Nonlinear Nonstationary Time Series Research Articles

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

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.

Reconstruction of driving forces from nonstationary time series including stationary regions and application to climate change

A new method based on cross-prediction errors is proposed for the rebuilding of the driving forces exerting influence on nonstationary systems. This method can retrieve driving force from nonlinear nonstationary time series including stationary regions. The numerical tests show that the method is very accurate. Generally, the correlation coefficients between the original and reconstructed driving force are larger than 0.96 for three commonly used maps for single-species discrete chaotic ecosystems. By using the above method, we reconstructed the driving force for an observed time series of the northern hemisphere monthly mean surface air temperature anomalies. Wavelet transformation method is applied to analyze the calculated driving forces. The results illustrated that the driving forces are characterized by the Hale’s solar cycle and the Atlantic Multi-decadal Oscillation, which are the two main separate degrees of freedom on the climate.

Processing Uncertainties in Modeling Nonstationary Time Series Using Decision Support Systems

Background . Forecasting of nonlinear nonstationary time series (NNTS) is important problem in economics, marketing, industry, ecology and many other branches of science and practical activities. Successful solution of the problem requires development of modern computer based decision support systems (DSS) capable to generate reliable estimates of forecasts in conditions of uncertainty of various type and origin. Objective . The purpose of the research is as follows: development of requirements to the modern DSS and their formal representation; analysis of uncertainty types characteristic for model building and forecasting; selection of techniques for taking into consideration of the uncertainties; and illustration of the system application to solving the problem of forecasts estimation for heteroscedastic NNTS using statistical data. Methods . To reach the objectives stated the following methods were used: systemic approach to statistical data analysis; statistical approach to identification and taking into consideration of possible uncertainties; Kalman filtering techniques; Bayesian programming approach and statistical criteria of model adequacy and quality of forecasts. Results . Formal description of the DSS is provided, and requirements to its development are given; the classes of mathematical methods necessary for DSS implementation are proposed; some approaches to formal taking into consideration of probabilistic, statistical and parametric uncertainties are discussed; and illustrating example of the DSS application is considered. Conclusions . Systemic approach to DSS constructing for solving the problem of nonlinear nonstationary time series forecasting turned out to be very fruitful. Using the system proposed it is possible to take into consideration various uncertainties of probabilistic, statistical and parametric type, and to compute high quality estimates of short and medium term forecasts for NNTS. The approach proposed has good perspectives for the future improvements and enhancement.

Noise Reduction for Nonlinear Nonstationary Time Series Data using Averaging Intrinsic Mode Function

A novel noise filtering algorithm based on averaging Intrinsic Mode Function (aIMF), which is a derivation of Empirical Mode Decomposition (EMD), is proposed to remove white-Gaussian noise of foreign currency exchange rates that are nonlinear nonstationary times series signals. Noise patterns with different amplitudes and frequencies were randomly mixed into the five exchange rates. A number of filters, namely; Extended Kalman Filter (EKF), Wavelet Transform (WT), Particle Filter (PF) and the averaging Intrinsic Mode Function (aIMF) algorithm were used to compare filtering and smoothing performance. The aIMF algorithm demonstrated high noise reduction among the performance of these filters.

A comment on “An analysis of global warming in the Alpine Region based on nonlinear nonstationary time series models” by F. Battaglia and M. K. Protopapas

We discuss the paper by Battaglia and Protopapas concerning the analysis of global warming phenomenon in the Alpine Region. The Authors consider a nonlinear model which takes into account regimes in time and levels. In this contribution some of the statistical results presented in the paper are commented and a different approach to the problem is proposed. It is based on a temporal aggregation analysis and it can help to highlight some features in the data.

