Comparison of Two Estimation Methods of Missing Values Using Pitman-Closeness Criterion

In this article, a proof is given that the linear interpolator is Pitman-closer than the linear predictor with respect to a missing value of a stationary first-order autoregressive process.     

Testing time series for interpolability and whiteness

We propose a test to decide if a time series is represented by its linear interpolator better than by its mean value. The same test can be employed to decide if a time series has to be considered white noise. The test is based on a new estimate of the index of linear determinism (Battaglia, 1983, Inverse autocovariances and a measure of linear determinism for a stationary process, J. Time Series Anal. 4, 79-87) and its asymptotic distribution is derived. Comparison with the popular Ljung-Box portmanteau test has been performed based on both asymptotic power and a simulation experiment. The new test     

The “Caterpillar”-SSA method for analysis of time series with missing values

The paper concerns the problem of applying singular spectrum analysis to time series with missing data. A method of filling in the missing data is proposed and is applied to time series of finite rank. Conditions of exact reconstruction of missing data are developed and versions of the algorithm applicable to real-life time series are presented. The proposed algorithms result in the extraction of additive components of time series such as trends and periodic components, with simultaneous filling in of the missing data. An example is presented.     

Interpolation,outliers mid inverse autocorrelations

The paper addresses the problem of estimating missing observations in an infinite realization of a linear, possibly nonstationary, stochastic processes when the model is known. The general case of any possible distribution of missing observations in the time series is considered, and analytical expressions for the optimal estimators and their associated mean squared errors are obtained. These expressions involve solely the elements of the inverse or dual autocorrelation function of the series.

This optimal estimator -the conditional expectation of the missing observations given the available ones- is equal to the estimator that results from filling the missing values in the series with arbitrary numbers, treating these numbers as additive outliers, and removing with intervention analysis the outlier effects from the invented numbers.     

Bayesian Multivariate Spatial Interpolation with Data Missing by Design

In a network of s _g sites, responses like levels of airborne pollutant concentrations may be monitored over time. The sites need not all measure the same set of response items and unmeasured items are considered as data missing by design . We propose a hierarchical Bayesian approach to interpolate the levels of, say, k responses at s _u other locations called ungauged sites and also the unmeasured levels of the k responses at the gauged sites. Our method involves two steps. First, when all hyperparameters are assumed to be known, a predictive distribution is derived. In turn, an interpolator, its variance and a simultaneous interpolation region are obtained. In step two, we propose the use of an empirical Bayesian approach to estimate the hyperparameters through an EM algorithm. We base our theory on a linear Gaussian model and the relationship between a multivariate normal and matrix T -distribution. Our theory allows us to pool data from several existing networks that measure different subsets of response items for interpolation.     

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.     

Spectral estimation for locally stationary time series with missing observations

Time series arising in practice often have an inherently irregular sampling structure or missing values, that can arise for example due to a faulty measuring device or complex time-dependent nature. Spectral decomposition of time series is a traditionally useful tool for data variability analysis. However, existing methods for spectral estimation often assume a regularly-sampled time series, or require modifications to cope with irregular or ‘gappy’ data. Additionally, many techniques also assume that the time series are stationary, which in the majority of cases is demonstrably not appropriate. This article addresses the topic of spectral estimation of a non-stationary time series sampled with missing data. The time series is modelled as a locally stationary wavelet process in the sense introduced by Nason et al. (J. R. Stat. Soc. B 62(2):271–292, 2000) and its realization is assumed to feature missing observations. Our work proposes an estimator (the periodogram) for the process wavelet spectrum, which copes with the missing data whilst relaxing the strong assumption of stationarity. At the centre of our construction are second generation wavelets built by means of the lifting scheme (Sweldens, Wavelet Applications in Signal and Image Processing III, Proc. SPIE, vol. 2569, pp. 68–79, 1995), designed to cope with irregular data. We investigate the theoretical properties of our proposed periodogram, and show that it can be smoothed to produce a bias-corrected spectral estimate by adopting a penalized least squares criterion. We demonstrate our method with real data and simulated examples.     

Quasi-maximum likelihood estimation of GARCH models in the presence of missing values

This work presents a new method to deal with missing values in financial time series. Previous works are generally based in state-space models and Kalman filter and few consider ARCH family models. The traditional approach is to bound the data together and perform the estimation without considering the presence of missing values. The existing methods generally consider missing values in the returns. The proposed method considers the presence of missing values in the price of the assets instead of in the returns. The performance of the method in estimating the parameters and the volatilities is evaluated through a Monte Carlo simulation. Value at risk is also considered in the simulation. An empirical application to NASDAQ 100 Index series is presented.     

Forecasting Vector ARMA Processes With Systematically Missing Observations

The following two predictors are compared for time series with systematically missing observations: (a) A time series model is fitted to the full series X_t , and forecasts are based on this model, (b) A time series model is fitted to the series with systematically missing observations Y _τ, and forecasts are based on the resulting model. If the data generation processes are known vector autoregressive moving average (ARMA) processes, the first predictor is at least as efficient as the second one in a mean squared error sense. Conditions are given for the two predictors to be identical. If only the ARMA orders of the generation processes are known and the coefficients are estimated, or if the process orders and coefficients are estimated, the first predictor is again, in general, superior. There are, however, exceptions in which the second predictor, using seemingly less information, may be better. These results are discussed, using both asymptotic theory and small sample simulations. Some economic time series are used as illustrative examples.     

