conditional expectations of sufficient statistics for...

Paula Tataru

Conditional expectations of sufficient statistics for continuous time Markov chains

Joint work with Asger Hobolth

November 28, 2012

Motivation

Methods

Results

Paula Tataru

2/30

Conditional expectations of sufficient statistics for CTMCs

Continuous-time Markov chains

A stochastic process {X(t) | t ≥ 0}

that fulfills the Markov property

Motivation

Methods

Results

Paula Tataru

3/30



• Credit risk

• States: different types of ratings

• Chemical reactions

• States: different states of a molecule

• DNA data

• States:• Nucleotides• Amino acids• Codons

Motivation

Methods

Results

Paula Tataru

4/30


Modeling DNA data

1st 2nd base 3rdbase T C A G base

T

TTTPhe

TCT

Ser

TATTyr

TGTCys

TTTC TCC TAC TGC CTTA

Leu

TCA TAA Stop TGA Stop ATTG TCG TAG Stop TGG Trp G

C

CTT CCT

Pro

CATHis

CGT

Arg

TCTC CCC CAC CGC CCTA CCA CAA

GlnCGA A

CTG CCG CAG CGG G

A

ATTIle

ACT

Thr

AATAsn

AGTSer

TATC ACC AAC AGC CATA ACA AAA

LysAGA

ArgA

ATG Met ACG AAG AGG G

G

GTT

Val

GCT

Ala

GATAsp

GGT

Gly

TGTC GCC GAC GGC CGTA GCA GAA

GluGGA A

GTG GCG GAG GGG G

Motivation

Methods

Results

Paula Tataru

5/30


Modeling DNA data

• Nucleotides (n = 4)

• Amino acids (n = 20)

• Codons (n = 64)

Motivation

Methods

Results

Paula Tataru

6/30



• Characterized by a rate matrix

with properties

Motivation

Methods

Results

Paula Tataru

7/30


Modeling DNA data

• Nucleotides (n = 4)

• Jukes Cantor (JC)

Motivation

Methods

Results

Paula Tataru

8/30



• Waiting time is exponentially distributed

• Jumps are discretely distributed

Motivation

Methods

Results

Paula Tataru

9/30


General EM

Motivation

Methods

Results

Paula Tataru

10/30


General EM

• Full log likelihood

• E-step

• M-step

Motivation

Methods

Results

Paula Tataru

11/30


EM for Jukes Cantor

• Full log likelihood

• M-step

Motivation

Methods

Results

Paula Tataru


12/30

Calculating expectations

Motivation

Methods

Results

Paula Tataru


13/30

Linear combinations

Motivation

Methods

Results

Paula Tataru


14/30

Linear combinations

Motivation

Methods

Results

Paula Tataru


15/30

Linear combinations

Motivation

Methods

Results

Paula Tataru


16/30

Methods

• In Hobolth&Jensen (2010), three methods are presented to evaluate the necessary statistics

• Eigenvalue decomposition (EVD)

• Uniformization (UNI)

• Exponentiation (EXPM)

• Extend efficiently to linear combinations

• Which method is more accurate?

• Which method is faster?

Motivation

Methods

Results

Paula Tataru


17/30

Eigenvalue decomposition

Motivation

Methods

Results

Paula Tataru


18/30

Uniformization

Motivation

Methods

Results

Paula Tataru


19/30

Uniformization

• Choose truncation point s(μt)

• Bound error using the tail of Pois(m+1; μt)

• s(μt) = 4 + 6 √μt + μt

Motivation

Methods

Results

Paula Tataru


20/30

Exponentiation

Motivation

Methods

Results

Paula Tataru


21/30

Algorithms

EVD UNI EXPM*

1 A

Order

2 J(t)

Order

3 (C; t)Σ (C; t)Σ (C; t)Σ

Order

Influenced by Q

, U, UΛ -1 μ, s(μt), R

O(n3) O(n2) O(n2)

Rm eAt

O(n2) O(s(μt)n3) O(n3)


μt —

* expm package in R

Motivation

Methods

Results

Paula Tataru


22/30

Accuracy

• Use two models for which ∑(C; t) is available in analytical form

• JC, varying n

• HKY, varying t

• Measure accuracy as the normalized difference: approximation / true - 1

• as a function of n

• as a function of t

Motivation

Methods

Results

Paula Tataru


23/30

Motivation

Methods

Results

Paula Tataru


24/30

Speed

EVD UNI EXPM*

1 A

Order

2 J(t)

Order

3 (C; t)Σ (C; t)Σ (C; t)Σ

Order

Influenced by Q

, U, UΛ -1 μ, s(μt), R

O(n3) O(n2) O(n2)

Rm eAt



μt —

Motivation

Methods

Results

Paula Tataru


25/30

Speed

• Partition of computation

• Precomputation• EVD• UNI

• Main computation

• Use three models and different time points to asses the speed

• GY, n = 61

• GTR, n = 4

• UNR, n = 4

Motivation

Methods

Results

Paula Tataru


26/30

Speed

• 10 equidistant time points

Motivation

Methods

Results

Paula Tataru


27/30

Motivation

Methods

Results

Paula Tataru


28/30

Results

• Accuracy

• Similar accuracy

• EXPM is the most accurate one

• Speed

• EXPM is the slowest

• EVD vs UNI

Motivation

Methods

Results

Paula Tataru


29/30

Results

• UNI has potential

• Better cut off point s(μt)

• Ameliorate effect of μt by

• Adaptive uniformization

• On-the-fly variant of adaptive uniformization

Thank you!

Comparison of methods for calculating conditional expectations of sufficient statistics

for continuous time Markov chains

BMC Bioinformatics 12(1):465, 2011

conditional expectations of sufficient statistics for...

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