Simulating chi-square data through algorithms in the presence of uncertainty

1.   Introduction

Among the statistical distributions, the chi-square distribution is very popular and has been used in many areas, including medical science ^[1] and engineering ^[2]. This distribution has been widely used in the goodness of fit test to see whether the data series is independent or not. The chi-square distribution is also been used for testing the variation in the variance of the variable, see ^[3]. The chi-square random variable is the sum of squares of a standard normal random variable. Due to the complexity of the systems, it may not possible to note the real data. In such cases, there is a need to generate the simulated data that can be applied for estimation and forecasting. The analysis of the simulated data is very close to the real data phenomena. As mentioned by ^[4]. "The simulation depends on the application of the study on systems similar to the real systems, and then projecting these results if they are appropriate on the real system. The simulation based on generating a series of random numbers that are subject to a uniform probability distribution". In addition, ^[5] suggested generating random variables from the underlying statistical distributions. The random numbers are generated using algorithms that are based on statistical distributions. Monahan ^[6] worked on generating chi-square random numbers. Shmerling ^[7] used the rational probability function in generating the random variables. Ortigosa et al. ^[8] presented the algorithms for the modified chi-square distribution. Devroye ^[9] proposed the simple algorithm for many distributions. Devroye ^[10] discussed the methods to generate non-uniform random variates. Devroye ^[11] presented the algorithm for generalized inverse Gaussian distribution. Luengo ^[12] worked on Pseudo random variate from the gamma distribution. Yao and Taimre ^[13] proposed the method to generate mixed random variables. More algorithms can be seen in ^[14]. Pereira ^[15] presented the simple method to generate a Pseudo random variate.

Smarandache ^[16] introduced descriptive neutrosophic statistics to deal with the data having imprecise observations. Neutrosophic statistics is found to be more efficient than classical statistics in terms of information obtained from the analysis of imprecise data. The results obtained from the neutrosophic statistical analysis reduce to the results of classical statistics when no imprecise observation is found in the data. Neutrosophic statistics offers greater information richness compared to classical statistics by providing an additional measure known as the degree of indeterminacy. Smarandache ^[17] demonstrated the superior efficiency of neutrosophic statistics over interval statistics. Chen et al. ^[18] and ^[19] provided the methodology to analyze neutrosophic numbers in engineering. Aslam ^[20] provided the algorithm for neutrosophic DUS-Weibull distribution. Smarandache ^[21] showed that neutrosophic statistics is more efficient than interval statistics. Alhabib et al. ^[22] worked on some statistical distribution under neutrosophic statistics. Khan et al. ^[23] worked on the gamma distribution using neutrosophic statistics. Sherwani et al. ^[24] presented spine test using neutrosophic normal distribution. Granados ^[25] and Granados et al. ^[26] proposed several discrete and continuous distributions using the idea of neutrosophy. Various algorithms within neutrosophic statistics have been introduced in the literature. Guo and Sengur ^[27] introduced an algorithm for neutrosophic c-means clustering. Garg ^[28] proposed an algorithm incorporating clustering techniques along with a novel distance measure. Aslam ^[29] introduced algorithms utilizing sine-cosine and convolution methods within neutrosophic statistics. Aslam ^[30] presented an algorithm for generating imprecise data from the Weibull distribution. Aslam and Alamri ^[31] introduced an algorithm employing the accept-reject method to generate neutrosophic data.

The existing methods for generating chi-square random variates are limited to deterministic environments, rendering them unsuitable for complex scenarios or uncertainty simulations. A thorough review of the literature indicates a dearth of algorithms for generating chi-square variates using neutrosophic statistics. To address this gap, this paper will introduce the chi-square distribution within the framework of neutrosophic statistics. Additionally, algorithms for generating chi-square data under neutrosophic statistics will be presented. Simulation methods will be provided for scenarios with both small and large degrees of freedom, generating neutrosophic chi-square random variates across varying degrees of indeterminacy/uncertainty. Furthermore, the application of the generated data will be discussed. It is anticipated that the degree of uncertainty will significantly influence the computation of neutrosophic chi-square variates. The proposed neutrosophic chi-square variate is expected to find application in various fields where obtaining original data is impractical or prohibitively expensive.

2.   Methods

In this section, we will introduce normal distribution, standard normal distribution and chi-square distribution under neutrosophic statistics.

