Wang et al. (2014) define Fishers discrimination ratio as a method used in dimension data for pattern reductions and differentiating between two sample points. The mean of two sample points is calculated and then squared. The value of the squared mean and the size of the samples within- class scatters then analyzed to find the line between the two sample points. A better line is formed when the mean squared bigger, and the within-class scatters smaller. FDR can further define using the following equation,) J F (w) = (m1 m 2 2 2 S 1 + S 2 w represents the direction of the expected line mi and Si represent the mean. ( i = 1, 2 ) represent the i th class of the within-class scatter (Wang et al., 2014). Cheng and Bingyu (2014) explained further using the following diagram how Fishers discrimination works.

The diagrams above are histograms representation of two class samples. The classes are represented by the red and blue colors. The green line between them represents the mean. The diagram on the right demonstrates the effectiveness of Fishers Linear Discriminant Analysis. FLDA is used to separate two feature classes to get a clear distinction between them. Cheng and Bingyu (2014) demonstrates how LDA uses mathematical equations to differentiate between two classes. The equation y = thT X is used to map X samples on an axis so that the value of y can be obtained. The ratio between-class variance to within-class variance is determined by modifying th.

Welling (2015) shows how to gather useful information from two classes. Classes have sample data scattered within and between the classes. It is very hard to derived information from such samples. The solution to this challenge is to employ LDA with a formula to reduce the overlap between classes. The following equation can be used to maximize the ratio between-lasses to within-classes.

J(w) = wT SBwwT SW w where SB is the between classes scatter matrix and SW is the within classes scatter matrix (Welling, 2015).

Wang et al. (2014) further demonstrates that the FDR can be modified to evaluate a feature. To evaluate the capability of a feature, W is replaced by its dimensions as shown by the following equation,

F (tk) = (E (t k | P) E (t k | N))2 (2) D (t k | P) + D (t k | N) where E (t k | P) and E (t k | N) are the conditional mean of the feature tk with respect to the categories P and N respectively, D (t k | P) and D (t k | N) are the conditional variances of the feature tk with respect to the categories P and N respectively (Wang et al., 2014).

The Kruskal-Wallis method is a nonparametric method analysis. Nonparametric means that it does not assume that the data come from a distribution completely described by two parameters, mean and standard deviation. Kruskal-Wallis method applies well where there are two types of variables, nominal and measurement. The two variable produce a better result. Kruskal-Wallis is used to determine Samples that are from the same population. Kruskal-Wallis method uses the several steps in analysis; the first step is combining the data from the groups. It then ranks the data to organize them in ascending order from highest to lowest. Data numbering depends on the numerical value assigned to them. Data are assigned a numeric value from the from the smallest to the biggest. The numbers are assigned in ascending order. The least data is assigned the smallest number and the highest Data assigned the largest number. The mean position in the list replaces data of the same score. H is then computed using the following equation. The degrees of freedom are K-1.

H=12N(N+1)R2ni-3(N+1) The Kruskal-Wallis analysis method is used when the data collected from ordinal nature. (Peterson et al. 2017). An example of a research that can be carried out and analyzed by Kruskal-Wallis method is when finding out the effect of coffee in driving. For this experiment, three substances are used, coffee water and decaf. The data was collected and analyzed for the different type of drink. It was reported that what the drivers drink has a direct effect on their performance. The result showed that drivers who drank coffee or decaf drove better than those drivers that took water. The drivers who drank coffee had no difference with those that drink decaffeinated beverages. The disadvantage of Kruskal-Wallis is that it will show no difference when samples from the same populations are used. The reason for not differentiating between samples from the same group is because it assumes that the different groups have the same distribution, and groups with different standard deviations have different distributions (Peterson et al. 2017).

Kruskal-Wallis method has a null hypothesis. The null hypothesis states that the groups have the same mean ranks. The normal mean level depends only on the total number of observations (for n observations, the expected mean position in each group is (n+1)/2). The advantage of KruskalWallis is that it does not assume that the data are normally distributed (Schwarz et al. 2016). If you're using it to test whether the medians are different, it does assume that the observations in each group come from populations with the same shape of distribution, so if the various groups have different ways (one is skewed to the right, and another is skewed to the left, for example, or they have different variances), the KruskalWallis test may give inaccurate results. KruskalWallis has proven to be very challenging when it comes to using graphs or pie chart to represent the analysis (Ponlawat et al. 2016). Usually, when one is plotting a graph, two variables are used for the X and Y axis. An equation is then generated that will relate the two variable together. At any point on the axis, the Y and X values can be calculated. KruskalWallis analysis uses nominal values. Plotting the observation on a y and x-axis can easily be made using the bar graph. However, the bar graphs will not be of the same pattern due to the nature of the variables. For better representation of KruskalWallis analysis, the use of tables replaces graphs. The tables are easier to populate with data and are easily interpreted (Ponlawat et al. 2016).

References

Ponlawat et al. (2016) Ponlawat Susceptibility of Adult Aedes aegypti and Anopheles dirus to BaitContaining Bacillus thuringiensis israelensis. IOSR Journal of Pharmacy 6.

Schwarz et al. (2016). Concatenative Sound Texture Synthesis Methods and Evaluation.

Peterson et al (2017). Drunk bugs: Chronic vapor alcohol exposure induces marked changes in the gut microbiome in mice. Behavioral brain.

Wang et al. (2014). A Feature Selection Method Based on Fishers Discriminant Ratio for Text Sentiment Classification. Science Press, Beijing 100717, China.

Cheng Li, Wang B. (2014) Fisher Linear Discriminant Analysis.

Welling M. (2014). Fisher Linear Discriminant Analysis

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