statistics man page
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::basicstats data
::math::statistics::histogram limits values ?weights?
::math::statistics::histogramalt limits values ?weights?
::math::statistics::corr data1 data2
::math::statistics::intervalmeanstdev data confidence
::math::statistics::ttestmean data est_mean est_stdev alpha
::math::statistics::testnormal data significance
::math::statistics::lillieforsFit data
::math::statistics::testDuckworth list1 list2 significance
::math::statistics::quantiles data confidence
::math::statistics::quantiles limits counts confidence
::math::statistics::autocorr data
::math::statistics::crosscorr data1 data2
::math::statistics::meanhistogramlimits mean stdev number
::math::statistics::minmaxhistogramlimits min max number
::math::statistics::linearmodel xdata ydata intercept
::math::statistics::linearresiduals xdata ydata intercept
::math::statistics::test2x2 n11 n21 n12 n22
::math::statistics::print2x2 n11 n21 n12 n22
::math::statistics::controlxbar data ?nsamples?
::math::statistics::controlRchart data ?nsamples?
::math::statistics::testxbar control data
::math::statistics::testRchart control data
::math::statistics::testKruskalWallis confidence args
::math::statistics::analyseKruskalWallis args
::math::statistics::grouprank args
::math::statistics::testWilcoxon sample_a sample_b
::math::statistics::spearmanrank sample_a sample_b
::math::statistics::spearmanrankextended sample_a sample_b
::math::statistics::kerneldensity data opt option value ...
::math::statistics::tstat dof ?alpha?
::math::statistics::mvwls wt1 weights_and_values
::math::statistics::mvols values
::math::statistics::pdfnormal mean stdev value
::math::statistics::pdflognormal mean stdev value
::math::statistics::pdfexponential mean value
::math::statistics::pdfuniform xmin xmax value
::math::statistics::pdfgamma alpha beta value
::math::statistics::pdfpoisson mu k
::math::statistics::pdfchisquare df value
::math::statistics::pdfstudentt df value
::math::statistics::pdfgamma a b value
::math::statistics::pdfbeta a b value
::math::statistics::pdfweibull scale shape value
::math::statistics::pdfgumbel location scale value
::math::statistics::pdfpareto scale shape value
::math::statistics::pdfcauchy location scale value
::math::statistics::cdfnormal mean stdev value
::math::statistics::cdflognormal mean stdev value
::math::statistics::cdfexponential mean value
::math::statistics::cdfuniform xmin xmax value
::math::statistics::cdfstudentst degrees value
::math::statistics::cdfgamma alpha beta value
::math::statistics::cdfpoisson mu k
::math::statistics::cdfbeta a b value
::math::statistics::cdfweibull scale shape value
::math::statistics::cdfgumbel location scale value
::math::statistics::cdfpareto scale shape value
::math::statistics::cdfcauchy location scale value
::math::statistics::empiricaldistribution values
::math::statistics::randomnormal mean stdev number
::math::statistics::randomlognormal mean stdev number
::math::statistics::randomexponential mean number
::math::statistics::randomuniform xmin xmax number
::math::statistics::randomgamma alpha beta number
::math::statistics::randompoisson mu number
::math::statistics::randomchisquare df number
::math::statistics::randomstudentt df number
::math::statistics::randombeta a b number
::math::statistics::randomweibull scale shape number
::math::statistics::randomgumbel location scale number
::math::statistics::randompareto scale shape number
::math::statistics::randomcauchy location scale number
::math::statistics::histogramuniform xmin xmax limits number
::math::statistics::incompleteGamma x p ?tol?
::math::statistics::incompleteBeta a b x ?tol?
::math::statistics::estimatepareto values
::math::statistics::filter varname data expression
::math::statistics::map varname data expression
::math::statistics::samplescount varname list expression
::math::statistics::subdivide
::math::statistics::plotscale canvas xmin xmax ymin ymax
::math::statistics::plotxydata canvas xdata ydata tag
::math::statistics::plotxyline canvas xdata ydata tag
::math::statistics::plottdata canvas tdata tag
::math::statistics::plottline canvas tdata tag
::math::statistics::plothistogram 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.
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
 ::math::statistics::min data

Determine the minimum value of the given list of data.
 list data
 List of data
 ::math::statistics::max data

