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Invariance properties for the error function used for multilinear regression.

Mark H Holmes1, Michael Caiola2

  • 1Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York, United States of America.

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Investigating multilinear regression error functions reveals scale and rotational invariance are incompatible. A geometric mean-based error function offers scale and reflective invariance, approximating minimizers with the centroid of the error simplex.

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Area of Science:

  • Statistics
  • Data Analysis
  • Multivariate Analysis

Background:

  • Multilinear regression is a statistical method for analyzing relationships between multiple independent variables and a dependent variable.
  • Commonly used error functions in regression analysis often assume desirable data properties like scale and rotational invariance.
  • The compatibility of these assumed data properties with the choice of error function is crucial for robust model performance.

Purpose of the Study:

  • To investigate the relationship between the error function in multilinear regression and fundamental data properties.
  • To determine the compatibility of scale and rotational invariance with standard error functions.
  • To introduce and evaluate a novel error function with desirable invariance properties.

Main Methods:

  • Theoretical analysis of error functions in multilinear regression.
  • Mathematical derivation of scale and reflective invariance properties.
  • Comparison of a geometric mean-derived error function with existing methods.
  • Application and validation on multidimensional real-world datasets.

Main Results:

  • Scale and rotational invariance are mathematically shown to be incompatible in standard multilinear regression error functions.
  • A novel error function derived from the geometric mean demonstrates both scale and reflective invariance.
  • The minimizer of this new error function can be approximated using the centroid of the error simplex under specific conditions.
  • Empirical application to real-world data shows competitive performance compared to other regression techniques.

Conclusions:

  • The choice of error function significantly impacts the properties and interpretability of multilinear regression models.
  • The proposed geometric mean-based error function offers enhanced invariance properties, making it suitable for diverse data types.
  • This approach provides a robust alternative for multidimensional data analysis where scale and reflective invariance are important.