Perbandingan Metode Holt Winter’s Exponential Smoothing dan ARIMA Pada Peramalan Data Jumlah Pengunjung di Lokawisata Baturraden
Nadia Putri Anggraeni
2024-05-14
IMPORT LIBRARY
#Import library yang dibutuhkan
library(readxl)
library(ggplot2)
library(forecast)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(pastecs) #stat.desc
library(tseries) #adf.test
library(Metrics)##
## Attaching package: 'Metrics'## The following object is masked from 'package:forecast':
##
## accuracyLOAD DATA
Data Jumlah Pengunjung di Lokawisata Baturaden per bulan (Januari - Desember) dari tahun 2014-2023.
df <- read_excel("C:/Users/naura/Downloads/FOLDER NADIA/SEMESTER 8/Data Terbaru.xlsx")
df## # A tibble: 120 × 2
## Bulan Jumlah_Pengunjung
## <chr> <dbl>
## 1 Januari 47237
## 2 Februari 18009
## 3 Maret 17462
## 4 April 17974
## 5 Mei 24137
## 6 Juni 28620
## 7 Juli 66517
## 8 Agustus 57905
## 9 September 15975
## 10 Oktober 21374
## # ℹ 110 more rowsView(df)EXPLORATORY DATA ANALYSIS (EDA)
#Mengecek dimensi data
dim(df)## [1] 120 2#Mengecek variabel data
names(df)## [1] "Bulan" "Jumlah_Pengunjung"#Mengecek tipe data tiap variabel
str(df)## tibble [120 × 2] (S3: tbl_df/tbl/data.frame)
## $ Bulan : chr [1:120] "Januari" "Februari" "Maret" "April" ...
## $ Jumlah_Pengunjung: num [1:120] 47237 18009 17462 17974 24137 ...#Statistika deskriptif
stat.desc(df)## Bulan Jumlah_Pengunjung
## nbr.val NA 1.200000e+02
## nbr.null NA 3.000000e+00
## nbr.na NA 0.000000e+00
## min NA 0.000000e+00
## max NA 1.929000e+05
## range NA 1.929000e+05
## sum NA 4.634130e+06
## median NA 3.056200e+04
## mean NA 3.861775e+04
## SE.mean NA 2.947106e+03
## CI.mean NA 5.835563e+03
## var NA 1.042252e+09
## std.dev NA 3.228392e+04
## coef.var NA 8.359867e-01summary(df)## Bulan Jumlah_Pengunjung
## Length:120 Min. : 0
## Class :character 1st Qu.: 19870
## Mode :character Median : 30562
## Mean : 38618
## 3rd Qu.: 45631
## Max. :192900#Menghitung nilai variansi
var(df$Jumlah_Pengunjung)## [1] 1042251770#Menghitung nilai standar deviasi
sqrt(var(df$Jumlah_Pengunjung))## [1] 32283.92#Mengubah data ke data time series
tsdata <- ts(df$Jumlah_Pengunjung, start = c(2014, 1), end = c(2023, 12), frequency = 12)
tsdata## Jan Feb Mar Apr May Jun Jul Aug Sep Oct
## 2014 47237 18009 17462 17974 24137 28620 66517 57905 15975 21374
## 2015 50717 22139 22571 24334 39411 24253 118361 34141 24731 25585
## 2016 71349 27932 27068 24150 52206 13814 140578 26706 30904 30941
## 2017 91912 30414 33408 45319 41580 106245 100674 30121 40614 27611
## 2018 71755 34453 39112 48870 37354 176599 77002 37200 45403 33180
## 2019 64179 35357 41663 51756 16184 192900 83148 33242 39699 40015
## 2020 60961 30710 17779 0 0 0 2481 29679 14428 17690
## 2021 12787 5718 13184 10387 46598 24281 336 175 5416 16706
## 2022 39566 21653 22694 4233 115430 45257 41466 19804 19892 18482
## 2023 38087 23004 20936 53182 43658 37191 37456 13467 21674 20818
## Nov Dec
## 2014 22328 46315
## 2015 21433 53813
## 2016 26229 66107
## 2017 20591 64931
## 2018 36639 78096
## 2019 40052 108701
## 2020 15964 10926
## 2021 15982 27019
## 2022 15320 38087
## 2023 16524 47737#Plot data time series
plot.ts(tsdata, xlab = "Tahun", ylab = "Jumlah Pengunjung")
#Menambahkan axis untuk tahun
axis(1, at = seq(2014, 2023, by = 1), labels = seq(2014, 2023, by = 1))
PREPROCESSING DATA
#Untuk data time series hanya digunakan variabel jumlah pengunjung
data <- df$Jumlah_Pengunjung
data## [1] 47237 18009 17462 17974 24137 28620 66517 57905 15975 21374
## [11] 22328 46315 50717 22139 22571 24334 39411 24253 118361 34141
## [21] 24731 25585 21433 53813 71349 27932 27068 24150 52206 13814
## [31] 140578 26706 30904 30941 26229 66107 91912 30414 33408 45319
## [41] 41580 106245 100674 30121 40614 27611 20591 64931 71755 34453
## [51] 39112 48870 37354 176599 77002 37200 45403 33180 36639 78096
## [61] 64179 35357 41663 51756 16184 192900 83148 33242 39699 40015
## [71] 40052 108701 60961 30710 17779 0 0 0 2481 29679
## [81] 14428 17690 15964 10926 12787 5718 13184 10387 46598 24281
## [91] 336 175 5416 16706 15982 27019 39566 21653 22694 4233
## [101] 115430 45257 41466 19804 19892 18482 15320 38087 38087 23004
## [111] 20936 53182 43658 37191 37456 13467 21674 20818 16524 47737#Mengecek missing value
is.na(data)## [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [49] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [61] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [73] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [85] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [97] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [109] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE#Cek outlier pada data
boxplot(data, ylab = "Jumlah Pengunjung")
#Identifikasi outlier (contoh menggunakan metode IQR)
Q1 <- quantile(data, 0.25)
Q3 <- quantile(data, 0.75)
IQR <- IQR(data)
lower_bound <- Q1 - 1.5 * IQR
upper_bound <- Q3 + 1.5 * IQR
#Menampilkan data outlier
outliers <- data[data < lower_bound | data > upper_bound]
outliers## [1] 118361 140578 91912 106245 100674 176599 192900 108701 115430#Transformasi data
transform <- sqrt(data)
transform## [1] 217.34075 134.19762 132.14386 134.06715 155.36087 169.17447 257.90890
## [8] 240.63458 126.39225 146.19850 149.42557 215.20920 225.20435 148.79180
## [15] 150.23648 155.99359 198.52204 155.73375 344.03634 184.77283 157.26093
## [22] 159.95312 146.40014 231.97629 267.11234 167.12869 164.52355 155.40270
## [29] 228.48632 117.53297 374.93733 163.41971 175.79534 175.90054 161.95370
## [36] 257.11282 303.16992 174.39610 182.77855 212.88260 203.91175 325.95245
## [43] 317.29166 173.55403 201.52915 166.16558 143.49564 254.81562 267.87124
## [50] 185.61519 197.76754 221.06560 193.27183 420.23684 277.49234 192.87302
## [57] 213.07980 182.15378 191.41317 279.45662 253.33575 188.03457 204.11516
## [64] 227.49945 127.21635 439.20383 288.35395 182.32389 199.24608 200.03750
## [71] 200.12996 329.69835 246.90281 175.24269 133.33792 0.00000 0.00000
## [78] 0.00000 49.80964 172.27594 120.11661 133.00376 126.34872 104.52751
## [85] 113.07962 75.61746 114.82160 101.91663 215.86570 155.82362 18.33030
## [92] 13.22876 73.59348 129.25169 126.41994 164.37457 198.91204 147.14958
## [99] 150.64528 65.06151 339.74991 212.73693 203.63202 140.72669 141.03900
## [106] 135.94852 123.77399 195.15891 195.15891 151.67070 144.69278 230.61223
## [113] 208.94497 192.84968 193.53553 116.04740 147.22092 144.28444 128.54571
## [120] 218.48799#Simpan hasil gabungan data ke dalam data frame baru
result <- data.frame(transform = transform)
#Tentukan lokasi dan nama file untuk menyimpan hasil
file_name <- "Hasil_transformasidata_akarkuadrat.csv"
#Menyimpan hasil ke dalam file CSV
write.csv(result, file = file_name, row.names = FALSE)
#Tampilkan pesan konfirmasi
print(paste("Hasil telah disimpan dalam file", file_name))## [1] "Hasil telah disimpan dalam file Hasil_transformasidata_akarkuadrat.csv"#Cek outlier pada data setelah transformasi data
boxplot(transform, ylab = "Jumlah Pengunjung")
#Identifikasi outlier setelah transformasi data (contoh menggunakan metode IQR)
Q1_t <- quantile(transform, 0.25)
Q3_t <- quantile(transform, 0.75)
IQR_t <- IQR(transform)
lower_bound_t <- Q1_t - 1.5 * IQR_t
upper_bound_t <- Q3_t + 1.5 * IQR_t
#Menampilkan data outlier
outliers1 <- transform[transform < lower_bound_t | transform > upper_bound_t]
outliers1## [1] 344.03634 374.93733 325.95245 420.23684 439.20383 329.69835 0.00000
## [8] 0.00000 0.00000 18.33030 13.22876 339.74991#Mengubah data ke data time series
datats <- ts(transform, start = c(2014, 1), end = c(2023, 12), frequency = 12)
datats## Jan Feb Mar Apr May Jun Jul
## 2014 217.34075 134.19762 132.14386 134.06715 155.36087 169.17447 257.90890
## 2015 225.20435 148.79180 150.23648 155.99359 198.52204 155.73375 344.03634
## 2016 267.11234 167.12869 164.52355 155.40270 228.48632 117.53297 374.93733
## 2017 303.16992 174.39610 182.77855 212.88260 203.91175 325.95245 317.29166
## 2018 267.87124 185.61519 197.76754 221.06560 193.27183 420.23684 277.49234
## 2019 253.33575 188.03457 204.11516 227.49945 127.21635 439.20383 288.35395
## 2020 246.90281 175.24269 133.33792 0.00000 0.00000 0.00000 49.80964
## 2021 113.07962 75.61746 114.82160 101.91663 215.86570 155.82362 18.33030
## 2022 198.91204 147.14958 150.64528 65.06151 339.74991 212.73693 203.63202
## 2023 195.15891 151.67070 144.69278 230.61223 208.94497 192.84968 193.53553
## Aug Sep Oct Nov Dec
## 2014 240.63458 126.39225 146.19850 149.42557 215.20920
## 2015 184.77283 157.26093 159.95312 146.40014 231.97629
## 2016 163.41971 175.79534 175.90054 161.95370 257.11282
## 2017 173.55403 201.52915 166.16558 143.49564 254.81562
## 2018 192.87302 213.07980 182.15378 191.41317 279.45662
## 2019 182.32389 199.24608 200.03750 200.12996 329.69835
## 2020 172.27594 120.11661 133.00376 126.34872 104.52751
## 2021 13.22876 73.59348 129.25169 126.41994 164.37457
## 2022 140.72669 141.03900 135.94852 123.77399 195.15891
## 2023 116.04740 147.22092 144.28444 128.54571 218.48799#Plot data time series
plot.ts(datats, xlab = "Tahun", ylab = "Jumlah Pengunjung")
#Menambahkan axis untuk tahun
axis(1, at = seq(2014, 2023, by = 1), labels = seq(2014, 2023, by = 1))
#Statistika deskriptif
stat.desc(datats)## x
## nbr.val 120.000000
## nbr.null 3.000000
## nbr.na 0.000000
## min 0.000000
## max 439.203825
## range 439.203825
## sum 21758.150781
## median 174.819395
## mean 181.317923
## SE.mean 6.946108
## CI.mean.0.95 13.753987
## var 5789.809145
## std.dev 76.090795
## coef.var 0.419654summary(datats)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0 141.0 174.8 181.3 213.6 439.2MODEL PERAMALAN HOLT WINTER’S EXPONENTIAL SMOOTHING ADITIF
vis.data <- decompose(datats)
plot(vis.data)
PERBANDINGAN EVALUASI MODEL (RMSE dan MAD) SETIAP MODEL HOLT WINTER’S EXPONENTIAL SMOOTHING ADITIF
Model HoltWinters yang digunakan yaitu model aditif karena data tersebut memiliki nilai-nol atau nilai yang sangat mendekati nol.
