A reparameterization to improve numerical optimization in multivariate REML (co)variance component estimation

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A reparameterization to improve numerical optimization

in multivariate REML (co)variance component

estimation

E Groeneveld

To cite this version:

Original

article

A

_{reparameterization}

to

_improve

numerical

_optimization

in

multivariate

REML

_(co)variance

_component

estimation

E

Groeneveld

Institute

_of

Animal

_Husbandry

and Animal

_Behaviour,

Federal

_Agricultural

Research

Center,

Höltystr

10, 31535

_{Neustadt, Germany}

(Received

28

_February

_1994;

_accepted

15 June

₁₉₉₄₎

Summary - Multivariate restricted maximum likelihood

_(REML)

_(co)variance

_component

estimation

_using

numerical

optimization

on the basis of

_{Downhill-Simplex}

_(DS)

or

quasi-Newton

_(QN)

_procedures

suffers from the

_problem

of undefined ’covariance matrices’ as are

_{produced by}

the

_optimizers.

So

far,

this

_problem

has been dealt with

_{by assigning}

’bad’ function values. For this

procedure

to

work,

it is

_implied

that the information this ’bad’ function value conveys is sufficient to avoid

_going

in the same direction in the

_following

optimization

step. To a limited

_degree

DS can _{cope with this situation. On the other hand}

QN usually

breaks down if this situation occurs too

_frequently.

This contribution

_analyzes

the

_problem

and _proposes a _{reparameterization} of the covariance matrices to solve it. As

a

_result,

faster

_{converging QN optimizers}

can be

_used,

as

_they

no

_longer

suffer from lack of robustness. Four real data sets were

_{analyzed using}

a multivariate model

_estimating

between 17 and 30

_(co)variance

_components

_{simultaneously. Optimizing}

on the

_Cholesky

factor instead of on the

_(co)variance

_components themselves reduced the

_computing

time

by

a factor of 2.5 to more than

_250,

when

_comparing

the robust modified DS

_optimizer

operating

on the

_original

covariance matrices to a

QN

_optimizer

_using

_{reparameterized} covariance matrices.

multivariate REML

_/

optimization

/

quasi-Newton

/

Downhill-Simplex

/

reparame-terization

Résumé - Un _{reparamétrage pour améliorer l’optimisation numérique} dans une

estimation REML multivariate de _composantes de variance-covariance. L’estimation

du ma!imum de vraisemblance restreinte

_(REML)

des _composantes de variance-covariance

à l’aide des

_{procédures numériques d’optimisation Simple!-Descendant}

_(SD)

ou quasi-Newton

_(QN)

se heurte à la

_difficulté

résultant de la

production

de matrices de covariances

non

_{définies. Jusqu’à présent,}

cette

_difficulté

a été résolue en attribuant une vraisemblance

arbitraitement mauvaise à de telles _matrices, de

_façon

à éviter de revenir dans cette

converge

plus lorsque

cette situation se

_reproduit

_trop souvent. Cette note propose un

reparamétrage

pour résoudre le

problème.

Il devient ainsi

possible

d’utiliser la

procédure

QN

dont la _{convergence est}

_rapide

et la robustesse assurée.

_{Quatre fichiers}

de données

ont été

_analysés

pour estimer simultanément de 17 à 30 composantes de

_{(co)-variance.}

L’optimisation

du

_facteur

de

_Cholesky

au lieu des _composantes elles-mêmes réduit le _temps de calcul d’un

_facteur

compris entre 2,5 et

_plus

de

_500,

quand

on

compare la procédure

QN

avec

_{reparamétrage}

des matrices de covariance à la

_procédure

SD

_{modifiée appliquée}

aux matrices de covariance

_d’origine.

REML multivariate

_/

_optimisation

_/

_quasi-Newton

_/

_{Simplex-Descendant}

_/

repara-métrage

THE PROBLEM

In restricted maximum likelihood

_(REML)

_(Patterson

and

_Thompson,

_1971),

maximization of the likelihood is done

_using

either the EM

_algorithm

_(Dempster

et

al,

1977)

or

_procedures

that do not

_require

_explicit

derivatives. A

_problem

_specific

to

the latter class of

optimizers

is addressed in this _paper. Graser et al

₍₁₉₈₇₎

_proposed

a

sampling technique

that

spanned

the

complete

_parameter

_space for a

_single

trait

analysis.

