set.seed(123)
n<-100
x<-seq(0,10,length.out=n)
y<-5+2*x-0.3*x^2 +rnorm(n,mean=0,sd=2)
data<-data.frame(x=x,y=y)
#Menampilkan 10 baris pertama dari 100 pengamatan
head(data,10)
## x y
## 1 0.0000000 3.879049
## 2 0.1010101 4.738604
## 3 0.2020202 8.509213
## 4 0.3030303 5.719529
## 5 0.4040404 6.017682
## 6 0.5050505 9.363708
## 7 0.6060606 7.023761
## 8 0.7070707 3.734034
## 9 0.8080808 5.046558
## 10 0.9090909 5.678924
model_linier <- lm(y~x, data=data)
model_poli2<- lm(y~poly(x,2,raw=TRUE),data=data)
model_poli3<- lm(y~poly(x,3,raw=TRUE),data=data)
summary(model_linier)
##
## Call:
## lm(formula = y ~ x, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.4317 -1.4051 -0.0466 2.2166 6.6109
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.88171 0.55369 17.847 <2e-16 ***
## x -0.95028 0.09566 -9.934 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.789 on 98 degrees of freedom
## Multiple R-squared: 0.5017, Adjusted R-squared: 0.4967
## F-statistic: 98.68 on 1 and 98 DF, p-value: < 2.2e-16
summary(model_poli2)
##
## Call:
## lm(formula = y ~ poly(x, 2, raw = TRUE), data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.8136 -1.1977 -0.0533 1.3549 4.3891
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.33982 0.53777 9.929 <2e-16 ***
## poly(x, 2, raw = TRUE)1 1.80266 0.24855 7.253 1e-10 ***
## poly(x, 2, raw = TRUE)2 -0.27529 0.02405 -11.447 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.829 on 97 degrees of freedom
## Multiple R-squared: 0.788, Adjusted R-squared: 0.7837
## F-statistic: 180.3 on 2 and 97 DF, p-value: < 2.2e-16
summary(model_poli3)
##
## Call:
## lm(formula = y ~ poly(x, 3, raw = TRUE), data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.7866 -1.1996 -0.0497 1.3386 4.3777
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.282801 0.708433 7.457 3.93e-11 ***
## poly(x, 3, raw = TRUE)1 1.872854 0.616613 3.037 0.00307 **
## poly(x, 3, raw = TRUE)2 -0.292931 0.143693 -2.039 0.04424 *
## poly(x, 3, raw = TRUE)3 0.001176 0.009443 0.125 0.90117
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.838 on 96 degrees of freedom
## Multiple R-squared: 0.7881, Adjusted R-squared: 0.7815
## F-statistic: 119 on 3 and 96 DF, p-value: < 2.2e-16
anova(model_linier, model_poli2, model_poli3)
## Analysis of Variance Table
##
## Model 1: y ~ x
## Model 2: y ~ poly(x, 2, raw = TRUE)
## Model 3: y ~ poly(x, 3, raw = TRUE)
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 98 762.42
## 2 97 324.33 1 438.09 129.6931 <2e-16 ***
## 3 96 324.28 1 0.05 0.0155 0.9012
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
AIC(model_linier, model_poli2, model_poli3)
## df AIC
## model_linier 3 492.9204
## model_poli2 4 409.4469
## model_poli3 5 411.4307
library(ggplot2)
data$pred_linier <- predict(model_linier)
data$pred_poli2 <- predict(model_poli2)
data$pred_poli3 <- predict(model_poli3)
ggplot(data, aes(x = x, y = y)) +
geom_point(alpha = 0.5) +
geom_line(aes(y = pred_linier, color = "Linier"), linewidth = 1) +
geom_line(aes(y = pred_poli2, color = "Polinomial derajat 2"), linewidth
= 1) +
geom_line(aes(y = pred_poli3, color = "Polinomial derajat 3"), linewidth
= 1) +
labs(title = "Perbandingan Regresi Linier vs Polinomial",
