Scaled PCA factor extraction

Implements scaled principal component analysis (sPCA): predictors are first standardized, then each standardized predictor is scaled by its univariate predictive slope on the target, and finally principal components are extracted from the scaled predictors.

Usage

spca_est(target, X, nfac, winsorize = FALSE, winsor_probs = c(0, 99))

Arguments

target

A numeric vector of length T_reg (T_reg <= T).

X

A numeric matrix or data frame with T rows and N columns. When length(target) < nrow(X), the first length(target) rows of the standardized X are used for the scaling regression while all T rows are used for standardization and factor extraction. This matches the out-of-sample workflow in Huang et al. (2022), where the predictive regression y_{t+1} ~ X_t uses fewer rows than the full training window.

nfac

A positive integer giving the number of factors to extract.

winsorize

Logical; if TRUE, winsorize absolute slope estimates before scaling predictors.

winsor_probs

Numeric vector of length 2 giving winsorization percentiles. Used only when winsorize = TRUE.

Value

An object of class "sdim_spca" with components:

factors

A T x nfac matrix of extracted sPCA factors.

beta

A numeric vector of predictor-specific predictive slopes.

beta_scaled

A numeric vector of scaling coefficients actually used.

col_means

Column means of X (used by predict).

col_sds

Column standard deviations of X (used by predict).

Xs

The standardized predictor matrix.

scaleXs

The scaled standardized predictor matrix.

lambda

The estimated loading matrix.

residuals

Residual matrix from the PCA reconstruction step.

ve2

Average squared residual by row.

eigvals

Singular values from the decomposition of scaleXs %*% t(scaleXs).

call

The matched function call.

Details

The function follows the MATLAB implementation of Huang, Jiang, Li, Tong, and Zhou (2022).

References

Huang, D., Jiang, F., Li, K., Tong, G., and Zhou, G. (2022). Scaled PCA: A New Approach to Dimension Reduction. Management Science, 68(3), 1678–1695. doi:10.1287/mnsc.2021.4020

Examples

set.seed(123)
X <- matrix(rnorm(200 * 10), nrow = 200, ncol = 10)
y <- rnorm(200)

fit <- spca_est(target = y, X = X, nfac = 3)
dim(fit$factors)
#> [1] 200   3
head(fit$beta)
#> [1]  0.0007544186  0.0307199352  0.0074016978 -0.0217043884  0.0892871951
#> [6]  0.0260282689

# Predictive alignment: target has fewer rows than X
fit2 <- spca_est(target = y[1:199], X = X, nfac = 3)
dim(fit2$factors)  # 200 x 3 (factors for all T rows)
#> [1] 200   3