Mixture Transition Distribution Model Research Articles

An Intention-Aware Markov Chain Based Method for Top-K Recommendation

Recommender systems play significant roles in business, especially in e-commerce. Nevertheless, users’ behaviors are usually mixing and drifting, which is hard to tackle. Current sequential methods of item-wise interest extracting suffer from the intricacy and sparsity of data. Inspired by that category-wise interests may be intrinsic drivers of user behaviors and heterogeneous actions may reveal certain behavior patterns, we introduce intention as a tuple of category and action to address the data issues. In this paper, an intention-aware Markov chain based sequential recommendation model (IMRec) is proposed. We model the overall preferences of users as the integration of long-term preferences and short-term intents. In particular, the matrix factorization method is adopted to extract the long-term user-item preference. For the modeling of the short-term intents transition, we adopt high-order Markov chain based methods. A factorized mixture transition distribution model for high-order Markov chain approximation is leveraged in this paper to reduce the algorithm complexity. An auxiliary loss on intention representation is utilized, which brings considerable performance improvements. Experiments on real-world datasets demonstrate that our model outperforms state-of-the-art baseline models in terms of three common metrics, and shows superior stability, scalability, and training efficiency. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In e-commerce, sequential recommenders are essential to facilitate decision-making and promote business. A big challenge is to capture the sequential patterns from the mixing and drifting user-item interaction sequences. Customer behaviors are often driven by their inherent intentions, which may present more stable and reliable patterns. Motivated by that, we propose a novel intention-aware next-item recommendation algorithm with high performance. Our method models the user-item preferences as the integration of long-term preferences and short-term intents. More specifically, long-term preferences show the general tastes of users, which are modeled by latent factors of users and items. Additionally, the intention is defined as a tuple of category and action. We model the short-term intents as the intention transition from the past intentions by a high-order Markov chain. We further leverage a factorization-based mixture transition distribution model for high-order Markov chain approximation and reduce the algorithm complexity. The proposed model is validated in 4 real-world datasets and shows good recommendation performance, performance stability, model scalability, and training efficiency. Our model provides an effective recommendation method at low computational costs for e-commerce companies.

The Mixture Transition Distribution approach to networks: Evidence from stock markets

Networks can be built by using correlations between time series. The approach based on correlations has many advantages which are essentially related to its simplicity. Nevertheless, it is well known that time series may show strong dependence even if they are uncorrelated. In this paper, we will advance a multivariate Markov chain model based on the Mixture Transition Distribution (MTD) model to build networks between time series. The multivariate MTD is able to consider the dependence between time series and, at the same time, reduce the number of parameters to be estimated compared to the classical multivariate Markov chain. We show, by a numerical example, that the multivariate MTD outperforms the classical correlation approach. Moreover, using the same model, we build the network of the 30 constituents of the Dow Jones index showing the usefulness of the methodology in real problems in financial markets.

Estimation and Selection for High-Order Markov Chains with Bayesian Mixture Transition Distribution Models

We develop a mixture model and diagnostic for Bayesian estimation and selection in high-order, discrete-state Markov chains. Both extend the mixture transition distribution, which constructs a transition probability tensor by aggregating probabilities from a set of single-lag transition matrices, through inclusion of mixture components dependent on multiple lags. We demonstrate two uses for the proposed model: identification of relevant lags through over-specification and shrinkage via priors for sparse probability vectors, and parsimonious approximation of multi-lag dynamics by mixing low-order transition models. The diagnostic yields a general and interpretable mixture decomposition for transition probability tensors estimated by any means. We demonstrate the utility of the model and diagnostic with simulation studies, and further apply the methodology to a data analysis from the high-order Markov chain literature, and to a time series of pink salmon abundance in Alaska, United States. Supplemental files for this article are available online.

On Construction and Estimation of Stationary Mixture Transition Distribution Models

Mixture transition distribution (MTD) time series models build high-order dependence through a weighted combination of first-order transition densities for each one of a specified number of lags. We present a framework to construct stationary MTD models that extend beyond linear, Gaussian dynamics. We study conditions for first-order strict stationarity which allow for different constructions with either continuous or discrete families for the first-order transition densities given a prespecified family for the marginal density, and with general forms for the resulting conditional expectations. Inference and prediction are developed under the Bayesian framework with particular emphasis on flexible, structured priors for the mixture weights. Model properties are investigated both analytically and through synthetic data examples. Finally, Poisson and Lomax examples are illustrated through real data applications. Supplementary files for this article are available online.

