Re: [R] skewness and kurtosis in e1071 correct?
To me your answer is opaque (but that seems to be rather a problem of language ;-) ). Perhaps my question has not been expressed clearly enough. Let me state it differently: In the R package e1071 the formulas (implicit) used are (3) and (4) (see below), the standard deviation used in these formulas, however is based on (2) (see below). This seems to be inconsistent and my question is, whether there is a commonly used third definition of skewness and kurtosis in which the formulas for the biased skewness and kurtosis _but_ with the unbiased standard deviation are employed. The standard deviation can be defined as the _sample_ statistic: sd = 1/n * sum( (x - mean(x))^2 ) # (1) and as the estimated population parameter: sd = 1/(n-1) * sum( (x-mean(x))^2 ) # (2). In R the function sd() calculates the latter. In the same way, expressed via z-values skewness and kurtosis can be defined as the _sample_ statistic (also called biased estimator , see: http://www.mathdaily.com/lessons/Skewness ): skewness = mean(z^3) # (3) kurtosis = mean(z^4)-3 # (4) with z = (x - mean(x))/sd(x) with sd = 1/n * sum( (x - mean(x)^2 ) (thus: here sd is the _sample_ statistic, see (1) above!) but they can also be defined as the estimated population parameters (also called unbiased, see: http://www.mathdaily.com/lessons/Kurtosis#Sample_kurtosis ): skewness = n/((n-1)*(n-2)) * sum(z^3) # (5) kurtosis = n*(n+1)/((n-1)*(n-2)*(n-3)) * sum(z^4) - 3*(n-1)^2/((n-2)*(n-3)) # (6) with z = (x - mean(x))/sd(x) with sd = 1/(n-1) * sum( (x - mean(x)^2 ) (thus: here sd is the estimated population parameter, see (2) above!. BTW: The R function scale() calculates the z-values based on this definition, as well.) Campbell wrote: This is probably an issue over definitions rather than the correct answer. To me skewness and kurtosis are functions of the distribution rather than the population, they are equivalent to expectation rather than mean. For the normal distribution it makes no sense to estimate them as the distribution is uniquely defined by its first two moments. However there are two defnitions of kurotsis as it is often standardized such that the expectation is 0. * Dr. Dirk Enzmann Institute of Criminal Sciences Dept. of Criminology Edmund-Siemers-Allee 1 D-20146 Hamburg Germany phone: +49-040-42838.7498 (office) +49-040-42838.4591 (Billon) fax: +49-040-42838.2344 email: [EMAIL PROTECTED] www: http://www2.jura.uni-hamburg.de/instkrim/kriminologie/Mitarbeiter/Enzmann/Enzmann.html __ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
Re: [R] skewness and kurtosis in e1071 correct? (correction)
I'm sorry, but my previous message, as often happens, some brackets were wrong: Here are the correct formulas: sd = 1/n * sum((x-mean(x))^2) # (1) sd = 1/(n-1) * sum((x-mean(x))^2) # (2) This also occured in the last paragraph. Dirk * Dr. Dirk Enzmann Institute of Criminal Sciences Dept. of Criminology Edmund-Siemers-Allee 1 D-20146 Hamburg Germany phone: +49-040-42838.7498 (office) +49-040-42838.4591 (Billon) fax: +49-040-42838.2344 email: [EMAIL PROTECTED] www: http://www2.jura.uni-hamburg.de/instkrim/kriminologie/Mitarbeiter/Enzmann/Enzmann.html __ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
Re: [R] skewness and kurtosis in e1071 correct?
My last question to the R list is another example of asking too fast before reading (sorry). But by the way and because it might be interesting for others, an answer can be found in: Joanes, D. N. Gill, C. A. (1998) Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society (Series D): The Statistician 47 (1), 183189. The measures discussed in this article are: g1 : skewness as defined in formula (3) of my mail (and in STATA) G1 : skewness as defined in formula (5) of my mail (and in SAS, SPSS, and EXCEL) b1 : skewness as defined in the package e1071 (and in MINITAB and BMDP) g2 : kurtosis as defined in formula (4) of my mail (and in STATA) G2 : kurtosis as defined in formula (6) of my mail (and in SAS, SPSS, and EXCEL) b2 : kurtosis as defined in the package e1071 (and in MINITAB and BMDP) Off topic: I don't know why the thread of these mails are not included in the correct thread. Dirk -- * Dr. Dirk Enzmann Institute of Criminal Sciences Dept. of Criminology Edmund-Siemers-Allee 1 D-20146 Hamburg Germany phone: +49-040-42838.7498 (office) +49-040-42838.4591 (Billon) fax: +49-040-42838.2344 email: [EMAIL PROTECTED] www: http://www2.jura.uni-hamburg.de/instkrim/kriminologie/Mitarbeiter/Enzmann/Enzmann.html __ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
Re: [R] skewness and kurtosis in e1071 correct?
