The covariation method of estimation Add_my_pet

Dina LikaDept of Biology

The covariation method of estimationAdd_my_pet

UNIVERSITY OF CRETETexel, 15/4/2013

Contents• The covariation method for parameter

estimation– DEB parameters – Auxiliary theory– Real & pseudo data– Zero & variate data– Estimation criteria– Numerical implementation– Evaluation of the estimation

The standard DEB model

variables • structure, reserve, maturity, density of damage inducing

compounds, and density of damage compounds

parameters• Core parameters

– Control changes of the state variables– Linked to the concepts on which the model is based on

• Auxiliary parameters– Convert measurement (e.g. from dry to wet mass, length to volume

etc.)

– Quantify effects of temperature on rates and time

• Primary parameters– Connected to a single underlying process

• Compound parameters– Depend on several underlying processes

1 food type, 1 reserve, 1 structure, isomorphExtended: V1-morphic early juvenile stage

Core parametersassimilation {pAm} max surface-specific assim rate Lm ( 22.5z J cm-2 d-1)

feeding {Fm} surface- specific searching rate (6.5 l d-1 cm-2)

digestion κX digestion efficiency (0.8)

product formation κXP defecation efficiency (0.1)

mobilisation v energy conductance (0.02 cm d-1)

allocation allocation fraction to soma (0.8)

reproduction R reproduction efficiency (0.95)

turnover,activity [pM] volume-specific somatic maint. costs ( 18 d-1cm-3)

heating,osmosis {pT} surface-specific somatic maint. costs (0 d-1cm-2)

development kJ maturity maintenance rate coefficient (0.002 d-1)

Growth [EG] specific growth for structure (2800 J cm-3)

life cycle EHb maturity at birth (0.275z3 J)

life cycle EHj maturity at metamorphosis ( z3 J)

life cycle EHp maturity at puberty (166z3 J)

aging ha Weibul aging acceleration (10-6z d-2)

aging sG Gompertz stress coefficient (0.01)maximum length Lm = {pAm} / [pM]

z zoom factor z= Lm / Lmref, with Lm

ref =1

Auxiliary parameters

Conversion parametersδM shape coefficient (-)

dO =(dX, dV, dE, dP) specific densities (g/cm3)

μO =(μX, μV, μE, μP) chemical potentials (J/mol)μM =(μC, μH, μO, μN) chemical potentials (J/mol)nO =(nX, nV, nE, nP) chemical indices (-)nM =(nC, nH, nO, nN) chemical indices (-)wO=(12 1 16 14) nO molecular weights (-)

Temperature parametersTref reference temperature (273 K)TA Arrhenius temperature (8000 K)TL, TH temperature tolerance range (277 K, 318 K)TAL, TAH Arrhenius temperatures for transitions to inert state (20 kK, 190kK)

Assumptions of auxiliary theory• A well-chosen physical length (volumetric) structural

length for isomorphs– Physical length Lw is the actual length of a body, defined for a

particular shape– Structural length L is the volumetric length of structure, where

the individual is assumed to consist of structure, reserve and the reproduction buffer.

δM = L/ Lw

• Volume, wet/dry weight have contributions from structure, reserve, reproduction buffer

• Constant specific mass & volume of structure, reserve, reproduction buffer

• Constant chemical composition of juvenile growing at constant food

Data

• Real-data Empirical observations of physiological process

– zero-variate– uni-variate

• Pseudo-data Prior knowledge of a selection of parameter values

– zero-variate

Zero-variate data

Life history events: hatching, birth, metamorphosis, puberty, death

Real data: age, length, dry-, wet-weight at life history events max rates: reproduction, respiration, feeding, growth

Modified by food, temperature

Pseudo-dataTypical parameter values of the generalized animal

Species specific parameters should not be included as pseudo-data (e.g., z, δM, EH

b, EHp)

Growth efficiency κG vary less than the specific cost for structure [EG], and should be preferred for pseudo-data

[EG] = μV [MV] / κG with [MV] =dV /wV

Typical values for the ash-free-dry-weight over wet-weight ratio.Scyphomedusa 0.04 Ctenophora 0.04 Ascidia 0.06 Ectoprocta 0.07Priapulida 0.07 Cheatognata 0.07 Actinaria 0.08 Bivalvia 0.09Echinodermata 0.09 Porifera 0.11 Sipuncula 0.11 Gastropoda 0.15Polychaeta 0.16 Crustacea 0.17 Cephalopoda 0.21 Pisces 0.22Turbellaria 0.25 Aves 0.28 Reptilia 0.30 Mammalia 0.30

Uni-variate data

• length, weight, reproduction, respiration, feeding as functions of time, temperature, food

• incubation time, juvenile period, life span as functions of time, temperature, food

• weight as function of length • egg number as function of weight/length

Completeness of Real-data 0 maximum length and body weight; weight as function of length

1 age, length and weight at birth and puberty for one food level;

mean life span (due to ageing)

2 growth (curve) at one food level: length and weight as function of age at constant (or abundant) food level

3 reproduction and feeding as function of age, length and/or weight at one food level

4 growth (curve) at several (>1) food levels;

age, length and weight at birth and puberty at several food levels

5 reproduction and feeding as function of age, length and/or weight at several (>1) food levels

6 respiration as function of length or weight and life span at several (>1) food levels

7 elemental composition at one food level, survival due to ageing as function of age

8 elemental composition at several (>1) food levels, including composition of food

9 elemental balances for C, H, O and N at several body sizes and several food levels

10

energy balance at several body sizes and several food levels (including heat)

Each level includes all lower levels

Core Primary Parameters

{pAm}[pM]

Mapping Functions

f[EG] v ...