x = "X", y = "Y", color = "Model") +
theme_minimal()
rmse <- function(actual, predicted) sqrt(mean((actual - predicted)^2))
data.frame(
Model = c("Linier", "Polinomial derajat 2", "Polinomial derajat 3"),
RMSE = c(rmse(data$y, data$pred_linier),
rmse(data$y, data$pred_poli2),
rmse(data$y, data$pred_poli3))
)
## Model RMSE
## 1 Linier 2.761195
## 2 Polinomial derajat 2 1.800917
## 3 Polinomial derajat 3 1.800772
library(caret)
## Loading required package: lattice
kontrol <- trainControl(method = "repeatedcv",number = 10,repeats = 5)
derajat_max <- 6
hasil_cv <- data.frame(
derajat = integer(),
RMSE = numeric()
)
for (d in 1:derajat_max) {
data_cv <- data[, c("x", "y")]
# Membuat variabel polynomial
if (d == 1) {
data_cv$x_poly <- data_cv$x
} else {
poly_x <- poly(data_cv$x, degree = d, raw = TRUE)
data_cv$x_poly <- poly_x[, 1]
for (j in 2:d) {
data_cv[[paste0("x_poly", j)]] <- poly_x[, j]
}
}
# Formula sesuai derajat
if (d == 1) {
formula_model <- y ~ x_poly
} else {
variabel <- paste0("x_poly", 2:d)
formula_model <- as.formula(
paste("y ~ x_poly +", paste(variabel, collapse = " + "))
)
}
# Seed yang sama untuk setiap derajat agar pembagian fold-nya identik
set.seed(42)
model_cv <- train(
formula_model,
data = data_cv,
method = "lm",
trControl = kontrol
)
hasil_cv <- rbind(
hasil_cv,
data.frame(
derajat = d,
RMSE = min(model_cv$results$RMSE)
)
)
}
hasil_cv
## derajat RMSE
## 1 1 2.757805
## 2 2 1.831405
## 3 3 1.857737
## 4 4 1.887214
## 5 5 1.901895
## 6 6 1.886359
derajat_optimal <- hasil_cv$derajat[which.min(hasil_cv$RMSE)]
cat("Derajat optimal berdasarkan CV:", derajat_optimal, "\n")
## Derajat optimal berdasarkan CV: 2
## Derajat optimal berdasarkan CV: 2
ggplot(hasil_cv, aes(x = derajat, y = RMSE)) +
geom_line(
color = "steelblue",
linewidth = 1
) +
geom_point(
size = 3,
color = "steelblue"
) +
geom_vline(
xintercept = derajat_optimal,
linetype = "dashed",
color = "firebrick"
) +
scale_x_continuous(
breaks = 1:derajat_max
) +
labs(
title = "Pemilihan Derajat Optimal via 10-Fold Cross-Validation",
x = "Derajat Polinomial",
y = "RMSE Rata-rata (Validasi Silang)"
) +
theme_minimal()
model_final <- lm(y ~ poly(x, derajat_optimal, raw = TRUE), data = data)
summary(model_final)
##
## Call:
## lm(formula = y ~ poly(x, derajat_optimal, raw = TRUE), data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.8136 -1.1977 -0.0533 1.3549 4.3891
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.33982 0.53777 9.929 <2e-16 ***
## poly(x, derajat_optimal, raw = TRUE)1 1.80266 0.24855 7.253 1e-10 ***
## poly(x, derajat_optimal, raw = TRUE)2 -0.27529 0.02405 -11.447 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.829 on 97 degrees of freedom
## Multiple R-squared: 0.788, Adjusted R-squared: 0.7837
## F-statistic: 180.3 on 2 and 97 DF, p-value: < 2.2e-16
model_linier <- lm(y ~ x, data = data)
model_poli2 <- lm(y ~ x + I(x^2), data = data)
model_poli3 <- lm(y ~ x + I(x^2) + I(x^3), data = data)
data$y_topi_linier <- predict(model_linier)
data$y_topi_poli2 <- predict(model_poli2)
data$y_topi_poli3 <- predict(model_poli3)
data
