Obtaining reliable estimates of conditional covariance matrices is an important task of heteroskedastic multivariate time series. In portfolio optimization and financial risk management, it is crucial to provide measures of uncertainty and risk as accurately as possible. We propose using mixture vector autoregressive (MVAR) models for portfolio optimization. Combining a mixture of distributions that depend on the recent history of the process, MVAR models can accommodate asymmetry, multimodality, heteroskedasticity and cross-correlation in multivariate time series data. For mixtures of Normal components, we exploit a property of the multivariate Normal distribution to obtain explicit formulas of conditional predictive distributions of returns on a portfolio of assets. After showing how the method works, we perform a comparison with other relevant multivariate time series models on real stock return data.
This paper introduces a fully Bayesian analysis of mixture autoregressive models with Student t components. With the capacity of capturing the behaviour in the tails of the distribution, the Student t MAR model provides a more flexible modelling framework than its Gaussian counterpart, leading to fitted models with fewer parameters and of easier interpretation. The degrees of freedom are also treated as random variables, and hence are included in the estimation process.
Analysis of multivariate time series is a common problem in areas like finance and economics. The classical tool 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 which allows for non-linear and asymmetric 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 demonstrate its usefulness in three relevant applicati…
Lung tumor tracking for radiotherapy requires real-time, multiple-step ahead forecasting of a quasi-periodic time series recording instantaneous tumor locations. We introduce a location-mixture autoregressive (LMAR) process that admits multimodal conditional distributions, fast approximate inference using the EM algorithm and accurate multiple-step ahead predictive distributions. LMAR outperforms several commonly used methods in terms of out-of-sample prediction accuracy using clinical data from lung tumor patients. With its superior predictive performance and real-time computation, the LMAR m…
This report introduces a parsimonious structure for mixture of autoregressive models, where the weighting coefficients are determined through latent random variables as functions of all past observations. These variables follow a hidden Markov model. We modify EM and Baum-Welch algorithms to estimate the parameters of the model.
Testing for regime switching when the regime switching probabilities are specified either as constants (`mixture models') or are governed by a finite-state Markov chain (`Markov switching models') are long-standing problems that have also attracted recent interest. This paper considers testing for regime switching when the regime switching probabilities are time-varying and depend on observed data (`observation-dependent regime switching'). Specifically, we consider the likelihood ratio test for observation-dependent regime switching in mixture autoregressive models. The testing problem is hig…
A new mixture autoregressive model based on Student's $t$-distribution is proposed. A key feature of our model is that the conditional $t$-distributions of the component models are based on autoregressions that have multivariate $t$-distributions as their (low-dimensional) stationary distributions. That autoregressions with such stationary distributions exist is not immediate. Our formulation implies that the conditional mean of each component model is a linear function of past observations and the conditional variance is also time varying. Compared to previous mixture autoregressive models ou…
Mixture autoregressive (MAR) models provide a flexible way to model time series with predictive distributions which depend on the recent history of the process and are able to accommodate asymmetry and multimodality. Bayesian inference for such models offers the additional advantage of incorporating the uncertainty in the estimated models into the predictions. We introduce a new way of sampling from the posterior distribution of the parameters of MAR models which allows for covering the complete parameter space of the models, unlike previous approaches. We also propose a relabelling algorithm …
We propose a new class of models specifically tailored for spatio-temporal data analysis. To this end, we generalize the spatial autoregressive model with autoregressive and heteroskedastic disturbances, i.e. SARAR(1,1), by exploiting the recent advancements in Score Driven (SD) models typically used in time series econometrics. In particular, we allow for time-varying spatial autoregressive coefficients as well as time-varying regressor coefficients and cross-sectional standard deviations. We report an extensive Monte Carlo simulation study in order to investigate the finite sample properties…
This paper introduces a new parsimonious structure for mixture of autoregressive models. the weighting coefficients are determined through latent random variables, following a hidden Markov model. We propose a dynamic programming algorithm for the application of forecasting. We also derive the limiting behavior of unconditional first moment of the process and an appropriate upper bound for the limiting value of the variance. This can be considered as long run behavior of the process. Finally we show convergence and stability of the second moment. Further, we illustrate the efficacy of the prop…
This paper develops tests of the null hypothesis of linearity in the context of autoregressive models with Markov-switching means and variances. These tests are robust to the identification failures that plague conventional likelihood-based inference methods. The approach exploits the moments of normal mixtures implied by the regime-switching process and uses Monte Carlo test techniques to deal with the presence of an autoregressive component in the model specification. The proposed tests have very respectable power in comparison to the optimal tests for Markov-switching parameters of Carrasco…
Using a proper model to characterize a time series is crucial in making accurate predictions. In this work we use time-varying autoregressive process (TVAR) to describe non-stationary time series and model it as a mixture of multiple stable autoregressive (AR) processes. We introduce a new model selection technique based on Gap statistics to learn the appropriate number of AR filters needed to model a time series. We define a new distance measure between stable AR filters and draw a reference curve that is used to measure how much adding a new AR filter improves the performance of the model, a…
A regularized vector autoregressive hidden semi-Markov model is developed to analyze multivariate financial time series with switching data generating regimes. Furthermore, an augmented EM algorithm is proposed for parameter estimation by embedding regularized estimators for the state-dependent covariance matrices and autoregression matrices in the M-step. The performance of the proposed regularized estimators is evaluated both in the simulation experiments and on the New York Stock Exchange financial portfolio data.
