Phasing via the Expectation Maximization (EM) Algorithm

Size: px
Start display at page:

Download "Phasing via the Expectation Maximization (EM) Algorithm"

Transcription

1 Computing Haplotype Frequencies and Haplotype Phasing via the Expectation Maximization (EM) Algorithm Department of Computer Science Brown University, Providence September 14, 2010

2 Outline 1 Outline 2 Problem definition 3 The Input The number of haplotypes in the input Computing θ (t+1) 4

3 EM by one Example Problem definition Problem: Consider two loci with two allele 0 and 1 at each locus. Given: (We observe) the genotypes of the individuals at both loci. Find: The estimate at the haplotype frequencies.

4 Solution Outline The Input The number of haplotypes in the input Computing θ (t+1) There are a total of four possible haplotypes, 01, 10, 11 at the two loci. Let us denote their frequencies by θ, θ 01, θ 10, θ 11. Suppose that we have computed already θ (t), θ(t) 01, θ(t) 10, θ(t) 11. We want to compute θ (t+1) as a function of θ (t), θ(t) 01, θ(t) 10, θ(t) 11.

5 The Input The number of haplotypes in the input Computing θ (t+1) The Genotype Sample: several types A, B, C, D, E, F There are n A genotypes or individuals of type 22 - we denote Y A the set of such genotypes There are n B genotypes or individuals of type 02 There are n C genotypes or individuals of type 20 There are n D genotypes or individuals of type There are n E genotypes or individuals of type 11

6 The Input The number of haplotypes in the input Computing θ (t+1) The fraction of the genotypes in each category that contains the haplotype (A) For the A group of n A individuals the possible haplotypes show as follows in explanations of the genotypes: or 10 (the fractions represent the separation of mother-father) chromosomes. P(Y A ) = 2θ (t) θ(t) θ(t) 01 θ(t) 10 P( 11 Y A) = 2θ (t) θ(t) 11 2θ (t) θ(t) 11 +2θ(t) 01 θ(t) 10

7 The Input The number of haplotypes in the input Computing θ (t+1) The fraction of the genotypes in each category that contains the haplotype (continued) For group B one haplotype is and the other one is 01 For group C one haplotype is and the other one is 10 For group D both haplotypes are For group E both haplotypes are 11

8 Computing θ (t+1) Outline The Input The number of haplotypes in the input Computing θ (t+1) Therefore the total expected number of haplotypes are: n (t+1) = n A P( 11 Y A) + n B + n C + 2n D so we update θ (t+1) = n(t+1) 2n where n = n A + n B + n C + n D + n E

9 The EM algorithm is an iterative method to compute successive sets of haplotype frequencies p 1, p 2,..., p T starting with some initial arbitrary values p (0) 1, p(0) 2,..., p(0) T Those initial values are used as used as if they were the unknown true frequencies to estimate the explanation frequencies P(h k h l ) (0). This is the Expectation step. These expected explanation frequencies are used in turn to estimate haplotype frequencies at the next iteration p (1) 1, p(1) 2,..., p(1) T. This is the Maximization step.... and so on until convergence is reached (i.e., when the changes in haplotype frequency in consecutive iterations are less than some small value (ɛ).

10 EM Algorithm initialization 1 All explanations are equally likely P j (h k h l ) (0) = 1 c j, 1 j m where m is the total number of genotypes in the input; and n 1, n 2,..., n m are the counts for each genotype type. 2 All haplotypes are equally frequent in the sample. 3 Complete Linkage Equilibrium: Haplotype frequencies = the product of single locus allele frequencies 4 Initial haplotype frequencies are picked at random.

11 The E Step The Expectation step at the tth iteration consists of using the haplotype frequencies in the previous iteration to calculate the probability of resolving each genotype into different possible explanations: P j = c j i=1 P(explanation i) = c j i=1 P(h ikh il ) if k = l then P(h k h l ) = pk 2 if k l then P(h k h l ) = 2p k p l where a 1 is a constant term and p ik and p il are the population frequencies of the corresponding haplotypes.

12 The E Step (continued) The likelihood of the haplotype frequencies given the genotype counts n 1, n 2,..., n m is m L(p 1,..., p T ) = a 1 ( P(h ik h il )) n j where T i=1 = 1, and(h ikh il ), 1 i c j are the set of explanations of the jth genotype that occurs n j times in the input. Let P (t) j = c j i=1 P(h ikh il ) (t) j=1 c j i=1

13 The E Step formula The E Step formula is: P j (h k h l ) (t) = P(h kh l ) (t) cj i=1 P(t) j

14 The M Step Haplotype frequencies are then computed for each Maximization step: for 1 r T p (t+1) r = 1 2 c m j δ ir P j (h ik h il ) (t) j=1 i=1 where δ it is an indicator variable equal to the number of times haplotype t is present in explanation i; and this number can be 0, 1 or 2.

