Animal growth in random environments: estimation with several paths

Animal Growth in Random Environments: Estimation with Several Paths

Patrícia A. Filipe

Carlos A. Braumann

Bulletin of the International Statistical Institute, LXII

(Proc. 56

th

Session of the International Statistical Institute)

Paths

Filipe,Patrí ia A.

Universidadede Évora, CIMA(Centro deInvestigação em Matemáti a e Apli ações) Rua Romão Ramalho59

7000-671Évora, Portugal E-mail: pasfuevora.pt

Braumann,Carlos A.

Universidadede Évora, CIMA(Centro deInvestigação em Matemáti a e Apli ações) E-mail: braumannuevora.pt

1. Introdu tion

Classi almodelsforthegrowthofanindividualanimal(orplant)intermsofitssize

X(t)

(some measureof weight, volume or height)at time

t

have assumedthe form ofa dierential equation

dY (t) = b (A − Y (t)) dt,

Y (t

0

) = y

0

,

(1)

whit

Y (t) = g(X(t))

, where

g

is a stri tly in reasing fun tion. We have

y

0

= g(x

0

)

, where

x

0

is the size at birth,and

A = g(a)

, where

a

isthe asymptoti size or size at maturity of the individual. The parameter

b > 0

isa rateof approa h to maturity.

TheBertalany-Ri hardsmodel,proposedbyvonBertalany(1957)andalsostudiedbyRi hards (1959), hasbeen extensively usedand orrespondsto

g(x) = x

c

for

c > 0

and to

g(x) = ln x

for

c = 0

(for

c = 0

, is also known as the Gompertz model, appropriate if one assumes growth to be basi ally a multipli ative pro ess). The spe ial ase

c = 1

is also known as the Mits herli h model and has been usedin agri ulture (see, for instan e, Goldsworthy and Colegrove, 1974), parti ularly for linear measurementslike lengthor height. Ifone,however, onsiderssize measuredasa volume ora weight,

c = 1/3

isquitea ommon hoi e (making

Y (t)

a kind of"length"). Other hoi es of

c

(in luding the

hoi e providingthe bestadjustment) have been proposed.

The Bertalany-Ri hards model has been applied to animal growth data extensively. See, for instan e: Freitas(2005),Mazinietal. (2003),OhnishiandAkamine(2006),Oliveira,Lbo,andPereira (2000). For anappli ation to tumor growth, see Kozuskoand Bajzer(2003).

When one onsiders the ee ts of environmental random u tuations on the growth pro ess, it isnatural to propose(see Gar ia, 1983)the sto hasti dierential equation (SDE) model

dY (t) = b (A − Y (t)) dt + σdW (t),

Y (t

0

) = y

0

,

(2)

where

W (t)

isastandardWienerpro essand

σ > 0

measurestheintensityoftheee tofenvironmental u tuations (internal and external) on growth. The solutionis a homogeneous diusion pro ess with drift

b (A − y)

and diusion oe ient

σ

2

. The solutionis(see, forinstan e, Braumann, 2005)

Y (t) = A + e

−bt

(y

0

− A) + σe

−bt

Z

t

0

e

bs

dW (s).

(3)

Thedistribution of

Y (t)

isGaussian with mean

A + e

−bt

_(y

0

− A)

and varian e

σ

2

2b

(1 − e

−2bt

)

.

Usually, random variations in data have been treated by lassi al regressionmodels. Regression models assume that the observed deviations from a deterministi urve are independent at dierent

random u tuations. For instan e, in a regression model, a delayin growth at a ertain time hasno reper ussionsonfutureweights, makingregressionmodelsinappropriate tomodelgrowthinarandom environment. TheSDE model(2) doesnot have theseshort omings.

In Patrí ia, Braumann, and Roquete (2007), we have onsidered, for a single path (a single animal), the statisti al problems of parameter estimationand of predi tion of future population sizes for model (2) and have illustrated the methods with data on the weight of bovine growth. Se tion 2 givesa briefsummaryofthe estimationpart. Here, we extend(see Se tion 3)the estimationmethods tothe aseofseveralpaths(severalanimalsofthesametyperaisedundersimilar onditions),assumed tobe independent,and also illustratewith bovine data.

