Tugas Praktikum Sesi UTS STA561 Pemrograman Statistika

1 STUDI KASUS 1

Buatlah program R untuk melakukan proses berikut ini!

Bangkitkanlah data populasi yang terdiri atas 1000 amatan dengan peubah:

1.1 No

No : 1 sd 500, 9001 sd 9500 (integer)

a<-seq(500)
b<-seq(9001,9500)
No<-c(a,b)
head(No)

## [1] 1 2 3 4 5 6

1.2 ID

ID: G0001 sd G0500, G9001 sd G9500 (diperoleh berdasarkan kolom No, dengan penambahan huruf “G” didepan No dan jumlah digitnya harus 4)

ID_a <- paste("G000",1:9,sep="")
ID_b <- paste("G00",10:99,sep="")
ID_c <- paste("G0",100:500,sep="")
ID_d <- paste("G",9001:9500,sep="")

ID <- c(ID_a, ID_b, ID_c, ID_d)
length(ID)

## [1] 1000

head(ID)

## [1] "G0001" "G0002" "G0003" "G0004" "G0005" "G0006"

1.3 Kelas

Kelas : “Paralel 1” jika ID genap, dan “Paralel 2” jika ID ganjil (factor)

Kelas <- ifelse((No%%2 == 1), "Paralel 2", "Paralel 1" )
head(Kelas)

## [1] "Paralel 2" "Paralel 1" "Paralel 2" "Paralel 1" "Paralel 2" "Paralel 1"

1.4 UTS

UTS : Diperoleh dengan membangkitkan bilangan acak yang menyebar \(𝑁(𝜇 = 70, 𝜎^2 = 25)\)

set.seed(100)
UTS <- rnorm(1000,70,5) #membangkitkan bilangan acak normal sebaran normal sebanyak 1000, dengan mean=70, st.dev=5
head(UTS)

## [1] 67.48904 70.65766 69.60541 74.43392 70.58486 71.59315

hist(UTS)

1.5 Indeks

Indeks : Diperoleh dengan membangkitkan bilangan acak yang menyebar exponential(𝜃 = 1)

set.seed(100)
Indeks <- rexp(1000,rate = 1) #membangkitkan bilangan acak sebaran eksponential
head(Indeks)

## [1] 0.9242116 0.7238372 0.1046449 3.0973623 0.6248052 1.1744293

hist(Indeks)

1.6 Hilangkan data peubah Indeks yang bernilai lebih dari 1 (menjadi missing value)

Indeks <- ifelse(Indeks>1, NA, Indeks)
head(Indeks)

## [1] 0.9242116 0.7238372 0.1046449        NA 0.6248052        NA

1.7 Hitunglah rata-rata dari UTS yang Indeks-nya ada

Mengubah bentuk data dari vektor menjadi data frame

data_gabungan <- data.frame(No, ID, Kelas, UTS, Indeks)
head(data_gabungan)

##   No    ID     Kelas      UTS    Indeks
## 1  1 G0001 Paralel 2 67.48904 0.9242116
## 2  2 G0002 Paralel 1 70.65766 0.7238372
## 3  3 G0003 Paralel 2 69.60541 0.1046449
## 4  4 G0004 Paralel 1 74.43392        NA
## 5  5 G0005 Paralel 2 70.58486 0.6248052
## 6  6 G0006 Paralel 1 71.59315        NA

Menghapus NA yang ada di data frame

data_gabungan <- na.omit(data_gabungan)
dim(data_gabungan)

## [1] 651   5

head(data_gabungan)

##   No    ID     Kelas      UTS     Indeks
## 1  1 G0001 Paralel 2 67.48904 0.92421162
## 2  2 G0002 Paralel 1 70.65766 0.72383718
## 3  3 G0003 Paralel 2 69.60541 0.10464487
## 5  5 G0005 Paralel 2 70.58486 0.62480524
## 7  7 G0007 Paralel 2 67.09105 0.09311719
## 9  9 G0009 Paralel 2 65.87370 0.24999295

rata-rata dari UTS yang Indeks-nya ada

mean(data_gabungan$UTS)