The E-M Algorithm in Genetics. Biostatistics 666 Lecture 8

The E-M Algorithm in Genetics. Biostatistics 666 Lecture 8 The E-M Algorithm in Genetics Biostatistics 666 Lecture 8 Maximum Likelihood Estimation of Allele Frequencies Find parameter estimates which make observed data most likely General approach, as long as

More information

Introduction to Linkage Disequilibrium

Introduction to Linkage Disequilibrium Introduction to September 10, 2014 Suppose we have two genes on a single chromosome gene A and gene B such that each gene has only two alleles Aalleles : A 1 and A 2 Balleles : B 1 and B 2 Suppose we have

More information

2. Map genetic distance between markers

2. Map genetic distance between markers Chapter 5. Linkage Analysis Linkage is an important tool for the mapping of genetic loci and a method for mapping disease loci. With the availability of numerous DNA markers throughout the human genome,

More information

1.5.1 ESTIMATION OF HAPLOTYPE FREQUENCIES:

1.5.1 ESTIMATION OF HAPLOTYPE FREQUENCIES: .5. ESTIMATION OF HAPLOTYPE FREQUENCIES: Chapter - 8 For SNPs, alleles A j,b j at locus j there are 4 haplotypes: A A, A B, B A and B B frequencies q,q,q 3,q 4. Assume HWE at haplotype level. Only the

More information

Lab 12. Linkage Disequilibrium. November 28, 2012

Lab 12. Linkage Disequilibrium. November 28, 2012 Lab 12. Linkage Disequilibrium November 28, 2012 Goals 1. Es

More information

Chapter 6 Linkage Disequilibrium & Gene Mapping (Recombination)

Chapter 6 Linkage Disequilibrium & Gene Mapping (Recombination) 12/5/14 Chapter 6 Linkage Disequilibrium & Gene Mapping (Recombination) Linkage Disequilibrium Genealogical Interpretation of LD Association Mapping 1 Linkage and Recombination v linkage equilibrium ²

More information

Affected Sibling Pairs. Biostatistics 666

Affected Sibling Pairs. Biostatistics 666 Affected Sibling airs Biostatistics 666 Today Discussion of linkage analysis using affected sibling pairs Our exploration will include several components we have seen before: A simple disease model IBD

More information

Expression QTLs and Mapping of Complex Trait Loci. Paul Schliekelman Statistics Department University of Georgia

Expression QTLs and Mapping of Complex Trait Loci. Paul Schliekelman Statistics Department University of Georgia Expression QTLs and Mapping of Complex Trait Loci Paul Schliekelman Statistics Department University of Georgia Definitions: Genes, Loci and Alleles A gene codes for a protein. Proteins due everything.

More information

An introduction to PRISM and its applications

An introduction to PRISM and its applications An introduction to PRISM and its applications Yoshitaka Kameya Tokyo Institute of Technology 2007/9/17 FJ-2007 1 Contents What is PRISM? Two examples: from population genetics from statistical natural

More information

Solutions to Even-Numbered Exercises to accompany An Introduction to Population Genetics: Theory and Applications Rasmus Nielsen Montgomery Slatkin

Solutions to Even-Numbered Exercises to accompany An Introduction to Population Genetics: Theory and Applications Rasmus Nielsen Montgomery Slatkin Solutions to Even-Numbered Exercises to accompany An Introduction to Population Genetics: Theory and Applications Rasmus Nielsen Montgomery Slatkin CHAPTER 1 1.2 The expected homozygosity, given allele

More information

AUTHORIZATION TO LEND AND REPRODUCE THE THESIS. Date Jong Wha Joanne Joo, Author

AUTHORIZATION TO LEND AND REPRODUCE THE THESIS. Date Jong Wha Joanne Joo, Author AUTHORIZATION TO LEND AND REPRODUCE THE THESIS As the sole author of this thesis, I authorize Brown University to lend it to other institutions or individuals for the purpose of scholarly research. Date

More information

Computational statistics

Computational statistics Computational statistics Combinatorial optimization Thierry Denœux February 2017 Thierry Denœux Computational statistics February 2017 1 / 37 Combinatorial optimization Assume we seek the maximum of f

More information

Learning gene regulatory networks Statistical methods for haplotype inference Part I

Learning gene regulatory networks Statistical methods for haplotype inference Part I Learning gene regulatory networks Statistical methods for haplotype inference Part I Input: Measurement of mrn levels of all genes from microarray or rna sequencing Samples (e.g. 200 patients with lung

More information

The Quantitative TDT

The Quantitative TDT The Quantitative TDT (Quantitative Transmission Disequilibrium Test) Warren J. Ewens NUS, Singapore 10 June, 2009 The initial aim of the (QUALITATIVE) TDT was to test for linkage between a marker locus

More information

Click-Through Rate prediction: TOP-5 solution for the Avazu contest

Click-Through Rate prediction: TOP-5 solution for the Avazu contest Click-Through Rate prediction: TOP-5 solution for the Avazu contest Dmitry Efimov Petrovac, Montenegro June 04, 2015 Outline Provided data Likelihood features FTRL-Proximal Batch algorithm Factorization

More information

Calculation of IBD probabilities

Calculation of IBD probabilities Calculation of IBD probabilities David Evans University of Bristol This Session Identity by Descent (IBD) vs Identity by state (IBS) Why is IBD important? Calculating IBD probabilities Lander-Green Algorithm

More information

For 5% confidence χ 2 with 1 degree of freedom should exceed 3.841, so there is clear evidence for disequilibrium between S and M.

