1. Dina Lika Dept of Biology TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA The covariation method of estimation Add_my_pet UNIVERSITY OF CRETE Texel, 15/4/2013

2. 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

3. 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, isomorph Extended: V1-morphic early juvenile stage

4. Core parameters assimilation {p Am } max surface-specific assim rate  L m ( 22.5z J cm -2 d -1 ) feeding {F m } surface- specific searching rate (6.5 l d -1 cm -2 ) digestion κ X digestion efficiency (0.8) product formation κ X P defecation efficiency (0.1) mobilisation venergy conductance (0.02 cm d -1 ) allocation  allocation fraction to soma (0.8) reproduction  R reproduction efficiency (0.95) turnover,activity [p M ] volume-specific somatic maint. costs ( 18 d -1 cm -3 ) heating,osmosis {p T } surface-specific somatic maint. costs (0 d -1 cm -2 ) development k J maturity maintenance rate coefficient (0.002 d -1 ) Growth[E G ] specific growth for structure (2800 J cm -3 ) life cycleE H b maturity at birth (0.275z 3 J) life cycleE H j maturity at metamorphosis (  z 3 J) life cycle E H p maturity at puberty (166z 3 J) aging h a Weibul aging acceleration (10 -6 z d -2 ) agings G Gompertz stress coefficient (0.01) maximum length L m =  {p Am } / [p M ] z zoom factor z= L m / L m ref, with L m ref =1

5. Auxiliary parameters Conversion parameters δ M shape coefficient (-) d O =(d X, d V, d E, d P )specific densities (g/cm 3 ) μ O =(μ X, μ V, μ E, μ P )chemical potentials (J/mol) μ M =(μ C, μ H, μ O, μ N )chemical potentials (J/mol) n O =(n X, n V, n E, n P ) chemical indices (-) n M =(n C, n H, n O, n N ) chemical indices (-) w O =(12 1 16 14) n O molecular weights (-) Temperature parameters T ref reference temperature (273 K) T A Arrhenius temperature (8000 K) T L, T H temperature tolerance range (277 K, 318 K) T AL, T AH Arrhenius temperatures for transitions to inert state (20 kK, 190kK)

6. Assumptions of auxiliary theory A well-chosen physical length  (volumetric) structural length for isomorphs –Physical length L w 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. δ M = L / L w 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

7. Data Real-data Empirical observations of physiological process –zero-variate –uni-variate Pseudo-data Prior knowledge of a selection of parameter values –zero-variate

8. 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

9. Pseudo-data Typical parameter values of the generalized animal Species specific parameters should not be included as pseudo-data (e.g., z, δ M, E H b, E H p ) Growth efficiency κ G vary less than the specific cost for structure [E G ], and should be preferred for pseudo-data [E G ] = μ V [M V ] / κ G with [M V ] =d V / w V 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

10. 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

11. 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

12. Core Primary Parameters {p Am }[pM][pM]  Mapping Functions f[EG][EG] v... L m =  {p Am }/[p M ] Auxiliary Parameters δMδM dVdV y EV... W m = L m 3 d V (1+fy EV [E m ]/[E G ])r B = 1/(3/ [p M ]/[E G ] + 3 * f * L m / v) Zero-variate Observations W m maximum dry mass (g) Uni-variate Observations L W (body lenght,cm)t (time, days) L Wm = L m /δ M r b von Bertalanffy growth rate (1/day) L Wm maximum body length (cm) L w (t)= L wm - (L wm - L wb ) exp(-r B t)... Abstract World Real World prediction estimation Zero-variate Pseudo-data [p M ] ref  ref [E G ] ref v ref LW(t1)LW(t1)t1t1 LW(t2)LW(t2)t2t2 LW(t3)LW(t3)t3t3... [E m ] = {p Am }/v  ref =  Lika et al., 2011 J. Sea Research 22:270-277

13. 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)

14. WLS criterion Minimization of a weighted sum of squared deviations between observations y ij and predictions f ij The weight coefficients : w ij / y ij 2 account for differences in units of the various data The dimensionless weight factor w ij account for the certainty of the individual data point

15. ML criterion For independently normally distributed dependent variables, the ln-likelihood function is The ML estimator for the squared variation coeff The ML estimates minimize

16. Numerical implementation ReflectionExpansion Contraction outside Contraction inside 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 x h, where f is largest, is rejected and replaced with a new vertex x C obtained via a sequence of transformations (reflect, expand or contract) or shrink the triangle towards the best. Does not require any derivative info Shrinking

17. Numerical implementation Nelder-Mead simplex method debtool/lib/regr/nmregr (WLS) debtool/lib/regr/nmvcregr (ML)

18. Numerical implementation Newton-Raphson A 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

19. Evaluation of the estimation Effects of pseudo-data –Elasticity coefficients θ a core parameter to be estimated estimate of θ given the pseudo data θ 0 α percentage increase in pseudo-value estimate of θ given the pseudo data θ 0 (1+α)

20. Evaluation of the estimation Goodness of fit –Mean relative error for the real data estimation criterion WLSML MRE function debtool/lib/regr/mredebtool/lib/regr/mrevc FIT =10 (1-MRE)

21. Parameter identifiability κ data on growth and reproduction and size at birth and puberty are required simultaneously z, δ M zero-variate data and growth data, while additional uni-variate data reduce the standard deviation of the estimate. κ Χ, {F m } feeding data k J, E H p, κ R reproduction at several food levels h a mean life span s G survival as a function of age Lika et al., 2011 J. Sea Research 22:278-288 Kooijman et al. 2008 Biol. Rev., 83:533-552.

22. 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 [p M ] the effects of the pseudo-value [p M ] are reduced as information on real data increases the least effect is obtained when information on respiration is included the estimation of [E G ] the effects of the pseudo-data κ G are reduced as information on real data increases estimation of the parameter k J the pseudo-value for k J does not play significant role

23. 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

24. Add_my_pet collection 2011 : ~ 60 species 2013 : 240 species

25. Max specific assimilation rate Before acceleration After acceleration Kooijman, 2013 Oikos 122:348-357

26. Maturity levels

27. Energy conductance Before acceleration After acceleration

28. Thank you for your attention