An Introduction to the EM AlgorithmBy Naala Brewer and Kehinde Salau

Project Advisor – Prof. Randy EubankAdvisor – Prof. Carlos Castillo-ChavezMTBI, Arizona State University

An Introduction to the EM AlgorithmOutline•History of the EM Algorithm

•Theory behind the EM Algorithm

•Biological Examples including derivations, coding in R, Matlab, C++

•Graphs of iterations and convergence

Brief History of the EM Algorithm

•Method frequently referenced throughout field of statistics

•Term coined in 1977 paper by Arthur Dempster, Nan Laird, and Donald Rubin

Breakdown of the EM Task•To compute MLEs of latent variables

and unknown parameters in probabilistic models

•E-step: computes expectation of complete/unobserved data

•M-step: computes MLEs of unknown parameters

•Repeat!!

Generalization of the EM Algorithm•X- Full sample (latent variable) ~ f(x; θ) Y - Observed sample (incomplete data) ~

f(y;θ) such that y(x) = y

•We define Q(θ;θp) = E[lnf(x;θ)|Y, θp]

•θp+1 obtained by solving, = 0

Generalization (cont.)

•Iterations continue until |θp+1 - θp| or |Q(θp+1;θp) - Q(θp;θp)| are sufficiently small

•Thus, optimal values for Q(θ;θp) and θ are obtained

•Likelihood nondecreasing with each iteration:

Q(θp+1;θp) ≥ Q(θp;θp)

Binomial Distribution – Bin(n,p)

Example 1 – Household Model•n-people, p-probability of getting disease•Derivation•Graphs

Binomial Distribution - Derivation

Binomial Derivation (cont.)

Binomial Derivation (cont.)

Example 2 – Population of Animals

Rao (1965, pp.368-369), Genetic Linkage Model• Suppose 197 animals are distributed multinomially into

four categories, y = (125, 18, 20, 34) = (y1, y2, y3, y4)

• A genetic model for the population specifies cell probabilities (1/2, ¼ – ¼л, ¼ – ¼л, ¼л)

• Represent y as incomplete data, y1=x1+x2, y2=x3, y3=x4, y4=x5.

Multinomial Distribution-Derivation

Multinomial Derivation (cont.)

Multinomial Derivation (cont.)

Multinomial Coding

Example 2 – Population of Animals•R Coding•Matlab Coding•C++ Coding

R Coding

#initial vector of data

y <- c(125, 18, 20, 34)

#Initial value for unknown parameter

pik <- .5

for(k in 1:10){

x2k <-y[1]*(.25*pik)/(.5 +.25*pik)

pik <- (x2k + y[4])/(x2k + sum(y[2:4]))

print(c(x2k,pik)) #Convergent values

}

Matlab Coding

%initial vector of data

y = [125, 18, 20, 34];

%Initial value for unknown parameter

pik = .5;

for k = 1:10

x2k = y(1)*(.25*pik)/(.5 + .25*pik)

pik = (x2k + y(4))/(x2k + sum(y(2:4)))

end

%Convergent values

[x2k,pik]

Multinomial Coding

C++ Coding

#include <iostream>

int main () {

int x1, x2, x3, x4;

float pik, x2k;

std::cout << "enter vector of values, there should be four inputs\n";

std::cin >> x1 >> x2 >> x3 >> x4;

std::cout << "enter value for pik\n";

std::cin >> pik;

for (int counter = 0; counter < 10; counter++){

x2k = x1*((0.25)*pik)/((0.5) + (0.25)*pik);

pik = (x2k + x4)/(x2k + x2 + x3 + x4);

std::cout << "x2k is " << x2k << " and " << " pik is " << pik << std::endl;

}

 

return 0;

}

Matlab Coding

%initial vector of data

y = [125, 18, 20, 34];

%Initial value for unknown parameter

pik = .5;

for k = 1:10

x2k = y(1)*(.25*pik)/(.5 + .25*pik)

pik = (x2k + y(4))/(x2k + sum(y(2:4)))

end

%Convergent values

[x2k,pik]

Multinomial Coding

Graph of Convergence of Unknowns,πk and x2

k

Multinomial Distribution

Example 2 -Failure TimesFlury and Zoppè (2000)▫Suppose the lifetime of bulbs follows an

exponential distribution with mean θ

▫The failure times (u1,...,un) are known for n light bulbs

▫In another experiment, m light bulbs (v1,...,vm) are tested; no individual recordings The number of bulbs, r, that fail at time t0 are

recorded

Exponential Distribution - Derivation

Exponential Derivation (cont.)

•Example 2 – Failure Times Graphs

Future Work

•More Biological Examples

An Introduction to the EM AlgorithmReferences[1] Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum

Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society. Series B (Methodological), Vol. 39, No. 1, , pp. 1-38

[2] Redner, R.A., Walker, H.F. (Apr., 1984). Mixture Densities, Maximum Likelihood and the EM Algorithm. SIAM Review, Vol. 26, No. 2., pp. 195-239.

[3] Tanner, A.T. (1996). Tools for Statistical Inference. Springer-Verlag New York, Inc. Third Edition