| fuzzy.p.value {FPV} | R Documentation |
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. (1) crisp (real) value, (2) triangular fuzzy number using function (3) trapezoidal fuzzy number using function (4) fuzzy number using function (5) power fuzzy number using function More details about these functions are presented in (Gagolewski and Caha, 2015). |
H0b |
the boundary of the null hypothesis. (1) crisp (real) value, (2) triangular fuzzy number using function (3) trapezoidal fuzzy number using function (4) fuzzy number using function (5) power fuzzy number using function |
sig |
the significance level of the test with defult (1) crisp (real) value, (2) triangular fuzzy number using function (3) trapezoidal fuzzy number using function (4) fuzzy number using function (5) power fuzzy number using function 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 |
knot.n |
the number of knots with defult |
fig |
a numeric argument which can tack only values 1, 2 or 3. If If If |
... |
additional arguments passed from |
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 |
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 |
Delta_SP |
returns a numeric value which is need for computing |
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 |
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 |
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)))
}