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Author's title

Author

*Unverified author*

R Software Module

rwasp_smp.wasp

Title produced by software

Standard Deviation-Mean Plot

Date of computation

Sun, 30 Nov 2008 11:33:39 -0700

Cite this page as follows

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2008/Nov/30/t1228070248wik62j3eq5ccyn6.htm/, Retrieved Sun, 19 May 2024 12:15:12 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=26679, Retrieved Sun, 19 May 2024 12:15:12 +0000

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Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords

Estimated Impact

162

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)

F     [Univariate Data Series] [Airline data] [2007-10-18 09:58:47] [42daae401fd3def69a25014f2252b4c2]
F RMPD    [Standard Deviation-Mean Plot] [Non-stationary ti...] [2008-11-30 18:33:39] [d41d8cd98f00b204e9800998ecf8427e] [Current]

Feedback Forum

2008-12-04 10:44:37 [Steven Vercammen] [reply] 
Deze vraag heb ik correct beantwoord. Er wordt goed naar de theorie verwezen: 
 
1)“The SMP is often used to identify the quasi-optimal Box-Cox transformation parameter that induces stationarity of the variance. “  
 
2)To achieve a constant variance over time a variance stabilizing transformation has to be applied to the measurements. The range of variance stabilizing transformations that can be used is very wide. However for most of the practical situations the power transformation has been found of considerable value. This transformation is given by : G(Zt)= Zt ^ lambda when lambda is not 0 and ln Zt when lambda is 0.  
 
De optimale lambda om een transformatie uit te voeren blijkt hier inderdaad -0.3 te zijn. Wanneer deze transformatie wordt toegepast zal de variantie stationair worden. 
 
2008-12-07 09:48:34 [Käthe Vanderheggen] [reply] 
De student gebruikt veel theorie maar legt niet duidelijk in woorden uit wat er gebeurt: 
We gaan proberen de spreiding eruit te halen. Hiervoor moeten we lambda bepalen. In de tweede tabel zien we dat de p waarden kleiner is dan 1%. Dit wil zeggen dat we minder dan 1 % kans hebben dat we ons vergissen bij het verwerpen van de nulhypothese. Er is een verband tussen het gemiddelde niveau en de standaardfout. Daarom mogen we kijken naar de derde tabel. Hieruit kunnen we lambda aflezen. deze is -0.3. 
Op de standard deviation mean plot zien we dat de spreiding niet constant is. We moeten dit dus modelleren. Om het model stationair te maken moet er een horizontale trend zijn en een gelijke spreiding. 
2008-12-08 19:00:11 [Koen Van Baelen] [reply] 
Q5: Correct. De lamba berekenen doen we door de standard deviation plot. Deze verdeelt de reeks onder in perioden. In de grafiek zie je punten die de jaren voorstellen. Dit keer heeft de student wel de seasonal period op 12 geplaatst. De x-as = het gemiddelde en y-as = standard deviation. Door een lambda toe te voegen kan men de tijdreeks transformeren waardoor de spreiding harder naar een diagonaal zal deinen. Een lambda van 1 geeft bijvoorbeeld een perfecte rechte. De bedoeling van deze transformatie is om een tijdreeks met gelijke spreidingen te bekomen en dus een stationaire tijdreeks te bekomen. Hier voor moet je zien dat de spreiding gelijk loopt door de tijd en hierdoor wordt de trend eruit gehaald. De optimale lambda is inderdaad -0.312592539725757. Ook kan er nog een opmerking worden gemaakt over de heterokedasticiteit. We hebben gezien op de run sequence plot dat er sprake was van heteroskedasticiteit : de variantie werd groter naarmate de tijd vordert. De gegevens in de standard deviation mean plot bevestigden dit. Ook de p value (6.19e-11) is kleiner dan 0.05 wat de heteroskedasticiteit bevestigt. We gaan de tijdreeks tot de 0.3e macht vereffenen om deze heteroskedastische trend weg te werken.

