Type:	Package
Title:	Sparse Canonical Correlation Analysis for High-Dimensional Mixed Data
Version:	1.6.2
Date:	2022-09-09
Maintainer:	Irina Gaynanova <irinag@stat.tamu.edu>
Description:	Semi-parametric approach for sparse canonical correlation analysis which can handle mixed data types: continuous, binary and truncated continuous. Bridge functions are provided to connect Kendall's tau to latent correlation under the Gaussian copula model. The methods are described in Yoon, Carroll and Gaynanova (2020) <doi:10.1093/biomet/asaa007> and Yoon, Mueller and Gaynanova (2021) <doi:10.1080/10618600.2021.1882468>.
License:	GPL-3
Encoding:	UTF-8
Depends:	R (≥ 3.0.1), stats, MASS
Imports:	Rcpp, pcaPP, Matrix, fMultivar, mnormt, irlba, latentcor (≥ 2.0.1)
NeedsCompilation:	yes
RoxygenNote:	7.2.1
LinkingTo:	Rcpp, RcppArmadillo
Packaged:	2022-09-09 21:19:49 UTC; irinag
Author:	Grace Yoon [aut], Mingze Huang [ctb], Irina Gaynanova [aut, cre]
Repository:	CRAN
Date/Publication:	2022-09-09 21:50:01 UTC

Construct a correlation matrix

Description

Functions to create autocorrelation matrix (p by p) with parameter rho and block correlation matrix (p by p) using group index (of length p) and (possibly) different parameter rho for each group.

Usage

autocor(p, rho)

blockcor(blockind, rho)

Arguments

Examples

# For p = 8,
# auto correlation matrix
autocor(8, 0.8)
# block correlation matrix: two blocks with the same correlation within each block
blockcor(c(rep(1,3), rep(2,5)), 0.8)
# block correlation matrix: two blocks with different correlation within each block
blockcor(c(rep(1,3), rep(2,5)), c(0.8, 0.3))

Mixed type simulation data generator for sparse CCA

Description

GenerateData is used to generate two sets of data of mixed types for sparse CCA under the Gaussian copula model.

Usage

GenerateData(
  n,
  trueidx1,
  trueidx2,
  Sigma1,
  Sigma2,
  maxcancor,
  copula1 = "no",
  copula2 = "no",
  type1 = "continuous",
  type2 = "continuous",
  muZ = NULL,
  c1 = NULL,
  c2 = NULL
)

Arguments

Value

GenerateData returns a list containing

• Z1: latent numeric data matrix (n by p1).

• Z2: latent numeric data matrix (n by p2).

• X1: observed numeric data matrix (n by p1).

• X2: observed numeric data matrix (n by p2).

• true_w1: normalized true canonical direction of length p1 for X1.

• true_w2: normalized true canonical direction of length p2 for X2.

• type: a vector containing types of two datasets.

• maxcancor: true canonical correlation between Z1 and Z2.

• c1: constant threshold for X1 for "trunc" and "binary" data type.

• c2: constant threshold for X2 for "trunc" and "binary" data type.

• Sigma: true latent correlation matrix of Z1 and Z2 ((p1+p2) by (p1+p2)).

Examples

### Simple example

# Data setting
n <- 100; p1 <- 15; p2 <- 10 # sample size and dimensions for two datasets.
maxcancor <- 0.9 # true canonical correlation

# Correlation structure within each data set
set.seed(0)
perm1 <- sample(1:p1, size = p1);
Sigma1 <- autocor(p1, 0.7)[perm1, perm1]
blockind <- sample(1:3, size = p2, replace = TRUE);
Sigma2 <- blockcor(blockind, 0.7)
mu <- rbinom(p1+p2, 1, 0.5)

