crit.odpc: Automatic Choice of Tuning Parameters for One-Sided Dynamic...

In odpc: One-Sided Dynamic Principal Components

Automatic Choice of Tuning Parameters for One-Sided Dynamic Principal Components via the Minimization of an Information Criterion

Description

Computes One-Sided Dynamic Principal Components, choosing the number of components and lags automatically, to minimize an information criterion.

Usage

crit.odpc(
  Z,
  k_list = 1:5,
  max_num_comp = 5,
  ncores = 1,
  method,
  tol = 1e-04,
  niter_max = 500
)

Arguments

Z	Data matrix. Each column is a different time series.
k_list	List of values of k to choose from.
max_num_comp	Maximum possible number of components to compute.
ncores	Number of cores to use in parallel computations.
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.

Details

We apply the same stepwise approach taken in cv.odpc, but now to minimize an information criterion instead of the cross-validated forecasting error. The criterion is inspired by the IC_{p3} criterion proposed in Bai and Ng (2002). Let \widehat{σ}^{2}_{1,k} be the reconstruction mean squared error for the first ODPC defined using k lags. Let T^{\ast,1,k}=T-2k. Then we choose the value k^{\ast,1} in k_list that minimizes

{BNG}_{1,k}=\log≤ft( \widehat{σ}^{2}_{1,k} \right) + ( k+1 ) \frac{\log≤ft(\min(T^{\ast,1,k},m)\right)}{\min(T^{\ast,1,k},m)}.

Suppose now that max_num_comp ≥q 2 and we have computed q-1 dynamic principal components, q-1 < max_num_comp, each with k_{1}^{i}=k_{2}^{i}=k^{\ast, i} lags, i=1,…,q-1. Let \widehat{σ}^{2}_{q,k} be the reconstruction mean squared error for the fit obtained using q components, where the first q-1 components are defined using k^{\ast, i}, i=1,…,q-1 and the last component is defined using k lags. Let T^{\ast,q,k}=T-\max\lbrace 2k^{\ast,1},…,2k^{\ast,q-1},2k \rbrace. Let k^{\ast,q} be the value in k_list that minimizes

{BNG}_{q,k}=\log≤ft( \widehat{σ}^{2}_{q,k} \right) + ≤ft(∑_{i=1}^{q-1}(k^{\ast,i}+1) + k+1 \right) \frac{\log≤ft(\min(T^{\ast,q,k},m)\right)}{\min(T^{\ast,q,k},m)} .

If {BNG}_{q,k^{\ast,q}} is larger than {BNG}_{q-1,k^{\ast,q-1}} we stop and the final model is the ODPC with q-1 components. Else we add the q-th component defined using k^{\ast,q} and continue as before.

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,…,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.

References

Peña D., Smucler E. and Yohai V.J. (2017). “Forecasting Multiple Time Series with One-Sided Dynamic Principal Components.” Available at https://arxiv.org/abs/1708.04705.

Bai J. and Ng S. (2002). “Determining the Number of Factors in Approximate Factor Models.” Econometrica, 70(1), 191–221.

See Also

odpc, cv.odpc, forecast.odpcs

Examples

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 
fit <- crit.odpc(x, k_list = 1:2, max_num_comp = 1)

odpc documentation built on March 18, 2022, 7:32 p.m.