroups prints some of the components of the kgroups object.

Expect that count is 0 if the algorithm converged to a local min (that is, 0 moves happened on the last iteration). If iterations equals iter.max and count is positive, then the algorithm did not converge to a local min.

Author(s)

Maria Rizzo and Songzi Li

References

Li, Songzi (2015). "K-groups: A Generalization of K-means by Energy Distance." Ph.D. thesis, Bowling Green State University.

Li, S. and Rizzo, M. L. (2017). "K-groups: A Generalization of K-means Clustering". ArXiv e-print 1711.04359. https://arxiv.org/abs/1711.04359

Szekely, G. J., and M. L. Rizzo. "Testing for equal distributions in high dimension." InterStat 5, no. 16.10 (2004).

Rizzo, M. L., and G. J. Szekely. "Disco analysis: A nonparametric extension of analysis of variance." The Annals of Applied Statistics (2010): 1034-1055.

Hartigan, J. A. and Wong, M. A. (1979). "Algorithm AS 136: A K-means clustering algorithm." Applied Statistics, 28, 100-108. doi: 10.2307/2346830.

Examples

  x <- as.matrix(iris[ ,1:4])
  set.seed(123)
  kg <- kgroups(x, k = 3, iter.max = 5, nstart = 2)
  kg
  fitted(kg)
  
  
    d <- dist(x)
    set.seed(123)
    kg <- kgroups(d, k = 3, iter.max = 5, nstart = 2)
    kg
    
    kg$cluster
  
    fitted(kg)
    fitted(kg, method = "groups")
    
x <- as.matrix(iris[ ,1:4])
  set.seed(123)
  kg <- kgroups(x, k = 3, iter.max = 5, nstart = 2)
  kg
  fitted(kg)
  
  
    d <- dist(x)
    set.seed(123)
    kg <- kgroups(d, k = 3, iter.max = 5, nstart = 2)
    kg
    
    kg$cluster
  
    fitted(kg)
    fitted(kg, method = "groups")

Energy Test of Mutual Independence

Description

The test statistic is the sum of d-1 bias-corrected squared dcor statistics where the number of variables is d. Implementation is by permuation test.

Usage

mutualIndep.test(x, R)
mutualIndep.test(x, R)

Arguments

`x`	data matrix or data frame
`R`	number of permutation replicates

Details

A population coefficient for mutual independence of d random variables, $d \geq 2$ , is

$\sum_{k=1}^{d-1} \mathcal R^2(X_k, [X_{k+1},\dots,X_d]).$

which is non-negative and equals zero iff mutual independence holds. For example, if d=4 the population coefficient is

$\mathcal R^2(X_1, [X_2,X_3,X_4]) + \mathcal R^2(X_2, [X_3,X_4]) + \mathcal R^2(X_3, X_4),$

A permutation test is implemented based on the corresponding sample coefficient. To test mutual independence of

$X_1,\dots,X_d$

the test statistic is the sum of the d-1 statistics (bias-corrected $dcor^2$ statistics):

$\sum_{k=1}^{d-1} \mathcal R_n^*(X_k, [X_{k+1},\dots,X_d])$

Value

mutualIndep.test returns an object of class power.htest.

Note

See Szekely and Rizzo (2014) for details on unbiased $dCov^2$ and bias-corrected $dCor^2$ (bcdcor) statistics.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2014), Partial Distance Correlation with Methods for Dissimilarities. Annals of Statistics, Vol. 42 No. 6, 2382-2412.

Examples

x <- matrix(rnorm(100), nrow=20, ncol=5)
mutualIndep.test(x, 199)
x <- matrix(rnorm(100), nrow=20, ncol=5)
mutualIndep.test(x, 199)

Independence Coefficient and Test

Description

Computes a type of multivariate nonparametric E-statistic and test of independence based on independence coefficient $\mathcal I_n$ . This coefficient pre-dates and is different from distance covariance or distance correlation.

