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Cost of surrogates In linear regression, the process of fitting involves solving a set of linear equations once. For moving least squares, we need to form and solve the system at every prediction point. With radial basis neural networks we have to optimize the selection of neurons, which will again entail multiple solutions of the linear system. We may find the best spread by minimizing cross-validation errors. Kriging, our next surrogate is even more expensive, we have a spread constant in every direction and we have to perform optimization to calculate the best set of constants. With many hundreds of data points this can become significant computational burden.

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Kriging - Introduction Method invented in the 1950s by South African geologist Daniel Krige (1919-) for predicting distribution of minerals. Became very popular for fitting surrogates to expensive computer simulations in the 21 st century. It is one of the best surrogates available. It probably became popular late mostly because of the high computer cost of fitting it to data.

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Kriging philosophy We assume that the data is sampled from an unknown function that obeys simple correlation rules. The value of the function at a point is correlated to the values at neighboring points based on their separation in different directions. The correlation is strong to nearby points and weak with far away points, but strength does not change based on location. Normally Kriging is used with the assumption that there is no noise so that it interpolates exactly the function values. It works out to be a local surrogate, and it uses functions that are very similar to radial basis functions.

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Reminder: Covariance and Correlation Covariance of two random variables X and Y The covariance of a random variable with itself is the square of the standard deviation Covariance matrix for a vector contains the covariances of the components Correlation The correlation matrix has 1 on the diagonal.

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Correlation between function values at nearby points for sine(x) Generate 10 random numbers, translate them by a bit (0.1), and by more (1.0) x=10*rand(1,10) 8.147 9.058 1.267 9.134 6.324 0.975 2.785 5.469 9.575 9.649 xnear=x+0.1; xfar=x+1; Calculate the sine function at the three sets. ynear=sin(xnear) 0.9237 0.2637 0.9799 0.1899 0.1399 0.8798 0.2538 -0.6551 -0.2477 -0.3185 y=sin(x) 0.9573 0.3587 0.9551 0.2869 0.0404 0.8279 0.3491 -0.7273 -0.1497 -0.2222 yfar=sin(xfar) 0.2740 -0.5917 0.7654 -0.6511 0.8626 0.9193 -0.5999 0.1846 -0.9129 -0.9405 Compare corelations. r=corrcoef(y,ynear) 0.9894; rfar=corrcoef(y,yfar) 0.4229 Decay to about 0.4 over one sixth of the wavelength.

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Gaussian correlation function

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Linear trend function is most often a low order polynomial We will cover ordinary kriging, where linear trend is just a constant to be estimated by data. There is also simple kriging, where constant is assumed to be known. Assumption: Systematic departures Z(x) are correlated. Kriging prediction comes with a normal distribution of the uncertainty in the prediction. Universal Kriging x y Kriging Sampling data points Systematic Departure Linear Trend Model Linear trend model Systematic departure

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Notation

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Prediction and shape functions

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Fitting the data

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Prediction variance Square root of variance is called standard error The uncertainty at any x is normally distributed.

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Kriging fitting problems The maximum likelihood or cross-validation optimization problem solved to obtain the kriging fit is often ill-conditioned leading to poor fit, or poor estimate of the prediction variance. Poor estimate of the prediction variance can be checked by comparing it to the cross validation error. Poor fits are often characterized by the kriging surrogate having large curvature near data points (see example on next slide). It is recommended to visualize by plotting the kriging fit and its standard error.

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Example of poor fits.

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SE: standard error

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Problems Fit the quadratic function of Slide 13 with kriging using different options, like different covariance and trend function and compare the accuracy of the fit. For this problems compare the standard error with the actual error.

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