# true variable indices for each dataset
trueidx1 <- c(rep(1, 3), rep(0, p1-3))
trueidx2 <- c(rep(1, 2), rep(0, p2-2))

# Data generation
simdata <- GenerateData(n=n, trueidx1 = trueidx1, trueidx2 = trueidx2, maxcancor = maxcancor,
                        Sigma1 = Sigma1, Sigma2 = Sigma2,
                        copula1 = "exp", copula2 = "cube",
                        muZ = mu,
                        type1 = "trunc", type2 = "trunc",
                        c1 = rep(1, p1), c2 =  rep(0, p2)
)
X1 <- simdata$X1
X2 <- simdata$X2

# Check the range of truncation levels of variables
range(colMeans(X1 == 0))
range(colMeans(X2 == 0))

Kendall's tau correlation

Description

Calculate Kendall's tau correlation.

\hat{\tau}_{jk} = \frac{2}{n(n-1)}\sum_{1\le i<i'\le n} sign(X_{ji}-X_{ji'}) sign(X_{ki}-X_{ki'})

The function KendallTau calculates Kendall's tau correlation between two variables, returning a single correlation value. The function Kendall_matrix returns a correlation matrix.

Usage

KendallTau(x, y)

Kendall_matrix(X, Y = NULL)

Arguments

Value

KendallTau(x, y) returns one Kendall's tau correlation value between two vectors, x and y.

Kendall_matrix(X) returns a p1 by p1 matrix of Kendall's tau correlation coefficients. Kendall_matrix(X, Y) returns a p1 by p2 matrix of Kendall's tau correlation coefficients.

Examples

n <- 100 # sample size
r <- 0.8 # true correlation

### vector input
# Data generation (X1: truncated continuous, X2: continuous)
Z <- mvrnorm(n, mu = c(0, 0), Sigma = matrix(c(1, r, r, 1), nrow = 2))
X1 <- Z[,1]
X1[Z[,1] < 1] <- 0
X2 <- Z[,2]

KendallTau(X1, X2)
Kendall_matrix(X1, X2)

### matrix data input
p1 <- 3; p2 <- 4 # dimension of X1 and X2
JSigma <- matrix(r, nrow = p1+p2, ncol = p1+p2); diag(JSigma) <- 1
Z <- mvrnorm(n, mu = rep(0, p1+p2), Sigma = JSigma)
X1 <- Z[,1:p1]
X1[Z[,1:p1] < 0] <- 0
X2 <- Z[,(p1+1):(p1+p2)]

Kendall_matrix(X1, X2)

Estimate latent correlation matrix

Description

Estimation of latent correlation matrix from observed data of (possibly) mixed types (continuous/binary/truncated continuous) based on the latent Gaussian copula model.

Usage

estimateR(
  X,
  type = "trunc",
  method = "original",
  use.nearPD = TRUE,
  nu = 0.01,
  tol = 0.001,
  verbose = FALSE
)

estimateR_mixed(
  X1,
  X2,
  type1 = "trunc",
  type2 = "continuous",
  method = "original",
  use.nearPD = TRUE,
  nu = 0.01,
  tol = 0.001,
  verbose = FALSE
)

Arguments

Value

estimateR returns

• type: Type of the data matrix X

• R: Estimated p by p latent correlation matrix of X

estimateR_mixed returns

• type1: Type of the data matrix X1

• type2: Type of the data matrix X2

• R: Estimated latent correlation matrix of whole X = (X1, X2) (p1+p2 by p1+p2)

• R1: Estimated latent correlation matrix of X1 (p1 by p1)

• R2: Estimated latent correlation matrix of X2 (p2 by p2)

• R12: Estimated latent correlation matrix between X1 and X2 (p1 by p2)

References

Fan J., Liu H., Ning Y. and Zou H. (2017) "High dimensional semiparametric latent graphicalmodel for mixed data" <doi:10.1111/rssb.12168>.

Yoon G., Carroll R.J. and Gaynanova I. (2020) "Sparse semiparametric canonical correlation analysis for data of mixed types" <doi:10.1093/biomet/asaa007>.

Yoon G., Mueller C.L., Gaynanova I. (2020) "Fast computation of latent correlations" <arXiv:2006.13875>.

