uas pss2
Asma Adzkia Mafaza
2025-06-16
UAS Nomor 1
Tujuan
Melakukan simulasi pemodelan regresi linier berganda untuk memprediksi harga rumah berdasarkan:
Luas rumah (X1)
Jumlah kamar tidur (X2)
Lokasi (X3) dengan kategori: cluster, town house, dan residence Semua variabel disimulasikan dari distribusi normal sesuai dengan ketentuan soal.
Pembangkitan Data
set.seed(123)Set.seed digunakan untuk memastikan bahwa hasil simulasi bersifat tetap (reproducible), yang jika dijalankan ulang akan menghasilkan data yang sama.
Jumlah data
n <- 200
n## [1] 200Menentukan jumlah data rumah yang disimulasikan, yaitu sebanyak 200 data.
X1 = Luas rumah
X1 <- rnorm(n, mean = 100, sd = 20)
X1## [1] 88.79049 95.39645 131.17417 101.41017 102.58575 134.30130 109.21832
## [8] 74.69878 86.26294 91.08676 124.48164 107.19628 108.01543 102.21365
## [15] 88.88318 135.73826 109.95701 60.66766 114.02712 90.54417 78.64353
## [22] 95.64050 79.47991 85.42218 87.49921 66.26613 116.75574 103.06746
## [29] 77.23726 125.07630 108.52928 94.09857 117.90251 117.56267 116.43162
## [36] 113.77281 111.07835 98.76177 93.88075 92.39058 86.10586 95.84165
## [43] 74.69207 143.37912 124.15924 77.53783 91.94230 90.66689 115.59930
## [50] 98.33262 105.06637 99.42906 99.14259 127.37205 95.48458 130.32941
## [57] 69.02494 111.69227 102.47708 104.31883 107.59279 89.95353 93.33585
## [64] 79.62849 78.56418 106.07057 108.96420 101.06008 118.44535 141.00169
## [71] 90.17938 53.81662 120.11477 85.81598 86.23983 120.51143 94.30454
## [78] 75.58565 103.62607 97.22217 100.11528 107.70561 92.58680 112.88753
## [85] 95.59027 106.63564 121.93678 108.70363 93.48137 122.97615 119.87008
## [92] 110.96794 104.77463 87.44188 127.21305 87.99481 143.74666 130.65221
## [99] 95.28599 79.47158 85.79187 105.13767 95.06616 93.04915 80.96763
## [106] 99.09945 84.30191 66.64116 92.39547 118.37993 88.49306 112.15929
## [113] 67.64235 98.88876 110.38814 106.02307 102.11352 87.18588 83.00591
## [120] 79.51742 102.35293 81.05051 90.18885 94.87816 136.87724 86.96100
## [127] 104.70773 101.55922 80.76287 98.57384 128.89102 109.03008 100.82466
## [134] 91.55006 58.93506 122.62674 70.78720 114.79895 138.18207 71.12214
## [141] 114.03569 94.75605 68.55712 69.70665 67.96928 89.38187 70.76489
## [148] 113.75834 142.00218 74.25939 115.75478 115.38084 106.64405 79.83247
## [155] 97.61095 94.39209 111.25979 92.55122 119.53947 92.50838 121.05423
## [162] 79.01646 74.79690 164.82080 91.66285 105.96455 112.73139 90.32439
## [169] 110.33724 107.37929 95.69239 101.30586 99.31865 142.56904 85.17328
## [176] 78.08007 100.75577 106.20961 108.73047 90.83269 78.73348 125.26370
## [183] 93.00699 82.68974 95.27441 96.05648 122.19841 101.69475 115.08108
## [190] 90.01416 104.28891 93.50628 101.89167 82.09273 73.78397 139.94427
## [197] 112.01418 74.97457 87.77668 76.29040Simulasi luas rumah dalam meter persegi menggunakan distribusi normal dengan rata-rata 100 dan standar deviasi 20.
X2 = Jumlah kamar tidur
X2 <- round(rnorm(n, mean = 3, sd = 1))
