Empirical properties of forecasts with the functional autoregressive model

Abstract

We study the finite sample performance of predictors in the functional (Hilbertian) autoregressive model X n+1 = Ψ(X n ) + ε n . Our extensive empirical study based on simulated and real data reveals that predictors of the formΨ(X n ) are practically optimal in a sense that their prediction errors are comparable with those of the infeasible perfect predictor Ψ(X n ). The predictionsΨ(X n ) cannot be improved by an improved estimation of Ψ , nor by a more refined prediction approach which uses predictive factors rather than the functional principal components. We also discuss the practical limits of predictions that are feasible using the functional autoregressive model. These findings have not been established by theoretical work currently available, and may serve as a practical reference to the properties of predictors of functional data.

Detecting a structural change in functional time series using local Wilcoxon statistic

“…location and or scale. Similarly as Horváth et al (2014) and Didericksen et al (2012) we considered samples with functional errors being generated by Wiener process and Brownian bridge divided into 1440 and 120 time points (24 hours divided into 1min and 12min time segments). We considered samples of equal and different sizes.…”

“…. , N. We used the following (Didericksen et al, 2012) designs of a simulation study. (5) under alternative in which the first sample is generated from population related to fig.…”

“…In experiment 1 we generated 100 curves using a Gaussian kernel ψ(t, s) = C exp{− 1 2 (t 2 + s 2 )}, and errors of type (8) from Didericksen et al (2012) and then 100 curves using a kernel ψ(t, s) = C. Fig. 7 presents an illustration for the experiment 1.…”

“…2. In experiment 2 we generated 100 curves using a Gaussian kernel with appropriate constant C and 100 curves from a mixture of two processes considered in the experiment 1 but differing with respect to parameters of the error term (8) taken from Didericksen et al (2012). Fig.…”

Dependent Functional Data

“…Predictive factors are interesting in that they are functions which can be used to expand the curves, very much like the FPCs are used, but they define directions in the space L 2 which are most relevant for prediction. Using a finite sample implementation, Didericksen et al 43 found that it does not lead to more accurate predictions. In general, predicted curves that use formula 3.10 tend to be closer to the mean curve and smoother than the actual observations.…”