Usage

    mvI.test(x, y, R)
    mvI(x, y)
mvI.test(x, y, R)
    mvI(x, y)

Arguments

`x`	matrix: first sample, observations in rows
`y`	matrix: second sample, observations in rows
`R`	number of replicates

Details

mvI computes the coefficient $\mathcal I_n$ and mvI.test performs a nonparametric test of independence. The test decision is obtained via permutation bootstrap, with R replicates. The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values.

Historically this is the first energy test of independence. The distance covariance test dcov.test, distance correlation dcor, and related methods are more recent (2007, 2009).

The distance covariance test dcov.test and distance correlation test dcor.test are much faster and have different properties than mvI.test. All are based on a population independence coefficient that characterizes independence and all of these tests are statistically consistent. However, dCor is scale invariant while $\mathcal I_n$ is not. In applications dcor.test or dcov.test are the recommended tests.

Computing formula from Bakirov, Rizzo, and Szekely (2006), equation (2):

Suppose the two samples are $X_1,\dots,X_n \in R^p$ and $Y_1,\dots,Y_n \in R^q$ . Define $Z_{kl} = (X_k, Y_l) \in R^{p+q}.$

The independence coefficient $\mathcal I_n$ is defined

$\mathcal I_n = \sqrt{\frac{2\bar z - z_d - z}{x + y - z}},$

where

$z_d= \frac{1}{n^2} \sum_{k,l=1}^n |Z_{kk}-Z_{ll}|_{p+q},$

$z= \frac{1}{n^4} \sum_{k,l=1}^n \sum_{i,j=1}^n |Z_{kl}-Z_{ij}|_{p+q},$

$\bar z= \frac{1}{n^3} \sum_{k=1}^n \sum_{i,j=1}^n |Z_{kk}-Z_{ij}|_{p+q},$

$x= \frac{1}{n^2} \sum_{k,l=1}^n |X_{k}-X_{l}|_p,$

$y= \frac{1}{n^2} \sum_{k,l=1}^n |Y_{k}-Y_{l}|_q.$

Some properties:

$0 \leq \mathcal I_n \leq 1$ (Theorem 1).
Large values of $n \mathcal I_n^2$ (or $\mathcal I_n$ ) support the alternative hypothesis that the sampled random variables are dependent.
$\mathcal I_n$ is invariant to shifts and orthogonal transformations of X and Y.
$\sqrt{n} \, \mathcal I_n$ determines a statistically consistent test of independence against all fixed dependent alternatives (Corollary 1).
The population independence coefficient $\mathcal I$ is a normalized distance between the joint characteristic function and the product of the marginal characteristic functions. $\mathcal I_n$ converges almost surely to $\mathcal I$ as $n \to \infty$ . X and Y are independent if and only if $\mathcal I(X, Y) = 0$ . See the 2006 reference below for more details.

Value

mvI returns the statistic. mvI.test returns a list with class htest containing

`method`	description of test
`statistic`	observed value of the test statistic $n\mathcal I_n^2$
`estimate`	$\mathcal I_n$
`replicates`	permutation replicates
`p.value`	p-value of the test
`data.name`	description of data

Note

On scale invariance: Distance correlation (dcor) has the property that if we change the scale of X from e.g., meters to kilometers, and the scale of Y from e.g. grams to ounces, the statistic and the test are not changed. $\mathcal I_n$ does not have this property; it is invariant only under a common rescaling of X and Y by the same constant. Thus, if the units of measurement change for either or both variables, dCor is invariant, but $\mathcal I_n$ and possibly the mvI.test decision changes.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Bakirov, N.K., Rizzo, M.L., and Szekely, G.J. (2006), A Multivariate Nonparametric Test of Independence, Journal of Multivariate Analysis 93/1, 58-80.

Szekely, G.J., Rizzo, M.L., and Bakirov, N.K. (2007), Measuring and Testing Dependence by Correlation of Distances, Annals of Statistics, Vol. 35 No. 6, pp. 2769-2794.

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1236-1265.

