Provided by: libmath-gsl-perl_0.43-4build1_amd64 
NAME
Math::GSL::Statistics - Statistical functions
SYNOPSIS
use Math::GSL::Statistics qw /:all/;
my $data = [17.2, 18.1, 16.5, 18.3, 12.6];
my $mean = gsl_stats_mean($data, 1, 5);
my $variance = gsl_stats_variance($data, 1, 5);
my $largest = gsl_stats_max($data, 1, 5);
my $smallest = gsl_stats_min($data, 1, 5);
print qq{
Dataset : @$data
Sample mean $mean
Estimated variance $variance
Largest value $largest
Smallest value $smallest
};
DESCRIPTION
Here is a list of all the functions in this module :
• "gsl_stats_mean($data, $stride, $n)" - This function returns the arithmetic mean of the
array reference $data, a dataset of length $n with stride $stride. The arithmetic mean,
or sample mean, is denoted by \Hat\mu and defined as, \Hat\mu = (1/N) \sum x_i where x_i
are the elements of the dataset $data. For samples drawn from a gaussian distribution
the variance of \Hat\mu is \sigma^2 / N.
• "gsl_stats_variance($data, $stride, $n)" - This function returns the estimated, or
sample, variance of data, an array reference of length $n with stride $stride. The
estimated variance is denoted by \Hat\sigma^2 and is defined by, \Hat\sigma^2 =
(1/(N-1)) \sum (x_i - \Hat\mu)^2 where x_i are the elements of the dataset data. Note
that the normalization factor of 1/(N-1) results from the derivation of \Hat\sigma^2 as
an unbiased estimator of the population variance \sigma^2. For samples drawn from a
gaussian distribution the variance of \Hat\sigma^2 itself is 2 \sigma^4 / N. This
function computes the mean via a call to gsl_stats_mean. If you have already computed
the mean then you can pass it directly to gsl_stats_variance_m.
• "gsl_stats_sd($data, $stride, $n)"
• "gsl_stats_sd_m($data, $stride, $n, $mean)"
The standard deviation is defined as the square root of the variance. These functions
return the square root of the corresponding variance functions above.
• "gsl_stats_variance_with_fixed_mean($data, $stride, $n, $mean)" - This function
calculates the standard deviation of the array reference $data for a fixed population
mean $mean. The result is the square root of the corresponding variance function.
• "gsl_stats_sd_with_fixed_mean($data, $stride, $n, $mean)" - This function computes an
unbiased estimate of the variance of data when the population mean $mean of the
underlying distribution is known a priori. In this case the estimator for the variance
uses the factor 1/N and the sample mean \Hat\mu is replaced by the known population mean
\mu, \Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2
• "gsl_stats_tss($data, $stride, $n)"
• "gsl_stats_tss_m($data, $stride, $n, $mean)"
These functions return the total sum of squares (TSS) of data about the mean. For
gsl_stats_tss_m the user-supplied value of mean is used, and for gsl_stats_tss it is
computed using gsl_stats_mean. TSS = \sum (x_i - mean)^2
• "gsl_stats_absdev($data, $stride, $n)" - This function computes the absolute deviation
from the mean of data, a dataset of length $n with stride $stride. The absolute
deviation from the mean is defined as, absdev = (1/N) \sum |x_i - \Hat\mu| where x_i
are the elements of the array reference $data. The absolute deviation from the mean
provides a more robust measure of the width of a distribution than the variance. This
function computes the mean of data via a call to gsl_stats_mean.
• "gsl_stats_skew($data, $stride, $n)" - This function computes the skewness of $data, a
dataset in the form of an array reference of length $n with stride $stride. The skewness
is defined as, skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3 where x_i are the
elements of the dataset $data. The skewness measures the asymmetry of the tails of a
distribution. The function computes the mean and estimated standard deviation of data
via calls to gsl_stats_mean and gsl_stats_sd.
• "gsl_stats_skew_m_sd($data, $stride, $n, $mean, $sd)" - This function computes the
skewness of the array reference $data using the given values of the mean $mean and
standard deviation $sd, skew = (1/N) \sum ((x_i - mean)/sd)^3. These functions are
useful if you have already computed the mean and standard deviation of $data and want to
avoid recomputing them.
• "gsl_stats_kurtosis($data, $stride, $n)" - This function computes the kurtosis of data,
an array reference of length $n with stride $stride. The kurtosis is defined as,
kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4) - 3. The kurtosis measures how
sharply peaked a distribution is, relative to its width. The kurtosis is normalized to
zero for a gaussian distribution.
