The frequency-domain properties of nonstationary functional time series often contain valuable information. These properties are characterized through its time-varying power spectrum. Practitioners seeking low-dimensional summary measures of the power spectrum often partition frequencies into bands and create collapsed measures of power within bands. However, standard frequency bands have largely been developed through manual inspection of time series data and may not adequately summarize power spectra. In this article, we propose a framework for adaptive frequency band estimation of nonstationary functional time series that optimally summarizes the time-varying dynamics of the series. We develop a scan statistic and search algorithm to detect changes in the frequency domain. We establish theoretical properties of this framework and develop a computationally-efficient implementation. The validity of our method is also justified through numerous simulation studies and an application to analyzing electroencephalogram data in participants alternating between eyes open and eyes closed conditions.
The Portmanteau test provides the vanilla method for detecting serial correlations in classical univariate time series analysis. The method is extended to the case of observations from a locally stationary functional time series. Asymptotic critical values are obtained by a suitable block multiplier bootstrap procedure. The test is shown to asymptotically hold its level and to be consistent against general alternatives.
An approach is presented for making predictions about functional time series. The method is applied to data coming from periodically correlated processes and electricity demand, obtaining accurate point forecasts and narrow prediction bands that cover high proportions of the forecasted functional datum, for a given confidence level. The method is computationally efficient and substantially different to other functional time series methods, offering a new insight for the analysis of these data structures.
Due to the surge of data storage techniques, the need for the development of appropriate techniques to identify patterns and to extract knowledge from the resulting enormous data sets, which can be viewed as collections of dependent functional data, is of increasing interest in many scientific areas. We develop a similarity measure for spectral density operators of a collection of functional time series, which is based on the aggregation of Hilbert-Schmidt differences of the individual time-varying spectral density operators. Under fairly general conditions, the asymptotic properties of the co…
We study the importance of group structure in grouped functional time series. Due to the non-uniqueness of group structure, we investigate different disaggregation structures in grouped functional time series. We address a practical question on whether or not the group structure can affect forecast accuracy. Using a dynamic multivariate functional time series method, we consider joint modeling and forecasting multiple series. Illustrated by Japanese sub-national age-specific mortality rates from 1975 to 2016, we investigate one- to 15-step-ahead point and interval forecast accuracies for the t…
A bootstrap procedure for functional time series is proposed which exploits a general vector autoregressive representation of the time series of Fourier coefficients appearing in the Karhunen-Loève expansion of the functional process. A double sieve-type bootstrap method is developed which avoids the estimation of process operators and generates functional pseudo-time series that appropriately mimic the dependence structure of the functional time series at hand. The method uses a finite set of functional principal components to capture the essential driving parts of the infinite dimensional pr…
Statistical analysis of high-dimensional functional times series arises in various applications. Under this scenario, in addition to the intrinsic infinite-dimensionality of functional data, the number of functional variables can grow with the number of serially dependent observations. In this paper, we focus on the theoretical analysis of relevant estimated cross-(auto)covariance terms between two multivariate functional time series or a mixture of multivariate functional and scalar time series beyond the Gaussianity assumption. We introduce a new perspective on dependence by proposing functi…
Functional times series have become an integral part of both functional data and time series analysis. This paper deals with the functional autoregressive model of order 1 and the autoregression bootstrap for smooth functions. The regression operator is estimated in the framework developed by Ferraty and Vieu [2004] and Ferraty et al. [2007] which is here extended to the double functional case under an assumption of stationary ergodic data which dates back to Laib and Louani [2010]. The main result of this article is the characterization of the asymptotic consistency of the bootstrapped regres…
Forecasting the evolution of complex systems is one of the grand challenges of modern data science. The fundamental difficulty lies in understanding the structure of the observed stochastic process. In this paper, we show that every uniformly-positive-definite-in-covariance and sufficiently short-range dependent non-stationary and nonlinear time series can be well approximated globally by an auto-regressive process of slowly diverging order. When linear prediction with ${\cal L}^2$ loss is concerned, the latter result facilitates a unified globally-optimal short-term forecasting theory for a w…
We propose a new autocorrelation measure for functional time series that we term spherical autocorrelation. It is based on measuring the average angle between lagged pairs of series after having been projected onto the unit sphere. This new measure enjoys several complimentary advantages compared to existing autocorrelation measures for functional data, since it both 1) describes a notion of sign or direction of serial dependence in the series, and 2) is more robust to outliers. The asymptotic properties of estimators of the spherical autocorrelation are established, and are used to construct …
The paper offers a unified approach to the study of three locally adaptive estimation methods in the context of univariate time series from both theoretical and empirical points of view. A general procedure for the computation of critical values is given. The underlying model encompasses all distributions from the exponential family providing for great flexibility. The procedures are applied to simulated and real financial data distributed according to the Gaussian, volatility, Poisson, exponential and Bernoulli models. Numerical results exhibit a very reasonable performance of the methods.
