Regresi Polinomial dengan R
Contoh Penerapan dan Pemilihan Derajat via Cross-Validation
2026-09-29
1. Simulasi Data
Data disimulasikan dengan hubungan non-linier (kuadratik) antara
x dan y.
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)
head(data)## 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 9.3637082. Pencocokan Model
Tiga model dicocokkan sebagai pembanding: linier, polinomial derajat 2, dan derajat 3.
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_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-163. Perbandingan Model
3.1 Uji ANOVA Bertingkat
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 ' ' 13.2 Perbandingan AIC
AIC(model_linier, model_poli2, model_poli3)## df AIC
## model_linier 3 492.9204
## model_poli2 4 409.4469
## model_poli3 5 411.43074. Visualisasi Perbandingan Model
data$pred_linier <- predict(model_linier)
data$pred_poli2 <- predict(model_poli2)
data$pred_poli3 <- predict(model_poli3)
ggplot(data, aes(x = x, y = y)) +
geom_point(alpha = 0.5) +
geom_line(aes(y = pred_linier, color = "Linier"), linewidth = 1) +
geom_line(aes(y = pred_poli2, color = "Polinomial derajat 2"), linewidth = 1) +
geom_line(aes(y = pred_poli3, color = "Polinomial derajat 3"), linewidth = 1) +
labs(title = "Perbandingan Regresi Linier vs Polinomial",
x = "X", y = "Y", color = "Model") +
theme_minimal()
5. Evaluasi dengan RMSE
rmse <- function(actual, predicted) sqrt(mean((actual - predicted)^2))
data.frame(
Model = c("Linier", "Polinomial derajat 2", "Polinomial derajat 3"),
RMSE = c(rmse(data$y, data$pred_linier),
rmse(data$y, data$pred_poli2),
rmse(data$y, data$pred_poli3))
)## Model RMSE
## 1 Linier 2.761195
## 2 Polinomial derajat 2 1.800917
## 3 Polinomial derajat 3 1.8007726. Pemilihan Derajat Optimal via Cross-Validation
Skema 10-fold cross-validation diulang 5 kali untuk menguji derajat polinomial 1 sampai 6.
names(data)## [1] "x" "y" "pred_linier" "pred_poli2" "pred_poli3"library(caret)
set.seed(42)
kontrol <- trainControl(
method = "repeatedcv",
number = 10,
repeats = 5
)
derajat_max <- 6
hasil_cv <- data.frame(
derajat = integer(),
RMSE = numeric()
)
for (d in 1:derajat_max) {
data_cv <- data
# Membuat variabel polynomial
if (d == 1) {
data_cv$x_poly <- data_cv$x
} else {
poly_x <- poly(data_cv$x, degree = d, raw = TRUE)
data_cv$x_poly <- poly_x[, 1]
for (j in 2:d) {
data_cv[[paste0("x_poly", j)]] <- poly_x[, j]
}
}
# Formula sesuai derajat
if (d == 1) {
formula_model <- y ~ x_poly
} else {
variabel <- paste0("x_poly", 2:d)
formula_model <- as.formula(
paste("y ~ x_poly +", paste(variabel, collapse = " + "))
)
}
model_cv <- train(
formula_model,
data = data_cv,
method = "lm",
trControl = kontrol
)
hasil_cv <- rbind(
hasil_cv,
data.frame(
derajat = d,
RMSE = min(model_cv$results$RMSE)
)
)
}
hasil_cv## derajat RMSE
## 1 1 2.757805
## 2 2 1.825068
## 3 3 1.859968
## 4 4 1.877628
## 5 5 1.871701
## 6 6 1.880719derajat_optimal <- hasil_cv$derajat[which.min(hasil_cv$RMSE)]
cat("Derajat optimal berdasarkan CV:", derajat_optimal, "\n")## Derajat optimal berdasarkan CV: 2ggplot(hasil_cv, aes(x = derajat, y = RMSE)) +
geom_line(
color = "steelblue",
linewidth = 1
) +
geom_point(
size = 3,
color = "steelblue"
) +
geom_vline(
xintercept = derajat_optimal,
linetype = "dashed",
color = "firebrick"
) +
scale_x_continuous(
breaks = 1:derajat_max
) +
labs(
title = "Pemilihan Derajat Optimal via 10-Fold Cross-Validation",
x = "Derajat Polinomial",
y = "RMSE Rata-rata (Validasi Silang)"
) +
theme_minimal()
7. Model Final
model_final <- lm(y ~ poly(x, derajat_optimal, raw = TRUE), data = data)
summary(model_final)##
## Call:
## lm(formula = y ~ poly(x, derajat_optimal, raw = TRUE), data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.8136 -1.1977 -0.0533 1.3549 4.3891
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.33982 0.53777 9.929 <2e-16 ***
## poly(x, derajat_optimal, raw = TRUE)1 1.80266 0.24855 7.253 1e-10 ***
## poly(x, derajat_optimal, raw = TRUE)2 -0.27529 0.02405 -11.447 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.829 on 97 degrees of freedom
## Multiple R-squared: 0.788, Adjusted R-squared: 0.7837
## F-statistic: 180.3 on 2 and 97 DF, p-value: < 2.2e-16# Memunculkan nilai Y topi (Y prediksi)
data$Y_topi <- predict(model_final)
i=(data[, c("x", "y", "Y_topi")])
