RETRACTED ARTICLE: Neutrosophic statistical test for counts in climatology

The existing F-test for two counts data from the Poisson distribution under classical statistics can be applied only when the counts in the data are exact or not intervals. The existing test cannot be applied when the count data is indeterminate, in the interval, and uncertain. In this paper, the F-test for two counts data from the Poisson distribution under classical statistics is designed. The test for two counts recording at the same time or different times is presented. The daily and the monthly number of records broken data in the U.S from the weather department is selected for the application of the proposed test. The application and comparison studies show the efficiency of the proposed test. The proposed test was found to be informative, flexible, and appropriate to be applied in an uncertain environment.


Methods
The existing F-test for two counts data from the Poisson distribution under classical statistics can be applied only when all counts in the data are determined, clear, and exact, see 1 . When the count data is the interval, the existing F-test for count data cannot be applied for testing the significance between two counted results. In this situation, the F-test for two count data under neutrosophic statistics can be applied. In this section, the methodology of the proposed F-test under indeterminacy will be presented. The main objective of the proposed F-test for the count data is to investigate the difference between two counted results having minimum and maximum counts in the data. The proposed F-test for the count data will be applicable under the assumptions that counts are from the Poisson distribution (rare events) and in addition, both samples of count data are recorded under uniform conditions. Let us assume that be neutrosophic forms of count data from the first and second populations, respectively. Note that N 1L and N 2L are the determined parts in neutrosophic forms and N 1U I 1N and N 2U I 2N are the indeterminate parts of neutrosophic forms. Note also that I 1N ǫ[I 1L , I 1U ] and I 2N ǫ[I 1L , I 2U ] are the measure of indeterminacy associated with counts in the first and second population, respectively. The information about the neutrosophic numbers can be seen in [43][44][45][46][47][48][49] . Suppose that µ 1N and µ 2N be the means of the first and second population, respectively. To test the null hypothesis that H 0N : The statistic F N1 ǫ[F L1 , F U1 ] follows the neutrosophic F-distribution with (2(N 2N + 1), 2N 1N ) degree of freedom, see Aslam 40 . It is worth noting that the statistic was given in Eq. (1) can be applied when two counts are recorded in the same period of time (t 1 = t 2 ) . The neutrosophic form of the proposed statistic F N1 ǫ[F L1 , F U1 ] can be expressed as The proposed statistic F N1 ǫ[F L1 , F U1 ] is a generalization of the existing F-test for two counts data. The statistic F L1 presents the existing F-test for two counts data. Note that F U1 I F N1 ; I F N1 ǫ I F L1 , I F U1 present the indeterminate part and I F N1 ǫ I F L1 , I F U1 is a measure of uncertainty associated with F N1 ǫ[F L1 , F U1 ] . The proposed statistic F N1 ǫ[F L1 , F U1 ] becomes the existing statistic when I F L1 = 0.
When the counts are noted over the different period's time t 1 and t 2 , the counting rates N 1N /t 1 and N 2N /t 2 are obtained. For this situation, the proposed statistic is a generalization of the existing F-test for two counts data. The statistic F L2 presents the existing F-test for two counts data. Note that F U2 I F N2 ; I F N2 ǫ I F L2 , I F U2 present the indeterminate part and I F N1 ǫ I F L1 , I F U1 is a measure of uncertainty associated with F N2 ǫ[F L2 , F U2 ] . The proposed statistic F N1 ǫ[F L1 , F U1 ] become the existing statistic when I F L2 = 0.

Application
Now, we will discuss the application of the proposed F-test for count data recorded from a subset of stations in the Global Historical Climatological Network. The weather data is selected from https:// www. ncdc. noaa. gov/ cdo-web/ datat ools/ recor ds on January 07, 2021. The U.S daily records broken are shown in Table 1 and U.S monthly records broken are shown in Table 2. From Tables 1-2, it can be seen that record counting is in intervals Step Step 2 Set the level of significance at α = 5% and select the critical value from F-table at α = 5% which is 1.
Step  Table 1 and Table 2 can be computed.
From the study, it is concluded that there is no statistical difference between the two counts of U.S daily records and U.S monthly records.

Comparative study
The proposed F-test for two counts data is reduced to F-test for two counts data under classical statistics when the counts are determined or not in intervals and no indeterminacy is recorded in counts. The comparison of the proposed test is given over the existing F-test for two counts data in terms of chance of uncertainty. The neutrosophic analyses of F N1 ǫ[F L1 , F U1 ] of both data sets along with the measures of indeterminacy are shown in Table 3. The neutrosophic forms consist of the statistic of the existing test and indeterminate part. Note that the symbols of statistic F N1 ǫ[F L1 , F U1 ] represent the corresponding number of days. For example, for the records in the last 30 days, the neutrosophic form is F N1 = 0.1923 − 0.1024I N130D ; I N130D [0, 0.87] , where the value of statistic 0.1923 presents the existing test when F L1 = 0 and 0.1024I N130D is an indeterminate part and the measure of uncertainty associated with F N1 ǫ[F L1 , F U1 ] is 0.87. It means that for the proposed test, the value of F N1 can be expected from 0.1923 to 0.1024. From the analysis, it can be seen that under uncertainty, the proposed test gives the values of statistic F N1 ǫ[F L1 , F U1 ] is a range rather than the exact value. Therefore, the proposed test is quite effective and flexible to apply in uncertainty. Similarly, other neutrosophic forms given in Table 3 can be interpreted. Based on the information, the proposed test can be interpreted as for α = 5%, the chance of accepting H 0N : µ 1N = µ 2N is 0.95, the chance of committing a type-I error (the probability of rejecting H 0N when it is true) is 0.05 and the chance of uncertainty about the acceptance of H 0N : µ 1N = µ 2N is 0.87. It is clear that for the real example, the chance of indeterminacy is high; therefore, the decision-makers should be careful in making the decision about the acceptance of H 0N : µ 1N = µ 2N . The proposed test under neutrosophic statistics is also a generalization of interval-based analysis. The interval analysis uses intervals instead of crisp numbers in order to approximate/capture the data inside the intervals. On the other hand, the neutrosophic statistics analysis uses set analysis (any type of set, not only intervals) in order to approximate/capture the data inside intervals. The results obtained from the proposed test can also be compared with the results obtained from the interval data analysis. From the data analysis, the value of F N1 from the interval analysis is 0.1923 to 0.1024. In addition, from the neutrosophic form F N1 = 0.1923 − 0.1024I N130D ; I N130D [0, 0.87] , it can be seen that F N1 is also better structured since we know that 0.1923 is the determinate part and 0.1024I N130D is the fluctuating part around 0.1923 . From the comparison, it can be seen that the interval-based analysis provides the results in an interval without the information about the measure of indeterminacy. From the study, it can be concluded that the proposed F-test is efficient than the existing F-test under classical statistics and interval-based analysis in terms of information and flexibility. Therefore, under indeterminacy, it is recommended to apply the proposed test for testing the daily recodes and monthly records data.

Concluding remarks
In this paper, the F-test for two counts data from the Poisson distribution under classical statistics was designed.
The tests for two counts time are the same or different was presented. The procedure to test two counts from the same or different times is equal or not was discussed. The application of the proposed was given using the weather records data. The application of the proposed test showed that the proposed test is flexible and informative to apply in uncertainty. In addition, the proposed test gives the results in indeterminate intervals. Based www.nature.com/scientificreports/ on the study, it is recommended to apply the proposed test when the counts are recorded in an indeterminate environment. The proposed test using double sampling can be considered as future research. The application of the proposed test for big data can be considered as future research.

Data availability
The data is given in the paper.