The data in the illustrations,provided byCarlos Roquete (ICAM-University of Évora), isfrom "mertolengo" attleofthe"rosilho"strandraisedinthe"HerdadedaAbóboda"intheSerparegion,at theleftmarginoftheGuadianariver. Theanimalswereraisedin pasture,together withtheirmothers duringnursing and latersupplemented withsilagewhen pastureisin shortage(Autumn andWinter).

2. Parameter estimation for a single path

Assumeweobservetheevolutionofthesizeofoneanimal(onepath)bymeasuringitssizeatthe times( ounted frombirth)

0 = t

0

< t

1

< ... < t

n

andwant to estimate

p

= (A, b, σ)

. Let

X

k

= X(t

k

)

be the animal size at time

t

k

(

k = 1, 2, ..., n

) and let

Y

k

= Y (t

k

) = g(X(t

k

))

. Let

x

= (x

0

, x

1

, ..., x

n

)

be the ve tor of observed values of

X

= (X

0

, X

1

, ..., X

n

)

and let

y

= (y

0

, y

1

, ..., y

n

)

, with

y

k

= g(x

k

)

(

k = 1, 2, ..., n

). Weassumethat

g

isaknownfun tion,sothatwe an omputethe

y

k

(

k = 1, 2, ..., n

).

For

k = 1, 2, ..., n

,one anseefrom(3)that

Y

k

= A+e

−b(t

k

−t

k−1

)

_(Y

k−1

−A)+σe

−bt

k

R

t

t

k−1

k

e

bs

_dW

s

. Therefore, onditionedon having

Y

k−1

= y

k−1

, the probabilitydensityfun tion (p.d.f.) of

Y

k

is

f

Y

k

|Y

k−1

=y

k−1

(y

k

) =

1

q

2π

σ

2

2b

(1 − e

−2b∆t

k

)

exp











−



y

k

− A − (y

k−1

− A) e

−b∆t

k



2

2

σ

2

2b

(1 − e

−2b∆t

k

)











,

where

∆t

k

= t

k

− t

k−1

. Sin e

Y (t)

isa Markovpro ess,the joint densityof

Y

1

= Y (t

1

), ..., Y

n

= Y (t

n

)

(given

Y

0

= y

0

, assumed known) is the produ t of the above onditional p.d.f. for

k = 1, 2, ..., n

and sothe log-likelihood fun tion in termsof the

Y

variables isgivenby

L

Y

(p) = −

n

2

ln

2πσ

2

2b

!

−

1

2

n

X

k=1

ln



1 − e

−2b∆t

k



₋

b

σ

2

n

X

k=1



y

k

− A − (y

k−1

− A) e

−b∆t

k



2

1 − e

−2b∆t

k

.

(4)

Intermsof the

X

variables,the log-likelihoodis

L

X

(p) = L

Y

(p) +

P

n

k=1

ln



dY

dX







_x=x

k



.

Themaximumlikelihood estimator

p

ˆ

= ( ˆ

A, ˆb, ˆ

σ)

isobtainedbymaximizationof

L

Y

(equivalent tothe maximization of

L

X

),usingnumeri al te hniques(in the appli ationswe have usedthe nlminb routine from the softwareS-PLUS). The estimators are asymptoti allyGaussian with mean ve tor

p

and varian e- ovarian e matrix

Σ

= F

−1

, where

F

is the Fisher information matrix with elements

F

ij

= −

E

[∂

2

_L

Y

/∂p

i

∂p

j

]

. Theexpressions ofthe

F

ij

an beexpli itly obtainedusing the propertiesof thepro ess

Y (t)

.

F

,andtherefore

Σ

, an beestimatedbyrepla ing

p

by

ˆ

p

onthoseexpressions,thus allowing the onstru tion ofapproximate onden e intervals for the parameters.

