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Art of modellingDEB course 2013
Bas KooijmanDept theoretical biology
Vrije Universiteit [email protected]
http://www.bio.vu.nl/thb
Texel, 2013/04/16
Modelling 1• model: scientific statement in mathematical language “all models are wrong, some are useful”
• aims: structuring thought; the single most useful property of models: “a model is not more than you put into it” how do factors interact? (machanisms/consequences) design of experiments, interpretation of results inter-, extra-polation (prediction) decision/management (risk analysis)
Modelling 2• language errors: mathematical, dimensions, conservation laws
• properties: generic (with respect to application) realistic (precision) simple (math. analysis, aid in thinking) plasticity in parameters (support, testability)
• ideals: assumptions for mechanisms (coherence, consistency) distinction action variables/meausered quantities core/auxiliary theory
Causation
Cause and effect sequences can work in chains A B C
But are problematic in networks A
B C Framework of dynamic systems allow for holistic approach
Dynamic systemsDefined by simultaneous behaviour of input, state variable, outputSupply systems: input + state variables outputDemand systems input state variables + outputReal systems: mixtures between supply & demand systemsConstraints: mass, energy balance equationsState variables: span a state space behaviour: usually set of ode’s with parametersTrajectory: map of behaviour state vars in state spaceParameters: constant, functions of time, functions of modifying variables compound parameters: functions of parameters
Empirical cycle
Modelling criteria
• Consistency dimensions, conservation laws, realism (consistency with data)
• Coherence consistency with neighbouring fields of interest, levels of
organisation
• Efficiency comparable level of detail, all vars and pars are effective
numerical behaviour
• Testability amount of support, hidden variables
Dimension rules
• quantities left and right of = must have equal dimensions
• + and – only defined for quantities with same dimension
• ratio’s of variables with similar dimensions are only dimensionless if addition of these variables has a meaning within the model context
• never apply transcendental functions to quantities with a dimension log, exp, sin, … What about pH, and pH1 – pH2?
• don’t replace parameters by their values in model representations y(x) = a x + b, with a = 0.2 M-1, b = 5 y(x) = 0.2 x + 5 What dimensions have y and x? Distinguish dimensions and units!
Model without dimension problem
Arrhenius model: ln k = a – T0 /Tk: some rate T: absolute temperaturea: parameter T0: Arrhenius temperature
Alternative form: k = k0 exp{1 – T0 /T}, with k0 = exp{a – 1}
Difference with allometric model: no reference value required to solve dimension problem
T-1ln r
ate
Arrhenius plot
Models with dimension problems 1.2.3
• Allometric model: y = a W b
y: some quantity a: proportionality constant W: body weight b: allometric parameter in (2/3, 1) Usual form ln y = ln a + b ln W Alternative form: y = y0 (W/W0 )b, with y0 = a W0
b
Alternative model: y = a L2 + b L3, where L W1/3
• Freundlich’s model: C = k c1/n
C: density of compound in soil k: proportionality constant c: concentration in liquid n: parameter in (1.4, 5) Alternative form: C = C0 (c/c0 )1/n, with C0 = kc0
1/n
Alternative model: C = 2C0 c(c0+c)-1 (Langmuir’s model)
Problem: No natural reference values W0 , c0
Values of y0 , C0 depend on the arbitrary choice
Allometric functions
Length, mmO2 c
onsu
mpt
ion,
μl/
h
Two curves fitted:
a L2 + b L3
with a = 0.0336 μl h-1 mm-2
b = 0.01845 μl h-1 mm-3
a Lb
with a = 0.0156 μl h-1 mm-2.437
b = 2.437
Kleber’s lawO2 consumption weight3/4
O2 consumption has contributions from• maintenance & development• overheads of assimilation, growth & reproduction
These are all functions of weight that should be added
But:
sum of functions of weight allometric function of weightProblem in relating respiration to other activities
Egg development time
Bottrell, H. H., Duncan, A., Gliwicz, Z. M. , Grygierek, E., Herzig, A., Hillbricht-Ilkowska, A., Kurasawa, H. Larsson, P., Weglenska, T. 1976 A review of some problems in zooplankton production studies.Norw. J. Zool. 24: 419-456
)))(ln(3414.0)ln(2193.03956.3exp( 2TTD
Kelvinin etemperatur t timedevelopmen egg
TD
2
2
)(ln
ln)dim(
ln
ln)dim(
ln)dim()))(ln()ln(exp(
K
tc
K
tb
taTcTbaD
molecule
cell
individual
population
ecosystem
system earth
time
spac
e
Space-time scales
When changing the space-time scale, new processes will become important other will become less importantIndividuals are special because of straightforward energy/mass balances
Each process has its characteristic domain of space-time scales
Complex models
• hardly contribute to insight• hardly allow parameter estimation• hardly allow falsification
Avoid complexity by• delineating modules• linking modules in simple ways• estimate parameters of modules only
Biodegradation of compoundsn-th order model Monod model
nkXXdt
d
1)1(10 )1()(
nn ktnXtX
ktXtXn
0
0
)( kXt /0
}exp{)( 0
1
ktXtXn
n
akXaXt
nn
1
1)(
111
00
XK
XkX
dt
d
ktXtXKXtX }/)(ln{)(0 00
ktXtXXK
0
0
)( }/exp{0 KktXt
}/exp{)( 0
0
KktXtXXK
aKkakXaXt ln)1()( 1100
; ;
X : conc. of compound, X0 : X at time 0 t : time k : degradation rate n : order K : saturation constant
Biodegradation of compoundsn-th order model Monod model
scaled time scaled time
scal
ed c
onc.
scal
ed c
onc.
