# Radar Circular Data Analysis Using a New Watson’s Goodness of Test under Complexity

Authors: Muhammad Aslam; Muhammad Saleem
Journal: Journal of Sensors
 (2021)

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## Abstract

Watson’s test is applied to test either the given angular data follows the specified distribution or not. The existing Watson’s test under classical statistics is applied when there is no uncertainty and indeterminacy in sample size or angular data. Under indeterminacy, the existing Watson’s test cannot be applied for testing purposes. Neutrosophic statistics is an alternative to classical statistics for this kind of situation. The Watson’s test under neutrosophic statistics is proposed in this paper. The test statistic of Watson’s test is introduced first. The operational procedure of the proposed Watson’s test is discussed with the help of radar data. From the data analysis and simulation study, it is concluded the proposed Watson’s test is efficient than the existing Watson’s test.


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## Body

## 1. Introduction

The data obtained from devices or systems follows any statistical distribution. For efficient prediction and estimation, the decision-makers are interested to investigate which statistical distribution is suitable for the data. [1] suggested, “when we assume that our data follows a specific distribution, we take a serious risk. If our assumption is wrong, then the results obtained may invalid.” Therefore, the goodness of fit tests is performed for testing either the data follows the specified theoretical distribution or not. These tests are performed using cumulative distribution function or probability density function of the theoretical distribution. [2–10] worked on the goodness of fit tests with applications using various types of data sets.Usually, the statistical tests are applied to the data obtained from the linear scale. In practice, the data obtained from metrology, space, radars, and ecology are circular observations that are measured in radian or degree. [11] mentioned, “Circular data need special treatment in data analysis: consider that an angle of 355° is much nearer to an angle of 5° than it is to an angle of 330°, and so simple arithmetic mean for example can be quite misleading.” The circular tests are applied for testing the randomness of circular data. [11–22] contributed in designing statistical tests for circular data.The presence of uncertainty and indeterminacy in the sample size, parameters, and the observations lead to apply the fuzzy-based tests for testing purpose. The fuzzy-based tests are quite flexible and widely applied under the uncertain environment in a variety of fields. [23, 24–29] presented various tests using the fuzzy approach.[30] introduced neutrosophic logic as a generalization of fuzzy logic. The efficiency of neutrosophic over fuzzy logic and interval-based analysis can be read in [31]. The applications of neutrosophic can be seen [32–36]. The classical statistics cannot be applied when vague, uncertain, and indeterminate observations are presented in the data. To overcome the issue, [37] used neutrosophic logic to introduce neutrosophic statistics. More information of neutrosophic statistics can be seen on the websites https://archive.org/details/neutrosophic-statistics?tab=about and https://archive.org/details/neutrosophic-statistics?tab=collection. The neutrosophic statistics is more informative than the classical statistics. [38–44] worked on neutrosophic statistics with applications. Recently, [45] introduced a neutrosophic test for circular data.The existing Watson’s test under classical statistics cannot be applied when uncertainty is presented. By exploring the literature and best of our knowledge, no work on Watson’s test under neutrosophic statistics is done so for. In this paper, we will introduce Watson’s test under neutrosophic statistics. The proposed Watson’s test is mainly aimed at testing either the given circular data follows the given theoretical distribution. The application of the proposed Watson’s test will be given with the help of data measured from radars. It is expected that the proposed Watson’s test will be efficient, flexible, and informative than the existing Watson’s test.