This is probably an issue over definitions rather than the correct
answer. To me skewness and kurtosis are functions of the distribution
rather than the population, they are equivalent to expectation rather
than mean. For the normal distribution it makes no sense to estimate
them as the distribution is uniquely defined by its first two moments.
However there are two defnitions of kurotsis as it is often
standardized such that the expectation is 0.
HTH
Phineas
www.pwp.phineas.blueyonder.co.uk
Dirk Enzmann [EMAIL PROTECTED] 05/23/05 3:17 AM
I wonder whether the functions for skewness and kurtosis in the e1071
package are based on correct formulas.
The functions in the package e1071 are:
#
skewness - function (x, na.rm = FALSE)
{
if (na.rm)
x - x[!is.na(x)]
sum((x - mean(x))^3)/(length(x) * sd(x)^3)
}
#
and
#
kurtosis - function (x, na.rm = FALSE)
{
if (na.rm)
x - x[!is.na(x)]
sum((x - mean(x))^4)/(length(x) * var(x)^2) - 3
}
#
However, sd() and var() are the estimated population parameters. To
calculate the sample statistics of skewness and kurtosis, shouldn't one
use the sample statistics of the standard deviation (and variance), as
well? For example:
#
# Function to calculate the sample statistic of skewness:
skew_s=function(x)
{
x = x[!is.na(x)]
n = length(x)
if (n 3)
{
cat('valid cases = ',n,'\nskewness is not defined for less than 3
valid cases!\n')
}
else
{
z = sqrt(n/(n-1))*scale(x)
mean(z^3)
}
}
#
and
#
# Function to calculate the sample statistic of kurtosis:
kurt_s=function(x)
{
x = x[!is.na(x)]
n = length(x)
if (n 4)
{
cat('valid cases = ',n,'\nkurtosis is not defined for less than 4
valid cases!\n')
}
else
{
z = sqrt(n/(n-1))*scale(x)
mean(z^4)-3
}
}
#
Whereas, to calculate the (unbiased) estimated population parameter of
skewness and kurtosis, the correction should also include the number of
cases in the following way:
#
# Function to calculate the unbiased populataion estimate of skewness:
skew=function(x)
{
x = x[!is.na(x)]
n = length(x)
if (n 3)
{
cat('valid cases = ',n,'\nskewness is not defined for less than 3
valid cases!\n')
}
else
{
z = scale(x)
sum(z^3)*n/((n-1)*(n-2))
}
}
#
and
#
# Function to calculate the unbiased population estimate of kurtosis:
kurt=function(x)
{
x = x[!is.na(x)]
n = length(x)
if (n 4)
{
cat('valid cases = ',n,'\nkurtosis is not defined for less than 4
valid cases!\n')
}
else
{
z = scale(x)
sum(z^4)*n*(n+1)/((n-1)*(n-2)*(n-3))-3*(n-1)^2/((n-2)*(n-3))
}
}
#
Thus, it seems that the formulas used in the e1071 package are neither
formulas for the sample statistics nor for the (unbiased) estimates of
the population parameters. Is there another definition of kurtosis and
skewness that I am not aware of?
Dirk
--
*
Dr. Dirk Enzmann
Institute of Criminal Sciences
Dept. of Criminology
Edmund-Siemers-Allee 1
D-20146 Hamburg
Germany
phone: +49-040-42838.7498 (office)
+49-040-42838.4591 (Billon)
fax: +49-040-42838.2344
email: [EMAIL PROTECTED]
www:
http://www2.jura.uni-hamburg.de/instkrim/kriminologie/Mitarbeiter/Enzmann/Enzmann.html
__
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[R] skewness and kurtosis in e1071 correct?
I wonder whether the functions for skewness and kurtosis in the e1071
package are based on correct formulas.