## [1] 70.0757

1.8 Buatlah data sampel berukuran 100 amatan dari Kelas “Paralel 1” yang Indeks-nya ada

Memilih data yang hanya dari Kelas==Paralel 1

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

data_par1 <- data_gabungan %>% 
                filter(Kelas=="Paralel 1") 
dim(data_par1)

## [1] 318   5

Mengambil data sampel berukuran 100 amatan dari Kelas “Paralel 1” yang Indeks-nya ada

set.seed(100)
sampel_UTS<- sample(data_par1$UTS, 100)
sampel_UTS

##   [1] 72.03224 73.68307 70.24721 69.85342 76.43773 69.31303 73.86702 71.79728
##   [9] 62.19770 65.08704 75.04278 65.54269 70.51459 73.06763 70.29516 71.30588
##  [17] 73.80459 64.89304 66.70254 67.80494 72.03314 66.27498 80.04539 63.39988
##  [25] 74.45597 71.29926 74.45308 76.17411 74.21438 65.27406 70.77307 68.85965
##  [33] 65.38940 64.80458 66.74356 66.83848 70.69492 67.21910 74.85101 64.40177
##  [41] 79.48733 66.79467 74.28642 69.87898 68.18616 77.44552 64.43178 61.63752
##  [49] 67.46451 66.51611 68.86291 68.43413 67.38146 67.48137 77.14151 75.40639
##  [57] 75.93661 71.05317 76.70719 75.03729 70.78187 70.27840 62.15416 83.06659
##  [65] 68.94523 78.21674 63.16334 69.47023 65.51144 69.10837 79.40989 67.96253
##  [73] 72.55428 72.37405 59.31753 76.97966 56.69920 73.70164 75.68852 79.62010
##  [81] 82.68280 64.92915 65.46754 66.38606 69.59750 74.82147 65.90582 70.93626
##  [89] 72.19329 67.04893 67.16239 71.77510 72.87518 59.29286 74.23191 64.33025
##  [97] 74.08418 67.85005 68.31438 86.52076

1.9 Hitunglah rata-rata dari UTS pada data sampel yang terambil, lalu bandingkan dengan populasi

mean(sampel_UTS) #rata-rata sampel UTS yang terambil (kelas paralel 1) 

## [1] 70.4267

mean(data_par1$UTS) #rata-rata UTS dari kelas paralel 1 

## [1] 70.4562

Mean dari sampel mendekati mean dari populasi

2 STUDI KASUS 2

Buatlah fungsi di R yang bernama RKU yang dapat digunakan untuk menghitung regresi komponen utama (principal component regression) dengan output utama adalah koefisien penduga parameter regresi komponen utama yang sudah ditransformasi balik untuk peubah X!

Petunjuk dan ketentuan:

1. Untuk menghindari kesalahpahaman konsep yang dimaksud, berikut gambaran umum regresi komponen utama yang dimaksud beserta output yang diminta

• Terdapat peubah \(𝑌, 𝑋_1, 𝑋_2, …, 𝑋_𝑝\)

• Transformasi ke komponen utama \(𝑃𝐶_1, 𝑃𝐶_2, …, 𝑃𝐶_𝑚\) \[𝑃𝐶_𝑗=Σ^p _{i=1} 𝑎_{𝑖𝑗}𝑋_𝑖=𝑎_{1𝑗}𝑋_1+𝑎_{2𝑗}𝑋_2+⋯+𝑎_{𝑝𝑚}𝑋_𝑚 ; 𝑗=1,…,𝑚\]

• Transformasi balik model regresi komponen utama ke peubah X \[𝑌=𝛽^∗ _0+𝛽^∗ _1𝑋_1+𝛽^∗ _2𝑋_2+⋯+𝛽^∗ _p𝑋_𝑝+𝜀^∗\]

• Output utama yang diminta: penduga koefisien \(𝜷^∗ _i\)

• Jika kurang jelas, dipersilakan merujuk ke referensi tertentu

2. Fungsi yang dapat digunakan adalah yang terdapat di R base tanpa menggunakan fungsi “tambahan” yang sudah terdapat pada package tertentu di R (tanpa mengaktifkan package tertentu), seperti lm, eigen, **%*%**, dsb.