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Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	2 seconds
R Server	'Herman Ole Andreas Wold' @ 193.190.124.10:1001

\begin{tabular}{lllllllll}
\hline
Summary of computational transaction \tabularnewline
Raw Input & view raw input (R code)  \tabularnewline
Raw Output & view raw output of R engine  \tabularnewline
Computing time & 2 seconds \tabularnewline
R Server & 'Herman Ole Andreas Wold' @ 193.190.124.10:1001 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=26679&T=0

[TABLE]
[ROW][C]Summary of computational transaction[/C][/ROW]
[ROW][C]Raw Input[/C][C]view raw input (R code) [/C][/ROW]
[ROW][C]Raw Output[/C][C]view raw output of R engine [/C][/ROW]
[ROW][C]Computing time[/C][C]2 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Herman Ole Andreas Wold' @ 193.190.124.10:1001[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=26679&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=26679&T=0

As an alternative you can also use a QR Code:

The GUIDs for individual cells are displayed in the table below:

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	2 seconds
R Server	'Herman Ole Andreas Wold' @ 193.190.124.10:1001

Standard Deviation-Mean Plot
Section	Mean	Standard Deviation	Range
1	126.666666666667	13.7201466552812	44
2	139.666666666667	19.0708408230201	56
3	170.166666666667	18.4382671899964	54
4	197	22.9663785881568	71
5	225	28.4668866643972	92
6	238.916666666667	34.9244856364370	114
7	284	42.1404577789347	131
8	328.25	47.8617801591207	142
9	368.416666666667	57.8908979081	166
10	381	64.5304720126997	195
11	428.333333333333	69.8300968368398	217
12	476.166666666667	77.7371250179771	232

\begin{tabular}{lllllllll}
\hline
Standard Deviation-Mean Plot \tabularnewline
Section & Mean & Standard Deviation & Range \tabularnewline
1 & 126.666666666667 & 13.7201466552812 & 44 \tabularnewline
2 & 139.666666666667 & 19.0708408230201 & 56 \tabularnewline
3 & 170.166666666667 & 18.4382671899964 & 54 \tabularnewline
4 & 197 & 22.9663785881568 & 71 \tabularnewline
5 & 225 & 28.4668866643972 & 92 \tabularnewline
6 & 238.916666666667 & 34.9244856364370 & 114 \tabularnewline
7 & 284 & 42.1404577789347 & 131 \tabularnewline
8 & 328.25 & 47.8617801591207 & 142 \tabularnewline
9 & 368.416666666667 & 57.8908979081 & 166 \tabularnewline
10 & 381 & 64.5304720126997 & 195 \tabularnewline
11 & 428.333333333333 & 69.8300968368398 & 217 \tabularnewline
12 & 476.166666666667 & 77.7371250179771 & 232 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=26679&T=1

[TABLE]
[ROW][C]Standard Deviation-Mean Plot[/C][/ROW]
[ROW][C]Section[/C][C]Mean[/C][C]Standard Deviation[/C][C]Range[/C][/ROW]
[ROW][C]1[/C][C]126.666666666667[/C][C]13.7201466552812[/C][C]44[/C][/ROW]
[ROW][C]2[/C][C]139.666666666667[/C][C]19.0708408230201[/C][C]56[/C][/ROW]
[ROW][C]3[/C][C]170.166666666667[/C][C]18.4382671899964[/C][C]54[/C][/ROW]
[ROW][C]4[/C][C]197[/C][C]22.9663785881568[/C][C]71[/C][/ROW]
[ROW][C]5[/C][C]225[/C][C]28.4668866643972[/C][C]92[/C][/ROW]
[ROW][C]6[/C][C]238.916666666667[/C][C]34.9244856364370[/C][C]114[/C][/ROW]
[ROW][C]7[/C][C]284[/C][C]42.1404577789347[/C][C]131[/C][/ROW]
[ROW][C]8[/C][C]328.25[/C][C]47.8617801591207[/C][C]142[/C][/ROW]
[ROW][C]9[/C][C]368.416666666667[/C][C]57.8908979081[/C][C]166[/C][/ROW]
[ROW][C]10[/C][C]381[/C][C]64.5304720126997[/C][C]195[/C][/ROW]
[ROW][C]11[/C][C]428.333333333333[/C][C]69.8300968368398[/C][C]217[/C][/ROW]
[ROW][C]12[/C][C]476.166666666667[/C][C]77.7371250179771[/C][C]232[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=26679&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=26679&T=1