X2## [1] 5 4 3 4 3 3 2 2 5 3 3 3 4 2 2 5 3 2 2 2 2 4 4 4 3 3 2 2 4 2 5 3 3 2 2 2 3
## [38] 3 3 2 2 2 4 2 3 5 3 2 2 3 3 2 3 3 5 3 4 4 3 1 2 3 3 4 5 5 3 1 3 3 4 4 4 2
## [75] 4 3 3 3 3 3 1 4 3 3 3 3 3 5 3 3 4 4 4 2 5 3 5 2 3 4 2 2 2 2 3 3 1 3 4 5 4
## [112] 4 1 2 3 4 3 2 5 4 3 4 2 4 2 4 3 4 4 3 4 2 2 5 3 1 2 3 4 3 4 4 5 3 3 1 3 2
## [149] 2 3 4 1 3 3 2 3 3 3 1 6 3 4 3 4 4 3 3 2 4 3 5 1 3 4 4 2 2 3 3 1 3 3 3 2 3
## [186] 4 3 2 3 3 2 1 4 2 2 5 2 4 2 3Simulasi jumlah kamar tidur, dibulatkan karena kamar tidak bisa pecahan.
Lokasi
X3 <- factor(sample(c("cluster", "town house", "residence"), size = n, replace = TRUE))
X3## [1] residence cluster residence cluster residence residence
## [7] cluster residence residence residence residence residence
## [13] town house residence town house residence town house cluster
## [19] cluster residence cluster town house cluster town house
## [25] town house cluster cluster town house residence cluster
## [31] town house cluster town house residence town house town house
## [37] residence cluster town house residence residence cluster
## [43] residence cluster cluster residence cluster cluster
## [49] residence cluster cluster cluster town house town house
## [55] town house town house residence residence town house cluster
## [61] residence cluster residence residence residence cluster
## [67] town house residence residence cluster cluster cluster
## [73] residence residence cluster cluster town house residence
## [79] residence town house cluster residence cluster cluster
## [85] residence town house residence residence residence town house
## [91] cluster residence residence town house town house town house
## [97] residence residence town house cluster cluster cluster
## [103] town house cluster town house town house residence residence
## [109] residence residence town house town house town house residence
## [115] residence residence cluster residence cluster residence
## [121] town house cluster cluster cluster residence cluster
## [127] town house cluster residence residence residence town house
## [133] residence town house cluster residence town house town house
## [139] cluster town house cluster town house residence residence
## [145] town house town house cluster town house residence town house
## [151] residence cluster town house residence cluster town house
## [157] residence town house cluster residence residence residence
## [163] cluster residence town house town house town house residence
## [169] cluster cluster town house town house residence town house
## [175] town house cluster residence cluster residence residence
## [181] residence residence town house town house cluster cluster
## [187] residence town house cluster town house town house town house
## [193] cluster residence residence town house town house town house
## [199] cluster residence
## Levels: cluster residence town houseLokasi disimulasikan secara acak dari tiga kategori menggunakan distribusi uniform (masing-masing punya peluang yang sama).
Koefisien simulasi
beta0 <- 50000
beta1 <- 3000
beta2 <- 10000
beta3 <- c("cluster" = 0, "town house" = 15000, "residence" = 30000) # efek lokasi terhadap hargaKoefisien yang menggambarkan pengaruh masing-masing variabel terhadap harga rumah. Lokasi memiliki efek tetap sesuai kategorinya.
Bangkitkan harga rumah
Y <- beta0 + beta1 * X1 + beta2 * X2 + beta3[X3] + rnorm(n, mean = 0, sd = 20000)
Y## town house cluster town house cluster town house town house cluster
## 384490.4 396825.1 505389.0 400810.8 365123.7 490490.1 378724.2
## town house town house town house town house town house residence town house
## 304728.1 361646.7 330447.6 467448.2 388353.0 421463.2 405581.0
## residence town house residence cluster cluster town house cluster
## 372998.7 505376.1 451992.2 265132.3 413556.9 387336.5 314208.9
## residence cluster residence residence cluster cluster residence
## 425716.9 322917.4 362416.3 403224.1 282253.0 402915.1 399484.3
## town house cluster residence cluster residence town house residence
## 305281.3 448671.8 458850.6 380578.8 467120.4 451332.0 438110.7
## residence town house cluster residence town house town house cluster
## 428429.7 423445.3 409934.2 386866.0 368724.4 321612.9 362546.2
## town house cluster cluster town house cluster cluster town house
## 357037.7 493605.0 473584.9 331484.9 339515.0 327971.1 402645.0
## cluster cluster cluster residence residence residence residence
## 363752.0 433555.2 363314.1 435040.6 494528.3 411244.1 507885.2
## town house town house residence cluster town house cluster town house
## 295982.7 462247.9 395369.7 388979.1 412549.4 352630.6 389533.2
## town house town house cluster residence town house town house cluster
## 326705.6 326140.8 412374.6 461147.3 389723.9 495944.6 482922.9
## cluster cluster town house town house cluster cluster residence
## 362657.1 312592.4 465739.2 360277.9 365708.4 473752.4 394019.6
## town house town house residence cluster town house cluster cluster
## 345864.9 370952.5 386629.1 361898.3 416314.7 362244.8 384350.5
## town house residence town house town house town house residence cluster
## 392475.4 420056.1 451553.8 438103.8 363744.7 444398.3 461690.7
## town house town house residence residence residence town house town house
## 438441.4 358384.6 384476.2 497805.6 377898.6 552085.1 455918.7
## residence cluster cluster cluster residence cluster residence
## 365664.1 359069.1 365849.5 397381.9 388531.7 351891.4 359177.1
## residence town house town house town house town house residence residence
## 401826.4 348874.4 308799.6 370285.0 472616.6 386229.6 457795.9
## residence town house town house town house cluster town house cluster
## 313494.1 371369.1 438630.0 426548.3 385645.0 366174.1 351789.6
## town house residence cluster cluster cluster town house cluster
## 321628.9 391384.6 326301.1 326238.6 349847.6 482461.5 314356.7
## residence cluster town house town house town house residence town house
## 415751.1 387353.6 341408.6 390259.1 498921.0 417756.0 393444.2
## residence cluster town house residence residence cluster residence
## 385596.0 270156.0 444625.0 310177.4 440847.0 538522.7 331083.0
## cluster residence town house town house residence residence cluster
## 437042.2 352529.5 281409.3 302893.8 331812.9 357584.4 272299.7
## residence town house residence town house cluster residence town house
## 438082.9 500452.7 337101.3 433885.2 388130.6 435704.3 353075.1
## cluster residence town house residence cluster town house town house
## 396293.8 398886.6 421327.7 358976.0 400929.0 425008.8 443706.6
## town house cluster town house residence residence residence town house
## 351302.7 298139.6 616427.5 393282.2 413586.8 433460.3 355828.7
## cluster cluster residence residence town house residence residence
## 435021.3 434665.7 449570.4 422291.3 373146.1 559974.5 365743.3
## cluster town house cluster town house town house town house town house
## 295788.5 391332.2 428533.4 395485.7 390428.3 327947.5 460323.5
## residence residence cluster cluster town house residence cluster
## 405889.9 342043.1 387580.1 378988.1 467325.9 405002.0 438979.5
## residence residence residence cluster town house town house residence
## 377148.9 400625.2 383130.2 371821.0 347977.5 318298.5 526216.5
## residence residence cluster town house
## 449392.0 327359.7 355279.5 302050.4Harga rumah dihitung berdasarkan model linear dengan tambahan error acak dari distribusi normal untuk mensimulasikan variasi alami.