Lm = {pAm}/[pM]

Auxiliary Parameters

δM dV yEV ...

Wm = Lm3dV(1+fyEV [Em]/[EG])rB = 1/(3/ [pM]/[EG] + 3 * f * Lm/ v)

Zero-variate Observations

Wm maximum dry mass (g)

Uni-variate Observations

LW (body lenght,cm)t (time, days)

LWm = Lm/δM

rb von Bertalanffy growth rate (1/day)LWm maximum body length (cm)

Lw (t)= Lwm - (Lwm - Lwb) exp(-rBt)

...

Abstract World

Real World

pred

ictio

n

estim

atio

n

Zero-variate Pseudo-data

[pM]ref ref[EG]refvrefLW(t1)t1

LW(t2)t2

LW(t3)t3

...

[Em] = {pAm}/v ref =

Lika et al., 2011J. Sea Research 22:270-277

The covariation methodEstimates all parameters simultaneously

using all data: single-step-procedure

Independently normally distributed error with constant variation coefficient

Estimation criteria

• Weighted Least Square (WLS)

• Maximum Likelihood (ML)

WLS criterion

Minimization of a weighted sum of squared deviations between observations yij and predictions fij

The weight coefficients : wij / yij2

account for differences in units of the various dataThe dimensionless weight factor wij

account for the certainty of the individual data point

j i ij

ijijij y

fyw

2

2)(

ML criterionFor independently normally distributed dependent

variables, the ln-likelihood function is

The ML estimator for the squared variation coeff

The ML estimates minimize

i i

iic

icc xfyxfnn 2

2)1);(/(

2

1);(lnln)2ln(

2),(

i

iic xfyn

22 )1);(/(1

ˆ

);(ln1

ˆln i

ic xfn

Numerical implementation

Reflection Expansion

Contractionoutside

Contractioninside

Nelder-Mead method

A simplex method for finding a local minimum of a function of several variables

For 2 variables, a simplex is a triangle

The function is evaluated at the vertices of the triangle.

The worst vertex xh , where f is largest, is rejected and replaced with a new vertex xC obtained via a sequence of transformations (reflect, expand or contract) or shrink the triangle towards the best.

Does not require any derivative info

Shrinking

Numerical implementationNelder-Mead simplex methoddebtool/lib/regr/nmregr (WLS)

debtool/lib/regr/nmvcregr (ML)

Numerical implementationNewton-RaphsonA method for finding successively the roots of

an equation f(x)=0.

The iteration scheme:

debtool/lib/regr/nrregr (WLS)

debtool/lib/regr/nrvcregr (ML)Source wikipedia

)(

)(1

n

nnn xf

xfxx

Evaluation of the estimation

• Effects of pseudo-data– Elasticity coefficients

θ a core parameter to be estimated estimate of θ given the pseudo data θ0

α percentage increase in pseudo-valueestimate of θ given the pseudo data

θ0(1+α)

0

01

ˆ

ˆˆ

e

0

1

Evaluation of the estimation

• Goodness of fit– Mean relative error for the real data

n

i i

i

n 1

2

obs

exp1

1

n

i i

i

n 1

2

exp

obs1

1

estimationcriterion

WLS ML

MRE

function debtool/lib/regr/mre debtool/lib/regr/mrevc

FIT =10 (1-MRE)

Parameter identifiability

κ data on growth and reproduction and size at birth and puberty are required simultaneously

z, δM zero-variate data and growth data, while additional uni-variate data reduce the

standard deviation of the estimate.κΧ, {Fm} feeding data

kJ, EHp , κR reproduction at several food levels

ha mean life span

sG survival as a function of age

Lika et al., 2011J. Sea Research 22:278-288

Kooijman et al. 2008 Biol. Rev., 83:533-552.

Properties of the covariation method

estimation of parameter κ The effect of the pseudo-value κ is reduced only when there is

informationfor both growth and reproduction

estimation of parameterthe effect of the pseudo-value is reduced only when information on

age at birth and puberty is given

estimation of parameter [pM]the effects of the pseudo-value [pM] are reduced as information onreal data increasesthe least effect is obtained when information on respiration is included

the estimation of [EG] the effects of the pseudo-data κG are reduced as information on real data increases

estimation of the parameter kJthe pseudo-value for kJ does not play significant role

The covariation method for parameter estimation

• Estimation of all parameters of the standard DEB model simultaneously

• Real-data and pseudo-data, exploiting the rules for the covariation of parameter values among species implied by the standard DEB model

• The least required information is the maximum size, but the pseudo-data fully control the result

• Increasing the number of type of data decreases the role of pseudo data

Add_my_pet collection2011 : ~ 60 species 2013 : 240 species

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

COMPLETE mark

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

FIT mark

1 2 3 4 50

2

4

6

8

10

COMPLETE mark

FIT

mar

k

Max specific assimilation rate

Before accelerationAfter acceleration

-2 -1 0 1 2

0

1

2

3

4

10log L, cm

10lo

g {p

Am

}, J

cm

-2 d

-1

Kooijman, 2013Oikos 122:348-357

Maturity levels

-2 -1 0 1 2-10

-5

0

5

10

10log L, cm

10lo

g E

Hb

-2 -1 0 1 2-7

-4

-1

2

5

8

10log L, cm

10lo

g E

Hj

-2 -1 0 1 2-6

-4

-2

0

2

4

6

8

10

10log L, cm

10lo

g E

Hp

Energy conductance

-2 -1 0 1 2-4

-3

-2

-1

0

1

2

10log L, cm

10lo

g v,

cm

d-1

Before accelerationAfter acceleration

Thank you for your attention