abc

ABC

The method: practical overview

1. Applications of ABC in population genetics2. Motivation for the application of ABC3. ABC approach

1. Characteristics of an ABC methodology2. Algorithm of an ABC inference3. Limitations of the ABC approach4. Typical ABC run

4. Present work1. Compare the ABC algorithm with MCMC2. Study the use of different summary statistics3. Study the use of ABC in more complex scenario4. “State of art” of the software

5. Future developments

Index

1. Application of ABC in population genetics

Popanc

Pop3 Pop4Pop2Pop1





Index

Two processes are usually considered important in determining population structure:

- Gene flow;

- Population splitting.

Most often these processes are modelled and inferred separately;

Recent advances by Nielsen and Wakeley (2001) and Hey and Nielsen (2004) for two-population scenario using Markov Chain Monte Carlo (MCMC) can study both processes at the same time;

An Approximate Bayesian Computation (ABC) method developed by (Beaumont, 2006) deals with the same problem but in a three-population scenario.

The idea is to avoid problems associated with MCMC such as poor-mixing and long convergence times. But it relies in a couple of approximations.

The aim of this study is to see how good these approximations are.

2. Motivation for the application of ABC

Wakeley, Hey (1997, Genetics) - developed an algorithm to estimate historic demographic parameters.

Nielsen, Wakeley (2001, Genetics) - developed a MCMC algorithm to infer about demographic parameters in a “Isolation with Migration” model.

Hey, Nielsen (2004, Genetics) - presents the IM program (software that uses the MCMC algorithm previously developed).

Hey et al (2004, Mol. Ecol.) - introduce changes in IM software (HapSTR data can be used).

Won, Hey (2005, Mol. Biol. Evol.) - presents a case study in 3 populations of chimpanzees.

Hey (2005, PLoS. Biol.) – the peopling of the Americas. Introduce changes in IM software (founder population size can be inferred).

Background using MCMC:


Background using ABC:


Tavaré et al. (1997, Genetics) – presented a simulation based-algorithm to infer about specific demographic parameters

Pritchard et al. (1999, MBE) - introduce the first ABC approach with a rejection method step to estimate demographic parameters.

Beaumont et al. (2002, Genetics) – introduce a regression method within a ABC framework to estimate demographic parameters.

Marjoram et al (2003, PNAS) – uses MCMC without likelihoods within an ABC framework.

Beaumont (2006, “Simulation, Genetics, and Human Prehistory”) - uses regression based ABC to estimate demographic parameters within a “Isolation with Migration” model for microsatellites in three populations.

Hickerson et al (2006, in press) – compares ABC with IM in two-population studies for sequence data.





Index

Replace the data with summary statistics:

2. ABC approach

2. Characteristics of an ABC methodology

Get the posterior distribution by sampling values from it:1. Simulate samples i, Di from the joint density p(,D):

1. First sample from the prior: i ~ p()2. Then simulate the data, given i: Di ~ p(D | i)

2. The posterior distribution,

p(|D) = p(D,) / p(D) , for any given D,

can be estimate by the proportion of all simulated points that correspond to that particular D and divided by the proportion of points corresponding to D (ignoring ).

Summarize a large amount of data into a few representative values By replacing the data with summary statistics, it is easier to decide how ‘similar’ data sets are to each other.


1. Bayesian inference on population genetics2. Characteristics of an ABC methodology3. Algorithm of an ABC inference4. Limitations of the ABC approach5. Typical ABC run



Index

2. ABC approach

2. Algorithm of an ABC inference

SummStats, S

Para

met

er,

Joint distribution (S,)Set of priors (

Get summary statistics (S)

Obtained genetic data

s’in (Nordborg, 2001)

By extracting the points near the real data set we obtain the posterior:

2. Algorithm of an ABC inference

2. ABC approach

SummStats, S

Para

met

er,

Joint distribution (S,)Posterior distribution – p( | S=s’)

p

s’





