The spectral analysis of nonstationary categorical time series using local spectral envelope

Most classical methods for the spectral analysis are based on the assumption that the time series is stationary. However, many time series in practical problems shows nonstationary behaviors. The data from some fields are huge and have variance and spectrum which changes over time. Sometimes,we are interested in the cyclic behavior of the categorical-valued time series such as EEG sleep state data or DNA sequence, the general method is to scale the data, that is, assign numerical values to the categories and then use the periodogram to find the cyclic behavior. But there exists numerous possible scaling. If we arbitrarily assign the numerical values to the categories and proceed with a spectral analysis, then the results will depend on the particular assignment. We would like to find the all possible scaling that bring out all of the interesting features in the data. To overcome these problems, there have been many approaches in the spectral analysis. Our goal is to develop a statistical methodology for analyzing nonstationary categorical time series in the frequency domain. In this dissertation, the spectral envelope methodology is introduced for spectral analysis of categorical time series. This provides the general framework for the spectral analysis of the categorical time series and summarizes information from the spectrum matrix. To apply this method to nonstationary process, I used the TBAS(Tree-Based Adaptive Segmentation) and local spectral envelope based on the piecewise stationary process. In this dissertation,the TBAS(Tree-Based Adpative Segmentation) using distance function based on the Kullback-Leibler divergence was proposed to find the best segmentation

Spectral analysis for nonstationary audio

A new approach for the analysis of nonstationary signals is proposed, with a focus on audio applications. Following earlier contributions, nonstationarity is modeled via stationarity-breaking operators acting on Gaussian stationary random signals. The focus is on time warping and amplitude modulation, and an approximate maximum-likelihood approach based on suitable approximations in the wavelet transform domain is developed. This paper provides theoretical analysis of the approximations, and introduces JEFAS, a corresponding estimation algorithm. The latter is tested and validated on synthetic as well as real audio signal.Comment: IEEE/ACM Transactions on Audio, Speech and Language Processing, Institute of Electrical and Electronics Engineers, In pres

A Statistical Study of Wavelet Coherence for Stationary and Nonstationary Processes

The coherence function measures the correlation between a pair of random processes in the frequency domain. It is a well studied and understood concept, and the distributional properties of conventional coherence estimators for stationary processes have been derived and applied in a number of physical settings. In recent years the wavelet coherence measure has been used to analyse correlations between a pair of processes in the time-scale domain, typically in hypothesis testing scenarios, but it has proven resistant to analytic study with resort to simulations for statistical properties. As part of the null hypothesis being tested, such simulations invariably assume joint stationarity of the series. In this thesis two methods of calculating wavelet coherence have been developed and distributional properties of the wavelet coherence estimators have been fully derived. With the first method, in an analogous framework to multitapering, wavelet coherence is estimated using multiple orthogonal Morse wavelets. The second coherence estimator proposed uses time-domain smoothing and a single Morlet wavelet. Since both sets of wavelets are complex-valued, we consider the case of wavelet coherence calculated from discrete-time complex-valued and stationary time series. Under Gaussianity, the Goodman distribution is shown, for large samples, to be appropriate for wavelet coherence. The true wavelet coherence value is identified in terms of its frequency domain equivalent and degrees of freedom can be readily derived. The theoretical results are verified via simulations. The notion of a spectral function is considered for the nonstationary case. Particular focus is given to Priestley’s evolutionary process and a Wold-Cramér nonstationary representation where time-varying spectral functions can be clearly defined. Methods of estimating these spectra are discussed, including the continuous wavelet transform, which when performed with a Morlet wavelet and temporal smoothing is shown to bear close resemblance to Priestley’s own estimation procedure. The concept of coherence for bivariate evolutionary nonstationary processes is discussed in detail. In such situations it can be shown that the coherence function, as in the stationary case, is invariant of time. It is shown that for spectra that vary slowly in time the derived statistics of the temporally smoothed wavelet coherence estimator are appropriate. Further to this the similarities with Priestleys spectral estimator are exploited to derive distributional properties of the corresponding Priestley coherence estimator. A well known class of the evolutionary and Wold-Cramér nonstationary processes are the modulated stationary processes. Using these it is shown that bivariate processes can be constructed that exhibit coherence variation with time, frequency, and time-and-frequency. The temporally smoothed Morlet wavelet coherence estimator is applied to these processes. It is shown that accurate coherence estimates can be achieved for each type of coherence, and that the distributional properties derived under stationarity are applicable

Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition

In this paper, the adaptive chirplet decomposition combined with the Wigner-Ville transform and the empirical mode decomposition combined with the Hilbert transform are employed to process various non-stationary signals (strong ground motions and structural responses). The efficacy of these two adaptive techniques for capturing the temporal evolution of the frequency content of specific seismic signals is assessed. In this respect, two near-field and two far-field seismic accelerograms are analyzed. Further, a similar analysis is performed for records pertaining to the response of a 20-story steel frame benchmark building excited by one of the four accelerograms scaled by appropriate factors to simulate undamaged and severely damaged conditions for the structure. It is shown that the derived joint time–frequency representations of the response time histories capture quite effectively the influence of non-linearity on the variation of the effective natural frequencies of a structural system during the evolution of a seismic event; in this context, tracing the mean instantaneous frequency of records of critical structural responses is adopted. The study suggests, overall, that the aforementioned techniques are quite viable tools for detecting and monitoring damage to constructed facilities exposed to seismic excitations