Examples

### Data setting
n <- 100; p1 <- 15; p2 <- 10 # sample size and dimensions for two datasets.
maxcancor <- 0.9 # true canonical correlation

### Correlation structure within each data set
set.seed(0)
perm1 <- sample(1:p1, size = p1);
Sigma1 <- autocor(p1, 0.7)[perm1, perm1]
blockind <- sample(1:3, size = p2, replace = TRUE);
Sigma2 <- blockcor(blockind, 0.7)
mu <- rbinom(p1+p2, 1, 0.5)

### true variable indices for each dataset
trueidx1 <- c(rep(1, 3), rep(0, p1-3))
trueidx2 <- c(rep(1, 2), rep(0, p2-2))

### Data generation
simdata <- GenerateData(n=n, trueidx1 = trueidx1, trueidx2 = trueidx2, maxcancor = maxcancor,
                        Sigma1 = Sigma1, Sigma2 = Sigma2,
                        copula1 = "exp", copula2 = "cube",
                        muZ = mu,
                        type1 = "trunc", type2 = "continuous",
                        c1 = rep(1, p1), c2 =  rep(0, p2)
)
X1 <- simdata$X1
X2 <- simdata$X2

### Check the range of truncation levels of variables
range(colMeans(X1 == 0))
range(colMeans(X2 == 0))

### Estimate latent correlation matrix
# with original method
R1_org <- estimateR(X1, type = "trunc", method = "original")$R
# with faster approximation method
R1_approx <- estimateR(X1, type = "trunc", method = "approx")$R
R12_approx <- estimateR_mixed(X1, X2, type1 = "trunc", type2 = "continuous", method = "approx")$R12

Internal mixedCCA function finding w1 and w2 given R1, R2 and R12

Description

Internal mixedCCA function finding w1 and w2 given R1, R2 and R12

Usage

find_w12bic(
  n,
  R1,
  R2,
  R12,
  lamseq1,
  lamseq2,
  w1init,
  w2init,
  BICtype,
  maxiter = 100,
  tol = 0.01,
  trace = FALSE,
  lassoverbose = FALSE
)

Arguments

Value

find_w12bic returns a data.frame containing

• w1: estimated canonical direction w1.

• w2: estimated canonical direction w2.

• w1trace: a matrix, of which column is the estimated canonical direction w1 at each iteration. The number of columns is the number of iteration until the convergence.

• w2trace: a matrix, of which column is the estimated canonical direction w2 at each iteration. The number of columns is the number of iteration until the convergence.

• lam1.iter: For each iteration, what lambda value is selected for w1 is stored.

• lam2.iter: For each iteration, what lambda value is selected for w2 is stored.

• obj: objective function value without penalty: w1^T * R12 * w2. If lamseq1 and lamseq2 are scalar, then original objective function including penalty part will be used.

References

Yoon G., Carroll R.J. and Gaynanova I. (2020) "Sparse semiparametric canonical correlation analysis for data of mixed types" <doi:10.1093/biomet/asaa007>.

Internal data-driven lambda sequence generating function.

Description

This internal function generates lambda sequence of length nlamseq equally spaced on a logarithmic scale. Since this is for sparse CCA, it returns a list of two vectors. Each vector will be used for each data set X1 and X2. And w1 and w2 denote canonical vector for each data set.

Usage

lambdaseq_generate(
  nlamseq = 20,
  lam.eps = 0.01,
  Sigma1,
  Sigma2,
  Sigma12,
  w1init = NULL,
  w2init = NULL
)

Arguments

Value

lambdaseq_generate returns a list of length 2. Each vector is of the same length nlamseq and will be used for each data set separately.

Sparse CCA for data of mixed types with BIC criterion

Description

Applies sparse canonical correlation analysis (CCA) for high-dimensional data of mixed types (continuous/binary/truncated continuous). Derived rank-based estimator instead of sample correlation matrix is implemented. There are two types of BIC criteria for variable selection. We found that BIC1 works best for variable selection, whereas BIC2 works best for prediction.

