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’s
and non-stationarity in the construction of forecasting models are considered. The features of the Bayesian approach in the
processing of nonlinearities and nonstationary are 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 is Bayesian 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 Bayesian structural 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 series model is presented. Various components of the
BSTS model are considered and analysed, 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_share time 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 filter and
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