Maximum Likelihood Estimates and the EM

Algorithms I

Henry Horng-Shing LuInstitute of Statistics

National Chiao Tung Universityhslu@stat.nctu.edu.tw

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Part 1Computation Tools

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Computation Tools R (http://www.r-project.org/): good for

statistical computing C/C++: good for fast computation and large

data sets More:

http://www.stat.nctu.edu.tw/subhtml/source/teachers/hslu/course/statcomp/links.htm

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The R Project R is a free software environment for

statistical computing and graphics. It compiles and runs on a wide variety of UNIX platforms, Windows and MacOS.

Similar to the commercial software of Splus.

C/C++, Fortran and other codes can be linked and called at run time.

More: http://www.r-project.org/

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Download R from http://www.r-project.org/

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Choose one Mirror Site of R

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Choose the OS System

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Select the Base of R

8

Download the Setup Program

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Install R

Double click R-icon to install R

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Execute R

Interactive command window

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Download Add-on Packages

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Choose a Mirror Site

Choose a mirror site close to you

1.

2.

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Select One Package to Download

Choose one package to download, like rgl.

1.

2. 14

Load Packages There are two methods to load packages:

Method 1:

Click from the menu bar

Method 2:

Type “library(rgl)” in the command window

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Help in R (1) What is the loaded library?

help(rgl)

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Help in R (2) How to search functions for key words?

help.search(“key words”)It will show all functions has the key words.

help.search(“3D plot”)

Function name (belong to which package) description

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Help in R (3) How to find the illustration of function?

?function nameIt will show the usage, arguments, author, reference, related functions, and examples.

?plot3d

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R Operators (1)Mathematic operators:

+, -, *, /, ^ Mod: %% Sqrt, exp, log, log10, sin, cos, tan, …

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R Operators (2)Other operators:

: sequence operator %*% matrix algebra <, >, <=, >= inequality ==, != comparison &, &&, |, || and, or ~ formulas <-, = assignment

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Algebra, Operators and Functions

> 1+2[1] 3> 1>2[1] FALSE> 1>2|2>1[1] TRUE> A=1:3> A[1] 1 2 3> A*6[1] 6 12 18> A/10[1] 0.1 0.2 0.3> A%%2[1] 1 0 1

> B=4:6> A*B[1] 4 10 18> t(A)%*%B

[1][1] 32> A%*%t(B)

[1] [2] [3][1] 4 5 6 [2] 8 10 12[3] 12 15 18> sqrt(A)[1] 1.000 1.1414 1.7320> log(A)[1] 0.000 0.6931 1.0986

> round(sqrt(A),2)[1] 1.00 1.14 1.73> ceiling(sqrt(A))[1] 1 2 2> floor(sqrt(A))[1] 1 1 1> eigen(A%*%t(B))$values[1] 3.20e+01 5.83e-16

2.48e-16$vectors

[1] [2] [3][1] 0.2672 0.3273 -0.8890[2] 0.5345 -0.5217 0.2530[3] 0.8017 0.4665 0.3810

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Variable TypesItem Descriptions

VectorX=c(10.4,5.6,3.1,6.4) or

Z=array(data_vector, dim_vector)

MatricesX=matrix(1:8,2,4) or

Z=matrix(rnorm(30),5,6)

Factors Statef=factor(state)

Lists pts = list(x=cars[,1], y=cars[,2])

Data Frames

data.frame(cbind(x=1, y=1:10), fac=sample(LETTERS[1:3], 10, repl=TRUE))

Functions name=function(arg_1,arg_2,…) expression

Missing Values

NA or NAN22

Define Your Own Function (1) Use “fix(myfunction)” # a window will show up function (parameter){

statements;return (object);# if you want to return some values

} Save the document Use “myfunction(parameter)” in R

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Define Your Own Function (2) Example: Find all the factors of an integer

1.

2.3.

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Define Your Own Function (3)

When you leave the program, remember to save the work space for the next use, or the function you defined will disappear after you close R project.

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Read and Write Files

Write Data to a CSV File

Write Data to a TXT File

Read TXT and CSV Files

Demo

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Write Data to a TXT File Usage:

write(x,file,…)> X=matrix(1:6,2,3)> X

[,1] [,2] [,3][1,] 1 3 5[2,] 2 4 6> write(t(X),file=“d:/out2.txt”,ncolumns=3)> write(X,file=“d:/out3.txt”,ncolumns=3)

d:/out2.txt1 3 52 4 6

d:/out3.txt1 2 34 5 6

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Write Data to a CSV File

d:/out4.txt1,23,45,6

d:/out5.txt1,3,52,4,6

Usage: write.table(x,file=“foo.csv”,sep=“,”,…)> X=matrix(1:6,2,3)> X [,1] [,2] [,3][1,] 1 3 5[2,] 2 4 6>write.table(t(X),file=“d:/out4.txt”,sep=“,”,col.names=FALSE,row.names=FALSE)>write.table(X,file=“d:/out5.txt”,sep=“,”,col.names=FALSE,row.names=FALSE)

