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.     

Forecasting functional time series

We propose forecasting functional time series using weighted functional principal component regression and weighted functional partial least squares regression. These approaches allow for smooth functions, assign higher weights to more recent data, and provide a modeling scheme that is easily adapted to allow for constraints and other information. We illustrate our approaches using age-specific French female mortality rates from 1816 to 2006 and age-specific Australian fertility rates from 1921 to 2006, and show that these weighted methods improve forecast accuracy in comparison to their unweighted counterparts. We also propose two new bootstrap methods to construct prediction intervals, and evaluate and compare their empirical coverage probabilities.     

Forecasting spring flow time series

Summary The paper deals with a statistical analysis, carried out to define the underlying reason of some of the damage observed in many buildings of a southern Italian town. Engineering considerations, substantiated by specific measurements, attributed them to the lowering of the groundwater table in the area below the building locations. Due to two coinciding events which occurred in the preceding years, i.e. a persistent drought and the start up of a system of wells, it was not possible to define the cause of the former phenomenon. As shutting down the wells could generate additional problems, an accurate picture of the whole situation was necessary, before taking any action. By taking advantage of some fragmentary data belonging to the flow of a spring located in the area and on the basis of the knowledge of the rainfall data recorded in the Italian hydrographic service directory, two models have been developed which reproduce the spring flow time series in relation to the rainfall recorded in the surrounding area. By comparing the spring flow predictions with the actual data it has been possible to highlight the main role played by the wells.     

Forecasting contemporal time series aggregates

Forecast of a contemporal aggregate of several time series can be obtained from ‘1’ an aggregate series, ‘2’ individual component processes, or ‘3’ a joint multiple forecasting model. Through general Hilbert space theory and some illustrative examples, this paper establishes the relative efficiencies among the three methods     

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).     

Forecasting time series of economic processes by model averaging across data frames of various lengths

This paper presents an extension of mean-squared forecast error (MSFE) model averaging for integrating linear regression models computed on data frames of various lengths. Proposed method is considered to be a preferable alternative to best model selection by various efficiency criteria such as Bayesian information criterion (BIC), Akaike information criterion (AIC), F-statistics and mean-squared error (MSE) as well as to Bayesian model averaging (BMA) and naïve simple forecast average. The method is developed to deal with possibly non-nested models having different number of observations and selects forecast weights by minimizing the unbiased estimator of MSFE. Proposed method also yields forecast confidence intervals with a given significance level what is not possible when applying other model averaging methods. In addition, out-of-sample simulation and empirical testing proves efficiency of such kind of averaging when forecasting economic processes.     

A time series model for cohort data

A univariate time-series model is presented for projecting age- or duration-specific data for recent cohorts. The model is also capable of taking period effects into account, is illustrated using Dutch data on marital fertility rates.     

Modeling time series data with semi-reflective boundaries

Time series data are increasingly common in many areas of the health sciences, and in some instances, may have natural boundaries serving as performance guidelines or as thresholds associated with adverse outcomes. Such boundaries may be labeled as semi-reflective, in that the time series values have an increased chance of returning towards middle levels as the boundaries are approached, but boundaries can still be breached. In this paper we review a model that was previously proposed for such data and we investigate its statistical properties. Specifically, this model consists of a third-order auto-regressive projection component, parameterized as a constrained linear combination of linear, flat, and quadratic trends, and an error term that uses a logistic regression model for its sign. We describe and compare a previously-proposed estimation method with a modified version thereof, using computer simulations, as well as data examples from heart monitoring and from a driving simulator. We find that the two methods tend to give different results, with the modified technique having lower bias and more accurate confidence intervals than the previously-proposed method.     

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.     

Principal components analysis of nonstationary time series data

The effect of nonstationarity in time series columns of input data in principal components analysis is examined. Nonstationarity are very common among economic indicators collected over time. They are subsequently summarized into fewer indices for purposes of monitoring. Due to the simultaneous drifting of the nonstationary time series usually caused by the trend, the first component averages all the variables without necessarily reducing dimensionality. Sparse principal components analysis can be used, but attainment of sparsity among the loadings (hence, dimension-reduction is achieved) is influenced by the choice of parameter(s) (λ _1,i ). Simulated data with more variables than the number of observations and with different patterns of cross-correlations and autocorrelations were used to illustrate the advantages of sparse principal components analysis over ordinary principal components analysis. Sparse component loadings for nonstationary time series data can be achieved provided that appropriate values of λ _1,j are used. We provide the range of values of λ _1,j that will ensure convergence of the sparse principal components algorithm and consequently achieve sparsity of component loadings.     