For 5% confidence χ 2 with 1 degree of freedom should exceed 3.841, so there is clear evidence for disequilibrium between S and M. STAT 550 Howework 6 Anton Amirov 1. This question relates to the same study you saw in Homework-4, by Dr. Arno Motulsky and coworkers, and published in Thompson et al. (1988; Am.J.Hum.Genet, 42, 113-124).

More information

STAT 536: Genetic Statistics

STAT 536: Genetic Statistics STAT 536: Genetic Statistics Frequency Estimation Karin S. Dorman Department of Statistics Iowa State University August 28, 2006 Fundamental rules of genetics Law of Segregation a diploid parent is equally

More information

p(d g A,g B )p(g B ), g B

p(d g A,g B )p(g B ), g B Supplementary Note Marginal effects for two-locus models Here we derive the marginal effect size of the three models given in Figure 1 of the main text. For each model we assume the two loci (A and B)

More information

Humans have two copies of each chromosome. Inherited from mother and father. Genotyping technologies do not maintain the phase

Humans have two copies of each chromosome. Inherited from mother and father. Genotyping technologies do not maintain the phase Humans have two copies of each chromosome Inherited from mother and father. Genotyping technologies do not maintain the phase Genotyping technologies do not maintain the phase Recall that proximal SNPs

More information

Parameter Estimation in the Spatio-Temporal Mixed Effects Model Analysis of Massive Spatio-Temporal Data Sets

Parameter Estimation in the Spatio-Temporal Mixed Effects Model Analysis of Massive Spatio-Temporal Data Sets Parameter Estimation in the Spatio-Temporal Mixed Effects Model Analysis of Massive Spatio-Temporal Data Sets Matthias Katzfuß Advisor: Dr. Noel Cressie Department of Statistics The Ohio State University

More information

Calculation of IBD probabilities

Calculation of IBD probabilities Calculation of IBD probabilities David Evans and Stacey Cherny University of Oxford Wellcome Trust Centre for Human Genetics This Session IBD vs IBS Why is IBD important? Calculating IBD probabilities

More information

Statistical learning. Chapter 20, Sections 1 4 1

Statistical learning. Chapter 20, Sections 1 4 1 Statistical learning Chapter 20, Sections 1 4 Chapter 20, Sections 1 4 1 Outline Bayesian learning Maximum a posteriori and maximum likelihood learning Bayes net learning ML parameter learning with complete

More information

Normal distribution We have a random sample from N(m, υ). The sample mean is Ȳ and the corrected sum of squares is S yy. After some simplification,

Normal distribution We have a random sample from N(m, υ). The sample mean is Ȳ and the corrected sum of squares is S yy. After some simplification, Likelihood Let P (D H) be the probability an experiment produces data D, given hypothesis H. Usually H is regarded as fixed and D variable. Before the experiment, the data D are unknown, and the probability

More information

Haplotyping. Biostatistics 666

Haplotyping. Biostatistics 666 Haplotyping Biostatistics 666 Previously Introduction to te E-M algoritm Approac for likeliood optimization Examples related to gene counting Allele frequency estimation recessive disorder Allele frequency

More information

EM algorithm and applications Lecture #9

EM algorithm and applications Lecture #9 EM algorithm and applications Lecture #9 Bacground Readings: Chapters 11.2, 11.6 in the text boo, Biological Sequence Analysis, Durbin et al., 2001.. The EM algorithm This lecture plan: 1. Presentation

More information

Finite Singular Multivariate Gaussian Mixture

Finite Singular Multivariate Gaussian Mixture 21/06/2016 Plan 1 Basic definitions Singular Multivariate Normal Distribution 2 3 Plan Singular Multivariate Normal Distribution 1 Basic definitions Singular Multivariate Normal Distribution 2 3 Multivariate

More information

Last lecture 1/35. General optimization problems Newton Raphson Fisher scoring Quasi Newton

Last lecture 1/35. General optimization problems Newton Raphson Fisher scoring Quasi Newton EM Algorithm Last lecture 1/35 General optimization problems Newton Raphson Fisher scoring Quasi Newton Nonlinear regression models Gauss-Newton Generalized linear models Iteratively reweighted least squares

More information

Population Genetics. with implications for Linkage Disequilibrium. Chiara Sabatti, Human Genetics 6357a Gonda

Population Genetics. with implications for Linkage Disequilibrium. Chiara Sabatti, Human Genetics 6357a Gonda 1 Population Genetics with implications for Linkage Disequilibrium Chiara Sabatti, Human Genetics 6357a Gonda csabatti@mednet.ucla.edu 2 Hardy-Weinberg Hypotheses: infinite populations; no inbreeding;

