Nonparametric Estimation of Continuously Parametrized Families of Probability Density Functions—Computational Aspects

• Main Author: Wojciech Rafajłowicz
• Format: Article
• Language: English
• Published: MDPI AG 2020-07-01
• Series: Algorithms
• Subjects: nonparametric estimation; FFT; family of probability density functions
• Online Access: https://www.mdpi.com/1999-4893/13/7/164

We consider a rather general problem of nonparametric estimation of an uncountable set of probability density functions (p.d.f.’s) of the form: <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>;</mo> <mspace width="0.166667em"></mspace> <mi>r</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, where <i>r</i> is a non-random real variable and ranges from <inline-formula> <math display="inline"> <semantics> <msub> <mi>R</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> to <inline-formula> <math display="inline"> <semantics> <msub> <mi>R</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula>. We put emphasis on the algorithmic aspects of this problem, since they are crucial for exploratory analysis of big data that are needed for the estimation. A specialized learning algorithm, based on the 2D FFT, is proposed and tested on observations that allow for estimate p.d.f.’s of a jet engine temperatures as a function of its rotation speed. We also derive theoretical results concerning the convergence of the estimation procedure that contains hints on selecting parameters of the estimation algorithm.