Linearna regresija. 7. prosinca 2012.

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Justyna Wojciechowska
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1 Linearna regresija 7. prosinca > setwd("/home/marina/statisticki praktikum/vjezbe9") > forbes = read.table("forbes.dat") > hooker = read.table("hooker.dat") > forbes V1 V > hooker V1 V

2 > x = c(forbes[, 2], hooker[, 2]) > y = c(forbes[, 1], hooker[, 1]) > x [1] [11] [21] [31] [41] > y [1] [13] [25] [37] > n = length(x) > n [1] 48 kvadratični model za Forbesove i Hookerove podatke zajedno > X = cbind(rep(1, n), x, x^2) > thetakapa = solve(t(x) %*% X) %*% t(x) %*% y > thetakapa x

3 > ykapa = function(x) rbind(c(1, x, x^2)) %*% thetakapa > ykapa = function(x) thetakapa[1] + thetakapa[2] * x + thetakapa[3] * + x^2 > ykapa(x) [1] [9] [17] [25] [33] [41] > plot(x, y, xlab = "atmosferski tlak", ylab = "vrelište") > curve(ykapa, add = T) reziduali > e = y - ykapa(x) > e [1] [6] [11] [16] [21] [26] [31] [36] [41] [46] koeficijent determinacije > SSE = sum(e^2) > SSE [1] > Syy = (n - 1) * var(y) > Syy [1] > R2 = 1 - SSE/Syy > R2 [1] standardizirani reziduali > k = 2 > sigmakapa = sqrt(sse/(n - k - 1)) > sigmakapa 3

4 [1] > H = X %*% solve(t(x) %*% X) %*% t(x) > hii = diag(h) > hii [1] [7] [13] [19] [25] [31] [37] [43] > es = e/(sigmakapa * sqrt(1 - hii)) > es [1] [7] [13] [19] [25] [31] [37] [43] > par(mfrow = c(1, 2)) > plot(ykapa(x), e, main = "Graf reziduala") > abline(0, 0) > plot(ykapa(x), es, main = "Graf standardiziranih reziduala") > abline(0, 0) provjera pripadnosti standardiziranih reziduala N(0,1) distribuciji > i = 1:n > x_es = qnorm((i - 3/8)/(n + 1/4)) > x_es [1] [7] [13] [19] [25] [31] [37] [43] > y_es = sort(es) > y_es 4

5 [1] [7] [13] [19] [25] [31] [37] [43] > par(mfrow = c(1, 1)) > plot(x_es, y_es, main = "Normalni vjerojatnosni graf") > abline(0, 1) > d = max(max(abs((i - 1)/n - pnorm(y_es)), abs(i/n - pnorm(y_es)))) > d [1] > ks.test(es, pnorm, 0, 1) One-sample Kolmogorov-Smirnov test data: es D = , p-value = alternative hypothesis: two-sided test prihvatljivosti linearnog u odnosu na kvadratični model > Xr = X[, 1:2] > Xr x [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,] [13,] [14,] [15,] [16,] [17,] [18,] [19,] [20,] [21,]

6 [22,] [23,] [24,] [25,] [26,] [27,] [28,] [29,] [30,] [31,] [32,] [33,] [34,] [35,] [36,] [37,] [38,] [39,] [40,] [41,] [42,] [43,] [44,] [45,] [46,] [47,] [48,] > thetakapar = solve(t(xr) %*% Xr) %*% t(xr) %*% y > thetakapar x > ykapar = function(x) thetakapar[1] + thetakapar[2] * x > er = y - ykapar(x) > er [1] [7] [13] [19] [25] [31] [37] [43] > SSEr = sum(er^2) > SSEr [1]