Determine the maximum value of the given list of data.
 list data
 List of data
 ::math::statistics::number data

Determine the number of nonmissing data in the given list
 list data
 List of data
 ::math::statistics::stdev data

Determine the sample standard deviation of the data in the given list
 list data
 List of data
 ::math::statistics::var data

Determine the sample variance of the data in the given list
 list data
 List of data
 ::math::statistics::pstdev data

Determine the population standard deviation of the data in the given list
 list data
 List of data
 ::math::statistics::pvar data

Determine the population variance of the data in the given list
 list data
 List of data
 ::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
 ::math::statistics::basicstats 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
 ::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
 ::math::statistics::histogramalt 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
 ::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
 ::math::statistics::intervalmeanstdev 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)
 ::math::statistics::ttestmean 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)
 ::math::statistics::testnormal 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 "1significance", 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 1significance (see ticket 2812473fff). This has been corrected as of version 0.9.3.
 ::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
 ::math::statistics::testDuckworth list1 list2 significance

Determine if two data sets have the same median according to the TukeyDuckworth 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)
 ::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.
 ::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.
 ::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
 ::math::statistics::crosscorr data1 data2

Return the crosscorrelation 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
 ::math::statistics::meanhistogramlimits 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)
 ::math::statistics::minmaxhistogramlimits 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)
 ::math::statistics::linearmodel 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
 ::math::statistics::linearresiduals 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)
 ::math::statistics::test2x2 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 "chisquare" 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.
 ::math::statistics::print2x2 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.
 ::math::statistics::controlxbar 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
 ::math::statistics::controlRchart 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
 ::math::statistics::testxbar 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 "controlxbar" procedure
 list data
 List of observed data
 ::math::statistics::testRchart 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 "controlRchart" procedure
 list data
 List of observed data
 ::math::statistics::testKruskalWallis confidence args

Check if the population medians of two or more groups are equal with a given confidence level, using the KruskalWallis test.
 float confidence
 Confidence level to be used (01)
 list args
 Two or more lists of data
 ::math::statistics::analyseKruskalWallis args

Compute the statistical parameters for the KruskalWallis test. Returns the KruskalWallis 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
 ::math::statistics::grouprank 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
 ::math::statistics::testWilcoxon 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
 ::math::statistics::spearmanrank 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
 ::math::statistics::spearmanrankextended 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 zscore, 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
 ::math::statistics::kerneldensity 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

 Optionvalue 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)
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 KempBenedict.
In addition to these two, it provides a procedure (tstat) for calculating the value of the tstatistic 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 tdistribution t* satisfying
P(t*) = 1  alpha/2 P(t*) = alpha/2
for the number of degrees of freedom dof.
Given a sample of normallydistributed 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 tstatistic 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 tdistribution. Defaults to 0.05.
 ::math::statistics::mvwls 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 rsquared statistic
 The adjusted rsquared 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::mvols 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::mvols $data] # Prettyprint the results puts "Rsquared: [lindex $results 0]" puts "Adj Rsquared: [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 lognormal distribution
 The uniform distribution  equal probability for all data within a given interval
 The exponential distribution  useful as a model for certain extremevalue distributions.
 The gamma distribution  based on the incomplete Gamma integral
 The beta distribution
 The chisquare distribution
 The student's T distribution
 The Poisson distribution
 The Pareto distribution
 The Gumbel distribution
 The Weibull distribution
 The Cauchy 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:
 ::math::statistics::pdfnormal 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
 ::math::statistics::pdflognormal mean stdev value

Return the probability of a given value for a lognormal 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
 ::math::statistics::pdfexponential 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
 ::math::statistics::pdfuniform 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
 ::math::statistics::pdfgamma 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
 ::math::statistics::pdfpoisson 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
 ::math::statistics::pdfchisquare 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
 ::math::statistics::pdfstudentt 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
 ::math::statistics::pdfgamma 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
 ::math::statistics::pdfbeta 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
 ::math::statistics::pdfweibull 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
 ::math::statistics::pdfgumbel 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
 ::math::statistics::pdfpareto 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
 ::math::statistics::pdfcauchy 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 higherorder moments.
 float location
 Location parameter
 float scale
 Shape parameter
 float value
 Value for which the probability is required
 ::math::statistics::cdfnormal 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
 ::math::statistics::cdflognormal mean stdev value

