Development of the group method of data handling with information-based model evaluation criteria : a new approach to statistical modeling

A new approach to statistical modeling is studied which integrates the information-based statistical criteria into Ivakhnenko's (1966) polynomial theory, also known as the Group Method of Data Handling (GMDH). The GMDH algorithm was introduced as a hierarchical algorithm which constructs a mathematical model of a complex system by the composition of lower-order polynomials of the form: Z = A + Bx + Cy + Dx² + Ey² + Fxy. Under the assumptions that there exists a random error component additive with the above deterministic component, the polynomials generated at each level of the GMDH algorithm are statistically modeled as regression-type equations of the following form: y = β₀ + β₁x₁ + β₂x₂ + β₃x²₁ + β₄x²₂ + β₅x₁x₂ + ε.

In this research, the assumption of independence of error terms is relaxed by the derivation of information-based model selection criteria for the GMDH algorithm. These criteria inherently guard against multicollinearity by incorporating a measure of the complexity of each model being evaluated. In addition to deriving the information-based criteria for the GMDH, the consistency of the criteria when used within the GMDH algorithm is established. While consistency is a desirable asymptotic property for model evaluation, it does not, by itself, guarantee that the criteria are effective in terms of model selection. Thus, Monte Carlo experimentation is used to study the aptness (i.e., the ability of the model to describe the system of interest) of the model identified by the GMDH algorithm using information-based criteria.

At the end of the GMDH algorithm, the "best approximating" model is identified. In generations two and beyond, the models are written in terms of the previous generation's polynomials. Hence, the chosen model is typically not written in terms of the original set of predictor variables. Even with the original GMDH algorithm, existing computer programs do not provide a means of backtracing (i.e., back-substituting) to the original variables. The revised algorithm (using the information-based model evaluation criteria) is coded in MATLAB®. A backtracing algorithm is introduced and designed for the GMDH hierarchical tree-algorithm to determine the "best" fitting model in terms of the original variables using a symbolic mathematical language by borrowing on the rules of algebraic manipulation.