## 2. Neutrosophic Watson’sUNn2 Test

The existing Watson’sUn2 test under classical statistics is applied to test either the given random sample of angular values is fitted to the specified distribution. The existing Watson’s Un2 test is applied when all circular observations in the data are determined, exact, and uncertain. In the case when the circular observations in the data are uncertain, imprecise, and are in intervals, the existing Watson’s Un2 test under classical statistics cannot be applied for fitting the given distribution. In this section, neutrosophic Watson’s UNn2 test will be introduced to fit neutrosophic angular values to the given specified distribution. The methodology of the proposed Watson’s UNn2 test by following [14] is discussed as follows.Suppose thatΦ1N,Φ2N,Φ3N,⋯,ΦnN be a random sample of neutrosophic angular values of size nN. The neutrosophic forms of angular values and neutrosophic sample size are given as follows: ΦN=ΦL+ΦUINΦ;INΦϵILΦ,IUΦ and nN=nL+nUInN;InNϵInL,InU, respectively. Note that ΦL, nL and ΦUINΦ, nUInN denote the determined and indeterminate parts, respectively, and INΦϵILΦ,IUΦ and InNϵInL,InU are the associated measure of uncertainty. To implement the proposed test, the first step is to arrange neutrosophic angular data Φ1N,Φ2N,Φ3N,⋯,ΦnN in ascending order Φ1N≤Φ2N≤Φ3N≤⋯≤ΦnN with respect to midvalues of each interval of ΦNϵΦL,ΦU. Let FNΦN be a neutrosophic cumulative distribution function (ncdf) of the given theoretical distribution.Let(1)ViN=FNΦN,i=1,2,3,⋯,nN.The neutrosophic average ofV¯NϵV¯L,V¯U is computed as follows:(2)V¯N=∑i=1nNViNnN;nNϵnL,nU,V¯NϵV¯L,V¯U.The test statistic of the proposed Watson’sUNn2ϵULn2,UUn2 is given by(3)UNn2=∑i=1nNViN2−∑i=1nNCiNViNnN+nN13,3−V¯N−12,22;nNϵnL,nU,V¯NϵV¯L,V¯U,where CiN=2i−1.The statisticUNn2ϵULn2,UUn2 in neutrosophic form is given by(4)UNn2=ULn2+UUn2IUNn2;IUNn2ϵIULn2,IUUn2.In the given neutrosophic form, the statisticULn2 presents the test statistic under classical statistics. The value of statistic UUn2IUNn2 shows the indeterminate value under uncertainty and IUNn2ϵIULn2,IUUn2 is a measure of uncertainty associated with UNn2ϵULn2,UUn2. The proposed statistic reduces to ULn2 whenIULn2=0.


## 3. Application

The application of the proposed Watson’sUNn2 test is given with the aid of angle data obtained from the radar. The decision-makers are interested to investigate either the radar data obtained from the radar systems follow the given theoretical distribution or not. For testing the null hypothesis, H0: radar data follows the given theoretical distribution vs. H1: the radar data does not follow the given theoretical distribution. For testing this hypothesis, the decision-maker is uncertain about the sample size with the measure of uncertainty IU=0.13 with nL=13. The neutrosophic form of sample size is nN=13+15InN;InNϵ0,0.13. The following radar data is obtained from [45].(5)nL=13,Φ1=2500,Φ2=275°,Φ3=285°,Φ4=2850,Φ5=290°,Φ6=290°,Φ7=295°,Φ8=300°,Φ9=305°,Φ10=310°,Φ11=315°,Φ12=320°,Φ13=330°,nL=15,Φ1=2500,Φ2=275°,Φ3=285°,Φ4=2850,Φ5=290°,Φ6=290°,Φ7=295°,Φ8=300°,Φ9=305°,Φ10=310°,Φ11=315°,Φ12=320°,Φ13=330°,Φ14=330°,Φ15=5°.The proposed test for the given data is implemented as follows:(6)UNn2=∑i=1nNViN2−∑i=1nNCiNViNnN+nN13,3−V¯N−12,22=0.7221,0.7287.Step 1.
Arrange the angle data in ascending order and assign numberi.Step 2.
Compute the values ofVi=ΦN0/360 andV¯Nϵ0.8226,0.775.Step 3.
Generate the values ofViN2 and CiN. The values of ∑i=1nNViN2=8.83,9.67.Step 4.
Generate the values of∑i=1nNCiNViN/nN=11.09,12.81.Step 5.
Finally, compute the values of the proposed test statistic as follows.Letα=0.05, and the tabulated value from [14] is 0.184. By comparing the values of statistic UNn2 with the critical value, the null hypothesisH0: radar data follows the given theoretical distribution is rejected. Based on the study, it is concluded that angle data obtained from the radar does not follow the given theoretical distribution.