More information

Genetic Association Studies in the Presence of Population Structure and Admixture

Genetic Association Studies in the Presence of Population Structure and Admixture Genetic Association Studies in the Presence of Population Structure and Admixture Purushottam W. Laud and Nicholas M. Pajewski Division of Biostatistics Department of Population Health Medical College

More information

Statistical Genetics I: STAT/BIOST 550 Spring Quarter, 2014

Statistical Genetics I: STAT/BIOST 550 Spring Quarter, 2014 Overview - 1 Statistical Genetics I: STAT/BIOST 550 Spring Quarter, 2014 Elizabeth Thompson University of Washington Seattle, WA, USA MWF 8:30-9:20; THO 211 Web page: www.stat.washington.edu/ thompson/stat550/

More information

CSci 8980: Advanced Topics in Graphical Models Analysis of Genetic Variation

CSci 8980: Advanced Topics in Graphical Models Analysis of Genetic Variation CSci 8980: Advanced Topics in Graphical Models Analysis of Genetic Variation Instructor: Arindam Banerjee November 26, 2007 Genetic Polymorphism Single nucleotide polymorphism (SNP) Genetic Polymorphism

More information

The universal validity of the possible triangle constraint for Affected-Sib-Pairs

The universal validity of the possible triangle constraint for Affected-Sib-Pairs The Canadian Journal of Statistics Vol. 31, No.?, 2003, Pages???-??? La revue canadienne de statistique The universal validity of the possible triangle constraint for Affected-Sib-Pairs Zeny Z. Feng, Jiahua

More information

Lecture 6: Gaussian Mixture Models (GMM)

Lecture 6: Gaussian Mixture Models (GMM) Helsinki Institute for Information Technology Lecture 6: Gaussian Mixture Models (GMM) Pedram Daee 3.11.2015 Outline Gaussian Mixture Models (GMM) Models Model families and parameters Parameter learning

More information

A new algorithm for deriving optimal designs

A new algorithm for deriving optimal designs A new algorithm for deriving optimal designs Stefanie Biedermann, University of Southampton, UK Joint work with Min Yang, University of Illinois at Chicago 18 October 2012, DAE, University of Georgia,

More information

Problems for 3505 (2011)

Problems for 3505 (2011) Problems for 505 (2011) 1. In the simplex of genotype distributions x + y + z = 1, for two alleles, the Hardy- Weinberg distributions x = p 2, y = 2pq, z = q 2 (p + q = 1) are characterized by y 2 = 4xz.

More information

Association studies and regression

Association studies and regression Association studies and regression CM226: Machine Learning for Bioinformatics. Fall 2016 Sriram Sankararaman Acknowledgments: Fei Sha, Ameet Talwalkar Association studies and regression 1 / 104 Administration

More information

Genotype Imputation and Haplotype Inference for Genome-wide Association Studies

Genotype Imputation and Haplotype Inference for Genome-wide Association Studies Genotype Imputation and Haplotype Inference for Genome-wide Association Studies Nab Raj Roshyara Institut fuer Medizinische Informatik, Statistik und Epidemiologie (IMISE) Forschungsgruppe Genetische Statistik

More information

Statistical learning. Chapter 20, Sections 1 3 1

Statistical learning. Chapter 20, Sections 1 3 1 Statistical learning Chapter 20, Sections 1 3 Chapter 20, Sections 1 3 1 Outline Bayesian learning Maximum a posteriori and maximum likelihood learning Bayes net learning ML parameter learning with complete

More information

Population Genetics: a tutorial

Population Genetics: a tutorial : a tutorial Institute for Science and Technology Austria ThRaSh 2014 provides the basic mathematical foundation of evolutionary theory allows a better understanding of experiments allows the development

More information

MAXIMUM LIKELIHOOD ESTIMATION IN A MULTINOMIAL MIXTURE MODEL. Charles E. McCulloch Cornell University, Ithaca, N. Y.

MAXIMUM LIKELIHOOD ESTIMATION IN A MULTINOMIAL MIXTURE MODEL. Charles E. McCulloch Cornell University, Ithaca, N. Y. MAXIMUM LIKELIHOOD ESTIMATION IN A MULTINOMIAL MIXTURE MODEL By Charles E. McCulloch Cornell University, Ithaca, N. Y. BU-934-MA May, 1987 ABSTRACT Maximum likelihood estimation is evaluated for a multinomial

More information

Closed-form sampling formulas for the coalescent with recombination

Closed-form sampling formulas for the coalescent with recombination 0 / 21 Closed-form sampling formulas for the coalescent with recombination Yun S. Song CS Division and Department of Statistics University of California, Berkeley September 7, 2009 Joint work with Paul

More information

Linear Regression (1/1/17)

Linear Regression (1/1/17) STA613/CBB540: Statistical methods in computational biology Linear Regression (1/1/17) Lecturer: Barbara Engelhardt Scribe: Ethan Hada 1. Linear regression 1.1. Linear regression basics. Linear regression

