This chapter describes the statistical functions in the library. The
basic statistical functions include routines to compute the mean,
variance and standard deviation. More advanced functions allow you to
calculate absolute deviations, skewness, and kurtosis as well as the
median and arbitrary percentiles. The algorithms use recurrence
relations to compute average quantities in a stable way, without large
intermediate values that might overflow.
The functions are available in versions for datasets in the standard
floating-point and integer types. The versions for double precision
floating-point data have the prefix gsl_stats and are declared in
the header file gsl_statistics_double.h. The versions for integer
data have the prefix gsl_stats_int and are declared in the header
file gsl_statistics_int.h. All the functions operate on C
arrays with a stride parameter specifying the spacing between
elements.
Weighted Samples
The functions described in this section allow the computation of
statistics for weighted samples. The functions accept an array of
samples,
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, with associated weights,
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. Each sample
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is considered as having been drawn from a Gaussian
distribution with variance
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. The sample weight
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is defined as the reciprocal of this variance,
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.
Setting a weight to zero corresponds to removing a sample from a dataset.
-
double
gsl_stats_wmean(const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function returns the weighted mean of the dataset data with
stride stride and length n, using the set of weights w
with stride wstride and length n. The weighted mean is defined as,
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-
double
gsl_stats_wvariance(const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function returns the estimated variance of the dataset data
with stride stride and length n, using the set of weights
w with stride wstride and length n. The estimated
variance of a weighted dataset is calculated as,
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Note that this expression reduces to an unweighted variance with the
familiar
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factor when there are
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equal non-zero
weights.
-
double
gsl_stats_wvariance_m(const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean)
This function returns the estimated variance of the weighted dataset
data using the given weighted mean wmean.
-
double
gsl_stats_wsd(const double w[], size_t wstride, const double data[], size_t stride, size_t 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.
-
double
gsl_stats_wsd_m(const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean)
This function returns the square root of the corresponding variance
function gsl_stats_wvariance_m() above.
-
double
gsl_stats_wvariance_with_fixed_mean(const double w[], size_t wstride, const double data[], size_t stride, size_t n, const double mean)
This function computes an unbiased estimate of the variance of the weighted
dataset data 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
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by the known
population mean
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,
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-
double
gsl_stats_wsd_with_fixed_mean(const double w[], size_t wstride, const double data[], size_t stride, size_t n, const double 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.
-
double
gsl_stats_wtss(const double w[], const size_t wstride, const double data[], size_t stride, size_t n)
-
double
gsl_stats_wtss_m(const double w[], const size_t wstride, const double data[], size_t stride, size_t n, double 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().
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-
double
gsl_stats_wabsdev(const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function computes the weighted absolute deviation from the weighted
mean of data. The absolute deviation from the mean is defined as,
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-
double
gsl_stats_wabsdev_m(const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean)
This function computes the absolute deviation of the weighted dataset
data about the given weighted mean wmean.
-
double
gsl_stats_wskew(const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function computes the weighted skewness of the dataset data.
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-
double
gsl_stats_wskew_m_sd(const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean, double 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.
-
double
gsl_stats_wkurtosis(const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function computes the weighted kurtosis of the dataset data.
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-
double
gsl_stats_wkurtosis_m_sd(const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean, double wsd)
This function computes the weighted kurtosis of the dataset data
using the given values of the weighted mean and weighted standard
deviation, wmean and wsd.
Order Statistics
The
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-th
order statistic of a sample is equal to its
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-th smallest value.
The
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-th order statistic of a set of
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values
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is
denoted
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. The median of the set
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is equal to
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if
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is odd, or the average of
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and
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if
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is even. The
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-th smallest element of a length
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vector can be found
in average
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time using the quickselect algorithm.
-
gsl_stats_select(double data[], const size_t stride, const size_t n, const size_t k)
This function finds the k-th smallest element of the input array data
of length n and stride stride using the quickselect method. The
algorithm rearranges the elements of data and so the input array is not preserved
on output.
