Testing Periodicity in Short Series and Application to Gene Expression Data

Testing for periodicity in microarray time series encounters the challenges of short series length, missing values and presence of non-Fourier frequencies. In this article, a test method for such series has been proposed. The method is completely simulation based and finds p-values for test of periodicity through fitting Pearson Type VI distribution. The simulation results compare and reveal the excellence of this method over Fisher's g test for varying series length, frequencies, and error variance. This approach is applied to Caulobacter crescentus cell cycle data in order to demonstrate the practical performance.     

Least squates estimation of missing values in time series

The minimum mean square error linear interpolator for missing values in time series is extended to handle any pattern of nonconsecutive observations. The paper then develops evidence with simple ARMA models that the usefulness of either the"nonparametric"or the parametric form of the least squares interpolator depends on the time series model, the arrangement of the missing data and the objective for completing the series.     

A Note on the Estimation of Missing Values in Time Series

This paper derives an expression for the likelihood function of the parameters in an autoregressive-moving average model when some values are missing from the observed time series. The estimation of the missing values and their mean squared errors is discussed. Stationary as well as nonstationary models are considered.     

Modeling Interval Time Series with Space–Time Processes

We consider interval-valued time series, that is, series resulting from collecting real intervals as an ordered sequence through time. Since the lower and upper bounds of the observed intervals at each time point are in fact values of the same variable, they are naturally related. We propose modeling interval time series with space–time autoregressive models and, based on the process appropriate for the interval bounds, we derive the model for the intervals’ center and radius. A simulation study and an application with data of daily wind speed at different meteorological stations in Ireland illustrate that the proposed approach is appropriate and useful.     

Some Properties and Generalizations of Non-negative Bayesian Time Series Models

We study the most basic Bayesian forecasting model for exponential family time series, the power steady model (PSM) of Smith, in terms of observable properties of one-step forecast distributions and sample paths. The PSM implies a constraint between location and spread of the forecast distribution. Including a scale parameter in the models does not always give an exact solution free of this problem, but it does suggest how to define related models free of the constraint. We define such a class of models which contains the PSM. We concentrate on the case where observations are non-negative. Probability theory and simulation show that under very mild conditions almost all sample paths of these models converge to some constant, making them unsuitable for modelling in many situations. The results apply more generally to non-negative models defined in terms of exponentially weighted moving averages. We use these and related results to motivate, define and apply very simple models based on directly specifying the forecast distributions.     

Priors and component structures in autoregressive time series models

New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.     

Censored time series analysis with autoregressive moving average models

The authors consider time series observations with data irregularities such as censoring due to a detection limit. Practitioners commonly disregard censored data cases which often result in biased estimates. The authors present an attractive remedy for handling autocorrelated censored data based on a class of autoregressive and moving average (ARMA) models. In particular, they introduce an imputation method well suited for fitting ARMA models in the presence of censored data. They demonstrate the effectiveness of their technique in terms of bias, efficiency, and information loss. They also describe its adaptation to a specific context of meteorological time series data on cloud ceiling height, which are measured subject to the detection limit of the recording device.     

Indirect inference for fractional time series models

In this paper we present an indirect estimation procedure for (ARFIMA) fractional time series models.The estimation method is based on an ‘incorrect’criterion which does not directly provide a consistent estimator of the parameters of interest,but leads to correct inference by using simulations.

The main steps are the following. First,we consider an auxiliary model which can be easily estimated.Specifically,we choose the finite lag Autoregressive model.Then, this is estimated on the observations and simulated values drawn from the ARFIMA model associated with a given value of the parameters of interest.Finally,the latter is calibrated in order to obtain close values of the two estimators of the auxiliary parameters.

In this article,we describe the estimation procedure and compare the performance of the indirect estimator with some alternative estimators based on the likelihood function by a Monte Carlo study.     

Nonstationary nonlinear quantile regression

This study examines estimation and inference based on quantile regression for parametric nonlinear models with an integrated time series covariate. We first derive the limiting distribution of the nonlinear quantile regression estimator and then consider testing for parameter restrictions, when the regression function is specified as an asymptotically homogeneous function. We also study linear-in-parameter regression models when the regression function is given by integrable regression functions as well as asymptotically homogeneous regression functions. We, furthermore, propose a fully modified estimator to reduce the bias in the original estimator under a certain set of conditions. Finally, simulation studies show that the estimators behave well, especially when the regression error term has a fat-tailed distribution.     

Structural changes in autoregressive models for binary time series

We study autoregressive models for binary time series with possible changes in their parameters. A procedure for detection and testing of a single change is suggested. The limiting behavior of the test statistic is derived. The performance of the test is analyzed under the null hypothesis as well as under different alternatives via a simulation study. Application of the method to a real data set on US recession is provided as an illustration.     

A Bayesian model for measurement and misclassification errors alongside missing data,with an application to higher education participation in Australia

In this paper we consider the impact of both missing data and measurement errors on a longitudinal analysis of participation in higher education in Australia. We develop a general method for handling both discrete and continuous measurement errors that also allows for the incorporation of missing values and random effects in both binary and continuous response multilevel models. Measurement errors are allowed to be mutually dependent and their distribution may depend on further covariates. We show that our methodology works via two simple simulation studies. We then consider the impact of our measurement error assumptions on the analysis of the real data set.     