Autoregressive density modeling with the Gaussian process mixture transition distribution

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 active, and averages over multiple and competing single‐lag models to quantify/propagate uncertainty. Sparsity‐inducing priors on the mixture weights aid in selecting a subset of active lags. The hierarchical model specification follows conventions for both GP regression and MTD models, admitting a convenient Gibbs sampling scheme for posterior inference. We demonstrate properties of the proposed model with two simulated and two real time series, emphasizing approximation of lag‐dependent transition densities and model selection. In most cases, the model decisively recovers important features. The proposed model provides a simple, yet flexible framework that preserves useful and distinguishing characteristics of the MTD model class.

A Multivariate High-Order Markov Model for the Income Estimation of a Wind Farm

The energy produced by a wind farm in a given location and its associated income depends both on the wind characteristics in that location—i.e., speed and direction—and the dynamics of the electricity spot price. Because of the evidence of cross-correlations between wind speed, direction and price series and their lagged series, we aim to assess the income of a hypothetical wind farm located in central Italy when all interactions are considered. To model these cross and auto-correlations efficiently, we apply a high-order multivariate Markov model which includes dependencies from each time series and from a certain level of past values. Besides this, we used the Raftery Mixture Transition Distribution model (MTD) to reduce the number of parameters to get a more parsimonious model. Using data from the MERRA-2 project and from the electricity market in Italy, we estimate the model parameters and validate them through a Monte Carlo simulation. The results show that the simulated income faithfully reproduces the empirical income and that the multivariate model also closely reproduces the cross-correlations between the variables. Therefore, the model can be used to predict the income generated by a wind farm.

The Convex Mixture Distribution: Granger Causality for Categorical Time Series.

We present a framework for learning Granger causality networks for multivariate categorical time series based on the mixture transition distribution (MTD) model. Traditionally, MTD is plagued by a nonconvex objective, non-identifiability, and presence of local optima. To circumvent these problems, we recast inference in the MTD as a convex problem. The new formulation facilitates the application of MTD to high-dimensional multivariate time series. As a baseline, we also formulate a multi-output logistic autoregressive model (mLTD), which while a straightforward extension of autoregressive Bernoulli generalized linear models, has not been previously applied to the analysis of multivariate categorial time series. We establish identifiability conditions of the MTD model and compare them to those for mLTD. We further devise novel and efficient optimization algorithms for MTD based on our proposed convex formulation, and compare the MTD and mLTD in both simulated and real data experiments. Finally, we establish consistency of the convex MTD in high dimensions. Our approach simultaneously provides a comparison of methods for network inference in categorical time series and opens the door to modern, regularized inference with the MTD model.

Optimization of the Mixture Transition Distribution Model Using the March Package for R

Optimization of mixture models such as the mixture transition distribution (MTD) model is notoriously difficult because of the high complexity of their solution space. The best approach comprises combining features of two types of algorithms: an algorithm that can explore as completely as possible the whole solution space (e.g., an evolutionary algorithm), and another that can quickly identify an optimum starting from a set of initial conditions (for instance, an EM algorithm). The march package for the R environment is a library dedicated to the computation of Markovian models for categorical variables. It includes different algorithms that can manage the complexity of the MTD model, including an ad hoc hill-climbing procedure. In this article, we first discuss the problems related to the optimization of the MTD model, and then we show how march can be used to solve these problems; further, we provide different syntaxes for the computation of other models, including homogeneous Markov chains, hidden Markov models, and double chain Markov models.

The Use of a Hidden Mixture Transition Distribution Model in Clustering Few but Long Continuous Sequences: An Illustration with Cognitive Skills Data

In accordance with the theme of this special issue, we present a model that indirectly discovers symmetries and asymmetries between past and present assessments within continuous sequences. More specifically, we present an alternative use of a latent variable version of the Mixture Transition Distribution (MTD) model, which allows for clustering of continuous longitudinal data, called the Hidden MTD (HMTD) model. We compare the HMTD and its clustering performance to the popular Growth Mixture Model (GMM), as well as to the recently introduced GMM based on individual case residuals (ICR-GMM). The GMM and the ICR-GMM contrast with HMTD, because they are based on an explicit change function describing the individual sequences on the dependent variable (here, we implement a non-linear exponential change function). This paper has three objectives. First, it introduces the HMTD. Second, we present the GMM and the ICR-GMM and compare them to the HMTD. Finally, we apply the three models and comment on how the conclusions differ depending on the clustering model, when using a specific dataset in psychology, which is characterized by a small number of sequences (n = 102), but that are relatively long (for the domains of psychology and social sciences: t = 20). We use data from a learning experiment, in which healthy adults (19–80 years old) were asked to perform a perceptual–motor skills over 20 trials.

