Introduction
This chapter is concerned with calculating various statistics from
subsets of a given dataset. The main idea is to compute statistics
in the vicinity of a given data sample by defining a window which
includes the sample itself as well as some specified number of samples
before and after the sample in question. For a sample
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, we
define a window
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as
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The parameters
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and
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are non-negative integers specifying
the number of samples to include before and after the sample
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.
Statistics such as the mean and standard deviation of the window
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may be computed, and then the window is shifted forward by one sample to
focus on
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. The total number of samples in the window is
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. To define a symmetric window centered on
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,
one would set
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.
Handling Endpoints
When processing samples near the ends of the input signal, there will not
be enough samples to fill the window
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defined above.
Therefore the user must specify how to construct the windows near the end points.
This is done by passing an input argument of type
gsl_movstat_end_t:
-
gsl_movstat_end_t
This data type specifies how to construct windows near end points and can
be selected from the following choices:
-
GSL_MOVSTAT_END_PADZERO
With this option, a full window of length
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will be constructed
by inserting zeros into the window near the signal end points. Effectively,
the input signal is modified to
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to ensure a well-defined window for all
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.
-
GSL_MOVSTAT_END_PADVALUE
With this option, a full window of length
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will be constructed
by padding the window with the first and last sample in the input signal.
Effectively, the input signal is modified to
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-
GSL_MOVSTAT_END_TRUNCATE
With this option, no padding is performed, and the windows are simply truncated
as the end points are approached.
Moving Mean
The moving window mean calculates the mean of the values of each window
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represents the number of elements in the window
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. This will normally be
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, unless the
GSL_MOVSTAT_END_TRUNCATE
option is selected, in which case it could be less than
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near the signal end points.
-
int
gsl_movstat_mean(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace * w)
This function computes the moving window mean of the input vector x, storing
the output in y. The parameter endtype specifies how windows near
the ends of the input should be handled. It is allowed to have x = y
for an in-place moving mean.
Moving Variance and Standard Deviation
The moving window variance calculates the sample variance of the values of each window
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,
defined by
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)
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where
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is the mean of
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defined above. The standard deviation
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is the square root of the variance.
-
int
gsl_movstat_variance(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace * w)
This function computes the moving window variance of the input vector x, storing
the output in y. The parameter endtype specifies how windows near
the ends of the input should be handled. It is allowed to have x = y
for an in-place moving variance.
-
int
gsl_movstat_sd(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace * w)
This function computes the moving window standard deviation of the input vector x, storing
the output in y. The parameter endtype specifies how windows near
the ends of the input should be handled. It is allowed to have x = y
for an in-place moving standard deviation.
Moving Minimum and Maximum
The moving minimum/maximum calculates the minimum and maximum values of
each window
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.
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y_i^{max} &= \max \left( W_i^{H,J} \right))
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-
int
gsl_movstat_min(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace * w)
This function computes the moving minimum of the input vector x, storing
the result in y. The parameter endtype specifies how windows near
the ends of the input should be handled.
It is allowed to have x = y for an in-place moving minimum.
-
int
gsl_movstat_max(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace * w)
This function computes the moving maximum of the input vector x, storing
the result in y. The parameter endtype specifies how windows near
the ends of the input should be handled.
It is allowed to have x = y for an in-place moving maximum.
-
int
gsl_movstat_minmax(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * y_min, gsl_vector * y_max, gsl_movstat_workspace * w)
This function computes the moving minimum and maximum of the input vector x, storing
the window minimums in y_min and the window maximums in y_max.
The parameter endtype specifies how windows near the ends of the input should be handled.
Moving Sum
The moving window sum calculates the sum of the values of each window
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.
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)
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-
int
gsl_movstat_sum(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace * w)
This function computes the moving window sum of the input vector x, storing
the output in y. The parameter endtype specifies how windows near
the ends of the input should be handled. It is allowed to have x = y
for an in-place moving sum.
Robust Scale Estimation
A common problem in statistics is to quantify the dispersion (also known as the variability, scatter, and spread) of
a set of data. Often this is done by calculating the variance or standard deviation. However these statistics
are strongly influenced by outliers, and can often provide erroneous results when even a small number of outliers
are present.