#Model holt winter's exponential smoothing aditif yang mungkin
#Fungsi untuk menghitung RMSE
calculate_rmse <- function(actual, predicted) {
rmse <- sqrt(mean((actual - predicted)^2))
return(rmse)
}
#Fungsi untuk menghitung MAD
calculate_mad <- function(actual, predicted) {
mad <- mean(abs(mean(actual) - predicted))
return(mad)
}
#Buat vektor nilai alpha, beta, dan gamma yang akan diuji
alpha_values <- seq(0.1, 1, by = 0.1)
beta_values <- seq(0.1, 1, by = 0.1)
gamma_values <- seq(0.1, 1, by = 0.1)
#Buat dataframe untuk menyimpan hasil eksplorasi
parameter_combinations <- expand.grid(alpha = alpha_values, beta = beta_values, gamma = gamma_values)
#Inisialisasi vektor untuk menyimpan hasil evaluasi model
rmse_values <- numeric(nrow(parameter_combinations))
mad_values <- numeric(nrow(parameter_combinations))
#Data aktual
actual_data <- datats
#Loop melalui setiap kombinasi parameter
for (i in seq_len(nrow(parameter_combinations))) {
current_params <- parameter_combinations[i, ]
#Buat model Holt-Winters dengan parameter saat ini
current_model <- HoltWinters(datats,
alpha = current_params$alpha,
beta = current_params$beta,
gamma = current_params$gamma,
seasonal = "additive")
#Membuat peramalan (fitted values)
predicted_data <- fitted(current_model)[,1]
#Evaluasi model menggunakan RMSE
rmse_values[i] <- calculate_rmse(datats, predicted_data)
#Evaluasi model menggunakan MAD
mad_values[i] <- calculate_mad(datats, predicted_data)
}
#Gabungkan hasil eksplorasi dengan nilai RMSE dan MAD
results <- cbind(parameter_combinations, RMSE = rmse_values, MAD = mad_values)
results## alpha beta gamma RMSE MAD
## 1 0.1 0.1 0.1 75.17761 54.58271
## 2 0.2 0.1 0.1 73.26560 53.48102
## 3 0.3 0.1 0.1 74.02178 54.95973
## 4 0.4 0.1 0.1 75.46913 56.58491
## 5 0.5 0.1 0.1 77.45235 58.58377
## 6 0.6 0.1 0.1 80.04941 60.81435
## 7 0.7 0.1 0.1 83.36977 63.28968
## 8 0.8 0.1 0.1 87.53811 65.74034
## 9 0.9 0.1 0.1 92.69210 68.18868
## 10 1.0 0.1 0.1 99.00214 71.29286
## 11 0.1 0.2 0.1 76.39494 55.69816
## 12 0.2 0.2 0.1 74.98170 54.31915
## 13 0.3 0.2 0.1 76.35238 55.97396
## 14 0.4 0.2 0.1 77.92148 57.76655
## 15 0.5 0.2 0.1 79.99400 60.15916
## 16 0.6 0.2 0.1 82.76701 62.81181
## 17 0.7 0.2 0.1 86.38549 65.54342
## 18 0.8 0.2 0.1 91.00940 68.02892
## 19 0.9 0.2 0.1 96.80892 70.63449
## 20 1.0 0.2 0.1 103.99346 74.15299
## 21 0.1 0.3 0.1 76.67105 55.98796
## 22 0.2 0.3 0.1 77.23708 54.90750
## 23 0.3 0.3 0.1 78.98123 57.19151
## 24 0.4 0.3 0.1 80.37410 59.40439
## 25 0.5 0.3 0.1 82.47486 61.98294
## 26 0.6 0.3 0.1 85.43067 65.05119
## 27 0.7 0.3 0.1 89.37969 67.90228
## 28 0.8 0.3 0.1 94.52040 70.28062
## 29 0.9 0.3 0.1 101.05800 72.98724
## 30 1.0 0.3 0.1 109.24642 77.03412
## 31 0.1 0.4 0.1 77.16081 54.90267
## 32 0.2 0.4 0.1 80.20424 56.37940
## 33 0.3 0.4 0.1 81.57279 58.90431
## 34 0.4 0.4 0.1 82.63149 61.48700
## 35 0.5 0.4 0.1 84.84168 64.04856
## 36 0.6 0.4 0.1 88.02725 67.38970
## 37 0.7 0.4 0.1 92.35418 70.21934
## 38 0.8 0.4 0.1 98.08684 72.57426
## 39 0.9 0.4 0.1 105.47017 75.45472
## 40 1.0 0.4 0.1 114.81036 80.10395
## 41 0.1 0.5 0.1 78.20405 53.77808
## 42 0.2 0.5 0.1 83.70768 58.76919
## 43 0.3 0.5 0.1 83.73490 61.69003
## 44 0.4 0.5 0.1 84.69852 63.42619
## 45 0.5 0.5 0.1 87.14159 66.32913
## 46 0.6 0.5 0.1 90.58153 69.88586
## 47 0.7 0.5 0.1 95.32784 72.63360
## 48 0.8 0.5 0.1 101.73596 74.67819
## 49 0.9 0.5 0.1 110.08591 77.91491
## 50 1.0 0.5 0.1 120.74454 83.49360
## 51 0.1 0.6 0.1 79.76652 53.54777
## 52 0.2 0.6 0.1 87.26305 60.92373
## 53 0.3 0.6 0.1 85.38741 64.02352
## 54 0.4 0.6 0.1 86.71485 65.27133
## 55 0.5 0.6 0.1 89.42322 68.82671
## 56 0.6 0.6 0.1 93.10305 72.63561
## 57 0.7 0.6 0.1 98.31247 74.91551
## 58 0.8 0.6 0.1 105.49582 76.86426
## 59 0.9 0.6 0.1 114.94953 80.67400
## 60 1.0 0.6 0.1 127.11477 87.28601
## 61 0.1 0.7 0.1 81.97736 54.38865
## 62 0.2 0.7 0.1 90.14209 63.64713
## 63 0.3 0.7 0.1 86.80857 65.77427
## 64 0.4 0.7 0.1 88.78037 67.41447
## 65 0.5 0.7 0.1 91.69200 71.51229
## 66 0.6 0.7 0.1 95.57967 75.24894
## 67 0.7 0.7 0.1 101.31920 76.95133
## 68 0.8 0.7 0.1 109.40042 79.18679
## 69 0.9 0.7 0.1 120.11070 83.15751
## 70 1.0 0.7 0.1 133.99458 92.24392
## 71 0.1 0.8 0.1 85.01975 55.87772
## 72 0.2 0.8 0.1 91.74578 66.58534
## 73 0.3 0.8 0.1 88.31714 67.10195
## 74 0.4 0.8 0.1 90.91901 69.61167
## 75 0.5 0.8 0.1 93.91779 74.28773
## 76 0.6 0.8 0.1 97.99524 77.60641
## 77 0.7 0.8 0.1 104.37089 79.04755
## 78 0.8 0.8 0.1 113.49278 81.34623
## 79 0.9 0.8 0.1 125.62328 85.77113
## 80 1.0 0.8 0.1 141.46670 97.50899
## 81 0.1 0.9 0.1 89.09226 59.13850
## 82 0.2 0.9 0.1 92.06801 68.87260
## 83 0.3 0.9 0.1 90.03863 68.00336
## 84 0.4 0.9 0.1 93.11633 71.82634
## 85 0.5 0.9 0.1 96.05109 77.11039
## 86 0.6 0.9 0.1 100.34876 79.42551
## 87 0.7 0.9 0.1 107.50611 80.88356
## 88 0.8 0.9 0.1 117.82133 83.49175
## 89 0.9 0.9 0.1 131.54195 89.03611
## 90 1.0 0.9 0.1 149.62744 103.18043
## 91 0.1 1.0 0.1 94.26693 63.76344
## 92 0.2 1.0 0.1 91.71786 70.44204
## 93 0.3 1.0 0.1 91.95151 69.43890
## 94 0.4 1.0 0.1 95.33399 74.34351
## 95 0.5 1.0 0.1 98.04245 79.57944
## 96 0.6 1.0 0.1 102.66472 81.08391
## 97 0.7 1.0 0.1 110.77382 82.57476
## 98 0.8 1.0 0.1 122.43355 85.46969
## 99 0.9 1.0 0.1 137.92077 93.03219
## 100 1.0 1.0 0.1 158.59754 109.69688
## 101 0.1 0.1 0.2 73.92294 54.37081
## 102 0.2 0.1 0.2 72.04302 52.33925
## 103 0.3 0.1 0.2 72.42395 53.81869
## 104 0.4 0.1 0.2 73.37033 55.52738
## 105 0.5 0.1 0.2 74.91818 57.29973
## 106 0.6 0.1 0.2 77.21814 59.56861
## 107 0.7 0.1 0.2 80.46444 62.15317
## 108 0.8 0.1 0.2 84.92056 64.82101
## 109 0.9 0.1 0.2 90.93021 67.65509
## 110 1.0 0.1 0.2 99.00214 71.29286
## 111 0.1 0.2 0.2 75.21183 55.04151
## 112 0.2 0.2 0.2 74.12033 53.76164
## 113 0.3 0.2 0.2 75.01386 55.45982
## 114 0.4 0.2 0.2 75.88802 57.20371
## 115 0.5 0.2 0.2 77.40837 58.99736
## 116 0.6 0.2 0.2 79.80289 61.48179
## 117 0.7 0.2 0.2 83.28506 64.33781
## 118 0.8 0.2 0.2 88.17591 67.06562
## 119 0.9 0.2 0.2 94.87980 70.00268
## 120 1.0 0.2 0.2 103.99346 74.15299
## 121 0.1 0.3 0.2 75.76667 55.98871
## 122 0.2 0.3 0.2 76.93999 55.10523
## 123 0.3 0.3 0.2 77.93676 57.20036
## 124 0.4 0.3 0.2 78.34773 59.30127
## 125 0.5 0.3 0.2 79.80703 60.94245
## 126 0.6 0.3 0.2 82.31041 63.71736
## 127 0.7 0.3 0.2 86.05517 66.56025
## 128 0.8 0.3 0.2 91.44387 69.31853
## 129 0.9 0.3 0.2 98.94390 72.28301
## 130 1.0 0.3 0.2 109.24642 77.03412
## 131 0.1 0.4 0.2 76.65368 54.88221
## 132 0.2 0.4 0.2 80.72299 57.42582
## 133 0.3 0.4 0.2 80.70817 59.54215
## 134 0.4 0.4 0.2 80.52274 61.53704
## 135 0.5 0.4 0.2 82.08013 62.98674
## 136 0.6 0.4 0.2 84.73038 65.94678
## 137 0.7 0.4 0.2 88.77345 68.77596
## 138 0.8 0.4 0.2 94.74049 71.49696
## 139 0.9 0.4 0.2 103.15490 74.57693
## 140 1.0 0.4 0.2 114.81036 80.10395
## 141 0.1 0.5 0.2 78.14661 53.80855
## 142 0.2 0.5 0.2 85.27672 60.24691
## 143 0.3 0.5 0.2 82.76317 62.46881
## 144 0.4 0.5 0.2 82.48732 63.49920
## 145 0.5 0.5 0.2 84.30198 65.04821
## 146 0.6 0.5 0.2 87.08193 68.22095
## 147 0.7 0.5 0.2 91.45554 71.05704
## 148 0.8 0.5 0.2 98.09635 73.52146
## 149 0.9 0.5 0.2 107.55618 76.87495
## 150 1.0 0.5 0.2 120.74454 83.49360
## 151 0.1 0.6 0.2 80.33546 53.26683
## 152 0.2 0.6 0.2 89.89864 63.13986
## 153 0.3 0.6 0.2 84.03230 64.63230
## 154 0.4 0.6 0.2 84.48499 65.00362
## 155 0.5 0.6 0.2 86.51527 67.43743
## 156 0.6 0.6 0.2 89.36132 70.64907
## 157 0.7 0.6 0.2 94.11375 73.08296
## 158 0.8 0.6 0.2 101.54606 75.40518
## 159 0.9 0.6 0.2 112.19542 79.40513
## 160 1.0 0.6 0.2 127.11477 87.28601
## 161 0.1 0.7 0.2 83.48322 54.38636
## 162 0.2 0.7 0.2 93.41927 67.29939
## 163 0.3 0.7 0.2 85.04137 66.15463
## 164 0.4 0.7 0.2 86.64089 66.73812
## 165 0.5 0.7 0.2 88.69779 69.85790
## 166 0.6 0.7 0.2 91.54632 73.00530
## 167 0.7 0.7 0.2 96.76606 74.90888
## 168 0.8 0.7 0.2 105.13214 77.53558
## 169 0.9 0.7 0.2 117.12541 81.70326
## 170 1.0 0.7 0.2 133.99458 92.24392
## 171 0.1 0.8 0.2 87.93168 57.58854
## 172 0.2 0.8 0.2 94.81121 71.06063
## 173 0.3 0.8 0.2 86.38425 67.28856
## 174 0.4 0.8 0.2 88.92919 68.79206
## 175 0.5 0.8 0.2 90.79627 72.35329
## 176 0.6 0.8 0.2 93.61758 75.08631
## 177 0.7 0.8 0.2 99.44820 76.66333
## 178 0.8 0.8 0.2 108.90594 79.59852
## 179 0.9 0.8 0.2 122.40267 84.08024
## 180 1.0 0.8 0.2 141.46670 97.50899
## 181 0.1 0.9 0.2 94.00743 62.81656
## 182 0.2 0.9 0.2 94.00920 73.22041
## 183 0.3 0.9 0.2 88.27412 68.39849
## 184 0.4 0.9 0.2 91.28296 70.92329
## 185 0.5 0.9 0.2 92.74580 74.78151
## 186 0.6 0.9 0.2 95.57908 76.63700
## 187 0.7 0.9 0.2 102.21511 78.82918
## 188 0.8 0.9 0.2 112.92243 81.65941
## 189 0.9 0.9 0.2 128.08439 86.70441
## 190 1.0 0.9 0.2 149.62744 103.18043
## 191 0.1 1.0 0.2 101.82016 69.59944
## 192 0.2 1.0 0.2 91.93415 73.14815
## 193 0.3 1.0 0.2 90.58535 69.91976
## 194 0.4 1.0 0.2 93.64113 73.49997
## 195 0.5 1.0 0.2 94.48249 76.82985
## 196 0.6 1.0 0.2 97.47125 78.33146
## 197 0.7 1.0 0.2 105.13111 80.74912
## 198 0.8 1.0 0.2 117.23404 83.79753
## 199 0.9 1.0 0.2 134.22742 89.89233
## 200 1.0 1.0 0.2 158.59754 109.69688
## 201 0.1 0.1 0.3 73.84443 54.78227
## 202 0.2 0.1 0.3 72.22643 52.63551
## 203 0.3 0.1 0.3 72.32158 53.91915
## 204 0.4 0.1 0.3 72.75590 55.39606
## 205 0.5 0.1 0.3 73.80370 57.08008
## 206 0.6 0.1 0.3 75.66507 58.61642
## 207 0.7 0.1 0.3 78.58223 61.19588
## 208 0.8 0.1 0.3 82.95817 64.04422
## 209 0.9 0.1 0.3 89.40995 67.14219
## 210 1.0 0.1 0.3 99.00214 71.29286
## 211 0.1 0.2 0.3 75.26144 55.63640
## 212 0.2 0.2 0.3 74.80835 54.52555
## 213 0.3 0.2 0.3 75.32068 56.00232
## 214 0.4 0.2 0.3 75.44177 57.42454
## 215 0.5 0.2 0.3 76.33836 58.95900
## 216 0.6 0.2 0.3 78.22323 60.58705
## 217 0.7 0.2 0.3 81.30834 63.32258
## 218 0.8 0.2 0.3 86.07071 66.25959
## 219 0.9 0.2 0.3 93.22351 69.41096
## 220 1.0 0.2 0.3 103.99346 74.15299
## 221 0.1 0.3 0.3 76.17433 56.40632
## 222 0.2 0.3 0.3 78.42101 56.30721
## 223 0.3 0.3 0.3 78.72671 57.91301
## 224 