Package 'FPV'

Testing Hypotheses via Fuzzy P-Value in Fuzzy Environment

Description

Statistical testing hypotheses has an important rule for making decision in practical and applied problems. In traditional testing methods, data, parameters, hypotheses and other elements of problem are considered crisp. But in applied sciences such as economics, agriculture and social sciences, it may be confront with vague definitions and fuzzy concepts. In such situations, the classical methods can not solve the vague test and they need to generalize for using in fuzzy environments. The vagueness entrance in testing hypotheses problem can be done via data or/and hypotheses. Therefore, the following three major problems can be usually considered for a fuzzy environment: (1) testing crisp hypotheses based on fuzzy data, (2) testing fuzzy hypotheses based on crisp data, and (3) testing fuzzy hypotheses based on fuzzy data. Similar to the classical testing hypotheses, one can consider different procedure methods for solving the above mentioned problems such as Neyman-Pearson, Bayes, likelihood ratio, minimax and p-value. Computing Fuzzy p-Value package, i.e. Fuzzy.p.value package, is an open source (LGPL 3) package for R which investigate on the above three problems on the basis of fuzzy p-value approach. All formulas and given examples are match with (Parchami and Mashinchi, 2016) to easily show the performance of the proposed methods.

Author(s)

Abbas Parchami (Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran)

Maintainer: Abbas Parchami <[email protected]>

References

Filzmoser, P., and Viertl, R. (2004). Testing hypotheses with fuzzy data: the fuzzy p-value. Metrika 59: 21-29.

Gagolewski, M., Caha, J. (2015) FuzzyNumbers Package: Tools to deal with fuzzy numbers in R. R package version 0.4-1, https://cran.r-project.org/web/packages=FuzzyNumbers

Gagolewski, M., Caha, J. (2015) A guide to the FuzzyNumbers package for R (FuzzyNumbers version 0.4-1) http://FuzzyNumbers.rexamine.com

Holena, M. (2004). Fuzzy hypotheses testing in a framework of fuzzy logic. Fuzzy Sets and Systems 145: 229-252.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2010). Fuzzy p-value in testing fuzzy hypotheses with crisp data. Statistical Papers 51: 209-226.

Parchami, A., Taheri, S. M., and Mashinchi, M. (2012). Testing fuzzy hypotheses based on vague observations: a p-value approach. Statistical Papers 53: 469-484.

Wang, X., Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (II). Fuzzy Sets and Systems 118: 387-405.

Viertl, R. (2011) Statistical methods for fuzzy data. Wiley, Chichester.

Yuan, Y. (1991). Criteria for evaluating fuzzy ranking methods. Fuzzy Sets Syst 43: 139-157.

See Also

FuzzyNumbers FuzzyNumbers.Ext.2 Fuzzy.p.value

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Testing hypotheses based on fuzzy hypotheses and fuzzy data: A fuzzy p-value approach

Description

Function fuzzy.p.value can draw the membership function of the fuzzy p-value for the following three major problems which can be usually considered for the following tests in a fuzzy environment: (1) testing crisp hypotheses based on fuzzy data, (2) testing fuzzy hypotheses based on crisp data, and (3) testing fuzzy hypotheses based on fuzzy data. Also, one can consider a fuzzy significance of level for test by function fuzzy.p.value.

Usage

fuzzy.p.value(t, H0b, sig = 0.05, p.value, knot.n=10, fig=c("1", "2", "3"), ...)

Arguments

t	the observed value of the test statistic. t can take one of the following precise / non-precise values: (1) crisp (real) value, (2) triangular fuzzy number using function TriangularFuzzyNumber from package FuzzyNumbers, (3) trapezoidal fuzzy number using function TrapezoidalFuzzyNumber from package FuzzyNumbers, (4) fuzzy number using function FuzzyNumber from package FuzzyNumbers, and (5) power fuzzy number using function PowerFuzzyNumber from package FuzzyNumbers. More details about these functions are presented in (Gagolewski and Caha, 2015).
H0b	the boundary of the null hypothesis. H0b can take one of the following precise / non-precise values: (1) crisp (real) value, (2) triangular fuzzy number using function TriangularFuzzyNumber from package FuzzyNumbers, (3) trapezoidal fuzzy number using function TrapezoidalFuzzyNumber from package FuzzyNumbers, (4) fuzzy number using function FuzzyNumber from package FuzzyNumbers, and (5) power fuzzy number using function PowerFuzzyNumber from package FuzzyNumbers.
sig	the significance level of the test with defult sig = 0.05. sig can take one of the following precise / non-precise values: (1) crisp (real) value, (2) triangular fuzzy number using function TriangularFuzzyNumber from package FuzzyNumbers, (3) trapezoidal fuzzy number using function TrapezoidalFuzzyNumber from package FuzzyNumbers, (4) fuzzy number using function FuzzyNumber from package FuzzyNumbers, and (5) power fuzzy number using function PowerFuzzyNumber from package FuzzyNumbers. More details about these functions are presented in (Gagolewski and Caha, 2015).
p.value	the p-value of test in non-fuzzy environment which is a function from t and H0b
knot.n	the number of knots with defult knot.n = 10; see package FuzzyNumbers
fig	a numeric argument which can tack only values 1, 2 or 3. If fig = 1, then just the membership function of fuzzy p-value will be shown in figure. If fig = 2, then the membership functions of fuzzy p-value and fuzzy significance level will be shown in a figure. If fig = 3, then three membership functions of $t$, $H0b$ (inputted fuzzy numbers) and also the fuzzy p-value (outputted fuzzy number) are drawn in a figure.
...	additional arguments passed from plot

