odpc (version 2.0.5)

cv.odpc: Automatic Choice of Tuning Parameters for One-Sided Dynamic Principal Components via Cross-Validation

Description

Computes One-Sided Dynamic Principal Components, choosing the number of components and lags automatically, to minimize an estimate of the forecasting mean squared error.

Usage

cv.odpc(
  Z,
  h,
  k_list = 1:5,
  max_num_comp = 5,
  window_size,
  ncores_k = 1,
  ncores_w = 1,
  method,
  tol = 1e-04,
  niter_max = 500,
  train_tol = 0.01,
  train_niter_max = 100
)

Arguments

Z

Data matrix. Each column is a different time series.

h

Forecast horizon.

k_list

List of values of k to choose from.

max_num_comp

Maximum possible number of components to compute.

window_size

The size of the rolling window used to estimate the forecasting error.

ncores_k

Number of cores to parallelise over k_list.

ncores_w

Number of cores to parallelise over the rolling window (nested in k_list).

method

A string specifying the algorithm used. Options are 'ALS', 'mix' or 'gradient'. See details in odpc.

tol

Relative precision. Default is 1e-4.

niter_max

Integer. Maximum number of iterations. Default is 500.

train_tol

Relative precision used in cross-validation. Default is 1e-2.

train_niter_max

Integer. Maximum number of iterations used in cross-validation. Default is 100.

Value

An object of class odpcs, that is, a list of length equal to the number of computed components, each computed using the optimal value of k. The i-th entry of this list is an object of class odpc, that is, a list with entries

f

Coordinates of the i-th dynamic principal component corresponding to the periods \(k_1 + 1,\dots,T\).

mse

Mean squared error of the reconstruction using the first i components.

k1

Number of lags used to define the i-th dynamic principal component f.

k2

Number of lags of f used to reconstruct.

alpha

Vector of intercepts corresponding to f.

a

Vector that defines the i-th dynamic principal component

B

Matrix of loadings corresponding to f. Row number \(k\) is the vector of \(k-1\) lag loadings.

call

The matched call.

conv

Logical. Did the iterations converge?

components, fitted, plot and print methods are available for this class.

Details

We assume that for each component \(k_{1}^{i}=k_{2}^{i}\), that is, the number of lags of \(\mathbf{z}_{t}\) used to define the dynamic principal component and the number of lags of \(\widehat{f}^{i}_{t}\) used to reconstruct the original series are the same. The number of components and lags is chosen to minimize the cross-validated forecasting error in a stepwise fashion. Suppose we want to make \(h\)-steps ahead forecasts. Let \(w=\) window_size. Then given \(k\in\) k_list we compute the first ODPC defined using \(k\) lags, using periods \(1,\dots,T-h-t+1\) for \(t=1,\dots,w\), and for each of these fits we compute an h-steps ahead forecast and the corresponding mean squared error \(E_{t,h}\). The cross-validation estimate of the forecasting error is then $$ \widehat{MSE}_{1,k}=\frac{1}{w}\sum\limits_{t=1}^{w}E_{t,h}. $$ We choose for the first component the value \(k^{\ast,1}\) that minimizes \(\widehat{MSE}_{1,k}\). Then, we fix the first component computed with \(k^{\ast,1}\) lags and repeat the procedure with the second component. If the optimal cross-validated forecasting error using the two components, \(\widehat{MSE}_{2,k^{\ast,2}}\) is larger than the one using only one component, \(\widehat{MSE}_{1,k^{\ast,1}}\), we stop and output as a final model the ODPC computed using one component defined with \(k^{\ast,1}\) lags; otherwise, if max_num_comp \(\geq 2\) we add the second component defined using \(k^{\ast,2}\) lags and proceed as before.

This method can be computationally costly, especially for large values of the window_size. Ideally, the user should set n_cores_k equal to the length of k_list and n_cores_w equal to window_size; this would entail using n_cores_k times n_cores_w cores in total.

References

Pe<U+00F1>a D., Smucler E. and Yohai V.J. (2017). <U+201C>Forecasting Multiple Time Series with One-Sided Dynamic Principal Components.<U+201D> Available at https://arxiv.org/abs/1708.04705.

See Also

odpc, crit.odpc, forecast.odpcs

Examples

# NOT RUN {
T <- 50 #length of series
m <- 10 #number of series
set.seed(1234)
f <- rnorm(T + 1)
x <- matrix(0, T, m)
u <- matrix(rnorm(T * m), T, m)
for (i in 1:m) {
  x[, i] <- 10 * sin(2 * pi * (i/m)) * f[1:T] + 10 * cos(2 * pi * (i/m)) * f[2:(T + 1)] + u[, i]
}
# Choose parameters to perform a one step ahead forecast. Use 1 or 2 lags, only one component 
# and a window size of 2 (artificially small to keep computation time low). Use two cores for the
# loop over k, two cores for the loop over the window
fit <- cv.odpc(x, h=1, k_list = 1:2, max_num_comp = 1, window_size = 2, ncores_k = 2, ncores_w = 2)
# }