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

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
data$y_topi <- predict(model_final)
data$residual <- data$y - data$y_topi
data
##              x           y pred_linier  pred_poli2  pred_poli3      y_topi
## 1    0.0000000  3.87904871   9.8817094  5.33982287  5.28280126  5.33982287
## 2    0.1010101  4.73860431   9.7857215  5.51910079  5.46899089  5.51910079
## 3    0.2020202  8.50921338   9.6897335  5.69276104  5.64921021  5.69276104
## 4    0.3030303  5.71952918   9.5937456  5.86080361  5.82346649  5.86080361
## 5    0.4040404  6.01768168   9.4977577  6.02322852  5.99176701  6.02322852
## 6    0.5050505  9.36370818   9.4017697  6.18003576  6.15411903  6.18003576
## 7    0.6060606  7.02376079   9.3057818  6.33122532  6.31052982  6.33122532
## 8    0.7070707  3.73403425   9.2097939  6.47679722  6.46100666  6.47679722
## 9    0.8080808  5.04655753   9.1138060  6.61675145  6.60555682  6.61675145
## 10   0.9090909  5.67892399   9.0178180  6.75108800  6.74418756  6.75108800
## 11   1.0101010  9.16227440   8.9218301  6.87980689  6.87690616  6.87980689
## 12   1.1111111  7.57147951   8.8258422  7.00290810  7.00371990  7.00290810
## 13   1.2121212  7.78501398   8.7298542  7.12039165  7.12463603  7.12039165
## 14   1.3131313  7.33033390   8.6338663  7.23225752  7.23966183  7.23225752
## 15   1.4141414  6.11666178   8.5378784  7.33850572  7.34880458  7.33850572
## 16   1.5151515 10.91542407   8.4418904  7.43913626  7.45207154  7.43913626
## 17   1.6161616  8.44443068   8.3459025  7.53414912  7.54946998  7.53414912
## 18   1.7171717  3.61650551   8.2499146  7.62354431  7.64100717  7.62354431
## 19   1.8181818  9.04733990   8.1539266  7.70732183  7.72669040  7.70732183
## 20   1.9191919  6.78781174   8.0579387  7.78548168  7.80652691  7.78548168
## 21   2.0202020  5.68039177   7.9619508  7.85802386  7.88052400  7.85802386
## 22   2.1212121  7.45661215   7.8659628  7.92494837  7.94868893  7.92494837
## 23   2.2222222  5.91095407   7.7699749  7.98625521  8.01102896  7.98625521
## 24   2.3232323  6.56945966   7.6739870  8.04194438  8.06755137  8.04194438
## 25   2.4242424  6.83532091   7.5779990  8.09201588  8.11826344  8.09201588
## 26   2.5252525  4.76404833   7.4820111  8.13646971  8.16317242  8.13646971
## 27   2.6262626  9.85892273   7.3860232  8.17530587  8.20228560  8.17530587
## 28   2.7272727  8.52988673   7.2900352  8.20852436  8.23561024  8.20852436
## 29   2.8282828  5.98053666   7.1940473  8.23612517  8.26315362  8.23612517
## 30   2.9292929 10.79198858   7.0980594  8.25810832  8.28492300  8.25810832
## 31   3.0303030  9.15871357   7.0020714  8.27447380  8.30092566  8.27447380
## 32   3.1313131  7.73094672   6.9060835  8.28522160  8.31116887  8.28522160
## 33   3.2323232 10.12052374   6.8100956  8.29035174  8.31565989  8.29035174
## 34   3.3333333 10.08960031   6.7141076  8.28986420  8.31440601  8.28986420
## 35   3.4343434  9.97343458   6.6181197  8.28375900  8.30741448  8.28375900
## 36   3.5353535  9.69837019   6.5221318  8.27203612  8.29469258  8.27203612
## 37   3.6363636  9.41362043   6.4261439  8.25469558  8.27624759  8.25469558
## 38   