In Filipe,Braumann, and Roquete (2007), we have applied the sto hasti Bertalany-Ri hards model,for the parti ular ases

c = 0

and

c = 1/3

, to the weight in Kgof asingleanimal for whi h we had79 observationssin e birth tillabout5 tears ofage. Wehave onsideredalso the ases

c = 1

(not veryappropriateforweightdata)and

g(x) = x

c

with

c

unknown(also tobeestimatedfromdata); the last aseis mu h more umbersome(we an notuse

L

Y

be ause wedo notknow

g

, andsowe have to maximize

L

X

)and the improvement over the ases

c = 0

and

c = 1/3

wasnot signi ant.

The estimatedasymptoti varian e- ovarian e matri esfor

ˆ

p

are

V

c=0

=





0.00570 −0.00680 −0.00005

−0.00680 0.03266

0.00024

−0.00005 0.00024

0.00033





V

c=1/3

=





1731.7 −4.9989 −0.0882

−4.9989 0.0457

0.0008

−0.0882 0.0008

0.0018





.

Table 1 showsthe maximum likelihood estimates(and the orresponding valueof

L

X

) together with the approximate 95% onden eintervals. We usethe parameter

a = g

−1

_(A)

(averageweight at maturity) sothatwe may ompare the two ases;the other parameters,

b

and

σ

, arenot omparable. Figure 1showsthe graphsof the adjusted models

c = 0

and

c = 1/3

in the absen e of environ-mental u tuations (

σ = 0

). Filipe, Braumann, and Roquete (2007) also studied the adjustment of the modelsin terms oftheir abilityto predi t futureweightsof the animalunder study.

Table 1. Maximum likelihood estimates and 95% onden e intervals (one animal)

for

a

for

b

for

σ

L

X

c = 0

(Gompertz)

407.1 ± 60.5

1.472 ± 0.354

0.2259 ± 0.0355

−338.12

c = 1/3

422.4 ± 81.6

1.096 ± 0.525

0.5248 ± 0.0827

−337.88

Observed weight

Gompertz (c=0)

c=1/3

Age (years)

W

e

i

g

h

t

K

g

Figure 1. Estimated urves for =0 (Gompertz) and =1/3, when

σ = 0

3. Parameter estimation for several independent paths

Letus onsider nowseveralpathsof thesto hasti pro ess orrespondingto dierent animalsof the same type and raisedunder similar onditions. Assume we have data on

m

animals. The size of animalnumber

j

(

j = 1, 2, ..., m

)isobservedatthetimes( ountedfrombirth)

0 = t

j0

< t

j1

< ... < t

jn

j

and is, respe tively,

X

j0

= X(t

j0

), X

j1

= X(t

j1

), ..., X

jn

j

= X(t

jn

j

)

. Let

Y

jk

= Y (t

jk

) = g(X

jk

)

. Let

x

j

=



x

j0

, x

j1

, ..., x

jn

j



be the ve tor of of observed values of

X

j

=



X

j0

, X

j1

, ..., X

jn

j



and let

y

j

=



y

j0

, y

j1

, ..., y

jn

j



, with

y

jk

= g(x

jk

)

(

j = 1, 2, ..., m

;

k = 1, 2, ..., n

j

). Assumethat

g

isknown. For animal (traje tory) number

j

we an obtain its log-likelihood

L

Y

j

bypro eeding asin (4):

L

Y

j

(p) = −

n

j

2

ln

2πσ

2

2b

!

−

1

2

n

j

X

k=1

ln



1−e

−2b∆t

jk



₋

b

σ

2

n

j

X

k=1



y

jk

− A− (y

j,k−1

−A) e

−b∆t

jk



2

1 − e

−2b∆t

jk

,

(5)

with

∆t

jk

= t

jk

− t

j,k−1

. From the independen e, the overalllog-likelihood for the

m

animals is

L

Y

1

,...,Y

m

(p) =

m

X

j=1

L

Y

j

(p).

(6)

The maximum likelihood estimator

ˆ

p

is obtained bymaximization of

L

Y

1

,...,Y

m

, againusing nu-meri al te hniques. The estimators are asymptoti ally Gaussian with mean ve tor

p

and varian e- ovarian e matrix

Σ

= F

−1

matri esofthe individual traje tories,whi h expressions wealready know. Repla ing

p

by

ˆ

p

, we an againobtain anestimation of

Σ

and, therefore,approximate onden e intervalsfor the parameters.

