CS003 Time Series and Functional Data Analysis

In time series analysis, we may analyze the behavior of a multivariate time series in time domain using the autocovariance matrices or in frequency domain using the spectral density matrices. Today spectral analysis of time series is of considerable interest in a variety of fields. Many natural phenomena have frequency-dependent variability quantified through the power spectrum, which contains important information on the underlying physical mechanisms. The multivariate power spectrum is a positive-definite and Hermitian matrix. So due to these constraints, estimating the spectrum of a multivariate time series is more complicated than the univariate one. There are many nonparametric methods for estimating the spectrum of a multivariate time series.
In this paper, we developed an efficient nonparametric method for estimating the multivariate spectral density using basis expansions. Similar to Dai and Guo (2004), Rosen and Stoffer (2007), and Krafty and Collinge (2013), we applied a Cholesky decomposition to the power spectrum to ensure Hermitian positive definiteness, where the real parts and imaginary parts of the Cholesky elements are modeled by basis expansions. The basis coefficients were estimated by minimizing the penalized Whittle likelihood approximation with two different penalties, based on the second derivative of the basis functions and the difference operator. The penalty parameter is selected by minimizing the well-known Akaike information criterion (AIC). To minimize the penalized Whittle likelihood, we applied a Fisher-Scoring algorithm.
In a simulation study, we illustrated our proposed algorithm (NMSDE) using some simulated data and compared its performance with some other alternative approaches. According to the simulation results, our approach is competitive with the method proposed by Krafty and Collinge (2013) which is the state of the art in the Cholesky-based methods. Finally, we applied our approach to a real-world problem, Elnino cycle. 

Time series on the unit n-sphere arise in many areas, from ecology, to astronomy, to genetics. There are relatively few models for such data, and the ones that exist suffer from several limitations; they are based on insufficiently flexible distributions, they are often difficult to fit, and many of them apply only to the circular case of n=2. We propose a state space model based on the projected normal distribution that can be applied to spherical time series of arbitrary dimension. We describe how to perform fully Bayesian offline inference for this model using an efficient Gibbs sampling algorithm, and we also describe how to perform online inference for streaming data using a Rao-Blackwellized particle filter. In an analysis of wind direction time series, we show that the proposed model can outperform competing models in terms of point, interval, and density forecasting. 

In this paper we consider aggregated functional data composed by a linear combination of component curves and the problem of estimating these component curves. We propose the application of a bayesian wavelet shrinkage rule based on a mixture of a point mass function at zero and the logistic distribution as prior to wavelet coefficients to estimate mean curves of components. This procedure has the advantage of estimating component functions with important local characteristics such as discontinuities, spikes and oscillations for example, due the features of wavelet basis expansion of functions. Simulation studies were done to evaluate the performance of the proposed method and its results are compared with a spline-based method. An application on the so called tecator dataset is also provided.