# Testing hypotheses based on fuzzy hypotheses and fuzzy data: A fuzzy p-value approach 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`. ```r 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` ## Returns 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 ```r 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))) } ```

fuzzy.p.value function