math::statistics(n) Math math::statistics(n)
NAME
math::statistics - Basic statistical functions and procedures
SYNOPSIS
package require Tcl 8
package require math::::statistics 00.1.1
::::math::::statistics::::mean data
::::math::::statistics::::min data
::::math::::statistics::::max data
::::math::::statistics::::number data
::::math::::statistics::::stdev data
::::math::::statistics::::var data
::::math::::statistics::::basic-stats data
::::math::::statistics::::histogram limits values
::::math::::statistics::::corr data1 data2
::::math::::statistics::::interval-mean-stdev data confidence
::::math::::statistics::::t-test-mean data estmean eststdev confidence
::::math::::statistics::::quantiles data confidence
::::math::::statistics::::quantiles limits counts confidence
::::math::::statistics::::autocorr data
::::math::::statistics::::crosscorr data1 data2
::::math::::statistics::::mean-histogram-limits mean stdev number
::::math::::statistics::::minmax-histogram-limits min max number
::::math::::statistics::::pdf-normal mean stdev value
::::math::::statistics::::pdf-exponential mean value
::::math::::statistics::::pdf-uniform xmin xmax value
::::math::::statistics::::cdf-normal mean stdev value
::::math::::statistics::::cdf-exponential mean value
::::math::::statistics::::cdf-uniform xmin xmax value
::::math::::statistics::::cdf-students-t degrees value
::::math::::statistics::::random-normal mean stdev number
::::math::::statistics::::random-exponential mean number
::::math::::statistics::::random-uniform xmin xmax value
::::math::::statistics::::histogram-uniform xmin xmax limits number
::::math::::statistics::::filter varname data expression
::::math::::statistics::::map varname data expression
::::math::::statistics::::samplescount varname list expression
::::math::::statistics::::subdivide
::::math::::statistics::::plot-scale canvas xmin xmax ymin ymax
::::math::::statistics::::plot-xydata canvas xdata ydata tag
::::math::::statistics::::plot-xyline canvas xdata ydata tag
::::math::::statistics::::plot-tdata canvas tdata tag
::::math::::statistics::::plot-tline canvas tdata tag
::::math::::statistics::::plot-histogram canvas counts limits tag
DESCRIPTION
The math::::statistics package contains functions and procedures for
basic statistical data analysis, such as:
]o Descriptive statistical parameters (mean, minimum, maximum,
standard deviation)
]o Estimates of the distribution in the form of histograms and
quantiles
]o Basic testing of hypotheses
]o Probability and cumulative density functions It is meant to help
in developing data analysis applications or doing ad hoc data
analysis, it is not in itself a full application, nor is it
intended to rival with full (non-)commercial statistical pack-
ages.
The purpose of this document is to describe the implemented procedures
and provide some examples of their usage. As there is ample literature
on the algorithms involved, we refer to relevant text books for more
explanations. The package contains a fairly large number of public
procedures. They can be distinguished in three sets: general proce-
dures, procedures that deal with specific statistical distributions,
list procedures to select or transform data and simple plotting proce-
dures (these require Tk). Note: The data that need to be analyzed are
always contained in a simple list. Missing values are represented as
empty list elements.
GENERAL PROCEDURES
The general statistical procedures are:
::::math::::statistics::::mean data
Determine the mean value of the given list of data.
data - List of data
::::math::::statistics::::min data
Determine the minimum value of the given list of data.
data - List of data
::::math::::statistics::::max data
Determine the maximum value of the given list of data.
data - List of data
::::math::::statistics::::number data
Determine the number of non-missing data in the given list
data - List of data
::::math::::statistics::::stdev data
Determine the standard deviation of the data in the given list
data - List of data
::::math::::statistics::::var data
Determine the variance of the data in the given list
data - List of data
::::math::::statistics::::basic-stats data
Determine a list of all the descriptive parameters: mean, mini-
mum, maximum, number of data, standard deviation and variance.
(This routine is called whenever either or all of the basic sta-
tistical parameters are required. Hence all calculations are
done and the relevant values are returned.)
data - List of data
::::math::::statistics::::histogram limits values
Determine histogram information for the given list of data.