Data frame
data_rumah <- data.frame(Harga = Y, Luas = X1, Kamar = X2, Lokasi = X3)
data_rumah## Harga Luas Kamar Lokasi
## 1 384490.4 88.79049 5 residence
## 2 396825.1 95.39645 4 cluster
## 3 505389.0 131.17417 3 residence
## 4 400810.8 101.41017 4 cluster
## 5 365123.7 102.58575 3 residence
## 6 490490.1 134.30130 3 residence
## 7 378724.2 109.21832 2 cluster
## 8 304728.1 74.69878 2 residence
## 9 361646.7 86.26294 5 residence
## 10 330447.6 91.08676 3 residence
## 11 467448.2 124.48164 3 residence
## 12 388353.0 107.19628 3 residence
## 13 421463.2 108.01543 4 town house
## 14 405581.0 102.21365 2 residence
## 15 372998.7 88.88318 2 town house
## 16 505376.1 135.73826 5 residence
## 17 451992.2 109.95701 3 town house
## 18 265132.3 60.66766 2 cluster
## 19 413556.9 114.02712 2 cluster
## 20 387336.5 90.54417 2 residence
## 21 314208.9 78.64353 2 cluster
## 22 425716.9 95.64050 4 town house
## 23 322917.4 79.47991 4 cluster
## 24 362416.3 85.42218 4 town house
## 25 403224.1 87.49921 3 town house
## 26 282253.0 66.26613 3 cluster
## 27 402915.1 116.75574 2 cluster
## 28 399484.3 103.06746 2 town house
## 29 305281.3 77.23726 4 residence
## 30 448671.8 125.07630 2 cluster
## 31 458850.6 108.52928 5 town house
## 32 380578.8 94.09857 3 cluster
## 33 467120.4 117.90251 3 town house
## 34 451332.0 117.56267 2 residence
## 35 438110.7 116.43162 2 town house
## 36 428429.7 113.77281 2 town house
## 37 423445.3 111.07835 3 residence
## 38 409934.2 98.76177 3 cluster
## 39 386866.0 93.88075 3 town house
## 40 368724.4 92.39058 2 residence
## 41 321612.9 86.10586 2 residence
## 42 362546.2 95.84165 2 cluster
## 43 357037.7 74.69207 4 residence
## 44 493605.0 143.37912 2 cluster
## 45 473584.9 124.15924 3 cluster
## 46 331484.9 77.53783 5 residence
## 47 339515.0 91.94230 3 cluster
## 48 327971.1 90.66689 2 cluster
## 49 402645.0 115.59930 2 residence
## 50 363752.0 98.33262 3 cluster
## 51 433555.2 105.06637 3 cluster
## 52 363314.1 99.42906 2 cluster
## 53 435040.6 99.14259 3 town house
## 54 494528.3 127.37205 3 town house
## 55 411244.1 95.48458 5 town house
## 56 507885.2 130.32941 3 town house
## 57 295982.7 69.02494 4 residence
## 58 462247.9 111.69227 4 residence
## 59 395369.7 102.47708 3 town house
## 60 388979.1 104.31883 1 cluster
## 61 412549.4 107.59279 2 residence
## 62 352630.6 89.95353 3 cluster
## 63 389533.2 93.33585 3 residence
## 64 326705.6 79.62849 4 residence
## 65 326140.8 78.56418 5 residence
## 66 412374.6 106.07057 5 cluster
## 67 461147.3 108.96420 3 town house
## 68 389723.9 101.06008 1 residence
## 69 495944.6 118.44535 3 residence
## 70 482922.9 141.00169 3 cluster
## 71 362657.1 90.17938 4 cluster
## 72 312592.4 53.81662 4 cluster
## 73 465739.2 120.11477 4 residence
## 74 360277.9 85.81598 2 residence
## 75 365708.4 86.23983 4 cluster
## 76 473752.4 120.51143 3 cluster
## 77 394019.6 94.30454 3 town house
## 78 345864.9 75.58565 3 residence
## 79 370952.5 103.62607 3 residence
## 80 386629.1 97.22217 3 town house
## 81 361898.3 100.11528 1 cluster
## 82 416314.7 107.70561 4 residence
## 83 362244.8 92.58680 3 cluster
## 84 384350.5 112.88753 3 cluster
## 85 392475.4 95.59027 3 residence
## 86 420056.1 106.63564 3 town house
## 87 451553.8 121.93678 3 residence
## 88 438103.8 108.70363 5 residence
## 89 363744.7 93.48137 3 residence
## 90 444398.3 122.97615 3 town house
## 91 461690.7 119.87008 4 cluster
## 92 438441.4 110.96794 4 residence
## 93 358384.6 104.77463 4 residence
## 94 384476.2 87.44188 2 town house
## 95 497805.6 127.21305 5 town house
## 96 377898.6 87.99481 3 town house
## 97 552085.1 143.74666 5 residence
## 98 455918.7 130.65221 2 residence
## 99 365664.1 95.28599 3 town house
## 100 359069.1 79.47158 4 cluster
## 101 365849.5 85.79187 2 cluster
## 102 397381.9 105.13767 2 cluster
## 103 388531.7 95.06616 2 town house
## 104 351891.4 93.04915 2 cluster
## 105 359177.1 80.96763 3 town house
## 106 401826.4 99.09945 3 town house
## 107 348874.4 84.30191 1 residence
## 108 308799.6 66.64116 3 residence
## 109 370285.0 92.39547 4 residence
## 110 472616.6 118.37993 5 residence
## 111 386229.6 88.49306 4 town house
## 112 457795.9 112.15929 4 town house
## 113 313494.1 67.64235 1 town house
## 114 371369.1 98.88876 2 residence
## 115 438630.0 110.38814 3 residence
## 116 426548.3 106.02307 4 residence
## 117 385645.0 102.11352 3 cluster
## 118 366174.1 87.18588 2 residence
## 119 351789.6 83.00591 5 cluster
## 120 321628.9 79.51742 4 residence
## 121 391384.6 102.35293 3 town house
## 122 326301.1 81.05051 4 cluster
## 123 326238.6 90.18885 2 cluster
## 124 349847.6 94.87816 4 cluster
## 125 482461.5 136.87724 2 residence
## 126 314356.7 86.96100 4 cluster
