Demographic Inference with Coalescent Hidden Markov Model

Transcription

1 Demographic Inference with Coalescent Hidden Markov Model Jade Y. Cheng Thomas Mailund Bioinformatics Research Centre Aarhus University Denmark The Thirteenth Asia Pacific Bioinformatics Conference HsinChu, Taiwan, January 2015

2 Presentation Outline CoalHMM framework Overview Model construction Continuous time Markov chain (CTMC) Hidden Markov model (HMM) Numerical optimization Genetic algorithm Particle swarm optimzation Model construction and implementation Simulation case studies Isolation model IIM with nine epochs Simulation case studies

3 Framework Overview CoalHMM is a demorgraphic inference framework based on combining the sequential Markov coalescence with hidden Markov models. E.g. Ancestral Period Migration Period Isolation Period Migration Demorgraphic parameters: 1. Isolation duration 2. Migration duration 3. Coalescent rate 4. Recombination rate 5. Migration rate Our framework is available under open source licence GPLv2 at h ps://github.com/mailund/imcoalhmm

4 Presentation Outline CoalHMM framework Overview Model construction Continuous time Markov chain (CTMC) Hidden Markov model (HMM) Numerical optimization Genetic algorithm Particle swarm optimzation Model construction and implementation Simulation case studies Isolation model IIM with nine epochs Simulation case studies

5 Hidden Markov Model In the context of CoalHMM, hidden states are different coalescence trees. E.g. for two samples, the hidden states are the coalescence times. ^ t3 x /-\ t2 / \ x /-----\ /-\ t1 / \ / \ x / \ /-----\ /-\ t0 time o o o o o o HMM state #3 HMM state #2 HMM state #1

6 HMM Transition Probabilities Transition probability T ij is the normalized joint probability J ij, which is the probability of observing coalescence of the le nucleotide in time period i and coalescence of the right nucleotide in time period j. E.g. ^ t4 x x t3 x-x t2 o-o o o t1 o-o o-o t0 time o-o o-o E.g. ^ t4 x-x t3 o x-o t2 o o o-o t1 o-o o-o t0 time o-o o-o J 33 J 34

7 Continuous Time Markov Chain CTMC state space for two samples in two isolated populations C2 C1 0 1 R - 0 C1 2 R 0 - C2 3 0 R R - 0 [(1, ([1], [])), (1, ([], [1])), (2, ([2], [])), (2, ([], [2]))] 1 [(1, ([1], [])), (1, ([], [1])), (2, ([2], [2]))] 2 [(1, ([1], [1])), (2, ([2], [])), (2, ([], [2]))] 3 [(1, ([1], [1])), (2, ([2], [2]))]

8 Continuous Time Markov Chain CTMC state space for two samples in a single population C1 C1 C1 C1 0 0 C1 0 0 C R C1 0 0 C1 0 0 C R C1 0 0 C1 0 C R C1 0 C C R C C1 0 0 C R 0 0 R C R R C C1 C C R C R C C1 C1 C R C R 0-0 C C R -

9 Continuous Time Markov Chain e size of CTMC state space grows exponentially with the number of populations, samples, or loci. E.g. CTMC for a 3 samples 3 populations 2 loci 2 donor populations 1 receiver population demographic senario has 578 states.

10 CTMC Projections We need projection matrices to move samples between time slices that have different CTMC state spaces. No Match Match Single CTMC Isolation CTMC Isolation CTMC Single CTMC

11 HMM Transition Probabilities Transition probability T ij is the normalized joint probability J ij, which is the probability of observing coalescence of the le nucleotide in time period i and coalescence of the right nucleotide in time period j. E.g. ^ t4 x x t3 x-x t2 o-o o o t1 o-o o-o t0 time o-o o-o E.g. ^ t4 x-x t3 o x-o t2 o o o-o t1 o-o o-o t0 time o-o o-o J 33 J 34

12 HMM Transition Probabilities For a two-sample CTMC, we can split its rate and probability matrices into 16 sections using the 4 state types: begin (B), le (L), right (R), and end (E) E.g. for the two-sample single-population CTMC

13 HMM Transition Probabilities ere are three possible ways to reach an E state from a B state, which is the initial condition of two samples. Goal: Possible Paths: B to E Legal Moves: B to B B to L B to R B to E L to L L to E R to R R to E #1 B to B B to E #2 B to B B to L L to L L to E #3 B to B B to R R to R R to E