More information

Performance Comparison of K-Means and Expectation Maximization with Gaussian Mixture Models for Clustering EE6540 Final Project

Performance Comparison of K-Means and Expectation Maximization with Gaussian Mixture Models for Clustering EE6540 Final Project Performance Comparison of K-Means and Expectation Maximization with Gaussian Mixture Models for Clustering EE6540 Final Project Devin Cornell & Sushruth Sastry May 2015 1 Abstract In this article, we explore

More information

Computing the MLE and the EM Algorithm

Computing the MLE and the EM Algorithm ECE 830 Fall 0 Statistical Signal Processing instructor: R. Nowak Computing the MLE and the EM Algorithm If X p(x θ), θ Θ, then the MLE is the solution to the equations logp(x θ) θ 0. Sometimes these equations

More information

Expectation maximization tutorial

Expectation maximization tutorial Expectation maximization tutorial Octavian Ganea November 18, 2016 1/1 Today Expectation - maximization algorithm Topic modelling 2/1 ML & MAP Observed data: X = {x 1, x 2... x N } 3/1 ML & MAP Observed

More information

Expectation Maximization, and Learning from Partly Unobserved Data (part 2)

Expectation Maximization, and Learning from Partly Unobserved Data (part 2) Expectation Maximization, and Learning from Partly Unobserved Data (part 2) Machine Learning 10-701 April 2005 Tom M. Mitchell Carnegie Mellon University Clustering Outline K means EM: Mixture of Gaussians

More information

Stage-structured Populations

Stage-structured Populations Department of Biology New Mexico State University Las Cruces, New Mexico 88003 brook@nmsu.edu Fall 2009 Age-Structured Populations All individuals are not equivalent to each other Rates of survivorship

More information

Math 152. Rumbos Fall Solutions to Exam #2

Math 152. Rumbos Fall Solutions to Exam #2 Math 152. Rumbos Fall 2009 1 Solutions to Exam #2 1. Define the following terms: (a) Significance level of a hypothesis test. Answer: The significance level, α, of a hypothesis test is the largest probability

More information

Haplotype-based variant detection from short-read sequencing

Haplotype-based variant detection from short-read sequencing Haplotype-based variant detection from short-read sequencing Erik Garrison and Gabor Marth July 16, 2012 1 Motivation While statistical phasing approaches are necessary for the determination of large-scale

More information

Maximum likelihood in log-linear models

Maximum likelihood in log-linear models Graphical Models, Lecture 4, Michaelmas Term 2010 October 22, 2010 Generating class Dependence graph of log-linear model Conformal graphical models Factor graphs Let A denote an arbitrary set of subsets

More information

Frequency Spectra and Inference in Population Genetics

Frequency Spectra and Inference in Population Genetics Frequency Spectra and Inference in Population Genetics Although coalescent models have come to play a central role in population genetics, there are some situations where genealogies may not lead to efficient

More information

EXERCISES FOR CHAPTER 3. Exercise 3.2. Why is the random mating theorem so important?

EXERCISES FOR CHAPTER 3. Exercise 3.2. Why is the random mating theorem so important? Statistical Genetics Agronomy 65 W. E. Nyquist March 004 EXERCISES FOR CHAPTER 3 Exercise 3.. a. Define random mating. b. Discuss what random mating as defined in (a) above means in a single infinite population

More information

Computational Systems Biology: Biology X

Computational Systems Biology: Biology X Bud Mishra Room 1002, 715 Broadway, Courant Institute, NYU, New York, USA L#7:(Mar-23-2010) Genome Wide Association Studies 1 The law of causality... is a relic of a bygone age, surviving, like the monarchy,

More information

Tutorial Session 2. MCMC for the analysis of genetic data on pedigrees:

Tutorial Session 2. MCMC for the analysis of genetic data on pedigrees: MCMC for the analysis of genetic data on pedigrees: Tutorial Session 2 Elizabeth Thompson University of Washington Genetic mapping and linkage lod scores Monte Carlo likelihood and likelihood ratio estimation

More information

QTL model selection: key players

QTL model selection: key players Bayesian Interval Mapping. Bayesian strategy -9. Markov chain sampling 0-7. sampling genetic architectures 8-5 4. criteria for model selection 6-44 QTL : Bayes Seattle SISG: Yandell 008 QTL model selection:

More information

theta H H H H H H H H H H H K K K K K K K K K K centimorgans

theta H H H H H H H H H H H K K K K K K K K K K centimorgans Linkage Phase Recall that the recombination fraction ρ for two loci denotes the probability of a recombination event between those two loci. For loci on different chromosomes, ρ = 1=2. For loci on the

More information

EM algorithm. Rather than jumping into the details of the particular EM algorithm, we ll look at a simpler example to get the idea of how it works

EM algorithm. Rather than jumping into the details of the particular EM algorithm, we ll look at a simpler example to get the idea of how it works EM algorithm The example in the book for doing the EM algorithm is rather difficult, and was not available in software at the time that the authors wrote the book, but they implemented a SAS macro to implement

