Time series forecasting: problem of heavy-tailed distributed noise

Time series forecasting has been the area of intensive research for years. Statistical, machine learning or mixed approaches have been proposed to handle this one of the most challenging tasks. However, little research has been devoted to tackle the frequently appearing assumption of normality of given data. In our research, we aim to extend the time series forecasting models for heavy-tailed distribution of noise. In this paper, we focused on normal and Student’s t distributed time series. The SARIMAX model (with maximum likelihood approach) is compared with the regression tree-based method—random forest. The research covers not only forecasts but also prediction intervals, which often have hugely informative value as far as practical applications are concerned. Although our study is focused on the selected models, the presented problem is universal and the proposed approach can be discussed in the context of other systems.

Asymptotic behavior of dependence measures for Ornstein-Uhlenbeck model based on long memory processes

In this paper, we study the long memory property of two processes based on the Ornstein-Uhlenbeck model. Their are extensions of the Ornstein-Uhlenbeck system for which in the classic version we replace the standard Brownian motion (or other L$$\acute{e}$$ e ´ vy process) by long range dependent processes based on $$\alpha -$$ α - stable distribution. One way of characterizing long- and short-range dependence of second order processes is in terms of autocovariance function. However, for systems with infinite variance the classic measure is not defined, therefore there is a need to consider alternative measures on the basis of which the long range dependence can be recognized. In this paper, we study three alternative measures adequate for $$\alpha -$$ α - stable-based processes. We calculate them for examined processes and indicate their asymptotic behavior. We show that one of the analyzed Ornstein-Uhlenbeck process exhibits long memory property while the second does not. Moreover, we show the ratio of two introduced measures is limited which can be a starting point to introduction of a new estimation method of stability index for analyzed Ornstein-Uhlenbeck processes.

Spatial components dependence for bidimensional time-constant AR(1) model with $$\alpha $$-stable noise and triangular coefficients matrix

In this paper, we examine the bidimensional time-constant autoregressive model of order 1 with $$\alpha $$ α -stable noise. We focus on the case of the triangular coefficients matrix for which one of the spatial components of the model simplifies to the one-dimensional autoregressive time series. We study the asymptotic behaviour of the cross-codifference and the cross-covariation applied to describe the dependence in time between the spatial components of the model. As a result, we formulate the theorem about the asymptotic relation between both measures, which is consistent with the result that is correct for the case of the non-triangular coefficients matrix.

New estimation method for periodic autoregressive time series of order 1 with additive noise

The periodic behavior of real data can be manifested in the time series or in its characteristics. One of the characteristics that often manifests the periodic behavior is the sample autocovariance function. In this case, the periodically correlated (PC) behavior is considered. One of the main models that exhibits PC property is the periodic autoregressive (PARMA) model that is considered as the generalization of the classical autoregressive moving average (ARMA) process. However, when one considers the real data, practically the observed trajectory corresponds to the “pure” model with the additional noise which is a result of the noise of the measurement device or other external forces. Thus, in this paper we consider the model that is a sum of the periodic autoregressive (PAR) time series and the additive noise with finite-variance distribution. We present the main properties of the considered model indicating its PC property. One of the main goals of this paper is to introduce the new estimation method for the considered model’s parameters. The novel algorithm takes under consideration the additive noise in the model and can be considered as the modification of the classical Yule–Walker algorithm that utilizes the autocovariance function. Here, we propose two versions of the new method, namely the classical and the robust ones. The effectiveness of the proposed methodology is verified by Monte Carlo simulations. The comparison with the classical Yule–Walker method is presented. The approach proposed in this paper is universal and can be applied to any finite-variance models with the additive noise.

Fractional differentiation and its use in machine learning

This article covers the implementation of fractional (non-integer order) differentiation on real data of four datasets based on stock prices of main international stock indexes: WIG 20, S&P 500, DAX and Nikkei 225. This concept has been proposed by Lopez de Prado [5] to find the most appropriate balance between zero differentiation and fully differentiated time series. The aim is making time series stationary while keeping its memory and predictive power. In addition, this paper compares fractional and classical differentiation in terms of the effectiveness of artificial neural networks. Root mean square error (RMSE) and mean absolute error (MAE) are employed in this comparison. Our investigations have determined the conclusion that fractional differentiation plays an important role and leads to more accurate predictions in case of ANN.