regresi_polinomial.knit

itle: “Regresi Polinomial dengan R”

ubtitle: “Contoh Penerapan dan Pemilihan Derajat via Cross-Validation”

uthor: “Agustina Indriyani”

ate: “2026-10-05”

utput:

html_document:

toc: true

toc_float: true

theme: flatly

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 9.363708

2. 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-16

3. 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 ' ' 1

3.2 Perbandingan AIC

AIC(model_linier, model_poli2, model_poli3)

##              df      AIC
## model_linier  3 492.9204
## model_poli2   4 409.4469
## model_poli3   5 411.4307

4. 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 1.800772

6. 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 1.880719

derajat_optimal <- hasil_cv$derajat[which.min(hasil_cv$RMSE)]
cat("Derajat optimal berdasarkan CV:", derajat_optimal, "\n")

## 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()

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

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 x distandardisasi 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.

data$Y_topi_linier <- predict(model_linier)
data$Y_topi_poli2 <- predict(model_poli2)
data$Y_topi_poli3 <- predict(model_poli3)

data.frame(
  Y = data$y,
  Y_topi_linier = data$Y_topi_linier,
  Y_topi_poli2 = data$Y_topi_poli2,
  Y_topi_poli3 = data$Y_topi_poli3
)

##               Y Y_topi_linier Y_topi_poli2 Y_topi_poli3
## 1    3.87904871     9.8817094   5.33982287   5.28280126
## 2    4.73860431     9.7857215   5.51910079   5.46899089
## 3    8.50921338     9.6897335   5.69276104   5.64921021
## 4    5.71952918     9.5937456   5.86080361   5.82346649
## 5    6.01768168     9.4977577   6.02322852   5.99176701
## 6    9.36370818     9.4017697   6.18003576   6.15411903
## 7    7.02376079     9.3057818   6.33122532   6.31052982
## 8    3.73403425     9.2097939   6.47679722   6.46100666
## 9    5.04655753     9.1138060   6.61675145   6.60555682
## 10   5.67892399     9.0178180   6.75108800   6.74418756
## 11   9.16227440     8.9218301   6.87980689   6.87690616
## 12   7.57147951     8.8258422   7.00290810   7.00371990
## 13   7.78501398     8.7298542   7.12039165   7.12463603
## 14   7.33033390     8.6338663   7.23225752   7.23966183
## 15   6.11666178     8.5378784   7.33850572   7.34880458
## 16  10.91542407     8.4418904   7.43913626   7.45207154
## 17   8.44443068     8.3459025   7.53414912   7.54946998
## 18   3.61650551     8.2499146   7.62354431   7.64100717
## 19   9.04733990     8.1539266   7.70732183   7.72669040
## 20   6.78781174     8.0579387   7.78548168   7.80652691
## 21   5.68039177     7.9619508   7.85802386   7.88052400
## 22   7.45661215     7.8659628   7.92494837   7.94868893
## 23   5.91095407     7.7699749   7.98625521   8.01102896
## 24   6.56945966     7.6739870   8.04194438   8.06755137
## 25   6.83532091     7.5779990   8.09201588   8.11826344
## 26   4.76404833     7.4820111   8.13646971   8.16317242
## 27   9.85892273     7.3860232   8.17530587   8.20228560
## 28   8.52988673     7.2900352   8.20852436   8.23561024
## 29   5.98053666     7.1940473   8.23612517   8.26315362
## 30  10.79198858     7.0980594   8.25810832   8.28492300
## 31   9.15871357     7.0020714   8.27447380   8.30092566
## 32   7.73094672     6.9060835   8.28522160   8.31116887
## 33  10.12052374     6.8100956   8.29035174   8.31565989
## 34  10.08960031     6.7141076   8.28986420   8.31440601
## 35   9.97343458     6.6181197   8.28375900   8.30741448
## 36   9.69837019     6.5221318   8.27203612   8.29469258
## 37   9.41362043     6.4261439   8.25469558   8.27624759
## 38   8.16053532     6.3301559   8.23173736   