Package 'mixedCCA' reference manual

Title:	Sparse Canonical Correlation Analysis for High-Dimensional Mixed Data
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>.
Authors:	Grace Yoon [aut] , Mingze Huang [ctb] , Irina Gaynanova [aut, cre]
Maintainer:	Irina Gaynanova <[email protected]>
License:	GPL-3
Version:	1.6.2
Built:	2024-10-06 06:46:59 UTC
Source:	CRAN

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)
autocor(p, rho)

blockcor(blockind, rho)

Arguments

`p`	Specified matrix dimension.
`rho`	Correlation value(s), must be between -0.99 and 0.99. Should be a scalar for `autocor`, and either a scalar or a vector of the same length as the maximal `blockind` K for `blockcor`.
`blockind`	Block index 1,..., K for a positive integer K specifying which variable belongs to which block, the matrix dimension is equal to `length(blockind)`.

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))

# 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))

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
)
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

`X`	A numeric data matrix (n by p), n is the sample size and p is the number of variables.
`type`	A type of variables in `X`, must be one of "continuous", "binary" or "trunc".
`method`	The calculation method of latent correlation. Either "original" method or "approx". If `method = "approx"`, multilinear approximation method is used, which is much faster than the original method (requires `chebpol` R package). If `method = "original"`, optimization of the bridge inverse function is used. The default is "original".
`use.nearPD`	A logical value indicating whether to use nearPD or not when the resulting correlation estimator is not positive definite (have at least one negative eigenvalue).
`nu`	Shrinkage parameter for correlation matrix, must be between 0 and 1, the default value is 0.01.
`tol`	Desired accuracy when calculating the solution of bridge function.
`verbose`	If `verbose = FALSE`, printing information whether nearPD is used or not is disabled. The default value is FALSE.
`X1`	A numeric data matrix (n by p1).
`X2`	A numeric data matrix (n by p2).
`type1`	A type of variables in `X1`, must be one of "continuous", "binary" or "trunc".
`type2`	A type of variables in `X2`, must be one of "continuous", "binary" or "trunc".

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


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

Arguments

`n`	Sample size
`R1`	Correlation matrix of dataset `X1` (p1 by p1)
`R2`	Correlation matrix of dataset `X2` (p2 by p2)
`R12`	Correlation matrix between the dataset `X1` and the dataset `X2` (p1 by p2)
`lamseq1`	A sequence of lambda values for the datasets `X1`. It can be a scalar (a vector of one value). should be the same length with lamseq2.
`lamseq2`	A sequence of lambda values for the datasets `X2`. It can be a scalar (a vector of one value). should be the same length with lamseq1.
`w1init`	An initial vector of length p1 for canonical direction $w1$ .
`w2init`	An initial vector of length p1 for canonical direction $w2$ .
`BICtype`	Either 1 or 2: For more details for two options, see the reference.
`maxiter`	The maximum number of iterations allowed.
`tol`	The desired accuracy (convergence tolerance).
`trace`	If `trace = TRUE`, progress per each iteration will be printed. The default value is `FALSE`.
`lassoverbose`	If `lassoverbose = TRUE`, all warnings from lassobic optimization regarding convergence will be printed. The default value is `lassoverbose = FALSE`.

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>.

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

Arguments

`n`	Sample size
`trueidx1`	True canonical direction of length p1 for `X1`. It will be automatically normalized such that $w_1^T \Sigma_1 w_1 = 1$ .
`trueidx2`	True canonical direction of length p2 for `X2`. It will be automatically normalized such that $w_2^T \Sigma_2 w_2 = 1$ .
`Sigma1`	True correlation matrix of latent variable `Z1` (p1 by p1).
`Sigma2`	True correlation matrix of latent variable `Z2` (p2 by p2).
`maxcancor`	True canonical correlation between `Z1` and `Z2`.
`copula1`	Copula type for the first dataset. U1 = f(Z1), which could be either "exp", "cube".
`copula2`	Copula type for the second dataset. U2 = f(Z2), which could be either "exp", "cube".
`type1`	Type of the first dataset `X1`. Could be "continuous", "trunc" or "binary".
`type2`	Type of the second dataset `X2`. Could be "continuous", "trunc" or "binary".
`muZ`	Mean of latent multivariate normal.
`c1`	Constant threshold for `X1` needed for "trunc" and "binary" data type - the default is NULL.
`c2`	Constant threshold for `X2` needed for "trunc" and "binary" data type - the default is NULL.