Several useful statistics have emerged to provide robust estimates of scale which are not as susceptible to data outliers.
A few of these statistical scale estimators are described below.
Moving MAD
The median absolute deviation (MAD) for the window
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is defined
to be the median of the absolute deviations from the window’s median:
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)
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The factor of
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makes the MAD an unbiased estimator of the standard deviation
for Gaussian data. The MAD has an efficiency of 37%. See
here for more information.
-
int
gsl_movstat_mad0(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * xmedian, gsl_vector * xmad, gsl_movstat_workspace * w)
-
int
gsl_movstat_mad(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * xmedian, gsl_vector * xmad, gsl_movstat_workspace * w)
These functions compute the moving MAD of the input vector x and store the result
in xmad. The medians of each window
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are stored in xmedian
on output. The inputs x, xmedian, and xmad must all be the same length.
The parameter endtype specifies how windows near the ends of the input should be handled.
The function mad0 does not include the scale factor of
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, while the
function mad does include this factor.
Moving QQR
The q-quantile range (QQR) is the difference between the
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and
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quantiles
of a set of data,
System Message: WARNING/2 (QQR = Q_{1-q} - Q_q
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The case
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corresponds to the well-known
interquartile range (IQR), which
is the difference between the 75th and 25th percentiles of a set of data. The QQR is
a
trimmed estimator, the main idea being to discard the largest and smallest values in
a data window and compute a scale estimate from the remaining middle values. In the case
of the IQR, the largest and smallest 25% of the data are discarded and the scale is
estimated from the remaining (middle) 50%.
-
int
gsl_movstat_qqr(const gsl_movstat_end_t endtype, const gsl_vector * x, const double q, gsl_vector * xqqr, gsl_movstat_workspace * w)
This function computes the moving QQR of the input vector x and stores the q-quantile ranges
of each window
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in xqqr. The quantile parameter q must be between
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and
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. The input
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corresponds to the IQR.
The inputs x and xqqr must be the same length.
The parameter endtype specifies how windows near the ends of the input should be handled.
Moving
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The
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statistic proposed by Croux and Rousseeuw is based on pairwise differences between
all samples in the window. It has an efficiency of 58%, significantly higher than the MAD.
See
here for more information.
-
int
gsl_movstat_Sn(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * xscale, gsl_movstat_workspace * w)
This function computes the moving
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of the input vector x and stores the output
in xscale. The inputs x and xscale must be the same length.
The parameter endtype specifies how windows near the ends of the input should be handled.
It is allowed for x = xscale for an in-place moving window
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.
Moving
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The
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statistic proposed by Croux and Rousseeuw is loosely based on the Hodges-Lehmann location
estimator. It has a relatively high efficiency of 82%. See
here for more information.
-
int
gsl_movstat_Qn(const gsl_movstat_end_t endtype, const gsl_vector * x, gsl_vector * xscale, gsl_movstat_workspace * w)
This function computes the moving
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of the input vector x and stores the output
in xscale. The inputs x and xscale must be the same length.
The parameter endtype specifies how windows near the ends of the input should be handled.
It is allowed for x = xscale for an in-place moving window
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.
User-defined Moving Statistics
GSL offers an interface for users to define their own moving window statistics
functions, without needing to implement the edge-handling and accumulator
machinery. This can be done by explicitly constructing the windows
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for a given input signal (
gsl_movstat_fill()), or by calculating a user-defined
function for each window automatically. In order to apply a user-defined
function to each window, users must define a variable of type
gsl_movstat_function to pass into
gsl_movstat_apply().
This structure is defined as follows.
-
gsl_movstat_function
Structure specifying user-defined moving window statistical function:
typedef struct
{
double (* function) (const size_t n, double x[], void * params);
void * params;
} gsl_movstat_function;
This structure contains a pointer to the user-defined function as well
as possible parameters to pass to the function.
-
double
(* function)(const size_t n, double x[], void * params)
This function returns the user-defined statistic of the array x
of length n. User-specified parameters are passed in via params.
It is allowed to modify the array x.
-
void *
params
User-specified parameters to be passed into the function.
-
int
gsl_movstat_apply(const gsl_movstat_end_t endtype, const gsl_movstat_function * F, const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace * w)
This function applies the user-defined moving window statistic specified in F
to the input vector x, storing the output in y.