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Read TXT and CSV Files Usage:read.table(file,...)> X=read.table(file="d:/out2.txt")> X v1 v2 v31 1 3 52 2 4 6> Y=read.table(file="d:/out5.txt",sep=",",header=FALSE)> Y V1 V21 1 22 3 43 5 6

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Demo> Data=read.table(file="d:/01.csv",header=TRUE,sep=",")>Data Y X1 X21 2.651680 13.808990 26.758962 1.875039 17.734520 37.898573 1.523964 19.891030 26.036244 2.984314 15.574260 30.217545 10.423090 9.293612 28.914596 0.840065 8.830160 30.385787 8.126936 9.615875 32.69579>mean(Data$Y)[1] 4.060727>boxplot(Data$Y)

01.csv

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Part 2Motivation Examples

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Example 1 in Genetics (1) Two linked loci with alleles A and a, and B

and b A, B: dominant a, b: recessive

A double heterozygote AaBb will produce gametes of four types: AB, Ab, aB, ab

F ( Female) 1- r’ r’ (female recombination fraction)

M (Male) 1-r r (male recombination fraction)

A

B b

a B

A

b

a a

B

b

A

A

B b

a

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Example 1 in Genetics (2) r and r’ are the recombination rates for male

and female Suppose the parental origin of these

heterozygote is from the mating of . The problem is to estimate r and r’ from the offspring of selfed heterozygotes.

Fisher, R. A. and Balmukand, B. (1928). The estimation of linkage from the offspring of selfed heterozygotes. Journal of Genetics, 20, 79–92.

http://en.wikipedia.org/wiki/Genetics http://www2.isye.gatech.edu/~brani/isyebayes/bank/handout12.pdf

AABB aabb

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Example 1 in Genetics (3)

b

a

B

A

A

B b

a

a

b b

aA

B B

A

A

B

A

B b

a

b

a

1/2 1/2

a

B b

A

A

B b

a

1/2 1/2

AB ab aB Ab

Male (1-r)/2 (1-r)/2 r/2 r/2

Female (1-r’)/2 (1-r’)/2 r’/2 r’/2

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Example 1 in Genetics (4)MALE

AB (1-r)/2

ab(1-r)/2

aBr/2

Abr/2

FEMALE

AB (1-r’)/2

AABB (1-r) (1-r’)/4

aABb(1-r) (1-r’)/4

aABBr (1-r’)/4

AABbr (1-r’)/4

ab(1-r’)/2

AaBb(1-r) (1-r’)/4

aabb(1-r) (1-r’)/4

aaBbr (1-r’)/4

Aabbr (1-r’)/4

aB r’/2

AaBB(1-r) r’/4

aabB(1-r) r’/4

aaBBr r’/4

AabBr r’/4

Ab r’/2

AABb(1-r) r’/4

aAbb(1-r) r’/4

aABbr r’/4

AAbb r r’/4

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Example 1 in Genetics (5) Four distinct phenotypes: A*B*, A*b*, a*B* and

a*b*. A*: the dominant phenotype from (Aa, AA, aA). a*: the recessive phenotype from aa. B*: the dominant phenotype from (Bb, BB, bB). b* : the recessive phenotype from bb. A*B*: 9 gametic combinations. A*b*: 3 gametic combinations. a*B*: 3 gametic combinations. a*b*: 1 gametic combination. Total: 16 combinations.

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Example 1 in Genetics (6)

Let (1 )(1 '), then

2( * *) ,

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( * *) ( * *) ,4

( * *) .4

r r

P A B

P A b P a B

P a b

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Example 1 in Genetics (7)

2 1 1; , , , .

4 4 4 4Multinomial n

We know that (1 )(1 '), 0 1/ 2, and 0 ' 1/ 2.

So, 1 1/ 4.

r r r r

Hence, the random sample of n from the offspring of selfed heterozygotes will follow a multinomial distribution:

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Example 1 in Genetics (8)

2 1 1; , , , .

4 4 4 4Multinomial n

2 31 4

1 2 3 4

! 2 1( , ) ( ) ( ) ( ) .

! ! ! ! 4 4 4y yy yn

g yy y y y

Suppose that we observe the data of y = (y1, y2, y3, y4) = (125, 18, 20, 24), which is a random sample from

Then the probability mass function is

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Estimation Methods Frequentist Approaches:

http://en.wikipedia.org/wiki/Frequency_probability

Method of Moments Estimate (MME)http://en.wikipedia.org/wiki/Method_of_moments_%28statistics%29

Maximum Likelihood Estimate (MLE)http://en.wikipedia.org/wiki/Maximum_likelihood

Bayesian Approaches:http://en.wikipedia.org/wiki/Bayesian_probability

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Method of Moments Estimate (MME) Solve the equations when population means are

equal to sample means:

for k = 1, 2, …, t, where t is the number of parameters to be estimated.

