Robert H. Shumway

Time Series Analysis and Its Applications

Characteristics of Time Series * Time Series Regression and ARIMA Models * Dynamic Linear Models and Kalman Filtering * Spectral Analysis and Its Applications.

Dynamic Linear Models with Switching

Abstract The problem of modeling change in a vector time series is studied using a dynamic linear model with measurement matrices that switch according to a time-varying independent random process. We derive filtered estimators for the usual state vectors and also for the state occupancy probabilities of the underlying nonstationary measurement process. A maximum likelihood estimation procedure is given that uses a pseudo-expectation-maximization algorithm in the initial stages and nonlinear optimization. We relate the models to those considered previously in the literature and give an application involving the tracking of multiple targets.

Estimating Mean Concentrations Under Transformation for Environmental Data With Detection Limits

The reporting procedures for potentially toxic pollutants are complicated by the fact that concentrations are measured using small samples that include a number of observations lying below some detection limit. Furthermore, there is often a small number of high concentrations observed in combination with a substantial number of low concentrations. This results in small, nonnormally distributed censored samples. This article presents maximum likelihood estimators for the mean of a population, based on censored samples that can be transformed to normality. The method estimates the optimal power transformation in the Box-Cox family by searching the censored-data likelihood. Maximum likelihood estimators for the mean in the transformed scale are calculated via the expectation-maximization algorithm. Estimates for the mean in the original scale are functions of the estimated mean and variance in the transformed population. Confidence intervals are computed using the delta method and the nonparametric percentil...

Time-frequency clustering and discriminant analysis

We consider the use of time-varying spectra for classification and clustering of non-stationary time series. In particular, recent developments using local stationarity and Kullback-Leibler discrimination measures of distance are exploited for classifying earthquakes and mining explosions at regional distances.

Modeling mortality fluctuations in los angeles as functions of pollution and weather effects

Linear and nonlinear models are used to investigate possible associations between mortality and pollution and weather effects in Los Angeles County. State-space modeling and time and frequency domain regressions are used to modify the data base and to isolate significant weather factors and pollutants associated with increased daily mortality. Nonparametric and parametric regression methods are used to develop nonlinear dose-response profiles relating mortality to temperature and to the statistically significant pollutants. A parametric nonlinear time series model involving linear and squared terms in temperature and the logarithm of pollution provides a reasonable predictive model.

Early Childhood Lower Respiratory Illness and Air Pollution

Background Few studies of air pollutants address morbidity in preschool children. In this study we evaluated bronchitis in children from two Czech districts: Teplice, with high ambient air pollution, and Prachatice, characterized by lower exposures. Objectives Our goal was to examine rates of lower respiratory illnesses in preschool children in relation to ambient particles and hydrocarbons. Methods Air monitoring for particulate matter < 2.5 μm in diameter (PM2.5) and polycyclic aromatic hydrocarbons (PAHs) was conducted daily, every third day, or every sixth day. Children born May 1994 through December 1998 were followed to 3 or 4.5 years of age to ascertain illness diagnoses. Mothers completed questionnaires at birth and at follow-up regarding demographic, lifestyle, reproductive, and home environmental factors. Longitudinal multivariate repeated-measures analysis was used to quantify rate ratios for bronchitis and for total lower respiratory illnesses in 1,133 children. Results After adjustment for season, temperature, and other covariates, bronchitis rates increased with rising pollutant concentrations. Below 2 years of age, increments in 30-day averages of 100 ng/m3 PAHs and of 25 μg/m3 PM2.5 resulted in rate ratios (RRs) for bronchitis of 1.29 [95 % confidence interval (CI), 1.07–1.54] and 1.30 (95% CI, 1.08–1.58), respectively; from 2 to 4.5 years of age, these RRs were 1.56 (95% CI, 1.22–2.00) and 1.23 (95% CI, 0.94–1.62), respectively. Conclusion Ambient PAHs and fine particles were associated with early-life susceptibility to bronchitis. Associations were stronger for longer pollutant-averaging periods and, among children > 2 years of age, for PAHs compared with fine particles. Preschool-age children may be particularly vulnerable to air pollution–induced illnesses.

Linear Discriminant Functions for Stationary Time Series

Abstract Certain spectral approximations are applied to the problem of discriminating between two normal processes by linear filtering. Limiting values for the (1) Kullback-Leibler discrimination information rate, (2) J-divergence rate and (3) detection probability are expressed in terms of the spectral densities of the two populations and the Fourier-Stieltjes transform of the mean difference between them. Spectral approximations to (1), (2) and (3), convenient for computing, are shown to have the same limits. Linear discriminant filters maximizing (1), (2) and (3) are approximated by the same methods and applied to seismic records from selected earthquakes and nuclear explosions.

The model selection criterion AICu

For regression and time series model selection, Hurvich and Tsai (1989) obtained a bias correction Akaike information criterion, AICc, which provides better model order choices than the Akaike information criterion, AIC (Akaike, 1973). In this paper, we propose an alternative improved regression model selection criterion, AICu, which is an approximate unbiased estimator of Kullback-Leibler information. We show that AICu is neither a consistent (Shibata, 1986) nor an efficient (Shibata, 1980, 1981) criterion. Our simulation studies indicate that the behavior of AICu is a compromise between that of efficient (AICc) and consistent (BIC, Akaike, 1978) criteria. Specifically, AICu performs better than AICc for moderate to large sample sizes except when the true model is of infinite order. In addition, it outperforms BIC except when a true model exists and the sample size is large.

Estimation and tests of hypotheses for the initial mean and covariance in the kalman filter model

Kalman filtering techniques are widely used by engineers to recursively estimate random signal parameters which are essentially coefficients in a large-scale time series regression model. These Bayesian estimators depend on the values assumed for the mean and covariance parameters associated with the initial state of the random signal. This paper considers a likelihood approach to estimation and tests of hypotheses involving the critical initial means and covariances. A computationally simple convergent iterative algorithm is used to generate estimators which depend only on standard Kalman filter outputs at each successive stage. Conditions are given under which the maximum likelihood estimators are consistent and asymptotically normal. The procedure is illustrated using a typical large-scale data set involving 10-dimensional signal vectors.

On Detecting a Signal in N Stationarily Correlated Noise Series

A useful model in the analysis of multiple time series is to assume that each series is composed of a fixed signal common to all series and a wide sense stationary normal noise process. If the noise series are uncorrelated with identical autocorrelation functions, we have in the time domain a collection of independent normal vectors with a common covariance matrix. The likelihood ratio statistic for detecting the signal is then a singular version of Hotellings T 2. In this paper, a likelihood ratio test utilizing the asymptotic properties of the finite Fourier transform leads to an asymptotically non-singular frequency domain version of Hotellings T 2. It is shown that the F statistic for detecting the signal depends only upon certain spectral power measurements with the Type I error and signal detection probabilities determined from existing tables for the central and non-central F distributions. Several examples are presented illustrating the detection procedure.