capReg {cap}

Covariate Assisted Principal Regression for Covariance Matrix Outcomes

Description

This function identifies the first k projection directions that satisfies the log-linear model assumption.

Usage

capReg(Y, X, nD = 1, method = c("CAP", "CAP-C"), CAP.OC = FALSE, 
  max.itr = 1000, tol = 1e-04, trace = FALSE, score.return = TRUE, 
  gamma0.mat = NULL, ninitial = NULL)

Arguments

Y

a data list of length n. Each list element is a T\times p matrix, the data matrix of T observations from p features.

X

a n\times q data matrix, the covariate matrix of n subjects with q-1 predictors. The first column is all ones.

nD

an integer, the number of directions to be identified. Default is 1.

method

a character of optimization method. method = "CAP" considers a weighted L2-norm on the \gamma vector and solve for the optimizer by block coordinated descent; method = "CAP-C" assumes the complete common principal component assumption which identifies the common principal component first and then searches for the optimal PC.

CAP.OC

a logic variable. Whether the orthogonal constraint is imposed when identifying higher-order PCs. When method = "CAP-C", this is ignored. Default is FALSE.

max.itr

an integer, the maximum number of iterations.

tol

a numeric value of convergence tolerance.

trace

a logic variable. Whether the solution path is reported. Default is FALSE.

score.return

a logic variable. Whether the log-variance in the transformed space is reported. Default is TRUE.

gamma0.mat

a data matrix, the initial value of \gamma. Default is NULL, and initial value is randomly chosen.

ninitial

an integer, the number of different initial value is tested. When it is greater than 1, multiple initial values will be tested, and the one yields the minimum objective function will be reported. Default is NULL.

Details

Considering y_{it} are p-dimensional independent and identically distributed random samples from a multivariate normal distribution with mean zero and covariance matrix \Sigma_{i}. We assume there exits a p-dimensional vector \gamma such that z_{it}:=\gamma'y_{it} satisfies the multiplicative heteroscedasticity:

\log(\mathrm{Var}(z_{it}))=\log(\gamma'\Sigma_{i}\gamma)=\beta_{0}+x_{i}'\beta_{1}

, where x_{i} contains explanatory variables of subject i, and \beta_{0} and \beta_{1} are model coefficients.

Parameters \gamma and \beta=(\beta_{0},\beta_{1}')' are study of interest, and we propose to estimate them by maximizing the likelihood function,

\ell(\beta,\gamma)=-\frac{1}{2}\sum_{i=1}^{n}T_{i}(x_{i}'\beta)-\frac{1}{2}\sum_{i=1}^{n}\exp(-x_{i}'\beta)\gamma'S_{i}\gamma,

where S_{i}=\sum_{t=1}^{T_{i}}y_{it}y_{it}'. To estimate \gamma, we impose the following constraint

\gamma' H\gamma=1,

where H is a positive definite matrix. In this study, we consider the choice that

H=\bar{\Sigma}, \quad \bar{\Sigma}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{T_{i}}S_{i}.

For higher order projecting directions, an orthogonal constraint is imposed as well.

Value

When method = "CAP",

gamma

the estimate of \gamma vectors, which is a p\times nD matrix.

beta

the estimate of \beta for each projecting direction, which is a q\times nD matrix, where q-1 is the number of explanatory variables.

orthogonality

an ad hoc checking of the orthogonality between \gamma vectors.

DfD

output of both average (geometric mean) and individual level of “deviation from diagonality”.

score

an output when score.return = TRUE. A n\times nD matrix of \log(\hat{\gamma}'S_{i}\hat{\gamma}) value.

When method = "CAP-C",

gamma

the estimate of \gamma vectors, which is a p\times nD matrix.

beta

the estimate of \beta for each projecting direction, which is a q\times nD matrix, where q-1 is the number of explanatory variables.

orthogonality

an ad hoc checking of the orthogonality between \gamma vectors.

PC.idx

a vector of length nD, the order index of identified \gamma vectors among all the common principal components.

aPC.idx

the order index of all the principal components that satisfy the log-linear model and the eigenvalue condition.

minmax

a logic output, whether the identified \gamma vectors are estimated from the minmax approach. If FALSE, indicating the eigenvalue condition is not satisfied for any principal component.

score

an output when score.return = TRUE. A n\times nD matrix of \log(\hat{\gamma}'S_{i}\hat{\gamma}) value.

Author(s)

Yi Zhao, Johns Hopkins University, <[e-mail]>

Bingkai Wang, Johns Hopkins University, <[e-mail]>

Stewart Mostofsky, Johns Hopkins University, <[e-mail]>

Brian Caffo, Johns Hopkins University, <[e-mail]>

Xi Luo, Brown University, <[e-mail]>

References

Zhao et al. (2018) Covariate Assisted Principal Regression for Covariance Matrix Outcomes <doi:10.1101/425033>

Examples

#############################################
data(env.example)
X<-get("X",env.example)
Y<-get("Y",env.example)

# method = "CAP"
# without orthogonal constraint
re1<-capReg(Y,X,nD=2,method=c("CAP"),CAP.OC=FALSE)
# with orthogonal constraint
re2<-capReg(Y,X,nD=2,method=c("CAP"),CAP.OC=TRUE)

# method = "CAP-C"
re3<-capReg(Y,X,nD=2,method=c("CAP-C"))
#############################################