Type:	Package
Title:	(Robust) Canonical Correlation Analysis via Projection Pursuit
Version:	0.3.4
Date:	2024-09-04
Depends:	R (≥ 3.2.0), parallel, pcaPP (≥ 1.8-1), robustbase
Imports:	Rcpp (≥ 0.11.0)
LinkingTo:	Rcpp (≥ 0.11.0), RcppArmadillo (≥ 0.4.100.0)
Suggests:	knitr, mvtnorm
VignetteBuilder:	knitr
Description:	Canonical correlation analysis and maximum correlation via projection pursuit, as well as fast implementations of correlation estimators, with a focus on robust and nonparametric methods.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
URL:	https://github.com/aalfons/ccaPP
BugReports:	https://github.com/aalfons/ccaPP/issues
LazyLoad:	yes
Author:	Andreas Alfons [aut, cre], David Simcha [ctb] (O(n log(n)) implementation of Kendall correlation)
Maintainer:	Andreas Alfons <alfons@ese.eur.nl>
Encoding:	UTF-8
RoxygenNote:	7.3.2
NeedsCompilation:	yes
Packaged:	2024-09-04 18:34:57 UTC; alfons
Repository:	CRAN
Date/Publication:	2024-09-04 22:20:10 UTC

(Robust) Canonical Correlation Analysis via Projection Pursuit

Description

Canonical correlation analysis and maximum correlation via projection pursuit, as well as fast implementations of correlation estimators, with a focus on robust and nonparametric methods.

Details

The DESCRIPTION file:

Index of help topics:

ccaGrid                 (Robust) CCA via alternating series of grid
                        searches
ccaPP-package           (Robust) Canonical Correlation Analysis via
                        Projection Pursuit
ccaProj                 (Robust) CCA via projections through the data
                        points
corFunctions            Fast implementations of (robust) correlation
                        estimators
diabetes                Diabetes data
fastMAD                 Fast implementation of the median absolute
                        deviation
fastMedian              Fast implementation of the median
maxCorGrid              (Robust) maximum correlation via alternating
                        series of grid searches
maxCorProj              (Robust) maximum correlation via projections
                        through the data points
permTest                (Robust) permutation test for no association

Author(s)

Andreas Alfons [aut, cre] (<https://orcid.org/0000-0002-2513-3788>), David Simcha [ctb] (O(n log(n)) implementation of Kendall correlation)

Maintainer: Andreas Alfons <alfons@ese.eur.nl>

References

A. Alfons, C. Croux and P. Filzmoser (2016) Robust maximum association between data sets: The R Package ccaPP. Austrian Journal of Statistics, 45(1), 71–79.

A. Alfons, C. Croux and P. Filzmoser (2016) Robust maximum association estimators. Journal of the American Statistical Association, 112(517), 435–445.

(Robust) CCA via alternating series of grid searches

Description

Perform canoncial correlation analysis via projection pursuit based on alternating series of grid searches in two-dimensional subspaces of each data set, with a focus on robust and nonparametric methods.

Usage

ccaGrid(
  x,
  y,
  k = 1,
  method = c("spearman", "kendall", "quadrant", "M", "pearson"),
  control = list(...),
  nIterations = 10,
  nAlternate = 10,
  nGrid = 25,
  select = NULL,
  tol = 1e-06,
  standardize = TRUE,
  fallback = FALSE,
  seed = NULL,
  ...
)

CCAgrid(
  x,
  y,
  k = 1,
  method = c("spearman", "kendall", "quadrant", "M", "pearson"),
  maxiter = 10,
  maxalter = 10,
  splitcircle = 25,
  select = NULL,
  zero.tol = 1e-06,
  standardize = TRUE,
  fallback = FALSE,
  seed = NULL,
  ...
)

Arguments

Details

The algorithm is based on alternating series of grid searches in two-dimensional subspaces of each data set. In each grid search, nGrid grid points on the unit circle in the corresponding plane are obtained, and the directions from the center to each of the grid points are examined. In the first iteration, equispaced grid points in the interval [-\pi/2, \pi/2) are used. In each subsequent iteration, the angles are halved such that the interval [-\pi/4, \pi/4) is used in the second iteration and so on. If only one data set is multivariate, the algorithm simplifies to iterative grid searches in two-dimensional subspaces of the corresponding data set.