7 > m = 1 > f = ((SSEr - SSE)/(k - m))/(sse/(n - k - 1)) > f [1] > pv = 1 - pf(f, k - m, n - k - 1) > pv [1] 0 p.i. za parametre kvadratičnog modela > alfa = 0.05 > t = qt(1 - alfa/2, n - k - 1) > t [1] > C = solve(t(x) %*% X) > cjj = diag(c) > cjj x e e e-05 > cbind(thetakapa - t * sigmakapa * sqrt(cjj), thetakapa + t * + sigmakapa * sqrt(cjj)) [,2] x donja i gornja krivulja p.i. za srednju vrijednost od Y uz dano x > f_donji = function(x0) rbind(c(1, x0, x0^2)) %*% thetakapa - + t * sigmakapa * sqrt(rbind(c(1, x0, x0^2)) %*% solve(t(x) %*% + X) %*% t(rbind(c(1, x0, x0^2)))) > f_gornji = function(x0) rbind(c(1, x0, x0^2)) %*% thetakapa + + t * sigmakapa * sqrt(rbind(c(1, x0, x0^2)) %*% solve(t(x) %*% + X) %*% t(rbind(c(1, x0, x0^2)))) > plot(x, y, xlab = "atmosferski tlak", ylab = "vrelište", main = "Donja i gornja krivulja > curve(ykapa, add = T) > x_os = seq(min(x), max(x), length = 50) > pi_donji = c() > for (i in 1:50) pi_donji = c(pi_donji, f_donji(x_os[i])) > pi_donji [1] [9] [17] [25] [33] [41] [49]

8 > pi_gornji = c() > for (i in 1:50) pi_gornji = c(pi_gornji, f_gornji(x_os[i])) > pi_gornji [1] [9] [17] [25] [33] [41] [49] > lines(x_os, pi_donji, col = "red", lty = 3) > lines(x_os, pi_gornji, col = "green", lty = 3) donja i gornja krivulja p.i. za Y uz dano x > f_donji = function(x0) rbind(c(1, x0, x0^2)) %*% thetakapa - + t * sigmakapa * sqrt(1 + rbind(c(1, x0, x0^2)) %*% solve(t(x) %*% + X) %*% t(rbind(c(1, x0, x0^2)))) > f_gornji = function(x0) rbind(c(1, x0, x0^2)) %*% thetakapa + + t * sigmakapa * sqrt(1 + rbind(c(1, x0, x0^2)) %*% solve(t(x) %*% + X) %*% t(rbind(c(1, x0, x0^2)))) > pi_donji = c() > for (i in 1:50) pi_donji = c(pi_donji, f_donji(x_os[i])) > pi_donji [1] [9] [17] [25] [33] [41] [49] > pi_gornji = c() > for (i in 1:50) pi_gornji = c(pi_gornji, f_gornji(x_os[i])) > pi_gornji [1] [9] [17] [25] [33] [41] [49] > lines(x_os, pi_donji, col = "red", lty = 2) > lines(x_os, pi_gornji, col = "green", lty = 2) 2. nacin korištenjem naredbe lm 8

9 > x_sort = x[order(x)] > y_sort = y[order(x)] > x_sortkv = x_sort^2 > regr = lm(y_sort ~ x_sort + x_sortkv) > regr Call: lm(formula = y_sort ~ x_sort + x_sortkv) Coefficients: (Intercept) x_sort x_sortkv > coefficients(regr) (Intercept) x_sort x_sortkv > fitted(regr) > residuals(regr)

10 > rstandard(regr) > summary(regr) Call: lm(formula = y_sort ~ x_sort + x_sortkv) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** x_sort <2e-16 *** x_sortkv <2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 45 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 1.281e+04 on 2 and 45 DF, p-value: < 2.2e-16 > pc = predict(regr, int = "c") > pc fit lwr upr

11 > pp = predict(regr, int = "p") > pp fit lwr upr

12 > par(mfrow = c(1, 1)) > plot(x_sort, y_sort) > matlines(x_sort, pc, lty = c(1, 3, 3)) > matlines(x_sort, pp, lty = c(1, 2, 2)) usporedba triju modela za Forbesove podatke 12