As an alternative you can also use a QR Code:

The GUIDs for individual cells are displayed in the table below:

Standard Deviation-Mean Plot
Section	Mean	Standard Deviation	Range
1	126.666666666667	13.7201466552812	44
2	139.666666666667	19.0708408230201	56
3	170.166666666667	18.4382671899964	54
4	197	22.9663785881568	71
5	225	28.4668866643972	92
6	238.916666666667	34.9244856364370	114
7	284	42.1404577789347	131
8	328.25	47.8617801591207	142
9	368.416666666667	57.8908979081	166
10	381	64.5304720126997	195
11	428.333333333333	69.8300968368398	217
12	476.166666666667	77.7371250179771	232

Regression: S.E.(k) = alpha + beta * Mean(k)
alpha	-11.4032541425579
beta	0.188613398899484
S.D.	0.00657733180244678
T-STAT	28.6762785525460
p-value	6.1917170560278e-11

\begin{tabular}{lllllllll}
\hline
Regression: S.E.(k) = alpha + beta * Mean(k) \tabularnewline
alpha & -11.4032541425579 \tabularnewline
beta & 0.188613398899484 \tabularnewline
S.D. & 0.00657733180244678 \tabularnewline
T-STAT & 28.6762785525460 \tabularnewline
p-value & 6.1917170560278e-11 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=26679&T=2

[TABLE]
[ROW][C]Regression: S.E.(k) = alpha + beta * Mean(k)[/C][/ROW]
[ROW][C]alpha[/C][C]-11.4032541425579[/C][/ROW]
[ROW][C]beta[/C][C]0.188613398899484[/C][/ROW]
[ROW][C]S.D.[/C][C]0.00657733180244678[/C][/ROW]
[ROW][C]T-STAT[/C][C]28.6762785525460[/C][/ROW]
[ROW][C]p-value[/C][C]6.1917170560278e-11[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=26679&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=26679&T=2

As an alternative you can also use a QR Code:

The GUIDs for individual cells are displayed in the table below:

Regression: S.E.(k) = alpha + beta * Mean(k)
alpha	-11.4032541425579
beta	0.188613398899484
S.D.	0.00657733180244678
T-STAT	28.6762785525460
p-value	6.1917170560278e-11

Regression: ln S.E.(k) = alpha + beta * ln Mean(k)
alpha	-3.70703989322047
beta	1.31259253972576
S.D.	0.0574958902763329
T-STAT	22.8293280340083
p-value	5.8658934502009e-10
Lambda	-0.312592539725755

\begin{tabular}{lllllllll}
\hline
Regression: ln S.E.(k) = alpha + beta * ln Mean(k) \tabularnewline
alpha & -3.70703989322047 \tabularnewline
beta & 1.31259253972576 \tabularnewline
S.D. & 0.0574958902763329 \tabularnewline
T-STAT & 22.8293280340083 \tabularnewline
p-value & 5.8658934502009e-10 \tabularnewline
Lambda & -0.312592539725755 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=26679&T=3

[TABLE]
[ROW][C]Regression: ln S.E.(k) = alpha + beta * ln Mean(k)[/C][/ROW]
[ROW][C]alpha[/C][C]-3.70703989322047[/C][/ROW]
[ROW][C]beta[/C][C]1.31259253972576[/C][/ROW]
[ROW][C]S.D.[/C][C]0.0574958902763329[/C][/ROW]
[ROW][C]T-STAT[/C][C]22.8293280340083[/C][/ROW]
[ROW][C]p-value[/C][C]5.8658934502009e-10[/C][/ROW]
[ROW][C]Lambda[/C][C]-0.312592539725755[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=26679&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=26679&T=3

