Neutrosophic kernel regression and its application

Introduction

The parametric technique is commonly used to illustrate the relationship between the independent variables and the dependent variable. Nevertheless, in some scenarios, the parametric model is inappropriate, and the resulting estimators do not outperform pure estimators in terms of efficiency. Kuo (1988) [8] proposed a natural alternative for the distribution function that uses a nonparametric approach and does not put any constraints on the relationship between the independent variables and the dependent variable. Some noteworthy studies on this topic are Chambers et al. (1993) [5] , Drofman (1993) [5] , and Drofman and Hall (1993) [5] . In sometimes, we cannot compute or supply exact values for statistical characteristics in real life, thus we must approximate them. As a result, neutrosophic statistics is an extension of classical statistics in which one works with set values rather than specific values. Smarandache (2014) [11] is the first one introduced Neutrosophic Statistics as a generalization of classical statistics applied when the data under consideration is in neutrosophic numbers. Many researchers are interested with neutrosophic statistics because it is more realistic in many problems. For example, Aslam and Albassam (2019) [3] studied the statistical correlation between dietary fat level and prostate cancer risk. Alhabib and Salama (2020) [1] presented a linear model for neutrosophic time series and used Student's distribution to determine the significance of its coefficient. Salem et al (2021) [9] derived the mathematical properties of the neutrosophic lognormal distribution. Aslam and Saleem (2023) [4] introduced the F-test of testing linearity under neutrosophic statistics. Alomair and Shahzad (2023) [2] introduced the neutrosophic Hartley-Ross-type ratio estimators to estimate the population mean of neutrosophic data when the data is infected by the outliers. Some of the more popular nonparametric regression methods are those based on kernel functions, spline functions and wavelets. Each of these methods has their own strengths and weaknesses but kernel functions have the advantage of mathematical simplicity. So, in this study, the neutrosophic kernel regression will be studied and applied to prostate cancer data. As far as we know, the neutrosophic structure of the kernel regression has never been addressed in previous studies.

Preliminaries

The Gaussian kernel function is an example of the neutrosophic kernel function, and it has the form

All proofs in this study are based on the algebraic framework of the neutrosophic numbers. [Smarandache (2014)] [11] .

Assuming two bounded real intervals

Also, if we have D is a calculus operator (differentiation or integration( and f is any function, then

where 0 N h  is a smoothing neutrosophic parameter called the neutrosophic bandwidth.

Proof. Using the properties of the neutrosophic kernel function, then we have

Hence, the neutrosophic kernel density estimation more than or equal zero i.e.

 

From ( 3) and (4), theorem (1) is proved.

Proof. From the properties of the neutrosophic kernel function

Therefore, the neutrosophic joint kernel density estimation more than or equal zero i.e.  

From ( 6) and (8), theorem (2) is proved.

(4) Neutrosophic Kernel Regression

Suppose that there is a neutrosophic set of data consists of n pairs of observations,      

The objective of neutrosophic kernel nonparametric regression is to estimate   N mx using local averaging; the average will be formed in such a way that it is defined only in small neighborhoods around 1

Using Theorem (1) and ( 2), then we have

 

Real Application

Prostate cancer is one of the most frequent malignant illnesses in men and the second greatest cause of death from cancer (Van Booven et al., 2021) [12] . Recent studies have shown that an increased level of fat in the diet causes an increased incidence of prostate cancer. Dietary fat may be measured by measuring the fat in each ingredient in our daily food or by calculating the per percentage of calories in our daily food using glucose machines through blood. Measuring the diet using these approaches does not result in an exact number but can be recorded in an interval range. So, the relation between dietary level and prostate cancer death rate cannot be studied using the traditional regression model when the variables are given as intervals. When the dietary fat level and prostate cancer death rate are in the interval range in an uncertain environment, neutrosophic statistics can be used for studying the relationship between the dietary fat level and prostate cancer death rate. In this study, we study the relationship between the predictor variable (The neutrosophic dietary fat level) and the response variable (the prostate cancer neutrosophic death rate) was studied using neutrosophic kernel regression. The dataset from Aslam and Albassam (2019) [19] represented in table (1) shows the neutrosophic dietary fat level and prostate cancer neutrosophic death rate for 30 countries.

Where ˆN  and N IQR are the estimated value of standard deviation and interquartile range for predictor variable, respectively and n is sample size. (See Silverman 1986) [10] From the above dataset, we found that

Conclusion

In this work, we introduced a neutrosophic kernel regression model as a novel neutrosophic model that may be employed more successfully in an uncertain environment than the classic regression model. The neutrosophic kernel regression model is used for studying the relationship between dietary fat level as the neutrosophic independent variable and the death rate from prostate cancer as the neutrosophic dependent variable.

References

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