A Functional Wavelet–Kernel Approach for Time Series Prediction

Abstract

We consider the prediction problem of a time series on a whole time interval in terms of its past. The approach that we adopt is based on functional kernel nonparametric regression estimation techniques where observations are discrete recordings of segments of an underlying stochastic process considered as curves. These curves are assumed to lie within the space of continuous functions, and the discretized time series data set consists of a relatively small, compared with the number of segments, number of measurements made at regular times. We estimate conditional expectations by using appropriate wavelet decompositions of the segmented sample paths. A notion of similarity, based on wavelet decompositions, is used to calibrate the prediction. Asymptotic properties when the number of segments grows to 1 are investigated under mild conditions, and a nonparametric resampling procedure is used to generate, in a flexible way, valid asymptotic pointwise prediction intervals for the trajectories predicted. We illustrate the usefulness of the proposed functional wavelet-kernel methodology in finite sample situations by means of a simulated example and two real life data sets, and we compare the resulting predictions with those obtained by three other methods in the literature, in particular with a smoothing spline method, with an exponential smoothing procedure and with a seasonal autoregressive integrated moving average model.

Dependent Functional Data

“…There are proponents and opponents of this approach. In several applications, it has been shown that the functional approach yields superior results, see for example, Antoniadis et al 35 , a few examples are also given in Bosq 9 . But is clear that in many cases the more standard time series techniques are competitive, if not superior.…”

Detecting and Localizing Differences in Functional Time Series Dynamics: A Case Study in Molecular Biophysics

“…The covariance operator and its eigendecomposition are at the basis of the Karhunen-Loève expansion (Grenander 1981), which provides insight into the fluctuations of the random function X, and is the basis for optimal linear finite-dimensional approximations of X (through functional principal component analysis, or natural form of dependency in the data collection, such as when data are collected sequentially in time. Examples of such data include (but are not limited to) daily electricity consumption curves (Antoniadis, Paparoditis, and Sapatinas 2006), functional MRI data (Aston and Kirch 2012a), or molecular dynamics trajectories of DNA minicircles (see Section 2). In such cases, the data can be modeled as a stationary time series of functions, or stationary functional time series.…”

“…Functional data analysis (FDA; Ramsay and Silverman 2005;Ferraty and Vieu 2006;Horváth and Kokoszka 2012;Wang, Chiou, and Mueller 2016) deals with inferential situations where each data point that is best modeled as the realization of a stochastic process, understood as a random function or a random surface, such as weather data, neuroimages, electricity consumption curves, or phonetics, to name a few (e.g., Ramsay and Silverman 2002;Antoniadis, Paparoditis, and Sapatinas 2006;Aston and Kirch 2012a;Hadjipantelis et al 2015). In a typical setting, one is interested in drawing inferences on the law of a random function X ∈ L 2 ([0, 1], R) based on a sample X 1 , .…”

Scalable Clustering of Individual Electrical Curves for Profiling and Bottom-up Forecasting

“…[29] proposes for the same problem to use a predictor of a similar nature but applied to a multivariate process. Next, [30] provide an appropriate framework for stationary functional processes using the wavelet transform. The latter model is adapted and extended to the case of non-stationary functional processes ( [31]).…”