LEARNING MIXTURES OF MARKOV CHAINS FROM AGGREGATE DATA WITH STRUCTURAL CONSTRAINTS

CONTACT: PRAVEEN KUMAR. L (, +91 – 9791938249)
MAIL ID: sunsid1989@gmail.com, praveen@nexgenproject.com
Web: www.nexgenproject.com, www.finalyear-ieeeprojects.com
LEARNING MIXTURES OF MARKOV CHAINS FROM AGGREGATE DATA WITH
STRUCTURAL CONSTRAINTS
ABSTRACT
Statistical models based on Markov chains, especially mixtures of Markov
chains, have recently been studied and demonstrated to be effective in various
data mining applications such as tourist flow analysis, animal migration
modeling, and transportation administration. Nevertheless, the research so far
has mainly focused on analyzing data at individual levels. Due to security and
privacy reasons, however, the observations in practice usually consist of
coarse-grained statistics of individual data, a.k.a. aggregate data, rendering
learning mixtures of Markov chains an even more challenging problem. In this
work, we show that this challenging problem, although intractable in its
original form, can be solved approximately by posing structural constraints on
the transition matrices. The proposed structural constraints include specifying
active state sets corresponding to the chains and adding a pairwise sparse
regularization term on transition matrices. Based on these two structural
constraints, we propose a constrained least-squares method to learn mixtures
of Markov chains. We further develop a novel iterative algorithm that
decomposes the overall problem into a set of convex subproblems and solves
each subproblem efficiently, making it possible to effectively learn mixtures of
Markov chains from aggregate data. We propose a framework for generating
synthetic data and analyze the complexity of our algorithm. Additionally, the
empirical results of the convergence and the robustness of our algorithm are

also presented. These results demonstrate the effectiveness and efficiency of
the proposed algorithm, comparing with traditional methods. Experimental
results on real-world data sets further validate that our algorithm can be used
to solve practical problems
DISCUSSION
In this paper, we study the problem of learning a mixture of Markov chains
model from aggregate data. With the help of two structural constraints, we
propose a new formulation for model learning and develop an effective and
efficient optimization algorithm.We analyze the proposed algorithm in depth.
The analytic experiments show that our method is robust and has encouraging
performances on estimating Markov chains and predicting the aggregate
behaviors. Our method still has many problems, for example: _ Similar to other
learning methods of Markov chain [23], [26], our MMCs learning algorithm’s
complexity is still O(N3) w.r.t. the number of states N. How to improve the
scalability of our algorithm is an important problem of large scale applications.
_ In some situations, the active state sets are hard to obtain. To deal with
these situations, it is necessary to find a more easily available structural
constraint. These problems point out the direction of our future work. We will
further explore the theoretical aspects of our method and extend its
applications accordingly.

REFERENCES
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LEARNING MIXTURES OF MARKOV CHAINS FROM AGGREGATE DATA WITH STRUCTURAL CONSTRAINTS

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LEARNING MIXTURES OF MARKOV CHAINS FROM AGGREGATE DATA WITH STRUCTURAL CONSTRAINTS