math::statistics - Basic statistical functions and procedures
The math::statistics package contains functions and procedures for basic statistical data analysis, such as:
Descriptive statistical parameters (mean, minimum, maximum, standard deviation)
Estimates of the distribution in the form of histograms and quantiles
Basic testing of hypotheses
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 packages.
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 procedures, procedures that deal with specific statistical distributions, list procedures to select or transform data and simple plotting procedures (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.
The general statistical procedures are:
Determine the mean value of the given list of data.
- List of data
Determine the minimum value of the given list of data.
- List of data
Determine the maximum value of the given list of data.
- List of data
Determine the number of non-missing data in the given list
- List of data
Determine the sample standard deviation of the data in the given list
- List of data
Determine the sample variance of the data in the given list
- List of data
Determine the population standard deviation of the data in the given list
- List of data
Determine the population variance of the data in the given list
- List of data
Determine the median of the data in the given list (Note that this requires sorting the data, which may be a costly operation)
- List of data
Determine a list of all the descriptive parameters: mean, minimum, maximum, number of data, sample standard deviation, sample variance, population standard deviation and population variance.
(This routine is called whenever either or all of the basic statistical parameters are required. Hence all calculations are done and the relevant values are returned.)
- List of data
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.)
Optionally, you can use weights to influence the histogram.
- List of upper limits (in ascending order) for the intervals of the histogram.
- List of data
- List of weights, one weight per value
Alternative implementation of the histogram procedure: the open end of the intervals is at the lower bound instead of the upper bound.
- List of upper limits (in ascending order) for the intervals of the histogram.
- List of data
- List of weights, one weight per value
Determine the correlation coefficient between two sets of data.
- First list of data
- Second list of data
Return the interval containing the mean value and one containing the standard deviation with a certain level of confidence (assuming a normal distribution)
- List of raw data values (small sample)
- Confidence level (0.95 or 0.99 for instance)
Test whether the mean value of a sample is in accordance with the estimated normal distribution with a certain probability. Returns 1 if the test succeeds or 0 if the mean is unlikely to fit the given distribution.
- List of raw data values (small sample)
- Estimated mean of the distribution
- Estimated stdev of the distribution
- Probability level (0.95 or 0.99 for instance)
Test whether the given data follow a normal distribution with a certain level of significance. Returns 1 if the data are normally distributed within the level of significance, returns 0 if not. The underlying test is the Lilliefors test. Smaller values of the significance mean a stricter testing.
- List of raw data values
- Significance level (one of 0.01, 0.05, 0.10, 0.15 or 0.20). For compatibility reasons the values "1-significance", 0.80, 0.85, 0.90, 0.95 or 0.99 are also accepted.
Compatibility issue: the original implementation and documentation used the term "confidence" and used a value 1-significance (see ticket 2812473fff). This has been corrected as of version 0.9.3.
Returns the goodness of fit to a normal distribution according to Lilliefors. The higher the number, the more likely the data are indeed normally distributed. The test requires at least five data points.
- List of raw data values
Return the quantiles for a given set of data
- List of raw data values
- Confidence level (0.95 or 0.99 for instance) or a list of confidence levels.
Return the quantiles based on histogram information (alternative to the call with two arguments)
- List of upper limits from histogram
- List of counts for for each interval in histogram
- Confidence level (0.95 or 0.99 for instance) or a list of confidence levels.
Return the autocorrelation 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 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
- Raw data for which the autocorrelation must be determined
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 values) increases.
- First list of data
- Second list of data
Determine reasonable limits based on mean and standard deviation for a histogram Convenience function - the result is suitable for the histogram function.
- Mean of the data
- Standard deviation
- Number of limits to generate (defaults to 8)
Determine reasonable limits based on a minimum and maximum for a histogram
Convenience function - the result is suitable for the histogram function.
- Expected minimum
- Expected maximum
- Number of limits to generate (defaults to 8)
Determine the coefficients for a linear regression between two series of data (the model: Y = A + B*X). Returns a list of parameters describing the fit
- List of independent data
- List of dependent data to be fitted
- (Optional) compute the intercept (1, default) or fit to a line through the origin (0)
The result consists of the following list:
(Estimate of) Intercept A
(Estimate of) Slope B
Standard deviation of Y relative to fit
Correlation coefficient R2
Number of degrees of freedom df
Standard error of the intercept A
Significance level of A
Standard error of the slope B
Significance level of B
Determine the difference between actual data and predicted from the linear model.
Returns a list of the differences between the actual data and the predicted values.
- List of independent data
- List of dependent data to be fitted
- (Optional) compute the intercept (1, default) or fit to a line through the origin (0)
Determine if two set of samples, each from a binomial distribution, differ significantly or not (implying a different parameter).
