Package 'msPCA'

Feasibility violation

Description

Computes the feasibility violation defined as $\sum_{t > s} u_{t}^\top u_{s}$ if orthogonality constraints are enforced (feasibilityConstraintType = 0) and $\sum_{t > s} u_{t}^\top C u_{s}$ if zero-correlation constraints are enforced (feasibilityConstraintType = 1).

Usage

feasibility_violation_off(C, U, feasibilityConstraintType)

Arguments

C	A matrix. The correlation or covariance matrix (p x p).
U	A matrix. Each column correspond to an p-dimensional PC.
feasibilityConstraintType	An integer. Type of feasibility constraints to be enforced. 0: orthogonality constraints; 1: uncorrelatedness constraints.

Value

A float.

Examples

library(datasets)
TestMat <- cor(datasets::mtcars)
mspcares <- mspca(TestMat, 2, c(4,4))
feasibility_violation_off(TestMat, mspcares$x_best, 0)

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Fraction of variance explained

Description

Computes the fraction of variance explained (variance explained normalized by the trace of the covariance/correlation matrix) by a set of PCs.

Usage

fraction_variance_explained(C, U)

Arguments

C	A matrix. The correlation or covariance matrix (p x p).
U	A matrix. The matrix containing the r PCs (p x r).

Value

A float.

Examples

library(datasets)
TestMat <- cor(datasets::mtcars)
mspcares <- mspca(TestMat, 2, c(4,4))
fraction_variance_explained(TestMat, mspcares$x_best)

---

Fraction of variance explained per PC

Description

Computes the fraction of variance explained (variance explained normalized by the trace of the covariance/correlation matrix) by each PC.

Usage

fraction_variance_explained_perPC(C, U)

Arguments

C	A matrix. The correlation or covariance matrix (p x p).
U	A matrix. The matrix containing the r PCs (p x r).

Value

An array.

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Multiple Sparse PCA

Description

Returns multiple sparse principal component of a matrix using an iterative deflation heuristic.

Usage

mspca(
  Sigma,
  r,
  ks,
  feasibilityConstraintType = 0L,
  verbose = TRUE,
  maxIter = 200L,
  feasibilityTolerance = 1e-04,
  stallingTolerance = 1e-08,
  timeLimitTPM = 20L,
  maxRestartTPM = 20L,
  minRestartTPM = 10L
)

Arguments

Sigma	A matrix. The correlation or covariance matrix, whose sparse PCs will be computed.
r	An integer. Number of principal components (PCs) to be computed.
ks	A list of integers. Target sparsity of each PC.
feasibilityConstraintType	(optional) An integer. Type of feasibility constraints to be enforced. 0: orthogonality constraints; 1: uncorrelatedness constraints. Default 0.
verbose	(optional) A Boolean. Controls console output. Default TRUE.
maxIter	(optional) An integer. Maximum number of iterations of the algorithm. Default 200.
feasibilityTolerance	(optional) A float. Tolerance for the violation of the orthogonality constraints. Default 1e-4
stallingTolerance	(optional) A float. Controls the objective improvement below which the algorithm is considered to have stalled. Default 1e-8
timeLimitTPM	(optional) An integer. Maximum time in seconds for the truncated power method (inner iteration). Default 20.
maxRestartTPM	(optional) An integer. Number of random restarts of the truncated power method (inner iteration) for the first outer iteration. Default 20.
minRestartTPM	(optional) An integer. Number of random restarts of the truncated power method (inner iteration) for outer iterations >= 2. Default 10.

Value

An object with 4 fields: 'x_best' (p x r array containing the sparse PCs), 'objective_value', 'feasibility_violation', 'runtime'.

Examples

library(datasets)
TestMat <- cor(datasets::mtcars)
mspca(TestMat, 2, c(4,4))

---

Print mspca output

Description

Displays the output of the msPCA algorithm.

Usage

print_mspca(sol_object, C)

Arguments

sol_object	A list. The output of the mspca or tpm function.
C	A matrix. The correlation or covariance matrix (p x p).

Value

None. Prints output to console.

Examples

library(datasets)
TestMat <- cor(datasets::mtcars)
mspcares <- mspca(TestMat, 2, c(4,4))
print_mspca(mspcares, TestMat)

---

Truncated Power Method

Description

Returns the leading sparse principal component of a matrix using the truncated power method.

Usage

tpm(Sigma, k, maxIter = 200L, verbose = TRUE, timeLimit = 10L)

Arguments

Sigma	A matrix. The correlation or covariance matrix, whose sparse PCs will be computed.
k	An integer. Target sparsity of the PC.
maxIter	(optional) An integer. Maximum number of iterations of the algorithm. Default 200.
verbose	(optional) A Boolean. Controls console output. Default TRUE.
timeLimit	(optional) An integer. Maximum time in seconds. Default 10.

Value

An object with 3 fields: 'x_best' (p x 1 array containing the sparse PC), 'objective_value', 'runtime'.

References

Yuan, X. T., & Zhang, T. (2013). Truncated power method for sparse eigenvalue problems. The Journal of Machine Learning Research, 14(1), 899-925.

Examples

library(datasets)
TestMat <- cor(datasets::mtcars)
tpm(TestMat, 4)

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Variance explained per PC

Description

Computes the variance explained by each PC.

Usage

variance_explained_perPC(C, U)

Arguments

C	A matrix. The correlation or covariance matrix (p x p).
U	A matrix. The matrix containing the r PCs (p x r).

Value

An array.

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