Evolutionary learning of fuzzy grey cognitive maps for the forecasting of multivariate,interval-valued time series

Time series are built as a result of real-valued observations ordered in time; however, in some cases, the values of the observed variables change significantly, and those changes do not produce useful information. Therefore, within defined periods of time, only those bounds in which the variables change are considered. The temporal sequence of vectors with the interval-valued elements is called a ‘multivariate interval-valued time series.’ In this paper, the problem of forecasting such data is addressed. It is proposed to use fuzzy grey cognitive maps (FGCMs) as a nonlinear predictive model. Using interval arithmetic, an evolutionary algorithm for learning FGCMs is developed, and it is shown how the new algorithm can be applied to learn FGCMs on the basis of historical time series data. Experiments with real meteorological data provided evidence that, for properly-adjusted learning and prediction horizons, the proposed approach can be used effectively to the forecasting of multivariate, interval-valued time series. The domain-specific interpretability of the FGCM-based model that was obtained also is confirmed.     

A note on the time series which is the product of two stationary time series

The time series […,x_-1y_-1,x₀y₀,x₁y₁,…]> which is the product of two stationary time series x_t and y_t is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.     

A nonparametric procedure to detect periods in time series

Suppose that for a given time series the experimenter knows that it has a certain periodic property and that he wishes to find out the length of the period. For this problem a nonparametric procedure is proposed. It consists of a new smoothing technique based on Kendall's Tau and a specific counting method. The procedure is studied under a simple model of periodic time series which are composed of periodic (deterministic) functions, a linear trend and exchangeable (stochastic) sequences. The performance of the procedure is illustrated by a simple example.     

Canonical correlation and reduction of multiple time series

It is shown that a degenerate rankd-variate stationary time series can be reduced to a full rank time series of lower dimension via an orthogonal transformationT provided that , the canonical correlation between past and future of the time series is strictly less than one. Procedures for estimation of rank of the multiple time series,T and testing =1 are outlined, the latter is related to testing the unit root hypothesis in ARMA models.     

A note on bilinear time series models

Conditions for the existence of a stationary solution for certain forms of bilinear difference equations are derived.     

A test of conditional heteroscedasticity in time series

A new test of conditional heteroscedasticity for time series is proposed. The new testing method is based on a goodness of fit type test statistics and a Cramer-von Mises type test statistic. The asymptotic properties of the new test statistic is establised. The results demonstrate that such a test is consistent. Project supported by the National Natural Science Foundation of China (Grant No. 19231050) and Postdoctoral Fund of China.     

Detection of multiple change-points in multivariate time series

We consider the multiple change-point problem for multivariate time series, including strongly dependent processes, with an unknown number of change-points. We assume that the covariance structure of the series changes abruptly at some unknown common change-point times. The proposed adaptive method is able to detect changes in multivariate i.i.d., weakly and strongly dependent series. This adaptive method outperforms the Schwarz criteria, mainly for the case of weakly dependent data. We consider applications to multivariate series of daily stock indices returns and series generated by an artificial financial market. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 351–376, July–September, 2006.     

一类随机环境下非线性时间序列模型的遍历性分析

本文主要讨论由模型Xn 1=h(en -q(n 1),-en,Xn 1-p(n 1),…Xn) en 1所确定的序列{Xn,n 1}的极限行为.     

COPAR—multivariate time series modeling using the copula autoregressive model

The analysis of multivariate time series is a common problem in areas like finance and economics. The classical tools for this purpose are vector autoregressive models. These however are limited to the modeling of linear and symmetric dependence. We propose a novel copula‐based model that allows for the non‐linear and non‐symmetric modeling of serial as well as between‐series dependencies. The model exploits the flexibility of vine copulas, which are built up by bivariate copulas only. We describe statistical inference techniques for the new model and discuss how it can be used for testing Granger causality. Finally, we use the model to investigate inflation effects on industrial production, stock returns and interest rates. In addition, the out‐of‐sample predictive ability is compared with relevant benchmark models. Copyright © 2014 John Wiley & Sons, Ltd.     

