# spca.cavi.Laplace: Function for the PX-CAVI algorithm using the Laplace slab In VBsparsePCA: The Variational Bayesian Method for Sparse PCA

## Description

This function employs the PX-CAVI algorithm proposed in Ning (2020). The g in the slab density of the spike and slab prior is chosen to be the Laplace density, i.e., N(0, σ^2/λ_1 I_r). Details of the model and the prior can be found in the Details section in the description of the 'VBsparsePCA()' function. This function is not capable of handling the case when r > 1. In that case, we recommend to use the multivariate distribution instead.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```spca.cavi.Laplace( x, r = 1, lambda = 1, max.iter = 100, eps = 0.001, sig2.true = NA, threshold = 0.5, theta.int = NA, theta.var.int = NA, kappa.para1 = NA, kappa.para2 = NA, sigma.a = NA, sigma.b = NA ) ```

## Arguments

 `x` Data an n*p matrix. `r` Rank. `lambda` Tuning parameter for the density g. `max.iter` The maximum number of iterations for running the algorithm. `eps` The convergence threshold; the default is 10^{-4}. `sig2.true` The default is false, σ^2 will be estimated; if sig2 is known and its value is given, then σ^2 will not be estimated. `threshold` The threshold to determine whether γ_j is 0 or 1; the default value is 0.5. `theta.int` The initial value of theta mean; if not provided, the algorithm will estimate it using PCA. `theta.var.int` The initial value of theta.var; if not provided, the algorithm will set it to be 1e-3*diag(r). `kappa.para1` The value of α_1 of π(κ); default is 1. `kappa.para2` The value of α_2 of π(κ); default is p+1. `sigma.a` The value of σ_a of π(σ^2); default is 1. `sigma.b` The value of σ_b of π(σ^2); default is 2.

## Value

 `iter` The number of iterations to reach convergence. `selection` A vector (if r = 1 or with the jointly row-sparsity assumption) or a matrix (if otherwise) containing the estimated value for \boldsymbol γ. `theta.mean` The loadings matrix. `theta.var` The covariance of each non-zero rows in the loadings matrix. `sig2` Variance of the noise. `obj.fn` A vector contains the value of the objective function of each iteration. It can be used to check whether the algorithm converges

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23``` ```#In this example, the first 20 rows in the loadings matrix are nonzero, the rank is 1 set.seed(2021) library(MASS) library(pracma) n <- 200 p <- 1000 s <- 20 r <- 1 sig2 <- 0.1 # generate eigenvectors U.s <- randortho(s, type = c("orthonormal")) U <- rep(0, p) U[1:s] <- as.vector(U.s[, 1:r]) s.star <- rep(0, p) s.star[1:s] <- 1 eigenvalue <- seq(20, 10, length.out = r) # generate Sigma theta.true <- U * sqrt(eigenvalue) Sigma <- tcrossprod(theta.true) + sig2*diag(p) # generate n*p dataset X <- t(mvrnorm(n, mu = rep(0, p), Sigma = Sigma)) result <- spca.cavi.Laplace(x = X, r = 1) loadings <- result\$theta.mean ```

VBsparsePCA documentation built on Feb. 12, 2021, 5:06 p.m.