0.4 0.3 0.3 77.99338 59.90385
## 225 0.5 0.3 0.3 78.74978 61.07242
## 226 0.6 0.3 0.3 80.69540 62.85725
## 227 0.7 0.3 0.3 83.96888 65.51439
## 228 0.8 0.3 0.3 89.17757 68.43419
## 229 0.9 0.3 0.3 97.13798 71.61668
## 230 1.0 0.3 0.3 109.24642 77.03412
## 231 0.1 0.4 0.3 77.52653 55.12440
## 232 0.2 0.4 0.3 83.39793 59.28706
## 233 0.3 0.4 0.3 81.86180 61.00781
## 234 0.4 0.4 0.3 80.13283 62.07579
## 235 0.5 0.4 0.3 81.03723 63.06304
## 236 0.6 0.4 0.3 83.08337 65.05274
## 237 0.7 0.4 0.3 86.56060 67.65064
## 238 0.8 0.4 0.3 92.29467 70.50167
## 239 0.9 0.4 0.3 101.18768 73.76126
## 240 1.0 0.4 0.3 114.81036 80.10395
## 241 0.1 0.5 0.3 79.59318 54.22604
## 242 0.2 0.5 0.3 89.58350 63.13030
## 243 0.3 0.5 0.3 83.90173 64.20701
## 244 0.4 0.5 0.3 82.02231 63.96727
## 245 0.5 0.5 0.3 83.32642 64.93204
## 246 0.6 0.5 0.3 85.40473 67.24333
## 247 0.7 0.5 0.3 89.09349 69.78315
## 248 0.8 0.5 0.3 95.45486 72.47201
## 249 0.9 0.5 0.3 105.41938 75.95685
## 250 1.0 0.5 0.3 120.74454 83.49360
## 251 0.1 0.6 0.3 82.63868 54.61207
## 252 0.2 0.6 0.3 96.05212 67.53130
## 253 0.3 0.6 0.3 84.73123 66.29782
## 254 0.4 0.6 0.3 84.06456 65.53750
## 255 0.5 0.6 0.3 85.66655 67.27007
## 256 0.6 0.6 0.3 87.63858 69.32978
## 257 0.7 0.6 0.3 91.57581 71.71512
## 258 0.8 0.6 0.3 98.69863 74.21288
## 259 0.9 0.6 0.3 109.88493 78.36713
## 260 1.0 0.6 0.3 127.11477 87.28601
## 261 0.1 0.7 0.3 87.12054 58.03631
## 262 0.2 0.7 0.3 101.05227 73.34851
## 263 0.3 0.7 0.3 85.14499 66.79937
## 264 0.4 0.7 0.3 86.45555 67.29805
## 265 0.5 0.7 0.3 88.00336 69.68934
## 266 0.6 0.7 0.3 89.74544 71.51459
## 267 0.7 0.7 0.3 94.02793 73.54017
## 268 0.8 0.7 0.3 102.07745 76.29682
## 269 0.9 0.7 0.3 114.64133 80.62394
## 270 1.0 0.7 0.3 133.99458 92.24392
## 271 0.1 0.8 0.3 93.60500 63.53316
## 272 0.2 0.8 0.3 102.94086 77.93026
## 273 0.3 0.8 0.3 86.13570 67.60834
## 274 0.4 0.8 0.3 89.12220 69.26472
## 275 0.5 0.8 0.3 90.25288 72.12332
## 276 0.6 0.8 0.3 91.69489 73.53000
## 277 0.7 0.8 0.3 96.49539 75.45662
## 278 0.8 0.8 0.3 105.65258 78.66446
## 279 0.9 0.8 0.3 119.74893 83.03659
## 280 1.0 0.8 0.3 141.46670 97.50899
## 281 0.1 0.9 0.3 102.59533 70.37473
## 282 0.2 0.9 0.3 101.52239 80.68955
## 283 0.3 0.9 0.3 88.14933 68.79683
## 284 0.4 0.9 0.3 91.92244 71.77058
## 285 0.5 0.9 0.3 92.33268 74.21392
## 286 0.6 0.9 0.3 93.48603 75.36697
## 287 0.7 0.9 0.3 99.04958 77.68390
## 288 0.8 0.9 0.3 109.48775 80.92627
## 289 0.9 0.9 0.3 125.26853 85.76743
## 290 1.0 0.9 0.3 149.62744 103.18043
## 291 0.1 1.0 0.3 114.23553 79.44582
## 292 0.2 1.0 0.3 97.83777 79.23241
## 293 0.3 1.0 0.3 90.99947 70.41609
## 294 0.4 1.0 0.3 94.75850 74.63522
## 295 0.5 1.0 0.3 94.16402 76.09926
## 296 0.6 1.0 0.3 95.16407 77.26660
## 297 0.7 1.0 0.3 101.77402 79.93391
## 298 0.8 1.0 0.3 113.64216 83.21823
## 299 0.9 1.0 0.3 131.26078 88.52740
## 300 1.0 1.0 0.3 158.59754 109.69688
## 301 0.1 0.1 0.4 74.32134 55.24714
## 302 0.2 0.1 0.4 73.13292 53.30572
## 303 0.3 0.1 0.4 73.02622 54.70896
## 304 0.4 0.1 0.4 72.95567 55.55071
## 305 0.5 0.1 0.4 73.49938 57.12157
## 306 0.6 0.1 0.4 74.89976 58.37990
## 307 0.7 0.1 0.4 77.39739 60.42275
## 308 0.8 0.1 0.4 81.49645 63.38035
## 309 0.9 0.1 0.4 88.09935 66.64859
## 310 1.0 0.1 0.4 99.00214 71.29286
## 311 0.1 0.2 0.4 75.91603 56.70728
## 312 0.2 0.2 0.4 76.35287 55.82922
## 313 0.3 0.2 0.4 76.58335 56.68002
## 314 0.4 0.2 0.4 75.89048 57.93269
## 315 0.5 0.2 0.4 76.13203 59.25195
## 316 0.6 0.2 0.4 77.49086 60.41108
## 317 0.7 0.2 0.4 80.09222 62.54215
## 318 0.8 0.2 0.4 84.51821 65.56191
## 319 0.9 0.2 0.4 91.80265 68.87748
## 320 1.0 0.2 0.4 103.99346 74.15299
## 321 0.1 0.3 0.4 77.23789 56.84917
## 322 0.2 0.3 0.4 81.00748 58.53769
## 323 0.3 0.3 0.4 80.69535 59.20078
## 324 0.4 0.3 0.4 78.60125 60.67619
## 325 0.5 0.3 0.4 78.60167 61.49913
## 326 0.6 0.3 0.4 79.99465 62.67727
## 327 0.7 0.3 0.4 82.71564 64.72793
## 328 0.8 0.3 0.4 87.52178 67.64460
## 329 0.9 0.3 0.4 95.59636 71.01260
## 330 1.0 0.3 0.4 109.24642 77.03412
## 331 0.1 0.4 0.4 79.11336 55.81525
## 332 0.2 0.4 0.4 87.63462 62.97476
## 333 0.3 0.4 0.4 84.45220 63.27320
## 334 0.4 0.4 0.4 80.72821 62.83476
## 335 0.5 0.4 0.4 80.95064 63.52427
## 336 0.6 0.4 0.4 82.43446 64.89588
## 337 0.7 0.4 0.4 85.26423 66.85199
## 338 0.8 0.4 0.4 90.52229 69.65417
## 339 0.9 0.4 0.4 99.51681 73.12085
## 340 1.0 0.4 0.4 114.81036 80.10395
## 341 0.1 0.5 0.4 81.90033 55.47489
## 342 0.2 0.5 0.4 96.20969 68.67361
## 343 0.3 0.5 0.4 86.67191 66.93532
## 344 0.4 0.5 0.4 82.52574 64.64873
## 345 0.5 0.5 0.4 83.38057 65.47422
## 346 0.6 0.5 0.4 84.83255 66.96639
## 347 0.7 0.5 0.4 87.74050 68.83941
## 348 0.8 0.5 0.4 93.55328 71.61638
## 349 0.9 0.5 0.4 103.61430 75.20997
## 350 1.0 0.5 0.4 120.74454 83.49360
## 351 0.1 0.6 0.4 86.09703 58.15525
## 352 0.2 0.6 0.4 105.61797 73.99469
## 353 0.3 0.6 0.4 87.11446 68.95833
## 354 0.4 0.6 0.4 84.60711 66.25922
## 355 0.5 0.6 0.4 85.96470 67.83011
## 356 0.6 0.6 0.4 87.14938 69.06401
## 357 0.7 0.6 0.4 90.14461 70.78696
## 358 0.8 0.6 0.4 96.65968 73.56102
## 359 0.9 0.6 0.4 107.94527 77.46434
## 360 1.0 0.6 0.4 127.11477 87.28601
## 361 0.1 0.7 0.4 92.43549 63.98129
## 362 0.2 0.7 0.4 113.35455 81.28263
## 363 0.3 0.7 0.4 86.78872 68.49716
## 364 0.4 0.7 0.4 87.29575 68.02625
## 365 0.5 0.7 0.4 88.61730 70.36043
## 366 0.6 0.7 0.4 89.32341 71.07043
## 367 0.7 0.7 0.4 92.49479 73.05311
## 368 0.8 0.7 0.4 99.90125 75.76553
## 369 0.9 0.7 0.4 112.57157 79.63803
## 370 1.0 0.7 0.4 133.99458 92.24392
## 371 0.1 0.8 0.4 101.80049 71.80801
## 372 0.2 0.8 0.4 116.92691 88.55583
## 373 0.3 0.8 0.4 87.15954 68.65804
## 374 0.4 0.8 0.4 90.48756 70.40818
## 375 0.5 0.8 0.4 91.21860 73.02256
## 376 0.6 0.8 0.4 91.30826 73.14943
## 377 0.7 0.8 0.4 94.84265 75.12754
## 378 0.8 0.8 0.4 103.34960 78.30583
## 379 0.9 0.8 0.4 117.55804 82.29242
## 380 1.0 0.8 0.4 141.46670 97.50899
## 381 0.1 0.9 0.4 114.91590 81.09591
## 382 0.2 0.9 0.4 116.10519 92.09340
## 383 0.3 0.9 0.4 89.11561 69.24314
## 384 0.4 0.9 0.4 93.95352 73.22931
## 385 0.5 0.9 0.4 93.66843 75.24823
## 386 0.6 0.9 0.4 93.09188 75.20965
## 387 0.7 0.9 0.4 97.27434 77.28172
## 388 0.8 0.9 0.4 107.07838 80.64901
## 389 0.9 0.9 0.4 122.96975 85.02944
## 390 1.0 0.9 0.4 149.62744 103.18043
## 391 0.1 1.0 0.4 131.84727 93.11124
## 392 0.2 1.0 0.4 112.06667 90.57259
## 393 0.3 1.0 0.4 92.50311 70.89561
## 394 0.4 1.0 0.4 97.54821 76.07443
## 395 0.5 1.0 0.4 95.87822 77.60191
## 396 0.6 1.0 0.4 94.71442 77.23824
## 397 0.7 1.0 0.4 99.89472 79.37286
## 398 0.8 1.0 0.4 111.15479 83.12583
## 399 0.9 1.0 0.4 128.87157 87.87326
## 400 1.0 1.0 0.4 158.59754 109.69688
## 401 0.1 0.1 0.5 75.13370 55.97810
## 402 0.2 0.1 0.5 74.50298 54.67068
## 403 0.3 0.1 0.5 74.25890 55.74207
## 404 0.4 0.1 0.5 73.66700 55.96219
## 405 0.5 0.1 0.5 73.69052 57.36527
## 406 0.6 0.1 0.5 74.63240 58.52211
## 407 0.7 0.1 0.5 76.69015 59.89304
## 408 0.8 0.1 0.5 80.41658 62.77802
## 409 0.9 0.1 0.5 86.97038 66.17845
## 410 1.0 0.1 0.5 99.00214 71.29286
## 411 0.1 0.2 0.5 76.95374 57.70781
## 412 0.2 0.2 0.5 78.50480 57.31666
## 413 0.3 0.2 0.5 78.54649 58.15691
## 414 0.4 0.2 0.5 76.93817 58.69239
## 415 0.5 0.2 0.5 76.46141 59.76216
## 416 0.6 0.2 0.5 77.29330 60.68420
## 417 0.7 0.2 0.5 79.39476 62.09152
## 418 0.8 0.2 0.5 83.38476 64.92038
## 419 0.9 0.2 0.5 90.58464 68.39372
## 420 1.0 0.2 0.5 103.99346 74.15299
## 421 0.1 0.3 0.5 78.72455 57.92627
## 422 0.2 0.3 0.5 84.49361 61.47270
## 423 0.3 0.3 0.5 83.65173 61.24413
## 424 0.4 0.3 0.5 79.89143 61.59382
## 425 0.5 0.3 0.5 79.01895 62.00291
## 426 0.6 0.3 0.5 79.86960 63.04435
## 427 0.7 0.3 0.5 82.02896 64.31699
## 428 0.8 0.3 0.5 86.32630 66.96376
## 429 0.9 0.3 0.5 94.28109 70.51680
## 430 1.0 0.3 0.5 109.24642 77.03412
## 431 0.1 0.4 0.5 81.20040 57.26067
## 432 0.2 0.4 0.5 93.32407 67.74624
## 433 0.3 0.4 0.5 88.39893 66.02907
## 434 0.4 0.4 0.5 82.04629 63.78713
## 435 0.5 0.4 0.5 81.45453 63.98733
## 436 0.6 0.4 0.5 82.41575 65.27803
## 437 0.7 0.4 0.5 84.59159 66.33951
## 438 0.8 0.4 0.5 89.25494 68.96093
## 439 0.9 0.4 0.5 98.09794 72.57306
## 440 1.0 0.4 0.5 114.81036 80.10395
## 441 0.1 0.5 0.5 84.91245 58.22930
## 442 0.2 0.5 0.5 105.22209 76.02519
## 443 0.3 0.5 0.5 91.13835 70.84727
## 444 0.4 0.5 0.5 83.73821 65.45644
## 445 0.5 0.5 0.5 84.07073 66.13148
## 446 0.6 0.5 0.5 84.96704 67.34655
## 447 0.7 0.5 0.5 87.07703 68.20495
## 448 0.8 0.5 0.5 92.20332 71.05887
## 449 0.9 0.5 0.5 102.08899 74.54218
## 450 1.0 0.5 0.5 120.74454 83.49360
## 451 0.1 0.6 0.5 90.66096 63.50614
## 452 0.2 0.6 0.5 118.95172 83.80507
## 453 0.3 0.6 0.5 91.44074 72.56743
## 454 0.4 0.6 0.5 85.83419 67.24608
## 455 0.5 0.6 0.5 86.98728 68.64987
## 456 0.6 0.6 0.5 87.46646 69.56102
## 457 0.7 0.6 0.5 89.47420 70.61880
## 458 0.8 0.6 0.5 95.21918 73.45408
## 459 0.9 0.6 0.5 106.31539 76.70285
## 460 1.0 0.6 0.5 127.11477 87.28601
## 461 0.1 0.7 0.5 99.56572 71.20338
## 462 0.2 0.7 0.5 131.03722 92.37084
## 463 0.3 0.7 0.5 90.40189 71.67041
## 464 0.4 0.7 0.5 88.85584 69.00469
## 465 0.5 0.7 0.5 90.09306 71.73001
## 466 0.6 0.7 0.5 89.82785 71.80834
## 467 0.7 0.7 0.5 91.79556 72.89049
## 468 0.8 0.7 0.5 98.36963 75.63940
## 469 0.9 0.7 0.5 110.84414 79.14485
## 470 1.0 0.7 0.5 133.99458 92.24392
## 471 0.1 0.8 0.5 112.97501 81.91015
## 472 0.2 0.8 0.5 137.75225 102.17883
## 473 0.3 0.8 0.5 89.88285 70.68465
## 474 0.4 0.8 0.5 92.69639 71.85611
## 475 0.5 0.8 0.5 93.23016 74.51325
## 476 0.6 0.8 0.5 91.98707 73.78503
## 477 0.7 0.8 0.5 94.09544 74.93513
## 478 0.8 0.8 0.5 101.73679 78.03053
## 479 0.9 0.8 0.5 115.74506 81.91121
## 480 1.0 0.8 0.5 141.46670 97.50899
## 481 0.1 0.9 0.5 131.82568 94.72333
## 482 0.2 0.9 0.5 138.99772 107.12928
## 483 0.3 0.9 0.5 91.49019 70.33726
## 484 0.4 0.9 0.5 97.03720 75.24167
## 485 0.5 0.9 0.5 96.28051 77.48942
## 486 0.6 0.9 0.5 93.91923 76.04531
## 487 0.7 0.9 0.5 