The Law Of Uncoscious Statistician

The EIMSE Formula

• As for the last section, the concepts are potentially relevant for predicting the errors of any “empirical model” in the context of a given input pattern or design of experiments (DOE) array. Also, this formula is useful for comparing response surface method (RSM) designs and generating them using optimization.
• The parts of the name include the “mean squared error” which derives from the fact that empirical models generally predict “mean” or average response values.
• The term “integrated” was originally used by Box and Draper (1959) to refer to the fact that the experimenter is not interested in the prediction errors at one point and would rather take an expected value or integration of these areas of all prediction points of interest.
• The term “expected” was added by Allen et al. (2003) who derived the formula presented here. It was included to emphasize the additional expectation taken over the unknown true system model.
• Important advantages of the EIMSE compared with many other RSM design criteria such as so-called “D-efficiency” include:

1.       The sqrt(EIMSE) has the simple interpretation of being the expected plus or minus prediction errors.

2.       The EIMSE criteria offers a more accurate evaluation of performance because it addresses contributions from both random errors and “bias” or model-misspecification, i.e., the fact that the fitted model form is limited in its ability to mimic the true input-output performance of the system being studied.

Advantages:

An advantage of the EIMSE compared with some other criteria is that it does not require simulation for its evaluation. The primary reason that simulation of the EIMSE was described in the last section was to clarify related concepts. The following quantities are used in the derivation of the EIMSE formula:

1.xp is the prediction point in the decision space where prediction is desired.

2.ρ(xp) is the distribution of the prediction points.

3.R is the region of interest which describes the area in which ρ(xp) is nonzero.

4.βtrue is the vector of true model coefficients.

5.ε is a vector of random or repeatability errors.

6.σ is the standard deviation of the random or repeatability errors.

7.y_true (xp,βtrue) is the true average system response at the point xp.

8.yest(xp,β_true,ε,_DOE) is the predicted average from the empirical model.

9.f₁(x) is the model form to be fitted after the testing, e.g., a second order polynomial.

10.f₂(x) contains terms in the true model not in f1(x), e.g., all third order terms.

11.β₁ is a k₁ dimensional vector including the true coefficients corresponding to those terms in f1(x) that the experimenter is planning to estimate.

12.β₂ is a k₂ dimensional vector including the true coefficients corresponding to those terms in f2(x) that the experimenter is hoping equal 0 but might not. These are the source of bias or model mis-specification related errors.

13.X₁ is the design matrix made using f₁(x) and the DOE array.

14.X₂ is the design matrix made using f₂(x) and the DOE array.

15.R is the “region of interest” or all points where prediction might be desired.

16.μ₁₁, μ₁₂, and μ₂₂ are “moment matrices” which depend only on the distribution of the prediction points and the model forms f1(x) and f2(x).

17.“E” indicates the statistical expectation operation which is here taken over a large number of random variables, xp, βtrue, and ε.

18.XN,₁ is the design matrix made using f₁(x) and all the points in the candidate set.

19.XN,₂ is the design matrix made using f₂(x) and all the points in the candidate set.