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.     

Forecasting financial time series with Boltzmann entropy through neural networks

Neural networks have recently been established as state-of-the-art in forecasting financial time series. However, many studies show how one architecture, the Long-Short Term Memory, is the most widespread in financial sectors due to its high performance over time series. Considering some stocks traded in financial markets and a crypto ticker, this paper tries to study the effectiveness of the Boltzmann entropy as a financial indicator to improve forecasting, comparing it with financial analysts’ most commonly used indicators. The results show how Boltzmann’s entropy, born from an Agent-Based Model, is an efficient indicator that can also be applied to stocks and cryptocurrencies alone and in combination with some classic indicators. This critical fact allows obtaining good results in prediction ability using Network architecture that is not excessively complex.

     

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.     

A hybrid algorithm to optimize RBF network architecture and parameters for nonlinear time series prediction

This paper proposes a novel hybrid algorithm for automatic selection of the proper input variables, the number of hidden nodes of the radial basis function (RBF) network, and optimizing network parameters (weights, centers and widths) simultaneously. In the proposed algorithm, the inputs and the number of hidden nodes of the RBF network are represented by binary-coded strings and evolved by a genetic algorithm (GA). Simultaneously, for each chromosome with fixed inputs and number of hidden nodes, the corresponding parameters of the network are real-coded and optimized by a gradient-based fast-converging parameter estimation method. Performance of the presented hybrid approach is evaluated by several benchmark time series modeling and prediction problems. Experimental results show that the proposed approach produces parsimonious RBF networks, and obtains better modeling accuracy than some other algorithms.     

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.     

Forecasting non-stationary time series by wavelet process modelling

Many time series in the applied sciences display a time-varying second order structure. In this article, we address the problem of how to forecast these nonstationary time series by means of non-decimated wavelets. Using the class of Locally Stationary Wavelet processes, we introduce a new predictor based on wavelets and derive the prediction equations as a generalisation of the Yule-Walker equations. We propose an automatic computational procedure for choosing the parameters of the forecasting algorithm. Finally, we apply the prediction algorithm to a meteorological time series. Financial supports from the contract ‘Projet d'Actions de Recherche Concertées’ no. 98/03-217 of the Belgian Government and from the IAP research network No. P5/24 of the Belgian State (Federal Office for Scientific, Technical and Cultural Affairs) are gratefully acknowledged.     

Forecasting portfolio returns using weighted fuzzy time series methods

We propose using weighted fuzzy time series (FTS) methods to forecast the future performance of returns on portfolios. We model the uncertain parameters of the fuzzy portfolio selection models using a possibilistic interval-valued mean approach, and approximate the uncertain future return on a given portfolio by means of a trapezoidal fuzzy number. Introducing some modifications into the classical models of fuzzy time series, based on weighted operators, enables us to generate trapezoidal numbers as forecasts of the future performance of the portfolio returns. This fuzzy forecast makes it possible to approximate both the expected return and the risk of the investment through the value and ambiguity of a fuzzy number.We incorporate our proposals into classical fuzzy time series methods and analyze their effectiveness compared with classical weighted fuzzy time series models, using historical returns on assets from the Spanish stock market. When our weighted FTS proposals are used to point-wise forecast portfolio returns the one-step ahead accuracy is improved, also with respect to non-fuzzy forecasting methods.     

Estimation of fixed points for nonlinear time series

Identification of fixed points is very important in dynamic systems analysis. One method used is based on polynomial regression. In this article, we show that methods other than that of Aguirre and Souza can be more accurate if the classical assumptions for regression are violated. Simulation results reveal that an artificial neural network (ANN) is more precise than the Aguirre and Souza method, which is based on cluster expansion method. Overall, ANN is the best method for finding fixed (equilibrium) points of nonlinear time series, followed by nonparametric regression in terms of accuracy. For larger sample sizes, ANN estimates are generally accurate and the method is robust to changes in the signal/noise ratio. © 2013 Wiley Periodicals, Inc. Complexity 19: 30–39, 2014     

On pricing derivatives under nonlinear time series models

In this note, we summarize some of our recent works on pricing derivative securities under nonlinear time series models. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the estimation of nonlinear time series models

In the present paper, a framework for parametric estimation in nonlinear time series is developed. Strong consistency and asymptotic normality of minimum Hellinger distance estimates for a determined class of nonlinear models are investigated. The main Interest for these estimates is motivated by their robustness under perturbations as it has been emphazized in Beran [2]. The first part of the paper is devoted to the study of some probabilistic properties which ensure the existence and the optimal properties of the estimates     

Diagnostic checking of multivariate nonlinear time series models with martingale difference errors

In this article, we derive the asymptotic distribution of residual autocovariance and autocorrelation matrices for a general class of multivariate nonlinear time series models by assuming only that the error term is a martingale difference sequence. Two types of applications are developed: global test statistics of the portmanteau type and one-lag test statistics, which describe the residual correlation at individual lags. To illustrate the proposed methodology, simulation results are reported for diagnosing multivariate threshold time series models. The following test statistics are compared: the classical test statistics presuming independent errors and the proposed methodology which supposes only martingale difference errors.     