We haveapplied the pro edure forthe sto hasti Bertalany-Ri hardsmodel,for the parti ular ases

c = 0

and

c = 1/3

, to

m = 5

animals of the same strand raised under similar onditions. One of them was the animal onsidered in the previous se tion (with 79 observations) and the other four animalshave 38 observations ea h.

The estimatedasymptoti varian e- ovarian e matri esfor

ˆ

p

were

V

c=0

=





0.00201 −0.00257 −0.00002

−0.00257 0.00965

0.00008

−0.00002 0.00008

0.00014





V

c=1/3

=





555.65 −1.7451 −0.0311

−1.7451 0.0115

0.0002

−0.0311 0.0002

0.0006





.

Table 2 shows the maximum likelihood estimates (and the orresponding value of

L

X

1

,...,X

m

) together with the approximate 95% onden e intervals.

Table 2. Maximum likelihood estimates and 95% onden e intervals (ve animals)

for

a

for

b

for

σ

L

X

1

,...,X

5

c = 0

(Gompertz)

352.4 ± 28.3

1.708 ± 0.193

0.2534 ± 0.0234

−958.84

c = 1/3

384.1 ± 46.2

1.147 ± 0.211

0.5062 ± 0.0468

−941.85

4. Con lusions

Sto hasti dierentialequationmodelsforthe growthofindividualanimalswere onsideredand parameterestimation pro edures were developed for the ase ofseveral traje tories(severalanimals), whereobservationsoftheanimalsizemaybemadeatdierenttimesfordierentanimals. An illustra-tionisshown for attledata provided byC. Roqueteusing the sto hasti Bertalany-Ri hards model with

c = 0

(Gompertzmodel)and

c = 1/3

. Estimationpro eduresfor the asewhere dierentanimals mayhavedierent randomly hosen

A

parameters will appearshortly in anup oming paper.

A knowledgment. Thisworkwasdevelopedin CIMA,aresear h enternan edbyFCT(Fundaçãoparaa Ciên iaeaTe nologia)withinits'ProgramadeFinan iamentoPlutianual'(underFEDER funding).

REFERENCES

Bertalany, L. von (1957). Quantitative laws in metabolism and growth. The Quarterly Review of Biology,34(3): 786-795.

Braumann,C.A.(2005). IntroduçãoàsEquaçõesDiferen iaisEsto ásti as. EdiçõesSPE.

Filipe, P. A., Braumann C. A., and Roquete, C. J. (2007). Modelos de res imento de animais em ambiente aleatório. A tasdo XIVCongressoAnualda So iedadePortuguesade Estatísti a(submitted).

Freitas, A. (2005). Curvas de res imento na produção animal. Revista Brasileira de Zoote nia, 35: 1843-1851.

Gar ia,O. (1983). A sto hasti dierentialequation model fortheheightof foreststands. Biometri s, 39: 1059-1072.

Goldsworthy,P.andColegrove,M.(1974). GrowthandyieldofhighlandmaizeinMexi o. J.Agri ulture S ien e,83: 213-221.

Kozusko, F. and Bajzer, Z. (2003). Combining gompertzian growth and ell population dynami s. Mathemati al Bios ien es,Vol. 185,p. 153-167.

Mazini, A., Muniz, J., Aquino, L., and Silva, F. (2003). Análise da urva de res imento de ma hos Hereford. Ciên ia Agroté ni a,27(5): 1105-1112.

Ohnishi,S. and Akamine, T.(2006). Extensionof vonBertallany growthmodelin orporatinggrowth patternsofsoftandhardtissuesinbivalvemollus s. Fisheries S ien e,72(4): 787-795.

Oliveira, H., Lbo, R., and Pereira, C. (2000). Comparaçãode modelosnão-lineares para des revero res imentodefêmeasdaraçaGuzerá. Pesquisa Agrope uáriaBrasileira,35: 1843-1851.