This document discusses methods for calculating conditional expectations of sufficient statistics for continuous time Markov chains. It presents three methods - eigenvalue decomposition, uniformization, and exponentiation - and evaluates their accuracy and speed for different models and time points. For accuracy, it finds that exponentiation is most accurate. For speed, exponentiation is slowest while eigenvalue decomposition and uniformization trade off accuracy and speed depending on the model and time points. Uniformization has potential to be improved through better cutoff points or adaptive uniformization techniques.

1. Paula Tataru
Conditional expectations
of sufficient statistics
for continuous time Markov chains
Joint work with Asger Hobolth
November 28, 2012

2. 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

3. Motivation
Methods
Results
Paula
Tataru
3/30
Conditional
expectations
of sufficient
statistics
for CTMCs
Continuous-time Markov chains
• Credit risk
• States: different types of ratings
• Chemical reactions
• States: different states of a molecule
• DNA data
• States:
• Nucleotides
• Amino acids
• Codons

4. Motivation
Methods
Results
Paula
Tataru
4/30
Conditional
expectations
of sufficient
statistics
for CTMCs
Modeling DNA data
1st 2nd base 3rd
base T C A G base
T
TTT
Phe
TCT
Ser
TAT
Tyr
TGT
Cys
T
TTC TCC TAC TGC C
TTA
Leu
TCA TAA Stop TGA Stop A
TTG TCG TAG Stop TGG Trp G
C
CTT CCT
Pro
CAT
His
CGT
Arg
T
CTC CCC CAC CGC C
CTA CCA CAA
Gln
CGA A
CTG CCG CAG CGG G
A
ATT
Ile
ACT
Thr
AAT
Asn
AGT
Ser
T
ATC ACC AAC AGC C
ATA ACA AAA
Lys
AGA
Arg
A
ATG Met ACG AAG AGG G
G
GTT
Val
GCT
Ala
GAT
Asp
GGT
Gly
T
GTC GCC GAC GGC C
GTA GCA GAA
Glu
GGA A
GTG GCG GAG GGG G

5. Motivation
Methods
Results
Paula
Tataru
5/30
Conditional
expectations
of sufficient
statistics
for CTMCs
Modeling DNA data
• Nucleotides (n = 4)
• Amino acids (n = 20)
• Codons (n = 64)

6. Motivation
Methods
Results
Paula
Tataru
6/30
Conditional
expectations
of sufficient
statistics
for CTMCs
Continuous-time Markov chains
• Characterized by a rate matrix
with properties

7. Motivation
Methods
Results
Paula
Tataru
7/30
Conditional
expectations
of sufficient
statistics
for CTMCs
Modeling DNA data
• Nucleotides (n = 4)
• Jukes Cantor (JC)

8. Motivation
Methods
Results
Paula
Tataru
8/30
Conditional
expectations
of sufficient
statistics
for CTMCs
Continuous-time Markov chains
• Waiting time is exponentially distributed
• Jumps are discretely distributed

9. Motivation
Methods
Results
Paula
Tataru
9/30
Conditional
expectations
of sufficient
statistics
for CTMCs
General EM

10. Motivation
Methods
Results
Paula
Tataru
10/30
Conditional
expectations
of sufficient
statistics
for CTMCs
General EM
• Full log likelihood
• E-step
• M-step

11. Motivation
Methods
Results
Paula
Tataru
11/30
Conditional
expectations
of sufficient
statistics
for CTMCs
EM for Jukes Cantor
• Full log likelihood
• M-step

12. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
12/30
Calculating expectations

13. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
13/30
Linear combinations

14. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
14/30
Linear combinations

15. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
15/30
Linear combinations

16. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
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?

17. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
17/30
Eigenvalue decomposition

18. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
18/30
Uniformization

19. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
19/30
Uniformization
• Choose truncation point s(μt)
• Bound error using the tail of Pois(m+1; μt)
• s(μt) = 4 + 6 √μt + μt

20. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
20/30
Exponentiation

21. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
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
)
O(n3
) O(s(μt)n2
) O(n2
)
μt —
* expm package in R

22. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
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

23. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
23/30

24. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
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
O(n2
) O(s(μt)n3
) O(n3
)
O(n3
) O(s(μt)n2
) O(n2
)
μt —

25. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
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

26. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
26/30
Speed
• 10 equidistant time points

27. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
27/30

28. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
28/30
Results
• Accuracy
• Similar accuracy
• EXPM is the most accurate one
• Speed
• EXPM is the slowest
• EVD vs UNI

29. Motivation
Methods
Results
Paula
Tataru
Conditional
expectations
of sufficient
statistics
for CTMCs
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

30. Thank you!
Comparison of methods for calculating
conditional expectations of sufficient statistics
for continuous time Markov chains
BMC Bioinformatics 12(1):465, 2011