On detecting outliers in complex data using Dixon’s test under neutrosophic statistics

1 Introduction

The estimation and forecasting are done using the data collected from the respective fields of interest. The statistical techniques/methods are applied for the data analysis. The collection of data from the fields is not an easy task. A lot of money, time and effort are needed to collect data from the fields of interest. For example, the collection of the ocean data is not an easy task with the limited resources, see for example, (Jingang et al., 2017). The presence of the potential outliers in the data may mislead the policymakers. In the statistics, an observation that is away from the other observations or has many variations in the sample is known as the outlier. For an accurate analysis of the data, the extreme values should be identified and eliminated from the sample. The elimination of the extreme values of the data is an important field of data mining. According to (Manoj and Senthamarai Kannan, 2013), “The identification of outliers can lead to unexpected knowledge discovery in areas such as credit card fraud detection, criminal behaviors detection, computer intrusion detection, calling card fraud detection, etc. Applications such as outlier detection customized marketing, network intrusion detection, weather prediction; pharmaceutical research and exploration in science databases require the detection of outliers”. (Dixon, 1950; Dixon, 1951) proposed the test for the identification of the extreme value in the data. Several authors paid their attention to this issue and presented valuable work in the detection of the outlier using the Dixon’s test, see for example, (Verma and Ruiz, 2006; Böhrer, 2008; Manoj and Senthamarai Kannan, 2013; Jabbari Nooghabi, 2017; Jingang et al., 2017; Jabbari Nooghabi, 2019).

In practice, for the variable data, it may not always possible to have the exact observations and attribute data may have the categories. In these situations, data analysis may be overestimated or underestimated. The fuzzy logic, which is the alternative to be applied for the analysis of fuzzy data. Therefore, data mining using fuzzy logic helps to eliminate the outlier from the imprecise data. (Rajeswari et al, 2014) discussed the fuzzy-based approach method to detect the outlier in educational data. More details about the detection of outlier in fuzzy data can be seen in (Hung and Yang, 2006; Gładysz and Kuchta, 2007; Cateni et al., 2007; Yousri et al., 2007; Top et al., 2009; Ramli et al., 2010; D'Errico and Murru, 2012; Kim et al., 2014; Duraj et al., 2019).

The neutrosophic logic which is the generalization of the fuzzy logic is introduced by (Smarandache, 1998). The neutrosophic deals with the measure of the indeterminacy. The neutrosophic statistics, which is the extension of classical statistics, used when there is Neutrosophy in sample or data. The neutrosophic statistics is applied when the observations in the sample or the population are indeterminate, vogues and imprecise. The neutrosophic statistics is quite effective and adequate to be applied in the uncertainty system, see for example, (Smarandache, 2014; Aslam, 2018b). Chen et al. (2017a) and Chen et al. (2017b) presented the application of the neutrosophic numbers in the area of rock engineering. Aslam (2018b) and Aslam (2019) introduced neutrosophic statistical quality control (NSQC).

As we mentioned earlier, the presence of the outlier can affect the forecasting and estimation of the data. Several authors contributed much work for the detection of the outlier using Dixon’s test under classical statistics and using fuzzy logic. By exploring the literature and according to the best of our knowledge, there is no work on Dixon’s test under the neutrosophic statistics. In this paper, we will originally propose Dixon’s test under the neutrosophic statistics for the identification of the outlier in the data or sample. We will develop Dixon’s test under the neutrosophic statistics and explain with the help of an example from the Samsung Company. We expect that the proposed Dixon’s test under the neutrosophic statistics will be more effective and adequate to be applied under an uncertainty environment.

2 Preliminaries

Suppose that $X_{\mathit{nN}}=X_n+u{I}_N$; $X_{\mathit{nN}}∊\left[X_{\mathit{nL}},X_{\mathit{nU}}\right]$ be a neutrosophic random variable having the determinate part $X_n$ and indeterminate part $u{I}_N$; $I_N∊\left[inf,sup\right]$. Note here that the neutrosophic random variable becomes the random variable under classical statistics when $I_N∊\left[0,sup\right]$. We also assume that $X_{\mathit{nN}}∊\left[X_{\mathit{nL}},X_{\mathit{nU}}\right]$ is generated from the neutrosophic normal distribution with mean ${\mu }_N=\mu +{\mathit{uI}}_N$;${\mu }_N∊\left[{\mu }_L,{\mu }_U\right]$ and neutrosophic standard deviation ${\sigma }_N=\sigma +{\mathit{uI}}_N$;${\sigma }_N∊\left[{\sigma }_L,{\sigma }_U\right]$. The neutrosophic mean and standard deviation reduce to mean under classical statistics when $X_{\mathit{nL}}=X_{\mathit{nU}}$. More details about the neutrosophic distributions can be seen in Smarandache (2010) and Aslam (2018a).