Here, we address the problem of trend estimation for functional time series. Existing contributions either deal with detecting a functional trend or assuming a simple model. They consider neither the estimation of a general functional trend nor the analysis of functional time series with a functional trend component. Similarly to univariate time series, we propose an alternative methodology to analyze functional time series, taking into account a functional trend component. We propose to estimate the functional trend by using a tensor product surface that is easy to implement, to interpret, an…
In this paper we develop statistical inference tools for high dimensional functional time series. We introduce a new concept of physical dependent processes in the space of square integrable functions, which adopts the idea of basis decomposition of functional data in these spaces, and derive Gaussian and multiplier bootstrap approximations for sums of high dimensional functional time series. These results have numerous important statistical consequences. Exemplarily, we consider the development of joint simultaneous confidence bands for the mean functions and the construction of tests for the…
Frequently econometricians are interested in verifying a relationship between two or more time series. Such analysis is typically carried out by causality and/or independence tests which have been well studied when the data is univariate or multivariate. Modern data though is increasingly of a high dimensional or functional nature for which finite dimensional methods are not suitable. In the present paper we develop methodology to check the assumption that data obtained from two functional time series are independent. Our procedure is based on the norms of empirical cross covariance operators …
Cluster indices describe extremal behaviour of stationary time series. We consider their sliding blocks estimators. Using a modern theory of multivariate, regularly varying time series, we obtain central limit theorems under conditions that can be easily verified for a large class of models. In particular, we show that in the Peak over Threshold framework, sliding and disjoint blocks estimators have the same limiting variance.
We develop methodology allowing to simulate a stationary functional time series defined by means of its spectral density operators. Our framework is general, in that it encompasses any such stationary functional time series, whether linear or not. The methodology manifests particularly significant computational gains if the spectral density operators are specified by means of their eigendecomposition or as a filtering of white noise. In the special case of linear processes, we determine the analytical expressions for the spectral density operators of functional autoregressive (fractionally int…
Bootstrap methods for estimating the long-run covariance of stationary functional time series are considered. We introduce a versatile bootstrap method that relies on functional principal component analysis, where principal component scores can be bootstrapped by maximum entropy. Two other bootstrap methods resample error functions, after the dependence structure being modeled linearly by a sieve method or nonlinearly by a functional kernel regression. Through a series of Monte-Carlo simulation, we evaluate and compare the finite-sample performances of these three bootstrap methods for estimat…
Long Range Dependence (LRD) in functional sequences is characterized in the spectral domain under suitable conditions. Particularly, multifractionally integrated functional autoregressive moving averages processes can be introduced in this framework. The convergence to zero in the Hilbert-Schmidt operator norm of the integrated bias of the periodogram operator is proved. Under a Gaussian scenario, a weak--consistent parametric estimator of the long--memory operator is then obtained by minimizing, in the norm of bounded linear operators, a divergence information functional loss.
