Limit theorems on a linear explosive stochastic model for time series with moving average error

Summary LetX(t) be a linear autoregressively generated explosive time series, with autoregressive coefficientsb ₁,…,b_q, and a constant termb ₀, and an error term ; a₀=1. Where ε(t),t≧1 are independent, Eε(t)=0, and Eε ²(t)=σ² is positive and finite. In this paper two categories of -consisent and asymptotically singularly normal estimators are proposed for (b ₁,…,b_q, b₀) thus settling an open problem since the publication of the paper (Venkataraman [5]). Based on these estimators several additional limit theorems based on estimated error residuals are proved. The parameter-free limit theorems of Spectral and Quenouille types of this paper serve as asymptotic goodness of fit tests for the model generatingX(t).     

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.     

Least absolute deviation estimation for general fractionally integrated autoregressive moving average time series models

We introduce a new class of ARFIMA models, which removes the restrictions that the roots of AR and MA polynomials are outside the unit circle. We establish consistency and asymptotic normality of the least absolute deviation estimator under non-Gaussian setting.     

An introduction to forecasting with time series models

The use of ARIMA time series models in forecasting is reviewed. In connection with this, some important points about forecasting are discussed, including: (1) difficulties in forecasting by fitting and extrapolating a deterministic function of time; (2) the importance of providing reasonable measures of forecast accuracy; and (3) the need to incorporate subject matter knowledge with time series models when forecasting.     

Neural network approach to forecasting of quasiperiodic financial time series

A novel neural network approach to forecasting of financial time series based on the presentation of the series as a combination of quasiperiodic components is presented. Separate components may have aliquant, and possibly non-stationary frequencies. All their parameters are estimated in real time in an ensemble of predictors, whose outputs are then optimally combined to obtain the final forecast. Special architecture of artificial neural network and learning algorithms implementing this approach are developed.     

Asymptotic properties of autoregressive integrated moving average processes

In this paper we study the asymptotic behavior of so-called autoregressive integrated moving average processes. These processes constitute a large class of stochastic difference equations which includes among many other well-known processes the simple one-dimensional random walk. They were dubbed by G.E.P. Box and G.M. Jenkins who found them to provide useful models for studying and controlling the behavior of certain economic variables and various chemical processes. We show that autoregressive integrated moving average processes are asymptotically normally distributed, and that the sample paths of such processes satisfy a law of the iterated logarithm. We also establish a law which determines the time spent by a sample path on one or the other side of the “trend line” of the process.     

Handling forecasting problems using fuzzy time series

In [6–9], Song et al. proposed fuzzy time-series models to deal with forecasting problems. In [10], Sullivan and Woodall reviewed the first-order time-invariant fuzzy time series model and the first-order time-variant model proposed by Song and Chissom [6–8], where the models are compared with each other and with a time-invariant Markov model using linguistic labels with probability distributions. In this paper, we propose a new method to forecast university enrollments, where the historical enrollments of the University of Alabama shown in [7,8] are used to illustrate the forecasting process. The average forecasting errors and the time complexity of these methods are compared. The proposed method is more efficient than the ones presented in [7, 8, 10] due to the fact that the proposed method simplifies the arithmetic operation process. Furthermore, the average forecasting error of the proposed method is smaller than the ones presented in [2, 7, 8].     

Monotonicity properties of bargaining solutions when applied to economics

We are concerned with the application of bargaining solutions to economic problems of fair division, and, in particular, with the way they respond to changes in the amount to be divided. For instance, one may want an increase in that amount to benefit all agents. A variety of monotonicity properties have been studied in the abstract framework of bargaining theory. Most of the commonly studied solutions are known not to satisfy many of these properties. Here, we investigate the extent to which they do when applied to economics. We show that when there is only one good, they do in fact satisfy many monotonicity properties that they do not satisfy in general. However, this positive result fails as soon as the number of commodities is greater than 2.     

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.     

Using interval information granules to improve forecasting in fuzzy time series

In the process of modeling and forecasting of fuzzy time series, an issue on how to partition the universe of discourse impacts the quality of the forecasting performance of the constructed fuzzy time series model. In this paper, a novel method of partitioning the universe of discourse of time series based on interval information granules is proposed for improving forecasting accuracy of model. In the method, the universe of discourse of time series is first pre-divided into some intervals according to the predefined number of intervals to be partitioned, and then information granules are constructed in the amplitude-change space on the basis of data of time series belonging to each of intervals and their corresponding change (trends). In the sequel, optimal intervals are formed by continually adjusting width of these intervals to make information granules which associate with the corresponding intervals become most “informative”. Three benchmark time series are used to perform experiments to validate the feasibility and effectiveness of proposed method. The experimental results clearly show that the proposed method produces more reasonable intervals exhibiting sound semantics. When using the proposed partitioning method to determine intervals for modeling of fuzzy time series, forecasting accuracy of the constructed model are prominently enhanced.     