3.7373737  8.16053532   6.3301559  8.23173736  8.25208677  8.23173736
## 39   3.8383838  7.64488520   6.2341680  8.20316147  8.22221739  8.20316147
## 40   3.9393939  7.46219849   6.1381801  8.16896791  8.18664672  8.16896791
## 41   4.0404040  6.79393468   6.0421921  8.12915669  8.14538204  8.12915669
## 42   4.1414141  7.72160040   5.9462042  8.08372779  8.09843062  8.08372779
## 43   4.2424242  5.55460675   5.8502163  8.03268122  8.04579973  8.03268122
## 44   4.3434343 12.36515405   5.7542283  7.97601698  7.98749663  7.97601698
## 45   4.4444444 10.37888696   5.6582404  7.91373507  7.92352860  7.91373507
## 46   4.5454545  5.64634482   5.5622525  7.84583549  7.85390291  7.84583549
## 47   4.6464646  7.01026951   5.4662645  7.77231824  7.77862683  7.77231824
## 48   4.7474747  6.80008384   5.3702766  7.69318332  7.69770763  7.69318332
## 49   4.8484848  9.20455834   5.2742887  7.60843073  7.61115259  7.60843073
## 50   4.9494949  7.38300169   5.1783007  7.51806047  7.51896896  7.51806047
## 51   5.0505051  7.95536675   5.0823128  7.42207253  7.42116404  7.42207253
## 52   5.1515152  7.28450429   4.9863249  7.32046693  7.31774507  7.32046693
## 53   5.2525253  7.14260313   4.8903369  7.21324366  7.20871935  7.21324366
## 54   5.3535354  9.84617304   4.7943490  7.10040271  7.09409413  7.10040271
## 55   5.4545455  6.53192910   4.6983611  6.98194410  6.97387668  6.98194410
## 56   5.5555556  9.88479306   4.6023731  6.85786782  6.84807429  6.85786782
## 57   5.6565657  3.61660520   4.5063852  6.72817386  6.71669421  6.72817386
## 58   5.7575758  7.73947543   4.4103973  6.59286224  6.57974373  6.59286224
## 59   5.8585859  6.66797173   4.3144093  6.45193294  6.43723011  6.45193294
## 60   5.9595960  6.69603986   4.2184214  6.30538598  6.28916062  6.30538598
## 61   6.0606061  6.86120734   4.1224335  6.15322134  6.13554253  6.15322134
## 62   6.1616162  4.92893130   4.0264455  5.99543903  5.97638312  5.99543903
## 63   6.2626263  5.09269145   3.9304576  5.83203905  5.81168965  5.83203905
## 64   6.3636364  3.54136163   3.8344697  5.66302141  5.64146939  5.66302141
## 65   6.4646465  3.24821430   3.7384818  5.48838609  5.46572963  5.48838609
## 66   6.5656566  5.80601657   3.6424938  5.30813310  5.28447762  5.30813310
## 67   6.6666667  5.89641956   3.5465059  5.12226244  5.09772064  5.12226244
## 68   6.7676768  4.90092734   3.4505180  4.93077411  4.90546596  4.93077411
## 69   6.8686869  6.42825088   3.3545300  4.73366811  4.70772084  4.73366811
## 70   6.9696970  8.46656056   3.2585421  4.53094444  4.50449257  4.53094444
## 71   7.0707071  3.16088227   3.1625542  4.32260310  4.29578842  4.32260310
## 72   7.1717172 -0.70496157   3.0665662  4.10864409  4.08161564  4.10864409
## 73   7.2727273  5.68916300   2.9705783  3.88906741  3.86198152  3.88906741
## 74   7.3737374  2.01747237   2.8745904  3.66387306  3.63689332  3.66387306
## 75   7.4747475  1.81192277   2.7786024  3.43306103  3.40635832  3.43306103
## 76   7.5757576  4.98502704   2.6826145  3.19663134  3.17038378  3.19663134
## 77   7.6767677  2.10416075   2.5866266  2.95458398  2.92897699  2.95458398
## 78   7.7777778 -0.03402802   2.4906386  2.70691894  