Examples

mvI(iris[1:25, 1], iris[1:25, 2])

mvI.test(iris[1:25, 1], iris[1:25, 2], R=99)

mvI(iris[1:25, 1], iris[1:25, 2])

mvI.test(iris[1:25, 1], iris[1:25, 2], R=99)

E-statistic (Energy) Test of Multivariate Normality

Description

Performs the E-statistic (energy) test of multivariate or univariate normality.

Usage

mvnorm.test(x, R)
mvnorm.etest(x, R)
mvnorm.e(x)
mvnorm.test(x, R)
mvnorm.etest(x, R)
mvnorm.e(x)

Arguments

`x`	data matrix of multivariate sample, or univariate data vector
`R`	number of bootstrap replicates

Details

If x is a matrix, each row is a multivariate observation. The data will be standardized to zero mean and identity covariance matrix using the sample mean vector and sample covariance matrix. If x is a vector, mvnorm.e returns the univariate statistic normal.e(x). If the data contains missing values or the sample covariance matrix is singular, mvnorm.e returns NA.

The $\mathcal{E}$ -test of multivariate normality was proposed and implemented by Szekely and Rizzo (2005). The test statistic for d-variate normality is given by

$\mathcal{E} = n (\frac{2}{n} \sum_{i=1}^n E\|y_i-Z\| - E\|Z-Z'\| - \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \|y_i-y_j\|),$

where $y_1,\ldots,y_n$ is the standardized sample, $Z, Z'$ are iid standard d-variate normal, and $\| \cdot \|$ denotes Euclidean norm.

The $\mathcal{E}$ -test of multivariate (univariate) normality is implemented by parametric bootstrap with R replicates.

Value

The value of the $\mathcal{E}$ -statistic for multivariate normality is returned by mvnorm.e.

mvnorm.test returns a list with class htest containing

`method`	description of test
`statistic`	observed value of the test statistic
`p.value`	approximate p-value of the test
`data.name`	description of data

mvnorm.etest is replaced by mvnorm.test.

Note

If the data is univariate, the test statistic is formally the same as the multivariate case, but a more efficient computational formula is applied in normal.e.

normal.test also provides an optional method for the test based on the asymptotic sampling distribution of the test statistic.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G. J. and Rizzo, M. L. (2005) A New Test for Multivariate Normality, Journal of Multivariate Analysis, 93/1, 58-80, doi:10.1016/j.jmva.2003.12.002.

Mori, T. F., Szekely, G. J. and Rizzo, M. L. "On energy tests of normality." Journal of Statistical Planning and Inference 213 (2021): 1-15.

Rizzo, M. L. (2002). A New Rotation Invariant Goodness-of-Fit Test, Ph.D. dissertation, Bowling Green State University.

Szekely, G. J. (1989) Potential and Kinetic Energy in Statistics, Lecture Notes, Budapest Institute of Technology (Technical University).

Examples

 ## compute normality test statistic for iris Setosa data
 data(iris)
 mvnorm.e(iris[1:50, 1:4])

 ## test if the iris Setosa data has multivariate normal distribution
 mvnorm.test(iris[1:50,1:4], R = 199)
## compute normality test statistic for iris Setosa data
 data(iris)
 mvnorm.e(iris[1:50, 1:4])

 ## test if the iris Setosa data has multivariate normal distribution
 mvnorm.test(iris[1:50,1:4], R = 199)

Energy Test of Univariate Normality

Description

Performs the energy test of univariate normality for the composite hypothesis Case 4, estimated parameters.

Usage

normal.test(x, method=c("mc","limit"), R)
normal.e(x)
normal.test(x, method=c("mc","limit"), R)
normal.e(x)

Arguments

`x`	univariate data vector
`method`	method for p-value
`R`	number of replications if Monte Carlo method

Details

If method="mc" this test function applies the parametric bootstrap method implemented in mvnorm.test.

If method="limit", the p-value of the test is computed from the asymptotic distribution of the test statistic under the null hypothesis. The asymptotic distribution is a quadratic form of centered Gaussian random variables, which has the form

$\sum_{k=1}^\infty \lambda_k Z_k^2,$

where $\lambda_k$ are positive constants (eigenvalues) and $Z_k$ are iid standard normal variables. Eigenvalues are pre-computed and stored internally. A p-value is computed using Imhof's method as implemented in the CompQuadForm package.

