FuzzySTs: Fuzzy Statistical Tools: A Detailed Description

Abstract

We propose a R package called FuzzySTs, where numerous functions are implemented. This package aims to provide a complete framework of theoretical and applied fuzzy statistical tools as described in the previous chapters using a variety of calculation methods. It is completely self-sustaining and constitutes a coherent programming environment and user-friendly analytical procedures. From a technical point of view, we introduce gaussian and gaussian bell numbers to the class of fuzzy numbers. In terms of fuzzy arithmetic operations with respect to the extension principle, we propose a numerical procedure to compute the fuzzy difference and the fuzzy square, where the polynomial forms of their α-cuts are implemented. Note that our functions have the great advantage of being able to consider any shape of fuzzy numbers. We highlight that the functions described in this chapter are designed and developed from scratch in compliance with the needed concepts of the theoretical part. These functions have been validated on multiple data bases from different sources. For our current case, simple examples illustrate each proposed function.

Appendix

1.1 E Intersection Situations Between the Fuzzy Confidence Intervals and the Fuzzy Hypothesis

We express the possible situations of intersection between a given fuzzy confidence interval and its complements from one side and the fuzzy null hypothesis from another one. The different FCIs are calculated by the traditional expression. The proposed situations correspond to the one-sided and the two-sided tests. As such, Table 9.3 displays the cases of a left one-sided test, Table 9.4 displays the cases of a right one-sided test, and Table 9.5 displays the ones of a two-sided test. For all these cases, we note that the fuzzy null hypothesis $\tilde {\text{H}}_0$ (the red curve), its core θ ₀ (the green dotted line), the fuzzy confidence interval $\tilde {\varPi }$ (the blue curve), and its complement $\neg \tilde {\varPi }$ (the black curve) are proposed.

Table 9.3 The different situations of intersection between the fuzzy null hypothesis $\tilde {H}_0$ and the confidence intervals: case of left one-sided test

Table 9.4 The different situations of intersection between the fuzzy null hypothesis $\tilde {H}_0$ and the confidence intervals: case of right one-sided test

Table 9.5 The different situations of intersection between the fuzzy null hypothesis $\tilde {H}_0$ and the confidence intervals: case of two-sided test

	$\tilde {D}_0 = ( 0, \mu _{\tilde {\varPi }} (\theta _0) , \mu _{\tilde {\varPi }} (A_L) )$
	$\tilde {D}_1 = ( 0, \mu _{\neg \tilde {\pi }} (\theta _0) , \mu _{\neg \tilde {\pi }} (R_R))$
	$\tilde {D}_0 = ( 0 , \mu _{\tilde {\pi }} (\theta _0) , \mu _{\tilde {\pi }} (A_L) )$
	$\tilde {D}_1 = ( \mu _{\neg \tilde {\varPi }} (R_L) , \mu _{\neg \tilde {\varPi }} (\theta _0) , \mu _{\neg \tilde {\varPi }} (R_R) )$
	$\tilde {D}_0 = ( \mu _{\tilde {\pi }} (A_L) , \mu _{\tilde {\pi }} (\theta _0) , \mu _{\tilde {\pi }} (A_R) )$
	$\tilde {D}_1 = ( \mu _{\neg \tilde {\varPi }} (R_R) , \mu _{\neg \tilde {\varPi }} (\theta _0) , \mu _{\neg \tilde {\varPi }} (R_L) )$
	$\tilde {D}_0 = ( \mu _{\tilde {\pi }} (A_L) , \mu _{\tilde {\pi }} (\theta _0) , \mu _{\tilde {\pi }} (A_R) )$
	$\tilde {D}_1 = ( \mu _{\neg \tilde {\varPi }} (R_R) , \mu _{\neg \tilde {\varPi }} (\theta _0) , \mu _{\neg \tilde {\varPi }} (R_L) )$

	$\tilde {D}_0 = ( 1 , 1, 1 )$
	$\tilde {D}_1 = ( 0, 0, \mu _{\neg \tilde {\varPi }} (R_R) )$
	$\tilde {D}_0 = ( \mu _{\tilde {\pi }} (A_R) , \mu _{\tilde {\pi }} (\theta _0) , \mu _{\tilde {\pi }} (A_L) )$
	$\tilde {D}_1 = ( \mu _{\neg \tilde {\varPi }} (R_L) , \mu _{\neg \tilde {\varPi }} (\theta _0) , \mu _{\neg \tilde {\varPi }} (R_R) )$
Else	The fuzziness is high

1.2 F Possible Positions of the Areas A _l and A _r Under the MF of $\tilde {t}$ on the Left and Right Sides of the Median $\tilde {M}$

See Figs. 9.1 and 9.2.

Fig. 9.1

Positions (a) 1 and (b) 2 where A _l > A _r

Fig. 9.2

Positions (a) 3, (b) 4, and (c) 5 where A _l ≤ A _r