The parameter endtype specifies how windows near the ends of the input should be handled.
It is allowed for x = y for an in-place moving window calculation.
-
size_t
gsl_movstat_fill(const gsl_movstat_end_t endtype, const gsl_vector * x, const size_t idx, const size_t H, const size_t J, double * window)
This function explicitly constructs the sliding window for the input vector x which
is centered on the sample idx. On output, the array window will contain
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. The number of samples to the left and right
of the sample idx are specified by H and J respectively.
The parameter endtype specifies how windows near the ends of the input should be handled.
The function returns the size of the window.
Examples
Example 1
The following example program computes the moving mean, minimum and maximum of a noisy
sinusoid signal of length
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with a symmetric moving window of size
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.
The program is given below.
#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_movstat.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_vector.h>
int
main(void)
{
const size_t N = 500; /* length of time series */
const size_t K = 11; /* window size */
gsl_movstat_workspace * w = gsl_movstat_alloc(K);
gsl_vector *x = gsl_vector_alloc(N);
gsl_vector *xmean = gsl_vector_alloc(N);
gsl_vector *xmin = gsl_vector_alloc(N);
gsl_vector *xmax = gsl_vector_alloc(N);
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
size_t i;
for (i = 0; i < N; ++i)
{
double xi = cos(2.0 * M_PI * i / (double) N);
double ei = gsl_ran_gaussian(r, 0.1);
gsl_vector_set(x, i, xi + ei);
}
/* compute moving statistics */
gsl_movstat_mean(GSL_MOVSTAT_END_PADVALUE, x, xmean, w);
gsl_movstat_minmax(GSL_MOVSTAT_END_PADVALUE, x, xmin, xmax, w);
/* print results */
for (i = 0; i < N; ++i)
{
printf("%zu %f %f %f %f\n",
i,
gsl_vector_get(x, i),
gsl_vector_get(xmean, i),
gsl_vector_get(xmin, i),
gsl_vector_get(xmax, i));
}
gsl_vector_free(x);
gsl_vector_free(xmean);
gsl_rng_free(r);
gsl_movstat_free(w);
return 0;
}
Example 2: Robust Scale
The following example program analyzes a time series of length
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composed
of Gaussian random variates with zero mean whose standard deviation changes in a piecewise constant fashion
as shown in the table below.
| Sample Range |
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|
|---|
| 1-200 |
1.0 |
| 201-450 |
5.0 |
| 451-600 |
1.0 |
| 601-850 |
3.0 |
| 851-1000 |
5.0 |
Additionally, about 1% of the samples are perturbed to represent outliers by adding
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to the random Gaussian variate.
The program calculates the moving statistics MAD, IQR,
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,
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, and
the standard deviation using a symmetric moving window of length
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. The results are shown in
Fig. 6.
The robust statistics follow the true standard deviation piecewise changes well, without being
influenced by the outliers. The moving standard deviation (gray curve) is heavily influenced by
the presence of the outliers. The program is given below.