As an alternative you can also use a QR Code:

The GUIDs for individual cells are displayed in the table below:

Regression: ln S.E.(k) = alpha + beta * ln Mean(k)
alpha	-3.70703989322047
beta	1.31259253972576
S.D.	0.0574958902763329
T-STAT	22.8293280340083
p-value	5.8658934502009e-10
Lambda	-0.312592539725755

Figure 1

PNG link

Postscript link

PDF link

Figure 2

PNG link

Postscript link

PDF link

Parameters (Session):

par1 = 12 ;

Parameters (R input):

par1 = 12 ;

R code (references can be found in the software module):

par1 <- as.numeric(par1)
(n <- length(x))
(np <- floor(n / par1))
arr <- array(NA,dim=c(par1,np))
j <- 0
k <- 1
for (i in 1:(np*par1))
{
j = j + 1
arr[j,k] <- x[i]
if (j == par1) {
j = 0
k=k+1
}
}
arr
arr.mean <- array(NA,dim=np)
arr.sd <- array(NA,dim=np)
arr.range <- array(NA,dim=np)
for (j in 1:np)
{
arr.mean[j] <- mean(arr[,j],na.rm=TRUE)
arr.sd[j] <- sd(arr[,j],na.rm=TRUE)
arr.range[j] <- max(arr[,j],na.rm=TRUE) - min(arr[,j],na.rm=TRUE)
}
arr.mean
arr.sd
arr.range
(lm1 <- lm(arr.sd~arr.mean))
(lnlm1 <- lm(log(arr.sd)~log(arr.mean)))
(lm2 <- lm(arr.range~arr.mean))
bitmap(file='test1.png')
plot(arr.mean,arr.sd,main='Standard Deviation-Mean Plot',xlab='mean',ylab='standard deviation')
dev.off()
bitmap(file='test2.png')
plot(arr.mean,arr.range,main='Range-Mean Plot',xlab='mean',ylab='range')
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Standard Deviation-Mean Plot',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Section',header=TRUE)
a<-table.element(a,'Mean',header=TRUE)
a<-table.element(a,'Standard Deviation',header=TRUE)
a<-table.element(a,'Range',header=TRUE)
a<-table.row.end(a)
for (j in 1:np) {
a<-table.row.start(a)
a<-table.element(a,j,header=TRUE)
a<-table.element(a,arr.mean[j])
a<-table.element(a,arr.sd[j] )
a<-table.element(a,arr.range[j] )
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Regression: S.E.(k) = alpha + beta * Mean(k)',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,lm1$coefficients[[1]])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'beta',header=TRUE)
a<-table.element(a,lm1$coefficients[[2]])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'S.D.',header=TRUE)
a<-table.element(a,summary(lm1)$coefficients[2,2])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'T-STAT',header=TRUE)
a<-table.element(a,summary(lm1)$coefficients[2,3])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'p-value',header=TRUE)
a<-table.element(a,summary(lm1)$coefficients[2,4])
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable1.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Regression: ln S.E.(k) = alpha + beta * ln Mean(k)',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,lnlm1$coefficients[[1]])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'beta',header=TRUE)
a<-table.element(a,lnlm1$coefficients[[2]])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'S.D.',header=TRUE)
a<-table.element(a,summary(lnlm1)$coefficients[2,2])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'T-STAT',header=TRUE)
a<-table.element(a,summary(lnlm1)$coefficients[2,3])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'p-value',header=TRUE)
a<-table.element(a,summary(lnlm1)$coefficients[2,4])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Lambda',header=TRUE)
a<-table.element(a,1-lnlm1$coefficients[[2]])
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable2.tab')

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Dataset

Tables (Output of Computation)

Figures (Output of Computation)

Input Parameters & R Code