96.47273 76.83112
## 488 0.8 0.9 0.5 105.40525 80.56087
## 489 0.9 0.9 0.5 121.08786 84.70087
## 490 1.0 0.9 0.5 149.62744 103.18043
## 491 0.1 1.0 0.5 155.77027 109.49939
## 492 0.2 1.0 0.5 136.64599 107.41079
## 493 0.3 1.0 0.5 95.31555 71.68298
## 494 0.4 1.0 0.5 101.68036 78.64780
## 495 0.5 1.0 0.5 99.15095 80.52751
## 496 0.6 1.0 0.5 95.65382 78.09402
## 497 0.7 1.0 0.5 99.05523 78.75414
## 498 0.8 1.0 0.5 109.45170 83.27795
## 499 0.9 1.0 0.5 126.94197 87.52537
## 500 1.0 1.0 0.5 158.59754 109.69688
## 501 0.1 0.1 0.6 76.21598 56.97721
## 502 0.2 0.1 0.6 76.25043 55.86827
## 503 0.3 0.1 0.6 75.92030 56.83767
## 504 0.4 0.1 0.6 74.76761 56.95814
## 505 0.5 0.1 0.6 74.22604 57.64923
## 506 0.6 0.1 0.6 74.69976 58.74778
## 507 0.7 0.1 0.6 76.31685 59.79856
## 508 0.8 0.1 0.6 79.62881 62.22787
## 509 0.9 0.1 0.6 85.99854 65.74385
## 510 1.0 0.1 0.6 99.00214 71.29286
## 511 0.1 0.2 0.6 78.30764 58.85168
## 512 0.2 0.2 0.6 81.19786 59.59476
## 513 0.3 0.2 0.6 81.14260 60.02224
## 514 0.4 0.2 0.6 78.48098 59.68670
## 515 0.5 0.2 0.6 77.17609 60.26795
## 516 0.6 0.2 0.6 77.45820 60.99996
## 517 0.7 0.2 0.6 79.06003 62.00179
## 518 0.8 0.2 0.6 82.57056 64.32944
## 519 0.9 0.2 0.6 89.54122 67.98351
## 520 1.0 0.2 0.6 103.99346 74.15299
## 521 0.1 0.3 0.6 80.56971 58.72818
## 522 0.2 0.3 0.6 88.86753 65.19160
## 523 0.3 0.3 0.6 87.60439 64.32359
## 524 0.4 0.3 0.6 81.79021 62.77410
## 525 0.5 0.3 0.6 79.85182 62.57254
## 526 0.6 0.3 0.6 80.13720 63.46415
## 527 0.7 0.3 0.6 81.73961 64.22088
## 528 0.8 0.3 0.6 85.48040 66.39166
## 529 0.9 0.3 0.6 93.15954 70.06907
## 530 1.0 0.3 0.6 109.24642 77.03412
## 531 0.1 0.4 0.6 83.76315 58.85534
## 532 0.2 0.4 0.6 100.54627 73.13521
## 533 0.3 0.4 0.6 93.83230 69.80580
## 534 0.4 0.4 0.6 84.05076 65.09251
## 535 0.5 0.4 0.6 82.39592 64.44839
## 536 0.6 0.4 0.6 82.83220 65.79856
## 537 0.7 0.4 0.6 84.35989 66.32546
## 538 0.8 0.4 0.6 88.37032 68.50954
## 539 0.9 0.4 0.6 96.89337 72.05079
## 540 1.0 0.4 0.6 114.81036 80.10395
## 541 0.1 0.5 0.6 88.68096 62.86529
## 542 0.2 0.5 0.6 116.82308 84.44125
## 543 0.3 0.5 0.6 97.57230 75.67849
## 544 0.4 0.5 0.6 85.65452 66.73447
## 545 0.5 0.5 0.6 85.23694 66.79788
## 546 0.6 0.5 0.6 85.60191 68.11788
## 547 0.7 0.5 0.6 86.90760 68.49778
## 548 0.8 0.5 0.6 91.27071 70.97110
## 549 0.9 0.5 0.6 100.79976 73.97327
## 550 1.0 0.5 0.6 120.74454 83.49360
## 551 0.1 0.6 0.6 96.50650 69.58410
## 552 0.2 0.6 0.6 136.44717 95.75725
## 553 0.3 0.6 0.6 98.19524 77.67312
## 554 0.4 0.6 0.6 87.74743 68.16424
## 555 0.5 0.6 0.6 88.56824 69.73542
## 556 0.6 0.6 0.6 88.37545 70.63938
## 557 0.7 0.6 0.6 89.35844 70.75599
## 558 0.8 0.6 0.6 94.23064 73.31547
## 559 0.9 0.6 0.6 104.94455 76.22179
## 560 1.0 0.6 0.6 127.11477 87.28601
## 561 0.1 0.7 0.6 108.93400 79.26400
## 562 0.2 0.7 0.6 154.75946 106.98377
## 563 0.3 0.7 0.6 96.76312 76.82026
## 564 0.4 0.7 0.6 91.13606 70.11510
## 565 0.5 0.7 0.6 92.26369 73.16665
## 566 0.6 0.7 0.6 91.04146 73.10940
## 567 0.7 0.7 0.6 91.71669 72.92774
## 568 0.8 0.7 0.6 97.32382 75.52749
## 569 0.9 0.7 0.6 109.40006 78.93817
## 570 1.0 0.7 0.6 133.99458 92.24392
## 571 0.1 0.8 0.6 127.99205 93.16672
## 572 0.2 0.8 0.6 166.20303 119.88856
## 573 0.3 0.8 0.6 95.20869 75.46255
## 574 0.4 0.8 0.6 95.75234 73.52112
## 575 0.5 0.8 0.6 96.12722 76.77205
## 576 0.6 0.8 0.6 93.51864 75.20132
## 577 0.7 0.8 0.6 94.03684 74.79191
## 578 0.8 0.8 0.6 100.64311 77.81156
## 579 0.9 0.8 0.6 114.24139 81.93423
## 580 1.0 0.8 0.6 141.46670 97.50899
## 581 0.1 0.9 0.6 154.68809 110.68684
## 582 0.2 0.9 0.6 170.73462 125.89173
## 583 0.3 0.9 0.6 96.08697 74.17444
## 584 0.4 0.9 0.6 101.18544 77.70090
## 585 0.5 0.9 0.6 100.02056 80.37646
## 586 0.6 0.9 0.6 95.77220 77.63003
## 587 0.7 0.9 0.6 96.42836 76.53888
## 588 0.8 0.9 0.6 104.28560 80.32977
## 589 0.9 0.9 0.6 119.54327 84.70808
## 590 1.0 0.9 0.6 149.62744 103.18043
## 591 0.1 1.0 0.6 187.61052 129.33702
## 592 0.2 1.0 0.6 171.93215 128.54806
## 593 0.3 1.0 0.6 100.16694 75.49854
## 594 0.4 1.0 0.6 107.18208 82.60167
## 595 0.5 1.0 0.6 103.84639 84.20557
## 596 0.6 1.0 0.6 97.82199 79.67973
## 597 0.7 1.0 0.6 99.04282 78.08742
## 598 0.8 1.0 0.6 108.33962 83.11019
## 599 0.9 1.0 0.6 125.38009 87.37481
## 600 1.0 1.0 0.6 158.59754 109.69688
## 601 0.1 0.1 0.7 77.56081 58.17856
## 602 0.2 0.1 0.7 78.36004 57.79290
## 603 0.3 0.1 0.7 77.98501 58.43676
## 604 0.4 0.1 0.7 76.21798 58.16687
## 605 0.5 0.1 0.7 75.04009 58.10110
## 606 0.6 0.1 0.7 75.01424 58.99422
## 607 0.7 0.1 0.7 76.18611 59.94342
## 608 0.8 0.1 0.7 79.06634 61.71923
## 609 0.9 0.1 0.7 85.16259 65.39110
## 610 1.0 0.1 0.7 99.00214 71.29286
## 611 0.1 0.2 0.7 79.96944 59.87285
## 612 0.2 0.2 0.7 84.44549 62.25208
## 613 0.3 0.2 0.7 84.37982 63.00990
## 614 0.4 0.2 0.7 80.50253 61.30206
## 615 0.5 0.2 0.7 78.21689 60.78252
## 616 0.6 0.2 0.7 77.89586 61.37388
## 617 0.7 0.2 0.7 78.99043 62.26843
## 618 0.8 0.2 0.7 82.00222 63.81734
## 619 0.9 0.2 0.7 88.64810 67.59940
## 620 1.0 0.2 0.7 103.99346 74.15299
## 621 0.1 0.3 0.7 82.77996 59.71925
## 622 0.2 0.3 0.7 94.20836 69.60410
## 623 0.3 0.3 0.7 92.64071 68.66009
## 624 0.4 0.3 0.7 84.31632 64.44502
## 625 0.5 0.3 0.7 81.04936 63.12360
## 626 0.6 0.3 0.7 80.70504 63.90701
## 627 0.7 0.3 0.7 81.74353 64.50763
## 628 0.8 0.3 0.7 84.90377 66.06496
## 629 0.9 0.3 0.7 92.20388 69.64060
## 630 1.0 0.3 0.7 109.24642 77.03412
## 631 0.1 0.4 0.7 86.85878 62.19932
## 632 0.2 0.4 0.7 109.45920 80.01383
## 633 0.3 0.4 0.7 100.96278 75.01583
## 634 0.4 0.4 0.7 86.80653 66.70932
## 635 0.5 0.4 0.7 83.72995 65.12730
## 636 0.6 0.4 0.7 83.58816 66.30617
## 637 0.7 0.4 0.7 84.45902 66.64802
## 638 0.8 0.4 0.7 87.78116 68.36273
## 639 0.9 0.4 0.7 95.87122 71.57963
## 640 1.0 0.4 0.7 114.81036 80.10395
## 641 0.1 0.5 0.7 93.33250 67.61211
## 642 0.2 0.5 0.7 131.20721 94.32541
## 643 0.3 0.5 0.7 106.29500 81.50678
## 644 0.4 0.5 0.7 88.39143 68.35754
## 645 0.5 0.5 0.7 86.83707 67.61260
## 646 0.6 0.5 0.7 86.63958 69.06073
## 647 0.7 0.5 0.7 87.11755 68.99041
## 648 0.8 0.5 0.7 90.66154 70.88244
## 649 0.9 0.5 0.7 99.71012 73.55699
## 650 1.0 0.5 0.7 120.74454 83.49360
## 651 0.1 0.6 0.7 103.88728 76.01450
## 652 0.2 0.6 0.7 158.34828 109.62965
## 653 0.3 0.6 0.7 107.85325 84.28972
## 654 0.4 0.6 0.7 90.49066 70.10198
## 655 0.5 0.6 0.7 90.66858 70.73639
## 656 0.6 0.6 0.7 89.78036 72.08192
## 657 0.7 0.6 0.7 89.68090 71.17128
## 658 0.8 0.6 0.7 93.59416 73.18446
## 659 0.9 0.6 0.7 103.79096 76.09403
## 660 1.0 0.6 0.7 127.11477 87.28601
## 661 0.1 0.7 0.7 121.11629 89.01684
## 662 0.2 0.7 0.7 184.95457 124.82494
## 663 0.3 0.7 0.7 106.68837 84.92829
## 664 0.4 0.7 0.7 94.29455 72.09611
## 665 0.5 0.7 0.7 95.09593 74.91901
## 666 0.6 0.7 0.7 92.87484 74.85068
## 667 0.7 0.7 0.7 92.14400 73.24684
## 668 0.8 0.7 0.7 96.65877 75.39435
## 669 0.9 0.7 0.7 108.19147 78.90528
## 670 1.0 0.7 0.7 133.99458 92.24392
## 671 0.1 0.8 0.7 147.99818 106.86857
## 672 0.2 0.8 0.7 202.90074 140.71250
## 673 0.3 0.8 0.7 104.24678 83.22273
## 674 0.4 0.8 0.7 99.83382 76.43459
## 675 0.5 0.8 0.7 99.88454 79.39144
## 676 0.6 0.8 0.7 95.82636 77.31997
## 677 0.7 0.8 0.7 94.56009 75.06382
## 678 0.8 0.8 0.7 99.95974 77.63337
## 679 0.9 0.8 0.7 112.99227 81.96678
## 680 1.0 0.8 0.7 141.46670 97.50899
## 681 0.1 0.9 0.7 185.22278 130.15462
## 682 0.2 0.9 0.7 211.60725 148.53822
## 683 0.3 0.9 0.7 104.00404 80.74638
## 684 0.4 0.9 0.7 106.59126 81.71572
## 685 0.5 0.9 0.7 104.86884 83.84813
## 686 0.6 0.9 0.7 98.59661 80.10234
## 687 0.7 0.9 0.7 97.04705 76.57248
## 688 0.8 0.9 0.7 103.60883 80.12719
## 689 0.9 0.9 0.7 118.27350 84.81930
## 690 1.0 0.9 0.7 149.62744 103.18043
## 691 0.1 1.0 0.7 229.31535 153.23449
## 692 0.2 1.0 0.7 217.42330 153.13065
## 693 0.3 1.0 0.7 108.07717 83.05036
## 694 0.4 1.0 0.7 114.24724 87.94125
## 695 0.5 1.0 0.7 109.93950 88.44826
## 696 0.6 1.0 0.7 101.20463 82.28192
## 697 0.7 1.0 0.7 99.77948 77.58322
## 698 0.8 1.0 0.7 107.70925 83.13608
## 699 0.9 1.0 0.7 124.11508 87.79551
## 700 1.0 1.0 0.7 158.59754 109.69688
## 701 0.1 0.1 0.8 79.18393 59.31523
## 702 0.2 0.1 0.8 80.84989 60.24362
## 703 0.3 0.1 0.8 80.45743 60.76620
## 704 0.4 0.1 0.8 78.01487 59.70185
## 705 0.5 0.1 0.8 76.10905 58.78311
## 706 0.6 0.1 0.8 75.53162 59.24494
## 707 0.7 0.1 0.8 76.24130 60.13371
## 708 0.8 0.1 0.8 78.68015 61.42662
## 709 0.9 0.1 0.8 84.44421 65.05261
## 710 1.0 0.1 0.8 99.00214 71.29286
## 711 0.1 0.2 0.8 81.95584 60.79854
## 712 0.2 0.2 0.8 88.30535 65.28940
## 713 0.3 0.2 0.8 88.29476 66.38542
## 714 0.4 0.2 0.8 83.02525 63.03531
## 715 0.5 0.2 0.8 79.56926 61.48238
## 716 0.6 0.2 0.8 78.56338 61.76533
## 717 0.7 0.2 0.8 79.12675 62.55590
## 718 0.8 0.2 0.8 81.62656 63.54673
## 719 0.9 0.2 0.8 87.88444 67.24454
## 720 1.0 0.2 0.8 103.99346 74.15299
## 721 0.1 0.3 0.8 85.39647 61.44278
## 722 0.2 0.3 0.8 100.65562 74.73293
## 723 0.3 0.3 0.8 98.87275 73.50515
## 724 0.4 0.3 0.8 87.52844 66.62605
## 725 0.5 0.3 0.8 82.60878 63.97767
## 726 0.6 0.3 0.8 81.53143 64.34608
## 727 0.7 0.3 0.8 81.97892 64.94612
## 728 0.8 0.3 0.8 84.53905 65.82481
## 729 0.9 0.3 0.8 91.39048 69.22913
## 730 1.0 0.3 0.8 109.24642 77.03412
## 731 0.1 0.4 0.8 90.57136 65.59625
## 732 0.2 0.4 0.8 120.27089 87.77676
## 733 0.3 0.4 0.8 110.02236 81.34770
## 734 0.4 0.4 0.8 90.41887 69.22882
## 735 0.5 0.4 0.8 85.46511 66.04196
## 736 0.6 0.4 0.8 84.64283 66.86824
## 737 0.7 0.4 0.8 84.82534 67.29805
## 738 0.8 0.4 0.8 87.42623 68.29059
## 739 0.9 0.4 0.8 95.00475 71.16654
## 740 1.0 0.4 0.8 114.81036 80.10395
## 741 0.1 0.5 0.8 98.99044 72.09820
## 742 0.2 0.5 0.8 148.52302 105.31293
## 743 0.3 0.5 0.8 117.61822 88.69432
## 744 0.4 0.5 0.8 92.12138 70.93962
## 745 0.5 0.5 0.8 88.88738 68.97042
## 746 0.6 0.5 0.8 88.04131 70.02095
## 747 0.7 0.5 0.8 87.64320 69.72609
## 748 0.8 0.5 0.8 90.31136 70.79507
## 749 0.9 0.5 0.8 98.78984 73.48861
