Multivariate Polynomial Regression

The TAYLORFIT program contains an algorithm which can identify complex relationships among data, including nonlinearity and interactions, and present the result as a mathematical expression which is much easier to use and understand than competing methods, but equally or more powerful in its capabilities. It can be used for multivariate correlation analysis, time-series analysis or to generate discriminant functions.

The method is called Multivariate Polynomial Regression, or MPR. It produces a polynomial describing the relationship between any set of inputs and corresponding outputs. It can replace techniques such as multilinear regression and ARMA (Box-Jenkins) models, and can produce far superior results. It can also be used in applications to which artificial neural networks have been applied. However, MPR is easier to use and understand and avoids overfitting errors. MPR is essentially a multiple regression model with polynomial and cross-product (interaction) terms. For example, if Y is a function of Q, R, and S, terms can be included such as QR2S or Q3S. MPR models can be fitted using conventional multiple regression software, and only terms which are statistically significant are retained in the model. MPR models are applicable to low-to-moderate dimensionality problems as are encountered in economics.

MPR models can describe logical relationships between a dependent variable and multiple independent variables which linear ARMA models are incapable of capturing. Therefore MPR models can produce better fits and less bias in predictions.

If the number of independent variables is not too great, MPR models compare favorably in performance to artificial neural network (ANN) models. MPR models can provide a better fit with fewer coefficients. This reduces overfitting or "memorizing" of data. The fitting procedure converges absolutely and does not require a priori selection of model structure. The procedure gives a simple explicit equation for prediction or analysis, and standard statistical tests can be applied to all coefficients and forecast predictions.

We have shown how time-series data can be modeled by multivariable polynomials. MPR is applied to the Lorenz equations, a challenging nonlinear system with chaotic dynamics. MPR accurately captures the dynamic behavior.

Other work describes the use of MPR for multivariable analysis, similar to multilinear regression. The application involves determining the relationship between the profitability of a series of stores and various predictor variables, such as market share, age of store, etc.

For additional information about MPR and the TAYLORFIT software, go to


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