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# math::statistics(n) 1 tcllib "Tcl Math Library"

## Name

math::statistics - Basic statistical functions and procedures

## Synopsis

- package require
**Tcl 8.4** - package require
**math::statistics 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::pstdev***data***::math::statistics::pvar***data***::math::statistics::median***data***::math::statistics::basic-stats***data***::math::statistics::histogram***limits**values*?weights?**::math::statistics::histogram-alt***limits**values*?weights?**::math::statistics::corr***data1**data2***::math::statistics::interval-mean-stdev***data**confidence***::math::statistics::t-test-mean***data**est_mean**est_stdev**alpha***::math::statistics::test-normal***data**significance***::math::statistics::lillieforsFit***data***::math::statistics::test-Duckworth***list1**list2**significance***::math::statistics::test-anova-F***alpha**args***::math::statistics::test-Tukey-range***alpha**args***::math::statistics::test-Dunnett***alpha**control**args***::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::linear-model***xdata**ydata**intercept***::math::statistics::linear-residuals***xdata**ydata**intercept***::math::statistics::test-2x2***n11**n21**n12**n22***::math::statistics::print-2x2***n11**n21**n12**n22***::math::statistics::control-xbar***data*?nsamples?**::math::statistics::control-Rchart***data*?nsamples?**::math::statistics::test-xbar***control**data***::math::statistics::test-Rchart***control**data***::math::statistics::test-Kruskal-Wallis***confidence**args***::math::statistics::analyse-Kruskal-Wallis***args***::math::statistics::group-rank***args***::math::statistics::test-Wilcoxon***sample_a**sample_b***::math::statistics::spearman-rank***sample_a**sample_b***::math::statistics::spearman-rank-extended***sample_a**sample_b***::math::statistics::kernel-density***data*opt*-option value*...**::math::statistics::bootstrap***data**sampleSize*?numberSamples?**::math::statistics::tstat***dof*?alpha?**::math::statistics::mv-wls***wt1**weights_and_values***::math::statistics::mv-ols***values***::math::statistics::pdf-normal***mean**stdev**value***::math::statistics::pdf-lognormal***mean**stdev**value***::math::statistics::pdf-exponential***mean**value***::math::statistics::pdf-uniform***xmin**xmax**value***::math::statistics::pdf-gamma***alpha**beta**value***::math::statistics::pdf-poisson***mu**k***::math::statistics::pdf-chisquare***df**value***::math::statistics::pdf-student-t***df**value***::math::statistics::pdf-gamma***a**b**value***::math::statistics::pdf-beta***a**b**value***::math::statistics::pdf-weibull***scale**shape**value***::math::statistics::pdf-gumbel***location**scale**value***::math::statistics::pdf-pareto***scale**shape**value***::math::statistics::pdf-cauchy***location**scale**value***::math::statistics::cdf-normal***mean**stdev**value***::math::statistics::cdf-lognormal***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::cdf-gamma***alpha**beta**value***::math::statistics::cdf-poisson***mu**k***::math::statistics::cdf-beta***a**b**value***::math::statistics::cdf-weibull***scale**shape**value***::math::statistics::cdf-gumbel***location**scale**value***::math::statistics::cdf-pareto***scale**shape**value***::math::statistics::cdf-cauchy***location**scale**value***::math::statistics::cdf-F***nf1**nf2**value***::math::statistics::empirical-distribution***values***::math::statistics::random-normal***mean**stdev**number***::math::statistics::random-lognormal***mean**stdev**number***::math::statistics::random-exponential***mean**number***::math::statistics::random-uniform***xmin**xmax**number***::math::statistics::random-gamma***alpha**beta**number***::math::statistics::random-poisson***mu**number***::math::statistics::random-chisquare***df**number***::math::statistics::random-student-t***df**number***::math::statistics::random-beta***a**b**number***::math::statistics::random-weibull***scale**shape**number***::math::statistics::random-gumbel***location**scale**number***::math::statistics::random-pareto***scale**shape**number***::math::statistics::random-cauchy***location**scale**number***::math::statistics::histogram-uniform***xmin**xmax**limits**number***::math::statistics::incompleteGamma***x**p*?tol?**::math::statistics::incompleteBeta***a**b**x*?tol?**::math::statistics::estimate-pareto***values***::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:

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.
*Note:* With version 1.0.1 a mistake in the procs *pdf-lognormal*,
*cdf-lognormal* and *random-lognormal* has been corrected. In
previous versions the argument for the standard deviation was actually
used as if it was the variance.