For testing hypothesis on the covariance operator of functional time series, we suggest to use the full functional information and to avoid dimension reduction techniques. The limit distribution follows from the central limit theorem of the weak convergence of the partial sum process in general Hilbert space applied to the product space. In order to obtain critical values for tests, we generalize bootstrap results from the independent to the dependent case. This results can be applied to covariance operators, autocovariance operators and cross covariance operators. We discuss one sample and ch…
We introduce a novel functional time series methodology for short-term load forecasting. The prediction is performed by means of a weighted average of past daily load segments, the shape of which is similar to the expected shape of the load segment to be predicted. The past load segments are identified from the available history of the observed load segments by means of their closeness to a so-called reference load segment, the later being selected in a manner that captures the expected qualitative and quantitative characteristics of the load segment to be predicted. Weak consistency of the su…
Interest in functional time series has spiked in the recent past with papers covering both methodology and applications being published at a much increased pace. This article contributes to the research in this area by proposing a new stationarity test for functional time series based on frequency domain methods. The proposed test statistics is based on joint dimension reduction via functional principal components analysis across the spectral density operators at all Fourier frequencies, explicitly allowing for frequency-dependent levels of truncation to adapt to the dynamics of the underlying…
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…
This paper presents static and dynamic versions of univariate, multivariate, and multilevel functional time-series methods to forecast implied volatility surfaces in foreign exchange markets. We find that dynamic functional principal component analysis generally improves out-of-sample forecast accuracy. More specifically, the dynamic univariate functional time-series method shows the greatest improvement. Our models lead to multiple instances of statistically significant improvements in forecast accuracy for daily EUR-USD, EUR-GBP, and EUR-JPY implied volatility surfaces across various maturit…
We introduce three adaptive time series learning methods, called Dynamic Model Selection (DMS), Adaptive Ensemble (AE), and Dynamic Asset Allocation (DAA). The methods respectively handle model selection, ensembling, and contextual evaluation in financial time series. Empirically, we use the methods to forecast the returns of four key indices in the US market, incorporating information from the VIX and Yield curves. We present financial applications of the learning results, including fully-automated portfolios and dynamic hedging strategies. The strategies strongly outperform long-only benchma…
This paper is concerned with inference based on the mean function of a functional time series, which is defined as a collection of curves obtained by splitting a continuous time record, e.g. into daily or annual curves. We develop a normal approximation for the functional sample mean, and then focus on the estimation of the asymptotic variance kernel. Using these results, we develop and asymptotically justify a testing procedure for the equality of means in two functional samples exhibiting temporal dependence. Evaluated by means of a simulations study and application to real data sets, this t…
Functional time series have become an integral part of both functional data and time series analysis. Important contributions to methodology, theory and application for the prediction of future trajectories and the estimation of functional time series parameters have been made in the recent past. This paper continues this line of research by proposing a first principled approach to estimate invertible functional time series by fitting functional moving average processes. The idea is to estimate the coefficient operators in a functional linear filter. To do this a functional Innovations Algorit…
Cluster indices describe extremal behaviour of stationary time series. We consider runs estimators of cluster indices. Using a modern theory of multivariate, regularly varying time series, we obtain central limit theorems under conditions that can be easily verified for a large class of models. In particular, we show that blocks and runs estimators have the same limiting variance.
When generating social policies and pricing annuity at national and subnational levels, it is essential both to forecast mortality accurately and ensure that forecasts at the subnational level add up to the forecasts at the national level. This has motivated recent developments in forecasting functional time series in a group structure, where static principal component analysis is used. In the presence of moderate to strong temporal dependence, static principal component analysis designed for independent and identically distributed functional data may be inadequate. Thus, through using the dyn…
This paper reviews the main estimation and prediction results derived in the context of functional time series, when Hilbert and Banach spaces are considered, specially, in the context of autoregressive processes of order one (ARH(1) and ARB(1) processes, for H and B being a Hilbert and Banach space, respectively). Particularly, we pay attention to the estimation and prediction results, and statistical tests, derived in both parametric and non-parametric frameworks. A comparative study between different ARH(1) prediction approaches is developed in the simulation study undertaken.
We consider a strictly stationary and ergodic sequence of random elements taking values in some Hilbert space. Our target is to study the weak convergence of the discrete Fourier transforms under sharp conditions. As a side-result we obtain the regular CLT for partial sums under mild assumptions.
