the covariation method of estimation add_my_pet
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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
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