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Description of Statistical Computation

Author's title

Author*Unverified author*
R Software Modulerwasp_density.wasp
Title produced by softwareKernel Density Estimation
Date of computationThu, 05 Feb 2015 13:48:58 +0000
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2015/Feb/05/t14231441829kzldbg98ow7nwq.htm/, Retrieved Wed, 09 Sep 2026 12:27:06 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=276751, Retrieved Wed, 09 Sep 2026 12:27:06 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact451

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Kernel Density Estimation] [] [2015-02-05 13:48:58] [8a8c4bd73c288c653fbb494901d06130] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
2
2.2
1.9
2.3
2.2
2.3
2.1
2.4
2.3
1.9
1.6
1.8
1.8
2
2.3
2.2
2.2
2
2
1.9
1.5
1.6
1.5
2
1.5
1.5
1.9
1.1
1.5
2.1
2.3
2.6
2.9
3.2
3.2
3.1
3
3.3
2.7
3.6
3.1
2.7
2.6
2.2
2.7
2.1
1.8
1.7
1.7
1.2
1.2
1.2
1.5
1.3
1.1
1.2
1.3
1.6
1.9
1.6
2.1
2.2
2.3
2.1
1.7
1.7
2.2
2
1.5
1.5
1.7
2.2
2.6
2.6
2.3
2.3
2.7
2.7
2.5
2.5
2.7
2.6
2.6
2.4
1.4
1.8
2.1
1.7
1.6
1.7
1.8
2
1.9
2
2.1
2.3

Tables (Output of Computation)





Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time4 seconds
R Server'Sir Maurice George Kendall' @ kendall.wessa.net

\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 & 4 seconds \tabularnewline
R Server & 'Sir Maurice George Kendall' @ kendall.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=276751&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]4 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Sir Maurice George Kendall' @ kendall.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=276751&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=276751&T=0

As an alternative you can also use a QR Code:  
QR Code


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

Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time4 seconds
R Server'Sir Maurice George Kendall' @ kendall.wessa.net






Properties of Density Trace
Bandwidth0.168485868839601
#Observations96

\begin{tabular}{lllllllll}
\hline
Properties of Density Trace \tabularnewline
Bandwidth & 0.168485868839601 \tabularnewline
#Observations & 96 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=276751&T=1

[TABLE]
[ROW][C]Properties of Density Trace[/C][/ROW]
[ROW][C]Bandwidth[/C][C]0.168485868839601[/C][/ROW]
[ROW][C]#Observations[/C][C]96[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=276751&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=276751&T=1

As an alternative you can also use a QR Code:  
QR Code


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

Properties of Density Trace
Bandwidth0.168485868839601
#Observations96






Maximum Density Values
Kernelx-valuemax. density
Gaussian2.099220341925890.739529760946156
Epanechnikov2.037384261852820.738918739570902
Rectangular2.051125612980170.767437678469976
Triangular2.003030884034440.735135657710646
Biweight2.064866964107510.735933632851427
Cosine2.078608315234860.735743155487511
Optcosine2.044254937416490.738532882462954

\begin{tabular}{lllllllll}
\hline
Maximum Density Values \tabularnewline
Kernel & x-value & max. density \tabularnewline
Gaussian & 2.09922034192589 & 0.739529760946156 \tabularnewline
Epanechnikov & 2.03738426185282 & 0.738918739570902 \tabularnewline
Rectangular & 2.05112561298017 & 0.767437678469976 \tabularnewline
Triangular & 2.00303088403444 & 0.735135657710646 \tabularnewline
Biweight & 2.06486696410751 & 0.735933632851427 \tabularnewline
Cosine & 2.07860831523486 & 0.735743155487511 \tabularnewline
Optcosine & 2.04425493741649 & 0.738532882462954 \tabularnewline
 \hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=276751&T=2

[TABLE]
[ROW][C]Maximum Density Values[/C][/ROW]
[ROW][C]Kernel[/C][C]x-value[/C][C]max. density[/C][/ROW]
[ROW][C]Gaussian[/C][C]2.09922034192589[/C][C]0.739529760946156[/C][/ROW]
[ROW][C]Epanechnikov[/C][C]2.03738426185282[/C][C]0.738918739570902[/C][/ROW]
[ROW][C]Rectangular[/C][C]2.05112561298017[/C][C]0.767437678469976[/C][/ROW]
[ROW][C]Triangular[/C][C]2.00303088403444[/C][C]0.735135657710646[/C][/ROW]
[ROW][C]Biweight[/C][C]2.06486696410751[/C][C]0.735933632851427[/C][/ROW]
[ROW][C]Cosine[/C][C]2.07860831523486[/C][C]0.735743155487511[/C][/ROW]
[ROW][C]Optcosine[/C][C]2.04425493741649[/C][C]0.738532882462954[/C][/ROW]
 [/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=276751&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=276751&T=2