## GENERAL PROCEDURES

The general statistical procedures are:

**::math::statistics::mean***data*Determine the

*mean*value of the given list of data.- list
*data* - List of data

- list
**::math::statistics::min***data*Determine the

*minimum*value of the given list of data.- list
*data* - List of data

- list
**::math::statistics::max***data*Determine the

*maximum*value of the given list of data.- list
*data* - List of data

- list
**::math::statistics::number***data*Determine the

*number*of non-missing data in the given list- list
*data* - List of data

- list
**::math::statistics::stdev***data*Determine the

*sample standard deviation*of the data in the given list- list
*data* - List of data

- list
**::math::statistics::var***data*Determine the

*sample variance*of the data in the given list- list
*data* - List of data

- list
**::math::statistics::pstdev***data*Determine the

*population standard deviation*of the data in the given list- list
*data* - List of data

- list
**::math::statistics::pvar***data*Determine the

*population variance*of the data in the given list- list
*data* - List of data

- list
**::math::statistics::median***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
*data* - List of data

- list
**::math::statistics::basic-stats***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
*data* - List of data

- list
**::math::statistics::histogram***limits**values*?weights?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
*limits* - List of upper limits (in ascending order) for the intervals of the histogram.

- list
*values* - List of data

- list
*weights* - List of weights, one weight per value

- list
**::math::statistics::histogram-alt***limits**values*?weights?Alternative implementation of the histogram procedure: the open end of the intervals is at the lower bound instead of the upper bound.

- list
*limits* - List of upper limits (in ascending order) for the intervals of the histogram.

- list
*values* - List of data

- list
*weights* - List of weights, one weight per value

- list
**::math::statistics::corr***data1**data2*Determine the correlation coefficient between two sets of data.

- list
*data1* - First list of data

- list
*data2* - Second list of data

- list
**::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)

- list
*data* - List of raw data values (small sample)

- float
*confidence* - Confidence level (0.95 or 0.99 for instance)

- list
**::math::statistics::t-test-mean***data**est_mean**est_stdev**alpha*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
*data* - List of raw data values (small sample)

- float
*est_mean* - Estimated mean of the distribution

- float
*est_stdev* - Estimated stdev of the distribution

- float
*alpha* - Probability level (0.95 or 0.99 for instance)

- list
**::math::statistics::test-normal***data**significance*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
*data* - List of raw data values

- float
*significance* - 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.

- list
**::math::statistics::lillieforsFit***data*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
*data* - List of raw data values

- list
**::math::statistics::test-Duckworth***list1**list2**significance*Determine if two data sets have the same median according to the Tukey-Duckworth test. The procedure returns 0 if the medians are unequal, 1 if they are equal, -1 if the test can not be conducted (the smallest value must be in a different set than the greatest value). # # Arguments: # list1 Values in the first data set # list2 Values in the second data set # significance Significance level (either 0.05, 0.01 or 0.001) # # Returns: 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
*list1* - First list of data

- list
*list2* - Second list of data

- float
*significance* - Significance level (either 0.05, 0.01 or 0.001)

- list
**::math::statistics::test-anova-F***alpha**args*Determine if two or more groups with normally distributed data have the same means. The procedure returns 0 if the means are likely unequal, 1 if they are. This is a one-way ANOVA test. The groups may also be stored in a nested list: The procedure returns a list of the comparison results for each pair of groups. Each element of this list contains: the index of the first group and that of the second group, whether the means are likely to be different (1) or not (0) and the confidence interval the conclusion is based on. The groups may also be stored in a nested list:

test-anova-F 0.05 $A $B $C # # Or equivalently: # test-anova-F 0.05 [list $A $B $C]

