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*The author of this computation has been verified*
R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)
Title produced by software: Multiple Regression
Date of computation: Fri, 20 Nov 2009 10:44:44 -0700
 
Cite this page as follows:
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp.htm/, Retrieved Fri, 20 Nov 2009 19:04:26 +0100
 
BibTeX entries for LaTeX users:
@Manual{KEY,
    author = {{YOUR NAME}},
    publisher = {Office for Research Development and Education},
    title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp.htm/},
    year = {2009},
}
@Manual{R,
    title = {R: A Language and Environment for Statistical Computing},
    author = {{R Development Core Team}},
    organization = {R Foundation for Statistical Computing},
    address = {Vienna, Austria},
    year = {2009},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}
 
Original text written by user:
 
IsPrivate?
No (this computation is public)
 
User-defined keywords:
 
Dataseries X:
» Textbox « » Textfile « » CSV «
3,2 1 1,9 1 0 1 0,6 1 0,2 1 0,9 1 2,4 1 4,7 1 9,4 1 12,5 1 15,8 1 18,2 1 16,8 0 17,3 0 19,3 0 17,9 0 20,2 0 18,7 0 20,1 0 18,2 0 18,4 0 18,2 0 18,9 0 19,9 0 21,3 0 20 0 19,5 0 19,6 0 20,9 0 21 0 19,9 0 19,6 0 20,9 0 21,7 0 22,9 0 21,5 0 21,3 0 23,5 0 21,6 0 24,5 0 22,2 0 23,5 0 20,9 0 20,7 0 18,1 0 17,1 0 14,8 0 13,8 0 15,2 0 16 0 17,6 0 15 0 15 0 16,3 0 19,4 0 21,3 0 20,5 0 21,1 0 21,6 0 22,6 0
 
Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits!


Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time6 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
Y[t] = + 21.9379166666667 -13.6895833333334X[t] -3.64M1[t] -3.46000000000001M2[t] -3.59999999999999M3[t] -3.67999999999999M4[t] -3.50000000000000M5[t] -3.12M6[t] -2.66M7[t] -2.30000000000000M8[t] -1.73999999999999M9[t] -1.07999999999999M10[t] -0.399999999999995M11[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)21.93791666666671.69344212.954600
X-13.68958333333341.209602-11.317400
M1-3.642.370325-1.53570.1313280.065664
M2-3.460000000000012.370325-1.45970.151020.07551
M3-3.599999999999992.370325-1.51880.1355170.067758
M4-3.679999999999992.370325-1.55250.1272440.063622
M5-3.500000000000002.370325-1.47660.1464550.073227
M6-3.122.370325-1.31630.1944650.097233
M7-2.662.370325-1.12220.2674740.133737
M8-2.300000000000002.370325-0.97030.3368490.168424
M9-1.739999999999992.370325-0.73410.4665490.233274
M10-1.079999999999992.370325-0.45560.6507520.325376
M11-0.3999999999999952.370325-0.16880.8667150.433357


Multiple Linear Regression - Regression Statistics
Multiple R0.86126669839
R-squared0.741780325755611
Adjusted R-squared0.675851898288959
F-TEST (value)11.2512971150543
F-TEST (DF numerator)12
F-TEST (DF denominator)47
p-value3.96747745767811e-10
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation3.74781342871466
Sum Squared Residuals660.166958333334


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
13.24.60833333333334-1.40833333333334
21.94.78833333333339-2.88833333333339
304.64833333333333-4.64833333333333
40.64.56833333333332-3.96833333333332
50.24.74833333333334-4.54833333333334
60.95.12833333333333-4.22833333333333
72.45.58833333333333-3.18833333333333
84.75.94833333333333-1.24833333333333
99.46.508333333333322.89166666666668
1012.57.168333333333325.33166666666668
1115.87.848333333333337.95166666666667
1218.28.248333333333339.95166666666667
1316.818.2979166666667-1.49791666666667
1417.318.4779166666667-1.17791666666665
1519.318.33791666666670.962083333333332
1617.918.2579166666667-0.357916666666669
1720.218.43791666666671.76208333333333
1818.718.8179166666667-0.117916666666668
1920.119.27791666666670.822083333333333
2018.219.6379166666667-1.43791666666667
2118.420.1979166666667-1.79791666666667
2218.220.8579166666667-2.65791666666667
2318.921.5379166666667-2.63791666666667
2419.921.9379166666667-2.03791666666666
2521.318.29791666666673.00208333333334
262018.47791666666671.52208333333335
2719.518.33791666666671.16208333333333
2819.618.25791666666671.34208333333333
2920.918.43791666666672.46208333333333
302118.81791666666672.18208333333333
3119.919.27791666666670.62208333333333
3219.619.6379166666667-0.037916666666668
3320.920.19791666666670.702083333333327
3421.720.85791666666670.842083333333333
3522.921.53791666666671.36208333333333
3621.521.9379166666667-0.437916666666662
3721.318.29791666666673.00208333333334
3823.518.47791666666675.02208333333335
3921.618.33791666666673.26208333333333
4024.518.25791666666676.24208333333333
4122.218.43791666666673.76208333333333
4223.518.81791666666674.68208333333333
4320.919.27791666666671.62208333333333
4420.719.63791666666671.06208333333333
4518.120.1979166666667-2.09791666666667
4617.120.8579166666667-3.75791666666667
4714.821.5379166666667-6.73791666666667
4813.821.9379166666667-8.13791666666666
4915.218.2979166666667-3.09791666666666
501618.4779166666667-2.47791666666665
5117.618.3379166666667-0.737916666666667
521518.2579166666667-3.25791666666667
531518.4379166666667-3.43791666666667
5416.318.8179166666667-2.51791666666667
5519.419.27791666666670.12208333333333
5621.319.63791666666671.66208333333333
5720.520.19791666666670.302083333333329
5821.120.85791666666670.242083333333335
5921.621.53791666666670.0620833333333334
6022.621.93791666666670.66208333333334