Index

2. ABC approach

Natural limitation due to lack of information in data sets

Limitation on the number of summary statistics used

Limitation on the calculation of summary statistic (time consuming)

Limitation on the time consumption of the simulation step

3. Limitations

3. ABC approach


Ss’ (, S = s’)

s’ (, S1 = s’1, S2 = s’2)s’2

s’1

S1

S2

Summary Statistics = 1 Summary Statistics = 2

2. ABC approach

Natural limitation due to lack of information in data sets


Limitation on the calculation of summary statistic (time consuming)

Limitation on the time consumption of the simulation step

3. Limitations


1. Bayesian inference on population genetics 2. Algorithm of an ABC inference3. Limitations of the ABC approach4. Typical ABC run



Index

3. ABC approach

2. Typical ABC run

Draw parameter values from prior distributions

Simulate genetic data

Compute summary statistics from simulated data

Compute summary statistics in “real” data

Enough simulations?

Compute distance between “real” data and simulated data

Retain simulated data closest to “real” data

Estimate parameters from the posterior distributions obtained

from the retained simulated data

yesno

Step1 - simulation Step2 – getting posterior distribution

Step3 - estimation

a) Choosing the priors

b) Choosing the summary statistics

c) Choosing a “rejection” method of the simulated data

3. ABC approach

2. Typical ABC run

Rejection method (Pritchard et al, 1999):

SummStats, S

Para

met

er,

tolerance

s’ – “real” dataPosterior distribution – p( | S)

3. ABC approach

2. Typical ABC run

Local Linear Multiple Regression adjustment and Weighting (Beaumont et al, 2002):

SummStats, S

Para

met

er,

s’ - “real” data

Posterior distribution – p( | S)Weighting

Regression

t

ttctK

0

,1)(

21

d

i ixx1

2

where

Epanechnikov kernel

n

iii

Ti sSKsS

1

2)(

We want to minimize

3. ABC approach

2. Typical ABC run

Spherical acceptance region

Local weighting

iT

ii sS )(Linear multiple regression:Correlation coefficients vector

Vector of standardized summstats

E [P(|S=s)]

Least square error

* Tiii sS

3. ABC approach

2. Typical ABC run

),|( sSE

To obtain samples from the posterior distribution we adjust the parameter values as

I.e. we are assuming that the conditional mean of the parameter is a linear function of the summary statistics, but all other moments remain the same.

Least squares gives an estimate of the posterior mean



4. Present work1. Compare the ABC algorithm with MCMC 2. Study the use of different summary statistics3. Study the use of ABC in more complex scenario4. “State of art” of the software


Index

Popanc

Pop2Pop1

t

One simple case:

4. Present Work

m1

m2

Ne1Ne2

Neanc

tev1

6 parameters to be estimated

+

(mutation rate)

Summary Statistics used

Sequence Data:1. mean of pairwise differences

a) in each populationb) both populations joined together

2. number of segregating sitesa) in each populationb) both populations joined together

3. number of haplotypesa) in each populationb) both populations joined together

4. Present Work

Simulated “real” data and Prior information

0 10000

1000 1000 1000 500

0 10000 0 10000 0 0.05 0 0.05 0 5000

0.01 0.01

Ne1 Ne2 Neanc TevMig2Mig1

“real” data

prior distribution ABC method

MCMC method

4. Present Work

Ne1 – no migration:

sim1 sim3sim2 sim4 sim5


4. Present Work

Ne2 – no migration:



4. Present Work

Neanc – no migration:



4. Present Work

Te1 – no migration:



4. Present Work

ABC vs MCMC:Data 1 (no migration); Simulation 7:

Data 2 (migration = 0.01); Simulation 9:

Ne1 Ne2 Neanc Tev


4. Present Work

ABC vs MCMC (500 000 iter, tol=0.02):

Ne1 Ne2 Neanc Mig1 Mig2 TevABC 3.857 0.899 2.529 0.653 3.956 0.532 - - - - 3.532 0.695