- float
*alpha* - Significance level

- list
*args* - Two or more groups of data to be checked

- float
**::math::statistics::test-Tukey-range***alpha**args*Determine if two or more groups with normally distributed data have the same means, using Tukey's range test. It is complementary to the ANOVA test. The procedure returns a list of the comparison results for each pair of groups. Each element of this list contains: the index of the first group and that of the second group, whether the means are likely to be different (1) or not (0) and the confidence interval the conclusion is based on. The groups may also be stored in a nested list, just as with the ANOVA test.

- float
*alpha* - Significance level - either 0.05 or 0.01

- list
*args* - Two or more groups of data to be checked

- float
**::math::statistics::test-Dunnett***alpha**control**args*Determine if one or more groups with normally distributed data have the same means as the group of control data, using Dunnett's test. It is complementary to the ANOVA test. The procedure returns a list of the comparison results for each group with the control group. Each element of this list contains: whether the means are likely to be different (1) or not (0) and the confidence interval the conclusion is based on. The groups may also be stored in a nested list, just as with the ANOVA test.

Note: some care is required if there is only one group to compare the control with:

test-Dunnett-F 0.05 $control [list $A]

Otherwise the group A is split up into groups of one element - this is due to an ambiguity.

- float
*alpha* - Significance level - either 0.05 or 0.01

- list
*args* - One or more groups of data to be checked

- float
**::math::statistics::quantiles***data**confidence*Return the quantiles for a given set of data

- list
*data* - List of raw data values

- float
*confidence* - Confidence level (0.95 or 0.99 for instance) or a list of confidence levels.

- list
**::math::statistics::quantiles***limits**counts**confidence*Return the quantiles based on histogram information (alternative to the call with two arguments)

- list
*limits* - List of upper limits from histogram

- list
*counts* - List of counts for for each interval in histogram

- float
*confidence* - Confidence level (0.95 or 0.99 for instance) or a list of confidence levels.

- list
**::math::statistics::autocorr***data*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

- list
*data* - Raw data for which the autocorrelation must be determined

- list
**::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 values) increases.

- list
*data1* - First list of data

- list
*data2* - Second list of data

- list
**::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.

- float
*mean* - Mean of the data

- float
*stdev* - Standard deviation

- int
*number* - Number of limits to generate (defaults to 8)

- float
**::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.

- float
*min* - Expected minimum

- float
*max* - Expected maximum

- int
*number* - Number of limits to generate (defaults to 8)

- float
**::math::statistics::linear-model***xdata**ydata**intercept*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
*xdata* - List of independent data

- list
*ydata* - List of dependent data to be fitted

- boolean
*intercept* - (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

- list
**::math::statistics::linear-residuals***xdata**ydata**intercept*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
*xdata* - List of independent data

- list
*ydata* - List of dependent data to be fitted

- boolean
*intercept* - (Optional) compute the intercept (1, default) or fit to a line through the origin (0)

- list
**::math::statistics::test-2x2***n11**n21**n12**n22*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.

- int
*n11* - Number of outcomes with the first value from the first sample.

- int
*n21* - Number of outcomes with the first value from the second sample.

- int
*n12* - Number of outcomes with the second value from the first sample.

- int
*n22* - Number of outcomes with the second value from the second sample.

- int
**::math::statistics::print-2x2***n11**n21**n12**n22*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.

- int
*n11* - Number of outcomes with the first value from the first sample.

- int
*n21* - Number of outcomes with the first value from the second sample.

- int
*n12* - Number of outcomes with the second value from the first sample.

- int
*n22* - Number of outcomes with the second value from the second sample.