• "gsl_stats_kurtosis_m_sd($data, $stride, $n, $mean, $sd)" - This function computes the
kurtosis of the array reference $data using the given values of the mean $mean and
standard deviation $sd, kurtosis = ((1/N) \sum ((x_i - mean)/sd)^4) - 3. This function
is useful if you have already computed the mean and standard deviation of data and want
to avoid recomputing them.
• "gsl_stats_lag1_autocorrelation($data, $stride, $n)" - This function computes the lag-1
autocorrelation of the array reference data.
a_1 = {\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu)
\over
\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}
• "gsl_stats_lag1_autocorrelation_m($data, $stride, $n, $mean)" - This function computes
the lag-1 autocorrelation of the array reference $data using the given value of the mean
$mean.
• "gsl_stats_covariance($data1, $stride1, $data2, $stride2, $n)" - This function computes
the covariance of the array reference $data1 and $data2 which must both be of the same
length $n. covar = (1/(n - 1)) \sum_{i = 1}^{n} (x_i - \Hat x) (y_i - \Hat y)
• "gsl_stats_covariance_m($data1, $stride1, $data2, $stride2, $n, $mean1, $mean2)" - This
function computes the covariance of the array reference $data1 and $data2 using the
given values of the means, $mean1 and $mean2. This is useful if you have already
computed the means of $data1 and $data2 and want to avoid recomputing them.
• "gsl_stats_correlation($data1, $stride1, $data2, $stride2, $n)" - This function
efficiently computes the Pearson correlation coefficient between the array reference
$data1 and $data2 which must both be of the same length $n.
r = cov(x, y) / (\Hat\sigma_x \Hat\sigma_y)
= {1/(n-1) \sum (x_i - \Hat x) (y_i - \Hat y)
\over
\sqrt{1/(n-1) \sum (x_i - \Hat x)^2} \sqrt{1/(n-1) \sum (y_i - \Hat y)^2}
}
• "gsl_stats_variance_m($data, $stride, $n, $mean)" - This function returns the sample
variance of $data, an array reference, relative to the given value of $mean. The
function is computed with \Hat\mu replaced by the value of mean that you supply,
\Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2
• "gsl_stats_absdev_m($data, $stride, $n, $mean)" - This function computes the absolute
deviation of the dataset $data, an array reference, relative to the given value of
$mean, absdev = (1/N) \sum |x_i - mean|. This function is useful if you have already
computed the mean of data (and want to avoid recomputing it), or wish to calculate the
absolute deviation relative to another value (such as zero, or the median).
• "gsl_stats_wmean($w, $wstride, $data, $stride, $n)" - This function returns the weighted
mean of the dataset $data array reference with stride $stride and length $n, using the
set of weights $w, which is an array reference, with stride $wstride and length $n. The
weighted mean is defined as, \Hat\mu = (\sum w_i x_i) / (\sum w_i)
• "gsl_stats_wvariance($w, $wstride, $data, $stride, $n)" - This function returns the
estimated variance of the dataset $data, which is the dataset, with stride $stride and
length $n, using the set of weights $w (as an array reference) with stride $wstride and
length $n. The estimated variance of a weighted dataset is defined as, \Hat\sigma^2 =
((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2))) \sum w_i (x_i - \Hat\mu)^2. Note that this
expression reduces to an unweighted variance with the familiar 1/(N-1) factor when there
are N equal non-zero weights.
• "gsl_stats_wvariance_m($w, $wstride, $data, $stride, $n, $wmean, $wsd)" - This function
returns the estimated variance of the weighted dataset $data (which is an array
reference) using the given weighted mean $wmean.
• "gsl_stats_wsd($w, $wstride, $data, $stride, $n)" - The standard deviation is defined as
the square root of the variance. This function returns the square root of the
corresponding variance function gsl_stats_wvariance above.
• "gsl_stats_wsd_m($w, $wstride, $data, $stride, $n, $wmean)" - This function returns the
square root of the corresponding variance function gsl_stats_wvariance_m above.
• "gsl_stats_wvariance_with_fixed_mean($w, $wstride, $data, $stride, $n, $mean)" - This
function computes an unbiased estimate of the variance of weighted dataset $data (which
is an array reference) when the population mean $mean of the underlying distribution is
known a priori. In this case the estimator for the variance replaces the sample mean
\Hat\mu by the known population mean \mu, \Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) /
(\sum w_i)
• "gsl_stats_wsd_with_fixed_mean($w, $wstride, $data, $stride, $n, $mean)" - The standard
deviation is defined as the square root of the variance. This function returns the
square root of the corresponding variance function above.
