Computations in Animal Breeding Ignacy Misztal and Romdhane Rekaya - - PowerPoint PPT Presentation

Computations in Animal Breeding

Ignacy Misztal and Romdhane Rekaya University of Georgia

Animal Breeding

• Selection of more productive animals as parents
• Effect=> Next generation is more productive
• Tools: artificial insemination + embryo transfer

– A dairy sire can have >100,000 daughters! – Selection of “good” sires important!

Information for selection

• DNA (active research)
• Data collected on large populations of

farm animals

– Records contain combination of genetics, environment, and systematic effects – Need statistical methodology to obtain best prediction of genetic effects

Dairy

• Mostly family farms (50-1000 cows)
• Mostly Holsteins
• Recording on:

– Production (milk, fat, protein yields) – Conformation (size, legs, udder..) – Secondary traits (Somatic cell count in milk,Calving ease, reproduction)

• Information on > 20 million Holsteins (kept by

USDA and breed associations)

• Semen market global

Molecular genetics

Poultry

• Integrated system
• Large companies
• Hierarchical breeding structure
• Final product - crossbreds
• Useful records on up to 200k birds
• Recording on:

– Number and size of eggs – Growth (weights at certain ages) – Fertility – …

Swine

• Becoming more integrated
• Hierarchical breeding structure
• Final product - crossbreds
• Recording on:

– Growth – Litter size – Meat quality – …

• Populations > 200k animals

Beef cattle

• Mostly family farms (5 - 50,000)
• Many breeds
• Final product – purebreds and crossbreds
• Recording on:

– growth – fertility – meat quality

• Data size up to 2 million animals

Results of genetic selection

• Milk yield 2 times higher
• Chicken

– Time to maturity over 2 times shorter – Food efficiency 2 times higher

• Swine
• Beef
• Fish

Genetic value of individual

On a population level: g = a + “rest” Var (a)=A sa, A - matrix of relationships among animals, sa- additive variance A dense; A-1 sparse and easy to set up

Example of a model

• Litter weight =

– contemporary group + – age class + – genetic group + – animal + – e – Var(e) = Ise, var(animal)=Asa – se, sa- variance components

Mixed model

y – vector of records ß – vector of fixed effects u – vector of random effects e – vector of residuals X, Z – design matrices Fixed effects – usually few levels, lots of information Random effects – usually many levels, little information

Mixed model equations

R block diagonal with small blocks G block diagonal with small

• r large blocks

Matrices in Mixed Model

• Symmetric
• Semi-positive definite
• Sparse

– 3-200 nonzeros per row (on average)

• Can be constructed as a sum of outer

products

Σ Wi Qi Wi

Wi – 1-20 x 20k-300 million, < 100 nonzeroes Qi - small square matrix (less than 20 x 20)

Models

• Sire model

Y = cg +… sire + e

1000k animals ≈ 50k equations

• Animal model

Y= cg +..+ animal + e

1000k animals ≈ 1200k equations

• Mutiple trait model

y y1

1 = cg1 +…+ animal1 + …+ e1

1000k animals >3000k equations

….

y yn

n = cgn +…+ animaln + …+ en

• Random regression model

1000k animals >6000k equations

(Longitudinal)

y y = cg +…+ Σ fi(x)animali + …+ e

Tasks

• Estimate variance components

– Usually sample of 5-50k animals

• Solve mixed model equations

– Populations up to 20 million animals – Up to 60 unknowns per animal

Data structures for sparse matrices

• Linked list
• Triples as Hash
• IJA

– (row pointer to columns and values)

• Matrix not stored

Data structures

Solving strategies

• Sparse factorization
• Iteration

– Gauss Seidel – Gauss Seidel + 2nd order Jacobi – PCG

• Conditioners from diagonal to incomplete

factorization

Gauss-Seidel Iteration & SOR

Ax = b

• Simple
• Stable and self correcting
• Converges for semi-positive definite A
• For balanced cross-classified models converges in one round
• Small memory requirements
• Hard to implement matrix-free
• Slow convergence for complicated models

Preconditioned Conjugate Gradient

Large memory requirements Tricky implementation Easy to implement matrix-free Usually converges a few times faster than SOR even with diagonal preconditioner! Matrix-free iteration Let A = Σ (Wi’ Wi) nonzeros(W) << nonzeros(A) W easily generated from data Ax = Σ (Wi’ Wi x)

Methodologies to estimate variance components

• Restricted Maximum Likelihood (REML)
• Monte Carlo Markov Chain

REML

Φ- variance components (in R and G) C* - LHS converted to full rank Maximizations Derivative free (use sparse factorization) First derivative (Expectation maximization; use sparse inverse) Second derivative D2 and E(D2) hard to compute but [D2 and E(D2) ]/2 simpler – Average Information REML

Sparse matrix inversion

• Takahashi method:
• Can obtain inverse elements only for elements where L

≠0

• Inverses obtained for sparse matrices as large as

1000kx1000k

• Cost ≈ 2 x sparse factorization

REML, properties

• Derivative free methods reliable only for simple

problems

• Derivative methods

– Difficult formulas

• Nearly impossible for nonstandard models

– High computing cost ≈ quadratic – Easy determination of termination

Bayesian Methods and MCMC

Samples

Approximate P(se|y)

Approximate (p(sp|y)

Approximate (p(sa|y)

MCMC, properties

– Much simpler formulas – Can accommodate

• large models
• Complicated models

– Can take months if not optimized – Details important and hard decision when to stop

Optimization in sampling methods

• 10k-1 million samples
• Slow if equations regenerated each round
• If equations can be represented as:

[R⊗X1 + G⊗X2] = R⊗y

R,G – estimated small matrices, X1, X2, y – constant

X1, X2 and y can be created and stored once. Requires tricks if models different/trait and missing traits

Software

• SAS (fixed models or small mixed models)
• Custom
• Packages

– PEST, VCE (Groeneveld et al) – ASREML (Gilmour) – DMU (Jensen et al) – MATVEC (Wang et al) – Blupf90 etc. (Misztal et al)

Larger data, more complicated models, simpler computing?

Computing platforms

• Past

– Mainframe – Supercomputers

• Sparse computations vectorize!
• Current

– PCs + workstations – Windows/Linux,Unix

• Parallel and vector processing not important

Random regression model on a parallel processor

Madsen et al, 1999; Lidauer et all, 1999)

Goal: compute Σ (Wi‘Wix); Wi -large sparse vectors

• Approaches:

a) Distribute collection of Σ (Wi‘Wi)x t o separate processors b) Optimize scalar algorithm first to Σ (Wi‘(Wix)) If Wi has 30 nonzeros: a) 900 multiplications b) 60 multiplications Scalar optimization more important than brute force parallelization

Other Models

• Censored
• Survival
• Threshold
• …..

Issues

• 0.99 problem
• Sophistication of statistics vs.

understanding of problem vs. data editing

• Undesired responses of selection

– Less fitness,…. – Aggressiveness (swine, poultry)

• Challenges of molecular genetics

Molecular Genetics

• Attempts to identify effects of genes on

individual traits

• Simple statistical methodologies
• Methodology for joint analyses with

phenotypic and DNA data difficult

• Active research area

Conclusions

• Animal breeding compute intensive

– Large systems of equations – Matrices sparse

• Research has high large economic value