# Kriging Models

**Introduction:**

As an alternative of regressions we can utilize kriging models and also artificial neural nets (ANN) Linear regression models are not the only curve-fitting methods in wide use. Also, these methods are not useful for analyzing data for categorical responses.

**Kriging Models and Computer Experiments:**

1. Matheron (1963) proposed so-called “kriging” meta-models to make predictions in the context of modeling physical, geology-related data.

2. Recently, the application of these same techniques in the context of computer experiments has received significant attention in part because of the above-mentioned advantage that kriging models provide smooth interpolating functions passing through all of the output data, e.g., see Sacks et al. (1989) and Welch et al. (1992).

3. Kriging procedures are relatively difficult to apply because the curve fitting involved requires a nontrivial optimization and, therefore, specialized software. However, as the necessary software becomes increasingly available, there is reason to expect that the methods will enjoy even more widespread application.

4. Kriging models under common assumptions provide prediction models, y_{est}(β_{est}, **x**), that pass through all the data points. This is considered to be desirable in the context of certain kinds of experiments that are perfectly repeatable, i.e., the same inputs give the same outputs with σ_{0} = 0.

5. The phrase “**computer experiments**” is often used to refer to finite element method (FEM) and finite difference method (FDM) testing in which prototypes are virtual and no ,sources of variation are involved in there empirical evaluation.

6. Kriging models are sufficiently flexible that they can seamlessly extend to situations in which the number of tests grows much higher than those involved in response surface methods.

7. Kriging models can model input-output relationships with multiple twists and turns, they are considered particularly relevant in the context of optimization.

**Design of Experiments for Kriging Models:**

1. Deriving desirable experimental plans to foster accurate fitted kriging models is an active area of research. For simplicity, only so-called Latin hypercube designs (LHDs) and space-filling designs for the data collection are considered because these designs have received the most attention in the kriging literature. LHDs have the advantage that they are easy to generate for any number of runs.

2. The version of LHDs here is based on McKay et al. (1979). For n runs, each of the k factors takes on equally spaced values –1 1/n, –1 3/n… 1 – 1/n, in different random orders. Space-filling designs are derived by maximizing the minimum Euclidean distance between all pairs of design points.