Value

This function returns some information about the fuzzy p-value and also plot a figure for it.

result	returns the result of the test, i.e. returns the accepted hypothesis and also the acceptance degree of the accepted hypothesis
cuts	returns the $\alpha$-cuts of the computed fuzzy p-value
core	returns the core of the computed fuzzy p-value
support	returns the support of the computed fuzzy p-value
Delta_PS	returns a numeric value which is need for computing $D(P>S)$ and $D(S>P)$. For more details, see $\Delta _{PS}$ from (Parchami et al., 2012)
Delta_SP	returns a numeric value which is need for computing $D(P>S)$ and $D(S>P)$. For more details, see $\Delta _{SP}$ from (Parchami et al., 2012)
Degree_P_biger_than_S	returns a real number between zero and one which show the degree of believe to the sentence "fuzzy p-value is bigger than fuzzy significance level". For more details, see $D(P>S)$ in (Parchami et al., 2012)
Degree_S_biger_than_P	returns a real number between zero and one which show the degree of believe to the sentence "fuzzy significance level is bigger than fuzzy p-value". For more details, see $D(P>S)$ in (Parchami et al., 2012)
accepted_hypothesis	returns the accepted hypothesis in the test. Returns "H0" iff the null hypothesis accepted, and returns "H1" iff the althernative hypothesis accepted
acceptance_degree	returns only the degree of acceptance for the accepted hypothesis in the test

Author(s)

Abbas Parchami

See Also

FuzzyNumbers FuzzyNumbers.Ext.2 Fuzzy.p.value

Examples

library(FuzzyNumbers)
library(FuzzyNumbers.Ext.2)

# Example 1: 
t <- FuzzyNumber(-0.5, .6, .8, 1,
  lower=function(alpha) qbeta(alpha,0.4,3),
  upper=function(alpha) (1-alpha)^4
  )
H0b = PowerFuzzyNumber(.5,1.2,1.5,1.6, p.left=1, p.right=0.5) 
p.value = function(t,H0b) pt((t-H0b)/sqrt(1/9), df=8)
fuzzy.p.value(t, H0b, sig=.05, p.value, knot.n=20,  fig=1, lty=4, lwd=4, col=6)  
fuzzy.p.value(t, H0b, sig=.05, p.value, knot.n=20, fig=2)$result
sig = TriangularFuzzyNumber(0, .03, .30)
fuzzy.p.value(t, H0b, sig, p.value, knot.n=20, fig=2)$cuts #Only print alpha-cuts of fuzzy p-value

sig = TrapezoidalFuzzyNumber(0, .05, .05, .20)
fuzzy.p.value(t, H0b, sig, p.value, knot.n=20, fig=3, col=2)$accepted

fuzzy.p.value(t, H0b, sig=0.05, p.value, knot.n=20, fig=3)


# Example 2: For working by some elements of  fuzzy p-value (continue of Example 1)
Fuzzy.p.value <- fuzzy.p.value(t, H0b, sig=.05, fig=1, p.value, knot.n=4)
class(Fuzzy.p.value)
print( Fuzzy.p.value )

Fuzzy.p.value$core  #Core of fuzzy p-value
Fuzzy.p.value$support  #Support of fuzzy p-value