In the basic algorithm, the order of the variables in a series of grid searches for each of the data sets is determined by the average absolute correlations with the variables of the respective other data set. Since this requires to compute the full (p \times q) matrix of absolute correlations, where p denotes the number of variables of x and q the number of variables of y, a faster modification is available as well. In this modification, the average absolute correlations are computed over only a subset of the variables of the respective other data set. It is thereby possible to use randomly selected subsets of variables, or to specify the subsets of variables directly.

Note that also the data sets are ordered according to the maximum average absolute correlation with the respective other data set to ensure symmetry of the algorithm.

For higher order canonical correlations, the data are first transformed into suitable subspaces. Then the alternate grid algorithm is applied to the reduced data and the results are back-transformed to the original space.

Value

An object of class "cca" with the following components:

Note

CCAgrid is a simple wrapper function for ccaGrid for more compatibility with package pcaPP concerning function and argument names.

Author(s)

Andreas Alfons

See Also

ccaProj, maxCorGrid, corFunctions

Examples

data("diabetes")
x <- diabetes$x
y <- diabetes$y

## Spearman correlation
ccaGrid(x, y, method = "spearman")

## Pearson correlation
ccaGrid(x, y, method = "pearson")

(Robust) CCA via projections through the data points

Description

Perform canoncial correlation analysis via projection pursuit based on projections through the data points, with a focus on robust and nonparametric methods.

Usage

ccaProj(
  x,
  y,
  k = 1,
  method = c("spearman", "kendall", "quadrant", "M", "pearson"),
  control = list(...),
  standardize = TRUE,
  useL1Median = TRUE,
  fallback = FALSE,
  ...
)

CCAproj(
  x,
  y,
  k = 1,
  method = c("spearman", "kendall", "quadrant", "M", "pearson"),
  standardize = TRUE,
  useL1Median = TRUE,
  fallback = FALSE,
  ...
)

Arguments

Details

First the candidate projection directions are defined for each data set from the respective center through each data point. Then the algorithm scans all n^2 possible combinations for the maximum correlation, where n is the number of observations.

For higher order canonical correlations, the data are first transformed into suitable subspaces. Then the alternate grid algorithm is applied to the reduced data and the results are back-transformed to the original space.

Value

An object of class "cca" with the following components:

Note

CCAproj is a simple wrapper function for ccaProj for more compatibility with package pcaPP concerning function names.

Author(s)

Andreas Alfons

See Also

ccaGrid, maxCorProj, corFunctions

Examples

data("diabetes")
x <- diabetes$x
y <- diabetes$y

## Spearman correlation
ccaProj(x, y, method = "spearman")

## Pearson correlation
ccaProj(x, y, method = "pearson")

Fast implementations of (robust) correlation estimators

Description

Estimate the correlation of two vectors via fast C++ implementations, with a focus on robust and nonparametric methods.

Usage

corPearson(x, y)

corSpearman(x, y, consistent = FALSE)

corKendall(x, y, consistent = FALSE)

corQuadrant(x, y, consistent = FALSE)

corM(
  x,
  y,
  prob = 0.9,
  initial = c("quadrant", "spearman", "kendall", "pearson"),
  tol = 1e-06
)

Arguments

Details

corPearson estimates the classical Pearson correlation. corSpearman, corKendall and corQuadrant estimate the Spearman, Kendall and quadrant correlation, respectively, which are nonparametric correlation measures that are somewhat more robust. corM estimates the correlation based on a bivariate M-estimator of location and scatter with a Huber loss function, which is sufficiently robust in the bivariate case, but loses robustness with increasing dimension.

