Generalization ability of fractional polynomial models.

Area of Science:

• Machine Learning
• Statistical Modeling
• Data Analysis

Background:

• Learning functional dependencies from scattered data is crucial in various scientific fields.
• Traditional models may struggle with complex relationships and generalization.
• Fractional Polynomial Models (FPM) offer a potential alternative for modeling such data.

Purpose of the Study:

• To investigate the use of Fractional Polynomial Models (FPM) for learning functional dependencies from scattered data.
• To analyze the theoretical properties and practical performance of FPM.
• To compare FPM with existing methods like Sparse Polynomial Models (SPM) and cubic splines.

Main Methods:

• Calculating the pseudo-dimension of FPM to derive estimation error bounds.
• Deriving a structural risk criterion analogous to the Schwartz Criterion for model selection.
• Employing the variable projection method for efficient FPM construction.
• Empirical model selection comparison and generalization performance evaluation.

Main Results:

• The pseudo-dimension of FPM is found to be equal to that of SPM.
• A linear decay of approximation error is achieved for a class of continuous functions.
• Minimizing the derived structural risk balances estimation and approximation errors.
• FPM demonstrates better generalization performance than SPM and cubic splines in empirical studies.

Conclusions:

• Fractional Polynomial Models (FPM) provide a theoretically sound and practically effective approach for learning functional dependencies from scattered data.
• The proposed structural risk criterion aids in selecting optimal model complexity.
• FPM offers superior generalization capabilities compared to established methods.