Validation of uncertain predictions against uncertain observations

Scott Ferson, scott@ramas.com16 October 2007, Stony Brook University, MAR 550, Challenger 165

V & V

• Verification (checking the math)– Code testing– Interval analysis, probability bounds analysis– Units/dimension checking

• Validation (checking against data)

Goals

• Objectively measure the conformance of predictions with empirical data

• Use this measure to characterize the reliability of other predictions

Initial setting

• The model is fixed, at least for the time being– No changing it on the fly during validation

• A prediction is a probability distribution– Expressing stochastic uncertainty

• Observations are precise (scalar) numbers– Measurement uncertainty is negligible

relaxed later

Validation metric

• A measure of the mismatch between the observed data and the model’s predictions– Low value means a good match– High value means they disagree

• Distance between prediction and data

Desirable properties of a metric

• Should be expressed in physical units

• Should generalize deterministic comparisons

• Should reflect performance of full distribution

• Shouldn’t be too sensitive to long tails

• Should be a true mathematical metric

• Should be unbounded (you can be really off)

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How the data come

How we look at them

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One suggestion for a metric

Area or average horizontal distance between the empirical distribution Sn and the predicted distribution

Reflects full distribution

Matches in mean

Both mean and variance

Matches well overall

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Single observation

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Single observation

• A single datum can’t match an entire distribution (unless it’s degenerate)

• Single datum matches best if it’s at the median

• If the prediction is a uniform distribution over [a,b], a single observation can’t be any ‘closer’ to it than (b a)/4

When the prediction is really bad

• The metric degenerates to simple distance

• Probability is dimensionless, so units are the same

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Depends on the local scale

• The metric depends on the units

• Can standardize (divide by s.d.), but this means the metric will no longer be in physical units

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Why physical units?

• Distributions in the left graph don’t overlap but they seem closer than those on the right

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Why an unbounded metric?

• Neither overlaps, but left is better fit than right

• Smirnov’s metric Dmax considers these two cases indistinguishable (they’re both just ‘far’)

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The model says different things

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Pooling data comparisons

• When data are to be compared against a single distribution, they’re pooled into Sn

• When data are compared against different distributions, this isn’t possible

• Conformance must be expressed on some universal scale

Universal scale

ui=Fi (xi) where xi are the data and Fi are their

respective predictions

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Backtransforming to physical scale

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Backtransforming to physical scale

• The distribution of G1(Fi (xi)) represents the

empirical data (like Sn does) but in a common, transformed scale

• Could pick any of many scales, and each leads to a different value for the metric

• The distribution of interest is the one used for the regulatory statement

Number of function evaluations

• Some models are difficult to evaluate

• Extracting distributional predictions may be expensive in terms of function evaluations – Blame the modeler rather than the validator!

• Can our validation metric be applied when only very coarse predictions based on few function evaluations are available?

Coarse prediction

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Prediction can be expressed as an ‘empirical’ distribution too

Statistical test for model accuracy

• Kolmogorov-Smirnov test of distribution of ui’s against uniform over [0,1]

• This tests whether the empirical data are as though they were drawn from the respective prediction distributions

Probability integral transform theorem (Angus 1994) says the u’s will be distributed as uniform(0,1) if xi ~ Fi

• Assumes the empirical data are independent of each other

Generalizations

• Nonrandom sampling of observations

• Measurement uncertainty of empirical observations

• Imprecise predictions (intervals or p-boxes)

Epistemic uncertainty in predictions

• In left, the datum evidences no discrepancy at all• In middle, the discrepancy is relative to the edge• In right, the discrepancy is even smaller

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a = N([5,11],1)show a

b = 8.1show b in blue

b = 15

breadth(env(rightside(a),b)) 4.023263478773

b = 11breadth(env(rightside(a),b)) / 2 0.4087173895951 d = 0 d 4 d 0.4

Epistemic uncertainty in bothP

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z=0.0001; zz =9.999show z,zza = N([6,7],1)-1show a

b = -1+mix(1,[5,7], 1,[6.5,8], 1,[7.6,9.99], 1, [3.3,6], 1,[4,8], 1,[4.5,8], 1,[5,7], 1,[7.5,9], 1,[4,8], 1,[5,9], 1,[6,9.99])show b in blue

b = -0.2+mix(1, [9,9.6],1, [5.3,6.2], 1,[5.6,6], 1,[7.8,8.4], 1,[5.9,7.8], 1,[8.3,8.7], 1,[5,7], 1,[7.5,8], 1,[7.6,9.99], 1, [3.3,6], 1,[4,8], 1,[4.5,8], 1,[5,7], 1,[8.5,9], 1,[7,8], 1,[7,9], 1,[8,9.99])

breadth(env(rightside(a),b)) 2.137345705795

c = -4b = -0.2+mix(1, [9,9.6],1, [5.3,6.2]+c, 1,[5.6,6]+c, 1,[7.8,8.4], 1,[5.9,7.8], 1,[8.3,8.7], 1,[5,7], 1,[7.5,8], 1,[7.6,9.99], 1, [3.3,6], 1,[4,8], 1,[4.5,8]+c, 1,[5,7]+c, 1,[8.5,9], 1,[7,8], 1,[7,9], 1,[8,9.99])

breadth(env(rightside(a),b)) / 2 1.329372857714

d = 0 d 0.05 d 0.07

Predictions in whiteObservations in blue

Validation: summary

• Both assessment and reliability of extrapolation– How good is the model?– Should we trust its pronouncements?

• Need metric to be both ad hoc and universal

• Updating is a separate activity

• Epistemic uncertainty introduces some wrinkles– Full credit for being modest about predictions

Complexities we can handle

• Variability in the experimental data• Prediction is a probability distribution rather than a point• Large measurement uncertainty in the data• Multiple predictions (dimensions) to be assessed• Available data aren’t directly relevant to the predictions• Validation data collected under other conditions• Model’s predictions are extremely expensive to compute

End

Definition of a true metric

• Positive, d(x, y) 0

• Symmetric, d(x, y) = d(y, x)

• Identicals indistinguishable, d(x, y) = 0 x = y

• Triangle inequality, d(x, y) + d(y, z) d(x, z)

• Quasi-, semi-, pseudo-, ultra-metric

Other metrics

• Area is only one of many possible metrics

• Area favors central tendency (median)

• Could also use the medial distance from a datum to the distribution, or maybe the 95th percentile of distances

• Might prefer conformance in the tails, or one tail in particular

Degrees of impossibility

• If a datum is completely outside the range of the prediction, it’s ‘impossible’

• Transforming to the u scale makes it 0 or 1

• We’d like to preserve how far outside it is

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Extended distribution functions F <(x), x < 0

F*(x) = F(x), 0 x 1F

>(x), x > 1

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FF*Extension slopes can be set by the distribution’s dispersion, to mimic tails, or as just relocated 45 lines

Using extensions in the metric

• Extended functions Fi* can be used to get u’s (now no longer ranging only on [0,1])

• The common backtransformation scale can also be extended to G* to accept these u’s

• This allows values considered impossible by the prediction to be represented

Vector of outputs

• Usually want to treat dimensions separately

• Possible to unify (pool) prediction-observation pairs even if they’re from different dimensions– Degrees, seconds, pascals, meters, etc.

• But there’s no G for backcalculation and so there can’t be a physically meaningful scale

Comparing accuracies

• Questions like “Is the match for temperature as good as the match for conductivity?” also require a universal scale to which all physical dimensions must be transformed

• If we do this, the metric becomes a norm

Uncertainty about a distribution

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95% confidence bounds on the normal distribution