ECF-MRS: An Efficient and Collaborative Framework With Markov-Based Reputation Scheme for IDSs in Vehicular Networks

Vehicle Ad Hoc Networks (VANETs) are considered to be a next big thing that will remarkably change our lives, since this kind of technology is able to make our lives and roads safer. Intrusion Detection Systems (IDS) is an important technology that can mitigate both inner and outer threats for the vulnerable networks like VANETs. However, it is difficult to adopt the same IDSs that have been appropriately used in wired networks, due to the fast moving and highly dynamic nature of VANETs. Thus, in this article, an Efficient and Collaborative Framework with a Markov-based Reputation Scheme is proposed, namely ECF-MRS. In the proposed framework, the collaborative mechanism is achieved by using Non-dominant Sorting Genetic Algorithm-III (NSGA-III)-Collaboration to merge the advantages of IDSs in VANETs to generate a more superior IDS, while Non-Linear Programming (NLP)-Optimization is designed as the efficient mechanism to reduce the execution time of IDSs in VANETs. Moreover, considering the security risks of collaboration, a Reputation Scheme based on the Hidden Generalized Mixture Transition Distribution (HgMTD) model, namely RS-HgMTD, is proposed for each vehicle in VANETs to evaluate the creditworthiness of their neighbors. Experiments show that the IDS with ECF-MRS has better performance than other existing IDSs in terms of detection rate, detection time and overhead.

Confidence Intervals for the Mixture Transition Distribution (MTD) Model and Other Markovian Models

The Mixture Transition Distribution (MTD) model used for the approximation of high-order Markov chains does not allow a simple calculation of confidence intervals, and computationnally intensive methods based on bootstrap are generally used. We show here how standard methods can be extended to the MTD model as well as other models such as the Hidden Markov Model. Starting from existing methods used for multinomial distributions, we describe how the quantities required for their application can be obtained directly from the data or from one run of the E-step of an EM algorithm. Simulation results indicate that when the MTD model is estimated reliably, the resulting confidence intervals are comparable to those obtained from more demanding methods.

A Filter Model Based on Hidden Generalized Mixture Transition Distribution Model for Intrusion Detection System in Vehicle Ad Hoc Networks

Vehicle ad hoc networks (VANETs) are considered to be the next big thing that will remarkably change our lives, since this kind of technology is able to make our lives and roads safer. Due to the very fast move and high dynamic in VANETs, it is important to quickly ascertain the reliability of information. Although intrusion detection system (IDS) has been proposed as a reliable approach to protect VANETs against attacks, its overhead is serious, which spends too much time on detection, especially when the number of vehicles increases. Thus, in this paper, we propose a novel filter model based on a hidden generalized mixture transition distribution model (HgMTD) in VANETs, called FM-HgMTD, which can quickly filter the messages from neighboring vehicles so as to reduce the overhead and detection time. It adopts a well-known multi-objective optimization (NSGA-II) algorithm combined with an expectation-maximization (EM) algorithm to forecast the future states of neighboring vehicles and then to filter out malicious messages, by monitoring the change of the state pattern of each neighboring vehicle. In addition, a timeliness method is used to maintain the accuracy of the forecast. The experiments show that IDS with the proposed FM-HgMTD has better performance than other available IDSs in terms of detection rate, detection time, and overhead.

Unsupervised approach for detecting shilling attacks in collaborative recommender systems based on user rating behaviours

Collaborative recommender systems have been known to be extremely vulnerable to shilling attacks. To prevent such attacks, many detection approaches including supervised and unsupervised have been proposed. However, the supervised approaches are only suitable for detecting known types of attacks and the unsupervised approaches require a priori knowledge to ensure the detection performance. To address the limitations, the authors propose an unsupervised approach for detecting shilling attacks based on user rating behaviours. They first use Gibbs latent Dirichlet allocation model to extract latent topics of user preferences from user rating item sequences, then they use mixture transition distribution model to construct the user's preference model and present several metrics to capture the diversity between genuine and attack users in rating behaviours. In the case of unknown attack size, the number of attack users is obtained by analysing the critical point of rating behaviour suspicious degrees between genuine and attack users, and based on which the attack users are identified. The experimental results on the MovieLens 1 M dataset show that the proposed approach outperforms the baseline methods in terms of recall and precision metrics.

Exploring the randomness of mentally generated head–tail sequences

It is well known that people deviate from randomness as they attempt to mentally generate head–tail sequences as randomly as possible. This deviation from randomness is quantified by an excess of repetitions or alternations between successive responses more than would be expected by chance. We conducted an experiment in which a sample of students was asked to mentally simulate a sequence as if it is produced by a fair coin. We propose several models based on Markov chains for analysing the dynamic of head–tail outcomes in these sequences. First, we explore observed Markov chains and suggest some practical solutions to reduce the number of parameters. However, there is a need for more sophisticated models, and in this case, we propose latent Markov models and mixture of Markov chains to analyse these head–tail sequences. A generalization of the so-called mixture transition distribution (MTD) model is also considered.