3. Format bentuk argumen input dan output/return fungsi tidak ditentukan secara spesifik tipe dan banyaknya objek, dapat dikreasikan sesuai keinginan masing-masing kelompok.

4. Beberapa contoh variasi yang dapat digunakan (variasi diluar contoh berikut diizinkan untuk digunakan) :

• Argumen input hanya 1 berupa matriks \([𝑌|𝑋]\) (peubah Y gabung/gandeng peubah-peubah X)

• Argumen input ada 3, yaitu matriks peubah-peubah X, vektor peubah Y, skalar k (banyaknya peubah komponen utama yang digunakan dalam pemodelan regresi)

• Output berupa matriks koefisien RKU untuk X

• Output berupa list yang berisi vektor koefisien RKU untuk X, vektor koefisien RKU untuk PC, matriks skor komponen utama, dan list akar ciri-vektor ciri

• Output berupa list yang berisi seluruh proses algoritme mulai dari akar ciri hingga vektor koefisien RKU untuk X, kemudian dibuat fungsi generik sehingga ketika dipanggil hanya muncul tampilan tertentu walaupun sebenarnya elemennya banyak (seperti output dari fungsi lm)

Jawab :

RKU<-function(A){
if(!is.matrix(A))
stop("input must be on matrix")
X <- A[,-1] # Variabel X
Y <- A[,1]  # Variabel Y
#membangun objek PCA dengan `prcomp` dan melakukan standarisasi
  pca <- prcomp(X, scale.=TRUE, center=TRUE) 
  akar.ciri <- pca$sdev 
  vektor.ciri <- pca$rotation
  matriks.skor <- pca$x
  pcs <- as.data.frame(pca$x)
  data.pcr <- cbind(Y, pcs)
  pcr <- lm(Y ~ ., data = data.pcr)
  b.pc <- pcr$coefficients
  b.pcr <- as.matrix(pcr$coefficients[-1])
  W <- as.matrix(pca$rotation)
  b.X <- W %*% b.pcr
  b.X <- rbind(pcr$coefficients[1], b.X)
  output <- list(coef.X = b.X, coef.PC = b.pc, Akar.Ciri = akar.ciri, Vector.Ciri = vektor.ciri, Matriks.Skor.PC = matriks.skor) 
  return(output)
}

# Test
Mtrx <- matrix(rnorm(150, mean = 5, sd = 3), ncol = 5)
Mtrx[1:6,]

##          [,1]       [,2]      [,3]      [,4]      [,5]
## [1,] 5.806896  7.2445810  5.295800 -1.081028  7.247066
## [2,] 6.729229  6.7139754  6.182134  2.927421  5.084265
## [3,] 6.360782  6.6273890  5.894587 -0.526108  9.348044
## [4,] 3.828368  6.6799139 10.932454  6.426702  4.974849
## [5,] 1.290174 -0.2171603  6.197968  4.152456 12.560115
## [6,] 7.961260  3.2493512  9.865179 10.341262  2.658650