More information

Learning with Probabilities

Learning with Probabilities Learning with Probabilities CS194-10 Fall 2011 Lecture 15 CS194-10 Fall 2011 Lecture 15 1 Outline Bayesian learning eliminates arbitrary loss functions and regularizers facilitates incorporation of prior

More information

Discussion of Dimension Reduction... by Dennis Cook

Discussion of Dimension Reduction... by Dennis Cook Discussion of Dimension Reduction... by Dennis Cook Department of Statistics University of Chicago Anthony Atkinson 70th birthday celebration London, December 14, 2007 Outline The renaissance of PCA? State

More information

Mathematical models in population genetics II

Mathematical models in population genetics II Mathematical models in population genetics II Anand Bhaskar Evolutionary Biology and Theory of Computing Bootcamp January 1, 014 Quick recap Large discrete-time randomly mating Wright-Fisher population

More information

THE GENETICS OF CERTAIN COMMON VARIATIONS IN COLEUS 1

THE GENETICS OF CERTAIN COMMON VARIATIONS IN COLEUS 1 THE GENETICS OF CERTAIN COMMON VARIATIONS IN COLEUS DAVID C. RIFE, The Ohio State University, Columbus, Ohio Coleus are characterized by great variations in leaf color, and to a lesser degree by variations

More information

Statistical learning. Chapter 20, Sections 1 3 1

Statistical learning. Chapter 20, Sections 1 3 1 Statistical learning Chapter 20, Sections 1 3 Chapter 20, Sections 1 3 1 Outline Bayesian learning Maximum a posteriori and maximum likelihood learning Bayes net learning ML parameter learning with complete

More information

Lecture 7. Logistic Regression. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza. December 11, 2016

Lecture 7. Logistic Regression. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza. December 11, 2016 Lecture 7 Logistic Regression Luigi Freda ALCOR Lab DIAG University of Rome La Sapienza December 11, 2016 Luigi Freda ( La Sapienza University) Lecture 7 December 11, 2016 1 / 39 Outline 1 Intro Logistic

More information

The genomes of recombinant inbred lines

The genomes of recombinant inbred lines The genomes of recombinant inbred lines Karl W Broman Department of Biostatistics Johns Hopkins University http://www.biostat.jhsph.edu/~kbroman C57BL/6 2 1 Recombinant inbred lines (by sibling mating)

More information

Maximum Likelihood Estimation in Latent Class Models for Contingency Table Data

Maximum Likelihood Estimation in Latent Class Models for Contingency Table Data Maximum Likelihood Estimation in Latent Class Models for Contingency Table Data Stephen E. Fienberg Department of Statistics, Machine Learning Department, Cylab Carnegie Mellon University May 20, 2008

More information

A brief introduction to Conditional Random Fields

A brief introduction to Conditional Random Fields A brief introduction to Conditional Random Fields Mark Johnson Macquarie University April, 2005, updated October 2010 1 Talk outline Graphical models Maximum likelihood and maximum conditional likelihood

More information

CSE 150. Assignment 6 Summer Maximum likelihood estimation. Out: Thu Jul 14 Due: Tue Jul 19

CSE 150. Assignment 6 Summer Maximum likelihood estimation. Out: Thu Jul 14 Due: Tue Jul 19 SE 150. Assignment 6 Summer 2016 Out: Thu Jul 14 ue: Tue Jul 19 6.1 Maximum likelihood estimation A (a) omplete data onsider a complete data set of i.i.d. examples {a t, b t, c t, d t } T t=1 drawn from

More information

Linkage and Chromosome Mapping

Linkage and Chromosome Mapping Linkage and Chromosome Mapping I. 1 st year, 2 nd semester, week 11 2007 Aleš Panczak, ÚBLG 1. LF a VFN Terminology, definitions The term recombination ratio (fraction), Θ (Greek letter theta), is used

More information

The implications of neutral evolution for neutral ecology. Daniel Lawson Bioinformatics and Statistics Scotland Macaulay Institute, Aberdeen

The implications of neutral evolution for neutral ecology. Daniel Lawson Bioinformatics and Statistics Scotland Macaulay Institute, Aberdeen The implications of neutral evolution for neutral ecology Daniel Lawson Bioinformatics and Statistics Scotland Macaulay Institute, Aberdeen How is How is diversity Diversity maintained? maintained? Talk

More information

Introduction to population genetics & evolution

Introduction to population genetics & evolution Introduction to population genetics & evolution Course Organization Exam dates: Feb 19 March 1st Has everybody registered? Did you get the email with the exam schedule Summer seminar: Hot topics in Bioinformatics

More information

Testing for Homogeneity in Genetic Linkage Analysis

Testing for Homogeneity in Genetic Linkage Analysis Testing for Homogeneity in Genetic Linkage Analysis Yuejiao Fu, 1, Jiahua Chen 2 and John D. Kalbfleisch 3 1 Department of Mathematics and Statistics, York University Toronto, ON, M3J 1P3, Canada 2 Department