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))
### 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)
KendallTau(x, y)

Kendall_matrix(X, Y = NULL)

Arguments

`x`	A numeric vector.
`y`	A numeric vector.
`X`	A numeric matrix (n by p1).
`Y`	A numeric matrix (n by p2).

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)

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)

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

Arguments

`nlamseq`	The length of lambda sequence
`lam.eps`	The smallest value for lambda as a fraction of maximum lambda value
`Sigma1`	Covariance/correlation matrix of $X1$ (p1 by p1)
`Sigma2`	Covariance/correlation matrix of $X2$ (p2 by p2)
`Sigma12`	Covariance/correlation matrix between $X1$ and $X2$
`w1init`	Initial value for canonical vector $w1$
`w2init`	Initial value for canonical vector $w2$

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
)
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

`X1`	A numeric data matrix (n by p1).
`X2`	A numeric data matrix (n by p2).
`type1`	A type of data `X1` among "continuous", "binary", "trunc".
`type2`	A type of data `X2` among "continuous", "binary", "trunc".
`lamseq1`	A tuning parameter sequence for `X1`. The length should be the same as `lamseq2`.
`lamseq2`	A tuning parameter sequence for `X2`. The length should be the same as `lamseq1`.
`nlamseq`	The number of tuning parameter sequence lambda - default is 20.
`lam.eps`	A ratio of the smallest value for lambda to the maximum value of lambda.
`w1init`	An initial vector of length p1 for canonical direction $w1$ .
`w2init`	An initial vector of length p2 for canonical direction $w2$ .
`BICtype`	Either 1 or 2: For more details for two options, see the reference.
`KendallR`	An estimated Kendall $\tau$ matrix. The default is NULL, which means that it will be automatically estimated by Kendall's $\tau$ estimator unless the user supplies.
`maxiter`	The maximum number of iterations allowed.
`tol`	The desired accuracy (convergence tolerance).
`trace`	If `trace = TRUE`, progress per each iteration will be printed. The default value is `FALSE`.
`lassoverbose`	If `lassoverbose = TRUE`, all warnings from lassobic optimization regarding convergence will be printed. The default value is `lassoverbose = FALSE`.

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

Arguments

`R1`	correlation/covariance/rank-based correlation matrix of dataset `X1`.
`R2`	correlation/covariance/rank-based correlation matrix of dataset `X2`.
`R12`	correlation/covariance/rank-based correlation matrix between dataset `X1` and dataset `X2`.
`lambda1`	tuning parameter (a scalar value) for dataset `X1`.
`lambda2`	tuning parameter (a scalar value) for dataset `X2`.
`tol`	tolerance for eigenvalues. Refer to `standardCCA` function.

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

Arguments

`S1`	correlation/covariance matrix of dataset `X1`.
`S2`	correlation/covariance matrix of dataset `X2`.
`S12`	correlation/covariance matrix between dataset `X1` and dataset `X2`.
`tol`	tolerance for eigenvalues. `standardCCA` function only deals with positive eigenvalues larger than the tolerance.

Value

standardCCA returns a data.frame containing

cancor: estimated canonical correlation.
w1: estimated canonical direction $w1$ .
w2: estimated canonical direction $w2$ .

Package 'mixedCCA'

Help Index

Construct a correlation matrix

Description

Usage

Arguments

Examples

Estimate latent correlation matrix

Description

Usage

Arguments

Value

References

Examples

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

Description

Usage

Arguments

Value

References

Mixed type simulation data generator for sparse CCA

Description

Usage

Arguments

Value

Examples

Kendall's tau correlation

Description

Usage

Arguments

Value

Examples

Internal data-driven lambda sequence generating function.

Description

Usage

Arguments

Value

Sparse CCA for data of mixed types with BIC criterion

Description

Usage

Arguments

Value

References

Examples

Internal RidgeCCA function

Description

Usage

Arguments

Value

Internal standard CCA function.

Description

Usage

Arguments

Value