Meyer

(1989)

used a

_{Dowhnill-Simplex}

_(DS)

and

_quasi-Newton

_(QN)

technique,

Kovac

₍₁₉₉₂₎

modified the DS

procedure

and also

expanded

Powell’s method of

_{conjugate gradients.}

All these authors had to deal with

_problems

_arising

from

_parameters

and

_posed

by optimizers

that lie outside the

_parameter

_space.

_Despite

this

problem,

a host of

REML estimates have been

_reported

with a

_varying

number of simultaneous traits

(Groeneveld,

1991; Meyer, 1991;

Tixier-Boichard et

_{al, 1992;}

Ducos et

_{al, 1993;}

Spilke

et

_{al, 1993;}

Mielenz et

_al,

_1994).

Consider a mixed linear model with one random effect and no

_missing

values as

specified

in

_equations

[1-5].

With a residual and an additive

_genetic

effect the

log

likelihood in

_equation

_[6]

must be maximized. The

_{computational}

_procedure

requires setting

up and

solving

the mixed-model

equations

(MME).

where:

y - vector of observations

Z - incidence matrix for the animal effect

j3 -

vector of unknown

_parameters

for fixed effects

u - vector of unknown

parameters

for the animal effect

e - vector of residuals

A -

_relationship

matrix of order number of animals and their known ancestors

Ro -

residual

_(co)variance

_among traits

Go -

covariance matrix for additive

_genetic

effects _{among traits} - Kronecker

_product

This

_requires

a set of covariance matrices for both the residual and the additive

genetic

components

R

o

,

Go.

Their inverses are used in

_setting

_up the MME.

After

_setting

_up the MME the

_system

of

_equations

_may be solved

_{by Cholesky}

factorization and backward substitution.

where

LV - value

_proportional

to the

_logarithm

of the likelihood function

b° -

solution vector of the MME C

*

- inverse of the coefficient matrix of the MME W -

_(XIZ)

n a

- number of animals

n - number of observations

In

_DS,

a

_complete

vertex is

_computed

before the

_{optimization begins.}

The DS

procedure

used here follows Kovac

_(1992),

and

_is,

_thus,

_very different from the

original

DS as

_proposed

_by

Nelder and Mead

(1965).

_Initially,

it

_{performs frequent}

restarts,

terminating

the iteration at

_increasing

_accuracy. This

_procedure

alleviated the well-known

_problem

of the DS to

_get

stuck at

_{suboptimal points.}

In

_QN,

gradients

are

_{required. They}

_may either be

_supplied

in their

_analytical

form or

approximated

using

finite differences

_(Schnabel

et

_al,

_1982).

REML is an iterative

procedure

and so the

R

°

and

Go

matrices have to be valid

for each round when the MME are set up.

Thus,

initially,

valid covariance matrices have to be

_provided

to the

_algorithm.

For a 2-trait model with 1 additive

_genetic

component

the residual and additive

_genetic

covariance matrices

_R

_o

and

_Go

of dimension 2 have to be estimated

_amounting

to an

_optimization

in a 6-dimensional

parameter

space.

However,

in the

_following

iteration,

new sets of

_(co)variance

are

_provided

_by

the

_optimizers.

For both DS and

QN,

the covariance matrices are an unstructured

vector of

_parameters,

in this case a vector of 6 values. The constraints of covariance

matrices,

ie that

_eigenvalues

have to be

positive

(equation

[7]),

are therefore unknown to the

_optimizers.

Given this

_background

it is not

_surprising

that the

outcome of an

_optimization

_step,

ie a new set of

_{(co)variances,}

is not

_guaranteed

to meet the

_requirements

of covariance matrices. In

practical

terms,

this means that

the determinant of the coefficient matrix

_generated

on the basis of these covariance

matrices will become less than zero, thus

_aborting

the whole process, because the

log

of a

_negative

number cannot be taken.

Groeneveld,

1994).

Furthermore,

when the true correlation between the traits is

high,

the ’covariance matrices’

_{proposed by}

the

_optimizers

are more

likely

to lie outside the

parameter

space, and the true covariance matrices to be located close

to the

_edge

of the

_parameter

_space.

An obvious

_(at

least in the context of

_DS)

solution to the treatment of undefined covariance matrices is to

_assign

a ’bad’ likelihood

value,

should

negative

eigenvalues

occur. This will tell the

optimizer

(DS)

to avoid this direction in

subsequent

optimization

steps.

