Bootstrap methods for stationary functional time series

Journal of Computational and Graphical Statistics, 2009

Unlike traditional time series analysis that focuses on one long time series, in many biomedical experiments, it is common to collect multiple time series and focus on how the design covariates impact the patterns of stochastic variation over time. In this article, we propose a time-frequency functional model for a family of time series indexed by a set of covariates. This model can be used to compare groups of time series in terms of the patterns of stochastic variation and to estimate the covariate effects. We focus our development on locally stationary time series and propose the covariate-indexed locally stationary setting, which include stationary processes as special cases. We use smoothing spline ANOVA models for the time-frequency coefficients. A two-stage procedure is introduced for estimation. To reduce the computational demand, we develop an equivalent state space model to the proposed model with an efficient algorithm. We also propose a new simulation method to generate replicated time series from their design spectra. An epileptic intracranial electroencephalogram (IEEG) dataset is analyzed for illustration.

arXiv (Cornell University), 2020

We are concerned with nonparametric hypothesis testing of time series functionals. It is known that the popular autoregressive sieve bootstrap is, in general, not valid for statistics whose (asymptotic) distribution depends on moments of order higher than two, irrespective of whether the data come from a linear time series or a nonlinear one. Inspired by nonlinear system theory we circumvent this non-validity by introducing a higher-order bootstrap scheme based on the Volterra series representation of the process. In order to estimate coefficients of such a representation efficiently, we rely on the alternative formulation of Volterra operators in reproducing kernel Hilbert space. We perform polynomial kernel regression which scales linearly with the input dimensionality and is independent of the degree of nonlinearity. We illustrate the applicability of the suggested Volterra-representation-based bootstrap procedure in a simulation study where we consider strictly stationary linear and nonlinear processes.

Studies in Classification, Data Analysis, and Knowledge Organization, 2005

We introduce functional principal component techniques for the statistical analysis of a set of financial time series from an explorative point of view. We show that this approach highlights some relevant statistical features of such related datasets. A case study is here considered concerning the daily traded volumes of the shares in the MIB30 basket from January 3rd, 2000 to December 30th, 2002. Moreover, since the first functional principal component accounts for the 89.4% of the whole variabilitity, this approach suggests the construction of new financial indices based on functional indicators. Vichi M., Monari P., Mignani S., Montanari A. (Eds.)

Journal of Statistical Computation and Simulation, 2019

Functional time series whose sample elements are recorded sequentially over time are frequently encountered with increasing technology. Recent studies have shown that analyzing and forecasting of functional time series can be performed easily using functional principal component analysis and existing univariate/multivariate time series models. However, the forecasting performance of such functional time series models may be affected by the presence of outlying observations which are very common in many scientific fields. Outliers may distort the functional time series model structure, and thus, the underlying model may produce high forecast errors. We introduce a robust forecasting technique based on weighted likelihood methodology to obtain point and interval forecasts in functional time series in the presence of outliers. The finite sample performance of the proposed method is illustrated by Monte Carlo simulations and four real-data examples. Numerical results reveal that the proposed method exhibits superior performance compared with the existing method(s).

2020

Science Journal of Applied Mathematics and Statistics, 2015

The functional time series (FTS) models are used for analyzing, modeling and forecasting age-specific mortality rates. However, the application of these models in presence of two or more groups within similar populations needs some modification. In these cases, it is desirable for the disaggregated forecasts to be coherent with the overall forecast. The 'coherent' forecasts are the non-divergent forecasts of subgroups within a population. Reference [1] first proposed a coherent functional model based on product and ratios of mortality rates. In this paper, we relate some of the functional time series models to the common principal components (CPC) and partial common principal components (PCPC) models introduced by [2] and provide the methods to estimate these models. We call them common functional principal component (CFPC) models and use them for coherent mortality forecasting. Here, we propose a sequential procedure based on Johansen methodology to estimate the model parameters. We use vector approach and make use of error correction models to forecast the specific time series coefficient for each subgroup .

2016

Functional data analysis is a branch of statistics that deals with observations \(X_1,..., X_n\) which are curves. We are interested in particular in time series of dependent curves and, specifically, consider the functional autoregressive process of order one (FAR(1)), which is defined as \(X_{n+1}=\Psi(X_{n})+\epsilon_{n+1}\) with independent innovations \(\epsilon_t\). Estimates \(\hat{\Psi}\) for the autoregressive operator \(\Psi\) have been investigated a lot during the last two decades, and their asymptotic properties are well understood. Particularly difficult and different from scalar- or vector-valued autoregressions are the weak convergence properties which also form the basis of the bootstrap theory. Although the asymptotics for \(\hat{\Psi}{(X_{n})}\) are still tractable, they are only useful for large enough samples. In applications, however, frequently only small samples of data are available such that an alternative method for approximating the distribution of \(\hat...

Physica A: Statistical Mechanics and its Applications, 2007

This paper deals with different bootstrap approaches and bootstrap confidence intervals in the fractionally autoregressive moving average ðARFIMAðp; d; qÞÞ process [J. Hosking, Fractional differencing, Biometrika 68(1) (1981) 165-175] using parametric and semi-parametric estimation techniques for the memory parameter d. The bootstrap procedures considered are: the classical bootstrap in the residuals of the fitted model [B. Efron, R. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, New York, 1993], the bootstrap in the spectral density function [E. Paparoditis, D.N Politis, The local bootstrap for periodogram statistics. J. Time Ser. Anal. 20(2) (1999) 193-222], the bootstrap in the residuals resulting from the regression equation of the semi-parametric estimators [G.C Franco, V.A Reisen, Bootstrap techniques in semiparametric estimation methods for ARFIMA models: a comparison study, Comput. Statist. 19 (2004) 243-259] and the Sieve bootstrap [P. Bu¨hlmann, Sieve bootstrap for time series, Bernoulli 3 (1997) 123-148]. The performance of these procedures and confidence intervals for d in the stationary and non-stationary ranges are empirically obtained through Monte Carlo experiments. The bootstrap confidence intervals here proposed are alternative procedures with some accuracy to obtain confidence intervals for d. r

SSRN Electronic Journal, 2021

This paper proposes a new AR-sieve bootstrap approach on high-dimensional time series. The major challenge of classical bootstrap methods on high-dimensional time series is twofold: the curse dimensionality and temporal dependence. To tackle such difficulty, we utilise factor modelling to reduce dimension and capture temporal dependence simultaneously. A factor-based bootstrap procedure is constructed, which conducts AR-sieve bootstrap on the extracted low-dimensional common factor time series and then recovers the bootstrap samples for original data from the factor model. Asymptotic properties for bootstrap mean statistics and extreme eigenvalues are established. Various simulations further demonstrate the advantages of the new AR-sieve bootstrap under high-dimensional scenarios. Finally, an empirical application on particulate matter (PM) concentration data is studied, where bootstrap confidence intervals for mean vectors and autocovariance matrices are provided.

Submitted for publication, 2010

We develop time series analysis of functional data observed discretely, treating the whole curve as a random realization from a distribution on functions that evolve over time. The method consists of principal components analysis of functional data and subsequently modeling the principal component scores as vector ARMA process. We carry out the estimation of VARMA parameters using the equivalent state space representation. We derive asymptotic properties of the estimators and the fits. We apply the method to two different data sets. For term structures of interest rates, this provides a unified framework for studying the time and maturity components of interest rates under one set-up with few parametric assumptions. We compare our forecasts to the parametric model. Secondly, we apply this approach to hourly spot prices of electricity and obtain fits and forecasts that are better than those existing in the electricity literature.