Plasticity in parameters
If plasticity of shapes of y(x|a) is large as function of a:
• little problems in estimating value of a from {xi,yi}i
(small confidence intervals)
• little support from data for underlying assumptions
(if data were different: other parameter value results, but still a good fit, so no rejection of assumption)
Stochastic vs deterministic models
Only stochastic models can be tested against experimental data
Standard way to extend deterministic model to stochastic one: regression model: y(x| a,b,..) = f(x|a,b,..) + e, with e N(0,2)Originates from physics, where e stands for measurement error
Problem: deviations from model are frequently not measurement errorsAlternatives:• deterministic systems with stochastic inputs• differences in parameter values between individualsProblem: parameter estimation methods become very complex
StatisticsDeals with• estimation of parameter values, and confidence in these values• tests of hypothesis about parameter values differs a parameter value from a known value? differ parameter values between two samples?
Deals NOT with• does model 1 fit better than model 2 if model 1 is not a special case of model 2
Statistical methods assume that the model is given(Non-parametric methods only use some properties of the given model, rather than its full specification)
Large scatter
• complicates parameter estimation• complicates falsification
Avoid large scatter by• Standardization of factors
that contribute to measurements• Stratified sampling
Kinds of statistics 1.2.4
Descriptive statistics sometimes useful, frequently boring
Mathematical statistics beautiful mathematical construct rarely applicable due to assumptions to keep it simple
Scientific statistics still in its childhood due to research workers being specialised upcoming thanks to increase of computational power (Monte Carlo studies)
Nested models
2210)( xwxwwxy
xwwxy 10)( 0)( wxy 220)( xwwxy
Venn diagram
02 w 01 w
Error of the first kind: reject null hypothesis while it is true
Error of the second kind: accept null hypothesis while the alternative hypothesis is true
Level of significance of a statistical test: = probability on error of the first kind
Power of a statistical test: = 1 – probability on error of the second kind
Testing of hypothesis
true false
accept 1 -
reject 1 -
null hypothesis
dec
isio
nNo certainty in statistics
Statements to remember
• “proving” something statistically is absurd
• if you do not know the power of your test, do don’t know what you are doing while testing
• you need to specify the alternative hypothesis to know the power this involves knowledge about the subject (biology, chemistry, ..)
• parameters only have a meaning if the model is “true” this involves knowledge about the subject
Independent observations
IIf
If X and Y are independent
Central limit theorems
The sum of n independent identically (i.i.) distributed random variables becomes normally distributed for increasing n.
The sum of n independent point processes tends to behave as a Poisson process for increasing n.
yy
YXZ yYPyzXPzZPdyyfyzfzfYXZ )()()(;)()()(
Number of events in a time interval is i.i. Poisson distributedTime intervals between subsequent events is i.i. exponentially distributed
Sums of random variables
)λexp()λ()(
λ)(
)λexp(λ)(
1 yyn
yf
xxf
nY
X
)(Var)(Var;1
i
n
ii XnYXY
)λexp(!
λ)()(
)λexp(!
λ)(
ny
nyYP
xxXP
y
x
Exp
onen
tial p
rob
dens
Poi
sson
pro
b
Normal probability density
2
2 σ
μ
2
1exp
πσ2
1)(
xxf X
μ'μ
2
1exp
π2
1)( 1- xxxf
nX
μ)/σ(x-
σ
σ95%
Parameter estimation
Most frequently used method: Maximization of (log) Likelihood
likelihood: probability of finding observed data (given the model), considered as function of parameter values
If we repeat the collection of data many times (same conditions, same number of data)the resulting ML estimate
Profile likelihoodlarge sample
approximation
95% conf interval
Comparison of modelsAkaike Information Criterion for sample size n and K parameters
12)θ(log2
Kn
nKL
12σlog 2
Kn
nKn
in the case of a regression model
You can compare goodness of fit of different models to the same databut statistics will not help you to choose between the models
Confidence intervals
parameter
estimate
excluding
point 4
sd
excluding
point 4
estimate
including
point 4
sd
including
point 4
L, mm 6.46 1.08 3.37 0.096
rB,d-1 0.099 0.022 0.277 0.023
time, d
leng
th, m
m
ttrLLLtrLLLtL
B
B
smallfor)()exp()()(
00
0
10 LBr
95% conf intervals
correlations amongparameter estimatescan have big effectson sim conf intervals
excludespoint 4
includespoint 4
L
: These gouramis are from the same nest, These gouramis are from the same nest, they have the same age and lived in the same tank they have the same age and lived in the same tankSocial interaction during feeding caused the huge size differenceSocial interaction during feeding caused the huge size differenceAge-based models for growth are bound to fail;Age-based models for growth are bound to fail; growth depends on food intake growth depends on food intake
No age, but size:No age, but size:
Trichopsis vittatus
Rules for feeding
time
time time
rese
rve
dens
ityre
serv
e de
nsity
leng
thle
ngth
time 1 ind
2 ind
determinexpectation
Social interaction Feeding
Dependent observations
Conclusion
Dependences can work out in complex ways
The two growth curves look like von Bertalanffy curves with very different parameters
But in reality both individuals had the same parameters!