A REVIEW OF PROBABILISTIC GRAPH MODELS FOR FEATURE SELECTION WITH APPLICATIONS IN ECONOMIC AND FINANCIAL TIME SERIES FORECASTING

In every field of life, people are interested to be able to forecast future.  A number of techniques are available to predict and forecasting upto a certain level of accuracy. Many techniques involve statistical tools and techniques for forecasting, modeling and control. Use of statistical techniques is growing with time and new techniques are being developed very rapidly. Especially in the field of economics and finance, the estimation and forecasting of economic and financial indicators play a vital role in decision making. Many models are developed in the last 2 decades to get better accuracy and efficiency in time series analysis and still there is a scope of learning and getting betterment in this field is available. In this research we have reviewed probability graphs, directed acyclic graphs, Bayesian networks, feature selection algorithms and Markov blankets for time series forecasting on the economic and financial problems (like stock exchange forecasting, multi-objective business risk analysis, consumers’ analysis, portfolio optimization, credit scoring etc). This is a new dimension for adaptive modeling techniques in economics and finance modeling

Counting and Sampling from Markov Equivalent DAGs Using Clique Trees

A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure--roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. In this paper, we propose a new technique for counting the number of DAGs in a Markov equivalence class. Our approach is based on the clique tree representation of chordal graphs. We show that in the case of bounded degree graphs, the proposed algorithm is polynomial time. We further demonstrate that this technique can be utilized for uniform sampling from a Markov equivalence class, which provides a stochastic way to enumerate DAGs in the equivalence class and may be needed for finding the best DAG or for causal inference given the equivalence class as input. We also extend our counting and sampling method to the case where prior knowledge about the underlying DAG is available, and present applications of this extension in causal experiment design and estimating the causal effect of joint interventions

Uniform random generation of large acyclic digraphs

Directed acyclic graphs are the basic representation of the structure underlying Bayesian networks, which represent multivariate probability distributions. In many practical applications, such as the reverse engineering of gene regulatory networks, not only the estimation of model parameters but the reconstruction of the structure itself is of great interest. As well as for the assessment of different structure learning algorithms in simulation studies, a uniform sample from the space of directed acyclic graphs is required to evaluate the prevalence of certain structural features. Here we analyse how to sample acyclic digraphs uniformly at random through recursive enumeration, an approach previously thought too computationally involved. Based on complexity considerations, we discuss in particular how the enumeration directly provides an exact method, which avoids the convergence issues of the alternative Markov chain methods and is actually computationally much faster. The limiting behaviour of the distribution of acyclic digraphs then allows us to sample arbitrarily large graphs. Building on the ideas of recursive enumeration based sampling we also introduce a novel hybrid Markov chain with much faster convergence than current alternatives while still being easy to adapt to various restrictions. Finally we discuss how to include such restrictions in the combinatorial enumeration and the new hybrid Markov chain method for efficient uniform sampling of the corresponding graphs.Comment: 15 pages, 2 figures. To appear in Statistics and Computin

Learning Markov Equivalence Classes of Directed Acyclic Graphs: an Objective Bayes Approach

A Markov equivalence class contains all the Directed Acyclic Graphs (DAGs) encoding the same conditional independencies, and is represented by a Completed Partially Directed Acyclic Graph (CPDAG), also named Essential Graph (EG).We approach the problem of model selection among noncausal sparse Gaussian DAGs by directly scoring EGs, using an objective Bayes method. Specifically, we construct objective priors for model selection based on the Fractional Bayes Factor, leading to a closed form expression for the marginal likelihood of an EG. Next we propose an MCMC strategy to explore the space of EGs using sparsity constraints, and illustrate the performance of our method on simulation studies, as well as on a real dataset. Our method provides a coherent quantication of inferential uncertainty, requires minimal prior specication, and shows to be competitive in learning the structure of the data-generating EG when compared to alternative state-of-the-art algorithms

Equivalence class selection of categorical graphical models

Learning the structure of dependence relations between variables is a pervasive issue in the statistical literature. A directed acyclic graph (DAG) can represent a set of conditional independences, but different DAGs may encode the same set of relations and are indistinguishable using observational data. Equivalent DAGs can be collected into classes, each represented by a partially directed graph known as essential graph (EG). Structure learning directly conducted on the EG space, rather than on the allied space of DAGs, leads to theoretical and computational benefits. Still, the majority of efforts in the literature has been dedicated to Gaussian data, with less attention to methods designed for multivariate categorical data. We then propose a Bayesian methodology for structure learning of categorical EGs. Combining a constructive parameter prior elicitation with a graph-driven likelihood decomposition, we derive a closed-form expression for the marginal likelihood of a categorical EG model. Asymptotic properties are studied, and an MCMC sampler scheme developed for approximate posterior inference. We evaluate our methodology on both simulated scenarios and real data, with appreciable performance in comparison with state-of-the-art methods

Functions of random walks on hyperplane arrangements

Many seemingly disparate Markov chains are unified when viewed as random walks on the set of chambers of a hyperplane arrangement. These include the Tsetlin library of theoretical computer science and various shuffling schemes. If only selected features of the chains are of interest, then the mixing times may change. We study the behavior of hyperplane walks, viewed on a subarrangement of a hyperplane arrangement. These include many new examples, for instance a random walk on the set of acyclic orientations of a graph. All such walks can be treated in a uniform fashion, yielding diagonalizable matrices with known eigenvalues, stationary distribution and good rates of convergence to stationarity.Comment: Final version; Section 4 has been split into two section

Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs

Graphical models are popular statistical tools which are used to represent dependent or causal complex systems. Statistically equivalent causal or directed graphical models are said to belong to a Markov equivalent class. It is of great interest to describe and understand the space of such classes. However, with currently known algorithms, sampling over such classes is only feasible for graphs with fewer than approximately 20 vertices. In this paper, we design reversible irreducible Markov chains on the space of Markov equivalent classes by proposing a perfect set of operators that determine the transitions of the Markov chain. The stationary distribution of a proposed Markov chain has a closed form and can be computed easily. Specifically, we construct a concrete perfect set of operators on sparse Markov equivalence classes by introducing appropriate conditions on each possible operator. Algorithms and their accelerated versions are provided to efficiently generate Markov chains and to explore properties of Markov equivalence classes of sparse directed acyclic graphs (DAGs) with thousands of vertices. We find experimentally that in most Markov equivalence classes of sparse DAGs, (1) most edges are directed, (2) most undirected subgraphs are small and (3) the number of these undirected subgraphs grows approximately linearly with the number of vertices. The article contains supplement arXiv:1303.0632, http://dx.doi.org/10.1214/13-AOS1125SUPPComment: Published in at http://dx.doi.org/10.1214/13-AOS1125 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org