model_RKU <- RKU(Mtrx)
model_RKU

## $coef.X
##      (Intercept)
## [1,]   5.5928981
## [2,]   0.4699009
## [3,]  -0.3874903
## [4,]  -0.3271612
## [5,]  -0.2758287
## 
## $coef.PC
## (Intercept)         PC1         PC2         PC3         PC4 
##   5.5928981  -0.2062802   0.2608687  -0.6471945  -0.1568656 
## 
## $Akar.Ciri
## [1] 1.2816851 1.0372926 0.8856542 0.7049283
## 
## $Vector.Ciri
##             PC1        PC2        PC3        PC4
## [1,] -0.6182366  0.2869916 -0.2463262 -0.6890158
## [2,] -0.5079770  0.2259498  0.7865777  0.2687035
## [3,] -0.1462270 -0.8880823  0.2759130 -0.3373419
## [4,]  0.5816877  0.2790921  0.4944602 -0.5824570
## 
## $Matriks.Skor.PC
##                PC1         PC2         PC3         PC4
##  [1,]  0.285291829  2.20562198 -0.76253159  0.09727627
##  [2,] -0.385881078  0.66689716 -0.42313118  0.19250616
##  [3,]  0.682569963  2.19472051 -0.16212808 -0.18146906
##  [4,] -1.372687981 -0.18341807  1.13447703  0.18330047
##  [5,]  2.386202171  0.27702425  1.48470905  0.19506554
##  [6,] -1.153538906 -2.12261052  1.18044335  0.81283685
##  [7,] -2.306520160 -0.07957482 -0.41896722 -0.04371122
##  [8,] -0.267595231 -0.28213656 -0.89948264  1.15800671
##  [9,]  0.482819431  1.42576531  0.27074250 -1.17316554
## [10,]  1.328442355  0.30966428  1.22238287  0.35036820
## [11,] -0.324408868 -0.15363303  0.65798340 -0.93675973
## [12,] -0.983573893  0.58352418 -1.12847742 -0.12447968
## [13,] -1.323130226  1.44461904  1.19596679  0.41871577
## [14,] -1.642609189  0.83085941  1.73514901 -0.50607158
## [15,]  0.601266238 -0.96362000 -0.13790193  1.31974894
## [16,] -2.148658118 -0.09648855  0.01100401 -0.32388163
## [17,]  0.182999605  0.14670636 -1.37001671 -0.32231033
## [18,]  1.208656582 -0.09417281  0.26702325  0.75462745
## [19,] -0.038421517 -0.13478641 -0.62332193 -0.44830401
## [20,]  1.323311999  0.07200414 -0.81924445  0.74833390
## [21,]  0.116168171  0.04252496 -0.30432980  1.17923754
## [22,]  1.450931629  0.40556486  0.15627982  0.21551819
## [23,]  1.370237080  0.02022831 -0.16940485 -0.46558583
## [24,]  0.804923426 -0.13072791 -0.69582331 -0.96109884
## [25,] -2.383251544 -0.66650086  0.53622751 -0.46400274
## [26,] -0.184235232 -0.09160734 -0.01128700  1.02878151
## [27,]  0.057650851 -0.83776585 -1.21284891 -0.23224438
## [28,] -0.308023420 -2.03955982 -1.03717715 -0.51310765
## [29,]  2.537966525 -2.02133150  1.11895035 -1.16077560
## [30,]  0.003097509 -0.72779069 -0.79526478 -0.79735567

3 STUDI KASUS 3

Bangkitkan peubah \(𝑌, 𝑋_1, 𝑋_2, 𝑋_3\) sebanyak 1000 amatan berdasarkan model regresi linear berganda berikut ini :

\[𝑌=10+3𝑋_1+5𝑋_2+7𝑋_3+𝜀\]

dengan mengasumsikan bahwa 𝜀~𝑁(0,1) dan antara peubah bebas terjadi multikolinearitas (tidak ada batasan fungsi/package yang digunakan)! Kemudian hitung koefisien penduga parameter regresi komponen utama yang sudah ditransformasi balik menggunakan fungsi RKU yang sudah dibuat di nomor 2!