Forecasting functional time series using weighted likelihood methodology

Functional time series whose sample elements are recorded sequentially over time are frequently encountered with increasing technology. Recent studies have shown that analyzing and forecasting of functional time series can be performed easily using functional principal component analysis and existing univariate/multivariate time series models. However, the forecasting performance of such functional time series models may be affected by the presence of outlying observations which are very common in many scientific fields. Outliers may distort the functional time series model structure, and thus, the underlying model may produce high forecast errors. We introduce a robust forecasting technique based on weighted likelihood methodology to obtain point and interval forecasts in functional time series in the presence of outliers. The finite sample performance of the proposed method is illustrated by Monte Carlo simulations and four real-data examples. Numerical results reveal that the proposed method exhibits superior performance compared with the existing method(s).     

A bivariate first-order signed integer-valued autoregressive process

Bivariate integer-valued time series occur in many areas, such as finance, epidemiology, business etc. In this article, we present bivariate autoregressive integer-valued time-series models, based on the signed thinning operator. Compared to classical bivariate INAR models, the new processes have the advantage to allow for negative values for both the time series and the autocorrelation functions. Strict stationarity and ergodicity of the processes are established. The moments and the autocovariance functions are determined. The conditional least squares estimator of the model parameters is considered and the asymptotic properties of the obtained estimators are derived. An analysis of a real dataset from finance and a simulation study are carried out to assess the performance of the model.     

Statistical methods for regular monitoring data

Summary.  Meteorological and environmental data that are collected at regular time intervals on a fixed monitoring network can be usefully studied combining ideas from multiple time series and spatial statistics, particularly when there are little or no missing data. This work investigates methods for modelling such data and ways of approximating the associated likelihood functions. Models for processes on the sphere crossed with time are emphasized, especially models that are not fully symmetric in space–time. Two approaches to obtaining such models are described. The first is to consider a rotated version of fully symmetric models for which we have explicit expressions for the covariance function. The second is based on a representation of space–time covariance functions that is spectral in just the time domain and is shown to lead to natural partially nonparametric asymmetric models on the sphere crossed with time. Various models are applied to a data set of daily winds at 11 sites in Ireland over 18 years. Spectral and space–time domain diagnostic procedures are used to assess the quality of the fits. The spectral-in-time modelling approach is shown to yield a good fit to many properties of the data and can be applied in a routine fashion relative to finding elaborate parametric models that describe the space–time dependences of the data about as well.     

Comparison of alternative imputation methods for ordinal data

In this article, we compare alternative missing imputation methods in the presence of ordinal data, in the framework of CUB (Combination of Uniform and (shifted) Binomial random variable) models. Various imputation methods are considered, as are univariate and multivariate approaches. The first step consists of running a simulation study designed by varying the parameters of the CUB model, to consider and compare CUB models as well as other methods of missing imputation. We use real datasets on which to base the comparison between our approach and some general methods of missing imputation for various missing data mechanisms.     

Mixtures of regression models with incomplete and noisy data

The estimation of the mixtures of regression models is usually based on the normal assumption of components and maximum likelihood estimation of the normal components is sensitive to noise, outliers, or high-leverage points. Missing values are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this article, we propose the mixtures of regression models for contaminated incomplete heterogeneous data. The proposed models provide robust estimates of regression coefficients varying across latent subgroups even under the presence of missing values. The methodology is illustrated through simulation studies and a real data analysis.     

The Multiple Outlier Problem in Time Series Analysis

In this paper we consider the multiple outlier problem in time series analysis. The underlying undisturbed time series is assumed to be an autoregressive process. The location of the suspicious values is supposed to be known. We introduce conditional least squares estimators for the parameters. The estimates are shown to be strongly consistent. Using similar arguments as in the theory of linear models, we get a test statistic for the general linear hypothesis. Its asymptotic distribution is derived.     

Residual variance estimation in moving average models

We consider time series models of the MA (moving average) family, and deal with the estimation of the residual variance. Results are known for maximum likelihood estimates under normality, both for known or unknown mean, in which case the asymptotic biases depend on the number of parameters (including the mean), and do not depend on the values of the parameters. For moment estimates the situation is different, because we find that the asymptotic biases depend on the values of the parameters, and become large as they approach the boundary of the region of invertibility. Our approach is to use Taylor series expansions, and the objective is to obtain asymptotic biases with error of o(l/T), where T is the sample size. Simulation results are presented, and corrections for bias suggested.     

Testing for intercept‐scale switch in linear autoregression

Autoregressive models with switching regime are a frequently used class of nonlinear time series models, which are popular in finance, engineering, and other fields. We consider linear switching autoregressions in which the intercept and variance possibly switch simultaneously, while the autoregressive parameters are structural and hence the same in all states, and we propose quasi‐likelihood‐based tests for a regime switch in this class of models. Our motivation is from financial time series, where one expects states with high volatility and low mean together with states with low volatility and higher mean. We investigate the performance of our tests in a simulation study, and give an application to a series of IBM monthly stock returns. The Canadian Journal of Statistics 40: 427–446; 2012 © 2012 Statistical Society of Canada     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.