# Upper bounds of fuzzy p-value
Fuzzy.p.value$cuts[,"U"] #Or equivalently,  Fuzzy.p.value$cuts[,2]



# Example 3: (Exam 4.4 from persian p-value paper)
knot.n = 10
t = TriangularFuzzyNumber(1315, 1327, 1342)
H0b = TriangularFuzzyNumber(1275, 1300, 1325)
sig = TriangularFuzzyNumber(0, .05, .1)
p.value = function(t,H0b) 1-pnorm((t-H0b)/(120/6))
fuzzy.p.value(t, H0b, sig, p.value, knot.n, fig=3)



# Example 4: (Exam 4.5 from persian p-value paper, where X~P(12*lambda) )
knot.n = 200
t = TriangularFuzzyNumber(24, 27, 30)
H0b = TriangularFuzzyNumber(2.75, 3.25, 3.25)
sig = TriangularFuzzyNumber(0, .05, .1)
p.value = function(t,H0b) ppois(t, 12*H0b)
fuzzy.p.value(t, H0b, sig, p.value=p.value, knot.n, fig=2, lwd=2)
# Repeat example with knot.n=10 to give a non-precise result 



# Example 5: A new example
t <- FuzzyNumber(1, 1.4, 1.8, 2,
  lower=function(alpha) pbeta(alpha,2,1),
  upper=function(alpha) 1-sqrt(alpha)
  )
H0b = TriangularFuzzyNumber(4,5,7) 
p.value = function(t,H0b) pt( (t-H0b)/sqrt(1/4), df=4)
fuzzy.p.value(t, H0b, sig=.1^3, p.value, knot.n=10, fig=3, col=2, lwd=2, xlim=c(0,.012)) 


# ---------- Examples of Springer fuzzy p-value paper ------------------

# Example 1 (Springer fuzzy p-value).
T1 = TriangularFuzzyNumber(1257,1261,1278) 
T2 = TriangularFuzzyNumber(1251,1287,1302) 
T3 = TriangularFuzzyNumber(1315,1346,1372)
T4 = TriangularFuzzyNumber(1306,1330,1348) 
T5 = TriangularFuzzyNumber(1298,1329,1349) 
T6 = TriangularFuzzyNumber(1288,1301,1320)
T7 = TriangularFuzzyNumber(1298,1317,1333) 
T8 = TriangularFuzzyNumber(1241,1269,1284) 
T9 = TriangularFuzzyNumber(1325,1353,1369)
T10= TriangularFuzzyNumber(1301,1337,1355)

t = 10^(-1)*(T1+T2+T3+T4+T5+T6+T7+T8+T9+T10)
t # T(1288,1313,1331)

plot(T1, add=FALSE, lwd=2, xlim=c(1230,1380))
plot(T2, add=TRUE, lwd=2)
plot(T3, add=TRUE, lwd=2)
plot(T4, add=TRUE, lwd=2)
plot(T5, add=TRUE, lwd=2)
plot(T6, add=TRUE, lwd=2)
plot(T7, add=TRUE, lwd=2)
plot(T8, add=TRUE, lwd=2)
plot(T9, add=TRUE, lwd=2)
plot(T10, add=TRUE, lwd=2)
plot(t, add=TRUE, col=2, lwd=4)

H0b = 1300
# T ~ N(1300,30^2/10)
p.value = function(t,H0b) pnorm( t, mean=1300, sd=30/sqrt(10), lower.tail=FALSE)
# Or equivalently
p.value = function(t,H0b) pnorm( (t-1300)/(30/sqrt(10)), lower.tail=FALSE)
sig = TriangularFuzzyNumber(0,0.05,0.1)
fuzzy.p.value(t, H0b, sig, p.value, knot.n=50, fig=2, lwd=2, xlim=c(0,1)) 


# Example 2. (continue of Example 1)
t = TriangularFuzzyNumber(1300,1313,1321)
p.value = function(t,H0b) 2 * pnorm( t, mean=1300, sd=30/sqrt(10), lower.tail=FALSE)
sig = TriangularFuzzyNumber(0,0.15,0.3)
fuzzy.p.value(t, H0b, sig, p.value, knot.n=50, fig=3, lwd=2) 



# Example 4 (Springer fuzzy p-value)  X ~ N(mu,sigma^2).
sigma =120
n = 36
x.bar <- 1327
H0b = TriangularFuzzyNumber(1275, 1300, 1325)
sig = TriangularFuzzyNumber(0, 0.15, 0.3)
p.value = function(x.bar,H0b) pnorm( x.bar, mean=H0b, sd=sigma/sqrt(n), lower.tail=FALSE)
fuzzy.p.value(x.bar, H0b, sig, p.value, knot.n=10, fig=2, lwd=2, xlim=c(0,1)) 

