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

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2 Outline Background Complex Models Simulators and Emulators Building an emulator Examples: 1 Dimensional 2 Dimensions Future Work Use of Derivatives

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3 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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4 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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5 Building an Emulator Deterministic function: y = η(x) Choose n design points x 1,..., x n Provides training data y T = {y 1 = η(x 1 ),..., y n = η(x n )} 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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6 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 ' ) | σ 2 ) = σ 2 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 σ 2 p (β, σ 2 ) α σ -2

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7 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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8 1 Dimensional Example η(x) = 5 + x + cos(x) Choose n = 7 design points: (x 1 = -6, x 2 = -4,..., x 6 = 4, x 7 = 6) Training data is then: y T = {y 1 = η(x 1 ),..., y n = η(x 7 )} 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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9 1 Dimensional Example

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10 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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Dimensional Example x = (x 1, x 2 ) T η(x) = x 1 + x 2 + sin(x 1 x 2 ) + 2cos(x 1 ) 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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Dimensional Example

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13 Future Work How can derivative (and integral) information help?

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14 Without Derivative Information

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15 Derivative Information

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16 Using Derivative Information

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

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