Returns the "chi-square" value, which can be used to the determine the significance.
- Number of outcomes with the first value from the first sample.
- Number of outcomes with the first value from the second sample.
- Number of outcomes with the second value from the first sample.
- Number of outcomes with the second value from the second sample.
Determine if two set of samples, each from a binomial distribution, differ significantly or not (implying a different parameter).
Returns a short report, useful in an interactive session.
- Number of outcomes with the first value from the first sample.
- Number of outcomes with the first value from the second sample.
- Number of outcomes with the second value from the first sample.
- Number of outcomes with the second value from the second sample.
Determine the control limits for an xbar chart. The number of data in each subsample defaults to 4. At least 20 subsamples are required.
Returns the mean, the lower limit, the upper limit and the number of data per subsample.
- List of observed data
- Number of data per subsample
Determine the control limits for an R chart. The number of data in each subsample (nsamples) defaults to 4. At least 20 subsamples are required.
Returns the mean range, the lower limit, the upper limit and the number of data per subsample.
- List of observed data
- Number of data per subsample
Determine if the data exceed the control limits for the xbar chart.
Returns a list of subsamples (their indices) that indeed violate the limits.
- Control limits as returned by the "control-xbar" procedure
- List of observed data
Determine if the data exceed the control limits for the R chart.
Returns a list of subsamples (their indices) that indeed violate the limits.
- Control limits as returned by the "control-Rchart" procedure
- List of observed data
Check if the population medians of two or more groups are equal with a given confidence level, using the Kruskal-Wallis test.
- Confidence level to be used (0-1)
- Two or more lists of data
Compute the statistical parameters for the Kruskal-Wallis test. Returns the Kruskal-Wallis statistic and the probability that that value would occur assuming the medians of the populations are equal.
- Two or more lists of data
Rank the groups of data with respect to the complete set. Returns a list consisting of the group ID, the value and the rank (possibly a rational number, in case of ties) for each data item.
- Two or more lists of data
Compute the Wilcoxon test statistic to determine if two samples have the same median or not. (The statistic can be regarded as standard normal, if the sample sizes are both larger than 10. Returns the value of this statistic.
- List of data comprising the first sample
- List of data comprising the second sample
Return the Spearman rank correlation as an alternative to the ordinary (Pearson's) correlation coefficient. The two samples should have the same number of data.
- First list of data
- Second list of data
Return the Spearman rank correlation as an alternative to the ordinary (Pearson's) correlation coefficient as well as additional data. The two samples should have the same number of data. The procedure returns the correlation coefficient, the number of data pairs used and the z-score, an approximately standard normal statistic, indicating the significance of the correlation.
- First list of data
- Second list of data
] Return the density function based on kernel density estimation. The procedure is controlled by a small set of options, each of which is given a reasonable default.
The return value consists of three lists: the centres of the bins, the associated probability density and a list of computational parameters (begin and end of the interval, mean and standard deviation and the used bandwidth). The computational parameters can be used for further analysis.
- The data to be examined
- Option-value pairs:
Per data point the weight (default: 1 for all data)
Bandwidth to be used for the estimation (default: determined from standard deviation)
Number of bins to be returned (default: 100)
Begin and end of the interval for which the density is returned (default: mean +/- 3*standard deviation)
Kernel to be used (One of: gaussian, cosine, epanechnikov, uniform, triangular, biweight, logistic; default: gaussian)
Besides the linear regression with a single independent variable, the statistics package provides two procedures for doing ordinary least squares (OLS) and weighted least squares (WLS) linear regression with several variables. They were written by Eric Kemp-Benedict.
In addition to these two, it provides a procedure (tstat) for calculating the value of the t-statistic for the specified number of degrees of freedom that is required to demonstrate a given level of significance.
Note: These procedures depend on the math::linearalgebra package.
Description of the procedures
Returns the value of the t-distribution t* satisfying
P(t*) = 1 - alpha/2
P(-t*) = alpha/2
for the number of degrees of freedom dof.
Given a sample of normally-distributed data x, with an estimate xbar for the mean and sbar for the standard deviation, the alpha confidence interval for the estimate of the mean can be calculated as
( xbar - t* sbar , xbar + t* sbar)
The return values from this procedure can be compared to an estimated t-statistic to determine whether the estimated value of a parameter is significantly different from zero at the given confidence level.
Number of degrees of freedom
Confidence level of the t-distribution. Defaults to 0.05.
Carries out a weighted least squares linear regression for the data points provided, with weights assigned to each point.
The linear model is of the form
y = b0 + b1 * x1 + b2 * x2 ... + bN * xN + error
and each point satisfies
yi = b0 + b1 * xi1 + b2 * xi2 + ... + bN * xiN + Residual_i
The procedure returns a list with the following elements:
The r-squared statistic
The adjusted r-squared statistic
A list containing the estimated coefficients b1, ... bN, b0 (The constant b0 comes last in the list.)