14 HMM Transition Probabilities e probability of taking path 3 is the same as taking path 2. Joint probability matrix, J, is symmetric. B L R E B B-B B-L B-R B-E L L-B L-L L-R L-E P = R R-B R-L R-R R-E E E-B E-L E-R E-E P LE

15 HMM Transition Probabilities E.g. a simple joint probability calculation ^ t4 x x t3 o x o t2 o x-o t1 o o o-o t0 time o-o o-o

16 HMM Transition Probabilities E.g. another joint probability calculation involving CTMC projection. ^ t4 x-x t3 o x-o t2 o o o-o t1 o o o-o t0 time o-o o-o

17 Presentation Outline CoalHMM framework Overview Model construction Continuous time Markov chain (CTMC) Hidden Markov model (HMM) Numerical optimization Genetic algorithm Particle swarm optimzation Model construction and implementation Simulation case studies Isolation model IIM with nine epochs Simulation case studies

18 optimization optimization minimises an objective function in a manydimensional space by continuously refining a simplex. reflect p e contract p r expand p c p min shrink p max

19 A is a type of evolutionary algorithm. evaluation selection mutation crossover

20 Fitness Proportion Selection Selection: a GA chooses a relatively fit subset of individuals for breeding. E.g. fitness 40; fitness 25; fitness 18; fitness 12; fitness 5 Roule e Wheel Selection Universal Sampling Breeding Pool Breeding Pool

21 Rank Based Selection Tournament selection selects individuals with the highest fitness values from random subsets of the population. E.g. fitness 40; fitness 25; fitness 18; fitness 12; fitness 5 Tournament #1 Tournament #2 Original Population Tournament #3 Breeding Pool

22 Crossover & Mutation Crossover: it is a genetic operation used to combine pairs of individuals previously selected for breeding the following generation. One-point crossover Two-point crossover Uniform crossover Mutation: each position has a certain probability to mutate, mutation

23 Genetic Optimization PSO is another heuristic based search algorithm. velocity cognitive influence global influence

24 Presentation Outline CoalHMM framework Overview Model construction Continuous time Markov chain (CTMC) Hidden Markov model (HMM) Numerical optimization Genetic algorithm Particle swarm optimzation Model construction and implementation Simulation case studies Isolation model IIM with nine epochs Simulation case studies

25 Pipeline & Isolation Model Simulation study pipeline. O P T I M ms seq-gen CoalHMM ziphmm I S A T I O N e simplest demographic model we consider is the clean isolation model. It has three parameters. Ancestral Period Isolation Period True Value: Isolation Time True Value: Coalescent Rate True Value: Recombination Rate

26 IIM-Nine Epoch Model 10-2 Isolation Time Migration Time Recombination Rate 10 4 Coalescent Rate #1 True Value: (31.3% out of range) True Value: (3.2% out of range) True Value: True Value: (20.4% out of range) Ancestral Epoch 9 Ancestral Epoch 8 Ancestral Epoch 7 Migration Epoch 6 True Value: (23.5% out of range) Coalescent Rate #2 True Value: (21.9% out of range) Coalescent Rate #3 True Value: (15.7% out of range) Coalescent Rate #4 True Value: (39.1% out of range) Coalescent Rate #5 Migration Epoch 5 Migration Epoch 4 Isolation Epoch 3 Isolation Epoch 2 Isolation Epoch 1 Migration True Value: Coalescent Rate # (51.6% out of range) Migration Rate #1 (1.6% out of range) True Value: (28.2% out of range) Coalescent Rate #7 Migration Rate #2 True Value: (42.2% out of range) Coalescent Rate #8 Migration Rate #3 True Value: (34.4% out of range) Coalescent Rate #9 True Value: True Value: True Value: (9.4% out of range) (43.8% out of range) (42.2% out of range)

27 Discussion We discussed the construction of our statistic inference framework, CoalHMM and showed how to infer demographics with it. CTMC HMM Numerical Optimization We compared the estimation accuracy between the previously-used with newly developed optimization methods GA PSO We presented the inference results on two demographic models, a Ancestral Epoch 9 simple one and a more complex one. Ancestral Epoch 8 Ancestral Period Isolation Period Ancestral Epoch 7 Migration Epoch 6 Migration Epoch 5 Migration Epoch 4 Isolation Epoch 3 Migration Isolation Epoch 2 Isolation Epoch 1

28 Acknowledgement Thanks!