## 127 415751.1 104.70773 3 town house
## 128 387353.6 101.55922 4 cluster
## 129 341408.6 80.76287 4 residence
## 130 390259.1 98.57384 3 residence
## 131 498921.0 128.89102 4 residence
## 132 417756.0 109.03008 2 town house
## 133 393444.2 100.82466 2 residence
## 134 385596.0 91.55006 5 town house
## 135 270156.0 58.93506 3 cluster
## 136 444625.0 122.62674 1 residence
## 137 310177.4 70.78720 2 town house
## 138 440847.0 114.79895 3 town house
## 139 538522.7 138.18207 4 cluster
## 140 331083.0 71.12214 3 town house
## 141 437042.2 114.03569 4 cluster
## 142 352529.5 94.75605 4 town house
## 143 281409.3 68.55712 5 residence
## 144 302893.8 69.70665 3 residence
## 145 331812.9 67.96928 3 town house
## 146 357584.4 89.38187 1 town house
## 147 272299.7 70.76489 3 cluster
## 148 438082.9 113.75834 2 town house
## 149 500452.7 142.00218 2 residence
## 150 337101.3 74.25939 3 town house
## 151 433885.2 115.75478 4 residence
## 152 388130.6 115.38084 1 cluster
## 153 435704.3 106.64405 3 town house
## 154 353075.1 79.83247 3 residence
## 155 396293.8 97.61095 2 cluster
## 156 398886.6 94.39209 3 town house
## 157 421327.7 111.25979 3 residence
## 158 358976.0 92.55122 3 town house
## 159 400929.0 119.53947 1 cluster
## 160 425008.8 92.50838 6 residence
## 161 443706.6 121.05423 3 residence
## 162 351302.7 79.01646 4 residence
## 163 298139.6 74.79690 3 cluster
## 164 616427.5 164.82080 4 residence
## 165 393282.2 91.66285 4 town house
## 166 413586.8 105.96455 3 town house
## 167 433460.3 112.73139 3 town house
## 168 355828.7 90.32439 2 residence
## 169 435021.3 110.33724 4 cluster
## 170 434665.7 107.37929 3 cluster
## 171 449570.4 95.69239 5 town house
## 172 422291.3 101.30586 1 town house
## 173 373146.1 99.31865 3 residence
## 174 559974.5 142.56904 4 town house
## 175 365743.3 85.17328 4 town house
## 176 295788.5 78.08007 2 cluster
## 177 391332.2 100.75577 2 residence
## 178 428533.4 106.20961 3 cluster
## 179 395485.7 108.73047 3 residence
## 180 390428.3 90.83269 1 residence
## 181 327947.5 78.73348 3 residence
## 182 460323.5 125.26370 3 residence
## 183 405889.9 93.00699 3 town house
## 184 342043.1 82.68974 2 town house
## 185 387580.1 95.27441 3 cluster
## 186 378988.1 96.05648 4 cluster
## 187 467325.9 122.19841 3 residence
## 188 405002.0 101.69475 2 town house
## 189 438979.5 115.08108 3 cluster
## 190 377148.9 90.01416 3 town house
## 191 400625.2 104.28891 2 town house
## 192 383130.2 93.50628 1 town house
## 193 371821.0 101.89167 4 cluster
## 194 347977.5 82.09273 2 residence
## 195 318298.5 73.78397 2 residence
## 196 526216.5 139.94427 5 town house
## 197 449392.0 112.01418 2 town house
## 198 327359.7 74.97457 4 town house
## 199 355279.5 87.77668 2 cluster
## 200 302050.4 76.29040 3 residenceMenggabungkan seluruh variabel menjadi satu tabel agar mudah dianalisis.
Menampilkan sebagian data
head(data_rumah)## Harga Luas Kamar Lokasi
## 1 384490.4 88.79049 5 residence
## 2 396825.1 95.39645 4 cluster
## 3 505389.0 131.17417 3 residence
## 4 400810.8 101.41017 4 cluster
## 5 365123.7 102.58575 3 residence
## 6 490490.1 134.30130 3 residenceVisualisasi
Scatterplot luas rumah vs harga
plot(data_rumah$Luas, data_rumah$Harga,
main = "Scatterplot Luas vs Harga Rumah",
xlab = "Luas Rumah (m2)",
ylab = "Harga Rumah",
pch = 16, col = "darkgreen")
Scatterplot untuk melihat hubungan antara luas rumah dan harga rumah secara visual.
Model Regresi Linear Berganda
model_rumah <- lm(Harga ~ Luas + Kamar + Lokasi, data = data_rumah)
summary(model_rumah)##
## Call:
## lm(formula = Harga ~ Luas + Kamar + Lokasi, data = data_rumah)
##
## Residuals:
## Min 1Q Median 3Q Max
## -56052 -11127 -1056 12919 55462
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 70025.71 8230.88 8.508 4.65e-15 ***
## Luas 2920.53 70.56 41.393 < 2e-16 ***
## Kamar 7482.90 1274.52 5.871 1.84e-08 ***
## Lokasiresidence 8481.88 3243.99 2.615 0.00963 **
## Lokasitown house 23522.27 3383.15 6.953 5.26e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 18740 on 195 degrees of freedom
## Multiple R-squared: 0.9033, Adjusted R-squared: 0.9013
## F-statistic: 455.4 on 4 and 195 DF, p-value: < 2.2e-16Membangun model regresi linear berganda untuk memprediksi harga berdasarkan semua variabel input. Hasil summary menunjukkan signifikansi dan kontribusi masing-masing variabel.
Plot Prediksi vs Harga Aktual
plot(data_rumah$Harga, fitted(model_rumah),
main = "Harga Aktual vs Prediksi",
xlab = "Harga Aktual",
ylab = "Harga Prediksi",
pch = 19, col = "blue")
abline(a = 0, b = 1, col = "red", lwd = 2)
Visualisasi untuk membandingkan seberapa dekat prediksi model terhadap harga aktual. Garis merah y = x membantu melihat kesesuaian.