3.1 Pembangkitan data

library(MASS) 

## 
## Attaching package: 'MASS'

## The following object is masked from 'package:dplyr':
## 
##     select

# Menginputkan parameter sesungguhnya
b0 <- 10
b1 <- 3
b2 <- 5
b3 <- 7 

# Membuat matriks korelasi yang tinggi antara peubah bebas  sehingga terjadi multikolinearitas (dalam hal ini korelasi antar X sebesar 0.95)
mat_kor <- matrix(c(1,0.95,0.95,0.95,1,0.95,0.95,0.95,1),nrow=3,ncol=3) 

# Membangkitkan data sebanyak 1000 amatan
for (i in 1:1000){ 
  # Membangkitkan error berdistribusi normal baku sebanyak 1000 amatan
  error <- rnorm(1000) 
  # Membangkitkan peubah X sebanyak 1000 amatan dan parameter Sigma sesuai dengan matriks korelasi yang didefinisikan
  X <- mvrnorm(n=1000, mu = c(0,0,0),Sigma=mat_kor) 
  Y <- b0 + b1*X[,1] + b2*X[,2] + b3*X[,3]+ error 
}

#Membangkitkan peubah Y
Y[1:6] 

## [1]   8.896584 -23.771419   7.747851  16.447841  21.808628 -15.494382

# Membangkitkan peubah X1, X2, X3
X[1:6,]

##             [,1]       [,2]         [,3]
## [1,]  0.22025882 -0.2518747 -0.008844288
## [2,] -2.12060669 -2.1458691 -2.264658979
## [3,]  0.08442513 -0.1341272 -0.038446960
## [4,]  0.23340122  0.6398220  0.564553892
## [5,]  0.60667723  0.8604665  0.973255946
## [6,] -2.07399076 -1.4740647 -1.812810089

# Memberi label nama untuk dapat membedakan antar peubah X1 hingga X3
colnames(X) <- c("X1", "X2","X3") 

# Menggabungkan antara Y dan X dalam 1 matriks
data_matriks <- cbind(Y,X)
data_matriks[1:6,]

##               Y          X1         X2           X3
## [1,]   8.896584  0.22025882 -0.2518747 -0.008844288
## [2,] -23.771419 -2.12060669 -2.1458691 -2.264658979
## [3,]   7.747851  0.08442513 -0.1341272 -0.038446960
## [4,]  16.447841  0.23340122  0.6398220  0.564553892
## [5,]  21.808628  0.60667723  0.8604665  0.973255946
## [6,] -15.494382 -2.07399076 -1.4740647 -1.812810089

# Melihat nilai korelasi antar peubah 
cor(X)

##           X1        X2        X3
## X1 1.0000000 0.9511266 0.9540706
## X2 0.9511266 1.0000000 0.9551494
## X3 0.9540706 0.9551494 1.0000000

# Mengubah data matriks dalam bentuk data frame 
# untuk menguji kebenaran syarat dalam Regresi MKT
data_df <- as.data.frame(data_matriks)
head(data_df)

##            Y          X1         X2           X3
## 1   8.896584  0.22025882 -0.2518747 -0.008844288
## 2 -23.771419 -2.12060669 -2.1458691 -2.264658979
## 3   7.747851  0.08442513 -0.1341272 -0.038446960
## 4  16.447841  0.23340122  0.6398220  0.564553892
## 5  21.808628  0.60667723  0.8604665  0.973255946
## 6 -15.494382 -2.07399076 -1.4740647 -1.812810089

3.2 Pembuktian Data Bangkitan sesuai model Regresi MKT

Pembuktian Model Regresi MKT \(𝑌=10+3𝑋_1+5𝑋_2+7𝑋_3+𝜀\)

model_mkt <- lm(Y~X1+X2+X3, data=data_df)
summary(model_mkt)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X3, data = data_df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.1945 -0.6417  0.0176  0.6410  2.7607 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  9.94805    0.03108  320.03   <2e-16 ***
## X1           2.92580    0.11261   25.98   <2e-16 ***
## X2           5.04422    0.11313   44.59   <2e-16 ***
## X3           7.08685    0.11617   61.01   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9822 on 996 degrees of freedom
## Multiple R-squared:  0.996,  Adjusted R-squared:  0.996 
## F-statistic: 8.199e+04 on 3 and 996 DF,  p-value: < 2.2e-16

library(car)