The information matrices of the parameters of multiple mixed time series

Closed form matrix equations are given for the information matrix of the parameters of the vector mixed autoregressive moving average time series model.     

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.     

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.     

Improved time-variant fuzzy time series forecast

Since Song and Chissom (Fuzzy Set Syst 54:1–9, 1993a) first proposed the structure of fuzzy time series forecast, researchers have devoted themselves to related studies. Among these studies, Hwang et al. (Fuzzy Set Syst 100:217–228, 1998) revised Song and Chissom’s method, and generated better forecasted results. In their method, however, several factors that affect the accuracy of forecast are not taken into consideration, such as levels of window base, length of interval, degrees of membership values, and the existence of outliers. Focusing on these factors, this study proposes an improved fuzzy time series forecasting method. The improved method can provide decision-makers with more precise forecasted values. Two numerical examples are employed to illustrate the proposed method, as well as to compare the forecasting accuracy of the proposed method with that of two fuzzy forecasting methods. The results of the comparison indicate that the proposed method produces more accurate forecasting results.     

Linear relations in time series models. II

In this paper an asymptotic theory is developed for a new time series model which was introduced in a previous paper [5]. An algorithm for computing estimates of the parameters of this time series model is given, and it is shown that these estimators are asymptotically efficient in the sense that they have the same asymptotic distribution as the maximum likelihood estimators.     

Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction

Summary We have applied topological methods to analyze chaotic time series data from the Belousov-Zhabotinskii reaction. First, the periodic orbits shadowed by the data set were identified. Next, a three-dimensional embedding without self-intersections was constructed from the data set. The topological structure of that flow was visualized by constructing a branched manifold such that every periodic orbit in the flow could be held by the branched manifold. The branched manifold, or induced template, was computed using the three lowest-period orbits. The organization of the higher-period orbits predicted by this induced template was compared with the organization of the orbits reconstructed from the data set with excellent results. The consequences of the presence of certain knots found in the data are discussed.     

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.     

Linear relations in time series models. I

A multiple time series is defined as the sum of an autoregressive process on a line and independent Gaussian white noise on a hyperplane that goes through the origin and intersects the line at a single point. This process is a multiple autoregressive time series in which the regression matrices satisfy suitable conditions. It is shown that the maximum likelihood estimates of the line and the autoregression coefficients can be obtained as the values that minimize a given function, and that the remaining maximum likelihood estimates can be computed as simple functions of the first ones. It is also shown that the maximum likelihood estimates are equivariant with respect to the group of bijective linear transformations.     

Forecasting stock index price based on M-factors fuzzy time series and particle swarm optimization

In real time, one observation always relies on several observations. To improve the forecasting accuracy, all these observations can be incorporated in forecasting models. Therefore, in this study, we have intended to introduce a new Type-2 fuzzy time series model that can utilize more observations in forecasting. Later, this Type-2 model is enhanced by employing particle swarm optimization (PSO) technique. The main motive behind the utilization of the PSO with the Type-2 model is to adjust the lengths of intervals in the universe of discourse that are employed in forecasting, without increasing the number of intervals. The daily stock index price data set of SBI (State Bank of India) is used to evaluate the performance of the proposed model. The proposed model is also validated by forecasting the daily stock index price of Google. Our experimental results demonstrate the effectiveness and robustness of the proposed model in comparison with existing fuzzy time series models and conventional time series models.     

Prediction of multivariate time series by autoregressive model fitting

Suppose the stationary r-dimensional multivariate time series {y_t} is generated by an infinite autoregression. For lead times h≥1, the linear prediction of y_t+h based on y_t, y_t−1,… is considered using an autoregressive model of finite order k fitted to a realization of length T. Assuming that k → ∞ (at some rate) as T → ∞, the consistency and asymptotic normality of the estimated autoregressive coefficients are derived, and an asymptotic approximation to the mean square prediction error based on this autoregressive model fitting approach is obtained. The asymptotic effect of estimating autoregressive parameters is found to inflate the minimum mean square prediction error by a factor of (1 + kr/T).