Linear models for the impact of order flow on prices. II. The Mixture Transition Distribution model

Modelling the impact of the order flow on asset prices is of primary importance to understand the behaviour of financial markets. Part I of this paper reported the remarkable improvements in the description of the price dynamics which can be obtained when one incorporates the impact of past returns on the future order flow. However, impact models presented in Part I consider the order flow as an exogenous process, only characterised by its two-point correlations. This assumption seriously limits the forecasting ability of the model. Here we attempt to model directly the stream of discrete events with a so-called Mixture Transition Distribution (MTD) framework, introduced originally by Raftery [J. R. Stat. Soc. Ser. B, 1985, 528–539]. We distinguish between price-changing and non price-changing events and combine them with the order sign in order to reduce the order flow dynamics to the dynamics of a four-state discrete random variable. The MTD represents a parsimonious approximation of a full high-order Markov chain. The new approach captures with adequate realism the conditional correlation functions between signed events for both small and large tick stocks and signature plots. From a methodological point of view, constraining the MTD within the class of ergodic Markov models, and exploiting the buy–sell symmetry of the data, we propose a weak restriction on the transition matrices which solves the problem of identifiability of mixture models. In spite of the large number of parameters, this translates into a feasible and robust estimation procedure. Out-of-sample analyses demonstrate that the model does not overfit the data.

A Direct Local Search Method and its Application to a Markovian Model

While the hidden mixture transition distribution (HMTD) model is a powerful framework for the description, analysis, and classification of longitudinal sequences of continuous data, it is notoriously difficult to estimate because of the complexity of its solution space. In this paper, we explore how a new heuristic specifically developed for the HMTD performs compared to different standard optimization algorithms. This specific heuristic can be classified as a hill-climbing method, and different variants are proposed, including a jittering procedure to escape local maxima and measures to speed up the convergence. Different popular approaches are used for comparison, including PSO, SA, GA, NM, L-BFGS-B, and DE. The same HMTD model was optimized on different datasets and the results were compared in terms of both fit to the data and estimated parameters. Even if the complexity of the problem implies that no one algorithm can be considered as an overall best, our heuristic performed well in all situations, leading to useful solutions in terms of both fit and interpretability.

Linear Models for the Impact of Order Flow on Prices II. The Mixture Transition Distribution Model

Linear Models for the Impact of Order Flow on Prices II. The Mixture Transition Distribution Model

Coherent forecasting for stationary time series of discrete data

Coherent forecasting for discrete-valued stationary time series is considered in this article. In the context of count time series, different methods of coherent forecasting such as median forecasting and mode forecasting are used to obtain \(h\)-step ahead coherent forecasting. However, there are not many existing works in the context of categorical time series. Here, we consider the case of a finite number of categories with different possible models, such as the Pegram’s operator-based ARMA(\(p\),\(q\)) model, the mixture transition distribution model and the logistic regression model, and study their \(h\)-step ahead coherent forecasting. Some theoretical results are derived along with some numerical examples. To facilitate comparison among the three models, we use some forecasting measures. The procedure is illustrated using one real-life categorical data, namely the infant sleep status data.

General framework and model building in the class of Hidden Mixture Transition Distribution models

Modeling time series that present non-Gaussian features plays as central role in many fields, including finance, seismology, psychological, and life course studies. The Hidden Mixture Transition Distribution model is an answer to the complexity of such series. The observed heterogeneity can be induced by one or several latent factors, and each level of these factors is related to a different component of the observed process. The time series is then treated as a mixture and the relation between the components is governed by a Markovian latent transition process. This framework generalizes several specifications that appear separately in related literature. Both the expectation and the standard deviation of each component are allowed to be functions of the past of the process. The latent process can be of any order, and can be modeled using a discrete Mixture Transition Distribution. The effects of covariates at the visible and hidden levels are also investigated. One of the main difficulties lies in correctly specifying the structure of the model. Therefore, we propose a hierarchical model selection procedure that exploits the multilevel structure of our approach. Finally, we illustrate the model and the model selection procedure through a real application in social science.

A New Model for Multivariate Markov Chains

ABSTRACTWe propose a new model for multivariate Markov chains of order one or higher on the basis of the mixture transition distribution (MTD) model. We call it the MTD‐Probit. The proposed model presents two attractive features: it is completely free of constraints, thereby facilitating the estimation procedure, and it is more precise at estimating the transition probabilities of a multivariate or higher‐order Markov chain than the standard MTD model.