2.68214520  2.70691894
## 79   7.8787879  2.49759319   2.3946507  2.45363624  2.42989569  2.45363624
## 80   7.9797980  1.57866050   2.2986628  2.19473586  2.17223573  2.19473586
## 81   8.0808081  1.58330676   2.2026748  1.93021782  1.90917259  1.93021782
## 82   8.1818182  2.05155254   2.1066869  1.66008210  1.64071354  1.66008210
## 83   8.2828283  0.24276319   2.0106990  1.38432872  1.36686585  1.38432872
## 84   8.3838384  1.96980605   1.9147110  1.10295766  1.08763680  1.10295766
## 85   8.4848485 -0.06907230   1.8187231  0.81596894  0.80303366  0.81596894
## 86   8.5858586  0.72019080   1.7227352  0.52336254  0.51306368  0.52336254
## 87   8.6868687  1.92890913   1.6267472  0.22513847  0.21773416  0.22513847
## 88   8.7878788  0.27807648   1.5307593 -0.07870327 -0.08294765 -0.07870327
## 89   8.8888889 -1.57778910   1.4347714 -0.38816268 -0.38897447 -0.38816268
## 90   8.9898990  1.03192806   1.3387834 -0.70323975 -0.70033903 -0.70323975
## 91   9.0909091  0.37543746   1.2427955 -1.02393450 -1.01703406 -1.02393450
## 92   9.1919192 -0.86678123   1.1468076 -1.35024692 -1.33905229 -1.35024692
## 93   9.2929293 -1.84423840   1.0508196 -1.68217701 -1.66638645 -1.68217701
## 94   9.3939394 -3.94176257   0.9548317 -2.01972477 -1.99902927 -2.01972477
## 95   9.4949495 -0.33501589   0.8588438 -2.36289021 -2.33697348 -2.36289021
## 96   9.5959596 -4.63333215   0.7628559 -2.71167331 -2.68021179 -2.71167331
## 97   9.6969697  0.55923899   0.6668679 -3.06607408 -3.02873696 -3.06607408
## 98   9.7979798 -1.13894159   0.5708800 -3.42609252 -3.38254170 -3.42609252
## 99   9.8989899 -5.07042123   0.4748921 -3.79172863 -3.74161874 -3.79172863
## 100 10.0000000 -7.05284180   0.3789041 -4.16298242 -4.10596081 -4.16298242
##         residual
## 1   -1.460774162
## 2   -0.780496476
## 3    2.816452348
## 4   -0.141274434
## 5   -0.005546837
## 6    3.183672422
## 7    0.692535462
## 8   -2.742762972
## 9   -1.570193912
## 10  -1.072164008
## 11   2.282467513
## 12   0.568571404
## 13   0.664622330
## 14   0.098076385
## 15  -1.221843946
## 16   3.476287814
## 17   0.910281560
## 18  -4.007038800
## 19   1.340018071
## 20  -0.997669946
## 21  -2.177632095
## 22  -0.468336219
## 23  -2.075301147
## 24  -1.472484723
## 25  -1.256694968
## 26  -3.372421376
## 27   1.683616859
## 28   0.321362376
## 29  -2.255588517
## 30   2.533880261
## 31   0.884239771
## 32  -0.554274883
## 33   1.830172006
## 34   1.799736106
## 35   1.689675587
## 36   1.426334071
## 37   1.158924856
## 38  -0.071202041
## 39  -0.558276270
## 40  -0.706769420
## 41  -1.335222006
## 42  -0.362127388
## 43  -2.478074473
## 44   4.389137069
## 45   2.465151889
## 46  -2.199490673
## 47  -0.762048731
## 48  -0.893099475
## 49   1.596127608
## 50  -0.135058776
## 51   0.533294216
## 52  -0.035962646
## 53  -0.070640526
## 54   2.745770326
## 55  -0.450014998
## 56   3.026925244
## 57  -3.111568665
## 58   1.146613197
## 59   0.216038785
## 60   0.390653881
## 61   0.707986002
## 62  -1.066507732
## 63  -0.739347608
## 64  -2.121659776
## 65  -2.240171786
## 66   0.497883473
## 67   0.774157116
## 68  -0.029846772
## 69   1.694582772
## 70   3.935616115
## 71  -1.161720835
## 72  -4.813605655
## 73   1.800095591
## 74  -1.646400691
## 75  -1.621138260
## 76   1.788395696
## 77  -0.850423227
## 78  -2.740946961
## 79   0.043956946
## 80  -0.616075370
## 81  -0.346911058
## 82   0.391470433
## 83  -1.141565525
## 84   0.866848389
## 85  -0.885041233
## 86   0.196828264
## 87   1.703770654
## 88   0.356779747
## 89  -1.189626421
## 90   1.735167817
## 91   1.399371968
## 92   0.483465697
## 93  -0.162061383
## 94  -1.922037791
## 95   2.027874319
## 96  -1.921658846
## 97   3.625313067
## 98   2.287150932
## 99  -1.278692593
## 100 -2.889859384

#Y topi

data$Ytopi <- predict(model_final)
data$residual <- data$y - data$Ytopi

data.frame(
  X = data$x,
  Y = data$y,
  Ytopi = data$Ytopi,
  Residual = data$residual
)
##              X           Y       Ytopi     Residual
## 1    0.0000000  3.87904871  5.33982287 -1.460774162
## 2    0.1010101  4.73860431  5.51910079 -0.780496476
## 3    0.2020202  8.50921338  5.69276104  2.816452348
## 4    0.3030303  5.71952918  5.86080361 -0.141274434
## 5    0.4040404  6.01768168  6.02322852 -0.005546837
## 6    0.5050505  9.36370818  6.18003576  3.183672422
## 7    0.6060606  7.02376079  6.33122532  0.692535462
## 8    0.7070707  3.73403425  6.47679722 -2.742762972
## 9    0.8080808  5.04655753  6.61675145 -1.570193912
## 10   0.9090909  5.67892399  6.75108800 -1.072164008
## 11   1.0101010  9.16227440  6.87980689  2.282467513
## 12   1.1111111  7.57147951  7.00290810  0.568571404
## 13   1.2121212  7.78501398  7.12039165  0.664622330
## 14   1.3131313  7.33033390  7.23225752  0.098076385
## 15   1.4141414  6.11666178  7.33850572 -1.221843946
## 16   1.5151515 10.91542407  7.43913626  3.476287814
## 17   1.6161616  8.44443068  7.53414912  0.910281560
## 18   1.7171717  3.61650551  7.62354431 -4.007038800
## 19   1.8181818  9.04733990  7.70732183  1.340018071
## 20   1.9191919  6.78781174  7.78548168 -0.997669946
## 21   2.0202020  5.68039177  7.85802386 -2.177632095
## 22   2.1212121  7.45661215  7.92494837 -0.468336219
## 23   2.2222222  5.91095407  7.98625521 -2.075301147
## 24   2.3232323  6.56945966  8.04194438 -1.472484723
## 25   2.4242424  6.83532091  8.09201588 -1.256694968
## 26   2.5252525  4.76404833  8.13646971 -3.372421376
## 27   2.6262626  9.85892273  8.17530587  1.683616859
## 28   2.7272727  8.52988673  8.20852436  0.321362376
## 29   2.8282828  5.98053666  8.23612517 -2.255588517
## 30   2.9292929 10.79198858  8.25810832  2.533880261
## 31   3.0303030  9.15871357  8.27447380  0.884239771
## 32   3.1313131  7.73094672  8.28522160 -0.554274883
## 33   3.2323232 10.12052374  8.29035174  1.830172006
## 34   3.3333333 10.08960031  8.28986420  1.799736106
## 35   3.4343434  9.97343458  8.28375900  1.689675587
## 36   3.5353535  9.69837019  8.27203612  1.426334071
## 37   3.6363636  9.41362043  8.25469558  1.158924856
## 38   3.7373737  8.16053532  8.23173736 -0.071202041
## 39   3.8383838  7.64488520  8.20316147 -0.558276270
## 40   3.9393939  7.46219849  8.16896791 -0.706769420
## 41   4.0404040  6.79393468  8.12915669 -1.335222006
## 42   4.1414141  7.72160040  8.08372779 -0.362127388
## 43   4.2424242  5.55460675  