2.1.   Neutrosophic normal distribution

Let ${x}_{1N, }{x}_{2N, }{x}_{3N, }\dots, {x}_{nN}$ be a neutrosophic normal variable of size $n$. Let ${x}_{N} = {x}_{L}+{x}_{L}{I}_{{x}_{N}}; {I}_{{x}_{N}}ϵ\left[{I}_{{x}_{L}}, {I}_{{x}_{U}}\right]$ be a neutrosophic form of standard normal variate. Note that ${x}_{L}$ presents the determinate part (classical statistics) with mean $\mu$ and variance ${\sigma }^{2}$, ${x}_{L}{I}_{{x}_{N}}$ presents the indeterminate part, and ${I}_{{x}_{N}}ϵ\left[{I}_{{x}_{L}}, {I}_{{x}_{U}}\right]$ is the measure of indeterminacy. The expected value of the neutrosophic random variable is given by

The variance of neutrosophic random variable is given by

Note that ${I}_{{x}_{N}}^{2} = {I}_{{x}_{N}}$.

The neutrosophic probability distribution function (npdf) of the normal distribution is given by

where ${\mu }_{N} = \mu \left(1+{I}_{{x}_{N}}\right)$ and ${\sigma }_{N} = \sqrt{{\left(1+{I}_{{x}_{N}}\right)}^{2}{\sigma }^{2}}$.

2.2.   Neutrosophic standard normal distribution

Suppose that ${z}_{1N, }{z}_{2N, }{z}_{3N, }\dots, {z}_{kN}$ be neutrosophic standard normal variable. Let ${z}_{iN} = {z}_{iL}+{z}_{iL}{I}_{{z}_{N}}; {I}_{{z}_{N}}ϵ\left[{I}_{{z}_{L}}, {I}_{{z}_{U}}\right]$ $\left(1 = \mathrm{1, 2}, .., k\right)$ be a neutrosophic form of a standard normal variate. Note that ${z}_{iL}$ presents the determinate part (classical statistics), ${z}_{iL}{I}_{{z}_{N}}$ presents the indeterminate part, and ${I}_{{z}_{N}}ϵ\left[{I}_{{z}_{L}}, {I}_{{z}_{U}}\right]$ is the measure of indeterminacy.

When$L = U$, the neutrosophic standard normal variable can be expressed as

The neutrosophic mean of ${z}_{iN}$ is given by

The neutrosophic variance of ${z}_{iN}$ is given by

Based on this information, the neutrosophic probability density function (npdf) of standard normal distribution is given by

2.3.   Neutrosophic chi-square distribution

In the area of classical statistics, the chi-square distribution is denoted by ${\chi }^{2}$ with degree of freedom$k$. Let ${\chi }_{N}^{2}$ denotes the chi-square distribution for neutrosophic statistics with ${k}_{N}$ a degree of freedom. As mentioned before, ${z}_{1N, }{z}_{2N, }{z}_{3N, }\dots, {z}_{kN}$ be neutrosophic standard normal variable, then ${Q}_{N} = \sum _{i = 1}^{kN}{z}_{iN}^{2}$ is distributed as a neutrosophic chi-square distribution ${k}_{N}$ degree of freedom. When $L = U$, ${Q}_{N} = {\left(1+{I}_{{z}_{N}}\right)}^{2}\sum _{i = 1}^{kN}{z}_{iL}^{2}$. We will denote it as ${Q}_{N}~{\chi }_{kN}^{2}$. The neutrosophic pdf of a chi-square distribution is given by

The mean of ${Q}_{N}$ with ${k}_{N}$ degree of freedom is given by

The $E\left({Q}_{N}^{2}\right)$ will be computed as

The variance of ${Q}_{N}$ with ${k}_{N}$ degree of freedom is given by

It is important to note that these distributions represent a generalization of those found in classical statistics. They revert to classical distributions in the absence of imprecise or uncertain values in the data. The proposed distributions operate on the premise that data is acquired within an uncertain environment, allowing for their utilization in scenarios where uncertainty is present during data recording.

3.   Generating neutrosophic chi-square variate $\left({\mathit{k}}_{\mathit{N}} < 30\right)$

In this section, we will present the routine and algorithm to generate neutrosophic chi-square variate when ${k}_{N}$ is less than 30. The neutrosophic chi-square variate having ${k}_{N}$ a degree of freedom will be generated by squaring and adding neutrosophic standard normal variables. The routine is explained as follows:

Step 1: fix the value of ${k}_{N}$.

Step 2: Generate ${k}_{N}$ standard normal variable ${z}_{iN}$; for $i = 1$ to ${k}_{N}$.

Step 3: Fix the values of ${I}_{N}$.

Step 4: Compute the values of ${Q}_{N} = {\left(1+{I}_{{z}_{N}}\right)}^{2}\sum _{i = 1}^{kN}{z}_{iN}^{2}$ random variate.