Kesimpulan
Model regresi menunjukkan bahwa:
Semakin luas rumah dan semakin banyak kamar tidur, harga semakin tinggi.
Lokasi town house dan residence memberikan peningkatan harga dibanding cluster.
Model dapat digunakan untuk memperkirakan harga rumah berdasarkan kriteria simulasi.
UAS Nomor 2
Tujuan
Mensimulasikan sistem antrian dengan distribusi kedatangan pelanggan mengikuti Poisson (λ = 50 pelanggan per jam), serta menganalisis ukuran kinerja sistem antrian pada skenario jumlah pelayan yang berbeda (1 dan 2 pelayan)
Pembangkitan Data Poisson
set.seed(123)Parameter Poisson
lambda <- 50
pelanggan <- rpois(100, lambda = lambda)Data jumlah pelanggan perjam diasumsikan menggunakan distribusi poisson dengan rata-rata 50 pelanggan perjam. Sebanyak 100 observasi dibuat untuk mewakili data simulasi selama 100 jam.
Menampilkan ringkasan data
summary(pelanggan)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 37.00 46.00 49.00 50.11 53.00 71.00Visualisasi Distribusi Jumlah Pelanggan
hist(pelanggan, col = "lightblue", breaks = 15,
main = "Distribusi Jumlah Pelanggan per Jam",
xlab = "Jumlah Pelanggan")
Histogram menunjukkan sebaran jumlah pelanggan yang datang perjam. Bentuk distribusi poisson akan terlihat dari frekuensi kedatangan per nilai pelanggan.
a. Sistem Antrian M/M/1
mu <- 60
rho1 <- lambda / mu
L1 <- rho1 / (1 - rho1)
Lq1 <- rho1^2 / (1 - rho1)
W1 <- 1 / (mu - lambda)
Wq1 <- lambda / (mu * (mu - lambda))
cat("== M/M/1 ==\n")## == M/M/1 ==cat("Utilisasi:", round(rho1, 3), "\n")## Utilisasi: 0.833cat("L :", round(L1, 2), "\n")## L : 5cat("Lq:", round(Lq1, 2), "\n")## Lq: 4.17cat("W :", round(W1, 2), "jam\n")## W : 0.1 jamcat("Wq:", round(Wq1, 2), "jam\n")## Wq: 0.08 jamSistem M/M/1 memiliki satu pelayan dengan tingkat layanan rata-rata μ = 60. Dihitung beberapa ukuran performa antrian:
ρ (rho): tingkat utilisasi pelayan
L: rata-rata jumlah pelanggan dalam sistem
Lq: rata-rata jumlah pelanggan dalam antrian
W: rata-rata waktu dalam sistem
Wq: rata-rata waktu tunggu dalam antrian
Hasil cat() menampilkan ringkasan semua metrik kinerja sistem M/M/1.
b. Sistem Antrian M/M/2
c <- 2
rho2 <- lambda / (c * mu)
fact <- function(x) ifelse(x == 0, 1, prod(1:x))
sum_terms <- sum(sapply(0:(c-1), function(k) (lambda/mu)^k / fact(k)))
last_term <- ((lambda/mu)^c / (fact(c) * (1 - rho2)))
P0 <- 1 / (sum_terms + last_term)
Pw <- last_term * P0
Lq2 <- Pw * (lambda * mu) / ((c * mu - lambda)^2)
L2 <- Lq2 + lambda / mu
Wq2 <- Lq2 / lambda
W2 <- Wq2 + 1 / mu
cat("== M/M/2 ==\n")## == M/M/2 ==cat("Utilisasi per pelayan:", round(rho2, 3), "\n")## Utilisasi per pelayan: 0.417cat("P(w) =", round(Pw, 3), "\n")## P(w) = 0.245cat("L :", round(L2, 2), "\n")## L : 0.98cat("Lq:", round(Lq2, 2), "\n")## Lq: 0.15cat("W :", round(W2, 2), "jam\n")## W : 0.02 jamcat("Wq:", round(Wq2, 2), "jam\n")## Wq: 0 jamUntuk sistem M/M/2, digunakan dua pelayan. Perhitungannya sedikit lebih kompleks karena melibatkan:
P₀: probabilitas sistem kosong
P(w): probabilitas pelanggan harus menunggu
Lq: jumlah rata-rata pelanggan dalam antrian
L: jumlah rata-rata pelanggan dalam sistem
Wq: waktu tunggu rata-rata dalam antrian
W: total waktu rata-rata dalam sistem
Perhitungan ini menggunakan rumus dari teori antrian multi-server (c > 1), termasuk fungsi faktorial untuk menghitung pembagi.
c. Probabilitas Jumlah Pelanggan > 60
p_gt_60 <- 1 - ppois(60, lambda = 50)
cat("P(X > 60) =", round(p_gt_60, 4),
"atau", round(p_gt_60 * 100, 2), "%")## P(X > 60) = 0.0722 atau 7.22 %Kesimpulan
Dengan 1 pelayan, sistem memiliki utilisasi tinggi dan waktu tunggu lebih lama.
Dengan 2 pelayan, sistem menjadi lebih stabil dan waktu tunggu menurun.
Probabilitas lebih dari 60 pelanggan per jam sekitar r round(p_gt_60, 4) atau r round(p_gt_60*100, 2)%.
UAS Nomor 3
Tujuan
Membangun model regresi linear berganda pada data time series dengan karakteristik:
Variabel dependen memiliki tren dan musiman
Variabel independen 1 (X1) mengandung tren
Variabel independen 2 (X2) mengandung musiman
Pembangkitan Data
library(forecast)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(ggplot2)set.seed(123)Bangkitkan data 10 tahun (120 bulan)
t <- 1:120Membuat vektor waktu dari bulan ke-1 sampai ke-120 (10 tahun).
