Factors: Factor analysis for vector time series

In HDTSA: High Dimensional Time Series Analysis Tools

Factor analysis for vector time series

Description

Factors() deals with factor modeling for high-dimensional time series proposed in Lam and Yao (2012):

{\bf y}_t = {\bf Ax}_t + {\boldsymbol{\epsilon}}_t,

where {\bf x}_t is an r \times 1 latent process with (unknown) r \leq p, {\bf A} is a p \times r unknown constant matrix, and {\boldsymbol{\epsilon}}_t is a vector white noise process. The number of factors r and the factor loadings {\bf A} can be estimated in terms of an eigenanalysis for a nonnegative definite matrix, and is therefore applicable when the dimension of {\bf y}_t is on the order of a few thousands. This function aims to estimate the number of factors r and the factor loading matrix {\bf A}.

Usage

Factors(
  Y,
  lag.k = 5,
  thresh = FALSE,
  delta = 2 * sqrt(log(ncol(Y))/nrow(Y)),
  twostep = FALSE
)

Arguments

Y	An n \times p data matrix {\bf Y} = ({\bf y}_1, \dots , {\bf y}_n )', where n is the number of the observations of the p \times 1 time series \{{\bf y}_t\}_{t=1}^n.
lag.k	The time lag K used to calculate the nonnegative definite matrix \hat{\mathbf{M}}: \hat{\mathbf{M}}\ =\ \sum_{k=1}^{K} T_\delta\{\hat{\mathbf{\Sigma}}_y(k)\} T_\delta\{\hat{\mathbf{\Sigma}}_y(k)\}'\,, where \hat{\bf \Sigma}_y(k) is the sample autocovariance of {\bf y}_t at lag k and T_\delta(\cdot) is a threshold operator with the threshold level \delta \geq 0. See 'Details'. The default is 5.
thresh	Logical. If thresh = FALSE (the default), no thresholding will be applied to estimate \hat{\mathbf{M}}. If thresh = TRUE, \delta will be set through delta.
delta	The value of the threshold level \delta. The default is \delta = 2 \sqrt{n^{-1}\log p}.
twostep	Logical. If twostep = FALSE (the default), the standard procedure [See Section 2.2 in Lam and Yao (2012)] for estimating r and {\bf A} will be implemented. If twostep = TRUE, the two-step estimation procedure [See Section 4 in Lam and Yao (2012)] for estimating r and {\bf A} will be implemented.

Details

The threshold operator T_\delta(\cdot) is defined as T_\delta({\bf W}) = \{w_{i,j}1(|w_{i,j}|\geq \delta)\} for any matrix {\bf W}=(w_{i,j}), with the threshold level \delta \geq 0 and 1(\cdot) representing the indicator function. We recommend to choose \delta=0 when p is fixed and \delta>0 when p \gg n.

Value

An object of class "factors", which contains the following components:

factor_num	The estimated number of factors \hat{r}.
loading.mat	The estimated p \times \hat{r} factor loading matrix \hat{\bf A}.
X	The n\times \hat{r} matrix \hat{\bf X}=(\hat{\bf x}_1,\dots,\hat{\bf x}_n)' with \hat{\bf x}_t = \hat{\bf A}'\hat{\bf y}_t.
lag.k	The time lag used in function.

References

Lam, C., & Yao, Q. (2012). Factor modelling for high-dimensional time series: Inference for the number of factors. The Annals of Statistics, 40, 694–726. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1214/12-AOS970")}.

Examples

# Example 1 (Example in Section 3.3 of lam and Yao 2012)
## Generate y_t
p <- 200
n <- 400
r <- 3
X <- mat.or.vec(n, r)
A <- matrix(runif(p*r, -1, 1), ncol=r)
x1 <- arima.sim(model=list(ar=c(0.6)), n=n)
x2 <- arima.sim(model=list(ar=c(-0.5)), n=n)
x3 <- arima.sim(model=list(ar=c(0.3)), n=n)
eps <- matrix(rnorm(n*p), p, n)
X <- t(cbind(x1, x2, x3))
Y <- A %*% X + eps
Y <- t(Y)

fac <- Factors(Y,lag.k=2)
r_hat <- fac$factor_num
loading_Mat <- fac$loading.mat

HDTSA documentation built on April 3, 2025, 11:07 p.m.