A list containing the standard errors of the coefficients
A list containing the 95% confidence bounds of the coefficients, with each set of bounds returned as a list with two values
Arguments:
A list consisting of: the weight for the first observation, the data for the first observation (as a sublist), the weight for the second observation (as a sublist) and so on. The sublists of data are organised as lists of the value of the dependent variable y and the independent variables x1, x2 to xN.
Carries out an ordinary least squares linear regression for the data points provided.
This procedure simply calls ::mvlinreg::wls with the weights set to 1.0, and returns the same information.
Example of the use:
# Store the value of the unicode value for the "+/-" character
set pm "\u00B1"
# Provide some data
set data {{ -.67 14.18 60.03 -7.5 }
{ 36.97 15.52 34.24 14.61 }
{-29.57 21.85 83.36 -7. }
{-16.9 11.79 51.67 -6.56 }
{ 14.09 16.24 36.97 -12.84}
{ 31.52 20.93 45.99 -25.4 }
{ 24.05 20.69 50.27 17.27}
{ 22.23 16.91 45.07 -4.3 }
{ 40.79 20.49 38.92 -.73 }
{-10.35 17.24 58.77 18.78}}
# Call the ols routine
set results [::math::statistics::mv-ols $data]
# Pretty-print the results
puts "R-squared: [lindex $results 0]"
puts "Adj R-squared: [lindex $results 1]"
puts "Coefficients $pm s.e. -- \[95% confidence interval\]:"
foreach val [lindex $results 2] se [lindex $results 3] bounds [lindex $results 4] {
set lb [lindex $bounds 0]
set ub [lindex $bounds 1]
puts " $val $pm $se -- \[$lb to $ub\]"
}
In the literature a large number of probability distributions can be found. The statistics package supports:
The normal or Gaussian distribution
The uniform distribution - equal probability for all data within a given interval
The exponential distribution - useful as a model for certain extreme-value distributions.
The gamma distribution - based on the incomplete Gamma integral
The chi-square distribution
The student's T distribution
The Poisson distribution
PM - binomial,F.
In principle for each distribution one has procedures for:
The probability density (pdf-*)
The cumulative density (cdf-*)
Quantiles for the given distribution (quantiles-*)
Histograms for the given distribution (histogram-*)
List of random values with the given distribution (random-*)
The following procedures have been implemented:
Return the probability of a given value for a normal distribution with given mean and standard deviation.
- Mean value of the distribution
- Standard deviation of the distribution
- Value for which the probability is required
Return the probability of a given value for an exponential distribution with given mean.
- Mean value of the distribution
- Value for which the probability is required
Return the probability of a given value for a uniform distribution with given extremes.
- Minimum value of the distribution
- Maximum value of the distribution
- Value for which the probability is required
Return the probability of a given value for a Gamma distribution with given shape and rate parameters
- Shape parameter
- Rate parameter
- Value for which the probability is required
Return the probability of a given number of occurrences in the same interval (k) for a Poisson distribution with given mean (mu)
- Mean number of occurrences
- Number of occurences
Return the probability of a given value for a chi square distribution with given degrees of freedom
- Degrees of freedom
- Value for which the probability is required
Return the probability of a given value for a Student's t distribution with given degrees of freedom
- Degrees of freedom
- Value for which the probability is required
Return the probability of a given value for a Beta distribution with given shape parameters
- First shape parameter
- First shape parameter
- Value for which the probability is required
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 value of the distribution
- Standard deviation of the distribution
- Value for which the probability is required
Return the cumulative probability of a given value for an exponential distribution with given mean.
- Mean value of the distribution
- Value for which the probability is required
Return the cumulative probability of a given value for a uniform distribution with given extremes.
- Minimum value of the distribution
- Maximum value of the distribution
- Value for which the probability is required
Return the cumulative probability of a given value for a Student's t distribution with given number of degrees.
- Number of degrees of freedom
- Value for which the probability is required
Return the cumulative probability of a given value for a Gamma distribution with given shape and rate parameters
- Shape parameter
- Rate parameter
- Value for which the cumulative probability is required
Return the cumulative probability of a given number of occurrences in the same interval (k) for a Poisson distribution with given mean (mu)
- Mean number of occurrences
- Number of occurences
Return the cumulative probability of a given value for a Beta distribution with given shape parameters
- First shape parameter
- Second shape parameter
- Value for which the probability is required
Return a list of "number" random values satisfying a normal distribution with given mean and standard deviation.
- Mean value of the distribution
- Standard deviation of the distribution
- Number of values to be returned
Return a list of "number" random values satisfying an exponential distribution with given mean.
