Remark on the Accuracy of Recurrent Forecasting in Singular Spectrum Analysis

Abstract

Within the series of singular spectrum analysis (SSA) methods, there exist several versions of forecasting algorithms for signals corrupted by additive noise. In this paper, a technique is proposed to estimate the asymptotic accuracy of the recurrent version of such forecasting when the length of a series tends to infinity. Most elements of this construction can be reduced to already studied and published results, although some of them are hard to implement in specific situations. The article brings together all these elements and augments and comments on them. Several examples of determining estimates of accuracy for a recurrent forecast are given for specific signals and noises. The computational experiments carried out confirm the theoretical results.

APPENDIX. COMPUTER EXPERIMENTS

In this section we present several variants of computer experiments that illustrate the theoretical results of Subsection 3.4. As already mentioned, the considered series have the form x_n = f_n + δe_n, 1 ≤ n ≤ N, and the problem lies in forecasting the signal value ${{f}_{{N + k}}}$, k ≥ 1.

We consider three variants of the signal f_n:

1. Exponential signal (EXP): f_n = aⁿ, a = 1.01, and δ = 1.

2. Linear signal (LIN): f_n = an + b, a = 0.5, b = 1, and δ = 0.5.

3. Harmonic signal (COS): f_n = cos(2πω₀n) with ω₀ = $\sqrt 2 {\text{/2}}$, and δ = 0.5.

In all cases, the noise has the form e_n = cos(2πωn) with ω = $\sqrt 3 {\text{/2}}$.

As before, ${{\tilde {f}}_{{N + k}}}$ denotes the value of the recurrent forecast of ${{f}_{{N + k}}}$, and $\Delta _{k}^{{(f)}}(N)$ = $\left| {{{{\tilde {f}}}_{{N + k}}} - {{f}_{{N + k}}}} \right|$ is the k-step forecast error.

k-Step forecast, 1 ≤ k ≤ 50. Figure 1 is of preliminary character. It shows k-step forecasting errors for exponential (EXP), harmonic (COS), and linear (LIN) signals for N = 500, L = M = 250, and k = 1(1)50. We note that these errors exhibit very smooth behavior for EXP and LIN, while $\Delta _{k}^{{(f)}}(N)$ strongly oscillates for COS.

Fig. 1.

k-Step forecast errors for exponential (EXP), harmonic (COS) and linear (LIN) signals as a function of k, k = 1(1)50.

Therefore, below we illustrate the behavior of $\Delta _{1}^{{(f)}}(N)$ as a function of N for exponential and linear signals, and take the characteristic $\left( {\sum\nolimits_{k = 1}^{10} {\Delta _{k}^{{(f)}}(N)} } \right){\text{/}}10$ to illustrate the forecasting behavior for a harmonic signal.

We note that in all the following experiments, we use the series lengths N = 50(50)1000 and L = M = N/2.

Exponential signal. Figure 2 shows the behavior of one-step forecasting errors as a function of the series length N for an exponential signal and harmonic noise.

Fig. 2.

One-step forecast errors as a function of the series length N for the EXP signal.

The theoretical results of Section 3.4 show that $\Delta _{1}^{{(f)}}(N)$ = O(1). Judging by Fig. 2, this estimate is quite accurate: starting from about N = 300, the values of $\Delta _{1}^{{(f)}}(N)$ do not show an obvious tendency toward a decrease.

Linear signal. As already mentioned, for a linear signal, the expected estimate for $\Delta _{1}^{{(f)}}(N)$ is O(N^–1/2). Figures 3 and 4 confirm this theoretical result.

Fig. 3.

One-step forecast errors as a function of the series length N for the LIN signal.

Fig. 4.

$\sqrt N$multiplied by one-step forecast errors as a function of the series length N for the LIN signal.

Harmonic signal. According to the results of Subsection 3.4, the expected upper bound for one-step forecast errors for a harmonic signal is O(N^–1/2). Figures 5 and 6 show that the actual convergence rate may be O(N^–1).

Fig. 5.

Mean of the errors for k = 1(1)10 forecast steps as a function of the series length N for the COS signal.

Fig. 6.

N multiplied by the mean of the errors for k = 1(1)10 forecast steps as a function of the series length N for the COS signal.