Rejoinder to the discussion of “An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models”

Let us first of all thank the Editors for organizing such a wide, deep and interesting discussion, and the authoritative scientists who accepted to give their contribution. We are particularly proud that Professor Tong participated the discussion, since he is the father of the threshold principle, and also the first proposal of a nonlinear nonstationary threshold model is due to him. A first general suggestion is to widen the proposed analysis, taking into account at least monthly data for analyzing seasonality, as so convincingly argued by Professor Bohm, and spatial relationships possibly integrating a dynamic cluster analysis. This is undoubtedly a crucial point, and calls for further study. Moreover, the present paper is essentially a statistical analysis based on a particular model, by no means exhaustive, while a deeper study would require a more specific climate knowledge. We are particularly honored that two authoritative climatologists accepted to read our paper and to contribute this discussion, and hope this may be an occasion for future fruitful collaboration. A second general question, as raised by Professor Piccolo, is more delicate and concerns the concept of data generating process, and the extent a model should try to reproduce it rather than privileging fitting. This is a controversial issue: we reported Rissanen’s position, Professor Tong cites the famous Box’s dictum all models are wrong but some are useful, Professor Piccolo’s position seems partially different. Such an important issue, dating back at least to themeasurement without theory debate (Koopmans 1947), cannot be appropriately discussed here; we just confirm that our

Discussion of ‘An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models’ by Battaglia and Protopapas

This discussion does not go into specific statistical details. It expresses the impressions of a climatologist when reading a paper about statistical methodologies applied on climate timeseries. Thus the main goal of the discussion is a typical interdisciplinary one. Some of the statistical results of the Battaglia and Protopapas paper are commented versus findings and also questions arising from the underlying physics of the climate system including also some climate impacts.

Discussion of “An analysis of global warming in the Alpine region based of nonlinear nonstationary time series models” by F. Battaglia and M. K. Protopapas

We discuss the scientific contribution of Battaglia and Protopapas’ paper concerning the debate on global warming supported by an extensive analysis of temperature time series in the Alpine region. In the work, Authors use several exploratory and modelling tools for assessing and discriminating the presence of different patterns in the data. We add some general and specific considerations mainly devoted to the modelling stage of their analysis.

An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models

The annual temperatures recorded for the last two centuries in fifteen european stations around the Alps are analyzed. They show a global warming whose growth rate is not however constant in time. An analysis based on linear Arima models does not provide accurate results. Thus, we propose threshold nonlinear nonstationary models based on several regimes both in time and in levels. Such models fit all series satisfactorily, allow a closer description of the temperature changes evolution, and help to discover the essential differences in the behavior of the different stations.

Discussion of “An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models” by F. Battaglia and M. K. Protopapas

The paper by Battaglia and Protopapas (Stat Method Appl 2012) is stimulating. It gives an elegant mathematical generalization of autoregressive models (the nine types). It explains state-of-the-art model fitting techniques (genetic algorithm combined with fitness function and least squares). It is written in a fluent and authoritative manner. Important for having a wider impact: it is accessible to non-statisticians. Finally, it has interesting results on the temperature evolution over the instrumental period (roughly the past 200 years). These merits make this paper an important contribution to applied statistics as well as climatology. As a climate researcher, coming from Physics and having had an affiliation with a statistical institute only as postdoc, I re-analyse here three data series with the aim of providing motivation for model selection and interpreting the results from the climatological perspective.

Discussion of ‘An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models’ by Battaglia and Protopapas

This discussion focuses on threshold nonstationary–nonlinear time series modelling; it raises various issues to do with identifiability and model complexity. It also gives some background history concerning smooth threshold/transition autoregressive models and hidden Markov switching models.

Multi–regime models for nonlinear nonstationary time series

Nonlinear nonstationary models for time series are considered, where the series is generated from an autoregressive equation whose coefficients change both according to time and the delayed values of the series itself, switching between several regimes. The transition from one regime to the next one may be discontinuous (self-exciting threshold model), smooth (smooth transition model) or continuous linear (piecewise linear threshold model). A genetic algorithm for identifying and estimating such models is proposed, and its behavior is evaluated through a simulation study and application to temperature data and a financial index.

ITERATIVE FILTERING AS AN ALTERNATIVE ALGORITHM FOR EMPIRICAL MODE DECOMPOSITION

The empirical mode decomposition (EMD) was a method pioneered by (N. Huang et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear nonstationary time series analysis, Proc. Roy. Soc. Lond. A454 (1998) 903–995) as an alternative technique to the traditional Fourier and wavelet techniques for studying signals. It decomposes a signal into several components called intrinsic mode functions (IMFs), which have shown to admit better behaved instantaneous frequencies via Hilbert transforms. In this paper, we propose an alternative algorithm for EMD based on iterating certain filters, such as Toeplitz filters. This approach yields similar results as the more traditional sifting algorithm for EMD. In many cases the convergence can be rigorously proved.