The purpose of this paper is to propose a time-varying vector autoregressive model (TV-VAR) for forecasting multivariate time series. The model is casted into a state-space form that allows flexible description and analysis. The volatility covariance matrix of the time series is modelled via inverted Wishart and singular multivariate beta distributions allowing a fully conjugate Bayesian inference. Model performance and model comparison is done via the likelihood function, sequential Bayes factors, the mean of squared standardized forecast errors, the mean of absolute forecast errors (known al…
Many relevant statistical and econometric models for the analysis of longitudinal data include a latent process to account for the unobserved heterogeneity between subjects in a dynamic fashion. Such a process may be continuous (typically an AR(1)) or discrete (typically a Markov chain). In this paper, we propose a model for longitudinal data which is based on a mixture of AR(1) processes with different means and correlation coefficients, but with equal variances. This model belongs to the class of models based on a continuous latent process, and then it has a natural interpretation in many co…
A system identification approach to clustering vector autoregressive time series
School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, China（华中科技大学人工智能与自动化学院）；School of Electrical Engineering and Telecommunications, UNSW Sydney, NSW, Australia（新南威尔士大学悉尼分校电子与电信工程学院）
Clustering time series into similar groups can improve models by combining information across like time series. While there is a well developed body of literature for clustering of time series, these approaches tend to generate clusters independently of model training which can lead to poor model fit. We propose a novel distributed approach that simultaneously clusters and fits autoregression models for groups of similar individuals. We apply a Wishart mixture model so as to cluster individuals while modeling the corresponding autocovariance matrices at the same time. The fitted Wishart scale …
We develop a mixture model for transition density approximation, together with soft model selection, in the presence of noisy and heterogeneous nonlinear dynamics. Our model builds on the Gaussian mixture transition distribution (MTD) model for continuous state spaces, extending component means with nonlinear functions that are modeled using Gaussian process (GP) priors. The resulting model flexibly captures nonlinear and heterogeneous lag dependence when several mixture components are active, identifies low-order nonlinear dependence while inferring relevant lags when few components are activ…
In this paper, we propose multi-variable LSTM capable of accurate forecasting and variable importance interpretation for time series with exogenous variables. Current attention mechanism in recurrent neural networks mostly focuses on the temporal aspect of data and falls short of characterizing variable importance. To this end, the multi-variable LSTM equipped with tensorized hidden states is developed to learn hidden states for individual variables, which give rise to our mixture temporal and variable attention. Based on such attention mechanism, we infer and quantify variable importance. Ext…
In this thesis, we develop a comprehensive account of the expressive power, modelling efficiency, and performance advantages of so-called trading agents (i.e., Deep Soft Recurrent Q-Network (DSRQN) and Mixture of Score Machines (MSM)), based on both traditional system identification (model-based approach) as well as on context-independent agents (model-free approach). The analysis provides conclusive support for the ability of model-free reinforcement learning methods to act as universal trading agents, which are not only capable of reducing the computational and memory complexity (owing to th…
The standard approach to analyzing brain electrical activity is to examine the spectral density function (SDF) and identify predefined frequency bands that have the most substantial relative contributions to the overall variance of the signal. However, a limitation of this approach is that the precise frequency and bandwidth of oscillations vary with cognitive demands. Thus they should not be arbitrarily defined a priori in an experiment. In this paper, we develop a data-driven approach that identifies (i) the number of prominent peaks, (ii) the frequency peak locations, and (iii) their corres…
This paper considers the problem of measuring the credit risk in portfolios of loans, bonds, and other instruments subject to possible default under multi-factor models. Due to the amount of the portfolio, the heterogeneous effect of obligors, and the phenomena that default events are rare and mutually dependent, it is difficult to calculate portfolio credit risk either by means of direct analysis or crude Monte Carlo under such models. To capture the extreme dependence among obligors, we provide an efficient simulation method for multi-factor models with a normal mixture copula that allows th…
In this paper we consider the problem of dynamic clustering, where cluster memberships may change over time and clusters may split and merge over time, thus creating new clusters and destroying existing ones. We propose a Bayesian nonparametric approach to dynamic clustering via mixture modeling. Our approach relies on a novel time-dependent nonparametric prior defined by combining: i) a copula-based transformation of a Gaussian autoregressive process; ii) the stick-breaking construction of the Dirichlet process. Posterior inference is performed through a particle Markov chain Monte Carlo algo…
Ambulance demand estimation at fine time and location scales is critical for fleet management and dynamic deployment. We are motivated by the problem of estimating the spatial distribution of ambulance demand in Toronto, Canada, as it changes over discrete 2-hour intervals. This large-scale dataset is sparse at the desired temporal resolutions and exhibits location-specific serial dependence, daily and weekly seasonality. We address these challenges by introducing a novel characterization of time-varying Gaussian mixture models. We fix the mixture component distributions across all time period…
Spatio-temporal models for infection counts generally follow themes of the broader disease mapping literature, but may need to address specific features of spatio-temporal infection data including considerable time fluctuations (with epidemic phases) and spatial diffusion. Low order autoregression is a feature of several recent spatio-temporal studies of infection data, possibly with lags on both within area infections and on infections in adjacent areas. Many epidemic time series show a period of relatively stable infection levels (possibly characterized as endemicity), followed by a sudden s…
Bayesian Varying-Effects Vector Autoregressive Models for Inference of Brain Connectivity Networks and Covariate Effects in Pediatric Traumatic Brain Injury
Bayesian non-parametric methods based on Dirichlet process mixtures have seen tremendous success in various domains and are appealing in being able to borrow information by clustering samples that share identical parameters. However, such methods can face hurdles in heterogeneous settings where objects are expected to cluster only along a subset of axes or where clusters of samples share only a subset of identical parameters. We overcome such limitations by developing a novel class of product of Dirichlet process location-scale mixtures that enable independent clustering at multiple scales, wh…
