Statistical model building

Marian Scott Dept of Statistics, University of Glasgow

Glasgow, Sept 2007

Outline of presentation

Statistical models- what are the principles – describing variation– empiricism

Fitting models- calibration Testing models- validation or verification Quantifying and apportioning variation in model and

data. Stochastic and deterministic models. Model choice

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.)

a quote from a famous statistician, George Box

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

Why model?

Purposes of modelling:– Describe/summarise– Predict - what if….– Test hypotheses– Manage

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

Value judgements

Different criteria of unequal importance key comparison often comparison to

observational data

but such comparisons must include the model

uncertainties and the uncertainties on the observational data (touched on later).

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?

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

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?

ask what the objective of modelling is look at the data, and data quality try and understand the measurement

processes think about how the scientific knowledge,

conceptual model relates to what we have measured

think about uncertainty

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

19

97

Flu

xe

s

Index

Sulphur and Nitrogen EMEP: 15 stations in United Kingdom and 4 in Republic of Ireland from 1978 to 1998. (aggregated to monthly means).

The Monitoring Networks

UK National Air Quality Information Archive: 8 stations in the United Kingdom corresponding to some of EMEP stations from 1983 to 2007. (hourly data, subsequently aggregated to monthly day and night means).

Ozone

(O3)

(SO2, SO4, NO2, NO3, NH4, HNO3+NO3, NH3+NH4)

GB02 EskdalemuirGB03 GoonhillyGB04 Stoke ferryGB05 LudlowGB06 Lough NavarGB07 Barcombe MillsGB13 Yarner WoodGB14 High MufflesGB15 Starth Vaich DamGB16 Glen DyeGB36 HarwellGB37 LadybowerGB38 Lullington HeathGB43 NarberthGB45 Wicken FenIE01 Valentia Obs.IE02 Turlough HillIE03 The BurrenIE04 Ridge of Capard

THE DATA

What evidence is there of a trend in the atmospheric concentrations?

M o n t h s

SO4

in pr

ecipi

t. co

rr. (G

B03)

20 40 60 80 100

0

2

4

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M o n t h s

SO2

(GB0

2)

0 50 100 150 200 250

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•Outliers •Missing values •Discontinuities

so2 monitored in GB02

observations

so2

0 50 100 150 200 250

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46

81

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

months

so2

1980 1985 1990 1995

02

46

81

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Log SRP

Years

Lo

g S

RP

, m

ug

/l

1970 1980 1990 2000

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24

Log TP

Years

Lo

g T

P,

mu

g/l

1970 1980 1990 2000

3.5

4.0

4.5

5.0

Log Secchi

Years

Lo

g S

ecc

hi,

me

tre

s

1970 1980 1990 2000

-0.5

0.0

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1.0

Log Daphnia

Years

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g D

ap

hn

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ind

ivid

ua

ls/l

1970 1980 1990 2000

-4-2

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4

Log Chlorophyll

Years

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g C

hlo

rop

hyl

l, m

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/l

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01

23

45

Water Temperature

Years

Wa

ter

Tem

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oC

1970 1980 1990 20000

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Loch Leven

Log SRP

Years

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, m

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Log TP

YearsL

og

TP

, m

ug

/l1970 1980 1990 2000

3.5

4.0

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Log Secchi

Years

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g S

ecc

hi,

me

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s

1970 1980 1990 2000

-0.5

0.0

0.5

1.0

Log Daphnia

Years

Lo

g D

ap

hn

ia,

ind

ivid

ua

ls/l

1970 1980 1990 2000

-4-2

02

4

Log Chlorophyll

Years

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g C

hlo

rop

hyl

l, m

ug

/l

1970 1980 1990 2000

01

23

45

Water Temperature

Years

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ter

Tem

pe

ratu

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oC

1970 1980 1990 20000

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Loch Leven

Loch LevenLog SRP

Month

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g S

RP

, m

ug

/l

2 4 6 8 10 12

-20

24

Log TP

MonthL

og

TP

, m

ug

/l

2 4 6 8 10 12

3.5

4.0

4.5

5.0

Log Chlorophyll

Month

Lo

g C

hlo

rop

hyl

l, m

ug

/l

2 4 6 8 10 12

01

23

45

Log Daphnia

Month

Lo

g D

ap

hn

ia,

ind

ivid

ua

ls/l

2 4 6 8 10 12

-4-2

02

4

Log Secchi

Month

Lo

g S

ecc

hi,

me

tre

s

2 4 6 8 10 12

-0.5

0.0

0.5

1.0

Water Temperature

Month

Wa

ter

Tem

pe

ratu

re,

oC

2 4 6 8 10 120

51

01

52

0

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

Evaluation of precision

Analysis of the instrumentation method to make a single measurement, and the propagation of any errors

Repeat measurements (true replicates) – using homogeneous material, repeatedly subsampling, etc….

Precision is linked to Variance (standard deviation)

The nature of measurement

All measurement is subject to uncertainty Analytical uncertainty reflects that every time a

measurement is made (under identical conditions), the result is different.