## 750 1.0 0.5 0.8 120.74454 83.49360
## 751 0.1 0.6 0.8 113.06484 82.80861
## 752 0.2 0.6 0.8 184.68951 125.09591
## 753 0.3 0.6 0.8 120.76694 92.50368
## 754 0.4 0.6 0.8 94.28496 72.48911
## 755 0.5 0.6 0.8 93.31132 72.96151
## 756 0.6 0.6 0.8 91.64746 73.54194
## 757 0.7 0.6 0.8 90.38079 71.93317
## 758 0.8 0.6 0.8 93.24308 73.04389
## 759 0.9 0.6 0.8 102.82048 75.98202
## 760 1.0 0.6 0.8 127.11477 87.28601
## 761 0.1 0.7 0.8 136.80680 99.78465
## 762 0.2 0.7 0.8 221.75758 145.17819
## 763 0.3 0.7 0.8 120.79002 94.92712
## 764 0.4 0.7 0.8 98.58011 75.14210
## 765 0.5 0.7 0.8 98.61737 77.87704
## 766 0.6 0.7 0.8 95.30300 76.72620
## 767 0.7 0.7 0.8 93.02394 73.70414
## 768 0.8 0.7 0.8 96.30701 75.19189
## 769 0.9 0.7 0.8 107.17988 78.89932
## 770 1.0 0.7 0.8 133.99458 92.24392
## 771 0.1 0.8 0.8 174.37029 123.68627
## 772 0.2 0.8 0.8 248.33442 164.08365
## 773 0.3 0.8 0.8 118.02490 93.88413
## 774 0.4 0.8 0.8 105.22521 80.17068
## 775 0.5 0.8 0.8 104.52734 82.86631
## 776 0.6 0.8 0.8 98.89766 79.86407
## 777 0.7 0.8 0.8 95.62487 75.62320
## 778 0.8 0.8 0.8 99.62010 77.46227
## 779 0.9 0.8 0.8 111.95443 81.99878
## 780 1.0 0.8 0.8 141.46670 97.50899
## 781 0.1 0.9 0.8 225.40073 153.69188
## 782 0.2 0.9 0.8 262.09222 176.30453
## 783 0.3 0.9 0.8 116.40594 90.54694
## 784 0.4 0.9 0.8 113.56312 86.10239
## 785 0.5 0.9 0.8 110.83249 88.34570
## 786 0.6 0.9 0.8 102.39746 83.13071
## 787 0.7 0.9 0.8 98.30813 77.02238
## 788 0.8 0.9 0.8 103.31174 79.90807
## 789 0.9 0.9 0.8 117.23012 85.05118
## 790 1.0 0.9 0.8 149.62744 103.18043
## 791 0.1 1.0 0.8 283.04983 183.37157
## 792 0.2 1.0 0.8 272.73301 181.75288
## 793 0.3 1.0 0.8 120.15430 92.33256
## 794 0.4 1.0 0.8 123.17684 93.62273
## 795 0.5 1.0 0.8 117.39375 93.93323
## 796 0.6 1.0 0.8 105.83603 85.85764
## 797 0.7 1.0 0.8 101.26864 77.95782
## 798 0.8 1.0 0.8 107.50472 83.22956
## 799 0.9 1.0 0.8 123.09311 88.49843
## 800 1.0 1.0 0.8 158.59754 109.69688
## 801 0.1 0.1 0.9 81.11408 60.41985
## 802 0.2 0.1 0.9 83.75934 62.70306
## 803 0.3 0.1 0.9 83.35543 63.30877
## 804 0.4 0.1 0.9 80.17105 61.28917
## 805 0.5 0.1 0.9 77.42939 59.87986
## 806 0.6 0.1 0.9 76.23171 59.61135
## 807 0.7 0.1 0.9 76.44843 60.33783
## 808 0.8 0.1 0.9 78.43465 61.31163
## 809 0.9 0.1 0.9 83.82764 64.72377
## 810 1.0 0.1 0.9 99.00214 71.29286
## 811 0.1 0.2 0.9 84.30038 62.06862
## 812 0.2 0.2 0.9 92.86959 69.09924
## 813 0.3 0.2 0.9 92.93369 70.10424
## 814 0.4 0.2 0.9 86.08974 65.23884
## 815 0.5 0.2 0.9 81.23958 62.61405
## 816 0.6 0.2 0.9 79.44346 62.13444
## 817 0.7 0.2 0.9 79.43433 62.87177
## 818 0.8 0.2 0.9 81.40552 63.50685
## 819 0.9 0.2 0.9 87.23249 66.89933
## 820 1.0 0.2 0.9 103.99346 74.15299
## 821 0.1 0.3 0.9 88.47943 63.81310
## 822 0.2 0.3 0.9 108.40630 80.49182
## 823 0.3 0.3 0.9 106.41026 78.79089
## 824 0.4 0.3 0.9 91.50356 69.16905
## 825 0.5 0.3 0.9 84.54980 65.03483
## 826 0.6 0.3 0.9 82.60212 64.77086
## 827 0.7 0.3 0.9 82.41075 65.37059
## 828 0.8 0.3 0.9 84.34585 65.78969
## 829 0.9 0.3 0.9 90.69941 68.88451
## 830 1.0 0.3 0.9 109.24642 77.03412
## 831 0.1 0.4 0.9 94.97485 69.08892
## 832 0.2 0.4 0.9 133.24806 96.44059
## 833 0.3 0.4 0.9 121.23241 88.25355
## 834 0.4 0.4 0.9 95.00471 72.36375
## 835 0.5 0.4 0.9 87.63604 67.61150
## 836 0.6 0.4 0.9 85.98467 67.55642
## 837 0.7 0.4 0.9 85.42422 67.95365
## 838 0.8 0.4 0.9 87.26326 68.30029
## 839 0.9 0.4 0.9 94.27161 70.95554
## 840 1.0 0.4 0.9 114.81036 80.10395
## 841 0.1 0.5 0.9 105.73419 77.27802
## 842 0.2 0.5 0.9 168.89324 117.46896
## 843 0.3 0.5 0.9 131.83224 97.63958
## 844 0.4 0.5 0.9 97.03505 74.10392
## 845 0.5 0.5 0.9 91.43394 71.05970
## 846 0.6 0.5 0.9 89.79881 71.16993
## 847 0.7 0.5 0.9 88.45211 70.65730
## 848 0.8 0.5 0.9 90.17688 70.72063
## 849 0.9 0.5 0.9 98.01399 73.40138
## 850 1.0 0.5 0.9 120.74454 83.49360
## 851 0.1 0.6 0.9 124.32234 90.49845
## 852 0.2 0.6 0.9 215.31291 142.23953
## 853 0.3 0.6 0.9 137.15762 102.32506
## 854 0.4 0.6 0.9 99.39230 76.43523
## 855 0.5 0.6 0.9 96.55106 75.64021
## 856 0.6 0.6 0.9 93.97342 75.26065
## 857 0.7 0.6 0.9 91.43076 72.93545
## 858 0.8 0.6 0.9 93.13423 72.93591
## 859 0.9 0.6 0.9 102.00544 75.87534
## 860 1.0 0.6 0.9 127.11477 87.28601
## 861 0.1 0.7 0.9 156.87514 112.04531
## 862 0.2 0.7 0.9 264.97581 167.66617
## 863 0.3 0.7 0.9 139.37484 106.39084
## 864 0.4 0.7 0.9 104.29662 79.84972
## 865 0.5 0.7 0.9 102.88340 81.16674
## 866 0.6 0.7 0.9 98.32932 79.09416
## 867 0.7 0.7 0.9 94.33865 74.69244
## 868 0.8 0.7 0.9 96.22692 74.99689
## 869 0.9 0.7 0.9 106.33466 78.84559
## 870 1.0 0.7 0.9 133.99458 92.24392
## 871 0.1 0.8 0.9 208.72786 143.08773
## 872 0.2 0.8 0.9 302.78193 191.57309
## 873 0.3 0.8 0.9 137.25975 107.24852
## 874 0.4 0.8 0.9 112.28234 85.50011
## 875 0.5 0.8 0.9 110.09326 86.65658
## 876 0.6 0.8 0.9 102.74338 82.92660
## 877 0.7 0.8 0.9 97.22923 76.56415
## 878 0.8 0.8 0.9 99.58592 77.22794
## 879 0.9 0.8 0.9 111.09410 81.99672
## 880 1.0 0.8 0.9 141.46670 97.50899
## 881 0.1 0.9 0.9 277.40951 180.90533
## 882 0.2 0.9 0.9 322.92566 206.65652
## 883 0.3 0.9 0.9 134.30188 102.92446
## 884 0.4 0.9 0.9 122.50189 92.41378
## 885 0.5 0.9 0.9 117.89889 92.92288
## 886 0.6 0.9 0.9 107.19345 86.38755
## 887 0.7 0.9 0.9 100.23348 78.04133
## 888 0.8 0.9 0.9 103.36182 79.75893
## 889 0.9 0.9 0.9 116.37613 85.36824
## 890 1.0 0.9 0.9 149.62744 103.18043
## 891 0.1 1.0 0.9 351.12983 219.41280
## 892 0.2 1.0 0.9 338.01532 214.74824
## 893 0.3 1.0 0.9 137.40228 103.78369
## 894 0.4 1.0 0.9 134.38149 99.91238
## 895 0.5 1.0 0.9 126.10226 99.75533
## 896 0.6 1.0 0.9 111.74683 90.00870
## 897 0.7 1.0 0.9 103.56243 79.33469
## 898 0.8 1.0 0.9 107.70307 83.20741
## 899 0.9 1.0 0.9 122.27393 89.07869
## 900 1.0 1.0 0.9 158.59754 109.69688
## 901 0.1 0.1 1.0 83.39472 62.06236
## 902 0.2 0.1 1.0 87.14639 65.21546
## 903 0.3 0.1 1.0 86.70592 66.21568
## 904 0.4 0.1 1.0 82.70730 63.26452
## 905 0.5 0.1 1.0 79.00718 61.14380
## 906 0.6 0.1 1.0 77.10708 60.47185
## 907 0.7 0.1 1.0 76.78782 60.53040
## 908 0.8 0.1 1.0 78.30436 61.40483
## 909 0.9 0.1 1.0 83.29941 64.40586
## 910 1.0 0.1 1.0 99.00214 71.29286
## 911 0.1 0.2 1.0 87.05598 63.66100
## 912 0.2 0.2 1.0 98.26346 73.35637
## 913 0.3 0.2 1.0 98.34621 74.21505
## 914 0.4 0.2 1.0 89.74695 68.09168
## 915 0.5 0.2 1.0 83.24419 64.10274
## 916 0.6 0.2 1.0 80.53228 62.85285
## 917 0.7 0.2 1.0 79.89396 63.19041
## 918 0.8 0.2 1.0 81.31227 63.57181
## 919 0.9 0.2 1.0 86.67722 66.58508
## 920 1.0 0.2 1.0 103.99346 74.15299
## 921 0.1 0.3 1.0 92.09870 66.81757
## 922 0.2 0.3 1.0 117.72033 87.39855
## 923 0.3 0.3 1.0 115.34533 84.37369
## 924 0.4 0.3 1.0 96.32970 72.32560
## 925 0.5 0.3 1.0 86.90251 66.73251
## 926 0.6 0.3 1.0 83.91732 65.46667
## 927 0.7 0.3 1.0 83.02052 65.76830
## 928 0.8 0.3 1.0 84.29619 65.98819
## 929 0.9 0.3 1.0 90.11392 68.68445
## 930 1.0 0.3 1.0 109.24642 77.03412
## 931 0.1 0.4 1.0 100.10876 73.37926
## 932 0.2 0.4 1.0 148.73772 106.25553
## 933 0.3 0.4 1.0 134.78013 96.18274
## 934 0.4 0.4 1.0 100.68036 76.11089
## 935 0.5 0.4 1.0 90.29065 69.45769
## 936 0.6 0.4 1.0 87.61730 68.55711
## 937 0.7 0.4 1.0 86.23868 68.57660
## 938 0.8 0.4 1.0 87.26367 68.47998
## 939 0.9 0.4 1.0 93.65329 70.90407
## 940 1.0 0.4 1.0 114.81036 80.10395
## 941 0.1 0.5 1.0 113.61403 83.13642
## 942 0.2 0.5 1.0 192.46175 130.79694
## 943 0.3 0.5 1.0 149.20834 108.91650
## 944 0.4 0.5 1.0 103.31663 78.70690
## 945 0.5 0.5 1.0 94.53973 73.37883
## 946 0.6 0.5 1.0 91.91880 72.61937
## 947 0.7 0.5 1.0 89.53047 71.55482
## 948 0.8 0.5 1.0 90.22978 70.77808
## 949 0.9 0.5 1.0 97.36221 73.37693
## 950 1.0 0.5 1.0 120.74454 83.49360
## 951 0.1 0.6 1.0 138.08532 99.12961
## 952 0.2 0.6 1.0 249.92522 160.30822
## 953 0.3 0.6 1.0 157.17268 113.77369
## 954 0.4 0.6 1.0 106.08800 81.65717
## 955 0.5 0.6 1.0 100.46110 78.88932
## 956 0.6 0.6 1.0 96.76795 77.35528
## 957 0.7 0.6 1.0 92.82282 74.00282
## 958 0.8 0.6 1.0 93.24080 72.83202
## 959 0.9 0.6 1.0 101.32357 75.85062
## 960 1.0 0.6 1.0 127.11477 87.28601
## 961 0.1 0.7 1.0 182.50967 126.55417
## 962 0.2 0.7 1.0 314.09071 191.96917
## 963 0.3 0.7 1.0 162.48974 119.49158
## 964 0.4 0.7 1.0 111.77790 85.97143
## 965 0.5 0.7 1.0 107.96493 85.33392
## 966 0.6 0.7 1.0 101.96563 81.82370
## 967 0.7 0.7 1.0 96.08998 76.20328
## 968 0.8 0.7 1.0 96.39465 74.95682
## 969 0.9 0.7 1.0 105.63165 78.78874
## 970 1.0 0.7 1.0 133.99458 92.24392
## 971 0.1 0.8 1.0 253.01227 166.23238
## 972 0.2 0.8 1.0 366.18278 221.49783
## 973 0.3 0.8 1.0 162.29516 122.93290
## 974 0.4 0.8 1.0 121.40996 92.48950
## 975 0.5 0.8 1.0 116.62107 91.50023
## 976 0.6 0.8 1.0 107.37546 86.62665
## 977 0.7 0.8 1.0 99.39133 78.17858
## 978 0.8 0.8 1.0 99.83784 77.10645
## 979 0.9 0.8 1.0 110.38528 82.04446
## 980 1.0 0.8 1.0 141.46670 97.50899
## 981 0.1 0.9 1.0 343.71746 214.79112
## 982 0.2 0.9 1.0 394.93769 242.79202
## 983 0.3 0.9 1.0 158.41458 117.53224
## 984 0.4 0.9 1.0 133.89514 100.15998
## 985 0.5 0.9 1.0 126.02405 97.60984
## 986 0.6 0.9 1.0 112.99437 90.49168
## 987 0.7 0.9 1.0 102.86677 79.88984
## 988 0.8 0.9 1.0 103.74625 79.67944
## 989 0.9 0.9 1.0 115.68376 85.70336
## 990 1.0 0.9 1.0 149.62744 103.18043
## 991 0.1 1.0 1.0 436.05927 261.47826
## 992 0.2 1.0 1.0 414.03305 251.50251
## 993 0.3 1.0 1.0 160.60356 118.11119
## 994 0.4 1.0 1.0 148.41411 108.44885
## 995 0.5 1.0 1.0 135.87138 105.77260
## 996 0.6 1.0 1.0 118.94370 94.80452
## 997 0.7 1.0 1.0 106.73906 81.38865
## 998 0.8 1.0 1.0 108.30133 83.27631
## 999 0.9 1.0 1.0 121.62817 89.52571
## 1000 1.0 1.0 1.0 158.59754 109.69688VERIFIKASI MODEL HOLT WINTER’S EXPONENTIAL SMOOTHING ADITIF
Pemodelan terbaik, yaitu pada fit102(alpha = 0.2, beta = 0.1 dan gamma = 0.2) karena karena memiliki nilai RMSE dan MAD paling kecil dimananilai RMSE sebesar 72.04 dan MAD sebesar 52.33 artinya model HWES Aditif (alpha = 0.2, beta = 0.1 dan gamma = 0.2) merupakan model yang terbaik.