While this

_procedure

may work

reasonably

well with

sampling-based

_{optimization algorithms,}

it

_{produces major problems}

with

_QN,

which

_requires

the function to be continuous and differentiable. If the condition occurs

_during

the proces of

approximating

the

gradients

by

finite

differencing,

assigning

a ’bad’

likelihood will result in a nonsensical

gradient.

When used

during

the

following

optimization

step

obviously

likewise nonsensical directions are chosen. In

short,

assigning

’bad’ likelihood values when undefined covariance matrices occur will

often result in

_aborting

the

_QN

_optimization

_step.

We can thus observe that the

optimizers

that have super linear convergence

properties

(and

are thus much faster than

sampling-based

procedures

like DS

(Dennis

and

Torczon,

1991))

fail

_increasingly

as the

dimensionality

of the

problem

increases.

_Thus,

the

problem

of undefined

_parameters

_{arising during optimization}

leads to the

_paradoxical

situation that an efficient class of

optimizers

can

only

be

used with confidence on small

_problems

with 1 or 2

_traits,

where

computing

time is not

_important

and

efficiency

of

_optimization

not an

_issue,

whereas inefficient

sampling-based

optimizers,

which are at best

_only

linearly converging,

must be used on

larger

problems.

A SOLUTION

Part of the

_problem

can be solved

_by

performing

a constrained

optimization.

This is

relatively

easy for the variances in which the constraint is

only

that

positive

numbers

be chosen.

_However,

no

technique

seems to be available to

_impose

a set of constraints such that no

_negative

_{eigenvalues A}

occur for a subset of the

dimensionality

of the

optimization

space as

_given

in

equation

(7!.

Instead,

we _propose to

_perform

the

_optimization

on the

Cholesky

factor of the covariance matrices. Let:

Thus,

instead of

optimizing

on the

R

o

,

its

_Cholesky

factor

_C

_r

is used

applying

the same

_operation

to all covariance matrices in the model.

_{Operationally,}

this

implies

the

_following

steps:

1)

User

_supplies

initial covariance matrices.

2)

The

Cholesky

factorization is

_performed

on all covariance matrices.

4)

The function SMME is called

_by

the

_optimizer.

This function sets _up and solves MME and

_computes

the likelihood value

_passing

the

_Cholesky

factor as

parameters.

5)

SMME

_computes

the

_original

_R

_o

and

_Go

from the

_factors,

sets _up and solves the MME and

_computes

the likelihood value.

6)

Control refers back to the

_optimizer

_{(Step 4)}

to have it decide on the next

_step

based on the last LV and the current set of factors.

This process results in a matrix that

_always

has the

_properties

of a covariance

matrix, irrespective

of the values that the

_optimizers

_may come _up with. This is

because:

As a

result,

undefined ’covariance’ matrices cannot occur.

A

special

case arises when certain covariances are not estimable. This _may

hap-pen with residual covariances when measurements on different traits are

_mutually

exclusive,

a situation

_frequently

_occurring

in

_joint

_analyses

of data from 2 test

en-vironments.

_During

_{optimization,}

the non-defined

_component

is

_{skipped, thereby}

reducing

its

_{dimensionality.}

The current

_{implementation}

of the

_{reparameterization}

computes

the

_Cholesky

factor on the basis of the

_complete

covariance matrix with zeros inserted for the undefined

_parameters.

_Although

the values of the

_Cholesky

factor

_depend

on the zero inserted for undefined

_{convariances,}

the same

_optimum

has

_always

been reached for

_optimization

on the

_{reparameterized}

and on the

_original

scale.

RESULTS

Four mutivariate runs are

_given

to assess the effect of

_optimizing

on the

_Cholesky

factors. The

_timings

listed refer to a Hewlett Packard 7100

_{computer system}

with the 99 MHz processor.

Run 1 was an

_analysis

of a selection

_experiment

in chickens with 5

_traits,

2 fixed

effects,

year and

barn,

and the animal

component.

As no traits were

_missing

a

canonical transformation was

_performed,

which reduced the number of numerical

operations

dramatically.

Thirty

components

must be estimated

_{simultaneously.}

The data set was

kindly

_supplied

_by

C

_Hagger

_(Swiss

Federal Institute of

_Technology).