- int
**::math::statistics::control-xbar***data*?nsamples?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
*data* - List of observed data

- int
*nsamples* - Number of data per subsample

- list
**::math::statistics::control-Rchart***data*?nsamples?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
*data* - List of observed data

- int
*nsamples* - Number of data per subsample

- list
**::math::statistics::test-xbar***control**data*Determine if the data exceed the control limits for the xbar chart.

Returns a list of subsamples (their indices) that indeed violate the limits.

- list
*control* - Control limits as returned by the "control-xbar" procedure

- list
*data* - List of observed data

- list
**::math::statistics::test-Rchart***control**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.

- list
*control* - Control limits as returned by the "control-Rchart" procedure

- list
*data* - List of observed data

- list
**::math::statistics::test-Kruskal-Wallis***confidence**args*Check if the population medians of two or more groups are equal with a given confidence level, using the Kruskal-Wallis test.

- float
*confidence* - Confidence level to be used (0-1)

- list
*args* - Two or more lists of data

- float
**::math::statistics::analyse-Kruskal-Wallis***args*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.

- list
*args* - Two or more lists of data

- list
**::math::statistics::group-rank***args*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.

- list
*args* - Two or more lists of data

- list
**::math::statistics::test-Wilcoxon***sample_a**sample_b*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
*sample_a* - List of data comprising the first sample

- list
*sample_b* - List of data comprising the second sample

- list
**::math::statistics::spearman-rank***sample_a**sample_b*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.

- list
*sample_a* - First list of data

- list
*sample_b* - Second list of data

- list
**::math::statistics::spearman-rank-extended***sample_a**sample_b*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.

- list
*sample_a* - First list of data

- list
*sample_b* - Second list of data

- list
**::math::statistics::kernel-density***data*opt*-option value*...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.

- list
*data* - The data to be examined

- list
*args* - Option-value pairs:

**-weights***weights*Per data point the weight (default: 1 for all data)

**-bandwidth***value*Bandwidth to be used for the estimation (default: determined from standard deviation)

**-number***value*Number of bins to be returned (default: 100)

**-interval***{begin end}*Begin and end of the interval for which the density is returned (default: mean +/- 3*standard deviation)

**-kernel***function*Kernel to be used (One of: gaussian, cosine, epanechnikov, uniform, triangular, biweight, logistic; default: gaussian)

- list
**::math::statistics::bootstrap***data**sampleSize*?numberSamples?Create a subsample or subsamples from a given list of data. The data in the samples are chosen from this list - multiples may occur. If there is only one subsample, the sample itself is returned (as a list of "sampleSize" values), otherwise a list of samples is returned.

- list
*data* List of values to chose from

- int
*sampleSize* Number of values per sample

- int
*numberSamples* Number of samples (default: 1)

- list

## MULTIVARIATE LINEAR REGRESSION

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*

**::math::statistics::tstat***dof*?alpha?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.

- int
*dof* Number of degrees of freedom

- float
*alpha* Confidence level of the t-distribution. Defaults to 0.05.

- int
**::math::statistics::mv-wls***wt1**weights_and_values*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:

- list
*weights_and_values* 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.

**::math::statistics::mv-ols***values*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 -- \[$lb to $ub\]" }

## STATISTICAL DISTRIBUTIONS

In the literature a large number of probability distributions can be found. The statistics package supports:

The normal or Gaussian distribution as well as the log-normal 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 beta distribution

The chi-square distribution

The student's T distribution

The Poisson distribution

The Pareto distribution

The Gumbel distribution

The Weibull distribution

The Cauchy distribution

The F distribution (only the cumulative density function)

PM - binomial.

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:

**::math::statistics::pdf-normal***mean**stdev**value*Return the probability of a given value for a normal distribution with given mean and standard deviation.

- float
*mean* - Mean value of the distribution

- float
*stdev* - Standard deviation of the distribution

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-lognormal***mean**stdev**value*Return the probability of a given value for a log-normal distribution with given mean and standard deviation.

- float
*mean* - Mean value of the distribution

- float
*stdev* - Standard deviation of the distribution

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-exponential***mean**value*Return the probability of a given value for an exponential distribution with given mean.