8.03268122 -2.478074473
## 44   4.3434343 12.36515405  7.97601698  4.389137069
## 45   4.4444444 10.37888696  7.91373507  2.465151889
## 46   4.5454545  5.64634482  7.84583549 -2.199490673
## 47   4.6464646  7.01026951  7.77231824 -0.762048731
## 48   4.7474747  6.80008384  7.69318332 -0.893099475
## 49   4.8484848  9.20455834  7.60843073  1.596127608
## 50   4.9494949  7.38300169  7.51806047 -0.135058776
## 51   5.0505051  7.95536675  7.42207253  0.533294216
## 52   5.1515152  7.28450429  7.32046693 -0.035962646
## 53   5.2525253  7.14260313  7.21324366 -0.070640526
## 54   5.3535354  9.84617304  7.10040271  2.745770326
## 55   5.4545455  6.53192910  6.98194410 -0.450014998
## 56   5.5555556  9.88479306  6.85786782  3.026925244
## 57   5.6565657  3.61660520  6.72817386 -3.111568665
## 58   5.7575758  7.73947543  6.59286224  1.146613197
## 59   5.8585859  6.66797173  6.45193294  0.216038785
## 60   5.9595960  6.69603986  6.30538598  0.390653881
## 61   6.0606061  6.86120734  6.15322134  0.707986002
## 62   6.1616162  4.92893130  5.99543903 -1.066507732
## 63   6.2626263  5.09269145  5.83203905 -0.739347608
## 64   6.3636364  3.54136163  5.66302141 -2.121659776
## 65   6.4646465  3.24821430  5.48838609 -2.240171786
## 66   6.5656566  5.80601657  5.30813310  0.497883473
## 67   6.6666667  5.89641956  5.12226244  0.774157116
## 68   6.7676768  4.90092734  4.93077411 -0.029846772
## 69   6.8686869  6.42825088  4.73366811  1.694582772
## 70   6.9696970  8.46656056  4.53094444  3.935616115
## 71   7.0707071  3.16088227  4.32260310 -1.161720835
## 72   7.1717172 -0.70496157  4.10864409 -4.813605655
## 73   7.2727273  5.68916300  3.88906741  1.800095591
## 74   7.3737374  2.01747237  3.66387306 -1.646400691
## 75   7.4747475  1.81192277  3.43306103 -1.621138260
## 76   7.5757576  4.98502704  3.19663134  1.788395696
## 77   7.6767677  2.10416075  2.95458398 -0.850423227
## 78   7.7777778 -0.03402802  2.70691894 -2.740946961
## 79   7.8787879  2.49759319  2.45363624  0.043956946
## 80   7.9797980  1.57866050  2.19473586 -0.616075370
## 81   8.0808081  1.58330676  1.93021782 -0.346911058
## 82   8.1818182  2.05155254  1.66008210  0.391470433
## 83   8.2828283  0.24276319  1.38432872 -1.141565525
## 84   8.3838384  1.96980605  1.10295766  0.866848389
## 85   8.4848485 -0.06907230  0.81596894 -0.885041233
## 86   8.5858586  0.72019080  0.52336254  0.196828264
## 87   8.6868687  1.92890913  0.22513847  1.703770654
## 88   8.7878788  0.27807648 -0.07870327  0.356779747
## 89   8.8888889 -1.57778910 -0.38816268 -1.189626421
## 90   8.9898990  1.03192806 -0.70323975  1.735167817
## 91   9.0909091  0.37543746 -1.02393450  1.399371968
## 92   9.1919192 -0.86678123 -1.35024692  0.483465697
## 93   9.2929293 -1.84423840 -1.68217701 -0.162061383
## 94   9.3939394 -3.94176257 -2.01972477 -1.922037791
## 95   9.4949495 -0.33501589 -2.36289021  2.027874319
## 96   9.5959596 -4.63333215 -2.71167331 -1.921658846
## 97   9.6969697  0.55923899 -3.06607408  3.625313067
## 98   9.7979798 -1.13894159 -3.42609252  2.287150932
## 99   9.8989899 -5.07042123 -3.79172863 -1.278692593
## 100 10.0000000 -7.05284180 -4.16298242 -2.889859384

Catatan