• "gsl_stats_wtss($w, $wstride, $data, $stride, $n)"
• "gsl_stats_wtss_m($w, $wstride, $data, $stride, $n, $wmean)" - These functions return
the weighted total sum of squares (TSS) of data about the weighted mean. For
gsl_stats_wtss_m the user-supplied value of $wmean is used, and for gsl_stats_wtss it is
computed using gsl_stats_wmean. TSS = \sum w_i (x_i - wmean)^2
• "gsl_stats_wabsdev($w, $wstride, $data, $stride, $n)" - This function computes the
weighted absolute deviation from the weighted mean of $data, which is an array
reference. The absolute deviation from the mean is defined as, absdev = (\sum w_i |x_i -
\Hat\mu|) / (\sum w_i)
• "gsl_stats_wabsdev_m($w, $wstride, $data, $stride, $n, $wmean)" - This function computes
the absolute deviation of the weighted dataset $data (an array reference) about the
given weighted mean $wmean.
• "gsl_stats_wskew($w, $wstride, $data, $stride, $n)" - This function computes the
weighted skewness of the dataset $data, an array reference. skew = (\sum w_i ((x_i -
xbar)/\sigma)^3) / (\sum w_i)
• "gsl_stats_wskew_m_sd($w, $wstride, $data, $stride, $n, $wmean, $wsd)" - This function
computes the weighted skewness of the dataset $data using the given values of the
weighted mean and weighted standard deviation, $wmean and $wsd.
• "gsl_stats_wkurtosis($w, $wstride, $data, $stride, $n)" - This function computes the
weighted kurtosis of the dataset $data, an array reference. kurtosis = ((\sum w_i ((x_i
- xbar)/sigma)^4) / (\sum w_i)) - 3
• "gsl_stats_wkurtosis_m_sd($w, $wstride, $data, $stride, $n, $wmean, $wsd)" - This
function computes the weighted kurtosis of the dataset $data, an array reference, using
the given values of the weighted mean and weighted standard deviation, $wmean and $wsd.
• "gsl_stats_pvariance($data, $stride, $n, $data2, $stride2, $n2)"
• "gsl_stats_ttest($data1, $stride1, $n1, $data2, $stride2, $n2)"
• "gsl_stats_max($data, $stride, $n)" - This function returns the maximum value in the
$data array reference, a dataset of length $n with stride $stride. The maximum value is
defined as the value of the element x_i which satisfies x_i >= x_j for all j. If you
want instead to find the element with the largest absolute magnitude you will need to
apply fabs or abs to your data before calling this function.
• "gsl_stats_min($data, $stride, $n)" - This function returns the minimum value in $data
(which is an array reference) a dataset of length $n with stride $stride. The minimum
value is defined as the value of the element x_i which satisfies x_i <= x_j for all j.
If you want instead to find the element with the smallest absolute magnitude you will
need to apply fabs or abs to your data before calling this function.
• "gsl_stats_minmax($data, $stride, $n)" - This function finds both the minimum and
maximum values in $data, which is an array reference, in a single pass and returns them
in this order.
• "gsl_stats_max_index($data, $stride, $n)" - This function returns the index of the
maximum value in $data array reference, a dataset of length $n with stride $stride. The
maximum value is defined as the value of the element x_i which satisfies x_i >= x_j for
all j. When there are several equal maximum elements then the first one is chosen.
• "gsl_stats_min_index($data, $stride, $n)" - This function returns the index of the
minimum value in $data array reference, a dataset of length $n with stride $stride. The
minimum value is defined as the value of the element x_i which satisfies x_i <= x_j for
all j. When there are several equal minimum elements then the first one is chosen.
• "gsl_stats_minmax_index($data, $stride, $n)" - This function returns the indexes of the
minimum and maximum values in $data, an array reference in a single pass. The value are
returned in this order.
• "gsl_stats_median_from_sorted_data($sorted_data, $stride, $n)" - This function returns
the median value of $sorted_data (which is an array reference), a dataset of length $n
with stride $stride. The elements of the array must be in ascending numerical order.
There are no checks to see whether the data are sorted, so the function gsl_sort should
always be used first. This function can be found in the Math::GSL::Sort module. When
the dataset has an odd number of elements the median is the value of element (n-1)/2.