MME is simple. Under regular conditions, the MME is consistent! More:

http://en.wikipedia.org/wiki/Method_of_moments_%28statistics%29

' 'k km

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MME for Example 1

Note: MME can’t assure

11 1 1

22 2 2

1 2 3 4

33 3 3

44 4 4

2 1ˆ( ) 4( )

4 21

ˆ( ) 1 4ˆ ˆ ˆ ˆ4 ˆ

1 4ˆ( ) 1 4

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ˆ( ) 4

MME

yE Y n y

ny

E Y n yny

E Y n yn

yE Y n y

n

ˆ [1/ 4,1]!MME

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MME by R

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MME by C/C++

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Maximum Likelihood Estimate (MLE) Likelihood: Maximize likelihood: Solve the score

equations, which are setting the first derivates of likelihood to be zeros.

Under regular conditions, the MLE is consistent, asymptotic efficient and normal!

More: http://en.wikipedia.org/wiki/Maximum_likelihood

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Example 2 (1)

# of tossing head ( ) probability

0 (0,0,0) (1-p)3

1 (1,0,0) (0,1,0) (0,0,1) p(1-p)2

2 (0,1,1) (1,0,1) (1,1,0) p2(1-p)

3 (1,1,1) p3

1 2 3, ,x x x

1, if the ith trial is head;

0, if the ith trial is tail.iX

1 with probability ;

0 with probability 1- .i

pX

p

We toss an unfair coin 3 times and the random variable is

If p is the probability of tossing head, then

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Example 2 (2)

21 2 3

3( | , , ) (1 ) , where 0 p 1.

2L p x x x p p

2 2

2

3(1 ) 3(1 ) 6 (1 )

2

9 12 3 = 0.

p p p p pp

p p

Suppose we observe the toss of 1 heads and 2 tails, the likelihood function becomes

One way to maximize this likelihood function is by solving the score equation, which sets the first derivative to be zero:

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Example 2 (3) The solution of p for the score equation is

1/3 or 1.

One can check that p=1/3 is the maximum point. (How?)

Hence, the MLE of p is 1/3 for this example.

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MLE for Example 1 (1) Likelihood MLE:

2 31 4

1 2 3 4

! 2 1( ) ( ) ( ) ( )

! ! ! ! 4 4 4y yy yn

Ly y y y

ˆ ˆmax ( ) max log ( ) MLE MLEL L

2 3 41 2 3 4

( ) ( )

! 2 1 log( ) log( ) ( ) log( ) log( )

! ! ! ! 4 4 4

logL

ny y y y

y y y y

2 31 4log ( ) 02 1

y yy ydL

d

21 2 3 4 1 2 3 4 4( ) ( 2 2 ) 2 0y y y y y y y y y A B C

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MLE for Example 1 (2)

Checking:

(1)

(2)

(3)

2 4

2MLE

B B AC

A

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2

2ˆ

( )0?

MLE

d

d

ˆ1/ 4 1?MLE

ˆCompare log ( )?MLEL

Use R to find MLE (1)

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Use R to find MLE (2)

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Use C/C++ to find MLE (1)

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Use C/C++ to find MLE (2)

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Exercises Write your own programs for those

examples presented in this talk. Write programs for those examples

mentioned at the following web page:http://en.wikipedia.org/wiki/Maximum_likelihood

Write programs for the other examples that you know.

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More Exercises (1) Example 3 in genetics: The observed data

are (nO, nA, nB, nAB) = (176, 182, 60, 17) ~ Multinomial(r^2, p^2+2pr, q^2+2qr, 2pq), where p, q, and r fall in [0,1] such that p+q+r = 1. Find the likelihood function and score equations for p, q, and r.

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More Exercises (2) Example 4 in the positron emission tomography

(PET): The observed data are n*(d) ~Poisson(λ*(d)), d = 1, 2, …, D, and

The values of p(b,d) are known and the unknown parameters are λ(b), b = 1, 2, …, B.

Find the likelihood function and score equations for λ(b), b = 1, 2, …, B.

*

1

( ) ( , ) ( ).B

b

d p b d b

.

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More Exercises (3) Example 5 in the normal mixture: The observed

data xi, i = 1, 2, …, n, are random samples from the following probability density function:

Find the likelihood function and score equations for the following parameters:

2

1

1

( ) ( , ),

1, and 0 1 for all .

K

i k k kk

K

k kk

f x Normal

k

1 1 1( ,..., , ,..., , ,..., ).K K K 58