Returns a list consisting of the number of values that fall into
each interval. (The first interval consists of all values lower
than the first limit, the last interval consists of all values
greater than the last limit. There is one more interval than
there are limits.)
limits - List of upper limits (in ascending order) for the
intervals of the histogram.
values - List of data
::::math::::statistics::::corr data1 data2
Determine the correlation coefficient between two sets of data.
data1 - First list of data
data2 - Second list of data
::::math::::statistics::::interval-mean-stdev data confidence
Return the interval containing the mean value and one containing
the standard deviation with a certain level of confidence
(assuming a normal distribution)
data - List of raw data values (small sample)
confidence - Confidence level (0.95 or 0.99 for instance)
::::math::::statistics::::t-test-mean data estmean eststdev confidence
Test whether the mean value of a sample is in accordance with
the estimated normal distribution with a certain level of confi-
dence. Returns 1 if the test succeeds or 0 if the mean is
unlikely to fit the given distribution.
data - List of raw data values (small sample)
estmean - Estimated mean of the distribution
eststdev - Estimated stdev of the distribution
confidence - Confidence level (0.95 or 0.99 for instance)
::::math::::statistics::::quantiles data confidence
Return the quantiles for a given set of data
data - List of raw data values
confidence - Confidence level (0.95 or 0.99 for instance)
::::math::::statistics::::quantiles limits counts confidence
Return the quantiles based on histogram information (alternative
to the call with two arguments)
limits - List of upper limits from histogram
counts - List of counts for for each interval in histogram
confidence - Confidence level (0.95 or 0.99 for instance)
::::math::::statistics::::autocorr data
Return the autocorrelation function as a list of values (assum-
ing equidistance between samples, about 1/2 of the number of raw
data)
The correlation is determined in such a way that the first value
is always 1 and all others are equal to or smaller than 1. The
number of values involved will diminish as the "time" (the index
in the list of returned values) increases
data - Raw data for which the autocorrelation must be determined
::::math::::statistics::::crosscorr data1 data2
Return the cross-correlation function as a list of values
(assuming equidistance between samples, about 1/2 of the number
of raw data)
The correlation is determined in such a way that the values can
never exceed 1 in magnitude. The number of values involved will
diminish as the "time" (the index in the list of returned val-
ues) increases.
data1 - First list of data
data2 - Second list of data
::::math::::statistics::::mean-histogram-limits mean stdev number
Determine reasonable limits based on mean and standard deviation
for a histogram
Convenience function - the result is suitable for the histogram
function.
mean - Mean of the data
stdev - Standard deviation
number - Number of limits to generate (defaults to 8)
::::math::::statistics::::minmax-histogram-limits min max number
Determine reasonable limits based on a minimum and maximum for a
histogram
Convenience function - the result is suitable for the histogram
function.
min - Expected minimum
max - Expected maximum
number - Number of limits to generate (defaults to 8)
STATISTICAL DISTRIBUTIONS
In the literature a large number of probability distributions can be
found. The statistics package supports:
]o The normal or Gaussian distribution
]o The uniform distribution - equal probability for all data within
a given interval
]o The exponential distribution - useful as a model for certain
extreme-value distributions.
]o PM - binomial, Poisson, chi-squared, student's T, F. In princi-
ple for each distribution one has procedures for:
]o The probability density (pdf-*)
]o The cumulative density (cdf-*)
]o Quantiles for the given distribution (quantiles-*)
]o Histograms for the given distribution (histogram-*)
]o List of random values with the given distribution (random-*) The
following procedures have been implemented:
::::math::::statistics::::pdf-normal mean stdev value
Return the probability of a given value for a normal distribu-
tion with given mean and standard deviation.
mean - Mean value of the distribution
stdev - Standard deviation of the distribution
value - Value for which the probability is required
::::math::::statistics::::pdf-exponential mean value
Return the probability of a given value for an exponential dis-
tribution with given mean.
mean - Mean value of the distribution
value - Value for which the probability is required
::::math::::statistics::::pdf-uniform xmin xmax value
Return the probability of a given value for a uniform distribu-
tion with given extremes.
xmin - Minimum value of the distribution
xmin - Maximum value of the distribution
value - Value for which the probability is required
::::math::::statistics::::cdf-normal mean stdev value
Return the cumulative probability of a given value for a normal
distribution with given mean and standard deviation, that is the
probability for values up to the given one.
mean - Mean value of the distribution
stdev - Standard deviation of the distribution
value - Value for which the probability is required
::::math::::statistics::::cdf-exponential mean value
Return the cumulative probability of a given value for an expo-
nential distribution with given mean.