Robust Location Estimates
A location estimate refers to a typical or central value which best describes a given
dataset. The mean and median are both examples of location estimators. However, the
mean has a severe sensitivity to data outliers and can give erroneous values when
even a small number of outliers are present. The median on the other hand, has
a strong insensitivity to data outliers, but due to its non-smoothness it can
behave unexpectedly in certain situations. GSL offers the following alternative
location estimators, which are robust to the presence of outliers.
Trimmed Mean
The trimmed mean, or truncated mean, discards a certain number of smallest and largest
samples from the input vector before computing the mean of the remaining samples. The
amount of trimming is specified by a factor
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. Then the
number of samples discarded from both ends of the input vector is
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, where
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is the length of the input.
So to discard 25% of the samples from each end, one would set
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.
-
double
gsl_stats_trmean_from_sorted_data(const double alpha, const double sorted_data[], const size_t stride, const size_t n)
This function returns the trimmed mean of sorted_data, 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. The trimming factor
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is given in alpha.
If
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, then the median of the input is returned.
Gastwirth Estimator
Gastwirth’s location estimator is a weighted sum of three order statistics,
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where
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is the one-third quantile,
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is the one-half
quantile (i.e. median), and
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is the two-thirds quantile.
-
double
gsl_stats_gastwirth_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n)
This function returns the Gastwirth location estimator of sorted_data, 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.
Robust Scale Estimates
A robust scale estimate, also known as a robust measure of scale, attempts to quantify
the statistical dispersion (variability, scatter, spread) in a set of data which may contain outliers.
In such datasets, the usual variance or standard deviation scale estimate can be rendered useless
by even a single outlier.
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Statistic
The
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statistic developed by Croux and Rousseeuw is defined as
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For each sample
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, the median of the values
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is computed for all
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. This yields
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values, whose median then gives the final
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.
The factor
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makes
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an unbiased estimate of the standard deviation for Gaussian data.
The factor
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is a correction factor to correct bias in small sample sizes.
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has an asymptotic
efficiency of 58%.
-
double
gsl_stats_Sn0_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n, double work[])
-
double
gsl_stats_Sn_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n, double work[])
These functions return the
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statistic of sorted_data, 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. The Sn0 function calculates
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(i.e. the
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statistic without the bias correction scale factors).
These functions require additional workspace of size
n provided in work.
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Statistic
The
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statistic developed by Croux and Rousseeuw is defined as
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The factor
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makes
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an unbiased estimate of the standard deviation for Gaussian data.
The factor
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is a correction factor to correct bias in small sample sizes. The order statistic
is
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has an asymptotic efficiency of 82%.
-
double
gsl_stats_Qn0_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n, double work[], int work_int[])
-
double
gsl_stats_Qn_from_sorted_data(const double sorted_data[], const size_t stride, const size_t n, double work[], int work_int[])
These functions return the
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statistic of sorted_data, 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. The Qn0 function calculates
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without the bias correction scale factors).
These functions require additional workspace of size
3n provided in work and integer workspace of size 5n
provided in work_int.
References and Further Reading
The standard reference for almost any topic in statistics is the
multi-volume Advanced Theory of Statistics by Kendall and Stuart.
- Maurice Kendall, Alan Stuart, and J. Keith Ord.
The Advanced Theory of Statistics (multiple volumes)
reprinted as Kendall’s Advanced Theory of Statistics.
Wiley, ISBN 047023380X.
Many statistical concepts can be more easily understood by a Bayesian
approach. The following book by Gelman, Carlin, Stern and Rubin gives a
comprehensive coverage of the subject.
- Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin.
Bayesian Data Analysis.
Chapman & Hall, ISBN 0412039915.
For physicists the Particle Data Group provides useful reviews of
Probability and Statistics in the “Mathematical Tools” section of its
Annual Review of Particle Physics.
- Review of Particle Properties,
R.M. Barnett et al., Physical Review D54, 1 (1996)
The Review of Particle Physics is available online at
the website http://pdg.lbl.gov/.
The following papers describe robust scale estimation,
- C. Croux and P. J. Rousseeuw, Time-Efficient algorithms for two highly robust
estimators of scale, Comp. Stat., Physica, Heidelberg, 1992.
- P. J. Rousseeuw and C. Croux, Explicit scale estimators with high breakdown point,
L1-Statistical Analysis and Related Methods, pp. 77-92, 1992.