Bayesian Inference and Forecasting in Dynamic Neural Networks with Fully Markov Switching ARCH Noises

We deal with one-layer feed-forward neural network for the Bayesian analysis of nonlinear time series. Noises are modeled nonlinearly and nonnormally, by means of ARCH models whose parameters are all dependent on a hidden Markov chain. Parameter estimation is performed by sampling from the posterior distribution via Evolutionary Monte Carlo algorithm, in which two new crossover operators have been introduced. Unknown parameters of the model also include the missing values which can occur within the observed series, so, considering future values as missing, it is also possible to compute point and interval multi-step-ahead predictions.     

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.     

A recursive approach for estimating missing observations in an univariate time series

A linear recursive technique that does not use the Kalman filter approach is proposed to estimate missing observations in an univariate time series. It is assumed that the series follows an invertible ARIMA model. The procedure is based on the restricted forecasting approach, and the recursive linear estimators are optimal in terms of minimum mean-square error.     

Missing data and forecasting in multivariate time series: An application of the common components dynamic linear model

Summary The paper deals with missing data and forecasting problems in multivariate time series making use of the Common Components Dynamic Linear Model (DLMCC), presented in Quintana (1985), and West and Harrison (1989). Some results are presented and discussed: exploiting the correlation between series, estimated by the DLMCC, the paper shows as it is possible to update state vector posterior distributions for the unobserved series. This is realized on the base of the updating of the observed series state vectors, for which the usual Kalman filter equations can be applied. An application concerning some Italian private consumption series provides an example of the model capabilities.     

Robust estimation of AR coefficients under simultaneously influencing outliers and missing values

A family of robust estimators for coefficients of Gaussian AR(p) time series under simultaneously influencing distortions of two types: outliers and missing values, is proposed. The estimators are based on special properties of the Cauchy probability distribution; consistency and the asymptotic normality of these estimators are proven. An approximate solution of the problem of minimization of the asymptotic variance within the proposed family of estimators is found. Performance of the proposed estimators is illustrated for simulated time series and for real data sets.     

Multivariate non-linear time series modelling of exposure and risk in road safety research

Summary.  A multivariate non-linear time series model for road safety data is presented. The model is applied in a case-study into the development of a yearly time series of numbers of fatal accidents (inside and outside urban areas) and numbers of kilometres driven by motor vehicles in the Netherlands between 1961 and 2000. The model accounts for missing entries in the disaggregated numbers of kilometres driven although the aggregated numbers are observed throughout. We consider a multivariate non-linear time series model for the analysis of these data. The model consists of dynamic unobserved factors for exposure and risk that are related in a non-linear way to the number of fatal accidents. The multivariate dimension of the model is due to its inclusion of multiple time series for inside and outside urban areas. Approximate maximum likelihood methods based on the extended Kalman filter are utilized for the estimation of unknown parameters. The latent factors are estimated by extended smoothing methods. It is concluded that the salient features of the observed time series are captured by the model in a satisfactory way.     

Missing observations in time series

A method based on forecasting techniques is proposed to estimate missing observations in time series. Using mean squares, this method is compared to the minimum mean square estimate.     

Bayesian analysis of multivariate threshold autoregressive models with missing data

In some fields, we are forced to work with missing data in multivariate time series. Unfortunately, the data analysis in this context cannot be carried out in the same way as in the case of complete data. To deal with this problem, a Bayesian analysis of multivariate threshold autoregressive models with exogenous inputs and missing data is carried out. In this paper, Markov chain Monte Carlo methods are used to obtain samples from the involved posterior distributions, including threshold values and missing data. In order to identify autoregressive orders, we adapt the Bayesian variable selection method in this class of multivariate process. The number of regimes is estimated using marginal likelihood or product parameter-space strategies.     

Forecasting time series with missing data using Holt's model

This paper deals with the prediction of time series with missing data using an alternative formulation for Holt's model with additive errors. This formulation simplifies both the calculus of maximum likelihood estimators of all the unknowns in the model and the calculus of point forecasts. In the presence of missing data, the EM algorithm is used to obtain maximum likelihood estimates and point forecasts. Based on this application we propose a leave-one-out algorithm for the data transformation selection problem which allows us to analyse Holt's model with multiplicative errors. Some numerical results show the performance of these procedures for obtaining robust forecasts.     

Detecting year-of-birth mortality patterns with limited data

Summary.  Late life mortality patterns are of crucial interest to actuaries assessing risk of longevity, most obviously for annuities and defined benefit pension schemes. The stability of public finances is also affected, as the governments have very substantial risk of longevity in the form of state benefits and public sector pension schemes. One important explanatory variable for late life mortality patterns is year of birth. Previous work has demonstrated various techniques for detecting such patterns, but always with long time series of mortality rates. The paper describes two alternative ways to detect such patterns, even with missing population data or the absence of a time series. The paper finds support for the idea that different birth cohorts have different rates of aging.     

Bayesian analysis of time series Poisson data

This paper provides a practical simulation-based Bayesian analysis of parameter-driven models for time series Poisson data with the AR(1) latent process. The posterior distribution is simulated by a Gibbs sampling algorithm. Full conditional posterior distributions of unknown variables in the model are given in convenient forms for the Gibbs sampling algorithm. The case with missing observations is also discussed. The methods are applied to real polio data from 1970 to 1983.