Detection of outliers in bivariate time series data

In this article, we use the influence function matrix of auto and cross-correlations of a bivariate (multivariate) time series for detecting the outliers. The multivariate analog of the graphical method of Chernick et. al. (1982), to detect outliers and partial outliers is presented. A simulation study illustrating the method is also given.     

Nonparametric confidence intervals for location in time series data

We extend the confidence interval construction procedure for location for symmetric iid data using the one-sample Wilcoxon signed rank statistic (T⁺) to stationary time series data. We propose a normal approximation procedure when explicit knowledge of the underlying dependence structure/distribution is unknown. By conducting extensive simulations from linear and nonlinear time series models, we show that the extended procedure is a strong contender for use in the construction of confidence intervals in time series analysis. Finally we demonstrate real application implementations in two case studies.     

A numerical nonmetric approach for analyzing time series data

A numerical nonmetric approach to data analysis of periodic series with polytone trend is suggested. Estimation is made of thesmallest number of tone (monotone segments) possible for the trend. The seasonal component is estimated without need for first removing the (estimated) polytone trend. A computer program has been developed which enables analysis of arbitrary series, either by a prespecified length of period or by estimating the period length if not known in advance. Robustness of the proposed approach enables analysis of very short series, series with missing values, and other series with limitations that cannot be easily handled otherwise. In a separate appendix some empirical results obtained by this approach are compared with those from the X-ll program; this appendix will be sent upon request.     

Smoothed estimates for nonlinear time series models with irregular data

Time series data observed at unequal time intervals (irregular data) occur quite often and this usually poses problems in its analysis. A recursive form of the exponentially smoothed estimated is here proposed for a nonlinear model with irregularly observed data and its asymptotic properties are discussed An alternative smoother to that of Wright (1985) is also derived. Numerical comparison is made between the resulting estimates and other smoothed estimates.     

Fitting time series models for longitudinal survey data under informative sampling

The purpose of this paper is to account for informative sampling in fitting time series models, and in particular an autoregressive model of order one, for longitudinal survey data. The idea behind the proposed approach is to extract the model holding for the sample data as a function of the model in the population and the first-order inclusion probabilities, and then fit the sample model using maximum-likelihood, pseudo-maximum-likelihood and estimating equations methods. A new test for sampling ignorability is proposed based on the Kullback–Leibler information measure. Also, we investigate the issue of the sensitivity of the sample model to incorrect specification of the conditional expectations of the sample inclusion probabilities. The simulation study carried out shows that the sample-likelihood-based method produces better estimators than the pseudo-maximum-likelihood method, and that sensitivity to departures from the assumed model is low. Also, we find that both the conventional t-statistic and the Kullback–Leibler information statistic for testing of sampling ignorability perform well under both informative and noninformative sampling designs.     

On the transformation of raw time series data: A review

Effective time-series analysis is based on the assumption that the series under investigation is a realisation of a "stationary" stochastic process. In practice, such a stable series can generally only be obtained after some appropriate transformation of the raw data. Two types of non-stationarity can be removed by, respectively, linear and non-linear transformation. These are "homogeneous" non-stationarity and variance instability. The first can be dealt with by backshift operator methods, whilst the second is usually carried out by the approach of Box and Cox, though an easier way is given. The loss of optimal properties, on transforming back to the original situation, can be offset by suitably biasing the results.     

Forecasting of some; transformed series

The delta method is proposed as a. retransformation approach for coramputing forecasts for a nonstationary process (Z$inf:t$einf t, = 1, 2, ...), A derived variance stabilizing transformation technique is also used to compute forecasts, The performance of the two methods is compared to other techniques. Numerical results show that forecasts based on the variance stabilizing transformation method (VSTM) can lead to forecasts with a lower mean square error (MSE) as compared to other transformation techniqnes.