Uncertain models and modelling uncertainty Marian Scott Dept of Statistics, University of Glasgow EMS workshop, Nottingham, April 2004.

Uncertain models and modelling uncertainty

Marian ScottDept of Statistics, University of Glasgow

EMS workshop, Nottingham, April 2004

Outline of presentation

Model building and testing- is the environment special? Statistical models vs physical/process based models What is sensitivity/uncertainty analysis? Quantifying and apportioning variation in model and

data. General comments- relevance and implementation.

All models are wrong but some are useful

(and some are more useful than others)

(All data are useful, but some are more varied than others.)

Questions we ask about models

Is the model valid? Are the assumptions

reasonable? Does the model make

sense based on best scientific knowledge

Is the model credible? Do the model predictions

match the observed data?

How uncertain are the results?

What is a good model?Simple, realistic, efficient, useful, reliable, valid etc

Statistical models

Always includes an term to describe random variation

Empirical Descriptive and predictive Model building goal: simplest model which is

adequate used for inference

Physical/process based models

Uses best scientific knowledge May not explicitly include , or any random

variation Descriptive and predictive Goal may not be simplest model Not used for inference

Models

Mathematical (deterministic/process based) models tend

to be complex to ignore important sources of uncertaintyStatistical models tend to be empirical To ignore much of the

biological/physical/chemical knowledge

Stages in modelling

Design and conceptualisation:– Visualisation of structure– Identification of processes (variable selection)– Choice of parameterisation

Fitting and assessment– parameter estimation (calibration)– Goodness of fit

Model evaluation tools

Graphical procedures % variation explained in response Statistical model comparisons (F-tests,

ANOVA, GLRT)

well designed for statistical models, but what of the physical, process-driven models?– Comparability to measurements

The story of randomness and uncertainty

Randomness as the source of variability

– A source of variation, different animals range over different territory, eat different sources of ….

The effect is that we cannot be certain

Uncertainty due to lack of knowledge

– conflicting evidence

– ignorance

– effects of scale

– lack of observations Uncertainty due to variability

– Natural randomness

– behavioural variability

Effect of uncertainties

Uncertainty in model quantities/parameters/

inputs Uncertainty about model

form Uncertainty about model

completeness

Lack of observations contribute to

– uncertainties in input data– parameter uncertainties

Conflicting evidence contributes to

– uncertainty about model form

– Uncertainty about validity of assumptions

Making it difficult to judge how good a model is!!

Modelling tools - SA/UA

Sensitivity analysis

  determining the amount and kind of change produced in the model predictions by a change in a model parameter

  Uncertainty analysis

 an assessment/quantification of the uncertainties associated with the parameters, the data and the model structure.

Modellers conduct SA to determine

(a)    if a model resembles the system or processes under study,

(b)   the factors that mostly contribute to the output variability,

(c)    the model parameters (or parts of the model itself) that are insignificant,

(d)   if there is some region in the space of input factors for which the model variation is maximum,

and (e)     if and which (group of) factors interact with each

other.

SA flow chart (Saltelli, Chan and Scott, 2000)

Design of the SA experiment

Simple factorial designs (one at a time) Factorial designs (including potential

interaction terms) Fractional factorial designs Important difference: design in the context of

computer code experiments – random variation due to variation in experimental units does not exist.

SA techniques

Screening techniques– O(ne) A(t) T(ime), factorial, fractional factorial

designs used to isolate a set of important factors

Local/differential analysis Sampling-based (Monte Carlo) methods Variance based methods

– variance decomposition of output to compute sensitivity indices

Screening

screening experiments can be used to identify the parameter subset that controls most of the output variability with low computational effort.

Screening methods

Vary one factor at a time (NOT particularly recommended)

Morris OAT design (global)– Estimate the main effect of a factor by computing a

number r of local measures at different points x1,…,xr in the input space and then average them.

– Order the input factors

Local SA

Local SA concentrates on the local impact of the factors on the model. Local SA is usually carried out by computing partial derivatives of the output functions with respect to the input variables.

The input parameters are varied in a small interval around a nominal value. The interval is usually the same for all of the variables and is not related to the degree of knowledge of the variables.

Global SA

Global SA apportions the output uncertainty to the uncertainty in the input factors, covering their entire range space.

A global method evaluates the effect of xj while all other xi,ij are varied as well.

How is a sampling (global) based SA implemented?

Step 1: define model, input factors and outputs

Step 2: assign p.d.f.’s to input parameters/factors and if necessary covariance structure. DIFFICULT

Step 3:simulate realisations from the parameter pdfs to generate a set of model runs giving the set of output values.

Choice of sampling method

S(imple) or Stratified R(andom) S(ampling)– Each input factor sampled independently many times from

marginal distbns to create the set of input values (or randomly sampled from joint distbn.)

Expensive (relatively) in computational effort if model has many input factors, may not give good coverage of the entire range space

L(atin) H(ypercube) S(sampling)– The range of each input factor is categorised into N equal

probability intervals, one observation of each input factor made in each interval.

SA -analysis

At the end of the computer experiment, data is of the form (yij, x1i,x2i,….,xni), where x1,..,xn are the realisations of the input factors.

Analysis includes regression analysis (on raw and ranked values), standard hypothesis tests of distribution (mean and variance) for sub-samples corresponding to given percentiles of x and Analysis of Variance.

Some ‘new’ methods of analysis

Measures of importance

VarXi(E(Y|Xj =xj))/Var(Y)

HIM(Xj) =yiyi’/N

Sobol sensitivity indices Fourier Amplitude Sensitivity test (FAST)

So far so good

but how useful are these techniques in some real life problems?

Are there other complicating factors?

Do statisticians have too simple/complex a view of the world?

