Kriging - Introduction

1. 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 21st 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.

2. 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 radial basis functions.

3. 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.

4. Correlation between function values at nearby points for sin(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.

5. Gaussian correlation function • Correlation between point x and point s • We would like the correlation to decay to about 0.4 at one sixth of the wavelength . • Approximately • For the function we would like to estimate

6. Linear trend model Systematic departure Sampling data points y Linear Trend Model Systematic Departure Kriging x Universal Kriging • Linear trend function is most often a low order polynomial • We will cover ordinarykriging, 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.

7. Notation • The function values are given at points , with the point having components . • The function value at the ithpoint is =y(), and the vector of function values is denoted y. • Given decay rates , we form the covariance matrix of the data • The correlation matrix R above is formed from the covariance matrix, assuming a constant standard deviation which measures the uncertainty in function values. • For dense data, will be small, for sparse data will be large. • How do you decide whether the data is sparse or dense?

8. Prediction and shape functions • Ordinary Kriging prediction formula • The equation is linear in r, which means that the exponentials may be viewed as basis functions. • The equation is linear in the data y, in common with linear regression, but b is not calculated by minimizing rms. • Note that far away from data .

9. Fitting the data • Fitting means finding the parameters • We fit by maximizing the likelihood that the data comes from a Gaussian process defined by . • Once they are found, the estimate of the mean and standard deviation is obtained as • Maximum likelihood is a tough optimization problem. • Some kriging codes minimize the cross validation error.

10. Top hat question • Comparing linear regression with kriging, which of the following statements are correct? • Linear regression assumes that the response is a linear combination of given shape functions, kriging does not. • Linear regression minimizes rms of residuals, kriging does not. • Linear regression is much cheaper than kriging. • Linear regression typically works with fewer parameters than data points, while kriging has more unknown parameters than data points.

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

12. KRIGING FIT AND THE IMPROVEMENT QUESTION • First we sample the function and fit a krigingmodel. • We note the present best solution (PBS) • At every x there is some chance of improving on the PBS. • Then we ask: Assuming an improvement over the PBS, where is it likely be largest? 12

13. WHAT IS EXPECTED IMPROVEMENT? Consider the point x=0.8, and the random variable Y, which is the possible values of the function there. Its mean is the kriging prediction, which is slightly above zero. 13

14. EXPLORATION AND EXPLOITATION EGO maximizes E[I(x)] to find the next point to be sampled. • The expected improvement balances exploration and exploitation: it can be high either due to high uncertainty or low surrogate prediction. • When can we say that the next point is “exploration?” 14

15. Constraint boundary estimation • When we optimize subject to constraints, evaluating the constraints is often computationally expensive. • Following references in notes, we denote the constraint as • When we evaluate the constraint, we do not mind having poor accuracy when the constraint is far from its critical value, but accuracy is important when it is nearly critical.

16. Feasibility function • We define a feasibility function • G is random due to uncertainty in surrogate that is fitted to g; • represents uncertainty in surrogate. Here we will use twice the standard error. • We will add points to maximize expected feasibility • where

17. Branin-Hoo example • Constraint function • mf is fraction of points misclassified on a grid of 10,000 points. .

18. Convergence .