Note that the "limit" method is intended for moderately large samples because it applies the asymptotic distribution.

The energy test of normality was proposed and implemented by Szekely and Rizzo (2005). See mvnorm.test for more details.

Value

normal.e returns the energy goodness-of-fit statistic for a univariate sample.

normal.test returns a list with class htest containing

`statistic`	observed value of the test statistic
`p.value`	p-value of the test
`estimate`	sample estimates: mean, sd
`data.name`	description of data

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G. J. and Rizzo, M. L. (2005) A New Test for Multivariate Normality, Journal of Multivariate Analysis, 93/1, 58-80, doi:10.1016/j.jmva.2003.12.002.

Mori, T. F., Szekely, G. J. and Rizzo, M. L. "On energy tests of normality." Journal of Statistical Planning and Inference 213 (2021): 1-15.

Rizzo, M. L. (2002). A New Rotation Invariant Goodness-of-Fit Test, Ph.D. dissertation, Bowling Green State University.

J. P. Imhof (1961). Computing the Distribution of Quadratic Forms in Normal Variables, Biometrika, Volume 48, Issue 3/4, 419-426.

Examples

  x <- iris[1:50, 1]
  normal.e(x)
  normal.test(x, R=199)
  normal.test(x, method="limit")
x <- iris[1:50, 1]
  normal.e(x)
  normal.test(x, R=199)
  normal.test(x, method="limit")

Partial distance correlation and covariance

Description

Partial distance correlation pdcor, pdcov, and tests.

Usage

  pdcov.test(x, y, z, R)
  pdcor.test(x, y, z, R)
  pdcor(x, y, z)
  pdcov(x, y, z)
pdcov.test(x, y, z, R)
  pdcor.test(x, y, z, R)
  pdcor(x, y, z)
  pdcov(x, y, z)

Arguments

`x`	data or dist object of first sample
`y`	data or dist object of second sample
`z`	data or dist object of third sample
`R`	replicates for permutation test

Details

pdcor(x, y, z) and pdcov(x, y, z) compute the partial distance correlation and partial distance covariance, respectively, of x and y removing z.

A test for zero partial distance correlation (or zero partial distance covariance) is implemented in pdcor.test, and pdcov.test.

Argument types supported are numeric data matrix, data.frame, tibble, numeric vector, class "dist" object, or factor. For unordered factors a 0-1 distance matrix is computed.

Value

Each test returns an object of class htest.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2014), Partial Distance Correlation with Methods for Dissimilarities. Annals of Statistics, Vol. 42 No. 6, 2382-2412.

Examples

  n = 30
  R <- 199

  ## mutually independent standard normal vectors
  x <- rnorm(n)
  y <- rnorm(n)
  z <- rnorm(n)

  pdcor(x, y, z)
  pdcov(x, y, z)
  set.seed(1)
  pdcov.test(x, y, z, R=R)
  set.seed(1)
  pdcor.test(x, y, z, R=R)


  if (require(MASS)) {
    p = 4
    mu <- rep(0, p)
    Sigma <- diag(p)
  
    ## linear dependence
    y <- mvrnorm(n, mu, Sigma) + x
    print(pdcov.test(x, y, z, R=R))
  
    ## non-linear dependence
    y <- mvrnorm(n, mu, Sigma) * x
    print(pdcov.test(x, y, z, R=R))
    }
  
n = 30
  R <- 199

  ## mutually independent standard normal vectors
  x <- rnorm(n)
  y <- rnorm(n)
  z <- rnorm(n)

  pdcor(x, y, z)
  pdcov(x, y, z)
  set.seed(1)
  pdcov.test(x, y, z, R=R)
  set.seed(1)
  pdcor.test(x, y, z, R=R)


  if (require(MASS)) {
    p = 4
    mu <- rep(0, p)
    Sigma <- diag(p)
  
    ## linear dependence
    y <- mvrnorm(n, mu, Sigma) + x
    print(pdcov.test(x, y, z, R=R))
  
    ## non-linear dependence
    y <- mvrnorm(n, mu, Sigma) * x
    print(pdcov.test(x, y, z, R=R))
    }

Goodness-of-Fit Tests for Poisson Distribution

Description

Performs the mean distance goodness-of-fit test and the energy goodness-of-fit test of Poisson distribution with unknown parameter.