#Model holt winter's exponential smoothing aditif terbaik
fit102 <- HoltWinters(datats, alpha = 0.2, beta = 0.1, gamma = 0.2, seasonal = "additive")
fit102## Holt-Winters exponential smoothing with trend and additive seasonal component.
##
## Call:
## HoltWinters(x = datats, alpha = 0.2, beta = 0.1, gamma = 0.2, seasonal = "additive")
##
## Smoothing parameters:
## alpha: 0.2
## beta : 0.1
## gamma: 0.2
##
## Coefficients:
## [,1]
## a 171.1587263
## b 0.0333926
## s1 44.6877354
## s2 -21.7674022
## s3 -21.4373447
## s4 -29.8114214
## s5 18.6258431
## s6 18.7344995
## s7 24.4047886
## s8 -18.8932133
## s9 -23.2355604
## s10 -18.5673080
## s11 -26.5944149
## s12 42.9890974fitted(fit102)[,1]## Jan Feb Mar Apr May Jun Jul
## 2015 215.35836 141.24234 146.97767 153.33181 197.80336 156.45841 274.38903
## 2016 238.60656 168.23060 171.75169 175.90569 214.96289 175.73354 292.06456
## 2017 254.23764 183.26893 184.15939 187.37958 242.02302 180.89902 364.42578
## 2018 271.77448 178.79008 182.13304 191.51221 235.32295 201.17404 368.68957
## 2019 293.93951 195.58256 197.42397 207.02620 237.16728 229.46173 343.53019
## 2020 297.12834 202.46366 202.58298 197.61860 157.32713 184.40338 162.47906
## 2021 119.09582 29.39964 31.39754 33.39528 55.06929 150.09137 190.12765
## 2022 138.83859 78.07225 98.40671 98.97087 120.06208 208.36935 226.01783
## 2023 221.59029 146.30381 151.26797 125.17543 216.38806 221.95193 225.09053
## Aug Sep Oct Nov Dec
## 2015 273.35906 141.51643 164.59791 165.88494 227.84446
## 2016 267.26289 147.65723 168.79846 169.13870 236.66109
## 2017 285.12125 186.30637 202.45905 191.92662 254.88914
## 2018 269.31875 197.57264 205.30335 195.48433 275.96983
## 2019 252.50292 195.62704 193.46172 191.90150 275.56337
## 2020 41.62608 24.81364 31.38182 40.91982 140.73540
## 2021 107.65366 46.94275 46.19046 54.45316 136.87291
## 2022 195.08384 171.83803 177.55194 165.54376 222.26422
## 2023 183.48852 157.41479 161.95269 151.61530 211.70648#Mengakses residual dari model
residual <- residuals(fit102)
residual## Jan Feb Mar Apr May
## 2015 9.84599388 7.54946552 3.25880546 2.66178326 0.71867512
## 2016 28.50577396 -1.10191105 -7.22813343 -20.50299219 13.52343220
## 2017 48.93227641 -8.87282663 -1.38083161 25.50302137 -38.11127641
## 2018 -3.90324121 6.82511481 15.63450177 29.55338966 -42.05112273
## 2019 -40.60376359 -7.54799324 6.69119764 20.47325545 -109.95092801
## 2020 -50.22552039 -27.22097508 -69.24506472 -197.61860379 -157.32712601
## 2021 -6.01620457 46.21781652 83.42406164 68.52135345 160.79641206
## 2022 60.07344935 69.07732916 52.23856509 -33.90936169 219.68782713
## 2023 -26.43138159 5.36688575 -6.57519271 105.43680310 -7.44309029
## Jun Jul Aug Sep Oct
## 2015 -0.72466537 69.64730519 -88.58622293 15.74449612 -4.64479530
## 2016 -58.20056895 82.87276466 -103.84318953 28.13810403 7.10208348
## 2017 145.05343312 -47.13411317 -111.56722180 15.22278148 -36.29347082
## 2018 219.06279534 -91.19722337 -76.44573786 15.50715640 -23.14956564
## 2019 209.74209329 -55.17624440 -70.17902864 3.61903410 6.57577210
## 2020 -184.40337788 -112.66941971 130.64985948 95.30297151 101.62193683
## 2021 5.73224369 -171.79734969 -94.42490148 26.65073062 83.06123258
## 2022 4.36757815 -22.38580812 -54.35715084 -30.79903172 -41.60341541
## 2023 -29.10224333 -31.55499954 -67.44111786 -10.19386494 -17.66824565
## Nov Dec
## 2015 -19.48480255 4.13183053
## 2016 -7.18500767 20.45172476
## 2017 -48.43097885 -0.07352327
## 2018 -4.07116468 3.48678242
## 2019 8.22845790 54.13497554
## 2020 85.42890082 -36.20788871
## 2021 71.96677331 27.50166255
## 2022 -41.76977473 -27.10531061
## 2023 -23.06958827 6.78150341df <- forecast(fit102)
plot(df)
#Plot perbandingan fitted model dengan true model (Data Asli setelah di transformasi)
fitted_modela <- (fitted(fit102)[,1])
#Plot data aktual
plot(datats, type = "l", col = "blue",
xlab = "Tahun", ylab = "Jumlah Pengunjung",
main = "Fitted vs True (Setelah Transformasi Data) Model HWES Aditif")
#Tambahkan plot peramalan dari model
lines(fitted_modela, type = "l", col = "red")
#Tambahkan legenda
legend("topright", legend = c("Prediksi", "Aktual"),
col = c("blue", "red"), lty = c(2, 1))
se <- datats - fitted_modela
se## Jan Feb Mar Apr May
## 2015 9.84599388 7.54946552 3.25880546 2.66178326 0.71867512
## 2016 28.50577396 -1.10191105 -7.22813343 -20.50299219 13.52343220
## 2017 48.93227641 -8.87282663 -1.38083161 25.50302137 -38.11127641
## 2018 -3.90324121 6.82511481 15.63450177 29.55338966 -42.05112273
## 2019 -40.60376359 -7.54799324 6.69119764 20.47325545 -109.95092801
## 2020 -50.22552039 -27.22097508 -69.24506472 -197.61860379 -157.32712601
## 2021 -6.01620457 46.21781652 83.42406164 68.52135345 160.79641206
## 2022 60.07344935 69.07732916 52.23856509 -33.90936169 219.68782713
## 2023 -26.43138159 5.36688575 -6.57519271 105.43680310 -7.44309029
## Jun Jul Aug Sep Oct
## 2015 -0.72466537 69.64730519 -88.58622293 15.74449612 -4.64479530
## 2016 -58.20056895 82.87276466 -103.84318953 28.13810403 7.10208348
## 2017 145.05343312 -47.13411317 -111.56722180 15.22278148 -36.29347082
## 2018 219.06279534 -91.19722337 -76.44573786 15.50715640 -23.14956564
## 2019 209.74209329 -55.17624440 -70.17902864 3.61903410 6.57577210
## 2020 -184.40337788 -112.66941971 130.64985948 95.30297151 101.62193683
## 2021 5.73224369 -171.79734969 -94.42490148 26.65073062 83.06123258
## 2022 4.36757815 -22.38580812 -54.35715084 -30.79903172 -41.60341541
## 2023 -29.10224333 -31.55499954 -67.44111786 -10.19386494 -17.66824565
## Nov Dec
## 2015 -19.48480255 4.13183053
## 2016 -7.18500767 20.45172476
## 2017 -48.43097885 -0.07352327
## 2018 -4.07116468 3.48678242
## 2019 8.22845790 54.13497554
## 2020 85.42890082 -36.20788871
## 2021 71.96677331 27.50166255
## 2022 -41.76977473 -27.10531061
## 2023 -23.06958827 6.78150341se_t <- sqrt(mean((se)^2))
se_t## [1] 72.04302PERAMALAN DENGAN MODEL HOLT WINTER’S EXPONENTIAL SMOOTHING ADITIF
#Peramalan untuk periode satu tahun ke depan (tahun 2024)
a <- predict(fit102, n.ahead = 12)
a## Jan Feb Mar Apr May Jun Jul Aug
## 2024 215.8799 149.4581 149.8216 141.4809 189.9515 190.0936 195.7973 152.5327
## Sep Oct Nov Dec
## 2024 148.2237 152.9253 144.9316 214.5485MODEL PERAMALAN ARIMA
IDENTIFIKASI MODEL DERET WAKTU
#Plot data time series
plot.ts(datats, xlab = "Tahun", ylab = "Jumlah Pengunjung")
#Menambahkan axis untuk tahun
axis(1, at = seq(2014, 2023, by = 1), labels = seq(2014, 2023, by = 1))
#Plot ACF
#Menampilkan plot korelasi ACF
Acf(datats)
#Menampilkan hasil korelasi ACF
print(Acf(datats))
##
## Autocorrelations of series 'datats', by lag
##
## 0 1 2 3 4 5 6 7 8 9 10
## 1.000 0.396 0.237 0.140 0.066 0.238 0.426 0.244 0.121 0.085 -0.033
## 11 12 13 14 15 16 17 18 19 20 21
## 0.149 0.412 0.115 0.120 0.024 0.010 0.151 0.219 -0.004 -0.125 -0.088
## 22 23 24
## -0.193 0.090 0.054CEK STATIONERITAS TERHADAP VARIANSI
Untuk melakukan pemeriksaan stationeritas terhadap variansi, maka dapat dilakukan dengan pemeriksaan Box-Cox, dimana nilai lambda harus mendekati 1. Jika nilai lambda mendekati 1 maka data stationer secara variansi.
BoxCox.lambda(datats)## Warning in guerrero(x, lower, upper): Guerrero's method for selecting a Box-Cox
## parameter (lambda) is given for strictly positive data.## [1] 0.9597101Dikarenakan hasil nilai lambda menunjukkan angka 0.95, dimana angka tersebut mendekati 1 maka data dikatakan stationer secara variansi.
CEK STATIONERITAS TERHADAP RATA-RATA
Dalam pemeriksaan stationeritas rata-rata dapat dilakukan uji ADF (Augmented Dickey Fuller) dengan ketentuan :
H0 : Data tidak stationer
H1 : Data stationer
Pengambilan Keputusan : H0 diterima jika p-value > alpha (5% / 0.05)
adf.test(datats)##
## Augmented Dickey-Fuller Test
##
## data: datats
## Dickey-Fuller = -3.4139, Lag order = 4, p-value = 0.05559
## alternative hypothesis: stationaryDari hasil diatas diperoleh p-value > nilai alpha (0.055 > 0.05) sehingga H0 diterima yang berarti data tidak stationer secara rata-rata.
