The covariation method of estimation
Add_my_pet
Dina Lika
Dept of Biology
Texel, 15/4/2013
UNIVERSITY OF
CRETE
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
1 food type, 1 reserve, 1 structure, isomorph
Extended: V1-morphic early juvenile stage
• 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
Core parameters
assimilation
{pAm}
feeding
{Fm}
digestion
κX
product formation κXP
mobilisation
v
allocation

reproduction
R
turnover,activity [pM]
heating,osmosis {pT}
development
kJ
Growth
[EG]
life cycle
EHb
life cycle
EHj
life cycle
EHp
aging
ha
aging
sG
z
max surface-specific assim rate  Lm ( 22.5z J cm-2 d-1)
surface- specific searching rate (6.5 l d-1 cm-2)
digestion efficiency (0.8)
defecation efficiency (0.1)
energy conductance (0.02 cm d-1)
allocation fraction to soma (0.8)
reproduction efficiency (0.95)
volume-specific somatic maint. costs ( 18 d-1cm-3)
surface-specific somatic maint. costs (0 d-1cm-2)
maturity maintenance rate coefficient (0.002 d-1)
specific growth for structure (2800 J cm-3)
maturity at birth (0.275z3 J)
maturity at metamorphosis ( z3 J)
maturity at puberty (166z3 J)
Weibul aging acceleration (10-6z d-2)
Gompertz stress coefficient (0.01)
zoom factor z= Lm / Lmref, with Lmref =1
maximum length Lm =  {pAm} / [pM]
Auxiliary parameters
Conversion parameters
δM
dO =(dX, dV, dE, dP)
μO =(μX, μV, μE, μP)
μM =(μC, μH, μO, μN)
nO =(nX, nV, nE, nP)
nM =(nC, nH, nO, nN)
wO=(12 1 16 14) nO
shape coefficient (-)
specific densities (g/cm3)
chemical potentials (J/mol)
chemical potentials (J/mol)
chemical indices (-)
chemical indices (-)
molecular weights (-)
Temperature parameters
Tref
TA
TL, TH
TAL, TAH
reference temperature (273 K)
Arrhenius temperature (8000 K)
temperature tolerance range (277 K, 318 K)
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-data
Typical parameter values of the generalized animal
Species specific parameters should not be included as pseudo-data
(e.g., z, δM, EHb, 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 0.07
Priapulida
0.07 Cheatognata 0.07 Actinaria
0.08 Bivalvia
0.09
Echinodermata 0.09 Porifera
0.11 Sipuncula
0.11 Gastropoda 0.15
Polychaeta
0.16 Crustacea
0.17 Cephalopoda 0.21 Pisces
0.22
Turbellaria
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
Abstract World
Auxiliary Parameters
Core Primary Parameters
[pM] [EG] v f {pAm} 
...
δM dV yEV ...
Mapping Functions
estimation
prediction
Lm = {pAm}/[pM]
[Em] = {pAm}/v
 =
ref
rB = 1/(3/ [pM]/[EG] + 3 * f * Lm/ v)
Wm = Lm3dV(1+fyEV [Em]/[EG])
LWm = Lm/δM
Lw (t)= Lwm - (Lwm - Lwb) exp(-rBt)
Wm maximum dry mass (g)
LWm maximum body length (cm)
rb von Bertalanffy growth rate (1/day)
...
Zero-variate Observations
Real World
[pM]ref vref [EG]ref ref
Zero-variate Pseudo-data
t (time, days) LW (body lenght,cm)
t1
t2
t3
...
LW(t1)
LW(t2)
LW(t3)
Uni-variate Observations
Lika et al., 2011
J. Sea Research 22:270-277
The covariation method
Estimates 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
 w
ij
j
i
( yij  f ij ) 2
yij2
The weight coefficients : wij / yij2
account for differences in units of the various data
The dimensionless weight factor wij
account for the certainty of the individual data point
ML criterion
For independently normally distributed dependent
variables, the ln-likelihood function is
n
1
( ,c )   ln( 2 )  n ln c   ln f ( xi ; )  2
2
2c
i
2
(
y
/
f
(
x
;

)

1
)
 i
i
i
The ML estimator for the squared variation coeff
ˆc2 
1
2
(
y
/
f
(
x
;

)

1
)
 i
i
n i
The ML estimates ˆ minimize
1
ln ˆc   ln f ( xi ; )
n i
Numerical implementation
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
Reflection
Contraction
outside
Shrinking
Expansion
Contraction
inside
Numerical implementation
Nelder-Mead simplex method
debtool/lib/regr/nmregr (WLS)
debtool/lib/regr/nmvcregr (ML)
Numerical implementation
Newton-Raphson
A method for finding successively the roots of
an equation f(x)=0.
The iteration scheme:
xn1
f ( xn )
 xn 
f ( xn )
debtool/lib/regr/nrregr (WLS)
debtool/lib/regr/nrvcregr (ML)
Source wikipedia
Evaluation of the estimation
• Effects of pseudo-data
– Elasticity coefficients
ˆ1  ˆ0
e 
ˆ0
θ a core parameter to be estimated
ˆ0 estimate of θ given the pseudo data θ0
α percentage increase in pseudo-value
ˆ1 estimate of θ given the pseudo data θ0(1+α)
Evaluation of the estimation
• Goodness of fit
– Mean relative error for the real data
estimation
criterion
MRE
function
WLS
n
1

n i 1
 exp i
1 
 obs i
ML



2
debtool/lib/regr/mre
FIT =10 (1-MRE)
n
1

n i 1
 obs i
1 
 exp i



2
debtool/lib/regr/mrevc
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.
feeding data
reproduction at several food levels
mean life span
survival as a function of age
κΧ, {Fm}
kJ, EHp , κR
ha
sG
Kooijman et al. 2008
Biol. Rev., 83:533-552.
Lika et al., 2011
J. Sea Research 22:278-288
Properties of the covariation method
estimation of parameter κ
The effect of the pseudo-value κ is reduced only when there is information
for both growth and reproduction
estimation of parameter 
the 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 on
real data increases
the 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 kJ
the 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 collection
2011 : ~ 60 species
2013 : 240 species
1
0.8
0.6
10
0.4
8
FIT mark
0.2
6
0
0
1
2
3
4
COMPLETE mark
5
6
1
4
0.8
2
0.6
0
1
2
3
COMPLETE mark
4
5
0.4
0.2
0
0
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FIT mark
8
10
Max specific assimilation rate
-2
log {pAm}, J cm d
10
-1
4
3
2
1
0
-2
-1
0
log L, cm
10
1
2
Before acceleration
After acceleration
Kooijman, 2013
Oikos 122:348-357
Maturity levels
10
8
5
log Ej
10
0
10
log Eb
H
H
5
2
-1
-5
-4
0
, cm
L
log

10
-7
2
1
-1
-2
10
8
6
H
-1
log Ep
-2
10
-10
4
2
0
-2
-4
-6
-2
-1
0
log
L
, cm
10

1
2
0
, cm
L
log

10
1
2
Energy conductance
2
10
log v, cm d-1
1
0
-1
-2
-3
-4
Before acceleration
After acceleration
-2
-1
0
log L, cm
10
1
2
Thank you for your attention