Kriging Models
Fitting Kriging Models:
The following equation offers intuition about how kriging models work:
Where f(x) is a regression model that is potentially the same as a linear regression model and Z(x) is a function that models departures from the regression model.
A relevant concept is, therefore, an attempt to more aggressively model the unexplained variation compared with using regression models only.
Attempting to predict the departures, Z(x), from regression is motivated by the fact computer experiments with little or no random error. Sacks et al. (1989) argue that it is reasonable for computer experiments to treat the departures Z(x) as if they can be modeled and not merely considered to be random noise.
Following Matheron (1963), Sacks et al. (1989) proposed to model the departures as “realizations” from a Gaussian stochastic process. Further, they and other authors suggest that the regression component, f(x), should be omitted because of empirical evidence that this gives superior or comparable accuracy. Here also, we focus on the assumption that a constant term only is included in the model instead of a complicated regression model. The variables used in fitting include:
1) m is the number of factors.
2) Θ i ≥ 0 and 0 ≤ pi ≤ 2 for i = 1… m are fitted parameters similar to regression coefficients.
3) R(w,x) is the correlation between the random departures Z (w) and Z(x) for decision vectors w and x.
4) R is an n × n matrix of correlations between the n points in the input array, which is a function of the qi and pi.
5) βest is the estimated regression coefficient vector. Here, we focus on the assumption that β est is just one coefficient, i.e., the constant term.
6) σestis the estimated standard deviation of the response variation that is roughly proportional to the range of the response.
7) ln L is the “log likelihood” which is the fitting objective analogous to least squares for linear regression.
8) x1… xn are the n input combinations in the input pattern. These could be specified by an experimental design such as a Latin-Hypercube.
9) xi, k refers to the settings of the ith run for the kth factor.
10) y is the response vector corresponding to the n runs.