Nonparametric Multivariate Density Estimation: Case Study of Cauchy Mixture Model

Abstract

Estimation of probability density functions (pdf) is considered an essential part of statistical modelling. Heteroskedasticity and outliers are the problems that make data analysis harder. The Cauchy mixture model helps us to cover both of them. This paper studies five different significant types of non-parametric multivariate density estimation techniques algorithmically and empirically. At the same time, we do not make assumptions about the origin of data from any known parametric families of distribution. The method of the inversion formula is made when the cluster of noise is involved in the general mixture model. The effectiveness of the method is demonstrated through a simulation study. The relationship between the accuracy of evaluation and complicated multidimensional Cauchy mixture models (CMM) is analyzed using the Monte Carlo method. For larger dimensions (d ~ 5) and small samples (n ~ 50), the adaptive kernel method is recommended. If the sample is n ~ 100, it is recommended to use a modified inversion formula (MIDE). It is better for larger samples with overlapping distributions to use a semi-parametric kernel estimation and more isolated distribution-modified inversion methods. For the mean absolute percentage error, it is recommended to use a semi-parametric kernel estimation when the sample has overlapping distributions. In the smaller dimensions (d = 2) and a sample is with overlapping distributions, it is recommended to use the semi-parametric kernel method (SKDE) and for isolated distributions, it is recommended to use modified inversion formula (MIDE). The inversion formula algorithm shows that with noise cluster, the results of the inversion formula improved significantly.

A New Clustering Method Based on the Inversion Formula

“…There are different researches on this topic: Parzen neural networks [19], soft constrained neural networks [20], and others [21]. Some time ago, we presented a modified inversion formula for density estimation [22]. In this research, we found that this density estimation works better with different data than multiple density estimators.…”

An Elastic Energy Management Algorithm in a Hierarchical Control System with Distributed Control Devices

“…-5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4. 5 Using the 3σ rule [24][25][26], a distribution of time values was obtained with a probability of 99.7% falling within the ±3σ time intervals. It was anticipated that SAs would be launched from 12:30 to 15:00.…”

“…In order to verify the correctness of the Java implementation of the activation algorithm of different SAs, the Probability Density Function (PDF) f (x) was determined from Formula (10) and the estimator PDF fk (11) [24][25][26][27]…”

Application of the multivariate and univariate analyses to estimate the feed efficiency in beef cattle

“…According to Guidoni (1994), DMI and ADG are mutually correlated continuous random variables that follow a normal probability distribution. The formation of a new variable may not have a normal distribution, and in this case, it approximates a non-parametric Cauchy distribution (Ruzgas et al 2021).…”

“…Other calculations for determining FE using univariate analysis are also described, such as Kleiber ratio (Kleiber 1936) and residual feed intake (RFI) according to Martin et al (2021). On the other hand, the linear combination of two variables in normal distribution produces a new variable with a normal distribution (Ruzgas et al 2021). The multivariate techniques allow combining the multiple information of the experimental unit (Schmit et al 2016).…”