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
160.2012018310853240.4024036621706470.798798168914676
170.1827773490277330.3655546980554670.817222650972267
180.0936519668155560.1873039336311120.906348033184444
190.04324973597990720.08649947195981430.956750264020093
200.05083288126162450.1016657625232490.949167118738375
210.2153157225363640.4306314450727280.784684277463636
220.5298386185231080.9403227629537840.470161381476892
230.7748299448324080.4503401103351840.225170055167592
240.8830761239834370.2338477520331260.116923876016563
250.8670312697713250.2659374604573500.132968730228675
260.8256761524285430.3486476951429140.174323847571457
270.7697989771473560.4604020457052880.230201022852644
280.710131437449680.5797371251006390.289868562550320
290.657410069167490.6851798616650210.342589930832511
300.5982414206435870.8035171587128260.401758579356413
310.5049434181032370.9901131637935260.495056581896763
320.4124251939419080.8248503878838150.587574806058092
330.3233614404437410.6467228808874830.676638559556259
340.250791011662110.501582023324220.74920898833789
350.2136240631049080.4272481262098160.786375936895092
360.1843867663642510.3687735327285030.815613233635749
370.1700502904336410.3401005808672810.82994970956636
380.2121299374902410.4242598749804810.78787006250976
390.1758370118596000.3516740237192000.8241629881404
400.3077938534830.6155877069660.692206146517
410.3458963820412820.6917927640825640.654103617958718
420.4162895344377120.8325790688754250.583710465562288
430.2891203040560000.5782406081119990.710879695944
440.1694320613658600.3388641227317190.83056793863414


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level00OK
10% type I error level10.0344827586206897OK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/109z811258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/109z811258739076.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/1lrec1258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/1lrec1258739076.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/2mcck1258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/2mcck1258739076.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/3pzqg1258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/3pzqg1258739076.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/46yeq1258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/46yeq1258739076.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/5lez51258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/5lez51258739076.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/6ucrn1258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/6ucrn1258739076.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/7oix61258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/7oix61258739076.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/8tpxz1258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/8tpxz1258739076.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/9um9q1258739076.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258740254r5fn9au22f1abrp/9um9q1258739076.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;
 
Parameters (R input):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;
 