mean - Mean value of the distribution
value - Value for which the probability is required
::::math::::statistics::::cdf-uniform xmin xmax value
Return the cumulative probability of a given value for a uniform
distribution with given extremes.
xmin - Minimum value of the distribution
xmin - Maximum value of the distribution
value - Value for which the probability is required
::::math::::statistics::::cdf-students-t degrees value
Return the cumulative probability of a given value for a Stu-
dent's t distribution with given number of degrees.
degrees - Number of degrees of freedom
value - Value for which the probability is required
::::math::::statistics::::random-normal mean stdev number
Return a list of "number" random values satisfying a normal dis-
tribution with given mean and standard deviation.
mean - Mean value of the distribution
stdev - Standard deviation of the distribution
number - Number of values to be returned
::::math::::statistics::::random-exponential mean number
Return a list of "number" random values satisfying an exponen-
tial distribution with given mean.
mean - Mean value of the distribution
number - Number of values to be returned
::::math::::statistics::::random-uniform xmin xmax value
Return a list of "number" random values satisfying a uniform
distribution with given extremes.
xmin - Minimum value of the distribution
xmin - Maximum value of the distribution
number - Number of values to be returned
::::math::::statistics::::histogram-uniform xmin xmax limits number
Return the expected histogram for a uniform distribution.
xmin - Minimum value of the distribution
xmax - Maximum value of the distribution
limits - Upper limits for the buckets in the histogram
number - Total number of "observations" in the histogram
TO DO: more function descriptions to be added
DATA MANIPULATION
The data manipulation procedures act on lists or lists of lists:
::::math::::statistics::::filter varname data expression
Return a list consisting of the data for which the logical
expression is true (this command works analogously to the com-
mand foreach).
varname - Name of the variable used in the expression
data - List of data
expression - Logical expression using the variable name
::::math::::statistics::::map varname data expression
Return a list consisting of the data that are transformed via
the expression.
varname - Name of the variable used in the expression
data - List of data
expression - Expression to be used to transform (map) the data
::::math::::statistics::::samplescount varname list expression
Return a list consisting of the counts of all data in the sub-
lists of the "list" argument for which the expression is true.
varname - Name of the variable used in the expression
data - List of sublists, each containing the data
expression - Logical expression to test the data (defaults to
"true").
::::math::::statistics::::subdivide
Routine PM - not implemented yet
PLOT PROCEDURES
The following simple plotting procedures are available:
::::math::::statistics::::plot-scale canvas xmin xmax ymin ymax
Set the scale for a plot in the given canvas. All plot routines
expect this function to be called first. There is no automatic
scaling provided.
canvas - Canvas widget to use
xmin - Minimum x value
xmax - Maximum x value
ymin - Minimum y value
ymax - Maximum y value
::::math::::statistics::::plot-xydata canvas xdata ydata tag
Create a simple XY plot in the given canvas - the data are shown
as a collection of dots. The tag can be used to manipulate the
appearance.
canvas - Canvas widget to use
xdata - Series of independent data
ydata - Series of dependent data
tag - Tag to give to the plotted data (defaults to xyplot)
::::math::::statistics::::plot-xyline canvas xdata ydata tag
Create a simple XY plot in the given canvas - the data are shown
as a line through the data points. The tag can be used to manip-
ulate the appearance.
canvas - Canvas widget to use
xdata - Series of independent data
ydata - Series of dependent data
tag - Tag to give to the plotted data (defaults to xyplot)
::::math::::statistics::::plot-tdata canvas tdata tag
Create a simple XY plot in the given canvas - the data are shown
as a collection of dots. The horizontal coordinate is equal to
the index. The tag can be used to manipulate the appearance.
This type of presentation is suitable for autocorrelation func-
tions for instance or for inspecting the time-dependent behav-
iour.
canvas - Canvas widget to use
tdata - Series of dependent data
tag - Tag to give to the plotted data (defaults to xyplot)
::::math::::statistics::::plot-tline canvas tdata tag
Create a simple XY plot in the given canvas - the data are shown
as a line. See plot-tdata for an explanation.
