Uncertainty and sensitivity analysis- model and measurements
Marian Scott and Ron Smith and Clive AndersonUniversity of Glasgow/CEH/University of Sheffield
Glasgow, Sept 2006
Outline of presentation
Errors and uncertainties on measurements Sensitivity and uncertainty analysis of models Quantifying and apportioning variation in model
and data. A Bayesian approach Some general comments
Uncertainties on measurement
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
Key properties of any measurement
Accuracy refers to the deviation of the measurement from the ‘true’ value (bias)
Precision refers to the variation in a series of replicate measurements (obtained under identical conditions) (variance)
Accurate
Imprecise
Inaccurate
Precise
Accuracy and precision
Evaluation of accuracy
In an inter-laboratory study, known-age material is used to define the ‘true’ age
The figure shows a measure of accuracy for individual laboratories
Accuracy is linked to Bias
1009080706050403020100
500400300200100
0-100-200-300-400-500-600
laboratory identifier
Off
set (
yea
rs B
P)
Evaluation of precision
Analysis of the instrumentation method to make a single measurement, and the propagation of any errors (theory)
Repeat measurements (true replicates) – using homogeneous material, repeatedly subsampling, etc…. (experimental)
Precision is linked to Variance (standard deviation)
The uncertainty range
for a measurement of 4509 years with quoted error (1 sigma) 20 years, the measurement uncertainty at 2 sigma, would be 4509 40 years or 4469 to 4549 years. We would say that the true age is highly likely to lie within the uncertainty range (95% confidence)
The uncertainty range on the mean
From the series of 27 replicate measurements made in a single laboratory over a period of several months. The average age of the series is 4497 years. The standard deviation of the series is 30.2 years. The error on the mean is (30.2/27) or 6 years. So the uncertainty (at 2 sigma) on the true age is 4497 12 years or 4485 to 4509 years.
Is the quoted error realistic?
Commonly judged by making a series of repeat measurements (replicates) and calculating the standard deviation of the series. For the 27 measurements, the st.dev. is 30.2 years but the quoted errors on individual measurements range from 13 to 33 years. So 30 years might be a more realistic individual error.
Are two measurements significantly different?
Two examples of measurements of a sample. The measurements were made in two different laboratories and so are assumed statistically independent.
Case A
a) 2759 years 39 and 2811 years20 The difference is -52 years and the error is 44
years, ((392+202)) therefore the uncertainty range is –52 88 years and includes 0.
There is no evidence that these two samples do not have the same true age. These two measurements could therefore be legitimately combined in a weighted average .
Case B
a) 2885 years 37 and 2781years 30. The difference is 104 years and error is 48 years,
therefore the uncertainty range is 10496 years or 8 to 200 years and does not include 0.
We could conclude that within the individual uncertainties on the measurements, these two samples do not have the same true age. Therefore these two measurements could not be legitimately combined.
Can we combine a series of measurements?
The results for 6 samples taken from Skara Brae on the Orkney Islands. The samples consisted of single entities (i.e. individual organisms) that represented a relatively short growth interval. The terrestrial samples were either carbonised plant macrofossils (cereal grains or hazelnut shells) or terrestrial mammal bones (cattle or red deer).
The test of homogeneity,
series of measurements xi, with error si
Null hypothesis says measurements are the same (within error)
Calculated the weighted mean , xp
the test statistic T = (xi –xp)2/si
2
This should have a 2(n-1) distribution
Case A
455540, 460540, 452540, 4530 35, 427040, 4735 40
Using all 6 measurements, the weighted average is 4536.34 years, and T is 72.2789.
T compared with a 2 (5), for which the critical value is 11.07, thus we would reject the hypothesis that the samples all had the same true age, so they cannot be combined.
Case B
455540, 460540, 452540, 4530 35 the weighted average is 4552 years, and T is
2.612. T compared with a 2 (3), for which the critical value is 7.8,
thus we would not reject the hypothesis that the samples all had the same true age, and so the weighted average (with its error) could be calculated.