MCMC 1.153 0.505 0.724 0.295 3.594 0.602 - - - - 1.567 0.429

Priors 24.33 - 24.33 - 24.33 - - - - - 24.33 -

Ne1 Ne2 Neanc Mig1 Mig2 TevABC 8.242 2.194 10.41 2.240 19.15 0.604 3.977 0.316 3.986 0.259 27.17 0.904

MCMC 4.196 1.132 5.693 1.839 18.85 0.602 2.760 0.391 3.031 0.483 26.54 1.510

Priors 24.33 - 24.33 - 24.33 - 4.33 - 4.33 - 24.33 -

MISE: No migration

MISE: Migration = 0.01

4. Present Work





Index

Summary Statistics usedSequence Data:

1. mean of pairwise differencesa) in each populationb) both populations joined together

2. number of segregating sitesa) in each populationb) both populations joined together

3. number of haplotypesa) in each populationb) both populations joined together

4. variance of pairwise differencesa) in each populationb) both populations joined together

5. Shanon’s indexa) in each populationb) both populations joined together

6. number of singletonsa) in each populationb) both populations joined together

4. Present Work


0 10000

1000 1000 1000 500

0 10000 0 10000 0 0.05 0 0.05 0 5000

0.01 0.01


“real” data

prior distributionstandard

previous + Shanon’s

previous + var pairwise dif

previous + singletons

MCMC based method

4. Present Work

Summary Statistics (500 000 iter, tol=0.02):Data 1 (no migration); Simulation 7:


Ne1 Ne2 Neanc Tev


4. Present Work

Summary Statistics (7 000 000 iter, tol=0.02):Data 1 (no migration); Simulation 7:


Ne1 Ne2 Neanc Tev


4. Present Work

Summary Statistics (7 000 000 iter, tol=0.02):

Ne1 Ne2 Neanc Mig1 Mig2 TevABC I 3.861 0.903 2.548 0.654 3.992 0.525 - - - - 3.548 0.702

ABC II 3.538 0.857 2.353 0.614 4.007 0.552 - - - - 3.324 0.615

ABC III 2.160 0.869 1.818 0.577 4.241 0.700 - - - - 4.266 0.949

ABC IV 2.205 0.721 1.606 0.548 4.536 0.700 - - - - 4.698 0.989

MCMC 1.153 0.505 0.724 0.295 3.594 0.602 - - - - 1.567 0.429

MISE: No migration

MISE: Migration = 0.01

Ne1 Ne2 Neanc Mig1 Mig2 TevABC I 8.216 2.170 10.31 2.204 19.03 0.617 3.925 0.318 4.000 0.276 27.05 0.907

ABC II 7.021 2.182 9.664 2.371 19.40 0.540 3.600 0.270 3.755 0.322 28.42 0.951

ABC III 6.285 1.765 7.425 1.415 19.69 0.612 3.435 0.312 3.308 0.349 29.67 1.056

ABC IV 6.585 2.026 6.564 1.218 19.38 0.587 3.410 0.313 3.329 0.334 28.74 0.845

MCMC 4.196 1.132 5.693 1.839 18.85 0.602 2.760 0.391 3.031 0.483 26.54 1.510

4. Present Work

Summary Statistics (7 000 000 iter, tol=0.02):

Ne1 Ne2 Neanc Mig1 Mig2 TevABC I 0.49 0.50 0.27 - - 0.65

ABC II 0.51 0.52 0.27 - - 0.67

ABC III 0.60 0.59 0.30 - - 0.67

ABC IV 0.55 0.55 0.27 - - 0.63

Adjusted R2: No migration

Adjusted R2: Migration = 0.01

Ne1 Ne2 Neanc Mig1 Mig2 TevABC I 0.23 0.23 0.01 0.08 0.08 0.02

ABC II 0.25 0.24 0.01 0.09 0.10 0.02

ABC III 0.30 0.30 0.01 0.11 0.11 0.01

ABC IV 0.26 0.26 0.01 0.11 0.11 0.01

4. Present Work





Index

4. Three populations model

m1 m2

Ne1 Ne3

Neanc1

tev2

11 parameters to be estimated

+

topology

+

(mutation rate)