8.25208677
## 39   7.64488520     6.2341680   8.20316147   8.22221739
## 40   7.46219849     6.1381801   8.16896791   8.18664672
## 41   6.79393468     6.0421921   8.12915669   8.14538204
## 42   7.72160040     5.9462042   8.08372779   8.09843062
## 43   5.55460675     5.8502163   8.03268122   8.04579973
## 44  12.36515405     5.7542283   7.97601698   7.98749663
## 45  10.37888696     5.6582404   7.91373507   7.92352860
## 46   5.64634482     5.5622525   7.84583549   7.85390291
## 47   7.01026951     5.4662645   7.77231824   7.77862683
## 48   6.80008384     5.3702766   7.69318332   7.69770763
## 49   9.20455834     5.2742887   7.60843073   7.61115259
## 50   7.38300169     5.1783007   7.51806047   7.51896896
## 51   7.95536675     5.0823128   7.42207253   7.42116404
## 52   7.28450429     4.9863249   7.32046693   7.31774507
## 53   7.14260313     4.8903369   7.21324366   7.20871935
## 54   9.84617304     4.7943490   7.10040271   7.09409413
## 55   6.53192910     4.6983611   6.98194410   6.97387668
## 56   9.88479306     4.6023731   6.85786782   6.84807429
## 57   3.61660520     4.5063852   6.72817386   6.71669421
## 58   7.73947543     4.4103973   6.59286224   6.57974373
## 59   6.66797173     4.3144093   6.45193294   6.43723011
## 60   6.69603986     4.2184214   6.30538598   6.28916062
## 61   6.86120734     4.1224335   6.15322134   6.13554253
## 62   4.92893130     4.0264455   5.99543903   5.97638312
## 63   5.09269145     3.9304576   5.83203905   5.81168965
## 64   3.54136163     3.8344697   5.66302141   5.64146939
## 65   3.24821430     3.7384818   5.48838609   5.46572963
## 66   5.80601657     3.6424938   5.30813310   5.28447762
## 67   5.89641956     3.5465059   5.12226244   5.09772064
## 68   4.90092734     3.4505180   4.93077411   4.90546596
## 69   6.42825088     3.3545300   4.73366811   4.70772084
## 70   8.46656056     3.2585421   4.53094444   4.50449257
## 71   3.16088227     3.1625542   4.32260310   4.29578842
## 72  -0.70496157     3.0665662   4.10864409   4.08161564
## 73   5.68916300     2.9705783   3.88906741   3.86198152
## 74   2.01747237     2.8745904   3.66387306   3.63689332
## 75   1.81192277     2.7786024   3.43306103   3.40635832
## 76   4.98502704     2.6826145   3.19663134   3.17038378
## 77   2.10416075     2.5866266   2.95458398   2.92897699
## 78  -0.03402802     2.4906386   2.70691894   2.68214520
## 79   2.49759319     2.3946507   2.45363624   2.42989569
## 80   1.57866050     2.2986628   2.19473586   2.17223573
## 81   1.58330676     2.2026748   1.93021782   1.90917259
## 82   2.05155254     2.1066869   1.66008210   1.64071354
## 83   0.24276319     2.0106990   1.38432872   1.36686585
## 84   1.96980605     1.9147110   1.10295766   1.08763680
## 85  -0.06907230     1.8187231   0.81596894   0.80303366
## 86   0.72019080     1.7227352   0.52336254   0.51306368
## 87   1.92890913     1.6267472   0.22513847   0.21773416
## 88   0.27807648     1.5307593  -0.07870327  -0.08294765
## 89  -1.57778910     1.4347714  -0.38816268  -0.38897447
## 90   1.03192806     1.3387834  -0.70323975  -0.70033903
## 91   0.37543746     1.2427955  -1.02393450  -1.01703406
## 92  -0.86678123     1.1468076  -1.35024692  -1.33905229
## 93  -1.84423840     1.0508196  -1.68217701  -1.66638645
## 94  -3.94176257     0.9548317  -2.01972477  -1.99902927
## 95  -0.33501589     0.8588438  -2.36289021  -2.33697348
## 96  -4.63333215     0.7628559  -2.71167331  -2.68021179
## 97   0.55923899     0.6668679  -3.06607408  -3.02873696
## 98  -1.13894159     0.5708800  -3.42609252  -3.38254170
## 99  -5.07042123     0.4748921  -3.79172863  -3.74161874
## 100 -7.05284180     0.3789041  -4.16298242  -4.10596081