Model uncertainty
Uncertainties
uncertainties in input data– uncertainty in model parameter values
Conflicting evidence contributes to– uncertainty about model form– uncertainty about validity of
assumptions
Conceptual system
Data
Model
Policy
inputs & parameters
model results
feedbacks
Goals
1. Transparent approach to facilitate awareness/identification/inclusion of uncertainties within analysis
2. Provide useful/robust/relevant uncertainty assessments
3. Provide a means to assess consequences
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 subsamples 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)
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
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 uplandAND 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: (a) More measurements
X at high altitude are not routine and are complicated
(b) Derive relationship with altitudeX 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 rainfalls 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’ groundf 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)
r modelled rainfall to 5km squares provided by UKMO - unknown uncertainty
scale issue - rainfall a point measurementmeasurement issue - rain gauges difficult
touse at high altitude
optimistic 30% pessimistic 50%
how is the uncertainty represented?(not e.g. 30% everywhere)
An uncertainty example (8)
s 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
f campaign measurements indicate valuesbetween 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)
OriginalMean of 100 simulations
Standard deviation
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 so far
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.
Bayesian Approach
to
Model Uncertainty, Calibration,
Sensitivity Analysis ….
Bayes Essentials
Eg experimental determination of a constant
Prior ideas
about
Data
Posterior ideas
about
Bayes’ Rule
likelihood – from model for data generation
Bayes Essentials
General form:
Observations
Unknown
a (statistical) model describing
data generation,
specified in a likelihood
For inferences to be coherent they must work in this way.
+
Bayes Essentials
Computer/Numerical Models
Scientific understanding of environmental processes often expressed in a computer/numerical model …
Climate
CO2, N
Soil
PHYSIOLOGY
BIOPHYSICS
WATER & NUTRIENT
FLUXES
PLANT
STRUCTURE
&
PHENOLOGY
DISTURBANCE
VEGETATION
DYNAMICS
Sheffield Dynamic Global Vegetation Model, SDGVM
Computer/Numerical Models
CO2: emissions vs atmospheric increase
‘Sinks for Anthropogenic Carbon’, Physics Today 2002, J L Sarmiento & N Gruber
Computer/Numerical Models
• usually deterministic, always wrong
Computer/Numerical Models
Computer/Numerical Models
• how to quantify the uncertainty?
Statistical Viewpoint on Numerical Models
MODELINPUT OUTPUT
Uncertain as a representation of reality:
• may not be known
• may be inadequate
— uncertainty analysis
— model inadequacy
Numerical model: a function mapping inputs into outputs
Output
Input
x
If model outputs available only at a limited number of inputs?
How represent knowledge about the model?
EmulationStatistical Viewpoint on Numerical Models
Bayes Formulation
Put a distribution on the space of possible functions;
ie, treat as random
and use the Bayes machinery to update knowledge about it from
runs of the computer model/simulator.
(Bayes rule!)
Statistical Viewpoint on Numerical Models
called an emulator
The probability distribution of
Numerical Models and Reality - Calibration, Model Inadequacy, Predictive Uncertainty
Main goal of modelling: to learn about reality.
Relation of numerical model to reality: represent via a statistical model and use the inference machinery to learn about it.
One formulation:
observations, the true process, the numerical model
observational error regression parameter
model inadequacy
Treat also as an unknown function
Earlier, used runs of numerical model to learn about and build emulator.
Now in same way use observed data and the emulator to learn about
via Bayes rule
Calibration: using observed data to learn about model inputs .
Find
parameters of the two GPs
via Bayes rule
Can integrate out and use maximizing to get
summarizing information about .
Prediction and predictive uncertainty:
ie what is ?
Conditionally is a Gaussian process
Combine with
for inference about
and further combine with
for inference about
Hence predictions and their uncertainty.
GEM software (Gaussian Emulation Machine)
Generates a statistical emulator of a computer code from training data consisting of an arbitrary set of inputs and the resulting outputs.
Gives the following: – prediction of code output at any untried inputs, taking account of
uncertainty in one or more of the code inputs. – main effects of each individual input. – joint effects of each pair of inputs. – percentage allocation to the variance from each individual input.
Calibrates code to observations, quantifies model inadequacy & predictive uncertainty
GEM-SA, GEM-CAL
Kennedy, M. C. & O’Hagan, A. (2001) Bayesian calibration of computer models. J. Roy. Statist. Soc. B, 63, 425-464.
Kennedy, M. C., O’Hagan, A. & Higgins, N. (2002) Bayesian analysis of computer code outputs. In Quantitative Methods for Current Environmental Issues, eds CW Anderson, V Barnett, P Chatwin & AH El-Shaarawi. Springer, London.
Oakley, J. E. & O’Hagan, A. (2004) Probabilistic sensitivity analysis. J. Roy. Statist. Soc. B,
66, 751-769.
Saltelli A, Chan K, Scott E M (2000) Sensitivity Analysis. Wiley.
Royal Society of Chemistry, Analytical Methods Sub-committee (web)
Some References:
For GEM software see www.ctcd.shef.ac.uk
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