Investigating asymptotic properties of vector nonlinear time series models

Analyses and simulations of vector nonlinear time series typically run into weeks or even months because the methods used are computationally intensive. Statisticians have been known to base empirical results on a relatively small number of simulation replications, sacrificing precision and reliability of results in the interest of time and productivity. The simulations are amenable for parallelization. However, parallel computing technology has not yet been widely used in this specific research area. This paper proposes an approach to the parallelization of statistical simulation codes to address the challenge of long running times. Requiring minimal code revision, this approach takes advantage of recent advances in dynamic loop scheduling to achieve high performance on general-purpose clusters, even with the presence of unpredictable load imbalance factors. Preliminary results of applying this approach in the simulation of normal white noise and threshold autoregressive model obtains efficiencies in the range 95%–98% on 8–64 processors. Furthermore, previously unobserved properties of the statistical procedures for the models are uncovered by the simulation.     

Testing nonstationary and absolutely regular nonlinear time series models

Statistical Inference for Stochastic Processes - We study some general methods for testing the goodness-of-fit of a general nonstationary and absolutely regular nonlinear time series model. These...     

Investigating asymptotic properties of vector nonlinear time series models

Analyses and simulations of vector nonlinear time series typically run into weeks or even months because the methods used are computationally intensive. Statisticians have been known to base empirical results on a relatively small number of simulation replications, sacrificing precision and reliability of results in the interest of time and productivity. The simulations are amenable for parallelization. However, parallel computing technology has not yet been widely used in this specific research area. This paper proposes an approach to the parallelization of statistical simulation codes to address the challenge of long running times. Requiring minimal code revision, this approach takes advantage of recent advances in dynamic loop scheduling to achieve high performance on general-purpose clusters, even with the presence of unpredictable load imbalance factors. Preliminary results of applying this approach in the simulation of normal white noise and threshold autoregressive model obtains efficiencies in the range 95%-98% on 8-64 processors. Furthermore, previously unobserved properties of the statistical procedures for the models are uncovered by the simulation.     

Stock market forecasting by using a hybrid model of exponential fuzzy time series

The initial aim of this study is to propose a hybrid method based on exponential fuzzy time series and learning automata based optimization for stock market forecasting. For doing so, a two-phase approach is introduced. In the first phase, the optimal lengths of intervals are obtained by applying a conventional fuzzy time series together with learning automata swarm intelligence algorithm to tune the length of intervals properly. Subsequently, the obtained optimal lengths are applied to generate a new fuzzy time series, proposed in this study, named exponential fuzzy time series. In this final phase, due to the nature of exponential fuzzy time series, another round of optimization is required to estimate certain method parameters. Finally, this model is used for future forecasts. In order to validate the proposed hybrid method, forty-six case studies from five stock index databases are employed and the findings are compared with well-known fuzzy time series models and classic methods for time series. The proposed model has outperformed its counterparts in terms of accuracy.     

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 business cycle with chaotic time series based on neural network with weighted fuzzy membership functions

This study presents a forecasting model of cyclical fluctuations of the economy based on the time delay coordinate embedding method. The model uses a neuro-fuzzy network called neural network with weighted fuzzy membership functions (NEWFM). The preprocessed time series of the leading composite index using the time delay coordinate embedding method are used as input data to the NEWFM to forecast the business cycle. A comparative study is conducted using other methods based on wavelet transform and Principal Component Analysis for the performance comparison. The forecasting results are tested using a linear regression analysis to compare the approximation of the input data against the target class, gross domestic product (GDP). The chaos based model captures nonlinear dynamics and interactions within the system, which other two models ignore. The test results demonstrated that chaos based method significantly improved the prediction capability, thereby demonstrating superior performance to the other methods.     

The validity of least squares estimation in a time series model using the bootstrap methodology

The boostrap methodology may be used for estimating standard errors of the estimated parameters in a time series model. The idea is to approximate the theoretical error distribution by the residual distribution. The main objective of this article is to demonstrate the use of the bootstrap to attach standard errors to coefficient estimates in a second-order auto-regressive model fitted by least squares estimation. A comparison of the conventional and bootstrap methodology is made. A numerical result shows that the traditional least squares asymptotic formula for estimating standard errors appear to overestimate the true standard errors. But there are two problems in the simulation world of bootstrap for the autoregressive model of order two: (1) the first two observations y₁ and y₂ have been fixed, and (2) the residuals have not been inflated. After these two factors are considered in the trial and bootstrap experiment, both the conventional least squares and bootstrap estimates of the standard errors appear to be performing quite well.     

A nonlinear time series model and estimation of missing observations

This paper formulates a nonlinear time series model which encompasses several standard nonlinear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithms and the other using optimal estimating equation theory. Recursive estimation of missing observations in an autoregressive conditionally heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. Construction of optimal estimates of missing observations using estimating equation theory is discussed and applied to some nonlinear models.Authors were supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.