Multi-step prediction of time series with random missing data

Multi-step prediction is still an open challenge in time series prediction. Moreover, practical observations are often incomplete because of sensor failure or outliers causing missing data. Therefore, it is very important to carry out research on multi-step prediction of time series with random missing data. Based on nonlinear filters and multilayer perceptron artificial neural networks (ANNs), one novel approach for multi-step prediction of time series with random missing data is proposed in the study. With the basis of original nonlinear filters which do not consider the missing data, first we obtain the generalized nonlinear filters by using a sequence of independent Bernoulli random variables to model random interruptions. Then the multi-step prediction model of time series with random missing data, which can be fit for the online training of generalized nonlinear filters, is established by using the ANN’s weights to present the state vector and the ANN’s outputs to present the observation equation. The performance between the original nonlinear filters based ANN model for multi-step prediction of time series with missing data and the generalized nonlinear filters based ANN model for multi-step prediction of time series with missing data is compared. Numerical results have demonstrated that the generalized nonlinear filters based ANN are proportionally superior to the original nonlinear filters based ANN for multi-step prediction of time series with missing data.     

Risk models based on time series for count random variables

In this paper, we generalize the classical discrete time risk model by introducing a dependence relationship in time between the claim frequencies. The models used are the Poisson autoregressive model and the Poisson moving average model. In particular, the aggregate claim amount and related quantities such as the stop-loss premium, value at risk and tail value at risk are discussed within this framework.     

Forecasting seasonal time series data: a Bayesian model averaging approach

A flexible Bayesian periodic autoregressive model is used for the prediction of quarterly and monthly time series data. As the unknown autoregressive lag order, the occurrence of structural breaks and their respective break dates are common sources of uncertainty these are treated as random quantities within the Bayesian framework. Since no analytical expressions for the corresponding marginal posterior predictive distributions exist a Markov Chain Monte Carlo approach based on data augmentation is proposed. Its performance is demonstrated in Monte Carlo experiments. Instead of resorting to a model selection approach by choosing a particular candidate model for prediction, a forecasting approach based on Bayesian model averaging is used in order to account for model uncertainty and to improve forecasting accuracy. For model diagnosis a Bayesian sign test is introduced to compare the predictive accuracy of different forecasting models in terms of statistical significance. In an empirical application, using monthly unemployment rates of Germany, the performance of the model averaging prediction approach is compared to those of model selected Bayesian and classical (non)periodic time series models.     

Ergodicity of a class of nonlinear time series models in random environment domain

In this paper, we study the problem of a variety of p, onlinear time series model Xn＋ 1= TZn＋1（X（n）, … ,X（n - Zn＋l）, en＋1（Zn＋1）） in which {Zn} is a Markov chain with finite state space, and for every state i of the Markov chain, {en（i）} is a sequence of independent and identically distributed random variables. Also, the limit behavior of the sequence {Xn} defined by the above model is investigated. Some new novel results on the underlying models are presented.     

Forecasting time series with multiple seasonal patterns

A new approach is proposed for forecasting a time series with multiple seasonal patterns. A state space model is developed for the series using the innovations approach which enables us to develop explicit models for both additive and multiplicative seasonality. Parameter estimates may be obtained using methods from exponential smoothing. The proposed model is used to examine hourly and daily patterns in hourly data for both utility loads and traffic flows. Our formulation provides a model for several existing seasonal methods and also provides new options, which result in superior forecasting performance over a range of prediction horizons. In particular, seasonal components can be updated more frequently than once during a seasonal cycle. The approach is likely to be useful in a wide range of applications involving both high and low frequency data, and it handles missing values in a straightforward manner.     

Parsimonious modelling and forecasting of seasonal time series

Some seasonal time series models are considered which are appropriate for the univariate modelling and forecasting of many time series. The equivalent ARIMA forms of these models provide the basis for a critical examination of the Box-Jenkins approach to seasonal model-building. It is concluded that this approach is unsatisfactory and in particular can often result in over-differencing and the adoption of an inappropriate model. Two main reasons for this are discussed: (a) the inadequate class of models considered which rests on a restricted view of parsimony, and (b) the shortcomings of the basic approach to model identification.     