As an alternative you can also use a QR Code:  
QR Code


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

Maximum Density Values
Kernelx-valuemax. density
Gaussian2.099220341925890.739529760946156
Epanechnikov2.037384261852820.738918739570902
Rectangular2.051125612980170.767437678469976
Triangular2.003030884034440.735135657710646
Biweight2.064866964107510.735933632851427
Cosine2.078608315234860.735743155487511
Optcosine2.044254937416490.738532882462954






Maximum Density Values
Kernelx-valuemax. density
Gaussian2.099220341925890.739529760946156
Epanechnikov2.037384261852820.738918739570902
Rectangular2.057996288543840.767437678469976
Triangular2.003030884034440.735135657710646
Biweight2.064866964107510.735933632851427
Cosine2.078608315234860.735743155487511
Optcosine2.044254937416490.738532882462954

\begin{tabular}{lllllllll}
\hline
Maximum Density Values \tabularnewline
Kernel & x-value & max. density \tabularnewline
Gaussian & 2.09922034192589 & 0.739529760946156 \tabularnewline
Epanechnikov & 2.03738426185282 & 0.738918739570902 \tabularnewline
Rectangular & 2.05799628854384 & 0.767437678469976 \tabularnewline
Triangular & 2.00303088403444 & 0.735135657710646 \tabularnewline
Biweight & 2.06486696410751 & 0.735933632851427 \tabularnewline
Cosine & 2.07860831523486 & 0.735743155487511 \tabularnewline
Optcosine & 2.04425493741649 & 0.738532882462954 \tabularnewline
 \hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=276751&T=3

[TABLE]
[ROW][C]Maximum Density Values[/C][/ROW]
[ROW][C]Kernel[/C][C]x-value[/C][C]max. density[/C][/ROW]
[ROW][C]Gaussian[/C][C]2.09922034192589[/C][C]0.739529760946156[/C][/ROW]
[ROW][C]Epanechnikov[/C][C]2.03738426185282[/C][C]0.738918739570902[/C][/ROW]
[ROW][C]Rectangular[/C][C]2.05799628854384[/C][C]0.767437678469976[/C][/ROW]
[ROW][C]Triangular[/C][C]2.00303088403444[/C][C]0.735135657710646[/C][/ROW]
[ROW][C]Biweight[/C][C]2.06486696410751[/C][C]0.735933632851427[/C][/ROW]
[ROW][C]Cosine[/C][C]2.07860831523486[/C][C]0.735743155487511[/C][/ROW]
[ROW][C]Optcosine[/C][C]2.04425493741649[/C][C]0.738532882462954[/C][/ROW]
 [/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=276751&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=276751&T=3

As an alternative you can also use a QR Code:  
QR Code


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

Maximum Density Values
Kernelx-valuemax. density
Gaussian2.099220341925890.739529760946156
Epanechnikov2.037384261852820.738918739570902
Rectangular2.057996288543840.767437678469976
Triangular2.003030884034440.735135657710646
Biweight2.064866964107510.735933632851427
Cosine2.078608315234860.735743155487511
Optcosine2.044254937416490.738532882462954






Maximum Density Values
Kernelx-valuemax. density
Gaussian2.099220341925890.739529760946156
Epanechnikov2.037384261852820.738918739570902
Rectangular2.064866964107510.767437678469976
Triangular2.003030884034440.735135657710646
Biweight2.064866964107510.735933632851427
Cosine2.078608315234860.735743155487511
Optcosine2.044254937416490.738532882462954

\begin{tabular}{lllllllll}
\hline
Maximum Density Values \tabularnewline
Kernel & x-value & max. density \tabularnewline
Gaussian & 2.09922034192589 & 0.739529760946156 \tabularnewline
Epanechnikov & 2.03738426185282 & 0.738918739570902 \tabularnewline
Rectangular & 2.06486696410751 & 0.767437678469976 \tabularnewline
Triangular & 2.00303088403444 & 0.735135657710646 \tabularnewline
Biweight & 2.06486696410751 & 0.735933632851427 \tabularnewline
Cosine & 2.07860831523486 & 0.735743155487511 \tabularnewline
Optcosine & 2.04425493741649 & 0.738532882462954 \tabularnewline
 \hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=276751&T=4