Variabel dependen Y
trend_Y <- 0.5 * t
season_Y <- 10 * sin(2 * pi * t / 12)
noise_Y <- rnorm(120, mean = 0, sd = 5)
Y <- trend_Y + season_Y + noise_Y
Y_ts <- ts(Y, frequency = 12)
Y_ts## Jan Feb Mar Apr May Jun
## 1 2.6976218 8.5093666 19.2935416 11.0127960 8.1464387 11.5753249
## 2 13.5038573 16.2136676 14.7207943 25.5948197 15.9892524 -0.8330858
## 3 14.3748037 13.2267875 27.6889352 23.4271196 13.8093153 21.2690746
## 4 26.2695883 27.3506955 27.9701867 26.7578990 22.0264651 19.9604136
## 5 33.3998256 33.2434087 36.7665926 34.5175203 31.2856477 33.8430114
## 6 37.3981974 37.1486368 39.8339631 35.5673771 32.1410439 34.5176432
## 7 46.5286926 42.1142502 44.0599569 51.7881109 42.0761350 32.8964114
## 8 46.3975672 53.3191639 58.9841951 54.8361615 47.8703421 50.7440381
## 9 64.4366650 65.3233072 58.3214982 53.5281495 51.9479672 52.2844185
## 10 57.5988674 68.2552371 62.6232652 67.7000756 53.4105865 56.7221902
## Jul Aug Sep Oct Nov Dec
## 1 0.8045810 -10.9855602 -8.9342643 -5.8885639 6.6204090 7.7990691
## 2 8.0067795 -1.0242111 -4.8391185 1.2498714 1.3699778 8.3555439
## 3 12.6323211 5.8643885 10.9756283 12.7304134 16.6079054 21.4432013
## 4 10.1730182 24.1845258 18.5398100 8.7242030 16.4855758 21.6667232
## 5 21.3711451 26.9220990 10.7562360 23.2628147 25.1192712 31.0797078
## 6 30.7410489 25.6047671 29.1113373 36.5901694 28.0448442 24.4541556
## 7 35.4065174 30.6452891 30.5288209 34.2661480 34.6466998 45.2218827
## 8 45.4675193 40.0817308 37.6936587 35.2002156 49.3032622 44.9987021
## 9 45.2665406 41.6020330 37.7419072 44.1146073 44.5754777 45.6602903
## 10 55.0970360 50.8455128 49.0283810 47.1362159 50.2514783 54.8793560trend_Y: tren linier meningkat seiring waktu.
season_Y: pola musiman dengan periode 12 bulan (setahun).
noise_Y: error acak dari distribusi normal.
Y: gabungan ketiganya (model aditif).
Y_ts: diubah ke bentuk time series dengan frekuensi tahunan (12 bulan).
Variabel independen X1
X1 <- 0.3 * t + rnorm(120, mean = 0, sd = 3)
X1_ts <- ts(X1, frequency = 12)