#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_movstat.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_vector.h>
int
main(void)
{
const size_t N = 1000; /* length of time series */
const double sigma[] = { 1.0, 5.0, 1.0, 3.0, 5.0 }; /* variances */
const size_t N_sigma[] = { 200, 450, 600, 850, 1000 }; /* samples where variance changes */
const size_t K = 41; /* window size */
gsl_vector *x = gsl_vector_alloc(N);
gsl_vector *xmedian = gsl_vector_alloc(N);
gsl_vector *xmad = gsl_vector_alloc(N);
gsl_vector *xiqr = gsl_vector_alloc(N);
gsl_vector *xSn = gsl_vector_alloc(N);
gsl_vector *xQn = gsl_vector_alloc(N);
gsl_vector *xsd = gsl_vector_alloc(N);
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_movstat_workspace * w = gsl_movstat_alloc(K);
size_t idx = 0;
size_t i;
for (i = 0; i < N; ++i)
{
double gi = gsl_ran_gaussian(r, sigma[idx]);
double u = gsl_rng_uniform(r);
double outlier = (u < 0.01) ? 15.0*GSL_SIGN(gi) : 0.0;
double xi = gi + outlier;
gsl_vector_set(x, i, xi);
if (i == N_sigma[idx] - 1)
++idx;
}
/* compute moving statistics */
gsl_movstat_mad(GSL_MOVSTAT_END_TRUNCATE, x, xmedian, xmad, w);
gsl_movstat_qqr(GSL_MOVSTAT_END_TRUNCATE, x, 0.25, xiqr, w);
gsl_movstat_Sn(GSL_MOVSTAT_END_TRUNCATE, x, xSn, w);
gsl_movstat_Qn(GSL_MOVSTAT_END_TRUNCATE, x, xQn, w);
gsl_movstat_sd(GSL_MOVSTAT_END_TRUNCATE, x, xsd, w);
/* scale IQR by factor to approximate standard deviation */
gsl_vector_scale(xiqr, 0.7413);
/* print results */
idx = 0;
for (i = 0; i < N; ++i)
{
printf("%zu %f %f %f %f %f %f %f\n",
i,
gsl_vector_get(x, i),
sigma[idx],
gsl_vector_get(xmad, i),
gsl_vector_get(xiqr, i),
gsl_vector_get(xSn, i),
gsl_vector_get(xQn, i),
gsl_vector_get(xsd, i));
if (i == N_sigma[idx] - 1)
++idx;
}
gsl_vector_free(x);
gsl_vector_free(xmedian);
gsl_vector_free(xmad);
gsl_vector_free(xiqr);
gsl_vector_free(xSn);
gsl_vector_free(xQn);
gsl_vector_free(xsd);
gsl_rng_free(r);
gsl_movstat_free(w);
return 0;
}
Example 3: User-defined Moving Window
This example program illustrates how a user can define their own moving window function to apply
to an input vector. It constructs a random noisy time series of length
System Message: WARNING/2 (N = 1000)
latex exited with error
[stdout]
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restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2017-04-15>
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(/usr/share/texmf-dist/tex/latex/base/article.cls
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(/usr/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def' not found.
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or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
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^^M
No pages of output.
Transcript written on math.log.
with
some outliers added. Then it applies a moving window trimmed mean to the time series with
trim parameter
System Message: WARNING/2 (\alpha = 0.1)
latex exited with error
[stdout]
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restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2017-04-15>
Babel <3.10> and hyphenation patterns for 84 language(s) loaded.
(/usr/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.158 \endinput
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. The length of the moving window is
System Message: WARNING/2 (K = 11)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2017-04-15>
Babel <3.10> and hyphenation patterns for 84 language(s) loaded.
(/usr/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.158 \endinput
^^M
No pages of output.
Transcript written on math.log.
, so
the smallest and largest sample of each window is discarded prior to computing the mean.
The results are shown in
Fig. 7.
The program is given below.
#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_movstat.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_statistics.h>
double
func(const size_t n, double x[], void * params)
{
const double alpha = *(double *) params;
gsl_sort(x, 1, n);
return gsl_stats_trmean_from_sorted_data(alpha, x, 1, n);
}
int
main(void)
{
const size_t N = 1000; /* length of time series */
const size_t K = 11; /* window size */
double alpha = 0.1; /* trimmed mean parameter */
gsl_vector *x = gsl_vector_alloc(N); /* input vector */
gsl_vector *y = gsl_vector_alloc(N); /* filtered output vector for alpha1 */
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_movstat_workspace *w = gsl_movstat_alloc(K);
gsl_movstat_function F;
size_t i;
double sum = 0.0;
/* generate input signal */
for (i = 0; i < N; ++i)
{
double ui = gsl_ran_gaussian(r, 1.0);
double outlier = (gsl_rng_uniform(r) < 0.01) ? 10.0*GSL_SIGN(ui) : 0.0;
sum += ui;
gsl_vector_set(x, i, sum + outlier);
}
/* apply moving window function */
F.function = func;
F.params = α
gsl_movstat_apply(GSL_MOVSTAT_END_PADVALUE, &F, x, y, w);
/* print results */
for (i = 0; i < N; ++i)
{
double xi = gsl_vector_get(x, i);
double yi = gsl_vector_get(y, i);
printf("%f %f\n", xi, yi);
}
gsl_vector_free(x);
gsl_vector_free(y);
gsl_rng_free(r);
gsl_movstat_free(w);
return 0;
}