More information

URN MODELS: the Ewens Sampling Lemma

URN MODELS: the Ewens Sampling Lemma Department of Computer Science Brown University, Providence sorin@cs.brown.edu October 3, 2014 1 2 3 4 Mutation Mutation: typical values for parameters Equilibrium Probability of fixation 5 6 Ewens Sampling

More information

Modeling IBD for Pairs of Relatives. Biostatistics 666 Lecture 17

Modeling IBD for Pairs of Relatives. Biostatistics 666 Lecture 17 Modeling IBD for Pairs of Relatives Biostatistics 666 Lecture 7 Previously Linkage Analysis of Relative Pairs IBS Methods Compare observed and expected sharing IBD Methods Account for frequency of shared

More information

Estimation of Parameters in Random. Effect Models with Incidence Matrix. Uncertainty

Estimation of Parameters in Random. Effect Models with Incidence Matrix. Uncertainty Estimation of Parameters in Random Effect Models with Incidence Matrix Uncertainty Xia Shen 1,2 and Lars Rönnegård 2,3 1 The Linnaeus Centre for Bioinformatics, Uppsala University, Uppsala, Sweden; 2 School

More information

Empirical Risk Minimization

Empirical Risk Minimization Empirical Risk Minimization Fabrice Rossi SAMM Université Paris 1 Panthéon Sorbonne 2018 Outline Introduction PAC learning ERM in practice 2 General setting Data X the input space and Y the output space

More information

Ph.D. Qualifying Exam Friday Saturday, January 3 4, 2014

Ph.D. Qualifying Exam Friday Saturday, January 3 4, 2014 Ph.D. Qualifying Exam Friday Saturday, January 3 4, 2014 Put your solution to each problem on a separate sheet of paper. Problem 1. (5166) Assume that two random samples {x i } and {y i } are independently

More information

1.5 MM and EM Algorithms

1.5 MM and EM Algorithms 72 CHAPTER 1. SEQUENCE MODELS 1.5 MM and EM Algorithms The MM algorithm [1] is an iterative algorithm that can be used to minimize or maximize a function. We focus on using it to maximize the log likelihood

More information

ML Testing (Likelihood Ratio Testing) for non-gaussian models

ML Testing (Likelihood Ratio Testing) for non-gaussian models ML Testing (Likelihood Ratio Testing) for non-gaussian models Surya Tokdar ML test in a slightly different form Model X f (x θ), θ Θ. Hypothesist H 0 : θ Θ 0 Good set: B c (x) = {θ : l x (θ) max θ Θ l

More information

Lesson 4: Understanding Genetics

Lesson 4: Understanding Genetics Lesson 4: Understanding Genetics 1 Terms Alleles Chromosome Co dominance Crossover Deoxyribonucleic acid DNA Dominant Genetic code Genome Genotype Heredity Heritability Heritability estimate Heterozygous

More information

KEY: Chapter 9 Genetics of Animal Breeding.

KEY: Chapter 9 Genetics of Animal Breeding. KEY: Chapter 9 Genetics of Animal Breeding. Answer each question using the reading assigned to you. You can access this information by clicking on the following URL: https://drive.google.com/a/meeker.k12.co.us/file/d/0b1yf08xgyhnad08xugxsnfvba28/edit?usp=sh

More information

Algorithmic approaches to fitting ERG models

Algorithmic approaches to fitting ERG models Ruth Hummel, Penn State University Mark Handcock, University of Washington David Hunter, Penn State University Research funded by Office of Naval Research Award No. N00014-08-1-1015 MURI meeting, April

More information

QTL Mapping I: Overview and using Inbred Lines

QTL Mapping I: Overview and using Inbred Lines QTL Mapping I: Overview and using Inbred Lines Key idea: Looking for marker-trait associations in collections of relatives If (say) the mean trait value for marker genotype MM is statisically different

More information

CSCE 471/871 Lecture 3: Markov Chains and

CSCE 471/871 Lecture 3: Markov Chains and and and 1 / 26 sscott@cse.unl.edu 2 / 26 Outline and chains models (s) Formal definition Finding most probable state path (Viterbi algorithm) Forward and backward algorithms State sequence known State

More information

Quantitative trait evolution with mutations of large effect

Quantitative trait evolution with mutations of large effect Quantitative trait evolution with mutations of large effect May 1, 2014 Quantitative traits Traits that vary continuously in populations - Mass - Height - Bristle number (approx) Adaption - Low oxygen

More information

CS1820 Notes. hgupta1, kjline, smechery. April 3-April 5. output: plausible Ancestral Recombination Graph (ARG)

CS1820 Notes. hgupta1, kjline, smechery. April 3-April 5. output: plausible Ancestral Recombination Graph (ARG) CS1820 Notes hgupta1, kjline, smechery April 3-April 5 April 3 Notes 1 Minichiello-Durbin Algorithm input: set of sequences output: plausible Ancestral Recombination Graph (ARG) note: the optimal ARG is