i## x y Y_topi
## 1 0.0000000 3.87904871 5.33982287
## 2 0.1010101 4.73860431 5.51910079
## 3 0.2020202 8.50921338 5.69276104
## 4 0.3030303 5.71952918 5.86080361
## 5 0.4040404 6.01768168 6.02322852
## 6 0.5050505 9.36370818 6.18003576
## 7 0.6060606 7.02376079 6.33122532
## 8 0.7070707 3.73403425 6.47679722
## 9 0.8080808 5.04655753 6.61675145
## 10 0.9090909 5.67892399 6.75108800
## 11 1.0101010 9.16227440 6.87980689
## 12 1.1111111 7.57147951 7.00290810
## 13 1.2121212 7.78501398 7.12039165
## 14 1.3131313 7.33033390 7.23225752
## 15 1.4141414 6.11666178 7.33850572
## 16 1.5151515 10.91542407 7.43913626
## 17 1.6161616 8.44443068 7.53414912
## 18 1.7171717 3.61650551 7.62354431
## 19 1.8181818 9.04733990 7.70732183
## 20 1.9191919 6.78781174 7.78548168
## 21 2.0202020 5.68039177 7.85802386
## 22 2.1212121 7.45661215 7.92494837
## 23 2.2222222 5.91095407 7.98625521
## 24 2.3232323 6.56945966 8.04194438
## 25 2.4242424 6.83532091 8.09201588
## 26 2.5252525 4.76404833 8.13646971
## 27 2.6262626 9.85892273 8.17530587
## 28 2.7272727 8.52988673 8.20852436
## 29 2.8282828 5.98053666 8.23612517
## 30 2.9292929 10.79198858 8.25810832
## 31 3.0303030 9.15871357 8.27447380
## 32 3.1313131 7.73094672 8.28522160
## 33 3.2323232 10.12052374 8.29035174
## 34 3.3333333 10.08960031 8.28986420
## 35 3.4343434 9.97343458 8.28375900
## 36 3.5353535 9.69837019 8.27203612
## 37 3.6363636 9.41362043 8.25469558
## 38 3.7373737 8.16053532 8.23173736
## 39 3.8383838 7.64488520 8.20316147
## 40 3.9393939 7.46219849 8.16896791
## 41 4.0404040 6.79393468 8.12915669
## 42 4.1414141 7.72160040 8.08372779
## 43 4.2424242 5.55460675 8.03268122
## 44 4.3434343 12.36515405 7.97601698
## 45 4.4444444 10.37888696 7.91373507
## 46 4.5454545 5.64634482 7.84583549
## 47 4.6464646 7.01026951 7.77231824
## 48 4.7474747 6.80008384 7.69318332
## 49 4.8484848 9.20455834 7.60843073
## 50 4.9494949 7.38300169 7.51806047
## 51 5.0505051 7.95536675 7.42207253
## 52 5.1515152 7.28450429 7.32046693
## 53 5.2525253 7.14260313 7.21324366
## 54 5.3535354 9.84617304 7.10040271
## 55 5.4545455 6.53192910 6.98194410
## 56 5.5555556 9.88479306 6.85786782
## 57 5.6565657 3.61660520 6.72817386
## 58 5.7575758 7.73947543 6.59286224
## 59 5.8585859 6.66797173 6.45193294
## 60 5.9595960 6.69603986 6.30538598
## 61 6.0606061 6.86120734 6.15322134
## 62 6.1616162 4.92893130 5.99543903
## 63 6.2626263 5.09269145 5.83203905
## 64 6.3636364 3.54136163 5.66302141
## 65 6.4646465 3.24821430 5.48838609
## 66 6.5656566 5.80601657 5.30813310
## 67 6.6666667 5.89641956 5.12226244
## 68 6.7676768 4.90092734 4.93077411
## 69 6.8686869 6.42825088 4.73366811
## 70 6.9696970 8.46656056 4.53094444
## 71 7.0707071 3.16088227 4.32260310
## 72 7.1717172 -0.70496157 4.10864409
## 73 7.2727273 5.68916300 3.88906741
## 74 7.3737374 2.01747237 3.66387306
## 75 7.4747475 1.81192277 3.43306103
## 76 7.5757576 4.98502704 3.19663134
## 77 7.6767677 2.10416075 2.95458398
## 78 7.7777778 -0.03402802 2.70691894
## 79 7.8787879 2.49759319 2.45363624
## 80 7.9797980 1.57866050 2.19473586
## 81 8.0808081 1.58330676 1.93021782
## 82 8.1818182 2.05155254 1.66008210
## 83 8.2828283 0.24276319 1.38432872
## 84 8.3838384 1.96980605 1.10295766
## 85 8.4848485 -0.06907230 0.81596894
## 86 8.5858586 0.72019080 0.52336254
## 87 8.6868687 1.92890913 0.22513847
## 88 8.7878788 0.27807648 -0.07870327
## 89 8.8888889 -1.57778910 -0.38816268
## 90 8.9898990 1.03192806 -0.70323975
## 91 9.0909091 0.37543746 -1.02393450
## 92 9.1919192 -0.86678123 -1.35024692
## 93 9.2929293 -1.84423840 -1.68217701
## 94 9.3939394 -3.94176257 -2.01972477
## 95 9.4949495 -0.33501589 -2.36289021
## 96 9.5959596 -4.63333215 -2.71167331
## 97 9.6969697 0.55923899 -3.06607408
## 98 9.7979798 -1.13894159 -3.42609252
## 99 9.8989899 -5.07042123 -3.79172863
## 100 10.0000000 -7.05284180 -4.16298242Catatan
poly(x, derajat, raw = TRUE)digunakan agar koefisien dapat diinterpretasi langsung sebagai \(\beta_1 x + \beta_2 x^2 + \dots\)- Untuk derajat tinggi (\(\geq 4\)),
sebaiknya
xdistandardisasi terlebih dahulu (scale(x)) agar model lebih stabil secara numerik. - Pertimbangkan aturan one-standard-error rule: pilih derajat terkecil dengan RMSE \(\leq\) (RMSE minimum + 1 SD) untuk model yang lebih sederhana namun performanya setara.