Download DEB theory

Art of modelling
DEB course 2013
Bas Kooijman
Dept theoretical biology
Vrije Universiteit Amsterdam
[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 systems
Defined by simultaneous behaviour of
input, state variable, output
Supply systems: input + state variables  output
Demand systems input  state variables + output
Real systems: mixtures between supply & demand systems
Constraints: mass, energy balance equations
State variables: span a state space
behaviour: usually set of ode’s with parameters
Trajectory: map of behaviour state vars in state space
Parameters:
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
Alternative form:
k = k0 exp{1 – T0 /T}, with k0 = exp{a – 1}
ln rate
Arrhenius model: ln k = a – T0 /T
k: some rate
T: absolute temperature
a: parameter
T0: Arrhenius temperature
Arrhenius plot
Difference with allometric model:
no reference value required to solve dimension problem
T-1
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 W0b
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 = kc01/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
O2 consumption, 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
Length, mm
Kleber’s law
O2 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 weight
Problem in relating respiration to other activities
Egg development time
D  exp( 3.3956  0.2193 ln( T )  0.3414(ln( T )) 2 )
D  exp( a  b ln( T )  c(ln( T )) 2 )
dim( a) 
ln t
ln t
dim( b) 
ln K
ln t
dim( c) 
(ln K ) 2
D egg developmen t time
T temperatur e in Kelvin
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
Space-time scales
space
Each process has its characteristic domain of space-time scales
system earth
ecosystem
population
individual
cell
molecule
When changing the space-time scale,
new processes will become important
other will become less important
Individuals are special because of
straightforward energy/mass balances
time
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 compounds
n-th order model
d
X  kX n
dt
(1 n ) 1
1 n
X (t )  X 0  (1  n)kt


n 0
X (t )  X 0  kt ; t  X 0 / k
n 1
X (t )  X 0 exp{ kt}
1 n
1

a
t (aX 0 )  X 01n k 1
1 n
Monod model
d
X
X  k
dt
KX
0  X (t )  X 0  K ln{ X (t ) / X 0 }  kt
K  X 0
X (t )  X 0  kt ; t  X 0 exp{kt / K}
K  X 0
X (t )  X 0 exp{kt / K}
1
1
t (aX 0 )  X 0 k (a  1)  Kk ln a
X : conc. of compound,
t : time
n : order
X0 : X at time 0
k : degradation rate
K : saturation constant
Biodegradation of compounds
Monod model
scaled conc.
scaled conc.
n-th order model
scaled time
scaled time
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 errors
Alternatives:
• deterministic systems with stochastic inputs
• differences in parameter values between individuals
Problem:
parameter estimation methods become very complex
Statistics
Deals 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
y ( x)  w0  w1 x  w2 x 2
w2  0
y( x)  w0  w1 x
Venn diagram
w1  0
y( x)  w0
y ( x)  w0  w2 x 2
Testing of hypothesis
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
No certainty in statistics
decision
null hypothesis
true
false
accept
1-

reject

1-
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
If X and Y are independent
I
I
f
Central limit theorems
The sum of n independent identically (i.i.) distributed random variables
becomes normally distributed for increasing n.
Z  X Y

f Z ( z )   f X ( z  y) fY ( y) dy; P( Z  z )   P( X  z  y) P(Y  y)
y
y
The sum of n independent point processes tends to behave as a
Poisson process for increasing n.
Number of events in a time interval is i.i. Poisson distributed
Time intervals between subsequent events is i.i. exponentially distributed
Poisson prob
Exponential prob dens
Sums of random variables
n
Y   Xi;
i 1
Var (Y )  nVar ( X i )
f X ( x)  λ exp( λx)
λ
fY ( y ) 
(λy) n1 exp( λy )
 ( n)
λx
P( X  x)  exp( λ)
x!
(nλ) y
P(Y  y ) 
exp( nλ)
y!
Normal probability density
σ
σ
 95%
(x-μ)/σ
 1  x  μ 2 
f X ( x) 
exp   
 
2

2πσ
 2 σ  
1
f X ( x) 
 1

exp    x  μ '  -1 x  μ 

2π n   2
1
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 likelihood
large sample
approximation
95% conf interval
Comparison of models
Akaike Information Criterion
for sample size n and K parameters
n
 2 log L(θ)  2 K
n  K 1
in the case of a regression model
n
2
n log σ  2 K
n  K 1
You can compare goodness of fit of different models to the same data
but statistics will not help you to choose between the models
Confidence intervals
length, mm
L(t )  L  ( L  L0 ) exp( rB t )
 L0  ( L  L0 )rB t for small t
L0  1
excludes
point 4
95% conf intervals
rB
includes
point 4
time, d
L
correlations among
parameter estimates
can have big effects
on sim conf intervals
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
parameter
:No age, but size:These gouramis are from the same nest,
they have the same age and lived in the same tank
Social interaction during feeding caused the huge size difference
Age-based models for growth are bound to fail;
growth depends on food intake
Trichopsis vittatus
Rules for feeding
determin
expectation
length
reserve density
Social interaction  Feeding
time
time
1 ind
2 ind
length
reserve density
time
time
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!