canvas - Canvas widget to use
tdata - Series of dependent data
tag - Tag to give to the plotted data (defaults to xyplot)
::::math::::statistics::::plot-histogram canvas counts limits tag
Create a simple histogram in the given canvas
canvas - Canvas widget to use
counts - Series of bucket counts
limits - Series of upper limits for the buckets
tag - Tag to give to the plotted data (defaults to xyplot)
THINGS TO DO
The following procedures are yet to be implemented:
]o F-test-stdev
]o interval-mean-stdev
]o histogram-normal
]o histogram-exponential
]o test-histogram
]o linear-model
]o linear-residuals
]o test-corr
]o quantiles-*
]o fourier-coeffs
]o fourier-residuals
]o onepar-function-fit
]o onepar-function-residuals
]o plot-linear-model
]o subdivide
EXAMPLES
The code below is a small example of how you can examine a set of data:
# Simple example:
# - Generate data (as a cheap way of getting some)
# - Perform statistical analysis to describe the data
#
package require math::statistics
#
# Two auxiliary procs
#
proc pause {time} {
set wait 0
after [expr {$time*1000}] {set ::wait 1}
vwait wait
}
proc print-histogram {counts limits} {
foreach count $counts limit $limits {
if { $limit != {} } {
puts [format "<%12.4g\t%d" $limit $count]
set prevlimit $limit
} else {
puts [format ">%12.4g\t%d" $prevlimit $count]
}
}
}
#
# Our source of arbitrary data
#
proc generateData { data1 data2 } {
upvar 1 $data1 data1
upvar 1 $data2 data2
set d1 0.0
set d2 0.0
for { set i 0 } { $i < 100 } { incr i } {
set d1 [expr {10.0-2.0*cos(2.0*3.1415926*$i/24.0)]3.5*rand()}]
set d2 [expr {0.7*$d2]0.3*$d1]0.7*rand()}]
lappend data1 $d1
lappend data2 $d2
}
return {}
}
#
# The analysis session
#
package require Tk
console show
canvas .plot1
canvas .plot2
pack .plot1 .plot2 -fill both -side top
generateData data1 data2
puts "Basic statistics:"
set b1 [::math::statistics::basic-stats $data1]
set b2 [::math::statistics::basic-stats $data2]
foreach label {mean min max number stdev var} v1 $b1 v2 $b2 {
puts "$label\t$v1\t$v2"
}
puts "Plot the data as function of \"time\" and against each other"
::math::statistics::plot-scale .plot1 0 100 0 20
::math::statistics::plot-scale .plot2 0 20 0 20
::math::statistics::plot-tline .plot1 $data1
::math::statistics::plot-tline .plot1 $data2
::math::statistics::plot-xydata .plot2 $data1 $data2
puts "Correlation coefficient:"
puts [::math::statistics::corr $data1 $data2]
pause 2
puts "Plot histograms"
::math::statistics::plot-scale .plot2 0 20 0 100
set limits [::math::statistics::minmax-histogram-limits 7 16]
set histogramdata [::math::statistics::histogram $limits $data1]
::math::statistics::plot-histogram .plot2 $histogramdata $limits
puts "First series:"
print-histogram $histogramdata $limits
pause 2
set limits [::math::statistics::minmax-histogram-limits 0 15 10]
set histogramdata [::math::statistics::histogram $limits $data2]
::math::statistics::plot-histogram .plot2 $histogramdata $limits d2
puts "Second series:"
print-histogram $histogramdata $limits
puts "Autocorrelation function:"
set autoc [::math::statistics::autocorr $data1]
puts [::math::statistics::map $autoc {[format "%.2f" $x]}]
puts "Cross-correlation function:"
set crossc [::math::statistics::crosscorr $data1 $data2]
puts [::math::statistics::map $crossc {[format "%.2f" $x]}]
::math::statistics::plot-scale .plot1 0 100 -1 4
::math::statistics::plot-tline .plot1 $autoc "autoc"
::math::statistics::plot-tline .plot1 $crossc "crossc"
puts "Quantiles: 0.1, 0.2, 0.5, 0.8, 0.9"
puts "First: [::math::statistics::quantiles $data1 {0.1 0.2 0.5 0.8 0.9}]"
puts "Second: [::math::statistics::quantiles $data2 {0.1 0.2 0.5 0.8 0.9}]"
If you run this example, then the following should be clear:
]o There is a strong correlation between two time series, as dis-
played by the raw data and especially by the correlation func-
tions.
]o Both time series show a significant periodic component
]o The histograms are not very useful in identifying the nature of
the time series - they do not show the periodic nature.
KEYWORDS
data analysis, mathematics, statistics
math 0.1.1 math::statistics(n)
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