Statistical model building Marian Scott and Ron Smith Dept of Statistics, University of Glasgow, CEH Glasgow, Aug 2008

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. intro to uncertainty and sensitivity analysis

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 feedbacks Data Model inputs & parameters Policy model results

Why model? Purposes of modelling: What is a good model? 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.

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

The data- variation soil or sediment samples taken side-by-side, from different parts of the same plant, or from different animals in the same environment, exhibit different activity densities of a given radionuclide. The distribution of values observed will provide an estimate of the variability inherent in the population of samples that, theoretically, could be taken.

Variation Activity (log10) of particles (Bq Cs-137) with Normal or Gaussian density superimposed

measured atmsopheric fluxes for 1997 measured fluxes for 1997 are still noisy. Is there a statistical signal and at what timescale?

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)

Accuracy and precision Accurate Inaccurate Precise Imprecise

Evaluation of accuracy In a laboratory inter-comparison, known-concentration material is used to define the ‘true’ concentration The figure shows a measure of accuracy for individual laboratories Accuracy is linked to Bias

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) Precision, error, uncertainty, all the terminology again how to estimate

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 Responses these are the results of the practical work and are sometimes referred to as ‘dependent variables’. Causes or Explanations these 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.

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 models 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)

statistical model interpretation Metabolite = Temperature + Error The outcome Metabolite is explained by Temperature and other things that we have not recorded which we call Error. The task that we then have in terms of data analysis is simply to find out if the effect that Temperature has is ‘large’ in comparison to that which Error has so that we can say whether or not the Metabolite that we observe is explained by Temperature.

SS=(observed y- model fitted y)2 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 (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

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 other methods: bootstrap and jackknife

Example 4: Models- 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 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

Statistical models and process models Loch Leven, modelling nutrients process model based on differential equations statistical model based on empirically determined relationships

Loch Leven

Loch Leven

Loch Leven

Uncertainty and sensitivity analysis

Uncertainty (in variables, models, parameters, data) what are uncertainty and sensitivity analyses? an example.

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

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 Local/differential analysis 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,ij 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 subsamples corresponding to given percentiles of x, and Analysis of Variance.

Some ‘newer’ 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)

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 model averaging

An uncertainty example (1) Wet deposition is rainfall  ion concentration Rainfall is measured at approximately 4000 locations, map produced by UK Met Office. Rain ion concentrations are measured weekly (now fortnightly or monthly) at around 32 locations.

An uncertainty example (2) BUT almost all measurements are at low altitudes much of Britain is upland AND measurement campaigns show rain increases with altitude rain ion concentrations increase with altitude Seeder rain, falling through feeder rain on hills, scavenges cloud droplets with high pollutant concentrations.

An uncertainty example (3) Solutions: More measurements X at high altitude are not routine and are complicated (b) Derive relationship with altitude X rain shadow and wind drift (over about 10km down wind) confound any direct altitude relationships (c) Derive relationship from rainfall map  model rainfall in 2 separate components

An uncertainty example (4)

An uncertainty example (5) Wet deposition is modelled by r actual rainfall s rainfall on ‘low’ ground (r = s on ‘low’ ground, and (r-s) is excess rainfall caused by the hill) c rain ion concentration as measured on ‘low’ ground f enhancement factor (ratio of rain ion concentration in excess rainfall to rain ion concentration in ‘low’ground rainfall) deposition = s.c + (r-s).c.f

An uncertainty example (6) Rainfall Concentration Deposition

An uncertainty example (7) a) modelled rainfall to 5km squares provided by UKMO - unknown uncertainty scale issue - rainfall a point measurement measurement issue - rain gauges difficult to use at high altitude optimistic 30%  pessimistic 50% how is the uncertainty represented? (not e.g. 30% everywhere)

An uncertainty example (8) b) some sort of smoothed surface (change in prevalence of westerly winds means it alters between years) c) kriged interpolation of annual rainfall weighted mean concentrations (variogram not well specified) assume 90% of observations within ±10% of correct value d) campaign measurements indicate values between 1.5 and 3.5

An uncertainty example (9) Output measures in the sensitivity analysis are the average flux (kg S ha-1 y-1) for (a) GB, and (b) 3 sample areas

An uncertainty example (10) Morris indices are one way of determining which effects are more important than others, so reducing further work. but different parameters are important in different areas

An uncertainty example (11) 100 simulations Latin Hypercube Sampling of 3 uncertainty factors: enhancement ratio % error in rainfall map % error in concentration

An uncertainty example (12) Note skewed distributions for GB and for the 3 selected areas

An uncertainty example (13) Mean of 100 simulations Standard deviation Original

An uncertainty example (14) CV from 100 simulations Possible bias from 100 simulations

An uncertainty example (15) 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 for critical loads

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