Common features of environmental modelling and observations

Knowledge of the processes creating the observational record may be incomplete

The observational records may be incomplete (observed often irregularly in space and time)

involve extreme events involve quantification of risk

Issues and purpose of analysis

Global and local pollutant mapping from Chernobyl

Global carbon cycle – greenhouse gases, CO2 levels and global warming

Ocean modelling

Air pollution modelling (local and regional scale)

Chronologies for past environment studies

Decision making- Which areas should be restricted?

Prediction-What is the trend in temperature? Predict its level in 2050?

Decision making-is it safe to eat fish?

Regulatory- Have emission control agreements reduced air pollutants?

Understanding -when did things happen in the past

Questions we ask about observations

Do they result from observational or designed; laboratory or field experiments?

What scale are they collected over (time and space)? Are they representative? Are they qualitative or quantitative? How are they connected to processes, how well

understood are these connections? How varied are they?

Example 1: are atmospheric SO2 concentrations declining?

Measurements made at a monitoring station over a 20 year period: processes involve meteorology (local and long-range, source distribution, chemistry of sulphur)

Complex statistical model developed to describe the pattern, the model portions the variation to ‘trend’, seasonality, residual variation

Main objective

so2 monitored in GB02

observations

so2

0 50 100 150 200 250

02

46

81

0

Plot of so2 against time, monitored in GB02Lines = Model 3

months

so2

1980 1985 1990 1995

02

46

81

0

SO4 in air, monitored at Lough Navar (GB06)

observations

SO

4 in

air

0 50 100 150

0.0

0.5

1.0

1.5

2.0

2.5

Example 2

Discovery of radioactive particles on the foreshore of a nuclear facility since 1983

Is the rate of finds falling off? Are the particle characteristics changing with time? Processes: transport in the marine environment,

chemistry of the particles in the sea, interaction with source

What can we infer about the size of the source and its distribution?

Log activity and trend

Date

logact

ivity

20.0

17.5

15.0

12.5

10.0

7.5

5.0

Accuracy MeasuresMAPE 11.8851MAD 1.4229MSD 3.8787

VariableActualFits

Trend Analysis Plot for logactivityLinear Trend Model

Yt = 14.9899 - 0.00712072*t

Trend in number of finds

year

number of finds

2002200019981996199419921990198819861984

25

20

15

10

5

0

Accuracy MeasuresMAPE 108.951MAD 4.025MSD 28.222

VariableActualFits

Trend Analysis Plot for number of findsLinear Trend Model

Yt = 14.7476 - 0.401299*t

Cumulative number of finds

1612840

200

150

100

50

0

1612840

Scatterplot of cumulative finds pre 1998 and post 1997

Example 3: how well should models agree?

6 ocean models (process based-transport, sedimentary processes, numerical solution scheme, grid size) used to predict the dispersal of a pollutant

Results to be used to determine a remediation policy

The models differ in their detail and also in their spatial scale

Model agreement

Three different sites (local, regional and global relative to a source)

6 different models Level of agreement (high

values are poor).

site 1site 2site 3

654321

6

5

4

3

2

1

0

Modelle

vel o

f agr

eem

ent

Sensitivity measures for each model

Predictions of levels of cobalt-60

Different models, same input data

Predictions vary by considerable margins

Magnitude of variation a function of spatial distribution of sites

tiwtistcwtcsbiwbisbcwbcs

250

150

50

Simulation condition

CV

(%)

CV(%) for location 7

tiwtistcwtcsbiwbisbcwbcs

250

150

50

Simulation condition

CV

(%)

CV(%) for location 8

tiwtistcwtcsbiwbisbcwbcs

250

150

50

Simulation condition

CV

(%)

CV(%) for location 9

tiwtistcwtcsbiwbisbcwbcs

250

150

50

Simulation condition

CV

(%)

CV(%) for location 10

tiwtistcwtcsbiwbisbcwbcs

250

150

50

Simulation condition

CV

(%)

CV(%) for location 11

Environmental modelling

Modelling may involve– Understanding and handling variation– Dealing with unusual observations– Dealing with missing observations– Evaluating uncertainties

How well should the model reproduce the data?

anecdotal comments ‘agreement between model and measurement better than 1 (2 ) orders of magnitude is acceptable’.

But this needs to be moderated by the measurement variation and uncertainties

It also depends on the purpose (model fit for purpose)

How can SA/UA help?

SA/UA have a role to play in all modelling stages:– We learn about model behaviour and ‘robustness’ to

change;– We can generate an envelope of ‘outcomes’ and

see whether the observations fall within the envelope;

– We can ‘tune’ the model and identify reasons/causes for differences between model and observations

On the other hand - Uncertainty analysis

Parameter uncertainty– usually quantified in form of a distribution.

Model structural uncertainty– more than one model may be fit, expressed as a

prior on model structure.

Scenario uncertainty– uncertainty on future conditions.

Tools for handling uncertainty

Parameter uncertainty– Probability distributions and Sensitivity analysis

Structural uncertainty– Bayesian framework– one possibility to define a discrete set of models,

other possibility to use a Gaussian process

Conclusions

The world is rich and varied in its complexity Modelling is an uncertain activity

Model assessment is a difficult process SA/UA are an important tools in model assessment The setting of the problem in a unified Bayesian

framework allows all the sources of uncertainty to be quantified, so a fuller assessment to be performed.

Challenges

Some challenges: different terminologies in different subject areas. need more sophisticated tools to deal with multivariate

nature of problem. challenges in describing the distribution of input

parameters. challenges in dealing with the Bayesian formulation of

structural uncertainty for complex models. Computational challenges in simulations for large

and complex computer models with many factors.