Insert Date HereSlide 1 Using Derivative and Integral Information in the Statistical Analysis of Computer Models Gemma Stephenson March 2007.

Insert Date Here Slide 1

Using Derivative and Integral Information in the Statistical

Analysis of Computer Models

Gemma Stephenson

March 2007

Slide 2www.mucm.group.shef.ac.uk

Outline

Background Complex Models

Simulators and Emulators Building an emulator

Examples: 1 Dimensional 2 Dimensions

Future Work Use of Derivatives


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


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


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.


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


Posterior Information

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

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


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




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



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




Future Work

How can derivative (and integral) information help?


Without Derivative Information


Derivative Information


Using Derivative Information


Future Work

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

www.mucm.group.shef.ac.uk

Insert Date HereSlide 1 Using Derivative and Integral Information in the Statistical Analysis of Computer Models Gemma Stephenson March 2007.

Documents

x x t b x x

x inputs

x close

x outputs

x prior information

dimensional example

known function of x

prior knowledge e x