DIFFERENCING
Melakukan differencing karena plot time series sebelumnya terlihat bahwa data belum stationer dalam rata-rata maupun variansi dimana plot time series nya tidak berfluktuasi disekitar titik nol/konstan/masih mengandung tren naik turun.
Melakukan differencing karena pada pemeriksaan stationeritas rataan melalui uji ADF diperoleh p-value > nilai alpha (0.055 > 0.05) dimana artinya data belum stationer terhadap rata-rata.
#Differencing 1 kali
datadiff1 <- diff(datats, differences = 1)
datadiff1## Jan Feb Mar Apr May
## 2014 -83.14313082 -2.05375448 1.92328636 21.29372288
## 2015 9.99515110 -76.41255080 1.44467941 5.75710935 42.52844947
## 2016 35.13604403 -99.98364299 -2.60513838 -9.12085187 73.08362076
## 2017 46.05710365 -128.77381848 8.38245367 30.10404221 -8.97085114
## 2018 13.05561811 -82.25604438 12.15234697 23.29806081 -27.79377183
## 2019 -26.12086999 -65.30117431 16.08059228 23.38428698 -100.28309940
## 2020 -82.79553209 -71.66012603 -41.90477228 -133.33791659 0.00000000
## 2021 8.55210963 -37.46215967 39.20414245 -12.90496816 113.94906639
## 2022 34.53746862 -51.76245710 3.49569496 -85.58376936 274.68839866
## 2023 0.00000000 -43.48821369 -6.97791794 85.91945066 -21.66725981
## Jun Jul Aug Sep Oct
## 2014 13.81359592 88.73443249 -17.27432091 -114.24233113 19.80624861
## 2015 -42.78829193 188.30258814 -159.26350179 -27.51190383 2.69218846
## 2016 -110.95334935 257.40435400 -211.51762304 12.37563049 0.10520453
## 2017 122.04070490 -8.66078659 -143.73763776 27.97512805 -35.36357415
## 2018 226.96500869 -142.74449575 -84.61932702 20.20678204 -30.92601607
## 2019 311.98747397 -150.84987595 -106.03006154 16.92219138 0.79141748
## 2020 0.00000000 49.80963762 122.46630382 -52.15933143 12.88714934
## 2021 -60.04208075 -137.49331546 -5.10154622 60.36472142 55.65821447
## 2022 -127.01298128 -9.10490569 -62.90533560 0.31231624 -5.09048203
## 2023 -16.09528644 0.68584406 -77.48812235 31.17351832 -2.93648113
## Nov Dec
## 2014 3.22707155 65.78363379
## 2015 -13.55298152 85.57615528
## 2016 -13.94684299 95.15911862
## 2017 -22.66993554 111.31997508
## 2018 9.25938451 88.04344990
## 2019 0.09246129 129.56838920
## 2020 -6.65503557 -21.82121544
## 2021 -2.83175731 37.95463711
## 2022 -12.17453209 71.38492204
## 2023 -15.73872958 89.94227423#Menampilkan plot time series setelah di differencing 1 kali
plot.ts(datadiff1, xlab = "Tahun", ylab = "Jumlah Pengunjung")
#Menambahkan axis untuk tahun
axis(1, at = seq(2014, 2023, by = 1), labels = seq(2014, 2023, by = 1))
CEK STATIONERITAS TERHADAP RATA-RATA SETELAH DIFFERENCING
Dalam pemeriksaan stationeritas rata-rata dapat dilakukan uji ADF (Augmented Dickey Fuller) dengan ketentuan :
H0 : Data tidak stationer
H1 : Data stationer
Pengambilan Keputusan : H0 diterima jika p-value > alpha (5% / 0.05)
adf.test(datadiff1)## Warning in adf.test(datadiff1): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: datadiff1
## Dickey-Fuller = -10.619, Lag order = 4, p-value = 0.01
## alternative hypothesis: stationaryDari hasil diatas diperoleh p-value < nilai alpha (0.01 < 0.05) sehingga H0 ditolak yang berarti data stationer secara rata-rata.
PLOT DAN HASIL KORELASI ACF SETELAH DIFFERENCING
Lag yang signifikan ada 9 berarti parameter q = 9.
#Menampilkan plot korelasi ACF
Acf(datadiff1)
#Menampilkan hasil korelasi ACF
print(Acf(datadiff1))
##
## Autocorrelations of series 'datadiff1', by lag
##
## 0 1 2 3 4 5 6 7 8 9 10
## 1.000 -0.362 -0.052 -0.020 -0.204 -0.017 0.303 -0.049 -0.068 0.071 -0.250
## 11 12 13 14 15 16 17 18 19 20 21
## -0.072 0.465 -0.244 0.082 -0.068 -0.131 0.060 0.232 -0.083 -0.118 0.114
## 22 23 24
## -0.321 0.258 0.133PLOT DAN HASIL KORELASI PACF SETELAH DIFFERENCING
Lag yang signifikan ada 7 berarti parameter p = 7.
#Menampilkan plot korelasi PACF
Pacf(datadiff1)
#Menampilkan hasil korelasi PACF
print(Pacf(datadiff1))
##
## Partial autocorrelations of series 'datadiff1', by lag
##
## 1 2 3 4 5 6 7 8 9 10 11
## -0.362 -0.210 -0.143 -0.343 -0.375 0.036 0.053 -0.092 0.033 -0.138 -0.331
## 12 13 14 15 16 17 18 19 20 21 22
## 0.236 -0.018 -0.049 -0.135 -0.016 0.070 0.144 0.139 -0.158 0.014 -0.122
## 23 24
## 0.199 0.044PEMODELAN ARIMA
#Maka dari identifikasi plot ACF dan PACF diperoleh model ARIMA yang mungkin sebagai berikut
fit1 = Arima(datats, order = c(1, 1, 1))
fit2 = Arima(datats, order = c(1, 1, 2))
fit3 = Arima(datats, order = c(1, 1, 3))
fit4 = Arima(datats, order = c(1, 1, 4))
fit5 = Arima(datats, order = c(1, 1, 5))
fit6 = Arima(datats, order = c(1, 1, 6))
fit7 = Arima(datats, order = c(1, 1, 7))
fit8 = Arima(datats, order = c(1, 1, 8))
fit9 = Arima(datats, order = c(1, 1, 9))
fit10 = Arima(datats, order = c(2, 1, 1))
fit11 = Arima(datats, order = c(2, 1, 2))
fit12 = Arima(datats, order = c(2, 1, 3))
fit13 = Arima(datats, order = c(2, 1, 4))
fit14 = Arima(datats, order = c(2, 1, 5))
fit15 = Arima(datats, order = c(2, 1, 6))
fit16 = Arima(datats, order = c(2, 1, 7))
fit17 = Arima(datats, order = c(2, 1, 8))
fit18 = Arima(datats, order = c(2, 1, 9))
fit19 = Arima(datats, order = c(3, 1, 1))
fit20 = Arima(datats, order = c(3, 1, 2))
fit21 = Arima(datats, order = c(3, 1, 3))
fit22 = Arima(datats, order = c(3, 1, 4))
fit23 = Arima(datats, order = c(3, 1, 5))
fit24 = Arima(datats, order = c(3, 1, 6))
fit25 = Arima(datats, order = c(3, 1, 7))
fit26 = Arima(datats, order = c(3, 1, 8))
fit27 = Arima(datats, order = c(3, 1, 9))
fit28 = Arima(datats, order = c(4, 1, 1))
fit29 = Arima(datats, order = c(4, 1, 2))
fit30 = Arima(datats, order = c(4, 1, 3))
fit31 = Arima(datats, order = c(4, 1, 4))
fit32 = Arima(datats, order = c(4, 1, 5))
fit33 = Arima(datats, order = c(4, 1, 7))
fit34 = Arima(datats, order = c(4, 1, 8))
fit35 = Arima(datats, order = c(4, 1, 9))
fit36 = Arima(datats, order = c(5, 1, 1))
fit37 = Arima(datats, order = c(5, 1, 2))
fit38 = Arima(datats, order = c(5, 1, 3))
fit39 = Arima(datats, order = c(5, 1, 4))
fit40 = Arima(datats, order = c(5, 1, 5))
fit41 = Arima(datats, order = c(5, 1, 6))
fit42 = Arima(datats, order = c(5, 1, 8))
fit43 = Arima(datats, order = c(5, 1, 9))
fit44 = Arima(datats, order = c(6, 1, 1))
fit45 = Arima(datats, order = c(6, 1, 2))
fit46 = Arima(datats, order = c(6, 1, 3))
fit47 = Arima(datats, order = c(6, 1, 4))
fit48 = Arima(datats, order = c(6, 1, 5))
fit49 = Arima(datats, order = c(6, 1, 6))
fit50 = Arima(datats, order = c(6, 1, 7))
fit51 = Arima(datats, order = c(6, 1, 8))
fit52 = Arima(datats, order = c(6, 1, 9))
fit53 = Arima(datats, order = c(7, 1, 1))
fit54 = Arima(datats, order = c(7, 1, 2))
fit55 = Arima(datats, order = c(7, 1, 3))
fit56 = Arima(datats, order = c(7, 1, 4))
fit57 = Arima(datats, order = c(7, 1, 5))
fit58 = Arima(datats, order = c(7, 1, 6))
fit59 = Arima(datats, order = c(7, 1, 7))
fit60 = Arima(datats, order = c(7, 1, 8))
fit61 = Arima(datats, order = c(7, 1, 9))PERBANDINGAN EVALUASI MODEL (RMSE dan MAD) SETIAP MODEL ARIMA
#Definisikan vektor nilai p, d, dan q yang akan diuji
p_values <- 1:7
d_values <- 1
q_values <- 1:9
#Fungsi untuk menghitung RMSE
calculate_rmse <- function(actual, predicted) {
rmse <- sqrt(mean((actual - predicted)^2))
return(rmse)
}
#Fungsi untuk menghitung MAD
calculate_mad <- function(actual, predicted) {
mad <- mean(abs(mean(actual) - predicted))
return(mad)
}
#Buat dataframe untuk menyimpan hasil eksplorasi
parameter_combinations <- expand.grid(p = p_values, d = d_values, q = q_values)
#Inisialisasi vektor untuk menyimpan hasil evaluasi model
rmse_values <- numeric(nrow(parameter_combinations))
mad_values <- numeric(nrow(parameter_combinations))
#Loop melalui setiap kombinasi parameter
for (i in seq_len(nrow(parameter_combinations))) {
current_params <- parameter_combinations[i, ]
#Buat model ARIMA dengan parameter saat ini
current_model <- tryCatch({
Arima(datats, order = c(current_params$p, current_params$d, current_params$q))
}, error = function(e) NULL)
#Jika model berhasil dibuat, lakukan prediksi dan evaluasi
if (!is.null(current_model)) {
predictions <- fitted(current_model)
#Evaluasi model menggunakan RMSE
rmse_values[i] <- calculate_rmse(datats, predictions)
#Evaluasi model menggunakan MAD
mad_values[i] <- calculate_mad(datats, predictions)
} else {
rmse_values[i] <- NA
mad_values[i] <- NA
}
}
#Gabungkan hasil eksplorasi dengan nilai RMSE dan MAD
results <- cbind(parameter_combinations, RMSE = rmse_values, MAD = mad_values)
#Tampilkan hasil
print(results)## p d q RMSE MAD
## 1 1 1 1 69.00212 33.95178
## 2 2 1 1 68.98947 34.10615
## 3 3 1 1 68.58773 35.14527
## 4 4 1 1 66.42020 38.98994
## 5 5 1 1 64.34061 41.27053
## 6 6 1 1 64.31875 41.49869
## 7 7 1 1 64.07352 41.81151
## 8 1 1 2 68.99565 34.03410
## 9 2 1 2 68.64805 35.77672
## 10 3 1 2 68.03790 36.55480
## 11 4 1 2 61.31408 46.77781
## 12 5 1 2 64.23973 41.70429
## 13 6 1 2 58.98696 46.59279
## 14 7 1 2 61.59347 42.27346
## 15 1 1 3 68.95299 33.96865
## 16 2 1 3 59.62932 42.04598
## 17 3 1 3 68.52314 35.98390
## 18 4 1 3 58.46399 44.15657
## 19 5 1 3 61.93980 41.66188
## 20 6 1 3 59.60236 42.09288
## 21 7 1 3 59.55278 40.36973
## 22 1 1 4 68.59765 34.66014
## 23 2 1 4 65.19517 36.74222
## 24 3 1 4 59.02980 43.56929
## 25 4 1 4 58.38722 44.01002
## 26 5 1 4 58.31763 46.93744
## 27 6 1 4 57.92480 42.82780
## 28 7 1 4 58.44944 37.21757
## 29 1 1 5 66.75392 38.78541
## 30 2 1 5 61.89691 39.81516
## 31 3 1 5 58.73291 43.69367
## 32 4 1 5 56.46450 44.22608
## 33 5 1 5 53.91328 42.73758
## 34 6 1 5 54.19057 41.63036
## 35 7 1 5 54.23143 41.35103
## 36 1 1 6 65.63158 40.29717
## 37 2 1 6 61.42491 39.48057
## 38 3 1 6 64.84459 36.61325
## 39 4 1 6 NA NA
## 40 5 1 6 54.70048 42.87284
## 41 6 1 6 54.88826 42.50198
## 42 7 1 6 54.17904 41.44758
## 43 1 1 7 65.92893 33.34122
## 44 2 1 7 60.28051 41.28544
## 45 3 1 7 57.45271 42.72832
## 46 4 1 7 54.29460 41.47902
## 47 5 1 7 NA NA
## 48 6 1 7 53.73688 39.25705
## 49 7 1 7 52.19660 43.71067
## 50 1 1 8 66.04233 35.30986
## 51 2 1 8 63.74913 39.83803
## 52 3 1 8 57.42767 43.01071
## 53 4 1 8 55.87842 42.93365
## 54 5 1 8 54.02291 41.92867
## 55 6 1 8 54.13981 39.36167
## 56 7 1 8 53.46167 39.28114
## 57 1 1 9 61.93023 48.67402
## 58 2 1 9 56.23922 44.25212
## 59 3 1 9 57.47676 43.31299
## 60 4 1 9 55.89215 42.72536
## 61 5 1 9 53.85237 41.97292
## 62 6 1 9 53.54824 39.30829
## 63 7 1 9 53.46338 38.86756VERIFIKASI MODEL ARIMA
Pemodelan terbaik, yaitu pada fit59 ARIMA(7, 1, 7) karena memiliki nilai RMSE dan MAD paling kecil dimana nilai RMSE sebesar 52.19 dan MAD sebesar 43.71 artinya model ARIMA(7,1,7) merupakan model yang terbaik.
Diperoleh nilai p-value dari ljung-box statistic sebesar 0.95 dimana p-value = 0.9505 > alpha = 0.05 maka tidak terdapat cukup bukti yang cukup untuk menunjukkan adanya korelasi yang signifikan dalam residual.