## Loading required package: carData

## 
## Attaching package: 'car'

## The following object is masked from 'package:dplyr':
## 
##     recode

#tes multikolinieritas (multikolinieritas terjadi jika VIF>10) 
car::vif(model_mkt)

##       X1       X2       X3 
## 13.95816 14.28599 15.17880

#Pembuktian error model MKT tersebar normal P-value>0.05 sehingga dapat disimpulkan bahwa residual berdistribusi normal
shapiro.test(model_mkt$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  model_mkt$residuals
## W = 0.99812, p-value = 0.3379

3.3 Mencari Regresi Komponen Utama dengan fungsi RKU yang telah dibangun

#Mencari Regresi Komponen Utama dengan function RKU yang telah dibangun
RKU(data_matriks)

## $coef.X
##    (Intercept)
##       9.967546
## X1    3.016233
## X2    5.236716
## X3    7.385645
## 
## $coef.PC
## (Intercept)         PC1         PC2         PC3 
##    9.967546   -9.030713   -1.902064    2.428369 
## 
## $Akar.Ciri
## [1] 1.7049630 0.2212584 0.2101094
## 
## $Vector.Ciri
##           PC1        PC2        PC3
## X1 -0.5770069  0.7544877 -0.3127481
## X2 -0.5772250 -0.6476202 -0.4973926
## X3 -0.5778186 -0.1064729  0.8091905
## 
## $Matriks.Skor.PC
##                   PC1           PC2           PC3
##    [1,]  0.0263690329  0.3107143520  0.0497778497
##    [2,]  3.6403598982  0.0094821483 -0.0841766746
##    [3,]  0.0533406026  0.1408740967  0.0115868753
##    [4,] -0.7946926020 -0.2945020392  0.0637878502
##    [5,] -1.3528991805 -0.2007105003  0.1621720969
##    [6,]  2.9902166526 -0.4216459076 -0.0693465139
##    [7,]  0.7211148091 -0.3094906455  0.4489042898
##    [8,]  0.6966932714  0.1614008564 -0.0840106054
##    [9,]  0.1750654575 -0.4859240114 -0.2908292070
##   [10,] -0.0004030352 -0.1625033724 -0.2087057930
##   [11,] -1.9258080606  0.2732811010 -0.0706464392
##   [12,]  0.1028447110  0.1954388823  0.3895752067
##   [13,] -2.1652603695 -0.2592086334 -0.1432276958
##   [14,] -1.7384558490 -0.0291801332 -0.5152935019
##   [15,]  2.5420176062 -0.1527208191 -0.0214079538
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##  [838,]  3.5099967603  0.0982937535  0.0847257115
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##  [842,]  0.8989421993  0.0961012947  0.3423306559
##  [843,]  5.0535841478 -0.1508953025  0.2348936544
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##  [853,]  0.9545861988 -0.4089977282 -0.1904929913
##  [854,]  0.9594069060  0.1366881708  0.0180489470
##  [855,]  5.1412419389  0.3008214589 -0.0962345123
##  [856,]  0.3981997316  0.1646155051 -0.1576980102
##  [857,]  1.0094904580  0.0426292085 -0.3246892814
##  [858,]  1.1386454207  0.0246629402 -0.2662118901
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##  [860,] -0.9410059143  0.1574729045 -0.0518437776
##  [861,] -2.1695522301  0.0057469039 -0.1740151335
##  [862,]  2.3074244766  0.1673712088 -0.0411405005
##  [863,] -1.7416607347  0.1069293416 -0.4478459418
##  [864,]  0.4923684749  0.0221165906  0.1193866099
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##  [868,]  1.4835232843  0.1660081737  0.1999533626
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##  [871,]  3.0031201247 -0.0549238529  0.0858785592
