Math::GSL::Statistics - Statistical functions
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 $smallest
};
Here is a list of all the functions in this module :
- "gsl_stats_mean($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/>
Jonathan "Duke" Leto <jonathan@leto.net> and
Thierry Moisan <thierry.moisan@gmail.com>
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.