Usage

poisson.e(x)
poisson.m(x)
poisson.etest(x, R)
poisson.mtest(x, R)
poisson.tests(x, R, test="all")
poisson.e(x)
poisson.m(x)
poisson.etest(x, R)
poisson.mtest(x, R)
poisson.tests(x, R, test="all")

Arguments

`x`	vector of nonnegative integers, the sample data
`R`	number of bootstrap replicates
`test`	name of test(s)

Details

Two distance-based tests of Poissonity are applied in poisson.tests, "M" and "E". The default is to do all tests and return results in a data frame. Valid choices for test are "M", "E", or "all" with default "all".

If "all" tests, all tests are performed by a single parametric bootstrap computing all test statistics on each sample.

The "M" choice is two tests, one based on a Cramer-von Mises distance and the other an Anderson-Darling distance. The "E" choice is the energy goodness-of-fit test.

R must be a positive integer for a test. If R is missing or 0, a warning is printed but test statistics are computed (without testing).

The mean distance test of Poissonity (M-test) is based on the result that the sequence of expected values E|X-j|, j=0,1,2,... characterizes the distribution of the random variable X. As an application of this characterization one can get an estimator $\hat F(j)$ of the CDF. The test statistic (see poisson.m) is a Cramer-von Mises type of distance, with M-estimates replacing the usual EDF estimates of the CDF:

$M_n = n\sum_{j=0}^\infty (\hat F(j) - F(j\;; \hat \lambda))^2 f(j\;; \hat \lambda).$

In poisson.tests, an Anderson-Darling type of weight is also applied when test="M" or test="all".

The tests are implemented by parametric bootstrap with R replicates.

An energy goodness-of-fit test (E) is based on the test statistic

$Q_n = n (\frac{2}{n} \sum_{i=1}^n E|x_i - X| - E|X-X'| - \frac{1}{n^2} \sum_{i,j=1}^n |x_i - x_j|,$

where X and X' are iid with the hypothesized null distribution. For a test of H: X ~ Poisson( $\lambda$ ), we can express E|X-X'| in terms of Bessel functions, and E|x_i - X| in terms of the CDF of Poisson( $\lambda$ ).

If test=="all" or not specified, all tests are run with a single parametric bootstrap. poisson.mtest implements only the Poisson M-test with Cramer-von Mises type distance. poisson.etest implements only the Poisson energy test.

Value

The functions poisson.m and poisson.e return the test statistics. The function poisson.mtest or poisson.etest return an htest object containing

`method`	Description of test
`statistic`	observed value of the test statistic
`p.value`	approximate p-value of the test
`data.name`	replicates R
`estimate`	sample mean

poisson.tests returns "M-CvM test", "M-AD test" and "Energy test" results in a data frame with columns

`estimate`	sample mean
`statistic`	observed value of the test statistic
`p.value`	approximate p-value of the test
`method`	Description of test

which can be coerced to a tibble.

Note

The running time of the M test is much faster than the E-test.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G. J. and Rizzo, M. L. (2004) Mean Distance Test of Poisson Distribution, Statistics and Probability Letters, 67/3, 241-247. doi:10.1016/j.spl.2004.01.005.

Szekely, G. J. and Rizzo, M. L. (2005) A New Test for Multivariate Normality, Journal of Multivariate Analysis, 93/1, 58-80, doi:10.1016/j.jmva.2003.12.002.

