A methodology for the approximation of discrete values using a minimal set of continuous functions.

摘要

In many applications it is quite common to have an ordered set of discrete observations, y, and hypothesize that the value of an observation (y_i) is dependent on the corresponding values of one or more independent variables (x_i). If this relationship is not exact, i.e. y_i ≠ f(x_i), statistical techniques are utilized to fit a function, y_i = f(x_i) + e_i , that may then be used to estimate the y_i. Let y_i = f(x _i) be the fitted value and let e = (y _i − ŷ_i) be the vector of residuals. Employing only currently available estimation techniques, for example ordinary least squares regression, some of the residuals or errors of estimation, the e_i, may be larger than acceptable for the application under consideration. Even if the relationship were known and exact, i.e. y_i = f(x_i) for all i, direct calculation of the y_i may be complex and/or time consuming thus making development of an estimating function, with sufficiently small errors of estimation, a desirable alternative.; This research develops a methodology that can be used to determine a set of approximating functions such that each residual, e_i = (y_i − ŷ_i), is less than a user defined amount, ε. The system designed to conduct the research is modular in nature, with each model consisting of a stand alone computer program. First, additional variables that are themselves functions of the original independent variables are constructed. Second, a modified backward regression technique, using either OLS or SVD, is developed to reduce the number of predictor variables to those that have the greatest impact on minimizing the number of observations where the approximation error exceeds ε. Last, an algorithm using a minimax LP is developed to produce a minimal number of approximating functions such that each residual e_i = (y_i − ŷ_i) ≤ ε.; Although preliminary, the results indicate that this methodology is viable and provides an estimation technique with application in business research and social sciences as well as in the field of statistics.