Step 5: Next $i$.

Step 6: Return ${Q}_{N}$.

The algorithm to generate neutrosophic chi-square random variate is also shown with the help of Figure 1.

By following the algorithm, the neutrosophic chi-square random variate for various values of ${k}_{N}$ and ${I}_{N}$ is presented in Tables 1–2. Table 1 presents the values of a neutrosophic chi-square random variate when ${k}_{N}$ = 3. Table 2 presents the values of a neutrosophic chi-square random variate when ${k}_{N}$ = 4. From Tables 1–2, it can be noted that as the measure of indeterminacy ${I}_{N}$ increases, the values of neutrosophic chi-square random variate also increase. For example, from Table 1, when ${I}_{N}$ = 0.10, the neutrosophic chi-square random variate is 3.4578 and when ${I}_{N}$ = 0.80, the neutrosophic chi-square random variate is 9.2590. It is also interesting to note that when the values of ${k}_{N}$ increases, we note the increasing trend in neutrosophic chi-square random variate. For example, when ${k}_{N}$ = 3 and ${I}_{N}$ = 0.20, neutrosophic chi-square random variate is 4.1151, and when ${k}_{N}$ = 4 and ${I}_{N}$ = 0.20, neutrosophic chi-square random variate is 8.8635.

4.   Generating neutrosophic chi-square variate $\left({\mathit{k}}_{\mathit{N}}\ge 30\right)$

In this section, we will discuss the routine and algorithm to generate neutrosophic chi-square random variate when ${k}_{N}$ is larger than 30. The neutrosophic chi-square random variate with ${k}_{N}$ degree of freedom will be generated with the help of approximation. When ${k}_{N}\ge 30$, due to the central limit theorem, the neutrosophic ${\chi }_{kN}^{2}$ distribution is shaped like the neutrosophic normal distribution that is ${\chi }_{kN}^{2}~N\left({k}_{N}, 2{k}_{N}\right)$, see ^[3]. An approximation to $\alpha$-percent ${\chi }_{kN}^{2}$ value is given by

where ${z}_{{\alpha }_{N}}$ is neutrosophic standard normal variables with $P\left({z}_{N} > {z}_{{\alpha }_{N}}\right) = {\alpha }_{N}$. The approximation formulas used in practice is given by

Based on the given information, the routine is stated as follows:

Step 1: fix the value of ${k}_{N}$.

Step 2: Generate ${k}_{N}$ standard normal variable ${z}_{iN}$; for $i = 1$ to ${k}_{N}$.

Step 3: Fix the values of ${I}_{N}$.

Step 4: Compute the values ${\chi }_{kN}^{2} = int\left({k}_{N}+{z}_{{\alpha }_{N}}\sqrt{2{k}_{N}}+0.5\right)$ random variate using ${z}_{iN}$ obtained in Step 3.

Step 5: Next $i$.

Step 6: Return ${\chi }_{kN}^{2}$.

The algorithm to generate neutrosophic chi-square random variate is also shown with the help of Figure 2.

By following the algorithm, the neutrosophic chi-square random variate for various values of ${k}_{N}$ and ${I}_{N}$ is presented in Tables 3–5. Table 3 presents the values of neutrosophic chi-square random variate when ${k}_{N}$ = 35. Table 2 presents the values of neutrosophic chi-square random variate when ${k}_{N}$ = 40. Table 3 presents the values of neutrosophic chi-square random variate when ${k}_{N}$ = 239. From Tables 3–5, it can be noted that as the measure of indeterminacy ${I}_{N}$ increases, the values of a neutrosophic chi-square random variate also increase. For example, when ${I}_{N}$ = 0.10, from Table 3, the neutrosophic chi-square random variate is 39.427and when ${I}_{N}$ = 0.80, the neutrosophic chi-square random variate is 41.925. It is also interesting to note that when the values of ${k}_{N}$ increases, we note the increasing trend in neutrosophic chi-square random variate. For example, when ${k}_{N}$ = 35 and ${I}_{N}$ = 0.20, neutrosophic chi-square random variate is 39.784, and when ${k}_{N}$ = 239 and ${I}_{N}$ = 0.20, the neutrosophic chi-square random variate is 250.694.