## 4. Comparative Study

In this section, the efficiency of the proposed Watson’sUNn2 test will be compared with the existing Watson’s Un2 test in terms of flexibility, the measure of uncertainty, and information. As mentioned before, the existing Watson’s Un2 test is a special case of the proposed Watson’s UNn2 test. The proposed Watson’s UNn2 test becomes the existing Watson’s Un2 test when IULn2=0. The neutrosophic form of the statistic UNn2ϵULn2,UUn2 for the given data is as follows: UNn2=0.7221+0.7287IUNn2;IUNn2ϵ0,0.0091. From the neutrosophic form, it can be seen that the statistic UNn2ϵULn2,UUn2 adopts the value in an indeterminate interval. Under uncertainty, the proposed statistic UNn2ϵULn2,UUn2 takes the value from 0.7221 to 0.7287. On the other hand, the existing Watson’s Un2 test takes only a single value. Therefore, under indeterminacy, the proposed Watson’s UNn2 test is flexible than the existing Watson’s Un2 test. In addition, the proposed Watson’s Un2 test gives information about the measure of uncertainty that the existing test cannot provide. For the radar data, the measure of uncertainty associated with Watson’s UNn2 test is 0.0091. For testing H0: radar data follows the given theoretical distribution vs. H1: the radar data does not follow the given theoretical distribution when α=0.05, the proposed Watson’s UNn2 test indicates that the probability of accepting H0 is 0.95, the probability of committing type-1 error is 0.05, and the probability of in-decision is 0.0091. From the study, it can be seen that the proposed Watson’s UNn2 test is informative than the existing Watson’s Un2 test.


## 5. Simulation Study

A simulation study is performed to see the effect of the indeterminacy parameters on the sample and the proposed Watson’sUNn2 test. The various values of the indeterminacy parameters are considered to see the behavior of sample size and test statistic UNn2. The values of nN and UNn2 for various values of InL=IULn2 are shown in Table 1. From Table 1, it can be seen that when InL=IULn2=0, the values of nNϵ13,13 and UNn2=0.7221,0.7221. It can be seen that the values of nN do not change up to InL=0.10. The indeterminate parts of nN and UNn2 increase as the values of InL=IULn2 increase 0 to 1.00. From this simulation study, it can be observed that the measure of indeterminacy plays a significant role in determined nN and UNn2; therefore, the sample size should be selected keeping in mind the measure of indeterminacy.Table 1
Effect of measure of indeterminacy onnN and UNn2.
InL=IULn2nNUNn20[13,13][0.7221,0.7221]0.001[13,13][0.7221,0.7228]0.005[13,13][0.7221,0.7257]0.010[13,13][0.7221,0.7294]0.10[13,15][0.7221,0.7950]0.20[13,16][0.7221,0.8678]0.30[13,18][0.7221,0.9407]0.40[13,19][0.7221,1.0136]0.50[13,21][0.7221,1.0865]0.60[13,22][0.7221,1.1593]0.70[13,24][0.7221,1.2322]0.80[13,25][0.7221,1.3051]0.90[13,27][0.7221,1.3779]1.00[13,28][0.7221,1.4508]


## 6. Concluding Remarks

The existing Watson’s test under classical statistics was applied when there is no uncertainty and indeterminacy in sample size or angular data. An extension of the existing Watson’s test was presented in the paper. The proposed test can be applied when uncertainty is presented in circular data. From the radar data analysis and simulation study, it is concluded that the proposed outperforms the existing Watson’s test. The proposed test has some limitations that it can be very practical if the software is available to perform it. The test can be applied to test neutrosophic random data. The proposed test for big angular data can be considered as future research.



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*Source: 7961306-2021-09-15.xml*