- float
*mean* - Mean value of the distribution

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-uniform***xmin**xmax**value*Return the probability of a given value for a uniform distribution with given extremes.

- float
*xmin* - Minimum value of the distribution

- float
*xmin* - Maximum value of the distribution

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-gamma***alpha**beta**value*Return the probability of a given value for a Gamma distribution with given shape and rate parameters

- float
*alpha* - Shape parameter

- float
*beta* - Rate parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-poisson***mu**k*Return the probability of a given number of occurrences in the same interval (k) for a Poisson distribution with given mean (mu)

- float
*mu* - Mean number of occurrences

- int
*k* - Number of occurences

- float
**::math::statistics::pdf-chisquare***df**value*Return the probability of a given value for a chi square distribution with given degrees of freedom

- float
*df* - Degrees of freedom

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-student-t***df**value*Return the probability of a given value for a Student's t distribution with given degrees of freedom

- float
*df* - Degrees of freedom

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-gamma***a**b**value*Return the probability of a given value for a Gamma distribution with given shape and rate parameters

- float
*a* - Shape parameter

- float
*b* - Rate parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-beta***a**b**value*Return the probability of a given value for a Beta distribution with given shape parameters

- float
*a* - First shape parameter

- float
*b* - Second shape parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-weibull***scale**shape**value*Return the probability of a given value for a Weibull distribution with given scale and shape parameters

- float
*location* - Scale parameter

- float
*scale* - Shape parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-gumbel***location**scale**value*Return the probability of a given value for a Gumbel distribution with given location and shape parameters

- float
*location* - Location parameter

- float
*scale* - Shape parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-pareto***scale**shape**value*Return the probability of a given value for a Pareto distribution with given scale and shape parameters

- float
*scale* - Scale parameter

- float
*shape* - Shape parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::pdf-cauchy***location**scale**value*Return the probability of a given value for a Cauchy distribution with given location and shape parameters. Note that the Cauchy distribution has no finite higher-order moments.

- float
*location* - Location parameter

- float
*scale* - Shape parameter

- float
*value* - Value for which the probability is required

- float
**::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.

- float
*mean* - Mean value of the distribution

- float
*stdev* - Standard deviation of the distribution

- float
*value* - Value for which the probability is required

- float
**::math::statistics::cdf-lognormal***mean**stdev**value*Return the cumulative probability of a given value for a log-normal distribution with given mean and standard deviation, that is the probability for values up to the given one.

- float
*mean* - Mean value of the distribution

- float
*stdev* - Standard deviation of the distribution

- float
*value* - Value for which the probability is required

- float
**::math::statistics::cdf-exponential***mean**value*Return the cumulative probability of a given value for an exponential distribution with given mean.

- float
*mean* - Mean value of the distribution

- float
*value* - Value for which the probability is required

- float
**::math::statistics::cdf-uniform***xmin**xmax**value*Return the cumulative probability of a given value for a uniform distribution with given extremes.

- float
*xmin* - Minimum value of the distribution

- float
*xmin* - Maximum value of the distribution

- float
*value* - Value for which the probability is required

- float
**::math::statistics::cdf-students-t***degrees**value*Return the cumulative probability of a given value for a Student's t distribution with given number of degrees.

- int
*degrees* - Number of degrees of freedom

- float
*value* - Value for which the probability is required

- int
**::math::statistics::cdf-gamma***alpha**beta**value*Return the cumulative probability of a given value for a Gamma distribution with given shape and rate parameters.

- float
*alpha* - Shape parameter

- float
*beta* - Rate parameter

- float
*value* - Value for which the cumulative probability is required

- float
**::math::statistics::cdf-poisson***mu**k*Return the cumulative probability of a given number of occurrences in the same interval (k) for a Poisson distribution with given mean (mu).