P-value dari tes Ljung-Box adalah ukuran statistik yang menentukan seberapa kuat bukti yang Anda miliki untuk menolak hipotesis nol bahwa tidak ada korelasi dalam residual pada lag-lag yang diberikan. Jadi, Anda membandingkan nilai p-value dari tes Ljung-Box dengan tingkat signifikansi yang telah ditentukan sebelumnya (biasanya 0.05 atau 0.01) untuk membuat keputusan apakah ada korelasi yang signifikan dalam residual.
Secara umum, aturan praktisnya adalah:
H0 : Tidak terdapat bukti yang cukup untuk menunjukkan adanya korelasi yang signifikan dalam residual
H1 : Terdapat korelasi signifikan dalam residual
Pengambilan Keputusan : H0 diterima jika p-value > alpha (5% / 0.05)
Uji Shapiro Wilk: H0 : Residu memiliki distribusi normal H1 : Residu tidak memiliki distribusi normal
Pengambilan Keputusan : H0 diterima jika p-value > alpha (5% / 0.05)
#Menampilkan plot Ljung-Box Statistics
tsdiag(fit59)
#Menampilkan plot histogram
checkresiduals(fit59)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(7,1,7)
## Q* = 21.252, df = 10, p-value = 0.01941
##
## Model df: 14. Total lags used: 24fit59## Series: datats
## ARIMA(7,1,7)
##
## Coefficients:
## ar1 ar2 ar3 ar4 ar5 ar6 ar7 ma1
## -0.9546 -1.9540 -1.1615 -1.8476 -0.9949 -0.8807 -0.1771 0.3097
## s.e. 0.1311 0.1559 0.2499 0.1784 0.2412 0.1492 0.1204 0.1150
## ma2 ma3 ma4 ma5 ma6 ma7
## 1.5048 -0.2249 1.1002 -0.7525 0.1371 -0.7579
## s.e. 0.1392 0.2034 0.1974 0.2128 0.1389 0.1171
##
## sigma^2 = 3114: log likelihood = -646.88
## AIC=1323.75 AICc=1328.41 BIC=1365.44fit59$fitted## Jan Feb Mar Apr May Jun Jul
## 2014 217.12341 188.64346 155.96108 143.35014 149.77260 170.65723 196.37964
## 2015 226.26696 206.74926 152.90949 139.50340 147.95554 195.40677 243.91167
## 2016 256.65516 220.42719 157.67768 155.74495 152.67146 220.23902 265.14981
## 2017 279.77154 223.22323 170.70326 187.73648 176.16251 240.79934 332.08094
## 2018 294.32683 199.61530 177.66842 176.25248 185.45276 305.21971 316.39054
## 2019 277.65272 171.18981 219.45683 165.28300 207.50636 327.76479 257.12723
## 2020 262.01042 222.79981 210.14466 121.74661 116.56461 175.37040 93.34808
## 2021 74.72059 91.91920 126.57156 110.94276 141.24255 115.78648 126.01461
## 2022 110.27235 169.40381 80.56310 102.13159 196.51963 193.97273 208.85992
## 2023 187.16920 145.91937 130.90205 159.72492 197.41261 228.95425 198.63002
## Aug Sep Oct Nov Dec
## 2014 192.22985 180.01296 157.85238 136.48131 174.01156
## 2015 217.54216 177.38903 147.16506 158.87839 198.66146
## 2016 197.92367 187.68716 164.88456 164.97350 216.51399
## 2017 247.48438 168.40184 185.06238 176.91045 246.21262
## 2018 226.63838 209.31014 177.70772 185.05419 331.21227
## 2019 237.38298 229.89655 164.03867 213.42206 316.93893
## 2020 52.87310 145.78922 122.03840 107.63373 144.46159
## 2021 125.93813 21.76303 55.44547 177.89015 122.63757
## 2022 165.76630 137.72287 130.45053 168.20171 200.00299
## 2023 137.74477 126.98658 170.57952 191.69022 177.07798summary(fit59)## Series: datats
## ARIMA(7,1,7)
##
## Coefficients:
## ar1 ar2 ar3 ar4 ar5 ar6 ar7 ma1
## -0.9546 -1.9540 -1.1615 -1.8476 -0.9949 -0.8807 -0.1771 0.3097
## s.e. 0.1311 0.1559 0.2499 0.1784 0.2412 0.1492 0.1204 0.1150
## ma2 ma3 ma4 ma5 ma6 ma7
## 1.5048 -0.2249 1.1002 -0.7525 0.1371 -0.7579
## s.e. 0.1392 0.2034 0.1974 0.2128 0.1389 0.1171
##
## sigma^2 = 3114: log likelihood = -646.88
## AIC=1323.75 AICc=1328.41 BIC=1365.44
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.5548457 52.1966 38.82733 -Inf Inf 0.7657792 -0.005665773#Mengakses residual dari model
residual <- residuals(fit59)
residual## Jan Feb Mar Apr May
## 2014 0.2173405 -54.4458400 -23.8172209 -9.2829890 5.5882670
## 2015 -1.0626088 -57.9574583 -2.6730091 16.4901907 50.5665029
## 2016 10.4571758 -53.2985004 6.8458752 -0.3422442 75.8148596
## 2017 23.3983810 -48.8271317 12.0752917 25.1461205 27.7492394
## 2018 -26.4555901 -14.0001044 20.0991200 44.8131179 7.8190693
## 2019 -24.3169786 16.8447629 -15.3416651 62.2164549 -80.2900080
## 2020 -15.1076036 -47.5571225 -76.8067405 -121.7466062 -116.5646080
## 2021 38.3590306 -16.3017392 -11.7499567 -9.0261297 74.6231508
## 2022 88.6396891 -22.2542257 70.0821816 -37.0700828 143.2302737
## 2023 7.9897083 5.7513229 13.7907320 70.8873113 11.5323632
## Jun Jul Aug Sep Oct
## 2014 -1.4827667 61.5292632 48.4047280 -53.6207102 -11.6538860
## 2015 -39.6730188 100.1246644 -32.7693268 -20.1280974 12.7880606
## 2016 -102.7060488 109.7875215 -34.5039668 -11.8918240 11.0159839
## 2017 85.1531123 -14.7892790 -73.9303571 33.1273176 -18.8967966
## 2018 115.0171326 -38.8981940 -33.7653612 3.7696589 4.4460650
## 2019 111.4390347 31.2267207 -55.0590952 -30.6504759 35.9988312
## 2020 -175.3703967 -43.5384453 119.4028410 -25.6726064 10.9653625
## 2021 40.0371363 -107.6843115 -112.7093711 51.8304478 73.8062188
## 2022 18.7641941 -5.2279014 -25.0396168 3.3161318 5.4979895
## 2023 -36.1045698 -5.0944930 -21.6973618 20.2343408 -26.2950774
## Nov Dec
## 2014 12.9442603 41.1976444
## 2015 -12.4782523 33.3148285
## 2016 -3.0197985 40.5988283
## 2017 -33.4148101 8.6030015
## 2018 6.3589791 -51.7556545
## 2019 -13.2920985 12.7594175
## 2020 18.7149979 -39.9340835
## 2021 -51.4702118 41.7369991
## 2022 -44.4277265 -4.8440758
## 2023 -63.1445098 41.4100056df <- forecast(fit59)
plot(df)
#Menampilkan nilai p-value dari uji Ljung-Box test
Box.test(df$residuals)##
## Box-Pierce test
##
## data: df$residuals
## X-squared = 0.0038521, df = 1, p-value = 0.9505#Melakukan uji Shapiro-Wilk pada residu
shapiro.test(residuals(fit59))##
## Shapiro-Wilk normality test
##
## data: residuals(fit59)
## W = 0.98131, p-value = 0.09433#Histogram
hist(df$residuals)
#Plot perbandingan fitted model dengan true model (Data Asli setelah di transformasi)
fitted_model2b <- (fitted.values(fit59))
plot(fitted_model2b, type = "l", col = "red",
ylim = range(fitted_model2b, datats),
xlab = "Tahun", ylab = "Jumlah Pengunjung", main = "Fitted vs True (Setelah Transformasi Data) Model ARIMA")
lines(datats, col = "blue")
#Menambahkan axis untuk tahun
axis(1, at = seq(2014, 2023, by = 1), labels = seq(2014, 2023, by = 1))
legend("topright", legend = c("Prediksi", "Aktual"),
col = c("red", "blue"), lty = c(2, 1), cex = 0.8)
se2 <- datats - fitted_model2b
se2## Jan Feb Mar Apr May
## 2014 0.2173405 -54.4458400 -23.8172209 -9.2829890 5.5882670
## 2015 -1.0626088 -57.9574583 -2.6730091 16.4901907 50.5665029
## 2016 10.4571758 -53.2985004 6.8458752 -0.3422442 75.8148596
## 2017 23.3983810 -48.8271317 12.0752917 25.1461205 27.7492394
## 2018 -26.4555901 -14.0001044 20.0991200 44.8131179 7.8190693
## 2019 -24.3169786 16.8447629 -15.3416651 62.2164549 -80.2900080
## 2020 -15.1076036 -47.5571225 -76.8067405 -121.7466062 -116.5646080
## 2021 38.3590306 -16.3017392 -11.7499567 -9.0261297 74.6231508
## 2022 88.6396891 -22.2542257 70.0821816 -37.0700828 143.2302737
## 2023 7.9897083 5.7513229 13.7907320 70.8873113 11.5323632
## Jun Jul Aug Sep Oct
## 2014 -1.4827667 61.5292632 48.4047280 -53.6207102 -11.6538860
## 2015 -39.6730188 100.1246644 -32.7693268 -20.1280974 12.7880606
## 2016 -102.7060488 109.7875215 -34.5039668 -11.8918240 11.0159839
## 2017 85.1531123 -14.7892790 -73.9303571 33.1273176 -18.8967966
## 2018 115.0171326 -38.8981940 -33.7653612 3.7696589 4.4460650
## 2019 111.4390347 31.2267207 -55.0590952 -30.6504759 35.9988312
## 2020 -175.3703967 -43.5384453 119.4028410 -25.6726064 10.9653625
## 2021 40.0371363 -107.6843115 -112.7093711 51.8304478 73.8062188
## 2022 18.7641941 -5.2279014 -25.0396168 3.3161318 5.4979895
## 2023 -36.1045698 -5.0944930 -21.6973618 20.2343408 -26.2950774
## Nov Dec
## 2014 12.9442603 41.1976444
## 2015 -12.4782523 33.3148285
## 2016 -3.0197985 40.5988283
## 2017 -33.4148101 8.6030015
## 2018 6.3589791 -51.7556545
## 2019 -13.2920985 12.7594175
## 2020 18.7149979 -39.9340835
## 2021 -51.4702118 41.7369991
## 2022 -44.4277265 -4.8440758
## 2023 -63.1445098 41.4100056se_t2 <- sqrt(mean((se2)^2))
se_t2## [1] 52.1966Kesimpulan :
Pada plot Ljung-Box Statistic juga terlihat bahwa semua lag pada plot tidak ada yang di bawah nilai 0.05 dan melalui uji Ljung-Box test diperoleh nilai p-value sebesar 0.95 output p-value = 0.05 sehingga diterima artinya tidak terdapat bukti yang cukup untuk menunjukkan adanya korelasi yang signifikan dalam residual, maka dapat disimpulkan bahwa residual dari model ARIMA(7, 1, 7) sudah bersifat white-noise.
Plot histogram dari residual model ARIMA(7, 1, 7) sudah berdistribusi normal dan melalui pengujian Shapiro Wilk diperoleh nilai p-value sebesar 0.09 output p-value = 0.05 sehingga diterima yang menunjukkan residual dari model ARIMA(7, 1, 7) sudah berdistribusi normal.
Maka dapat disimpulkan bahwa ARIMA(7, 1, 7) merupakan model terbaik untuk meramalkan jumlah pengunjung di Lokawisata Baturraden karena residual dari model ARIMA(7, 1, 7) sudah bersifat white-noise dan berdistribusi normal.
PERAMALAN DENGAN MODEL ARIMA
Plot untuk data aktual dan hasil peramalan jumlah pengunjung tahun 2014-2023 pada model HWES aditif memiliki selisih atau nilai error berdasarkan MAD sebesar 52.33. Hal tersebut dibuktikan dengan plot hasil peramalan jumlah pengunjung tahun 2014-2023 cukup mendekati dengan data aktualnya.
Plot untuk data aktual dan hasil peramalan jumlah pengunjung tahun 2014-2023 pada model ARIMA(7, 1, 7) memiliki selisih atau nilai error berdasarkan MAD sebesar 43.71. Hal tersebut dibuktikan dengan plot hasil peramalan jumlah pengunjung cukup mendekati dengan data aktualnya.
Dari uraian penjelasan di atas, dapat disimpulkan bahwa model paling terbaik, yaitu model ARIMA(7, 1, 7) karena mempunyai nilai RMSE (52.19) dan MAD (43.71) yang paling kecil dan hasil peramalan jumlah pengunjung tahun 2014-2023 paling cukup mendekati dengan data aktualnya dengan nilai error berdasarkan MAD lebih kecil dibandingkan dengan model HWES aditif sebesar 43.71.
#Peramalan untuk periode satu tahun ke depan (tahun 2024)
b <- predict(fit59, n.ahead = 12)
b## $pred
## Jan Feb Mar Apr May Jun Jul Aug
## 2024 176.6889 136.6339 130.7420 183.4703 173.4229 178.5206 172.7962 131.0909
## Sep Oct Nov Dec
## 2024 154.5536 175.4812 163.1146 186.8139
##
## $se
## Jan Feb Mar Apr May Jun Jul Aug
## 2024 56.71966 60.11930 66.54118 67.79731 68.43032 68.48766 69.58702 72.73378
## Sep Oct Nov Dec
## 2024 76.23064 79.01868 79.12170 79.16692#Data hasil prediksi
pred <- c(176.6889, 136.6339, 130.7420, 183.4703, 173.4229, 178.5206,
172.7962, 131.0909, 154.5536, 175.4812, 163.1146, 186.8139)
#Nama-nama bulan
months <- c("Jan", "Feb", "Mar", "Apr", "May", "Jun",
"Jul", "Aug", "Sep", "Oct", "Nov", "Dec")
#Plot hasil prediksi
plot(pred, type = "o", xaxt = "n", xlab = "Bulan", ylab = "Jumlah Pengunjung",
main = "Peramalan Jumlah Pengunjung Tahun 2024")
#Menambahkan sumbu x dengan nama-nama bulan
axis(1, at = 1:12, labels = months)
Kesimpulan :
Hasil peramalan jumlah pengunjung di Lokawisata Baturraden menggunakan model ARIMA(7, 1, 7) mengalami penurunan dan kenaikan yang stabil dan konsisten. Terlihat bahwa kenaikan ataupun penurunan tersebut signifikan.