Matlab Simulations of Markov Models Yu Meng Department of Computer Science and Engineering Southern Methodist University

Outlines Markov models /process/chain/property/HMM Matlab simulations Applications Advantages and Limitations Conclusions.

Markov Process Markov process is a simple stochastic process in which the distribution of future states depends only on the present state and not on how it arrived in the present state.

Markov Models – A finite state representation

Markov Property Many systems in real world have the property that given present state, the past states have no influence on the future. This property is called Markov property.

State Space and Time Space State Space Time Space Discrete Continuous Discrete (Markov chain) X Continuous X X

Markov Chain Let {X t : t is in T} be a stochastic process with discrete-state space S and discrete-time space T satisfying Markov property P ( X n+1 = j|X n = i, X n-1 = i n-1, · · ·,X 0 = i 0 ) = P ( X n+1 = j|X n = i ) for any set of state i 0, i 1, · · ·, i n-1, i, j in S and n ≥ 0 is called a Markov Chain.

Hidden Markov Models (HMM) In an hidden Markov Model(HMM), we don’t know the state sequence. However we know some probabilistic function of it. In plain English, Markov model can be viewed as a probabilistic finite state engine. The state is changing over time. But we have no way to determine the exact changes of the state. We are able to observe some fuzzy reflections of the change. Our objective is to estimate the states of the machine via the (possibly fuzzy) observations.

Markov Model example The weather in Dallas of past 26 days STATES = { pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, rainy, rainy, rainy, rainy, rainy, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty }

Markov Model example The weather in Dallas of past 26 days STATES = { pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, rainy, rainy, rainy, rainy, rainy, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty, pretty } Tomorrow’s weather Today’s Weather PrettyRainy Pretty Rainy0.20.8

Markov Model example (2)

Markov Model example(3) Hidden model. Many time we don’t have a direct observation of the change of the states. Therefore we say the model is hidden. However we are still able to observe an emission of the state changes, and the emission could be fuzzy. For example, you are isolated in a closed room during the experiment. In the room, you have no direct observation of how weather changes. Everyday, an assistant delivers meals for you once a day. The only way you have clue of the weather is to observe the how the guy’s dress changes.

Markov Model example(4) In the room, you might observe that your assistant dresses with regular coat (CT), rain coat(RN), or brings an umbrella (UM). Your observation sequence may be SEQ = {CT, CT, CT, UM, CT, CT, CT, CT, CT, CT, CT, RN, CT, RN, UM, CT, CT, CT, RN, CT, CT, CT, CT, CT, CT, CT}

Markov Model example(5) Regular CoatRain CoatUmbrella Pretty5/61/12 Rainy1/3

Markov Model example(5) Regular CoatRain CoatUmbrella Pretty5/61/12 Rainy1/3

Mathematical Elements A set of states over time, denoted by STATES A set of emissions, or observations over time, denoted by SEQ An M-by-M transition matrix TRANS whose entry(i,j) is the probability of a transition from state i to state j. An M-by-N emission matrix EMIS whose i,k entry gives the probability of emitting symbol s k given that the model is in state i. EMIS

Outlines Markov models /process/chain/property/HMM Matlab simulations Applications Advantages and Limitations Conclusions.

Why Matlab? Matlab is a tool for doing numerical computations with matrices and vectors. It can also display information graphically. Combined with numerous mathematical libraries, Matlab has become one of the few tools that can catch up with my ideas.

Why Matlab? MATLAB Compiler translates MATLAB code to ANSI standard C code. With the MATLAB Compiler, we will be able to automatically generate optimized C and C++ code. By translating MATLAB code to C and C++, the compiler can significantly speed up MATLAB applications and development. >>mcc -t -L C myfun1 % yields myfun1.c mcc -t -L C myfun2 % yields myfun2.c mcc -W main -L C myfun1 myfun2 libmmfile.mlib % yields myfun1_main.c

Matlab simulations Matlab Statistics Toolbox 4.1 (Released in May 2003) hmmdecode hmmgenerate hmmestimate hmmtrain hmmviterbi

Matlab simulations Matlab scripts demo.m in Matlab 6.5 environment

Outlines Markov models /process/chain/property/HMM Matlab simulations Applications Advantages and Limitations Conclusions.

Applications of Markov Models Speech recognition, Modeling of coding/noncoding regions in DNA, Protein binding sites in DNA, Protein folding, Protein superfamilies, Multiple sequence alignment, Flood predictions, Ion channel recordings, Optical character recognition.

Outlines Markov models /process/chain/property/HMM Matlab simulations Applications Advantages and Limitations Conclusions.

Advantages of Markov Models MMs and HMMs have proved effective in a number of domains. The basic theory of HMMs is very elegant and easy to understand. This makes it easier to analyse and implement, with the help of Matlab. Because MM uses only positive data, their scalability is very good. Dr. Dunham’s research group is investigating an incremental extension algorithm of Markov chain, which fits for dynamic data processing. It is complementary of other non-linear models such as neural networks and time-series analysis.

Limitations of Markov Models It is a data hog. Markov property.

Conclusions 2.cs.cmu.edu/~awm/tutorials/

Any Questions?