insert date hereslide 1 using derivative and integral information in the statistical analysis of...
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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
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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
Slide 4www.mucm.group.shef.ac.uk
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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