3 Dixon’s test under NSIM

Let $X_{\mathit{nN}}=X_n+u{I}_N$ be neutrosophic random variable consists of a determined part $X_n$ and indeterminate part $u{I}_N$ and $I_N∊\left[I_L,I_U\right]$ is an indeterminacy interval. We assumed that $X_{\mathit{nN}}∊\left[X_{\mathit{nL}},X_{\mathit{nU}}\right]$ is selected from the neutrosophic normal distribution. The purpose of the proposed Dixon’s test under the neutrosophic statistics is called (NDT) is to detect either the extreme value belongs to the group of observations or not. Like the existing Dixon’s test under CS, we have the following assumptions for the proposed NDT

4 Example

In this section, we discuss the application of the proposed Dixon’s test statistic in the area of neutrosophic statistical quality control (NSQC). The Samsung Company is manufacturing several electronic equipment and mobile phones. This company is very popular in Saudi Arabia due to the high reliability of their product. The Samsung Company is interested to apply the proposed Dixon’s test statistic on cell phone batteries failure time. For the inspection of the batteries product, the experimenters at the company have selected a sample at regular interval during the manufacturing process. Due to the complexity of the lifetime measuring process, on several processes, the measurements are not accurate, precise and sometimes in an interval. Therefore, there is Neutrosophy in the failure time of the batteries product. Therefore, for the testing of the outlier in the failure time, the existing Dixon’s test statistic under classical statistics cannot be applied. The failure time of 12 batteries having neutrosophic observations is shown in Table 1. The company is expected to have failed time greater than 1000 h. From the data, they note that one failure time is from 973 to 975. The company is interested to test it either it is an outlier or not in the sample. Therefore, we apply the proposed Dixon’s test statistic to test the null hypothesis that $X_{1N}∊\left[973,975\right]$ is an outlier Vs the alternative hypothesis that it is not an outlier and belong to the same group. There are 12 observations in the sample, therefore, we will use $R_N^{\left(n_N\right)}∊\left[R_L^{\left(n_N\right)},R_U^{\left(n_N\right)}\right]$ based on $R_{21N}∊\left[R_{21L},R_{21U}\right]$. The null hypothesis will be rejected if the value of $R_N^{\left(n_N\right)}∊\left[R_L^{\left(n_N\right)},R_U^{\left(n_N\right)}\right]$ based on $R_{21N}∊\left[R_{21L},R_{21U}\right]$ will be greater than critical. Some necessary calculations for this data are shown as follows.

Step-1: Use the neutrosophic Kolmogrov-Smirnov (NK-S) test to see either the available data follows the neutrosophic normal distribution or not. The neutrosophic P-value of $X_{\mathit{nN}}∊\left[X_{\mathit{nL}},X_{\mathit{nU}}\right]$ is [0.79, 0.80] which indicates that given lifetime data follows the neutrosophic normal distribution.

Step-2: arrange the batteries failure time $n_N$ ∊ [12, 12] in ascending order, say $[973,975]\le$[1056, 1056] $\le \cdots \le [1946,1950]$ as in Table 1. Let$X_{12N}∊[973,975]$, $X_{10N}∊[1207,1210]$ and $X_{2N}∊[1921,1925]$.

Step-3: Compute $R_N^{\left(n_N\right)}=\frac{\left(X_{{12}_N}-X_{10N}\right)}{\left(X_{{12}_N}-X_{2N}\right)}=\frac{\left[973,975\right]-[1207,1210]}{\left[973,975\right]-[1921,1925]}=[0.2468,0.2473]$.

Step-4: From (Kanji, 2006), the tabulated value is [0.546, 0.546] when $\alpha$ = 0.05 and $n_N$ ∊ [12, 12].

Step-5: The computed $R_N^{\left(n_N\right)}∊[0.2468,0.2473]$ is smaller than the critical value [0.546, 0.546]. Therefore, it is concluded that the suspected value in the sample is not an outlier.

5 Discussion

We presented the testing procedure of the proposed Dixon’s test in Section 4. We note from the calculations of the real data under the neutrosophic statistics; the proposed testing procedure provides the analysis of the data in the indeterminacy interval rather than the exact or determined values as in classical statistics. Chen et al. (2017a) and Chen et al. (2017b) suggested that a method which provides the values in the indeterminacy interval under uncertainty is said to be a more effective and adequate method. The neutrosophic form of the proposed test for the data is: $R_N^{\left(n_N\right)}=0.2468+0.2473I_N;I_N∊\left[0,0.0020\right]$. Note here that there is measure of indeterminacy 0.0020, which is associated with the proposed test under uncertainty. The analysis of the lifetime of the data suggested that the values of the proposed statistic can be from $0.2468$ and 0.2473. On the other hand, the existing testing procedure given by Dixon (1950) and Dixon (1951) which is the special case of the proposed testing procedure provide the exact value of the statistic which is 0.2468. Furthermore, the existing test does not provide the measure of indeterminacy associated with the test. From this comparison, it is clear that the proposed testing procedure is more adequate and effective to be used in uncertainty environment than the testing procedure given by Dixon (1950) and Dixon (1951) under classical statistics.

6 Concluding remarks

In this paper, we introduced a new Dixon’s test under the neutrosophic statistics. We presented the testing procedure for the proposed test using the neutrosophic statistical interval method. The proposed testing procedure for the detection of the outlier in the data is more adequate and effective than the testing procedure available in the literature. The proposed testing procedure is more information to be applied when observations in the sample or the data are imprecise and fuzzy. The comparison between the proposed testing procedure and existing testing procedure support claim and coincides with the theory of Chen et al. (2017a) and Chen et al. (2017b). The proposed Dixon’s test under the neutrosophic statistics is an effective alternative of the existing Dixon’s test under classical statistics. The proposed Dixon’s test statistic can be used to detect the outlier in the data for the better inferences from it. We will introduce the proposed Dixon’s test in the area of control chart theory as future research.