The nonparametric correlation measures do not estimate the same population quantities as the Pearson correlation, the latter of which is consistent at the bivariate normal model. Let \rho denote the population correlation at the normal model. Then the Spearman correlation estimates (6/\pi) \arcsin(\rho/2), while the Kendall and quadrant correlation estimate (2/\pi) \arcsin(\rho). Consistent estimates are thus easily obtained by taking the corresponding inverse expressions.

The Huber M-estimator, on the other hand, is consistent at the bivariate normal model.

Value

The respective correlation estimate.

Note

The Kendall correlation uses a naive n^2 implementation if n < 30 and a fast O(n \log(n)) implementation for larger values, where n denotes the number of observations.

Functionality for removing observations with missing values is currently not implemented.

Author(s)

Andreas Alfons, O(n \log(n)) implementation of the Kendall correlation by David Simcha

See Also

ccaGrid, ccaProj, cor

Examples

## generate data
library("mvtnorm")
set.seed(1234)  # for reproducibility
sigma <- matrix(c(1, 0.6, 0.6, 1), 2, 2)
xy <- rmvnorm(100, sigma=sigma)
x <- xy[, 1]
y <- xy[, 2]

## compute correlations

# Pearson correlation
corPearson(x, y)

# Spearman correlation
corSpearman(x, y)
corSpearman(x, y, consistent=TRUE)

# Kendall correlation
corKendall(x, y)
corKendall(x, y, consistent=TRUE)

# quadrant correlation
corQuadrant(x, y)
corQuadrant(x, y, consistent=TRUE)

# Huber M-estimator
corM(x, y)

Diabetes data

Description

Subset of the diabetes data from Andrews & Herzberg (1985).

Usage

data(diabetes)

Format

A list with components x and y. Both components are matrices with observations on different variables for the same n = 76 persons.

Component x is a matrix containing the following p = 2 variables.

RelativeWeight

relative weight.

PlasmaGlucose

fasting plasma glucose.

Component y is a matrix containing the following q = 3 variables.

GlucoseIntolerance

glucose intolerance.

InsulinResponse

insulin response to oral glucose.

InsulinResistance

insulin resistance.

Source

Andrews, D.F. and Herzberg, A.M. (1985) Data. Springer-Verlag. Page 215.

Examples

data("diabetes")
x <- diabetes$x
y <- diabetes$y

## Spearman correlation
maxCorGrid(x, y, method = "spearman")
maxCorGrid(x, y, method = "spearman", consistent = TRUE)

## Pearson correlation
maxCorGrid(x, y, method = "pearson")

Fast implementation of the median absolute deviation

Description

Compute the median absolute deviation with a fast C++ implementation. By default, a multiplication factor is applied for consistency at the normal model.

Usage

fastMAD(x, constant = 1.4826)

Arguments

Value

A list with the following components:

Note

Functionality for removing observations with missing values is currently not implemented.

Author(s)

Andreas Alfons

See Also

fastMedian, mad

Examples

set.seed(1234)  # for reproducibility
x <- rnorm(100)
fastMAD(x)

Fast implementation of the median

Description

Compute the sample median with a fast C++ implementation.

Usage

fastMedian(x)

Arguments

Value

The sample median.

Note

Functionality for removing observations with missing values is currently not implemented.

Author(s)

Andreas Alfons

See Also

fastMAD, median

Examples

set.seed(1234)  # for reproducibility
x <- rnorm(100)
fastMedian(x)

(Robust) maximum correlation via alternating series of grid searches

Description

Compute the maximum correlation between two data sets via projection pursuit based on alternating series of grid searches in two-dimensional subspaces of each data set, with a focus on robust and nonparametric methods.

Usage

maxCorGrid(
  x,
  y,
  method = c("spearman", "kendall", "quadrant", "M", "pearson"),
  control = list(...),
  nIterations = 10,
  nAlternate = 10,
  nGrid = 25,
  select = NULL,
  tol = 1e-06,
  standardize = TRUE,
  fallback = FALSE,
  seed = NULL,
  ...
)

Arguments

Details

The algorithm is based on alternating series of grid searches in two-dimensional subspaces of each data set. In each grid search, nGrid grid points on the unit circle in the corresponding plane are obtained, and the directions from the center to each of the grid points are examined. In the first iteration, equispaced grid points in the interval [-\pi/2, \pi/2) are used. In each subsequent iteration, the angles are halved such that the interval [-\pi/4, \pi/4) is used in the second iteration and so on. If only one data set is multivariate, the algorithm simplifies to iterative grid searches in two-dimensional subspaces of the corresponding data set.