5.   Comparative study

The effect of the degree of uncertainty/indeterminacy on the chi-square variate will be discussed now. The chi-square variates under classical statistics are given in Tables 1–5. The chi-square variates under classical statistics when ${k}_{N} < 30$ are reported in Tables 1 and 2. The chi-square variates under classical statistics when ${k}_{N}\ge 30$ are reported in Tables 3–5. From Tables 1–5, we note that the values of chi-square variates are higher for the classical statistics. In general, there is an increasing trend in neutrosophic chi-square variates. For example, when ${k}_{N}$ = 35 and ${I}_{N}$ = 0, the chi-square variate under classical statistics provides the value that is 39.070. On the other hand, ${k}_{N}$ = 35, and ${I}_{N}$ = 0.10, the neutrosophic chi-square variate is 39.427. The trends in chi-square variates under classical statistics and neutrosophic statistics are given in Figures 3–6. Figures 3 and 4 show the curves of chi-square variates when ${k}_{N} < 30$. From Figures 3 and 4, it can be observed that the curve of chi-square variates under classical statistics is lower than the neutrosophic chi-square variates at various values of ${I}_{N}$. Figures 5 and 6 present the curves of chi-square variates when ${k}_{N}\ge 30$. From Figures 5 and 6, it can be observed that the curve of chi-square variates under classical statistics is lower than the neutrosophic chi-square variates at various values of ${I}_{N}$. In addition, it can be noted that when the values of ${k}_{N}$ is larger than 30, the neutrosophic chi-square variates are close to ${k}_{N}$. This statistical analysis highlights significant disparities between the data generated from the chi-square distribution under uncertainty and that obtained from the chi-square distribution under classical statistics. It is evident that the degree of indeterminacy/uncertainty significantly influences data generation. Consequently, based on this study, it is concluded that decision-makers should exercise caution when employing existing algorithms rooted in classical statistics for generating chi-square data. The utilization of such algorithms in uncertain contexts may lead to misleading outcomes in decision-making processes.

6.   Application using big data in transportation

In this section, we will discuss the application of the neutrosophic chi-square test for big data of transportation. ^[32] used the uncertainty based chi-square for the big data. According to ^[33], uncertainty is always presented in big data, therefore, it is always expected uncertainty or impression in transportation data. According to (https://www.mongodb.com/big-data-explained/examples) "Airplanes generate enormous volumes of data, on the order of 1,000 gigabytes for transatlantic flights. Aviation analytics systems ingest all of this to analyze fuel efficiency, passenger and cargo weights, and weather conditions, with a view toward optimizing safety and energy consumption". Suppose that there is uncertainty in this transportation data with a degree of uncertainty that is 0.10. Let the degree of freedom is 239. Let us define the null hypothesis ${H}_{0}:$ Flight operation is efficient vs. the alternative hypothesis ${H}_{1}:$ Flight operation is not efficient. From Table 5, the value of ${\chi }_{kN}^{2}$ is 249.761. Suppose that level of significance$\alpha = 0.10$, the tabulated value of the chi-square test is 267.412. By comparing ${\chi }_{kN}^{2}$ with the tabulated value, we will not reject the null hypothesis that the flight is efficient. Based on the analysis, it can be concluded that the flight operation is efficient. For the same level of significance$\alpha = 0.10$, the chi-square value under classical statistics is ${\chi }_{k}^{2}$ = 248.828. By comparing ${\chi }_{k}^{2}$ = 249.761 with the tabulated value, again, we do not reject the null hypothesis. But, from this comparison, it can be seen that the ${\chi }_{k}^{2}$ = 249.761 is close to 267.412 as compared to ${\chi }_{kN}^{2}$ = 248.828. The study suggests that the proposed test can effectively be applied to test hypotheses concerning flight efficiency within the transportation sector.

7.   Concluding remarks

The algorithms to generate chi-square random numbers were presented in this paper. The methods to generate random variables were presented when the degree of freedom is small and large. The results were presented by implementing the proposed algorithms. The results showed that the measure of indeterminacy plays an important role in determining chi-square random numbers. From the tables, it was concluded that as the measure of indeterminacy increases, the values of chi-square random numbers also increase. This random number generator can be used to generate chi-square random numbers under uncertainty. The proposed method holds applicability for generating chi-square random numbers across diverse domains such as medical science, engineering, quality control, and reliability analysis. Additionally, it can facilitate the application of goodness-of-fit tests and tests of variance homogeneity across a variety of fields. The proposed study has a limitation in that the data generated from the proposed algorithm is exclusively applicable within uncertain environments. Moreover, the proposed neutrosophic chi-square distribution is suitable solely for modeling imprecise data. Future research could explore modifications to the simulation method, incorporating alternative statistical distributions or sampling schemes. Furthermore, the potential application of the proposed test in handling large datasets within metrology and healthcare warrants investigation in future studies. Additionally, there is scope for further research into the proposed algorithm utilizing the accept-reject method.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality and presentation of the paper.

Conflict of interest

The authors declare no conflicts of interest.