Statistical / empirical models

• Model ‘validation’– should obtain biomass/NDVI measurements over

wide range of conditions– R2 quoted relates only to conditions under which

model was developed • i.e. no information on NDVI values outside of

range measured (0.11 to 0.18 in e.g. shown)

‘Physically-based’ models

• e.g. Simple population growth• Require:

– model of population Q over time t• Theory:

– in a ‘closed’ community, population change given by:

• increase due to births• decrease due to deaths

– over some given time period t

‘Physically-based’ models

• population change given by:• increase due to births• decrease due to deaths

)()()()( tdeathstbirthstQttQ [1]

‘Physically-based’ models

• Assume that for period t– rate of births per head of population is B– rate of deaths per head of population is D

• We can write:

– Implicit assumption that B,D are constant over time t

ttQDtdeaths

ttQBtbirths

*)(*)(

*)(*)(

[2]

‘Physically-based’ models

• If model parameters, B,D, constant

and ignoring age/sex distribution and environmental factors (food, disease etc)

Then ...

[3])()(

)()(

)()(

:so

)()()(

tQDB

t

tdeathstbirths

t

tQttQ

t

Q

tQttQtQ

‘Physically-based’ models

• As time period considered decreases, can write eqn [3] as a differential equation:

• i.e. rate of change of population with time equal to birth-rate minus death-rate multiplied by current population

• Solution is ...

QDBdt

dQ)(

‘Physically-based’ models

• Consider the following:

– what does Q0 mean?

– does the model work if the population is small?– What happens if B>D (and vice-versa)?– How might you ‘calibrate’ the model parameters?

[hint - think logarithmically]

tDBeQtQ )(

0)(

‘Physically-based’ models

tDBeQtQ )(

0)(

Q0

t

B > D

B < D

Q

Other Distinctions

• Another distinction– Analytical / Numerical

Other Distinctions

• Analytical– resolution of statement of model as combination of

mathematical variables and ‘analytical functions’– i.e. “something we can actually write down”

– e.g. biomass = a + b*NDVI– e.g.

aQdt

dQ ateQQ 0

Other Distinctions

• Numerical– solution to model statement found e.g. by

calculating various model components over discrete intervals

• e.g. for integration / differentiation

Which type of model to use?

• Statistical– advantages

• simple to formulate• generally quick to calculate• require little / no knowledge of underlying (e.g.

physical) principles• (often) easy to invert

– as have simple analytical formulation

Which type of model to use?

• Statistical– disadvantages

• may only be appropriate to limited range of parameter

• may only be applicable under limited observation conditions

• validity in extrapolation difficult to justify• does not improve general understanding of

process

Which type of model to use?

• Physical/Theoretical/Mechanistic– advantages

• if based on fundamental principles, applicable to wide range of conditions

• use of numerical solutions (& fast computers) allow great flexibility in modelling complexity

• may help to understand process– e.g. examine role of different assumptions

Which type of model to use?

• Physical/Theoretical/Mechanistic– disadvantages

• more complex models require more time to calculate

– get a bigger computer!

• Supposition of knowledge of all important processes and variables as well as mathematical formulation for process

• often difficult to obtain analytical solution• often tricky to invert

Summary

• Empirical (regression) vs theoretical (understanding)• uncertainties• validation

– Computerised Environmetal Modelling: A Practical Introduction Using Excel, Jack Hardisty, D. M. Taylor, S. E. Metcalfe, 1993 (Wiley)

– Computer Simulation in Physical Geography, M. J. Kirkby, P. S. Naden, T. P. Burt, D. P. Butcher, 1993 (Wiley)

‘Physically-based’ models

• e.g. Simple population growth• Require:

– model of population Q over time t• Theory:

– in a ‘closed’ community, population change given by:

• increase due to births• decrease due to deaths

– over some given time period t

‘Physically-based’ models

• population change given by:• increase due to births• decrease due to deaths

)()()()( tdeathstbirthstQttQ [1]

‘Physically-based’ models

• Assume that for period t– rate of births per head of population is B– rate of deaths per head of population is D

• We can write:

– Implicit assumption that B,D are constant over time t

ttQDtdeaths

ttQBtbirths

*)(*)(

*)(*)(

[2]

‘Physically-based’ models

• If model parameters, B,D, constant

and ignoring age/sex distribution and environmental factors (food, disease etc)

Then ...

[3])()(

)()(

)()(

:so

)()()(

tQDB

t

tdeathstbirths

t

tQttQ

t

Q

tQttQtQ

‘Physically-based’ models

• As time period considered decreases, can write eqn [3] as a differential equation:

• i.e. rate of change of population with time equal to birth-rate minus death-rate multiplied by current population

• Solution is ...

QDBdt

dQ)(

‘Physically-based’ models

• Consider the following:

– what does Q0 mean?

– does the model work if the population is small?– What happens if B>D (and vice-versa)?– How might you ‘calibrate’ the model parameters?

[hint - think logarithmically]

tDBeQtQ )(

0)(

‘Physically-based’ models

tDBeQtQ )(

0)(

Q0

t

B > D

B < D

Q

Other Distinctions

• Another distinction– Analytical / Numerical

Other Distinctions

• Analytical– resolution of statement of model as combination of

mathematical variables and ‘analytical functions’– i.e. “something we can actually write down”

– e.g. biomass = a + b*NDVI– e.g.

aQdt

dQ ateQQ 0

Other Distinctions

• Numerical– solution to model statement found e.g. by

calculating various model components over discrete intervals

• e.g. for integration / differentiation

Which type of model to use?

• Statistical– advantages

• simple to formulate• generally quick to calculate• require little / no knowledge of underlying (e.g.

physical) principles• (often) easy to invert

– as have simple analytical formulation

Which type of model to use?

• Statistical– disadvantages

• may only be appropriate to limited range of parameter

• may only be applicable under limited observation conditions

• validity in extrapolation difficult to justify• does not improve general understanding of

process

Which type of model to use?

• Physical/Theoretical/Mechanistic– advantages

• if based on fundamental principles, applicable to wide range of conditions

• use of numerical solutions (& fast computers) allow great flexibility in modelling complexity

• may help to understand process– e.g. examine role of different assumptions

Which type of model to use?

• Physical/Theoretical/Mechanistic– disadvantages

• more complex models require more time to calculate

– get a bigger computer!

• Supposition of knowledge of all important processes and variables as well as mathematical formulation for process

• often difficult to obtain analytical solution• often tricky to invert

Summary

• Empirical (regression) vs theoretical (understanding)• uncertainties• validation

– Computerised Environmetal Modelling: A Practical Introduction Using Excel, Jack Hardisty, D. M. Taylor, S. E. Metcalfe, 1993 (Wiley)

– Computer Simulation in Physical Geography, M. J. Kirkby, P. S. Naden, T. P. Burt, D. P. Butcher, 1993 (Wiley)