- float
*mu* - Mean number of occurrences

- int
*k* - Number of occurences

- float
**::math::statistics::cdf-beta***a**b**value*Return the cumulative probability of a given value for a Beta distribution with given shape parameters

- float
*a* - First shape parameter

- float
*b* - Second shape parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::cdf-weibull***scale**shape**value*Return the cumulative probability of a given value for a Weibull distribution with given scale and shape parameters.

- float
*scale* - Scale parameter

- float
*shape* - Shape parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::cdf-gumbel***location**scale**value*Return the cumulative probability of a given value for a Gumbel distribution with given location and scale parameters.

- float
*location* - Location parameter

- float
*scale* - Scale parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::cdf-pareto***scale**shape**value*Return the cumulative probability of a given value for a Pareto distribution with given scale and shape parameters

- float
*scale* - Scale parameter

- float
*shape* - Shape parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::cdf-cauchy***location**scale**value*Return the cumulative probability of a given value for a Cauchy distribution with given location and scale parameters.

- float
*location* - Location parameter

- float
*scale* - Scale parameter

- float
*value* - Value for which the probability is required

- float
**::math::statistics::cdf-F***nf1**nf2**value*Return the cumulative probability of a given value for an F distribution with nf1 and nf2 degrees of freedom.

- float
*nf1* - Degrees of freedom for the numerator

- float
*nf2* - Degrees of freedom for the denominator

- float
*value* - Value for which the probability is required

- float
**::math::statistics::empirical-distribution***values*Return a list of values and their empirical probability. The values are sorted in increasing order. (The implementation follows the description at the corresponding Wikipedia page)

- list
*values* - List of data to be examined

- list
**::math::statistics::random-normal***mean**stdev**number*Return a list of "number" random values satisfying a normal distribution with given mean and standard deviation.

- float
*mean* - Mean value of the distribution

- float
*stdev* - Standard deviation of the distribution

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-lognormal***mean**stdev**number*Return a list of "number" random values satisfying a log-normal distribution with given mean and standard deviation.

- float
*mean* - Mean value of the distribution

- float
*stdev* - Standard deviation of the distribution

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-exponential***mean**number*Return a list of "number" random values satisfying an exponential distribution with given mean.

- float
*mean* - Mean value of the distribution

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-uniform***xmin**xmax**number*Return a list of "number" random values satisfying a uniform distribution with given extremes.

- float
*xmin* - Minimum value of the distribution

- float
*xmax* - Maximum value of the distribution

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-gamma***alpha**beta**number*Return a list of "number" random values satisfying a Gamma distribution with given shape and rate parameters.

- float
*alpha* - Shape parameter

- float
*beta* - Rate parameter

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-poisson***mu**number*Return a list of "number" random values satisfying a Poisson distribution with given mean.

- float
*mu* - Mean of the distribution

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-chisquare***df**number*Return a list of "number" random values satisfying a chi square distribution with given degrees of freedom.

- float
*df* - Degrees of freedom

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-student-t***df**number*Return a list of "number" random values satisfying a Student's t distribution with given degrees of freedom.

- float
*df* - Degrees of freedom

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-beta***a**b**number*Return a list of "number" random values satisfying a Beta distribution with given shape parameters.

- float
*a* - First shape parameter

- float
*b* - Second shape parameter

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-weibull***scale**shape**number*Return a list of "number" random values satisfying a Weibull distribution with given scale and shape parameters.

- float
*scale* - Scale parameter

- float
*shape* - Shape parameter

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-gumbel***location**scale**number*Return a list of "number" random values satisfying a Gumbel distribution with given location and scale parameters.

- float
*location* - Location parameter

- float
*scale* - Scale parameter

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-pareto***scale**shape**number*Return a list of "number" random values satisfying a Pareto distribution with given scale and shape parameters.

- float
*scale* - Scale parameter

- float
*shape* - Shape parameter

- int
*number* - Number of values to be returned

- float
**::math::statistics::random-cauchy***location**scale**number*Return a list of "number" random values satisfying a Cauchy distribution with given location and scale parameters.