In the basic algorithm, the order of the variables in a series of grid searches for each of the data sets is determined by the average absolute correlations with the variables of the respective other data set. Since this requires to compute the full (p \times q) matrix of absolute correlations, where p denotes the number of variables of x and q the number of variables of y, a faster modification is available as well. In this modification, the average absolute correlations are computed over only a subset of the variables of the respective other data set. It is thereby possible to use randomly selected subsets of variables, or to specify the subsets of variables directly.

Note that also the data sets are ordered according to the maximum average absolute correlation with the respective other data set to ensure symmetry of the algorithm.

Value

An object of class "maxCor" with the following components:

Author(s)

Andreas Alfons

References

A. Alfons, C. Croux and P. Filzmoser (2016) Robust maximum association between data sets: The R Package ccaPP. Austrian Journal of Statistics, 45(1), 71–79.

A. Alfons, C. Croux and P. Filzmoser (2016) Robust maximum association estimators. Journal of the American Statistical Association, 112(517), 435–445.

See Also

maxCorProj, ccaGrid, corFunctions

Examples

data("diabetes")
x <- diabetes$x
y <- diabetes$y

## Spearman correlation
maxCorGrid(x, y, method = "spearman")
maxCorGrid(x, y, method = "spearman", consistent = TRUE)

## Pearson correlation
maxCorGrid(x, y, method = "pearson")

(Robust) maximum correlation via projections through the data points

Description

Compute the maximum correlation between two data sets via projection pursuit based on projections through the data points, with a focus on robust and nonparametric methods.

Usage

maxCorProj(
  x,
  y,
  method = c("spearman", "kendall", "quadrant", "M", "pearson"),
  control = list(...),
  standardize = TRUE,
  useL1Median = TRUE,
  fallback = FALSE,
  ...
)

Arguments

Details

First the candidate projection directions are defined for each data set from the respective center through each data point. Then the algorithm scans all n^2 possible combinations for the maximum correlation, where n is the number of observations.

Value

An object of class "maxCor" with the following components:

Author(s)

Andreas Alfons

See Also

maxCorGrid, ccaProj, corFunctions,

Examples

data("diabetes")
x <- diabetes$x
y <- diabetes$y

## Spearman correlation
maxCorProj(x, y, method = "spearman")
maxCorProj(x, y, method = "spearman", consistent = TRUE)

## Pearson correlation
maxCorProj(x, y, method = "pearson")

(Robust) permutation test for no association

Description

Test whether or not there is association betwenn two data sets, with a focus on robust and nonparametric correlation measures.

Usage

permTest(
  x,
  y,
  R = 1000,
  fun = maxCorGrid,
  permutations = NULL,
  nCores = 1,
  cl = NULL,
  seed = NULL,
  ...
)

Arguments

Details

The test generates R data sets by randomly permuting the observations of x, while keeping the observations of y fixed. In each replication, a function to compute a maximum correlation measure is applied to the permuted data sets. The p-value of the test is then given by the percentage of replicates of the maximum correlation measure that are larger than the maximum correlation measure computed from the original data.

Value

An object of class "permTest" with the following components:

Author(s)

Andreas Alfons

References

A. Alfons, C. Croux and P. Filzmoser (2016) Robust maximum association between data sets: The R Package ccaPP. Austrian Journal of Statistics, 45(1), 71–79.

See Also

maxCorGrid, maxCorProj

Examples

data("diabetes")
x <- diabetes$x
y <- diabetes$y

## Spearman correlation
permTest(x, y, R = 100, method = "spearman")
permTest(x, y, R = 100, method = "spearman", consistent = TRUE)

## Pearson correlation
permTest(x, y, R = 100, method = "pearson")