Run 2 was an

_analysis

of 3 meat

_quality

traits on around 2 000

_pigs.

Not all records were

_complete.

There were 6 class effects and 1 covariable in the

_model,

which was identical for all traits. Random

components

were common litter and

animal, resulting

in 18

_components

to be estimated

_(Dietl

et

_al,

_1993).

Run 3 was an

analysis

of 4 048 station test records from swine

_comprising

the 4 traits

_daily

_gain,

feed conversion

_efficiency,

valuable

cuts,

and a meat

_quality

parameter.

There were 2

covariables,

4 fixed class

_effects,

1 random litter

_component

and the random correlated animal effect. The covariable

_weight

at the end of test

the field or in test

_stations,

but not in both

_{environments,}

the

_{corresponding}

residual covariance

_component

was not estimable. This results in 17 variances and

(co)variances

to be estimated. All 4 models included the full

relationship

among

animals. A summary

description

of the runs is

_given

table I.

Table II shows results from DS and

_QN

_using

this

_procedure.

The number of

function evaluations declines

_{substantially}

for both.

The

_general

_picture

shows a much smaller number of function evaluations for the

QN optimizer

compared

with DS. This is to be

_expected

as

QN

_optimizers

have a

super linear rate of _{convergence, ie} the number of iterations decreases for a fixed

increase _{in accuracy} as _convergence is

_{approached. However, QN optimizers}

aborted in 3 out of the 4 models when

_optimization

was done on the

_original

(co)variance

matrices

attesting

to the

_problems

of undefined

_parameters

outlined above. With

_optimization

on the

_components

of the

_(co)variance

_matrices,

_only

the

modified DS succeeded in

_locating

the

_optimum

_(DS

column in table

_II).

The

costs,

however,

were

_large,

_particularly

for run 1

_(at

3.7 s _per function

evaluation).

This

_converged

at the number of rounds indicated with an _accuracy of

10-

6

on the distance between the worst and best

_parameter

set in the vertex, but had still not

_quite

reached the

_{optimum. Interestingly,}

_apart

from the 8 557

_illegal

points

that were

_produced

_by

the DS

_optimizer,

a loss of rank of the coefficient

system

occurred 23 120 times. This condition is encountered

_during

factorization when a zero

_pivot

is detected. In this _case, factorization is aborted

_{and, again,}

are

_partly

due to

_badly

conditioned covariance

_matrices,

that

_just

about _pass the

test for

_positive

_eigenvalues,

but still do not render the coefficient matrix

_positive

definite. It is thus an effect of limited _accuracy on

_digital

_computers

for these 2 tests.

_Obviously,

this

_phenomenon

also

_produced

a

_large

amount of directional

misinformation, wasting

a substantial amount of CPU time.

Runs 1-3 had _{to cope} with a

_large

number of undefined

_parameters

_nearly

1

undefined for the 3 or 4 that were within the

_parameter

_space,

_resulting

in a

substantial amount of

_conflicting

directional information. The situation was better

in run

_4,

where DS

_only

left the

_parameter

_space 13 times.

QN

aborted in the first 3 runs after a

_varying

number of function evaluations because of a discontinuous function surface introduced

_by

the ’bad’ function values

given

to undefined

_points.

_Only

run 4 was

completed successfully,

_despite

the occurrence of 96

_illegal

_points.

With

_optimization

on the

_Cholesky

factor of the

_(co)variance

matrices the

_picture

changes drastically.

The extent to which the DS

_optimizer

benefited was

_dependent

on the number of

_illegal

_points

_prior

to

_{reparameterization.}

_Accordingly,

run 1

finished in less than a 20th of the time

_converging

to the best

_point,

while run 2

and 3 finished twice as fast.

_Only

run 4 did not

_benefit,

which was to be

_expected.

In

_fact,

the

_{reparameterization}

resulted in more function evaluations. Whether this is a chance result or an indication of a more

_general

_pattern,

cannot be

_conclusively

decided at this

_{point. However, experience}

from a

_large

number of other runs has

shown a substantial

_variability

in the number of function evaluation till convergence

for

_seemingly

identical models in terms of number of

_parameters.

It is therefore assumed that the observed slow down is more

_likely

a chance result than a

_general

phenomenon.

QN

found the solution in all runs.