Examples

 x <- rpois(50, 2)
 poisson.m(x)
 poisson.e(x)
 
 poisson.etest(x, R=199)
 poisson.mtest(x, R=199)
 poisson.tests(x, R=199)
 
x <- rpois(50, 2)
 poisson.m(x)
 poisson.e(x)
 
 poisson.etest(x, R=199)
 poisson.mtest(x, R=199)
 poisson.tests(x, R=199)

Sort, order and rank a vector

Description

A utility that returns a list with the components equivalent to sort(x), order(x), rank(x, ties.method = "first").

Usage

sortrank(x)
sortrank(x)

Arguments

`x`	vector compatible with sort(x)

Details

This utility exists to save a little time on large vectors when two or all three of the sort(), order(), rank() results are required. In case of ties, the ranks component matches rank(x, ties.method = "first").

Value

A list with components

`x`	the sorted input vector x
`ix`	the permutation = order(x) which rearranges x into ascending order
`r`	the ranks of x

Note

This function was benchmarked faster than the combined calls to sort and rank.

Author(s)

Maria L. Rizzo [email protected]

References

See sort.

Examples

sortrank(rnorm(5))
sortrank(rnorm(5))

Inner product in the Hilbert space of U-centered distance matrices

Description

Stand-alone function to compute the inner product in the Hilbert space of U-centered distance matrices, as in the definition of partial distance covariance.

Usage

U_product(U, V)
U_product(U, V)

Arguments

`U`	U-centered distance matrix
`V`	U-centered distance matrix

Details

Note that pdcor, etc. functions include the centering and projection operations, so that these stand alone versions are not needed except in case one wants to check the internal computations.

Exported from U_product.cpp.

Value

U_product returns the inner product, a scalar.

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2014), Partial Distance Correlation with Methods for Dissimilarities, Annals of Statistics, Vol. 42, No. 6, pp. 2382-2412.

Examples

 x <- iris[1:10, 1:4]
 y <- iris[11:20, 1:4]
 M1 <- as.matrix(dist(x))
 M2 <- as.matrix(dist(y))
 U <- U_center(M1)
 V <- U_center(M2)

 U_product(U, V)
 dcovU_stats(M1, M2)
 x <- iris[1:10, 1:4]
 y <- iris[11:20, 1:4]
 M1 <- as.matrix(dist(x))
 M2 <- as.matrix(dist(y))
 U <- U_center(M1)
 V <- U_center(M2)

 U_product(U, V)
 dcovU_stats(M1, M2)

Unbiased dcov and bias-corrected dcor statistics

Description

These functions compute unbiased estimators of squared distance covariance and a bias-corrected estimator of (squared) distance correlation.

Usage

bcdcor(x, y)
dcovU(x, y)
bcdcor(x, y)
dcovU(x, y)

Arguments

`x`	data or dist object of first sample
`y`	data or dist object of second sample

Details

The unbiased (squared) dcov is inner product definition of dCov, in the Hilbert space of U-centered distance matrices.

The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values.

Value

dcovU returns the unbiased estimator of squared dcov. bcdcor returns a bias-corrected estimator of squared dcor.

Note

Author(s)

Maria L. Rizzo [email protected] and Gabor J. Szekely

References

Szekely, G.J. and Rizzo, M.L. (2014), Partial Distance Correlation with Methods for Dissimilarities. Annals of Statistics, Vol. 42 No. 6, 2382-2412.

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1236-1265.
doi:10.1214/09-AOAS312

Examples

 x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 dcovU(x, y)
 bcdcor(x, y)
x <- iris[1:50, 1:4]
 y <- iris[51:100, 1:4]
 dcovU(x, y)
 bcdcor(x, y)

Package 'energy'

Help Index

E-statistics: Multivariate Inference via the Energy of Data

Description

Author(s)

References

Double centering and U-centering

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

Examples

Distance Correlation t-test for High Dimensions

Description

Usage

Arguments

Details

Author(s)

Distance Correlation t-Test

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Distance Covariance Test and Distance Correlation test

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Fast dCor and dCov for bivariate data only

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Unbiased distance covariance statistics

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

Examples

distance components (DISCO)

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Distance Correlation and Covariance Statistics

Description

Usage

Arguments