Modeling and forecasting of nonlinear nonstationary processes based on the Bayesian structural time series

Abstract

The article describes an approach to modelling and forecasting non-linear non-stationary time series for various purposes using Bayesian structural time series. The concepts of non-linearity and non-stationarity, as well as methods for processing non-linearity’sand non-stationarity in the construction of forecasting models are considered. The features of the Bayesian approach in the processing of nonlinearities and nonstationaryare presented. An approach to the construction of probabilistic-statistical models based on Bayesian structural models of time series has been studied. Parametric and non-parametric methods for forecasting non-linear and non-stationary time series are considered. Parametric methods include methods: classical autoregressive models, neural networks, models of support vector machines, hidden Markov models. Non-parametric methods include methods: state-space models, functional decomposition models, Bayesian non-parametric models. One of the types of non-parametric models isBayesian structural time series. The main features of constructing structural time series are considered. Models of structural time series are presented. The process of learning the Bayesianstructural model of time series is described. Training is performed in four stages: setting the structure of the model and a priori probabilities; applying a Kalman filter to update state estimates based on observed data;application of the “spike-and-slab”method to select variables in a structural model; Bayesian averaging to combine the results to make a prediction. An algorithm for constructing a Bayesian structural time seriesmodel is presented. Various components of the BSTS model are considered andanalysed, with the help of which the structures of alternative predictive models are formed. As an example of the application of Bayesian structural time series, the problem of predicting Amazon stock prices is considered. The base dataset is amzn_share. After loading, the structure and data types were analysed, and missing values were processed. The data are characterized by irregular registration of observations, which leads to a large number of missing values and “masking” possible seasonal fluctuations. This makes the task of forecasting rather difficult. To restore gaps in the amzn_sharetime series, the linear interpolation method was used. Using a set of statistical tests (ADF, KPSS, PP), the series was tested for stationarity. The data set is divided into two parts: training and testing. The fitting of structural models of time series was performed using the Kalman filterand the Monte Carlo method according to the Markov chain scheme. To estimate and simultaneously regularize the regression coefficients, the spike-and-slab method was applied. The quality of predictive models was assessed.

Systematic use of nonlinear data filtering methods in forecasting tasks

“…ARIMA statistical models, the method of fitting generalized additive models (GAM), Bayesian structural time series models (BSTS) [33] and forward propagation artificial neural networks (NNAR) are used in the modeling block as basic forecasting models. These methods were chosen because of their ability to recognize complex patterns in time series.…”

“…If an improvement in forecast accuracy is not found, it is necessary to return to the stage of forming basic models, or to change their number and type of combination. Such a structural scheme fully corresponds to the process of building combined forecasts for time series based on simple averaging of forecasts, weighted combination of forecasts and regression [33]. To increase the accuracy of the combined forecast, the forecasting procedure is performed on the models with close variance values.…”