: One use of spectral analysis of time series is to determine if two different time series are realizations from the same process This thesis develops the theory behind Krishnaiah and Schuurmann's theoretical work reported in their report Approximations to the Distributions of the Traces of Complex Multivariate Beta and F Matrices We take the trace of a test statistic calculated from the spectral density matrices of the time series and test it The thesis applies the theory to two small sample simulations (Author)

Spectral Analysis of Time Series.

Functional Data Analysis With R And Matlab

This paper studies Quasi Maximum Likelihood estimation of dynamic factor models for large panels of time series. Specifically, we consider the case in which the autocorrelation of the factors is explicitly accounted for and therefore the model has a state-space form. Estimation of the factors and of their loadings is implemented by means of the Expectation Maximization (EM) algorithm, jointly with the Kalman smoother.~We prove that, as both the dimension of the panel $n$ and the sample size $T$ diverge to infinity: (i) the estimated loadings are $\sqrt T$-consistent and asymptotically normal if $\sqrt T/n\to 0$; (ii) the estimated factors are $\sqrt n$-consistent and asymptotically normal if $\sqrt n/T\to 0$; (iii) the estimated common component is $\min(\sqrt T,\sqrt n)$-consistent and asymptotically normal regardless of the relative rate of divergence of $n$ and $T$. Although the model is estimated as if the idiosyncratic terms were cross-sectionally and serially uncorrelated, we show that these mis-specifications do not affect consistency. Moreover, the estimated loadings are asymptotically as efficient as those obtained with the Principal Components estimator, whereas numerical results show that the loss in efficiency of the estimated factors becomes negligible as $n$ and $T$ increase.~We then propose robust estimators of the asymptotic covariances, which can be used to conduct inference on the loadings and to compute confidence intervals for the factors and common components. In a MonteCarlo simulation exercise and an analysis of US macroeconomic data, we study the performance of our estimators and we compare them with the traditional Principal Components approach.

Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM algorithm

We develop tests of normality for time series of functions. The tests are related to the commonly used Jarqueâ€“Bera test. The assumption of normality has played an important role in many methodological and theoretical developments in the field of functional data analysis. Yet, no inferential procedures to verify it have been proposed so far, even for i.i.d. functions. We propose several approaches which handle two paramount challenges: (i) the unknown temporal dependence structure and (ii) the estimation of the optimal finiteâ€ dimensional projection space. We evaluate the tests via simulations and establish their large sample validity under general conditions. We obtain useful insights by applying them to pollution and intraday price curves. While the pollution curves can be treated as normal, the normality of highâ€ frequency price curves is rejected.

Testing Normality of Functional Time Series

The paper develops inferential methodology for detecting a change in the annual pattern of an environmental variable measured at fixed locations in a spatial region. Using a framework built on functional data analysis, we model observations as a collection of function‐valued time sequences available at many sites. Each sequence is modelled as an annual mean function, which may change, plus a sequence of error functions, which are spatially correlated. The tests statistics extend the cumulative sum paradigm to this more complex setting. Their asymptotic distributions are not parameter free because of the spatial dependence but can be effectively approximated by Monte Carlo simulations. The new methodology is applied to precipitation data. Its finite sample performance is assessed by a simulation study.

Detection of change in the spatiotemporal mean function Series B Statistical methodology

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 weakly dependent contaminating process. 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. Generally, for the functional series that motivate this research, the fully functional tests exhibit a superior balance of size and power. Asymptotic null distributions of all tests are derived and their consistency is established. Their finite sample performance is examined and compared by numerical studies and application to pollution data.

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Testing for periodicity in functional time series

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 superior balance of size and power. Asymptotic null distributions of all tests are derived and their consistency is established. Their finite sample performance is examined and compared by numerical studies and application to pollution data.

Detection of periodicity in functional time series

Prediction of singular VARs and an application to generalized dynamic factor models

This article is the second one in a set of two laying the theoretical foundations for a high-dimensional functional factor model approach in the analysis of large cross-sections (panels) of functional time series (FTS). Part I establishes a representation result by which, under mild assumptions on the covariance operator of the cross-section, any FTS admits a canonical representation as the sum of a common and an idiosyncratic component; common components are driven by a finite and typically small number of scalar factors loaded via functional loadings, while idiosyncratic components are only mildly cross-correlated. Building on that representation result, Part II is dealing with the identification of the number of factors, their estimation, the estimation of their loadings and the common components, and the resulting forecasts. We provide a family of information criteria for identifying the number of factors, and prove their consistency. We provide average error bounds for the estimators of the factors, loadings, and common components; our results encompass the scalar case, for which they reproduce and extend, under weaker conditions, well-established similar results. Under slightly stronger assumptions, we also provide uniform bounds for the estimators of factors, loadings, and common components, thus extending existing scalar results. Our consistency results in the asymptotic regime where the number N of series and the number T of time points diverge thus extend to the functional context the ‘blessing of dimensionality’ that explains the success of factor models in the analysis of high-dimensional (scalar) time series. We provide numerical illustrations that corroborate the convergence rates predicted by the theory, and provide a finer understanding of the interplay between N and T for estimation purposes. We conclude with an application to forecasting mortality curves, where our approach outperforms existing methods. 

Gilles Nisol

Papers

Testing for periodicity in functional time series

Detection of periodicity in functional time series

Prediction of singular VARs and an application to generalized dynamic factor models

Factor Models for High‐Dimensional Functional Time Series II : Estimation and Forecasting