Some autoregressive moving average processes with generalized Poisson marginal distributions

Some simple models are introduced which may be used for modelling or generating sequences of dependent discrete random variables with generalized Poisson marginal distribution. Our approach for building these models is similar to that of the Poisson ARMA processes considered by Al-Osh and Alzaid (1987,J. Time Ser. Anal.,8, 261–275; 1988,Statist. Hefte,29, 281–300) and McKenzie (1988,Adv. in Appl. Probab.,20, 822–835). The models have the same autocorrelation structure as their counterparts of standard ARMA models. Various properties, such as joint distribution, time reversibility and regression behavior, for each model are investigated.     

Risk forecasting in the context of time series*

Lithuanian Mathematical Journal - We propose an approach for forecasting risk contained in future observations in a time series. We take into account both the shape parameter and the extremal index...     

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.     

A novel two-factor forecasting model for fuzzy time series

The study of fuzzy time series has increasingly attracted much attention due to its salient capabilities of tackling vague and incomplete data. A variety of forecasting models have devoted to improving forecasting accuracy, however, the issue of partitioning intervals has rarely been investigated. Recently, we proposed a novel deterministic forecasting model to eliminate the major overhead of determining the k-order issue in high-order models. This paper presents a continued work with focusing on handling the interval partitioning issue by applying the fuzzy c-means technology, which can take the distribution of data points into account and produce unequal-sized intervals. In addition, the forecasting model is extended to allow process twofactor problems. The accuracy superiority of the proposed model is demonstrated by conducting two empirical experiments and comparison to other existing models. The reliability of the forecasting model is further justified by using a Monte Carlo simulation and box plots. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some properties of generalized epicycloids applied to fracture mechanics

Résumé La réflexion partielle d'un faisceau lumineux sur le revers d'une plaque mince en plexiglas comportant une fissure intérieure symétrique formant un angle quelconque avec l'axe d'application de la charge extérieure, crée des déviations des rayons réfléchis dans la zone contractée qui entoure l'extrémité de la fissure. Ces déviations forment une caustique. On montre, dans cet article, que cette caustique prend la forme d'une courbe épicycloïdale générale, valable pour tous les matériaux qui ont un comportement élastique. Cette courbe est créée par l'extremité du vecteur résultant qui représente les rayons de lumière réfléchis et déviés sur la plaque fissurée. On étudie ensuite les proprietés caractéristiques de ce type d'épicycloïdes en relation avec le facteur complexe d'intensité des contraintesK ^*. Les résultats des essais faits à l'aide des plaques munies des fissures intérieures symétriques d'orientation différente par rapport à l'axe d'application de la charge extérieure sont en parfait accord avec les résultats de la théorie.     

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.     

The estimation of a nonlinear moving average model

We consider discrete-parameter stochastic processes that are the output of a nonlinear filter driven by white noise. For a simple model, we derive estimates of the unknown coefficients in the transfer function and the noise variance, and investigate their asymptotic properties. We prove some lemmas that can also be used to obtain rates of convergence in the weak and strong laws of large numbers, and central limit theorems, for estimates of more general nonlinear models.     

Potential inventory cost reductions using advanced time series forecasting techniques

This paper compares demand forecasts computed using the time series forecasting techniques of vector autoregression (VAR) and Bayesian VAR (BVAR) with forecasts computed using exponential smoothing and seasonal decomposition. These forecasts for three demand data series were used to determine three inventory management policies for each time series. The inventory costs associated with each of these policies were used as a further basis for comparison of the forecasting techniques. The results show that the BVAR technique, which uses mixed estimation, is particularly useful in reducing inventory costs in cases where the limited historical data offer little useful information for forecasting. The BVAR technique was effective in improving forecast accuracy and reducing inventory costs in two of the three cases tested. In the third case, unrestricted VAR and exponential smoothing produced the lowest experimental forecast errors and computed inventory costs. Furthermore, this research illustrates that improvements in demand forecasting can provide better cost reductions than relying on stochastic inventory models to provide cost reductions.