##  [872,]  0.9038251086 -0.1269648459  0.2191911669
##  [873,]  0.0182107018 -0.3122506037 -0.1779391150
##  [874,]  1.1532945479  0.1121760096 -0.2576242886
##  [875,]  0.9431186240 -0.0582238089 -0.3329679811
##  [876,]  0.6448212549  0.5536311405 -0.1828277316
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##  [880,] -0.4526171091 -0.5690873200 -0.0328787411
##  [881,] -0.6811270942 -0.1139266120 -0.2214585105
##  [882,] -0.6353005500  0.1908457510 -0.0441719016
##  [883,]  2.7795522921  0.3020948006  0.0602393932
##  [884,]  0.0170071762 -0.0940467840  0.0018625295
##  [885,]  1.9652828791 -0.0795262819 -0.3894317934
##  [886,] -3.8022370466 -0.1626008898 -0.1293236832
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##  [890,] -1.4789836872 -0.2023251611  0.0309499405
##  [891,]  2.2752785564 -0.1454898552 -0.0763533803
##  [892,]  1.1015860640 -0.1063226911 -0.5772846317
##  [893,]  1.1058906619  0.0541508978  0.0785024847
##  [894,]  2.2618177669 -0.1123000645 -0.2143791628
##  [895,] -1.3478678533 -0.1547771472  0.2744639104
##  [896,]  2.0051410976  0.1363576270 -0.1582370515
##  [897,] -4.4761355503 -0.0303840181  0.0418659520
##  [898,]  0.4134537104  0.0811489732  0.0585688652
##  [899,]  0.2968549779  0.0305591727  0.2752840825
##  [900,]  1.9054628093 -0.2433190065 -0.0909889679
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##  [902,] -2.5782510611 -0.1344898503 -0.1992951671
##  [903,] -4.4243096501 -0.1625225138 -0.3335800895
##  [904,]  1.1031365667 -0.0786475867 -0.1492984821
##  [905,]  1.3205483465 -0.2601291291 -0.3297669787
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##  [907,] -1.8086005848 -0.1390284591  0.2052146925
##  [908,]  1.4068143247 -0.0095549108  0.0982360614
##  [909,]  3.8711158901  0.1996197156 -0.2695747929
##  [910,] -0.8741772389 -0.0716709367  0.3974550209
##  [911,]  0.0284380006 -0.5483572428 -0.1846458741
##  [912,]  7.7413665258  0.0115865011 -0.1491332296
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##  [914,]  0.3856268085  0.2064905881 -0.1696202884
##  [915,]  0.1686189097 -0.2458894393  0.1173581340
##  [916,]  3.6903739565 -0.1754891245  0.0251478170
##  [917,] -3.0529264923  0.0128698026 -0.3762311362
##  [918,]  2.2980928054 -0.2587995921  0.4030404485
##  [919,] -2.6724192095  0.5818569020 -0.0301919434
##  [920,] -1.2748206775 -0.1574855217 -0.0963110622
##  [921,]  3.1629451258  0.1547750649 -0.1279875108
##  [922,]  0.6664668498  0.2454984096  0.0230384821
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##  [924,]  0.5661750141 -0.3166585768  0.1646454482
##  [925,]  0.4769431191  0.1093270262 -0.1136377170
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##  [927,]  0.8562541899  0.1940968139  0.1988837664
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##  [930,]  0.2094710280  0.0894299372  0.0473473076
##  [931,] -0.7159286940  0.0402470731  0.0082790560
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##  [933,]  1.5568841340  0.1701107111  0.1212628390
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## [1000,] -3.5395684630  0.0343145924  0.2916532719