This article introduces a nonparametric approach to spectral analysis of a high-dimensional multivariate nonstationary time series. The procedure is based on a novel frequency-domain factor model that provides a flexible yet parsimonious representation of spectral matrices from a large number of simultaneously observed time series. Real and imaginary parts of the factor loading matrices are modeled independently using a prior that is formulated from the tensor product of penalized splines and multiplicative gamma process shrinkage priors, allowing for infinitely many factors with loadings incr…
We study distributional properties of a quadratic form of a stationary functional time series under mild moment conditions. As an important application, we obtain consistency rates of estimators of spectral density operators and prove joint weak convergence to a vector of complex Gaussian random operators. Weak convergence is established based on an approximation of the form via transforms of Hilbert-valued martingale difference sequences. As a side-result, the distributional properties of the long-run covariance operator are established.
We address the problem of forecasting high-dimensional functional time series through a two-fold dimension reduction procedure. The difficulty of forecasting high-dimensional functional time series lies in the curse of dimensionality. In this paper, we propose a novel method to solve this problem. Dynamic functional principal component analysis is first applied to reduce each functional time series to a vector. We then use the factor model as a further dimension reduction technique so that only a small number of latent factors are preserved. Classic time series models can be used to forecast t…
Detecting structural changes in functional data is a prominent topic in statistical literature. However not all trends in the data are important in applications, but only those of large enough influence. In this paper we address the problem of identifying relevant changes in the eigenfunctions and eigenvalues of covariance kernels of $L^2[0,1]$-valued time series. By self-normalization techniques we derive pivotal, asymptotically consistent tests for relevant changes in these characteristics of the second order structure and investigate their finite sample properties in a simulation study. The…
We give a new large deviation inequality for sums of random variables of the form $Z_k = f(X_k,X_t)$ for $k,t\in \mathbb{N}$, $t$ fixed, where the underlying process $X$ is $β$-mixing. The inequality can be used to derive concentration inequalities. We demonstrate its usefulness in the functional kernel regression model of Ferraty et al. (2007) where we study the consistency of dynamic forecasts.
We derive several tests for the presence of a periodic component in a time series of functions. We consider both the traditional setting in which the periodic functional signal is contaminated by functional white noise, and a more general setting of a contaminating process which is weakly dependent. Several forms of the periodic component are considered. Our tests are motivated by the likelihood principle and fall into two broad categories, which we term multivariate and fully functional. Overall, for the functional series that motivate this research, the fully functional tests exhibit a super…
We propose a method for adaptive nonlinear sequential modeling of vector-time series data. Data is modeled as a nonlinear function of past values corrupted by noise, and the underlying non-linear function is assumed to be approximately expandable in a spline basis. We cast the modeling of data as finding a good fit representation in the linear span of multi-dimensional spline basis, and use a variant of l1-penalty regularization in order to reduce the dimensionality of representation. Using adaptive filtering techniques, we design our online algorithm to automatically tune the underlying param…
In this paper we introduce a micro-clustering strategy for Functional Boxplots. The aim is to summarize a set of streaming time series splitted in non overlapping windows. It is a two step strategy which performs at first, an on-line summarization by means of functional data structures, named Functional Boxplot micro-clusters; then it reveals the final summarization by processing, off-line, the functional data structures. Our main contribute consists in providing a new definition of micro-cluster based on Functional Boxplots and, in defining a proximity measure which allows to compare and upda…
In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in [8, 7]; more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. …
We consider $α$-mixing observations and deal with the estimation of the conditional mode of a scalar response variable $Y$ given a random variable $X$ taking values in a semi-metric space. We provide a convergence rate in $L^p$ norm of the estimator. A useful and typical application to functional times series prediction is given.
Univariate time series often take the form of a collection of curves observed sequentially over time. Examples of these include hourly ground-level ozone concentration curves. These curves can be viewed as a time series of functions observed at equally spaced intervals over a dense grid. Since functional time series may contain various types of outliers, we introduce a robust functional time series forecasting method to down-weigh the influence of outliers in forecasting. Through a robust principal component analysis based on projection pursuit, a time series of functions can be decomposed int…
