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![Page 1: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/1.jpg)
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)
![Page 2: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/2.jpg)
‘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
![Page 3: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/3.jpg)
‘Physically-based’ models
• population change given by:• increase due to births• decrease due to deaths
)()()()( tdeathstbirthstQttQ [1]
![Page 4: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/4.jpg)
‘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]
![Page 5: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/5.jpg)
‘Physically-based’ models
• If model parameters, B,D, constant
and ignoring age/sex distribution and environmental factors (food, disease etc)
Then ...
![Page 6: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/6.jpg)
[3])()(
)()(
)()(
:so
)()()(
tQDB
t
tdeathstbirths
t
tQttQ
t
Q
tQttQtQ
![Page 7: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/7.jpg)
‘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)(
![Page 8: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/8.jpg)
‘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)(
![Page 9: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/9.jpg)
‘Physically-based’ models
tDBeQtQ )(
0)(
Q0
t
B > D
B < D
Q
![Page 10: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/10.jpg)
Other Distinctions
• Another distinction– Analytical / Numerical
![Page 11: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/11.jpg)
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
![Page 12: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/12.jpg)
Other Distinctions
• Numerical– solution to model statement found e.g. by
calculating various model components over discrete intervals
• e.g. for integration / differentiation
![Page 13: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/13.jpg)
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
![Page 14: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/14.jpg)
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
![Page 15: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/15.jpg)
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
![Page 16: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/16.jpg)
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
![Page 17: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/17.jpg)
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)
![Page 18: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/18.jpg)
‘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
![Page 19: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/19.jpg)
‘Physically-based’ models
• population change given by:• increase due to births• decrease due to deaths
)()()()( tdeathstbirthstQttQ [1]
![Page 20: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/20.jpg)
‘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]
![Page 21: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/21.jpg)
‘Physically-based’ models
• If model parameters, B,D, constant
and ignoring age/sex distribution and environmental factors (food, disease etc)
Then ...
![Page 22: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/22.jpg)
[3])()(
)()(
)()(
:so
)()()(
tQDB
t
tdeathstbirths
t
tQttQ
t
Q
tQttQtQ
![Page 23: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/23.jpg)
‘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)(
![Page 24: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/24.jpg)
‘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)(
![Page 25: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/25.jpg)
‘Physically-based’ models
tDBeQtQ )(
0)(
Q0
t
B > D
B < D
Q
![Page 26: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/26.jpg)
Other Distinctions
• Another distinction– Analytical / Numerical
![Page 27: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/27.jpg)
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
![Page 28: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/28.jpg)
Other Distinctions
• Numerical– solution to model statement found e.g. by
calculating various model components over discrete intervals
• e.g. for integration / differentiation
![Page 29: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/29.jpg)
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
![Page 30: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/30.jpg)
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
![Page 31: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/31.jpg)
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
![Page 32: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/32.jpg)
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
![Page 33: Statistical / empirical models Model ‘validation’ –should obtain biomass/NDVI measurements over wide range of conditions –R 2 quoted relates only to conditions.](https://reader035.fdocuments.us/reader035/viewer/2022062806/56649f095503460f94c1dfd2/html5/thumbnails/33.jpg)
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)