R code (references can be found in the software module):
library(lattice)
library(lmtest)
n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test
par1 <- as.numeric(par1)
x <- t(y)
k <- length(x[1,])
n <- length(x[,1])
x1 <- cbind(x[,par1], x[,1:k!=par1])
mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1])
colnames(x1) <- mycolnames #colnames(x)[par1]
x <- x1
if (par3 == 'First Differences'){
x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep='')))
for (i in 1:n-1) {
for (j in 1:k) {
x2[i,j] <- x[i+1,j] - x[i,j]
}
}
x <- x2
}
if (par2 == 'Include Monthly Dummies'){
x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep ='')))
for (i in 1:11){
x2[seq(i,n,12),i] <- 1
}
x <- cbind(x, x2)
}
if (par2 == 'Include Quarterly Dummies'){
x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep ='')))
for (i in 1:3){
x2[seq(i,n,4),i] <- 1
}
x <- cbind(x, x2)
}
k <- length(x[1,])
if (par3 == 'Linear Trend'){
x <- cbind(x, c(1:n))
colnames(x)[k+1] <- 't'
}
x
k <- length(x[1,])
df <- as.data.frame(x)
(mylm <- lm(df))
(mysum <- summary(mylm))
if (n > n25) {
kp3 <- k + 3
nmkm3 <- n - k - 3
gqarr <- array(NA, dim=c(nmkm3-kp3+1,3))
numgqtests <- 0
numsignificant1 <- 0
numsignificant5 <- 0
numsignificant10 <- 0
for (mypoint in kp3:nmkm3) {
j <- 0
numgqtests <- numgqtests + 1
for (myalt in c('greater', 'two.sided', 'less')) {
j <- j + 1
gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value
}
if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1
if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1
if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1
}
gqarr
}
bitmap(file='test0.png')
plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index')
points(x[,1]-mysum$resid)
grid()
dev.off()
bitmap(file='test1.png')
plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index')
grid()
dev.off()
bitmap(file='test2.png')
hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals')
grid()
dev.off()
bitmap(file='test3.png')
densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals')
dev.off()
bitmap(file='test4.png')
qqnorm(mysum$resid, main='Residual Normal Q-Q Plot')
qqline(mysum$resid)
grid()
dev.off()
(myerror <- as.ts(mysum$resid))
bitmap(file='test5.png')
dum <- cbind(lag(myerror,k=1),myerror)
dum
dum1 <- dum[2:length(myerror),]
dum1
z <- as.data.frame(dum1)
z
plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals')
lines(lowess(z))
abline(lm(z))
grid()
dev.off()
bitmap(file='test6.png')
acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function')
grid()
dev.off()
bitmap(file='test7.png')
pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function')
grid()
dev.off()
bitmap(file='test8.png')
opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0))
plot(mylm, las = 1, sub='Residual Diagnostics')
par(opar)
dev.off()
if (n > n25) {
bitmap(file='test9.png')
plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint')
grid()
dev.off()
}
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE)
a<-table.row.end(a)
myeq <- colnames(x)[1]
myeq <- paste(myeq, '[t] = ', sep='')
for (i in 1:k){
if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '')
myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ')
if (rownames(mysum$coefficients)[i] != '(Intercept)') {
myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='')
if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='')
}
}
myeq <- paste(myeq, ' + e[t]')
a<-table.row.start(a)
a<-table.element(a, myeq)
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable1.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Variable',header=TRUE)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'S.D.',header=TRUE)
a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE)
a<-table.element(a,'2-tail p-value',header=TRUE)
a<-table.element(a,'1-tail p-value',header=TRUE)
a<-table.row.end(a)
for (i in 1:k){
a<-table.row.start(a)
a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE)
a<-table.element(a,mysum$coefficients[i,1])
a<-table.element(a, round(mysum$coefficients[i,2],6))
a<-table.element(a, round(mysum$coefficients[i,3],4))
a<-table.element(a, round(mysum$coefficients[i,4],6))
a<-table.element(a, round(mysum$coefficients[i,4]/2,6))
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable2.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Multiple R',1,TRUE)
a<-table.element(a, sqrt(mysum$r.squared))
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'R-squared',1,TRUE)
a<-table.element(a, mysum$r.squared)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Adjusted R-squared',1,TRUE)
a<-table.element(a, mysum$adj.r.squared)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'F-TEST (value)',1,TRUE)
a<-table.element(a, mysum$fstatistic[1])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE)
a<-table.element(a, mysum$fstatistic[2])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE)
a<-table.element(a, mysum$fstatistic[3])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'p-value',1,TRUE)
a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3]))
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Residual Standard Deviation',1,TRUE)
a<-table.element(a, mysum$sigma)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Sum Squared Residuals',1,TRUE)
a<-table.element(a, sum(myerror*myerror))
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable3.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Time or Index', 1, TRUE)
a<-table.element(a, 'Actuals', 1, TRUE)
a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE)
a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE)
a<-table.row.end(a)
for (i in 1:n) {
a<-table.row.start(a)
a<-table.element(a,i, 1, TRUE)
a<-table.element(a,x[i])
a<-table.element(a,x[i]-mysum$resid[i])
a<-table.element(a,mysum$resid[i])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable4.tab')
if (n > n25) {
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'p-values',header=TRUE)
a<-table.element(a,'Alternative Hypothesis',3,header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'breakpoint index',header=TRUE)
a<-table.element(a,'greater',header=TRUE)
a<-table.element(a,'2-sided',header=TRUE)
a<-table.element(a,'less',header=TRUE)
a<-table.row.end(a)
for (mypoint in kp3:nmkm3) {
a<-table.row.start(a)
a<-table.element(a,mypoint,header=TRUE)
a<-table.element(a,gqarr[mypoint-kp3+1,1])
a<-table.element(a,gqarr[mypoint-kp3+1,2])
a<-table.element(a,gqarr[mypoint-kp3+1,3])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable5.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Description',header=TRUE)
a<-table.element(a,'# significant tests',header=TRUE)
a<-table.element(a,'% significant tests',header=TRUE)
a<-table.element(a,'OK/NOK',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'1% type I error level',header=TRUE)
a<-table.element(a,numsignificant1)
a<-table.element(a,numsignificant1/numgqtests)
if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK'
a<-table.element(a,dum)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'5% type I error level',header=TRUE)
a<-table.element(a,numsignificant5)
a<-table.element(a,numsignificant5/numgqtests)
if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK'
a<-table.element(a,dum)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'10% type I error level',header=TRUE)
a<-table.element(a,numsignificant10)
a<-table.element(a,numsignificant10/numgqtests)
if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK'
a<-table.element(a,dum)
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable6.tab')
}
 





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Software written by Ed van Stee & Patrick Wessa


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