[TABLE]
[ROW][C]Maximum Density Values[/C][/ROW]
[ROW][C]Kernel[/C][C]x-value[/C][C]max. density[/C][/ROW]
[ROW][C]Gaussian[/C][C]2.09922034192589[/C][C]0.739529760946156[/C][/ROW]
[ROW][C]Epanechnikov[/C][C]2.03738426185282[/C][C]0.738918739570902[/C][/ROW]
[ROW][C]Rectangular[/C][C]2.06486696410751[/C][C]0.767437678469976[/C][/ROW]
[ROW][C]Triangular[/C][C]2.00303088403444[/C][C]0.735135657710646[/C][/ROW]
[ROW][C]Biweight[/C][C]2.06486696410751[/C][C]0.735933632851427[/C][/ROW]
[ROW][C]Cosine[/C][C]2.07860831523486[/C][C]0.735743155487511[/C][/ROW]
[ROW][C]Optcosine[/C][C]2.04425493741649[/C][C]0.738532882462954[/C][/ROW]
 [/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=276751&T=4

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=276751&T=4

As an alternative you can also use a QR Code:  
QR Code


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

Maximum Density Values
Kernelx-valuemax. density
Gaussian2.099220341925890.739529760946156
Epanechnikov2.037384261852820.738918739570902
Rectangular2.064866964107510.767437678469976
Triangular2.003030884034440.735135657710646
Biweight2.064866964107510.735933632851427
Cosine2.078608315234860.735743155487511
Optcosine2.044254937416490.738532882462954






Maximum Density Values
Kernelx-valuemax. density
Gaussian2.099220341925890.739529760946156
Epanechnikov2.037384261852820.738918739570902
Rectangular2.071737639671190.767437678469976
Triangular2.003030884034440.735135657710646
Biweight2.064866964107510.735933632851427
Cosine2.078608315234860.735743155487511
Optcosine2.044254937416490.738532882462954

\begin{tabular}{lllllllll}
\hline
Maximum Density Values \tabularnewline
Kernel & x-value & max. density \tabularnewline
Gaussian & 2.09922034192589 & 0.739529760946156 \tabularnewline
Epanechnikov & 2.03738426185282 & 0.738918739570902 \tabularnewline
Rectangular & 2.07173763967119 & 0.767437678469976 \tabularnewline
Triangular & 2.00303088403444 & 0.735135657710646 \tabularnewline
Biweight & 2.06486696410751 & 0.735933632851427 \tabularnewline
Cosine & 2.07860831523486 & 0.735743155487511 \tabularnewline
Optcosine & 2.04425493741649 & 0.738532882462954 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=276751&T=5

[TABLE]
[ROW][C]Maximum Density Values[/C][/ROW]
[ROW][C]Kernel[/C][C]x-value[/C][C]max. density[/C][/ROW]
[ROW][C]Gaussian[/C][C]2.09922034192589[/C][C]0.739529760946156[/C][/ROW]
[ROW][C]Epanechnikov[/C][C]2.03738426185282[/C][C]0.738918739570902[/C][/ROW]
[ROW][C]Rectangular[/C][C]2.07173763967119[/C][C]0.767437678469976[/C][/ROW]
[ROW][C]Triangular[/C][C]2.00303088403444[/C][C]0.735135657710646[/C][/ROW]
[ROW][C]Biweight[/C][C]2.06486696410751[/C][C]0.735933632851427[/C][/ROW]
[ROW][C]Cosine[/C][C]2.07860831523486[/C][C]0.735743155487511[/C][/ROW]
[ROW][C]Optcosine[/C][C]2.04425493741649[/C][C]0.738532882462954[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=276751&T=5

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=276751&T=5

As an alternative you can also use a QR Code:  
QR Code


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

Maximum Density Values
Kernelx-valuemax. density
Gaussian2.099220341925890.739529760946156
Epanechnikov2.037384261852820.738918739570902
Rectangular2.071737639671190.767437678469976
Triangular2.003030884034440.735135657710646
Biweight2.064866964107510.735933632851427
Cosine2.078608315234860.735743155487511
Optcosine2.044254937416490.738532882462954


