spca {sparsepca}

Sparse Principal Component Analysis (spca).

Description

Implementation of SPCA, using variable projection as an optimization strategy.

Usage

spca(X, k = NULL, alpha = 1e-04, beta = 1e-04, center = TRUE,
  scale = FALSE, max_iter = 1000, tol = 1e-05, verbose = TRUE)

Arguments

X

array_like;
a real (n, p) input matrix (or data frame) to be decomposed.

k

integer;
specifies the target rank, i.e., the number of components to be computed.

alpha

float;
Sparsity controlling parameter. Higher values lead to sparser components.

beta

float;
Amount of ridge shrinkage to apply in order to improve conditioning.

center

bool;
logical value which indicates whether the variables should be shifted to be zero centered (TRUE by default).

scale

bool;
logical value which indicates whether the variables should be scaled to have unit variance (FALSE by default).

max_iter

integer;
maximum number of iterations to perform before exiting.

tol

float;
stopping tolerance for the convergence criterion.

verbose

bool;
logical value which indicates whether progress is printed.

Details

Sparse principal component analysis is a modern variant of PCA. Specifically, SPCA attempts to find sparse weight vectors (loadings), i.e., a weight vector with only a few 'active' (nonzero) values. This approach leads to an improved interpretability of the model, because the principal components are formed as a linear combination of only a few of the original variables. Further, SPCA avoids overfitting in a high-dimensional data setting where the number of variables p is greater than the number of observations n.

Such a parsimonious model is obtained by introducing prior information like sparsity promoting regularizers. More concreatly, given an (n,p) data matrix X, SPCA attemps to minimize the following objective function:

f(A,B) = \frac{1}{2} \| X - X B A^\top \|^2_F + \psi(B)

where B is the sparse weight (loadings) matrix and A is an orthonormal matrix. \psi denotes a sparsity inducing regularizer such as the LASSO (\ell_1 norm) or the elastic net (a combination of the \ell_1 and \ell_2 norm). The principal components Z are formed as

Z = X B

and the data can be approximately rotated back as

\tilde{X} = Z A^\top

The print and summary method can be used to present the results in a nice format.

Value

spca returns a list containing the following three components:

loadings

array_like;
sparse loadings (weight) vector; (p, k) dimensional array.

transform

array_like;
the approximated inverse transform; (p, k) dimensional array.

scores

array_like;
the principal component scores; (n, k) dimensional array.

eigenvalues

array_like;
the approximated eigenvalues; (k) dimensional array.

center, scale

array_like;
the centering and scaling used.

Author(s)

N. Benjamin Erichson, Peng Zheng, and Sasha Aravkin

References

• [1] N. B. Erichson, P. Zheng, K. Manohar, S. Brunton, J. N. Kutz, A. Y. Aravkin. "Sparse Principal Component Analysis via Variable Projection." Submitted to IEEE Journal of Selected Topics on Signal Processing (2018). (available at 'arXiv https://arxiv.org/abs/1804.00341).

See Also

rspca, robspca

Examples

# Create artifical data
m <- 10000
V1 <- rnorm(m, 0, 290)
V2 <- rnorm(m, 0, 300)
V3 <- -0.1*V1 + 0.1*V2 + rnorm(m,0,100)

X <- cbind(V1,V1,V1,V1, V2,V2,V2,V2, V3,V3)
X <- X + matrix(rnorm(length(X),0,1), ncol = ncol(X), nrow = nrow(X))

# Compute SPCA
out <- spca(X, k=3, alpha=1e-3, beta=1e-3, center = TRUE, scale = FALSE, verbose=0)
print(out)
summary(out)