Return the cumulative probability of a given value for a lognormal 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
 ::math::statistics::cdfexponential 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
 ::math::statistics::cdfuniform 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
 ::math::statistics::cdfstudentst 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
 ::math::statistics::cdfgamma 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
 ::math::statistics::cdfpoisson 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
 ::math::statistics::cdfbeta 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
 ::math::statistics::cdfweibull 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
 ::math::statistics::cdfgumbel 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
 ::math::statistics::cdfpareto 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
 ::math::statistics::cdfcauchy 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
 ::math::statistics::empiricaldistribution 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
 ::math::statistics::randomnormal 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
 ::math::statistics::randomlognormal mean stdev number

Return a list of "number" random values satisfying a lognormal 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
 ::math::statistics::randomexponential 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
 ::math::statistics::randomuniform 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
 ::math::statistics::randomgamma 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
 ::math::statistics::randompoisson 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
 ::math::statistics::randomchisquare 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
 ::math::statistics::randomstudentt 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
 ::math::statistics::randombeta 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
 ::math::statistics::randomweibull 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
 ::math::statistics::randomgumbel 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
 ::math::statistics::randompareto 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
 ::math::statistics::randomcauchy 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
 ::math::statistics::histogramuniform 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
 ::math::statistics::incompleteGamma x p ?tol?

Evaluate the incomplete Gamma integral
1 / x p1 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: 1.0e9)
 ::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: 1.0e9)
 ::math::statistics::estimatepareto 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
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
 ::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
 ::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").
 ::math::statistics::subdivide
Routine PM  not implemented yet
Plot Procedures
The following simple plotting procedures are available:
 ::math::statistics::plotscale 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
 ::math::statistics::plotxydata 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)
 ::math::statistics::plotxyline 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)
 ::math::statistics::plottdata 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 timedependent 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)
 ::math::statistics::plottline canvas tdata tag

Create a simple XY plot in the given canvas  the data are shown as a line. See plottdata 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)
 ::math::statistics::plothistogram 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)
Things to Do
The following procedures are yet to be implemented:
 Fteststdev
 intervalmeanstdev
 histogramnormal
 histogramexponential
 testhistogram
 testcorr
 quantiles*
 fouriercoeffs
 fourierresiduals
 oneparfunctionfit
 oneparfunctionresiduals
 plotlinearmodel
 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 printhistogram {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 } { $i < 100 } { incr i } { set d1 [expr {10.02.0*cos(2.0*3.1415926*$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::basicstats $data1] set b2 [::math::statistics::basicstats $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::plotscale .plot1 0 100 0 20 ::math::statistics::plotscale .plot2 0 20 0 20 ::math::statistics::plottline .plot1 $data1 ::math::statistics::plottline .plot1 $data2 ::math::statistics::plotxydata .plot2 $data1 $data2 puts "Correlation coefficient:" puts [::math::statistics::corr $data1 $data2] pause 2 puts "Plot histograms" .plot2 delete all ::math::statistics::plotscale .plot2 0 20 0 100 set limits [::math::statistics::minmaxhistogramlimits 7 16] set histogram_data [::math::statistics::histogram $limits $data1] ::math::statistics::plothistogram .plot2 $histogram_data $limits puts "First series:" printhistogram $histogram_data $limits pause 2 set limits [::math::statistics::minmaxhistogramlimits 0 15 10] set histogram_data [::math::statistics::histogram $limits $data2] ::math::statistics::plothistogram .plot2 $histogram_data $limits d2 .plot2 itemconfigure d2 fill red puts "Second series:" printhistogram $histogram_data $limits puts "Autocorrelation function:" set autoc [::math::statistics::autocorr $data1] puts [::math::statistics::map $autoc {[format "%.2f" $x]}] puts "Crosscorrelation function:" set crossc [::math::statistics::crosscorr $data1 $data2] puts [::math::statistics::map $crossc {[format "%.2f" $x]}] ::math::statistics::plotscale .plot1 0 100 1 4 ::math::statistics::plottline .plot1 $autoc "autoc" ::math::statistics::plottline .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 [http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas for enhancements you may have for either package and/or documentation.
Keywords
data analysis, mathematics, statistics
Category
Mathematics