#Continue
sig1 = PowerFuzzyNumber(0, 0.15, 0.15, 0.3, p.left=2, p.right=2)
plot(sig1, xlim=c(0,.6))
sig2 = PowerFuzzyNumber(0, 0.15, 0.15, 0.3, p.left=.5, p.right=.5)
plot(sig2, col=2, add=TRUE)
fuzzy.p.value(x.bar, H0b, sig1, p.value, knot.n=10, fig=2, lwd=2, xlim=c(0,1)) 
fuzzy.p.value(x.bar, H0b, sig2, p.value, knot.n=10, fig=2, lwd=2, xlim=c(0,1)) 






## The function is currently defined as
function (t, H0b, sig = 0.05, p.value, knot.n = 10, fig = c("1", 
    "2", "3"), ...) 
{
    if (fig == 1) {
        P = f2apply(t, H0b, p.value, knot.n = knot.n, type = "l", 
            I.O.plot = FALSE, ...)
    }
    else {
        if (fig == 2) {
            P = f2apply(t, H0b, p.value, knot.n = knot.n, type = "l", 
                I.O.plot = FALSE, ...)
            if (class(sig) == "numeric") {
                abline(v = sig, col = 2, lty = 3)
            }
            else {
                plot(sig, col = 2, lty = 3, add = TRUE)
            }
            legend("topright", c("Fuzzy p-value", "Significance level"), 
                col = c(1, 2), text.col = 1, lwd = c(1, 1), lty = c(1, 3),
                bty = "n")
        }
        else {
            if (fig == 3) {
                P = f2apply(t, H0b, p.value, knot.n = knot.n, 
                  type = "l", I.O.plot = TRUE, ...)
                x = t
                y = H0b
            }
        }
    }
    if (class(sig) == "numeric") {
        sig <- TriangularFuzzyNumber(sig, sig, sig)
    }
    P_L = P$cuts[, "L"]
    P_L = P_L[length(P_L):1]
    P_U = P$cuts[, "U"]
    P_U = P_U[length(P_U):1]
    S_L = alphacut(sig, round(seq(0, 1, len = knot.n), 5))[, 
        "L"]
    S_U = alphacut(sig, round(seq(0, 1, len = knot.n), 5))[, 
        "U"]
    Int1 = (P_U - S_L) * (P_U > S_L)
    Int2 = (P_L - S_U) * (P_L > S_U)
    Arz = 1/(knot.n - 1)
    Integral1 <- (sum(Int1) - Int1[1]/2 - Int1[length(Int1)]/2) * 
        Arz
    Integral2 <- (sum(Int2) - Int2[1]/2 - Int2[length(Int2)]/2) * 
        Arz
    Delta_PS = Integral1 + Integral2
    Int3 = (S_U - P_L) * (S_U > P_L)
    Int4 = (S_L - P_U) * (S_L > P_U)
    Integral3 <- (sum(Int3) - Int3[1]/2 - Int3[length(Int3)]/2) * 
        Arz
    Integral4 <- (sum(Int4) - Int4[1]/2 - Int4[length(Int4)]/2) * 
        Arz
    Delta_SP = Integral3 + Integral4
    Degree_P_biger_than_S = Delta_PS/(Delta_PS + Delta_SP)
    Degree_S_biger_than_P = 1 - Degree_P_biger_than_S
    if (Degree_P_biger_than_S >= Degree_S_biger_than_P) {
        result = noquote(paste("The null hypothesis (H0) is accepted with degree D(P>S)=", 
            round(Degree_P_biger_than_S, 4), ", at  the considered significance level."))
        accepted_hypothesis = noquote("H0")
        acceptance_degree = Degree_P_biger_than_S
    }
    else {
        if (Degree_P_biger_than_S < Degree_S_biger_than_P) {
        result = noquote(paste("The althernative hypothesis (H1) is accepted with degree D(S>P)=",
                round(Degree_S_biger_than_P, 4), ", at  the considered significance level."))
            accepted_hypothesis = noquote("H1")
            acceptance_degree = Degree_S_biger_than_P
        }
        else {
            print(noquote(paste0("Impossible case")))
        }
    }
    return(list(result = result, cuts = P$cuts, core = P$core, 
        support = P$support, Delta_PS = as.numeric(Delta_PS), 
        Delta_SP = as.numeric(Delta_SP), Degree_P_biger_than_S = as.numeric(Degree_P_biger_than_S),
        Degree_S_biger_than_P = as.numeric(Degree_S_biger_than_P), 
     accepted_hypothesis = accepted_hypothesis, acceptance_degree = as.numeric(acceptance_degree)))
  }

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