The functions in the package e1071 are:
#
skewness - function (x, na.rm = FALSE)
{
if (na.rm)
x - x[!is.na(x)]
sum((x - mean(x))^3)/(length(x) * sd(x)^3)
}
#
and
#
kurtosis - function (x, na.rm = FALSE)
{
if (na.rm)
x - x[!is.na(x)]
sum((x - mean(x))^4)/(length(x) * var(x)^2) - 3
}
#
However, sd() and var() are the estimated population parameters. To
calculate the sample statistics of skewness and kurtosis, shouldn't one
use the sample statistics of the standard deviation (and variance), as
well? For example:
#
# Function to calculate the sample statistic of skewness:
skew_s=function(x)
{
x = x[!is.na(x)]
n = length(x)
if (n 3)
{
cat('valid cases = ',n,'\nskewness is not defined for less than 3
valid cases!\n')
}
else
{
z = sqrt(n/(n-1))*scale(x)
mean(z^3)
}
}
#
and
#
# Function to calculate the sample statistic of kurtosis:
kurt_s=function(x)
{
x = x[!is.na(x)]
n = length(x)
if (n 4)
{
cat('valid cases = ',n,'\nkurtosis is not defined for less than 4
valid cases!\n')
}
else
{
z = sqrt(n/(n-1))*scale(x)
mean(z^4)-3
}
}
#
Whereas, to calculate the (unbiased) estimated population parameter of
skewness and kurtosis, the correction should also include the number of
cases in the following way:
#
# Function to calculate the unbiased populataion estimate of skewness:
skew=function(x)
{
x = x[!is.na(x)]
n = length(x)
if (n 3)
{
cat('valid cases = ',n,'\nskewness is not defined for less than 3
valid cases!\n')
}
else
{
z = scale(x)
sum(z^3)*n/((n-1)*(n-2))
}
}
#
and
#
# Function to calculate the unbiased population estimate of kurtosis:
kurt=function(x)
{
x = x[!is.na(x)]
n = length(x)
if (n 4)
{
cat('valid cases = ',n,'\nkurtosis is not defined for less than 4
valid cases!\n')
}
else
{
z = scale(x)
sum(z^4)*n*(n+1)/((n-1)*(n-2)*(n-3))-3*(n-1)^2/((n-2)*(n-3))
}
}
#
Thus, it seems that the formulas used in the e1071 package are neither
formulas for the sample statistics nor for the (unbiased) estimates of
the population parameters. Is there another definition of kurtosis and
skewness that I am not aware of?
Dirk
--
*
Dr. Dirk Enzmann
Institute of Criminal Sciences
Dept. of Criminology
Edmund-Siemers-Allee 1
D-20146 Hamburg
Germany
phone: +49-040-42838.7498 (office)
+49-040-42838.4591 (Billon)
fax: +49-040-42838.2344
email: [EMAIL PROTECTED]
www:
http://www2.jura.uni-hamburg.de/instkrim/kriminologie/Mitarbeiter/Enzmann/Enzmann.html
__
R-help@stat.math.ethz.ch mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
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[R] Skewness and Kurtosis
Hi,
in which R-package I could find skewness and kurtosis
measures for a distribution?
I built some functions:
gamma1-function(x)
{
m=mean(x)
n=length(x)
s=sqrt(var(x))
m3=sum((x-m)^3)/n
g1=m3/(s^3)
return(g1)
}
skewness-function(x)
{
m=mean(x)
me=median(x)
s=sqrt(var(x))
sk=(m-me)/s
return(sk)
}
bowley-function(x)
{
q-as.vector(quantile(x,prob=c(.25,.50,.75)))
b=(q[3]+q[1]-2*q[2])/(q[3]-q[2])
return(b)
}
b3-function(x)
{
m=mean(x)
me=median(x)
n=length(x)
d=sum(abs(x-me))/n
b=(m-me)/d
return(b)
}
but I'm looking for some already included in a
package.
Thanks in advance.
Best
Vito
=
Diventare costruttori di soluzioni
The business of the statistician is to catalyze
the scientific learning process.
George E. P. Box
Visitate il portale http://www.modugno.it/
e in particolare la sezione su Palese http://www.modugno.it/archivio/cat_palese.shtml
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Re: [R] Skewness and Kurtosis
Vito Ricci wrote:
Hi,
in which R-package I could find skewness and kurtosis
measures for a distribution?
Both funcions are in package e1071.
