Insert Date HereSlide 1 Using Derivative and Integral Information in the Statistical Analysis of...

Insert Date Here Slide 1

Using Derivative and Integral Information in the Statistical

Analysis of Computer Models

Gemma Stephenson

March 2007

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Outline

Background Complex Models

Simulators and Emulators Building an emulator

Examples: 1 Dimensional 2 Dimensions

Future Work Use of Derivatives

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Complex Models

Simulate the behaviour of real-world systems

Simulator: deterministic function, y = η(x) Inputs: x Outputs: y are the predictions of the real-world system being

modelled

Uncertainty in x in η(.) in how well the emulator approximates the simulator

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Emulators Gaussian Process (GP) Emulation

A Gaussian Process is one where every finite linear combination of values of the process has a normal distribution

Emulator - Statistical approximation of the simulator

Mean used as an approximation to simulator

Approximation is simpler and quicker than original function

Used for any uncertainty analysis and sensitivity analysis

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Building an Emulator

Deterministic function: y = η(x)

Choose n design points x1 , . . . , xn

Provides training data yT = {y1 = η(x1), . . . , yn = η(xn)}

Aim: using the observations above we want to make Bayesian Inferences about η(x)

Prior information about η(.) is represented as a GP and after the training data is applied; the posterior distribution is a GP also.

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Prior Knowledge

E [η(x) | β] = h(x)T β h(x)T is a known function of x β is a vector comprising of unknown coefficients

Cov ( η(x), η(x') | σ2 ) = σ2 c(x, x') c(x, x') = exp {− (x − x')T B (x − x') }

B is a diagonal matrix of smoothing parameters

Weak prior distribution for β and σ2

p (β, σ2 ) α σ -2

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Posterior Information

m**(x) is the posterior mean used to predict the output at new points

c**(x, x) is the posterior covariance

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1 Dimensional Example

η(x) = 5 + x + cos(x)

Choose n = 7 design points: (x1 = -6, x2 = -4, . . . , x6 = 4, x7 = 6)

Training data is then: yT = {y1 = η(x1), . . . , yn = η(x7)}

Take h(x)T =(1 x) then emulator mean is derived.

Variance derived choosing c(x, x') = exp {− 0.5 (x − x')2 } as the correlation function

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1 Dimensional Example

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Smoothness

Assume that η(.) is a smooth, continuous function of the inputs.

Given we know y at x = i, smoothness implies y is close to the same value, for any x close enough to i.

The parameter, b, specifies how smooth the function is. b tells us how far a point can be from a design point before the

uncertainty becomes appreciable

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2 Dimensional Example

x = (x1, x2)T

η(x) = x1 + x2 + sin(x1x2) + 2cos(x1)

n = 20 design points chosen using Latin Hypercube Sampling

B estimated from the training data

Emulator mean used to predict the output at 100 new inputs

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2 Dimensional Example

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Future Work

How can derivative (and integral) information help?

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Without Derivative Information

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Derivative Information

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Using Derivative Information

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Future Work

Cost of using derivatives When already available When we have the capability to produce them

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