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 0 ; par2 = no ; par3 = 512 ;
Parameters (R input):
par1 = 0 ; par2 = no ; par3 = 512 ;
R code (references can be found in the software module):
if (par1 == '0') bw <- 'nrd0'
if (par1 != '0') bw <- as.numeric(par1)
par3 <- as.numeric(par3)
mydensity <- array(NA, dim=c(par3,8))
bitmap(file='density1.png')
mydensity1<-density(x,bw=bw,kernel='gaussian',na.rm=TRUE)
mydensity[,8] = signif(mydensity1$x,3)
mydensity[,1] = signif(mydensity1$y,3)
plot(mydensity1,main='Gaussian Kernel',xlab=xlab,ylab=ylab)
grid()
dev.off()
mydensity1
bitmap(file='density2.png')
mydensity2<-density(x,bw=bw,kernel='epanechnikov',na.rm=TRUE)
mydensity[,2] = signif(mydensity2$y,3)
plot(mydensity2,main='Epanechnikov Kernel',xlab=xlab,ylab=ylab)
grid()
dev.off()
bitmap(file='density3.png')
mydensity3<-density(x,bw=bw,kernel='rectangular',na.rm=TRUE)
mydensity[,3] = signif(mydensity3$y,3)
plot(mydensity3,main='Rectangular Kernel',xlab=xlab,ylab=ylab)
grid()
dev.off()
bitmap(file='density4.png')
mydensity4<-density(x,bw=bw,kernel='triangular',na.rm=TRUE)
mydensity[,4] = signif(mydensity4$y,3)
plot(mydensity4,main='Triangular Kernel',xlab=xlab,ylab=ylab)
grid()
dev.off()
bitmap(file='density5.png')
mydensity5<-density(x,bw=bw,kernel='biweight',na.rm=TRUE)
mydensity[,5] = signif(mydensity5$y,3)
plot(mydensity5,main='Biweight Kernel',xlab=xlab,ylab=ylab)
grid()
dev.off()
bitmap(file='density6.png')
mydensity6<-density(x,bw=bw,kernel='cosine',na.rm=TRUE)
mydensity[,6] = signif(mydensity6$y,3)
plot(mydensity6,main='Cosine Kernel',xlab=xlab,ylab=ylab)
grid()
dev.off()
bitmap(file='density7.png')
mydensity7<-density(x,bw=bw,kernel='optcosine',na.rm=TRUE)
mydensity[,7] = signif(mydensity7$y,3)
plot(mydensity7,main='Optcosine Kernel',xlab=xlab,ylab=ylab)
grid()
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Properties of Density Trace',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Bandwidth',header=TRUE)
a<-table.element(a,mydensity1$bw)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'#Observations',header=TRUE)
a<-table.element(a,mydensity1$n)
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,'Maximum Density Values',3,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Kernel',1,TRUE)
a<-table.element(a,'x-value',1,TRUE)
a<-table.element(a,'max. density',1,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Gaussian',1,TRUE)
a<-table.element(a,mydensity1$x[mydensity1$y==max(mydensity1$y)],1)
a<-table.element(a,mydensity1$y[mydensity1$y==max(mydensity1$y)],1)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Epanechnikov',1,TRUE)
a<-table.element(a,mydensity2$x[mydensity2$y==max(mydensity2$y)],1)
a<-table.element(a,mydensity2$y[mydensity2$y==max(mydensity2$y)],1)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Rectangular',1,TRUE)
a<-table.element(a,mydensity3$x[mydensity3$y==max(mydensity3$y)],1)
a<-table.element(a,mydensity3$y[mydensity3$y==max(mydensity3$y)],1)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Triangular',1,TRUE)
a<-table.element(a,mydensity4$x[mydensity4$y==max(mydensity4$y)],1)
a<-table.element(a,mydensity4$y[mydensity4$y==max(mydensity4$y)],1)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Biweight',1,TRUE)
a<-table.element(a,mydensity5$x[mydensity5$y==max(mydensity5$y)],1)
a<-table.element(a,mydensity5$y[mydensity5$y==max(mydensity5$y)],1)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Cosine',1,TRUE)
a<-table.element(a,mydensity6$x[mydensity6$y==max(mydensity6$y)],1)
a<-table.element(a,mydensity6$y[mydensity6$y==max(mydensity6$y)],1)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Optcosine',1,TRUE)
a<-table.element(a,mydensity7$x[mydensity7$y==max(mydensity7$y)],1)
a<-table.element(a,mydensity7$y[mydensity7$y==max(mydensity7$y)],1)
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable2.tab')
if (par2=='yes') {
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Kernel Density Values',8,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'x-value',1,TRUE)
a<-table.element(a,'Gaussian',1,TRUE)
a<-table.element(a,'Epanechnikov',1,TRUE)
a<-table.element(a,'Rectangular',1,TRUE)
a<-table.element(a,'Triangular',1,TRUE)
a<-table.element(a,'Biweight',1,TRUE)
a<-table.element(a,'Cosine',1,TRUE)
a<-table.element(a,'Optcosine',1,TRUE)
a<-table.row.end(a)
for(i in 1:par3) {
a<-table.row.start(a)
a<-table.element(a,mydensity[i,8],1,TRUE)
for(j in 1:7) {
a<-table.element(a,mydensity[i,j],1)
}
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable1.tab')
}