- Mean value of the distribution
- Number of values to be returned
Return a list of "number" random values satisfying a uniform distribution with given extremes.
- Minimum value of the distribution
- Maximum value of the distribution
- Number of values to be returned
Return a list of "number" random values satisfying a Gamma distribution with given shape and rate parameters
- Shape parameter
- Rate parameter
- Number of values to be returned
Return a list of "number" random values satisfying a Poisson distribution with given mean
- Mean of the distribution
- Number of values to be returned
Return a list of "number" random values satisfying a chi square distribution with given degrees of freedom
- Degrees of freedom
- Number of values to be returned
Return a list of "number" random values satisfying a Student's t distribution with given degrees of freedom
- Degrees of freedom
- Number of values to be returned
Return a list of "number" random values satisfying a Beta distribution with given shape parameters
- First shape parameter
- Second shape parameter
- Number of values to be returned
Return the expected histogram for a uniform distribution.
- Minimum value of the distribution
- Maximum value of the distribution
- Upper limits for the buckets in the histogram
- Total number of "observations" in the histogram
Evaluate the incomplete Gamma integral
1 / x p-1
P(p,x) = -------- | dt exp(-t) * t
Gamma(p) / 0
- Value of x (limit of the integral)
- Value of p in the integrand
- Required tolerance (default: 1.0e-9)
Evaluate the incomplete Beta integral
- First shape parameter
- Second shape parameter
- Value of x (limit of the integral)
- Required tolerance (default: 1.0e-9)
TO DO: more function descriptions to be added
The data manipulation procedures act on lists or lists of lists:
Return a list consisting of the data for which the logical expression is true (this command works analogously to the command foreach).
- Name of the variable used in the expression
- List of data
- Logical expression using the variable name
Return a list consisting of the data that are transformed via the expression.
- Name of the variable used in the expression
- List of data
- Expression to be used to transform (map) the data
Return a list consisting of the counts of all data in the sublists of the "list" argument for which the expression is true.
- Name of the variable used in the expression
- List of sublists, each containing the data
- Logical expression to test the data (defaults to "true").
Routine PM - not implemented yet
The following simple plotting procedures are available:
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 widget to use
- Minimum x value
- Maximum x value
- Minimum y value
- Maximum y value
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 widget to use
- Series of independent data
- Series of dependent data
- Tag to give to the plotted data (defaults to xyplot)
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 manipulate the appearance.
- Canvas widget to use
- Series of independent data
- Series of dependent data
- Tag to give to the plotted data (defaults to xyplot)
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 functions for instance or for inspecting the time-dependent behaviour.
- Canvas widget to use
- Series of dependent data
- Tag to give to the plotted data (defaults to xyplot)
Create a simple XY plot in the given canvas - the data are shown as a line. See plot-tdata for an explanation.
- Canvas widget to use
- Series of dependent data
- Tag to give to the plotted data (defaults to xyplot)
Create a simple histogram in the given canvas
- Canvas widget to use
- Series of bucket counts
- Series of upper limits for the buckets
- Tag to give to the plotted data (defaults to xyplot)
The following procedures are yet to be implemented:
F-test-stdev
interval-mean-stdev
histogram-normal
histogram-exponential
test-histogram
test-corr
quantiles-*
fourier-coeffs
fourier-residuals
onepar-function-fit
onepar-function-residuals
plot-linear-model
subdivide
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 prev_limit $limit
} else {
puts [format ">%12.4g\t%d" $prev_limit $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"
.plot2 delete all
::math::statistics::plot-scale .plot2 0 20 0 100
set limits [::math::statistics::minmax-histogram-limits 7 16]
set histogram_data [::math::statistics::histogram $limits $data1]
::math::statistics::plot-histogram .plot2 $histogram_data $limits
puts "First series:"
print-histogram $histogram_data $limits
pause 2
set limits [::math::statistics::minmax-histogram-limits 0 15 10]
set histogram_data [::math::statistics::histogram $limits $data2]
::math::statistics::plot-histogram .plot2 $histogram_data $limits d2
.plot2 itemconfigure d2 -fill red
puts "Second series:"
print-histogram $histogram_data $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"
.plot1 itemconfigure autoc -fill green
.plot1 itemconfigure crossc -fill yellow
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:
There is a strong correlation between two time series, as displayed by the raw data and especially by the correlation functions.
Both time series show a significant periodic component
The histograms are not very useful in identifying the nature of the time series - they do not show the periodic nature.
This document, and the package it describes, will undoubtedly contain bugs and other problems. Please report such in the category math :: statistics of the Tcllib Trackers. Please also report any ideas for enhancements you may have for either package and/or documentation.
Mathematics