Will Gersch

A smoothness priors time-varying AR coefficient modeling of nonstationary covariance time series

A smoothness priors time varying AR coefficient model approach for the modeling of nonstationary in the covariance time series is shown. Smoothness priors in the form of a difference equation constraint excited by an independent white noise are imposed on each AR coefficient. The unknown white noise variances are hyperparameters of the AR coefficient distribution. The critical computation is of the likelihood of the hyperparameters of the Bayesian model. This computation is facilitated by a state-space representation Kalman filter implementation. The best difference equation order-best AR model order-best hyperparameter model locally in time is selected using the minimum AIC method. Also, an instantaneous spectral density is defined in terms of the instantaneous AR model coefficients and a smoothed estimate of the instantaneous time series variance. An earthquake record is analyzed. The changing spectral analysis of the original data and of simulations from a time varying AR coefficient model of that data are shown.

A Smoothness Priors–State Space Modeling of Time Series with Trend and Seasonality

Abstract A smoothness priors modeling of time series with trends and seasonalities is shown. An observed time series is decomposed into local polynomial trend, seasonal, globally stationary autoregressive and observation error components. Each component is characterized by an unknown variance–white noise perturbed difference equation constraint. The constraints or Bayesian smoothness priors are expressed in state space model form. Trading day factors are also incorporated in the model. A Kalman predictor yields the likelihood for the unknown variances (hyperparameters). Likelihoods are computed for different constraint order models in different subsets of constraint equation model classes. Akaikes minimum AIC procedure is used to select the best model fitted to the data within and between the alternative model classes. Smoothing is achieved by using a fixed-interval smoother algorithm. Examples are shown.

The Prediction of Time Series With Trends and Seasonalities

The modeling and prediction of time series with trend and seasonal mean value functions and stationary covariances is approached from a maximization of the expected entropy of the predictive distribution interpretation of Akaikes minimum AIC procedure. The AIC criterion best one-step-ahead and best twelvestep-ahead prediction models are different. They exhibit the relative optimality properties for which they were designed. The results are related to open questions on optimal trend estimation and optimal seasonal adjustment of time series.

Estimation of power spectra with finite-order autoregressive models

Results of an empirical study of the application of Akaikes final predictor error (FPE) criterion to the estimation of the order of finite autoregressive models to infinite autoregressive model scheme data and the subsequent application of those models to spectral estimation are given.

A smoothness priors long AR model method for spectral estimation

A new smoothness priors long AR model method approach is taken to the short data span spectral estimation problem. An autoregressive (AR) model that is relatively long compared to the data length is considered. The smoothness priors are in the form of the integrated squared derivatives of the AR model whitening filter. A smoothness tradeoff parameter or Bayesian hyperparameter balances the tradeoff between the infidelity of the AR model to the data and the infidelity of the model to the smoothness constraint. The critical computation of the likelihood of the hyperparameters of the Bayesian model is realized by a constrained least squares computation. Numerical examples are shown. The results of simulation studies using entropy comparison evaluations of the Bayesian and minimum AIC-AR methods of spectral estimation are also shown.

Parametric time series models for multivariate EEG analysis

Abstract The autoregressive (AR) and mixed autoregressive-moving average (AR-MA) parametric models of stationary time series are of current interest for the purposes of spectral analysis and for the extraction of features for automatic EEG classification. Procedures for computing AR models and a new two-stage least-squares procedure for computing AR-MA models of multivariate time series are shown. The results of spectral estimation of EEGs using the multivariate AR, AR-MA, and conventional windowed periodogram analysis are compared.

Causality or driving in electrophysiological signal analysis

Abstract A single time series is defined to be causal relative to other stationary time series if it uniquely explains the pair-wise linear relationship between other time series. This definition lends itself to explicit statistical tests, to multivariate extensions of the definition of causality, to the case in which a linear combination of time series may be causal relative to other time series, and to the physical plausibility arguments that we think are essential in order to conclude that causality is present in time series. More specifically, consider three simultaneous time series {itxt}, {ityt} and {itzt}. Assume that (i) {itxt}, {ityt} and {itzt} have a “significant” amount of energy over a common physically relevant frequency interval and that over this frequency interval there is significant spectral coherence between each pair of time series. Roughly, the spectral coherence is required to be substantially different than zero over the frequency interval of interest and not be spurious or due to other nonmeasured variates. We say that one time series, say {itzt}, is causal relative to {itxt} and {ityt} if (ii) the partial spectral coherence between {itxt} and {ityt} conditioned {itzt} is zero over this frequency interval and the partial coherences between {itxt} and {itzt} conditioned on {ityt} and between {ityt} and {itzt} conditioned on {itxt} are not zero over this same interval. If the unconditional coherence between two time series is nonzero and the coherence conditioned on a third time series is zero, then formally the third time series accounts for the linear relationship between the first two time series. Illustrative examples and practical applications are demonstrated.

Least squares estimates of structural system parameters using covariance function data

A statistically efficient and computationally economical two-stage least squares procedure for the estimation of the natural frequencies and damping parameters of structural systems under stationary random vibration conditions is considered. The structural system is represented by the system of ordinary differential equations that is characteristic of lumped mass-spring-damper systems with a random forcing function. Emphasis is placed on the problem corresponding to the observation of the top story vibrations of a tall building under random wind excitation. In that case, the random excitation can be approximated by a white noise and the regularly sampled vibration record can be represented as a mixed autoregressive-moving average (ARMA) time series. The ARMA time series parameters are estimated by a two-stage least squares method using only the covariance function of the top story vibrations. The natural frequency and damping parameters of the structural system can be expressed in terms of the AR parameters. Estimates of the coefficient of variation of the structural system parameter estimates are expressed in terms of the ARMA parameter estimates. The numerical results of the least squares and maximum likelihood parameter estimation procedures worked on a real vibration data example are shown.

Automatic classification of multivariate EEGs using an amount of information measure and the eigenvalues of parametric time series model features

Abstract Two new classes of features are introduced for the automatic classification of multichannel stationary time series EEG data. The features are the Shannon-Gelfand-Yaglom measure of the amount of information between two sets of stationary Gaussian time series and the eigenvalues computed from a parametric model of the time series. The performance of these features for automatic sleep stage scoring from two EEG data channels, evaluated using the multinomial logistic function, is presented as an example. This parametric modeled EEG time series-two features for classification approach is a radical departure from the more conventional windowed periodogram spectral analysis-discriminant analysis packaged computer program approach.

NEAREST NEIGHBOR RULE CLASSIFICATION OF STATIONARY AND NONSTATIONARY TIME SERIES

A unified methodological nearest neighbor rule approach is introduced for the classification of stationary and nonstationary time series. In that approach a measure of dissimilarity is computed between a new to-be-classified time series and each of a set of categorically labeled sample time series. The new time series is classified with the label of the particular time series which is least dissimilar. The dissimilarity measure between the time series is computed as an estimate of the Kullback-Leibler number between the corresponding time series populations as if the time series were normally distributed. The population screening classification problem and the normalized baseline classification problems are distinguished. The probability of misclassification properties of the nearest neighbor rule method are developed for each variety of classication problem. The methodology is applied in depth to population screening problems including the classification of anesthesia levels of humans in surgery by the analysis of stationary time series electroencephalograms and a study of the information for discrimination in human evoked potentials that is based upon the analysis of ensembles of time series with mean value time functions and nonstationary covariances. The emphasis in the normalized baseline classification problem examples is in the automatic classification of faults in rotating machinery in which the time series may be considered to be stationary, locally stationary or have a mean value time function and a stationary covariance.