X1_ts## Jan Feb Mar Apr May Jun Jul
## 1 0.6529398 -2.2424238 -0.5716723 0.4317234 7.0315860 -0.1558497 2.8061597
## 2 4.0236988 2.9325095 -1.6597417 8.1940116 0.7180798 7.6198425 11.4273107
## 3 2.6953915 6.2072804 3.7147332 10.4637503 15.0003268 5.1389086 11.6632165
## 4 12.7889686 10.2826837 14.6309202 10.8762574 15.4581344 9.4524690 9.1195343
## 5 16.2505861 16.1068936 14.6538585 15.7958791 15.7977982 22.5853557 14.2759917
## 6 15.1100216 22.3895555 17.8510488 16.6034614 18.7911613 19.2084723 23.4297609
## 7 22.1837506 19.5139099 18.5675954 28.7916402 24.9021265 19.6461859 21.8665022
## 8 24.2569802 24.3712593 23.7341915 24.6161482 31.6527224 26.8379156 27.6577357
## 9 27.7765103 27.2308021 25.9911806 26.1458528 28.5780796 32.4539575 34.2295444
## 10 35.3539515 29.9532223 39.1658819 33.3290412 34.5436165 31.9844169 32.7768339
## Aug Sep Oct Nov Dec
## 1 2.6338825 -0.1855699 2.7860757 7.6336526 4.9545122
## 2 1.6683205 8.4053530 5.8134075 2.1835675 2.6559970
## 3 11.9071267 10.8966077 7.1748702 10.1416422 9.9588140
## 4 22.9231198 12.2494272 14.6946828 16.0097090 12.9486581
## 5 13.5120112 17.2133652 18.3314422 19.0095704 16.6249040
## 6 20.6542119 22.9621614 19.5021239 21.9433359 20.6259423
## 7 20.4435597 30.8964310 28.5372389 24.1045648 26.8295822
## 8 28.3310623 31.5974276 26.6518085 25.5224785 33.8270908
## 9 33.3227651 30.4090281 31.9792498 29.9862106 30.2483455
## 10 30.8489516 34.5512238 36.6569472 36.6729130 33.6553905Mengandung tren linier + noise, tanpa komponen musiman.
Variabel independen X2
X2 <- 8 * sin(2 * pi * t / 12) + rnorm(120, mean = 0, sd = 2)
X2_ts <- ts(X2, frequency = 12)
X2_ts## Jan Feb Mar Apr May
## 1 2.422756058 5.923805794 10.992121340 4.653595989 3.641896811
## 2 3.312165532 7.109196525 11.197017542 6.751073006 6.161598992
## 3 8.586157948 10.023365348 7.733698071 3.415148439 3.222440272
## 4 4.349605400 7.077305585 8.856333530 6.977553196 0.665049805
## 5 3.561899242 7.264333998 10.336767746 9.036565277 6.290526221
## 6 2.569515626 5.422825294 6.122922593 4.823176672 3.125680934
## 7 0.546539202 5.725189814 7.295907087 8.335251036 3.788657332
## 8 2.954175247 8.295694274 7.878356091 8.194124656 6.671035230
## 9 1.271925096 6.526641199 9.731558809 6.724436719 5.248374944
## 10 2.051980834 6.568390768 10.029886345 2.942706253 3.145441426
## Jun Jul Aug Sep Oct
## 1 3.804723643 -4.201949771 -9.647884638 -9.329538871 -5.957283272
## 2 1.261508231 -4.227279791 -9.994007236 -9.042234635 -7.907944137
## 3 0.178414446 -2.309973992 -5.003147293 -6.631381141 -9.718751930
## 4 1.472991930 -3.227946863 -7.459506481 -7.763710978 -6.660125940
## 5 -1.154936002 0.004965461 -6.794801488 -4.266296311 -9.630008602
## 6 0.662358346 -8.028420996 -6.504242364 -5.526649907 -2.853055194
## 7 -2.517297256 -0.631128584 -5.105420647 -7.525139455 -4.491986010
## 8 0.014580181 -1.964882726 -9.305071301 -9.443208881 -3.889767807
## 9 1.918010756 -0.657890342 -6.816169764 -8.103963812 -10.434677949
## 10 0.233274567 -5.786415140 -6.260397345 -7.177140159 -6.994275549
## Nov Dec
## 1 -4.751205743 -1.123752727
## 2 -3.905691134 2.600397355
## 3 -2.300713909 -0.893114433
## 4 -3.557961063 3.281692332
## 5 -3.958032827 2.499829142
## 6 -1.397648016 1.513549528
## 7 -6.677548574 1.321640596
## 8 -3.245224054 -4.104445641
## 9 -3.801344812 -1.143700116
## 10 -8.931796388 5.142916292Variabel musiman dengan variasi bulanan.
Data frame
data_ts <- data.frame(
Y = as.numeric(Y_ts),
X1 = as.numeric(X1_ts),
X2 = as.numeric(X2_ts),
trend = t,
season = as.factor(cycle(Y_ts))
)
data_ts## Y X1 X2 trend season
## 1 2.6976218 0.6529398 2.422756058 1 1
## 2 8.5093666 -2.2424238 5.923805794 2 2
## 3 19.2935416 -0.5716723 10.992121340 3 3
## 4 11.0127960 0.4317234 4.653595989 4 4
## 5 8.1464387 7.0315860 3.641896811 5 5
## 6 11.5753249 -0.1558497 3.804723643 6 6
## 7 0.8045810 2.8061597 -4.201949771 7 7
## 8 -10.9855602 2.6338825 -9.647884638 8 8
## 9 -8.9342643 -0.1855699 -9.329538871 9 9
## 10 -5.8885639 2.7860757 -5.957283272 10 10
## 11 6.6204090 7.6336526 -4.751205743 11 11
## 12 7.7990691 4.9545122 -1.123752727 12 12
## 13 13.5038573 4.0236988 3.312165532 13 1
## 14 16.2136676 2.9325095 7.109196525 14 2
## 15 14.7207943 -1.6597417 11.197017542 15 3
## 16 25.5948197 8.1940116 6.751073006 16 4
## 17 15.9892524 0.7180798 6.161598992 17 5
## 18 -0.8330858 7.6198425 1.261508231 18 6
## 19 8.0067795 11.4273107 -4.227279791 19 7
## 20 -1.0242111 1.6683205 -9.994007236 20 8
## 21 -4.8391185 8.4053530 -9.042234635 21 9
## 22 1.2498714 5.8134075 -7.907944137 22 10
## 23 1.3699778 2.1835675 -3.905691134 23 11
## 24 8.3555439 2.6559970 2.600397355 24 12
## 25 14.3748037 2.6953915 8.586157948 25 1
## 26 13.2267875 6.2072804 10.023365348 26 2
## 27 27.6889352 3.7147332 7.733698071 27 3
## 28 23.4271196 10.4637503 3.415148439 28 4
## 29 13.8093153 15.0003268 3.222440272 29 5
## 30 21.2690746 5.1389086 0.178414446 30 6
## 31 12.6323211 11.6632165 -2.309973992 31 7
## 32 5.8643885 11.9071267 -5.003147293 32 8
## 33 10.9756283 10.8966077 -6.631381141 33 9
## 34 12.7304134 7.1748702 -9.718751930 34 10
## 35 16.6079054 10.1416422 -2.300713909 35 11
## 36 21.4432013 9.9588140 -0.893114433 36 12
## 37 26.2695883 12.7889686 4.349605400 37 1
## 38 27.3506955 10.2826837 7.077305585 38 2
## 39 27.9701867 14.6309202 8.856333530 39 3
## 40 26.7578990 10.8762574 6.977553196 40 4
## 41 22.0264651 15.4581344 0.665049805 41 5
## 42 19.9604136 9.4524690 1.472991930 42 6
## 43 