## x y pred_linier pred_poli2 pred_poli3 y_topi_linier
## 1 0.0000000 3.87904871 9.8817094 5.33982287 5.28280126 9.8817094
## 2 0.1010101 4.73860431 9.7857215 5.51910079 5.46899089 9.7857215
## 3 0.2020202 8.50921338 9.6897335 5.69276104 5.64921021 9.6897335
## 4 0.3030303 5.71952918 9.5937456 5.86080361 5.82346649 9.5937456
## 5 0.4040404 6.01768168 9.4977577 6.02322852 5.99176701 9.4977577
## 6 0.5050505 9.36370818 9.4017697 6.18003576 6.15411903 9.4017697
## 7 0.6060606 7.02376079 9.3057818 6.33122532 6.31052982 9.3057818
## 8 0.7070707 3.73403425 9.2097939 6.47679722 6.46100666 9.2097939
## 9 0.8080808 5.04655753 9.1138060 6.61675145 6.60555682 9.1138060
## 10 0.9090909 5.67892399 9.0178180 6.75108800 6.74418756 9.0178180
## 11 1.0101010 9.16227440 8.9218301 6.87980689 6.87690616 8.9218301
## 12 1.1111111 7.57147951 8.8258422 7.00290810 7.00371990 8.8258422
## 13 1.2121212 7.78501398 8.7298542 7.12039165 7.12463603 8.7298542
## 14 1.3131313 7.33033390 8.6338663 7.23225752 7.23966183 8.6338663
## 15 1.4141414 6.11666178 8.5378784 7.33850572 7.34880458 8.5378784
## 16 1.5151515 10.91542407 8.4418904 7.43913626 7.45207154 8.4418904
## 17 1.6161616 8.44443068 8.3459025 7.53414912 7.54946998 8.3459025
## 18 1.7171717 3.61650551 8.2499146 7.62354431 7.64100717 8.2499146
## 19 1.8181818 9.04733990 8.1539266 7.70732183 7.72669040 8.1539266
## 20 1.9191919 6.78781174 8.0579387 7.78548168 7.80652691 8.0579387
## 21 2.0202020 5.68039177 7.9619508 7.85802386 7.88052400 7.9619508
## 22 2.1212121 7.45661215 7.8659628 7.92494837 7.94868893 7.8659628
## 23 2.2222222 5.91095407 7.7699749 7.98625521 8.01102896 7.7699749
## 24 2.3232323 6.56945966 7.6739870 8.04194438 8.06755137 7.6739870
## 25 2.4242424 6.83532091 7.5779990 8.09201588 8.11826344 7.5779990
## 26 2.5252525 4.76404833 7.4820111 8.13646971 8.16317242 7.4820111
## 27 2.6262626 9.85892273 7.3860232 8.17530587 8.20228560 7.3860232
## 28 2.7272727 8.52988673 7.2900352 8.20852436 8.23561024 7.2900352
## 29 2.8282828 5.98053666 7.1940473 8.23612517 8.26315362 7.1940473
## 30 2.9292929 10.79198858 7.0980594 8.25810832 8.28492300 7.0980594
## 31 3.0303030 9.15871357 7.0020714 8.27447380 8.30092566 7.0020714
## 32 3.1313131 7.73094672 6.9060835 8.28522160 8.31116887 6.9060835
## 33 3.2323232 10.12052374 6.8100956 8.29035174 8.31565989 6.8100956
## 34 3.3333333 10.08960031 6.7141076 8.28986420 8.31440601 6.7141076
## 35 3.4343434 9.97343458 6.6181197 8.28375900 8.30741448 6.6181197
## 36 3.5353535 9.69837019 6.5221318 8.27203612 8.29469258 6.5221318
## 37 3.6363636 9.41362043 6.4261439 8.25469558 8.27624759 6.4261439
## 38 3.7373737 8.16053532 6.3301559 8.23173736 8.25208677 6.3301559
## 39 3.8383838 7.64488520 6.2341680 8.20316147 8.22221739 6.2341680
## 40 3.9393939 7.46219849 6.1381801 8.16896791 8.18664672 6.1381801
## 41 4.0404040 6.79393468 6.0421921 8.12915669 8.14538204 6.0421921
## 42 4.1414141 7.72160040 5.9462042 8.08372779 8.09843062 5.9462042
## 43 4.2424242 5.55460675 5.8502163 8.03268122 8.04579973 5.8502163
## 44 4.3434343 12.36515405 5.7542283 7.97601698 7.98749663 5.7542283
## 45 4.4444444 10.37888696 5.6582404 7.91373507 7.92352860 5.6582404
## 46 4.5454545 5.64634482 5.5622525 7.84583549 7.85390291 5.5622525
## 47 4.6464646 7.01026951 5.4662645 7.77231824 7.77862683 5.4662645
## 48 4.7474747 6.80008384 5.3702766 7.69318332 7.69770763 5.3702766
## 49 4.8484848 9.20455834 