_Depending

on the number of

_illegal

_points

encountered

_before,

the

_speed

_up was substantial. This was

_computed

as the ratio

of the number of function evaluations of

_QN

with

_Cholesky

factor versus DS on the

original

scale. If

_QN

and DS can

_only

be used on the basis of the

_original

covariance

matrices,

only

DS will

_reliably

_give

results.

_Thus,

this is considered as the reference

point.

Run 1 was

_particularly

_impressive:

with DS

_operating

on the

_original

covariance

matrices, optimization

was

_basically

_{impracticable}

as

_computing

time was at around 3 weeks or more of CPU time

_{prohibitively high}

_(and

the best

_point

was still not

quite

reached).

QN,

on the other

_hand,

solved the

_problem

in less than 3 h. The other 3 runs showed a

_speed-up

factor of between 2.5 and 15.7.

CONCLUSIONS

Multivariate REML estimates for

_general

statistical models suffer from

_high

com-putational

demands and the rather low

_{dimensionality}

in terms of number of traits that

_they

could handle. In most _cases,

_only

bivariate

_analyses

could be done with

general

models

_{producing suboptimal}

covariance matrices of

_higher

order

_(Spilke

and

_Groeneveld,

_1994).

The indicated

_improvement

in

_speed

_by

_optimizing

on the

Cholesky

factor of the covariance matrices will have a 2-fold effect:

_firstly,

it will

much more efficient

_QN

_algorithms,

which will

_considerably

extend the _scope of

multivariate REML

_(co)variance

component

estimation. Most

_importantly,

it will allow users to increase the

_{dimensionality}

of the models that can be

handled,

thus

helping

close the gap between the number of traits used in

_genetic

evaluation and

(co)variance

component

estimation.

This

_analysis

was done with the VCE _program, a

multivariate,

multimodel

REML

_(co)variance

_component

estimation

_package

_{(Groeneveld, 1994),}

which is

available for research purposes on the _anonymous

ftp

server under the Internet

number 192.103.38.1.

ACKNOWLEDGMENT

The valuable

_suggestions

of the reviewers are

_{gratefully acknowledged.}

One reviewer

referred to a _paper

_by

Lindstrom and Bates

₍₁₉₈₈₎

who also

_{suggested reparameterization}

of covariance matrices in mixed models for

_{repeated-measures}

data. Their conclusion that

&dquo;reparameterization

is

_key

to

_ensuring

_{consistent convergence} of the

Newton-Raphson

algorithm&dquo;

for this class of models agrees with our

_findings.

REFERENCES

Dempster APN,

Laird

_NM,

Rubin DB

₍₁₉₇₇₎

Maximum likelihood from

_incomplete

data

via the EM

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J

R Stat Soc Ser B 1-38

Dennis

_JE,

Torczon V

₍₁₉₉₁₎

Direct search methods on

parallel

machines. SIAM J Sci

Comput 1, 448

Dietl

_G,

Groeneveld

E,

Fiedler I

₍₁₉₉₃₎

Genetic parameters of muscle structure traits in

_pigs.

In:

44th

Annual

_{Meeting of}

the

European

Associatiore

_for

Animal

Production,

Aarhus,

Denmark

Ducos

_A,

Bidanel

J, Ducrocq

V, Boichard

_D,

Groeneveld E

₍₁₉₉₃₎

Multivariate restricted maximum likelihood estimation of

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parameters for

_growth,

carcass and meat

quality

traits in French

Large

White and French Landrace

Pigs.

Genet Sel

Evol 25,

475

Graser

_HU,

Smith

_SP,

Tier B

₍₁₉₈₇₎

A derivative-free

_approach

for

_estimating

variance

components in animal models

_by

restricted maximum likelihood. J Anim Sci 64, 1362

Groeneveld E

₍₁₉₉₁₎

Simultaneous REML estimation of 60 covariance _components in

an animal model with

_missing

values

_using

a

_{Downhill-Simplex algorithm.}

In:

_42nd

Annual

_{Meeting of}

the

European

Association

_for

Animal

Prod!ctio!c, Berlin, Germany

Groeneveld E

₍₁₉₉₄₎

REML VCE - a multivariate multimodel restricted maximum

likelihood

_(co)variance

_component estimation

_package.

In:

_{Proceedings of}

an EC

Symposium

on

Application of

Mixed Linear Models in the Prediction

of

Genetic Merit

in

_Pigs

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