随机环境中的一类非线性时间序列的伴随几何遍历性分析

本文将AR(m)-ARCH(m)模型推广为REAR(m)-ARCH(m)模型,并给出了REAR(m)-ARCH(m)具有伴随几何遍历性的一个充分条件。     

On multiplicative seasonal modelling for vector time series

Many time series encountered in real applications display seasonal behavior. In this paper, we consider multiplicative seasonal vectorial autoregressive moving average (SVARMA) models to describe seasonal vector time series. We discuss conditional maximum likelihood estimation of the model parameters, allowing them to satisfy general linear constraints. Having fitted a model, residual autocovariances (or autocorrelations) have been found useful in checking time series models. Consequently, we obtain the asymptotic distributions of the residual autocovariance matrices. As applications of these results, Portmanteau test statistics are proposed and their asymptotic distributions are studied. The finite-sample properties of the test statistics are evaluated using Monte Carlo experiments.     

High-level dependence in time series models

We present several notions of high-level dependence for stochastic processes, which have appeared in the literature. We calculate such measures for discrete and continuous-time models, where we concentrate on time series with heavy-tailed marginals, where extremes are likely to occur in clusters. Such models include linear models and solutions to random recurrence equations; in particular, discrete and continuous-time moving average and (G)ARCH processes. To illustrate our results we present a small simulation study.     

Estimation in nonlinear time series models

A general framework for analyzing estimates in nonlinear time series is developed. General conditions for strong consistency and asymptotic normality are derived both for conditional least squares and maximum likelihood types estimates. Ergodie strictly stationary processes are studied in the first part and certain nonstationary processes in the last part of the paper. Examples are taken from most of the usual classes of nonlinear time series models.     

Estimation of a time series model from unequally spaced data

A process generated by a stochastic differential equation driven by pure noise is sampled at irregular intervals. A model for the sampled sequence is deduced. We describe a maximum likelihood procedure for estimating the parameters and establish the strong consistency and asymptotic normality of the estimates. The use of the model in prediction is considered. Simplifications in the case of periodic sampling are explored.     

Neural network forecasting for seasonal and trend time series

Neural networks have been widely used as a promising method for time series forecasting. However, limited empirical studies on seasonal time series forecasting with neural networks yield mixed results. While some find that neural networks are able to model seasonality directly and prior deseasonalization is not necessary, others conclude just the opposite. In this paper, we investigate the issue of how to effectively model time series with both seasonal and trend patterns. In particular, we study the effectiveness of data preprocessing, including deseasonalization and detrending, on neural network modeling and forecasting performance. Both simulation and real data are examined and results are compared to those obtained from the Box–Jenkins seasonal autoregressive integrated moving average models. We find that neural networks are not able to capture seasonal or trend variations effectively with the unpreprocessed raw data and either detrending or deseasonalization can dramatically reduce forecasting errors. Moreover, a combined detrending and deseasonalization is found to be the most effective data preprocessing approach.     

On the invertibility of time series models

A generalized definition of invertibility is proposed and applied to linear, non-linear and bilinear models. It is shown that some recently studied non-linear models are not invertible, but conditions for invertibility can be achieved for the other models.     

Estimators of covariances in time series models

The theory of Minimum Norm Quadratic Estimators for estimating variances and covariances is applied to show that some commonly used estimators of covariances in time series models are easily derived using the above principle. In applying the theory MINQE, it is observed that no unbiased estimator exists in the class of invariant quadratics.     

Some mixing properties of time series models

Sufficient conditions are given for linear processes and ARMA processes to have the Gaswirth and Rubin mixing condition. The mixing rates are also determined.     

Kolmogorov space in time series data

We provide the proof that the space of time series data is a Kolmogorov space with T₀‐separation axiom using the loop space of time series data. In our approach, we define a cyclic coordinate of intrinsic time scale of time series data after empirical mode decomposition. A spinor field of time series data comes from the rotation of data around price and time axis by defining a new extradimension to time series data. We show that there exist hidden eight dimensions in Kolmogorov space for time series data. Our concept is realized as the algorithm of empirical mode decomposition and intrinsic time scale decomposition, and it is subsequently used for preliminary analysis on the real time series data. Copyright © 2016 John Wiley & Sons, Ltd.