10.1730182 9.1195343 -3.227946863 43 7
## 44 24.1845258 22.9231198 -7.459506481 44 8
## 45 18.5398100 12.2494272 -7.763710978 45 9
## 46 8.7242030 14.6946828 -6.660125940 46 10
## 47 16.4855758 16.0097090 -3.557961063 47 11
## 48 21.6667232 12.9486581 3.281692332 48 12
## 49 33.3998256 16.2505861 3.561899242 49 1
## 50 33.2434087 16.1068936 7.264333998 50 2
## 51 36.7665926 14.6538585 10.336767746 51 3
## 52 34.5175203 15.7958791 9.036565277 52 4
## 53 31.2856477 15.7977982 6.290526221 53 5
## 54 33.8430114 22.5853557 -1.154936002 54 6
## 55 21.3711451 14.2759917 0.004965461 55 7
## 56 26.9220990 13.5120112 -6.794801488 56 8
## 57 10.7562360 17.2133652 -4.266296311 57 9
## 58 23.2628147 18.3314422 -9.630008602 58 10
## 59 25.1192712 19.0095704 -3.958032827 59 11
## 60 31.0797078 16.6249040 2.499829142 60 12
## 61 37.3981974 15.1100216 2.569515626 61 1
## 62 37.1486368 22.3895555 5.422825294 62 2
## 63 39.8339631 17.8510488 6.122922593 63 3
## 64 35.5673771 16.6034614 4.823176672 64 4
## 65 32.1410439 18.7911613 3.125680934 65 5
## 66 34.5176432 19.2084723 0.662358346 66 6
## 67 30.7410489 23.4297609 -8.028420996 67 7
## 68 25.6047671 20.6542119 -6.504242364 68 8
## 69 29.1113373 22.9621614 -5.526649907 69 9
## 70 36.5901694 19.5021239 -2.853055194 70 10
## 71 28.0448442 21.9433359 -1.397648016 71 11
## 72 24.4541556 20.6259423 1.513549528 72 12
## 73 46.5286926 22.1837506 0.546539202 73 1
## 74 42.1142502 19.5139099 5.725189814 74 2
## 75 44.0599569 18.5675954 7.295907087 75 3
## 76 51.7881109 28.7916402 8.335251036 76 4
## 77 42.0761350 24.9021265 3.788657332 77 5
## 78 32.8964114 19.6461859 -2.517297256 78 6
## 79 35.4065174 21.8665022 -0.631128584 79 7
## 80 30.6452891 20.4435597 -5.105420647 80 8
## 81 30.5288209 30.8964310 -7.525139455 81 9
## 82 34.2661480 28.5372389 -4.491986010 82 10
## 83 34.6466998 24.1045648 -6.677548574 83 11
## 84 45.2218827 26.8295822 1.321640596 84 12
## 85 46.3975672 24.2569802 2.954175247 85 1
## 86 53.3191639 24.3712593 8.295694274 86 2
## 87 58.9841951 23.7341915 7.878356091 87 3
## 88 54.8361615 24.6161482 8.194124656 88 4
## 89 47.8703421 31.6527224 6.671035230 89 5
## 90 50.7440381 26.8379156 0.014580181 90 6
## 91 45.4675193 27.6577357 -1.964882726 91 7
## 92 40.0817308 28.3310623 -9.305071301 92 8
## 93 37.6936587 31.5974276 -9.443208881 93 9
## 94 35.2002156 26.6518085 -3.889767807 94 10
## 95 49.3032622 25.5224785 -3.245224054 95 11
## 96 44.9987021 33.8270908 -4.104445641 96 12
## 97 64.4366650 27.7765103 1.271925096 97 1
## 98 65.3233072 27.2308021 6.526641199 98 2
## 99 58.3214982 25.9911806 9.731558809 99 3
## 100 53.5281495 26.1458528 6.724436719 100 4
## 101 51.9479672 28.5780796 5.248374944 101 5
## 102 52.2844185 32.4539575 1.918010756 102 6
## 103 45.2665406 34.2295444 -0.657890342 103 7
## 104 41.6020330 33.3227651 -6.816169764 104 8
## 105 37.7419072 30.4090281 -8.103963812 105 9
## 106 44.1146073 31.9792498 -10.434677949 106 10
## 107 44.5754777 29.9862106 -3.801344812 107 11
## 108 45.6602903 30.2483455 -1.143700116 108 12
## 109 57.5988674 35.3539515 2.051980834 109 1
## 110 68.2552371 29.9532223 6.568390768 110 2
## 111 62.6232652 39.1658819 10.029886345 111 3
## 112 67.7000756 33.3290412 2.942706253 112 4
## 113 53.4105865 34.5436165 3.145441426 113 5
## 114 56.7221902 31.9844169 0.233274567 114 6
## 115 55.0970360 32.7768339 -5.786415140 115 7
## 116 50.8455128 30.8489516 -6.260397345 116 8
## 117 49.0283810 34.5512238 -7.177140159 117 9
## 118 47.1362159 36.6569472 -6.994275549 118 10
## 119 50.2514783 36.6729130 -8.931796388 119 11
## 120 54.8793560 33.6553905 5.142916292 120 12Plot Time Series
autoplot(cbind(Y = Y_ts, X1 = X1_ts, X2 = X2_ts)) +
ggtitle("Visualisasi Time Series Y, X1, dan X2") +
xlab("Tahun") + ylab("Nilai") +
theme_minimal()
Visualisasi deret waktu untuk Y, X1, dan X2 untuk melihat tren dan pola musiman masing-masing variabel.
Model Regresi Time Series
ts_data <- ts(cbind(Y, X1, X2), start = c(1,1), frequency = 12)
model_ts <- tslm(ts_data[, "Y"] ~ trend + X1 + X2 + season, data = data_ts)
model_ts##
## Call:
## tslm(formula = ts_data[, "Y"] ~ trend + X1 + X2 + season, data = data_ts)
##
## Coefficients:
## (Intercept) trend X1 X2 season2 season3
## 6.8785 0.4130 0.2686 0.1080 1.5000 3.4422
## season4 season5 season6 season7 season8 season9
## 2.2669 -4.9967 -5.1168 -10.3215 -13.3242 -16.3728
## season10 season11 season12
## -13.9906 -11.1496 -8.8431Dibuat data frame yang menggabungkan semua variabel serta menambahkan:
trend sebagai variabel waktu eksplisit
season sebagai faktor musiman (bulan ke-1 sampai ke-12)
Model dibangun menggunakan fungsi tslm() dari paket forecast, yang cocok untuk regresi dengan komponen waktu dan musiman.
Visualisasi Hasil Fitting
autoplot(Y_ts, series = "Y (aktual)") +
autolayer(fitted(model_ts), series = "Y (fitted)", PI = FALSE) +
ggtitle("Hasil Fitting Regresi Time Series") +
ylab("Y") + xlab("Tahun") +
theme_minimal()## Warning in ggplot2::geom_line(ggplot2::aes(x = .data[["timeVal"]], y =
## .data[["seriesVal"]], : Ignoring unknown parameters: `PI`
Grafik ini menunjukkan seberapa baik model mengikuti nilai aktual Y selama 120 bulan. Garis hasil model dibandingkan dengan garis data aktual.
Kesimpulan
Model berhasil memodelkan variabel dependen Y menggunakan variabel X1, X2, serta komponen tren dan musiman. Koefisien model menunjukkan bahwa baik tren maupun musiman memberikan kontribusi signifikan terhadap variabel respon.