When the dataset has an even number of elements the median is the mean of the two
nearest middle values, elements (n-1)/2 and n/2. Since the algorithm for computing the
median involves interpolation this function always returns a floating-point number, even
for integer data types.
• "gsl_stats_quantile_from_sorted_data($sorted_data, $stride, $n, $f)" - This function
returns a quantile value of $sorted_data, a double-precision array reference of length
$n with stride $stride. The elements of the array must be in ascending numerical order.
The quantile is determined by the f, a fraction between 0 and 1. For example, to compute
the value of the 75th percentile f should have the value 0.75. There are no checks to
see whether the data are sorted, so the function gsl_sort should always be used first.
This function can be found in the Math::GSL::Sort module. The quantile is found by
interpolation, using the formula quantile = (1 - \delta) x_i + \delta x_{i+1} where i is
floor((n - 1)f) and \delta is (n-1)f - i. Thus the minimum value of the array
(data[0*stride]) is given by f equal to zero, the maximum value (data[(n-1)*stride]) is
given by f equal to one and the median value is given by f equal to 0.5. Since the
algorithm for computing quantiles involves interpolation this function always returns a
floating-point number, even for integer data types.
The following function are simply variants for int and char of the last functions:
• "gsl_stats_int_mean "
• "gsl_stats_int_variance "
• "gsl_stats_int_sd "
• "gsl_stats_int_variance_with_fixed_mean "
• "gsl_stats_int_sd_with_fixed_mean "
• "gsl_stats_int_tss "
• "gsl_stats_int_tss_m "
• "gsl_stats_int_absdev "
• "gsl_stats_int_skew "
• "gsl_stats_int_kurtosis "
• "gsl_stats_int_lag1_autocorrelation "
• "gsl_stats_int_covariance "
• "gsl_stats_int_correlation "
• "gsl_stats_int_variance_m "
• "gsl_stats_int_sd_m "
• "gsl_stats_int_absdev_m "
• "gsl_stats_int_skew_m_sd "
• "gsl_stats_int_kurtosis_m_sd "
• "gsl_stats_int_lag1_autocorrelation_m "
• "gsl_stats_int_covariance_m "
• "gsl_stats_int_pvariance "
• "gsl_stats_int_ttest "
• "gsl_stats_int_max "
• "gsl_stats_int_min "
• "gsl_stats_int_minmax "
• "gsl_stats_int_max_index "
• "gsl_stats_int_min_index "
• "gsl_stats_int_minmax_index "
• "gsl_stats_int_median_from_sorted_data "
• "gsl_stats_int_quantile_from_sorted_data "
• "gsl_stats_char_mean "
• "gsl_stats_char_variance "
• "gsl_stats_char_sd "
• "gsl_stats_char_variance_with_fixed_mean "
• "gsl_stats_char_sd_with_fixed_mean "
• "gsl_stats_char_tss "
• "gsl_stats_char_tss_m "
• "gsl_stats_char_absdev "
• "gsl_stats_char_skew "
• "gsl_stats_char_kurtosis "
• "gsl_stats_char_lag1_autocorrelation "
• "gsl_stats_char_covariance "
• "gsl_stats_char_correlation "
• "gsl_stats_char_variance_m "
• "gsl_stats_char_sd_m "
• "gsl_stats_char_absdev_m "
• "gsl_stats_char_skew_m_sd "
• "gsl_stats_char_kurtosis_m_sd "
• "gsl_stats_char_lag1_autocorrelation_m "
• "gsl_stats_char_covariance_m "
• "gsl_stats_char_pvariance "
• "gsl_stats_char_ttest "
• "gsl_stats_char_max "
• "gsl_stats_char_min "
• "gsl_stats_char_minmax "
• "gsl_stats_char_max_index "
• "gsl_stats_char_min_index "
• "gsl_stats_char_minmax_index "
• "gsl_stats_char_median_from_sorted_data "
• "gsl_stats_char_quantile_from_sorted_data "
You have to add the functions you want to use inside the qw /put_function_here /. You can
also write use Math::GSL::Statistics qw/:all/; to use all available functions of the
module. Other tags are also available, here is a complete list of all tags for this
module :
all
int
char
For more information on the functions, we refer you to the GSL official documentation:
<http://www.gnu.org/software/gsl/manual/html_node/>
AUTHORS
Jonathan "Duke" Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>
COPYRIGHT AND LICENSE
Copyright (C) 2008-2021 Jonathan "Duke" Leto and Thierry Moisan
This program is free software; you can redistribute it and/or modify it under the same
terms as Perl itself.