More information

Logistic Regression Review Fall 2012 Recitation. September 25, 2012 TA: Selen Uguroglu

Logistic Regression Review Fall 2012 Recitation. September 25, 2012 TA: Selen Uguroglu Logistic Regression Review 10-601 Fall 2012 Recitation September 25, 2012 TA: Selen Uguroglu!1 Outline Decision Theory Logistic regression Goal Loss function Inference Gradient Descent!2 Training Data

More information

STAT 536: Genetic Statistics

STAT 536: Genetic Statistics STAT 536: Genetic Statistics Tests for Hardy Weinberg Equilibrium Karin S. Dorman Department of Statistics Iowa State University September 7, 2006 Statistical Hypothesis Testing Identify a hypothesis,

More information

Clustering. Léon Bottou COS 424 3/4/2010. NEC Labs America

Clustering. Léon Bottou COS 424 3/4/2010. NEC Labs America Clustering Léon Bottou NEC Labs America COS 424 3/4/2010 Agenda Goals Representation Capacity Control Operational Considerations Computational Considerations Classification, clustering, regression, other.

More information

Part of Speech Tagging: Viterbi, Forward, Backward, Forward- Backward, Baum-Welch. COMP-599 Oct 1, 2015

Part of Speech Tagging: Viterbi, Forward, Backward, Forward- Backward, Baum-Welch. COMP-599 Oct 1, 2015 Part of Speech Tagging: Viterbi, Forward, Backward, Forward- Backward, Baum-Welch COMP-599 Oct 1, 2015 Announcements Research skills workshop today 3pm-4:30pm Schulich Library room 313 Start thinking about

More information

Lecture 2. Basic Population and Quantitative Genetics

Lecture 2. Basic Population and Quantitative Genetics Lecture Basic Population and Quantitative Genetics Bruce Walsh. Aug 003. Nordic Summer Course Allele and Genotype Frequencies The frequency p i for allele A i is just the frequency of A i A i homozygotes

More information

CSC411 Fall 2018 Homework 5

CSC411 Fall 2018 Homework 5 Homework 5 Deadline: Wednesday, Nov. 4, at :59pm. Submission: You need to submit two files:. Your solutions to Questions and 2 as a PDF file, hw5_writeup.pdf, through MarkUs. (If you submit answers to

More information

Machine Learning Tom M. Mitchell Machine Learning Department Carnegie Mellon University. September 20, 2012

Machine Learning Tom M. Mitchell Machine Learning Department Carnegie Mellon University. September 20, 2012 Machine Learning 10-601 Tom M. Mitchell Machine Learning Department Carnegie Mellon University September 20, 2012 Today: Logistic regression Generative/Discriminative classifiers Readings: (see class website)

More information

Stephen Scott.

Stephen Scott. 1 / 27 sscott@cse.unl.edu 2 / 27 Useful for modeling/making predictions on sequential data E.g., biological sequences, text, series of sounds/spoken words Will return to graphical models that are generative

More information

Lecture 4: Probabilistic Learning

Lecture 4: Probabilistic Learning DD2431 Autumn, 2015 1 Maximum Likelihood Methods Maximum A Posteriori Methods Bayesian methods 2 Classification vs Clustering Heuristic Example: K-means Expectation Maximization 3 Maximum Likelihood Methods

More information

CSE446: Clustering and EM Spring 2017

CSE446: Clustering and EM Spring 2017 CSE446: Clustering and EM Spring 2017 Ali Farhadi Slides adapted from Carlos Guestrin, Dan Klein, and Luke Zettlemoyer Clustering systems: Unsupervised learning Clustering Detect patterns in unlabeled

More information

CS446: Machine Learning Fall Final Exam. December 6 th, 2016

CS446: Machine Learning Fall Final Exam. December 6 th, 2016 CS446: Machine Learning Fall 2016 Final Exam December 6 th, 2016 This is a closed book exam. Everything you need in order to solve the problems is supplied in the body of this exam. This exam booklet contains

More information

EM-algorithm for motif discovery

EM-algorithm for motif discovery EM-algorithm for motif discovery Xiaohui Xie University of California, Irvine EM-algorithm for motif discovery p.1/19 Position weight matrix Position weight matrix representation of a motif with width

More information

Lecture 1 Hardy-Weinberg equilibrium and key forces affecting gene frequency

Lecture 1 Hardy-Weinberg equilibrium and key forces affecting gene frequency Lecture 1 Hardy-Weinberg equilibrium and key forces affecting gene frequency Bruce Walsh lecture notes Introduction to Quantitative Genetics SISG, Seattle 16 18 July 2018 1 Outline Genetics of complex

More information

Goodness of Fit Goodness of fit - 2 classes

Goodness of Fit Goodness of fit - 2 classes Goodness of Fit Goodness of fit - 2 classes A B 78 22 Do these data correspond reasonably to the proportions 3:1? We previously discussed options for testing p A = 0.75! Exact p-value Exact confidence

More information