Usage

mixedCCA(
  X1,
  X2,
  type1,
  type2,
  lamseq1 = NULL,
  lamseq2 = NULL,
  nlamseq = 20,
  lam.eps = 0.01,
  w1init = NULL,
  w2init = NULL,
  BICtype,
  KendallR = NULL,
  maxiter = 100,
  tol = 0.01,
  trace = FALSE,
  lassoverbose = FALSE
)

Arguments

Value

mixedCCA returns a data.frame containing

• KendallR: estimated Kendall's \tau matrix estimator.

• lambda_seq: the values of lamseq used for sparse CCA.

• w1: estimated canonical direction w1.

• w2: estimated canonical direction w2.

• cancor: estimated canonical correlation.

• fitresult: more details regarding the progress at each iteration.

References

Yoon G., Carroll R.J. and Gaynanova I. (2020) "Sparse semiparametric canonical correlation analysis for data of mixed types" <doi:10.1093/biomet/asaa007>.

Examples

### Simple example

# Data setting
n <- 100; p1 <- 15; p2 <- 10 # sample size and dimensions for two datasets.
maxcancor <- 0.9 # true canonical correlation

# Correlation structure within each data set
set.seed(0)
perm1 <- sample(1:p1, size = p1);
Sigma1 <- autocor(p1, 0.7)[perm1, perm1]
blockind <- sample(1:3, size = p2, replace = TRUE);
Sigma2 <- blockcor(blockind, 0.7)
mu <- rbinom(p1+p2, 1, 0.5)

# true variable indices for each dataset
trueidx1 <- c(rep(1, 3), rep(0, p1-3))
trueidx2 <- c(rep(1, 2), rep(0, p2-2))

# Data generation
simdata <- GenerateData(n=n, trueidx1 = trueidx1, trueidx2 = trueidx2, maxcancor = maxcancor,
                        Sigma1 = Sigma1, Sigma2 = Sigma2,
                        copula1 = "exp", copula2 = "cube",
                        muZ = mu,
                        type1 = "trunc", type2 = "trunc",
                        c1 = rep(1, p1), c2 =  rep(0, p2)
)
X1 <- simdata$X1
X2 <- simdata$X2

# Check the range of truncation levels of variables
range(colMeans(X1 == 0))
range(colMeans(X2 == 0))

# Kendall CCA with BIC1
kendallcca1 <- mixedCCA(X1, X2, type1 = "trunc", type2 = "trunc", BICtype = 1, nlamseq = 10)

# Kendall CCA with BIC2. Estimated correlation matrix is plugged in from the above result.
R <- kendallcca1$KendallR
kendallcca2 <- mixedCCA(X1, X2, type1 = "trunc", type2 = "trunc",
                        KendallR = R, BICtype = 2, nlamseq = 10)

Internal RidgeCCA function

Description

This function is modified from CCA package rcc function. This function is used for simulation. Inputs should be correlation or covariance matrices of each data set and between datasets.

Usage

myrcc(R1, R2, R12, lambda1, lambda2, tol = 1e-04)

Arguments

Value

myrcc returns a data.frame containing

• cancor: estimated canonical correlation.

• w1: estimated canonical direction w1.

• w2: estimated canonical direction w2.

Internal standard CCA function.

Description

This function is modified from original CCA function for two reasons: to deal with only positive eigenvalues larger than the tolerance when calculating the inverse of the matrices and to compute Singular Value Decomposition using irlba algorithm. Inputs should be correlation or covariance matrices of each data set and between datasets. This function returns only the first pair of canonical covariates.

Usage

standardCCA(S1, S2, S12, tol = 1e-04)

Arguments

Value

standardCCA returns a data.frame containing

• cancor: estimated canonical correlation.

• w1: estimated canonical direction w1.

• w2: estimated canonical direction w2.