Course round-upsubtitle- Statistical model building

Marian ScottUniversity of Glasgow

Glasgow, Aug 2013

Step 1

why do you want to build a model- what is your objective?

what data are available and how were they collected?

is there a natural response or outcome and other explanatory variables or covariates?

Modelling objectives

explore relationships make predictions improve understanding test hypotheses

Conceptual system

Data

Model

Policy

inputs & parameters

model results

feedbacks

Value judgements

Different criteria of unequal importance key comparison often comparison to

observational data (RSS, AIC......)

but such comparisons must include the model

uncertainties and the uncertainties on the observational data.

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?

Stages in modelling

Design and conceptualisation:– Visualisation of structure– Identification of processes– Choice of parameterisation

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

a visual model- atmospheric flux of pollutants

•Atmospheric pollutants dispersed over Europe

•In the 1970’ considerable environmental damage caused by acid rain

•International action

•Development of EMEP programme, models and measurements

The mathematical flux model

L: Monin-Obukhov length

u*: Friction velocity of wind

cp: constant (=1.01)

: constant (=1246 gm-3)

T: air temperature (in Kelvin)

k: constant (=0.41)

g: gravitational force (=9.81m/s)

H: the rate of heat transfer per unit area

gasht: Current height that measurements are taken at.

d: zero plane displacement

what would a statistician do if confronted with this problem?

Look at the data understand the measurement processes think about how the scientific knowledge,

conceptual model relates to what we have measured

Step 2- understand your data

study your data learn its properties tools- graphical

measured atmospheric fluxes for 1997

•measured fluxes for 1997 are still noisy.

•Is there a statistical signal and at what timescale?0

5

10

15

100 200 300

1997

Flu

xes

Index

Key properties of any measurement

Accuracy refers to the deviation of the measurement from the ‘true’ value

Precision refers to the variation in a series of replicate measurements (obtained under identical conditions)

Accurate

Imprecise

Inaccurate

Precise

Accuracy and precision

Data properties

Nature and distribution of the data- continuous, counts.... Normal, exponential, poisson, maybe need a transformation

Missing data- outliers- limits of detection

Use pictures to explore

Step 3- build the statistical model

Outcomes or Responses

Causes or Explanationsthese are the conditions or environment within which the outcomes or responses have been observed -the covariates.

This has very much been the focus of much of the week- whether a linear model, a smooth flexible model, a time series model, a bayesian model.....

Are you a bayesian?

What does that mean?

It means, you have prior information (belief) that you want to include in your statistical model

You need to find a way of capturing this in the prior distribution

Model output then a posterior distribution on the quantity of interest- automatically incorporates uncertainty

Calibration-using the data

A good idea, if possible to have a training and a test set of data-split the data (90%/10%)

Fit the model using the training set, evaluate the model using the test set.

why? because if we assess how well the model

performs on the data that were used to fit it, then we are being over optimistic

other methods: bootstrap and jackknife

Which variables to include

Use your science knowledge Use pictures to look for patterns Maybe use some of the more algorithmic

ways to select the set (stepwise, BSR...)

How to compare models? Nested models (ANOVA, likelihood ratio test)

Uncertainty and sensitivity analysis

Uncertainty (in variables, models, parameters,

data) what are uncertainty and sensitivity analyses?

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.

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.

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.

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 subsamples corresponding to given percentiles of x, and Analysis of Variance.

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.

An uncertainty example (Ron Smith)

OriginalMean of 100 simulations

Standard deviation

An uncertainty example

CV from 100 simulations

Possible bias from 100 simulations

An uncertainty example

• model sensitivity analysis identifies weak areas• lack of knowledge of accuracy of inputs a

significant problem• there may be biases in the model output which,

although probably small in this case, may be important

• Model emulators have become popular

Take home message

• Only able to give you a flavour of what might be possible

• Good environmental science and good statistical science is key for all problems

• Think critically- test and re-test your hypotheses and assumptions

Take home message

• Resources• Many good books (have seen some of these

over the sessions- not one size fits all

• JISC mail list- Envstat (worth joining)• Royal Statistical Society has an Environmental

Statistics section, sometimes holds tutorial meetings on topics.