5.2742887 7.60843073 7.61115259 5.2742887
## 50 4.9494949 7.38300169 5.1783007 7.51806047 7.51896896 5.1783007
## 51 5.0505051 7.95536675 5.0823128 7.42207253 7.42116404 5.0823128
## 52 5.1515152 7.28450429 4.9863249 7.32046693 7.31774507 4.9863249
## 53 5.2525253 7.14260313 4.8903369 7.21324366 7.20871935 4.8903369
## 54 5.3535354 9.84617304 4.7943490 7.10040271 7.09409413 4.7943490
## 55 5.4545455 6.53192910 4.6983611 6.98194410 6.97387668 4.6983611
## 56 5.5555556 9.88479306 4.6023731 6.85786782 6.84807429 4.6023731
## 57 5.6565657 3.61660520 4.5063852 6.72817386 6.71669421 4.5063852
## 58 5.7575758 7.73947543 4.4103973 6.59286224 6.57974373 4.4103973
## 59 5.8585859 6.66797173 4.3144093 6.45193294 6.43723011 4.3144093
## 60 5.9595960 6.69603986 4.2184214 6.30538598 6.28916062 4.2184214
## 61 6.0606061 6.86120734 4.1224335 6.15322134 6.13554253 4.1224335
## 62 6.1616162 4.92893130 4.0264455 5.99543903 5.97638312 4.0264455
## 63 6.2626263 5.09269145 3.9304576 5.83203905 5.81168965 3.9304576
## 64 6.3636364 3.54136163 3.8344697 5.66302141 5.64146939 3.8344697
## 65 6.4646465 3.24821430 3.7384818 5.48838609 5.46572963 3.7384818
## 66 6.5656566 5.80601657 3.6424938 5.30813310 5.28447762 3.6424938
## 67 6.6666667 5.89641956 3.5465059 5.12226244 5.09772064 3.5465059
## 68 6.7676768 4.90092734 3.4505180 4.93077411 4.90546596 3.4505180
## 69 6.8686869 6.42825088 3.3545300 4.73366811 4.70772084 3.3545300
## 70 6.9696970 8.46656056 3.2585421 4.53094444 4.50449257 3.2585421
## 71 7.0707071 3.16088227 3.1625542 4.32260310 4.29578842 3.1625542
## 72 7.1717172 -0.70496157 3.0665662 4.10864409 4.08161564 3.0665662
## 73 7.2727273 5.68916300 2.9705783 3.88906741 3.86198152 2.9705783
## 74 7.3737374 2.01747237 2.8745904 3.66387306 3.63689332 2.8745904
## 75 7.4747475 1.81192277 2.7786024 3.43306103 3.40635832 2.7786024
## 76 7.5757576 4.98502704 2.6826145 3.19663134 3.17038378 2.6826145
## 77 7.6767677 2.10416075 2.5866266 2.95458398 2.92897699 2.5866266
## 78 7.7777778 -0.03402802 2.4906386 2.70691894 2.68214520 2.4906386
## 79 7.8787879 2.49759319 2.3946507 2.45363624 2.42989569 2.3946507
## 80 7.9797980 1.57866050 2.2986628 2.19473586 2.17223573 2.2986628
## 81 8.0808081 1.58330676 2.2026748 1.93021782 1.90917259 2.2026748
## 82 8.1818182 2.05155254 2.1066869 1.66008210 1.64071354 2.1066869
## 83 8.2828283 0.24276319 2.0106990 1.38432872 1.36686585 2.0106990
## 84 8.3838384 1.96980605 1.9147110 1.10295766 1.08763680 1.9147110
## 85 8.4848485 -0.06907230 1.8187231 0.81596894 0.80303366 1.8187231
## 86 8.5858586 0.72019080 1.7227352 0.52336254 0.51306368 1.7227352
## 87 8.6868687 1.92890913 1.6267472 0.22513847 0.21773416 1.6267472
## 88 8.7878788 0.27807648 1.5307593 -0.07870327 -0.08294765 1.5307593
## 89 8.8888889 -1.57778910 1.4347714 -0.38816268 -0.38897447 1.4347714
## 90 8.9898990 1.03192806 1.3387834 -0.70323975 -0.70033903 1.3387834
## 91 9.0909091 0.37543746 1.2427955 -1.02393450 -1.01703406 1.2427955
## 92 9.1919192 -0.86678123 1.1468076 -1.35024692 -1.33905229 1.1468076
## 93 9.2929293 -1.84423840 1.0508196 -1.68217701 -1.66638645 1.0508196
## 94 9.3939394 -3.94176257 0.9548317 -2.01972477 -1.99902927 0.9548317
## 95 9.4949495 -0.33501589 0.8588438 -2.36289021 -2.33697348 0.8588438
## 96 9.5959596 -4.63333215 0.7628559 -2.71167331 -2.68021179 0.7628559
## 97 9.6969697 0.55923899 0.6668679 -3.06607408 -3.02873696 0.6668679
## 98 9.7979798 -1.13894159 0.5708800 -3.42609252 -3.38254170 0.5708800
## 99 9.8989899 -5.07042123 0.4748921 -3.79172863 -3.74161874 0.4748921
## 100 10.0000000 -7.05284180 0.3789041 -4.16298242 -4.10596081 0.3789041
## y_topi_poli2 y_topi_poli3
## 1 5.33982287 5.28280126
## 2 5.51910079 5.46899089
## 3 5.69276104 5.64921021
## 4 5.86080361 5.82346649
## 5 6.02322852 5.99176701
## 6 6.18003576 6.15411903
## 7 6.33122532 6.31052982
## 8 6.47679722 6.46100666
## 9 6.61675145 6.60555682
## 10 6.75108800 6.74418756
## 11 6.87980689 6.87690616
## 12 7.00290810 7.00371990
## 13 7.12039165 7.12463603
## 14 7.23225752 7.23966183
## 15 7.33850572 7.34880458
## 16 7.43913626 7.45207154
## 17 7.53414912 7.54946998
## 18 7.62354431 7.64100717
## 19 7.70732183 7.72669040
## 20 7.78548168 7.80652691
## 21 7.85802386 7.88052400
## 22 7.92494837 7.94868893
## 23 7.98625521 8.01102896
## 24 8.04194438 8.06755137
## 25 8.09201588 8.11826344
## 26 8.13646971 8.16317242
## 27 8.17530587 8.20228560
## 28 8.20852436 8.23561024
## 29 8.23612517 8.26315362
## 30 8.25810832 8.28492300
## 31 8.27447380 8.30092566
## 32 8.28522160 8.31116887
## 33 8.29035174 8.31565989
## 34 8.28986420 8.31440601
## 35 8.28375900 8.30741448
## 36 8.27203612 8.29469258
## 37 8.25469558 8.27624759
## 38 8.23173736 8.25208677
## 39 8.20316147 8.22221739
## 40 8.16896791 8.18664672
## 41 8.12915669 8.14538204
## 42 8.08372779 8.09843062
## 43 8.03268122 8.04579973
## 44 7.97601698 7.98749663
## 45 7.91373507 7.92352860
## 46 7.84583549 7.85390291
## 47 7.77231824 7.77862683
## 48 7.69318332 7.69770763
## 49 7.60843073 7.61115259
## 50 7.51806047 7.51896896
## 51 7.42207253 7.42116404
## 52 7.32046693 7.31774507
## 53 7.21324366 7.20871935
## 54 7.10040271 7.09409413
## 55 6.98194410 6.97387668
## 56 6.85786782 6.84807429
## 57 6.72817386 6.71669421
## 58 6.59286224 6.57974373
## 59 6.45193294 6.43723011
## 60 6.30538598 6.28916062
## 61 6.15322134 6.13554253
## 62 5.99543903 5.97638312
## 63 5.83203905 5.81168965
## 64 5.66302141 5.64146939
## 65 5.48838609 5.46572963
## 66 5.30813310 5.28447762
## 67 5.12226244 5.09772064
## 68 4.93077411 4.90546596
## 69 4.73366811 4.70772084
## 70 4.53094444 4.50449257
## 71 4.32260310 4.29578842
## 72 4.10864409 4.08161564
## 73 3.88906741 3.86198152
## 74 3.66387306 3.63689332
## 75 3.43306103 3.40635832
## 76 3.19663134 3.17038378
## 77 2.95458398 2.92897699
## 78 2.70691894 2.68214520
## 79 2.45363624 2.42989569
## 80 2.19473586 2.17223573
## 81 1.93021782 1.90917259
## 82 1.66008210 1.64071354
## 83 1.38432872 1.36686585
## 84 1.10295766 1.08763680
## 85 0.81596894 0.80303366
## 86 0.52336254 0.51306368
## 87 0.22513847 0.21773416
## 88 -0.07870327 -0.08294765
## 89 -0.38816268 -0.38897447
## 90 -0.70323975 -0.70033903
## 91 -1.02393450 -1.01703406
## 92 -1.35024692 -1.33905229
## 93 -1.68217701 -1.66638645
## 94 -2.01972477 -1.99902927
## 95 -2.36289021 -2.33697348
## 96 -2.71167331 -2.68021179
## 97 -3.06607408 -3.02873696
## 98 -3.42609252 -3.38254170
## 99 -3.79172863 -3.74161874
## 100 -4.16298242 -4.10596081