13 > xf = forbes[, 2] > yf = forbes[, 1] > zf = log10(xf) > nf = length(xf) > nf [1] 17 > X1 = cbind(rep(1, nf), xf) > X2 = cbind(rep(1, nf), xf, xf^2) > X3 = cbind(rep(1, nf), zf) > thetakapa1 = solve(t(x1) %*% X1) %*% t(x1) %*% yf > thetakapa2 = solve(t(x2) %*% X2) %*% t(x2) %*% yf > thetakapa3 = solve(t(x3) %*% X3) %*% t(x3) %*% yf > thetakapa xf > thetakapa xf > thetakapa zf > ykapa1 = function(x) thetakapa1[1] + thetakapa1[2] * x > ykapa2 = function(x) thetakapa2[1] + thetakapa2[2] * x + thetakapa2[3] * + x^2 > ykapa3 = function(x) thetakapa3[1] + thetakapa3[2] * x krivulje regresije > par(mfrow = c(1, 3)) > plot(xf, yf, xlab = "atmosferski tlak", ylab = "vrelište", main = "Linearni model") > curve(ykapa1, add = T) > plot(xf, yf, xlab = "atmosferski tlak", ylab = "vrelište", main = "Kvadratični model") > curve(ykapa2, add = T) > plot(zf, yf, xlab = "atmosferski tlak", ylab = "vrelište", main = "Linearni model na tran > curve(ykapa3, add = T) koeficijenti determinacije > e1 = yf - ykapa1(xf) > e2 = yf - ykapa2(xf) > e3 = yf - ykapa3(zf) 13

14 > SSE1 = sum(e1^2) > SSE2 = sum(e2^2) > SSE3 = sum(e3^2) > Syy = (nf - 1) * var(yf) > Rkv1 = 1 - SSE1/Syy > Rkv2 = 1 - SSE2/Syy > Rkv3 = 1 - SSE3/Syy > Rkv1 [1] > Rkv2 [1] > Rkv3 [1] grafovi reziduala > par(mfrow = c(1, 3)) > plot(ykapa1(xf), e1, main = "Linearni model\n graf reziduala") > abline(0, 0) > plot(ykapa2(xf), e2, main = "Kvadratični model\n graf reziduala") > abline(0, 0) > plot(ykapa3(zf), e3, main = "Linearni model na transformiranim podacima\n graf reziduala" > abline(0, 0) Durbin-Watsonov test o nezavisnosti sl. gresaka za linerni model na transformiranim podacima > i = 2:nf > d = sum((e3[i] - e3[i - 1])^2)/sum(e3^2) > d [1] > nf [1] 17 > r = 1 > alfa = 0.05 > c1 = 1.13 > d1 = 1.38 > c2 = 4 - c1 > c2 [1] 2.87 > d2 = 4 - d1 > d2 14

15 [1] 2.62 > sort(c(c1 = c1, d1 = d1, c2 = c2, d2 = d2, d = d)) c1 d1 d d2 c pouzdano područje za parametar linearnog modela na transformiranim podacima > q = qf(1-0.05, 2, nf - 2) > q [1] > B = t(x3) %*% X3 > B zf zf > sigmakapa3 = sqrt(sse3/(nf - 2)) > sigmakapa3 [1] > A = B/(2 * sigmakapa3^2) > A zf zf > D = diag(eigen(a)$values) > D [,2] [1,] [2,] > V = eigen(a)$vectors > V [,2] [1,] [2,] > V %*% D %*% t(v) [,2] [1,] [2,]

16 > b = -B %*% thetakapa3/(sigmakapa3^2) > b zf > bc = t(v) %*% b > bc [1,] [2,] > c = t(thetakapa3) %*% B %*% thetakapa3/(2 * sigmakapa3^2) - q > c [1,] > sx = -bc[1]/(2 * D[1, 1]) > sx [1] > sy = -bc[2]/(2 * D[2, 2]) > sy [1] > cnovi = c - D[1, 1] * sx^2 - D[2, 2] * sy^2 > cnovi [1,] > rx = sqrt(-cnovi/d[1, 1]) > rx [1,] > ry = sqrt(-cnovi/d[2, 2]) > ry [1,] > phi = seq(0, 2 * pi, length = 50) > xe = sx + rx * cos(phi) > ye = sy + ry * sin(phi) > par(mfrow = c(1, 1)) > plot(xe, ye, type = "l") > xp = V[1, 1] * xe + V[1, 2] * ye > yp = V[2, 1] * xe + V[2, 2] * ye > plot(xp, yp, type = "l", main = "Pouzdano podrucje") 16

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