Popanc1

Pop2Pop1

Popanc2

Pop3

tev1

Neanc2

Ne2

m3

manc


0 10000

1000 1000 1000

0 10000 0 10000 0 0.05

0 0.05

0.01

0.01

Ne1 Ne2 Ne3

Mig2

Mig1

free top

fixed top

500

0 0.05 0 0.05 0 5000

0.01 0.01

Tev2MigancMig3

1500

0 5000

Tev1

0 10000

1000 1000

0 10000

Neanc2 Neanc1

4. Present Work

Three Populations model (no migration):

Ne1 Ne2 Ne3

Tev2 Tev1

Neanc2 Neanc1

Topology:

Data 1 (no migration); Simulation 7:

(2,3)1)

4. Present Work

Three Populations model (migration = 0.01):Data 2 (migration = 0.01); Simulation 6:

Topology:

(1,2)3)

Ne1 Ne2 Ne3

Mig2

Mig1

Tev2MigancMig3 Tev1

Neanc2 Neanc1

4. Present Work

Three Populations model (500 000 iter, tol=0.02):

MISE

Ne Ne Ne* Neanc2 Neanc1 Mig Mig Mig* Miganc Tev2 Tev1

Free 5.700 5.438 3.739 4.781 0.886 - - - - 0.44 18.39

Fixed 5.467 5.282 3.815 4.511 0.264 - - - - 0.55 9.59

No migration:

Migration = 0.01:MISE

Ne Ne Ne* Neanc2 Neanc1 Mig Mig Mig* Miganc Tev2 Tev1

Free 5.415 5.521 4.339 4.864 0.837 4.18 4.03 4.11 4.32 0.51 23.32

Fixed 5.382 5.456 4.327 5.007 0.831 4.28 4.18 4.12 4.34 0.54 23.60

Topology

Free 0.76 0.05

Prior 0.33 -

Topology

Free 0.41 0.02

Prior 0.33 -

4. Present Work

Conclusions:

ABC up to 2 orders of magnitude faster for single locus

ABC modes are similar to MCMC but overall precision is lower

No substantial improvement with more summary statistics

No substantial improvement with more iterations

ABC is able to consider more complex scenarios,

but ability to infer parameters is reduced when considering migration



4. Present work1. Compare the ABC algorithm with a MCMC one2. Study the use of different summary statistics3. Study the use of ABC in more complex scenario4. “State of art” of the software


Index

The user-friendly version of the program (initial stage)

Features of the program Use of heredity scalars for each locus Use different types of DNA data at the same time (Microsatellite and DNA sequence) Use an unlimited number of populations within an IM model Use of different combinations of 7 different summary statistics for each DNA data type

Freeware and source code available (soon)

4. Present Work



4. Present work1. Compare the ABC algorithm with a MCMC one2. Study the use of different summary statistics3. Study the use of ABC in more complex scenario4. “State of art” of the software


Index

5. Future Developments

Current Goals Currently addressing the method to a published data set (Won & Hey, 2005) Continue to improve the accuracy of ABC (e.g. identify better summary statistics) Obtain better estimations for MISE (e.g. using more simulated ‘real’ data)

Future Goals Add recombination Create a user-friendly interface Use a variable migration rate through time Improve ABC: sequential method non-linear regression

Acknowledgements

I would like to acknowledge David Balding for helpful discussion on the methods used. And also a special thanks to Mark Beaumont for advice and comments on the work.

Support for this work was provided by EPSRC.

[email protected]://www.rdg.ac.uk/~sar05sal

mailto:[email protected]

http://www.rdg.ac.uk/~sar05sal

abc

Documents

abc algorithm

use of abc

abc inferencelimitations

abc methodologyalgorithm

abc framework

mcmc algorithm

population geneticsmotivation

population scenario