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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: Wed, 18 Nov 2009 13:49:00 -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/18/t1258578962urfs7ngk6nori5g.htm/, Retrieved Wed, 18 Nov 2009 22:16:14 +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/18/t1258578962urfs7ngk6nori5g.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 «
1.7 0 2.4 0 2.0 0 2.1 0 2.0 0 1.8 0 2.7 0 2.3 0 1.9 0 2.0 0 2.3 0 2.8 0 2.4 0 2.3 0 2.7 0 2.7 0 2.9 0 3.0 0 2.2 0 2.3 0 2.8 0 2.8 0 2.8 0 2.2 0 2.6 0 2.8 0 2.5 0 2.4 0 2.3 0 1.9 0 1.7 0 2.0 0 2.1 0 1.7 0 1.8 0 1.8 0 1.8 0 1.3 0 1.3 0 1.3 0 1.2 0 1.4 0 2.2 1 2.9 1 3.1 1 3.5 1 3.6 1 4.4 1 4.1 1 5.1 1 5.8 1 5.9 1 5.4 1 5.5 1 4.8 1 3.2 1 2.7 1 2.1 1 1.9 1 0.6 1
 
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 time3 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
Y[t] = + 2.19333333333333 + 2.31666666666667X[t] + 0.391111111111108M1[t] + 0.672222222222222M2[t] + 0.773333333333333M3[t] + 0.814444444444445M4[t] + 0.715555555555555M5[t] + 0.696666666666665M6[t] + 0.254444444444444M7[t] + 0.0955555555555554M8[t] + 0.0966666666666665M9[t] + 0.0177777777777775M10[t] + 0.0988888888888882M11[t] -0.0211111111111111t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)2.193333333333330.5157134.2530.0001025.1e-05
X2.316666666666670.442225.23874e-062e-06
M10.3911111111111080.598610.65340.5167740.258387
M20.6722222222222220.5975891.12490.2664710.133235
M30.7733333333333330.5967931.29580.2015030.100752
M40.8144444444444450.5962231.3660.1785790.089289
M50.7155555555555550.5958821.20080.2359620.117981
M60.6966666666666650.5957681.16940.2482830.124141
M70.2544444444444440.5961550.42680.6715090.335755
M80.09555555555555540.5951290.16060.8731410.43657
M90.09666666666666650.594330.16260.8715080.435754
M100.01777777777777750.5937580.02990.9762440.488122
M110.09888888888888820.5934150.16660.8683810.43419
t-0.02111111111111110.011654-1.81160.0765850.038293


Multiple Linear Regression - Regression Statistics
Multiple R0.698672819377352
R-squared0.488143708536698
Adjusted R-squared0.343488669644896
F-TEST (value)3.37453649922153
F-TEST (DF numerator)13
F-TEST (DF denominator)46
p-value0.00112376002416636
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.938090876596518
Sum Squared Residuals40.4806666666667


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
11.72.56333333333334-0.86333333333334
22.42.82333333333333-0.423333333333331
322.90333333333333-0.903333333333332
42.12.92333333333333-0.823333333333334
522.80333333333333-0.803333333333332
61.82.76333333333333-0.963333333333334
72.72.30.400000000000002
82.32.120.180000000000000
91.92.1-0.2
10221.66533453693773e-16
112.32.060.24
122.81.940.86
132.42.310.0900000000000015
142.32.57-0.270000000000001
152.72.650.0499999999999999
162.72.670.0300000000000005
172.92.550.35
1832.510.490000000000001
192.22.046666666666670.153333333333334
202.31.866666666666670.433333333333333
212.81.846666666666670.953333333333333
222.81.746666666666671.05333333333333
232.81.806666666666670.993333333333333
242.21.686666666666670.513333333333333
252.62.056666666666660.543333333333335
262.82.316666666666670.483333333333333
272.52.396666666666670.103333333333333
282.42.41666666666667-0.0166666666666666
292.32.296666666666670.00333333333333302
301.92.25666666666667-0.356666666666666
311.71.79333333333333-0.0933333333333333
3221.613333333333330.386666666666666
332.11.593333333333330.506666666666666
341.71.493333333333330.206666666666666
351.81.553333333333330.246666666666667
361.81.433333333333330.366666666666666
371.81.80333333333333-0.00333333333333195
381.32.06333333333333-0.763333333333334
391.32.14333333333333-0.843333333333334
401.32.16333333333333-0.863333333333333
411.22.04333333333333-0.843333333333334
421.42.00333333333333-0.603333333333333
432.23.85666666666667-1.65666666666667
442.93.67666666666667-0.776666666666667
453.13.65666666666667-0.556666666666667
463.53.55666666666667-0.0566666666666667
473.63.61666666666667-0.0166666666666663
484.43.496666666666670.903333333333333
494.13.866666666666660.233333333333335
505.14.126666666666670.973333333333333
515.84.206666666666671.59333333333333
525.94.226666666666671.67333333333333
535.44.106666666666671.29333333333333
545.54.066666666666671.43333333333333
554.83.603333333333331.19666666666667
563.23.42333333333333-0.223333333333334
572.73.40333333333333-0.703333333333334
582.13.30333333333333-1.20333333333333
591.93.36333333333333-1.46333333333333
600.63.24333333333333-2.64333333333333


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
170.0430370818095250.086074163619050.956962918190475
180.02620329929330390.05240659858660790.973796700706696
190.04512342899753540.09024685799507090.954876571002465
200.02218833984299260.04437667968598510.977811660157007
210.01050637150432770.02101274300865540.989493628495672
220.004406279131261740.008812558262523490.995593720868738
230.001605797447594990.003211594895189990.998394202552405
240.002478811082816840.004957622165633690.997521188917183
250.0008910194368757230.001782038873751450.999108980563124
260.000310608638991250.00062121727798250.99968939136101
270.0001415601110386140.0002831202220772280.999858439888961
287.69962916701483e-050.0001539925833402970.99992300370833
294.72658269742324e-059.45316539484647e-050.999952734173026
307.0002883637721e-050.0001400057672754420.999929997116362
319.36543726337404e-050.0001873087452674810.999906345627366
325.24932600055746e-050.0001049865200111490.999947506739994
333.35445007176023e-056.70890014352047e-050.999966455499282
343.80249155590266e-057.60498311180533e-050.99996197508444
356.157440915633e-050.000123148818312660.999938425590844
360.0002675906463087850.000535181292617570.999732409353691
370.0002814925098366610.0005629850196733230.999718507490163
380.0003344997593292830.0006689995186585660.99966550024067
390.0002325528592232090.0004651057184464180.999767447140777
400.0001374195991919410.0002748391983838830.999862580400808
417.34083723223419e-050.0001468167446446840.999926591627678
422.32832942647783e-054.65665885295566e-050.999976716705735
430.002221716299401080.004443432598802160.997778283700599


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level220.814814814814815NOK
5% type I error level240.888888888888889NOK
10% type I error level271NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/10usaw1258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/10usaw1258577335.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/15aar1258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/15aar1258577335.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/2iqwe1258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/2iqwe1258577335.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/3bfef1258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/3bfef1258577335.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/4fmpi1258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/4fmpi1258577335.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/5pvju1258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/5pvju1258577335.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/6h7sx1258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/6h7sx1258577335.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/7jmum1258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/7jmum1258577335.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/8hjy61258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/8hjy61258577335.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/9vfbz1258577335.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258578962urfs7ngk6nori5g/9vfbz1258577335.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;
 
Parameters (R input):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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