Uwe Ligges
I built some functions:
gamma1-function(x)
{
m=mean(x)
n=length(x)
s=sqrt(var(x))
m3=sum((x-m)^3)/n
g1=m3/(s^3)
return(g1)
}
skewness-function(x)
{
m=mean(x)
me=median(x)
s=sqrt(var(x))
sk=(m-me)/s
return(sk)
}
bowley-function(x)
{
q-as.vector(quantile(x,prob=c(.25,.50,.75)))
b=(q[3]+q[1]-2*q[2])/(q[3]-q[2])
return(b)
}
b3-function(x)
{
m=mean(x)
me=median(x)
n=length(x)
d=sum(abs(x-me))/n
b=(m-me)/d
return(b)
}
but I'm looking for some already included in a
package.
Thanks in advance.
Best
Vito
=
Diventare costruttori di soluzioni
The business of the statistician is to catalyze
the scientific learning process.
George E. P. Box
Visitate il portale http://www.modugno.it/
e in particolare la sezione su Palese http://www.modugno.it/archivio/cat_palese.shtml
__
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Re: [R] Skewness and Kurtosis
Hello,
fBasics has functions for skewness and kurtosis:
Comments on these functions are welcome!
Diethelm Wuertz
# **
kurtosis =
function (x, ...)
{ # A function implemented by D. Wuertz
# FUNCTION:
UseMethod(kurtosis)
}
#
--
kurtosis.default =
function (x, na.rm = FALSE, ...)
{ # A function implemented by D. Wuertz
# Description:
# Returns the value of the kurtosis of a
# distribution function. Missing values
# can be handled.
# FUNCTION:
# Warnings:
if (!is.numeric(x) !is.complex(x) !is.logical(x)) {
warning(argument is not numeric or logical: returning NA)
return(as.numeric(NA))}
# Remove NAs:
if (na.rm) x = x[!is.na(x)]
# Kurtosis:
n = length(x)
if (is.integer(x)) x = as.numeric(x)
kurtosis = sum((x-mean(x))^4/var(x)^2)/length(x) - 3
# Return Value:
kurtosis
}
#
--
kurtosis.data.frame =
function (x, ...)
{ # A function implemented by D. Wuertz
sapply(x, kurtosis, ...)
}
#
**
skewness =
function (x, ...)
{ # A function implemented by D. Wuertz
UseMethod(skewness)
}
#
--
skewness.default =
function (x, na.rm = FALSE, ...)
{ # A function implemented by D. Wuertz
# Description:
# Returns the value of the skewness of a
# distribution function. Missing values
# can be handled.
# FUNCTION:
# Warnings:
if (!is.numeric(x) !is.complex(x) !is.logical(x)) {
warning(argument is not numeric or logical: returning NA)
return(as.numeric(NA))}
# Remove NAs:
if (na.rm) x = x[!is.na(x)]
# Skewness:
n = length(x)
if (is.integer(x)) x = as.numeric(x)
skewness = sum((x-mean(x))^3/sqrt(var(x))^3)/length(x)
# Return Value:
skewness
}
#
--
skewness.data.frame =
function (x, ...)
{ # A function implemented by D. Wuertz
sapply(x, skewness, ...)
}
Vito Ricci wrote:
Hi,
in which R-package I could find skewness and kurtosis
measures for a distribution?
I built some functions:
gamma1-function(x)
{
m=mean(x)
n=length(x)
s=sqrt(var(x))
m3=sum((x-m)^3)/n
g1=m3/(s^3)
return(g1)
}
skewness-function(x)
{
m=mean(x)
me=median(x)
s=sqrt(var(x))
sk=(m-me)/s
return(sk)
}
bowley-function(x)
{
q-as.vector(quantile(x,prob=c(.25,.50,.75)))
b=(q[3]+q[1]-2*q[2])/(q[3]-q[2])
return(b)
}
b3-function(x)
{
m=mean(x)
me=median(x)
n=length(x)
d=sum(abs(x-me))/n
b=(m-me)/d
return(b)
}
but I'm looking for some already included in a
package.
Thanks in advance.
Best
Vito
=
Diventare costruttori di soluzioni
The business of the statistician is to catalyze
the scientific learning process.
George E. P. Box
Visitate il portale http://www.modugno.it/
e in particolare la sezione su Palese http://www.modugno.it/archivio/cat_palese.shtml
__
[EMAIL PROTECTED] mailing list
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