Sampling uncertainty represents the ‘natural’ variation in the organism within the environment.

The error and uncertainty in a measurement

The error is a single value, which represents the difference between the measured value and the true value

The uncertainty is a range of values, and describes the errors which might have been observed were the measurement repeated under IDENTICAL conditions

Error (and uncertainty) includes a combination of variance and bias

Effect of uncertainties

Lack of observations contribute to– uncertainties in input data– uncertainty in model parameter values

Conflicting evidence contributes to– uncertainty about model form– uncertainty about validity of

assumptions

Step 3- build the statistical model

Outcomes or Responsessometimes referred to as ‘dependent variables’.

Causes or Explanationsthese are the conditions or environment within which the outcomes or responses have been observed and are sometimes referred to as ‘independent variables’, but more commonly known as covariates.

Statistical models

In experiments many of the covariates have been determined by the experimenter but some may be aspects that the experimenter has no control over but that are relevant to the outcomes or responses.

In observational studies, these are usually not under the control of the experimenter but are recorded as possible explanations of the outcomes or responses.

we may not know which covariates are important. recognise that we may build several models before

making the final choice

Specifying a statistical models

Models specify the way in which outcomes and causes link together, eg.

Metabolite = Temperature The = sign does not indicate equality in a mathematical

sense and there should be an additional item on the right hand side giving a formula:-

Metabolite = Temperature + Error

Specifying a statistical model

Metabolite = Temperature + Error In mathematical terms, there will be some unknown

parameters to be estimated, and some assumptions will be made about the error distribution

Metabolite = + temperature + ~ N(0, σ2)- appropriate perhaps? σ,, are model parameters and are unknown

Model calibration

Statisticians tend to talk about model fitting, calibration means something else to them.

Methods- least squares or maximum likelihood

least squares:- find the parameter estimates that minimise the sum of squares (SS)

SS=(observed y- model fitted y)2

maximum likelihood- find the parameter estimates that maximise the likelihood of the data

Calibration-using the data

A good idea, if possible to have a training and a test set of data-split the data (e.g. 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

How good is my statistical model?

What criteria do we use to judge the value of our model?

– may depend on what the model was built to do

Obvious ones– Closeness to the observed data– Goodness of predictions at previously unobserved

covariate values– % variation in response explained (R2)

Model validation

what is validation? 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

assessment of goodness of fit?– residual sums of squares, mean square error for prediction

Model validation

splitting the data set, is it possible? cross-validation

– leave–one-out, leave-k-out

• split at random, a ‘small’ % kept aside for testing

• other methods: bootstrap and jack-knife

an aside- how well should models agree?

6 physical-deterministic 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 for an illegal dumping of “radioactive waste” The what if scenario investigation

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

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

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Simulation condition

CV

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CV(%) for location 7

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Simulation condition

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CV(%) for location 8

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Simulation condition

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CV(%) for location 9

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Simulation condition

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Simulation condition

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CV(%) for location 11

model ensembles

becoming increasingly common in climate, meteorology, to make ‘many’ model runs, different models, different starting conditions and then to ‘average’ the results.

why would we do this?

the statistical approach to model building and selection

in a regression situation, we may have many potential explanatory variables, how do we choose which to include in the final model?

– answer may depend on purpose, on how many explanatory variables there may be

identify variables that can be omitted on statistical grounds (no evidence of effect) (see regression sessions with Adrian for testing and CI approaches)- is an effect statistically significant?

the statistical approach to model building and selection

automatic selection procedures can be useful but also potentially dangerous (e.g. stepwise regression, best subset-regression)

they often identify the ‘best’ under a defined criterion in a family of models (like smallest residual sum of squares).

but this ‘best’ model could in an absolute sense be poor.

the statistical approach to model building and selection

other statistical criteria exist for model choice-AIC, DIC, BIC- based on likelihood approaches, can be used to compare non-nested models (ie parameter set of one model is not contained within parameter set of the ‘larger’ model)

need to be careful of ‘dredging’ for significance remember statistical significance is not always equal to

practical importance

Information criterion

In the general case, the AIC is– AIC=2k-2ln(L)

where k is the number of parameters, and L is the likelihood function.

if we assume that the model errors are normally and independently distributed. Let n be the number of observations and RSS be the residual sum of squares. Then AIC becomes– AIC=2k+nln(RSS/n)

AIC

Increasing the number of free parameters to be estimated improves the goodness of fit. Hence AIC not only rewards goodness of fit, but also includes a penalty that is an increasing function of the number of estimated parameters. This penalty discourages overfitting. The preferred model is the one with the lowest AIC value. The AIC methodology attempts to find the model that best explains the data with a minimum of free parameters.

blending the statistical modelling approach to deterministic models

relatively new area (at least for statisticians) phrased in a Bayesian framework (see later

session on Bayesian methods) makes use of data (very important) and data

modelling still at the research stage (probably most used

on climatology)

in summary

model building is iterative should combine statistical skills and scientific

knowledge think about your objectives, think about the

data model selection- many different approaches uncertainty is a factor at all stages and should

be considered.