- float
*location* - Location parameter

- float
*scale* - Scale parameter

- int
*number* - Number of values to be returned

- float
**::math::statistics::histogram-uniform***xmin**xmax**limits**number*Return the expected histogram for a uniform distribution.

- float
*xmin* - Minimum value of the distribution

- float
*xmax* - Maximum value of the distribution

- list
*limits* - Upper limits for the buckets in the histogram

- int
*number* - Total number of "observations" in the histogram

- float
**::math::statistics::incompleteGamma***x**p*?tol?Evaluate the incomplete Gamma integral

1 / x p-1 P(p,x) = -------- | dt exp(-t) * t Gamma(p) / 0

- float
*x* - Value of x (limit of the integral)

- float
*p* - Value of p in the integrand

- float
*tol* - Required tolerance (default: 1.0e-9)

- float
**::math::statistics::incompleteBeta***a**b**x*?tol?Evaluate the incomplete Beta integral

- float
*a* - First shape parameter

- float
*b* - Second shape parameter

- float
*x* - Value of x (limit of the integral)

- float
*tol* - Required tolerance (default: 1.0e-9)

- float
**::math::statistics::estimate-pareto***values*Estimate the parameters for the Pareto distribution that comes closest to the given values. Returns the estimated scale and shape parameters, as well as the standard error for the shape parameter.

- list
*values* - List of values, assumed to be distributed according to a Pareto distribution

- list

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 command

**foreach**).- string
*varname* - Name of the variable used in the expression

- list
*data* - List of data

- string
*expression* - Logical expression using the variable name

- string
**::math::statistics::map***varname**data**expression*Return a list consisting of the data that are transformed via the expression.

- string
*varname* - Name of the variable used in the expression

- list
*data* - List of data

- string
*expression* - Expression to be used to transform (map) the data

- string
**::math::statistics::samplescount***varname**list**expression*Return a list consisting of the

*counts*of all data in the sublists of the "list" argument for which the expression is true.- string
*varname* - Name of the variable used in the expression

- list
*data* - List of sublists, each containing the data

- string
*expression* - Logical expression to test the data (defaults to "true").

- string
**::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.

- widget
*canvas* - Canvas widget to use

- float
*xmin* - Minimum x value

- float
*xmax* - Maximum x value

- float
*ymin* - Minimum y value

- float
*ymax* - Maximum y value

- widget
**::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.

- widget
*canvas* - Canvas widget to use

- float
*xdata* - Series of independent data

- float
*ydata* - Series of dependent data

- string
*tag* - Tag to give to the plotted data (defaults to xyplot)

- widget
**::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 manipulate the appearance.

- widget
*canvas* - Canvas widget to use

- list
*xdata* - Series of independent data

- list
*ydata* - Series of dependent data

- string
*tag* - Tag to give to the plotted data (defaults to xyplot)

- widget
**::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 functions for instance or for inspecting the time-dependent behaviour.

- widget
*canvas* - Canvas widget to use

- list
*tdata* - Series of dependent data

- string
*tag* - Tag to give to the plotted data (defaults to xyplot)

- widget
**::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.

- widget
*canvas* - Canvas widget to use

- list
*tdata* - Series of dependent data

- string
*tag* - Tag to give to the plotted data (defaults to xyplot)

- widget
**::math::statistics::plot-histogram***canvas**counts**limits**tag*Create a simple histogram in the given canvas

- widget
*canvas* - Canvas widget to use

- list
*counts* - Series of bucket counts

- list
*limits* - Series of upper limits for the buckets

- string
*tag* - Tag to give to the plotted data (defaults to xyplot)

- widget

## THINGS TO DO

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

## 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 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.

## Bugs, Ideas, Feedback

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.

When proposing code changes, please provide *unified diffs*,
i.e the output of **diff -u**.

Note further that *attachments* are strongly preferred over
inlined patches. Attachments can be made by going to the **Edit**
form of the ticket immediately after its creation, and then using the
left-most button in the secondary navigation bar.

## Category

Mathematics