math::statistics(n)            Tcl Math Library            math::statistics(n)



______________________________________________________________________________

NAME
       math::statistics - Basic statistical functions and procedures

SYNOPSIS
       package require Tcl  8

       package require math::statistics  0.5

       ::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

       ::math::statistics::corr data1 data2

       ::math::statistics::interval-mean-stdev data confidence

       ::math::statistics::t-test-mean data est_mean est_stdev confidence

       ::math::statistics::test-normal data confidence

       ::math::statistics::lillieforsFit data

       ::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::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-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-beta a b value

       ::math::statistics::cdf-normal mean stdev value

       ::math::statistics::cdf-exponential mean value

       ::math::statistics::cdf-uniform xmin xmax value

       ::math::statistics::cdf-students-t degrees value

       ::math::statistics::cdf-gamma alpha beta value

       ::math::statistics::cdf-poisson mu k

       ::math::statistics::cdf-beta a b value

       ::math::statistics::random-normal 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-chisquare df number

       ::math::statistics::random-student-t df number

       ::math::statistics::random-beta a b number

       ::math::statistics::histogram-uniform xmin xmax limits number

       ::math::statistics::incompleteGamma x p ?tol?

       ::math::statistics::incompleteBeta a b x ?tol?

       ::math::statistics::filter varname data expression

       ::math::statistics::map varname data expression

       ::math::statistics::samplescount varname list expression

       ::math::statistics::subdivide

       ::math::statistics::test-Kruskal-Wallis confidence args

       ::math::statistics::analyse-Kruskal-Wallis args

       ::math::statistics::group-rank args

       ::math::statistics::plot-scale canvas xmin xmax ymin ymax

       ::math::statistics::plot-xydata canvas xdata ydata tag

       ::math::statistics::plot-xyline canvas xdata ydata tag

       ::math::statistics::plot-tdata canvas tdata tag

       ::math::statistics::plot-tline canvas tdata tag

       ::math::statistics::plot-histogram canvas counts limits tag

_________________________________________________________________

DESCRIPTION
       The math::statistics package contains functions and procedures for
       basic statistical data analysis, such as:

       o      Descriptive statistical parameters (mean, minimum, maximum,
              standard deviation)

       o      Estimates of the distribution in the form of histograms and
              quantiles

       o      Basic testing of hypotheses

       o      Probability and cumulative density functions

       It is meant to help in developing data analysis applications or doing
       ad hoc data analysis, it is not in itself a full application, nor is it
       intended to rival with full (non-)commercial statistical 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 non-missing 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::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


       ::math::statistics::histogram limits values
              Determine histogram information for the given list of data.
              Returns a list consisting of the number of values that fall into
              each interval.  (The first interval consists of all values lower
              than the first limit, the last interval consists of all values
              greater than the last limit.  There is one more interval than
              there are limits.)

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

              list values
                     - List of data


       ::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::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)


       ::math::statistics::t-test-mean data est_mean est_stdev confidence
              Test whether the mean value of a sample is in accordance with
              the estimated normal distribution with a certain level of
              confidence.  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 confidence
                     - Confidence level (0.95 or 0.99 for instance)


       ::math::statistics::test-normal data confidence
              Test whether the given data follow a normal distribution with a
              certain level of confidence.  Returns 1 if the data are normally
              distributed within the level of confidence, returns 0 if not.
              The underlying test is the Lilliefors test.

              list data
                     - List of raw data values

              float confidence
                     - Confidence level (one of 0.80, 0.90, 0.95 or 0.99)


       ::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::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)



       ::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)


       ::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 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


       ::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)


       ::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)


       ::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:

                     o      (Estimate of) Intercept A

                     o      (Estimate of) Slope B

                     o      Standard deviation of Y relative to fit

                     o      Correlation coefficient R2

                     o      Number of degrees of freedom df

                     o      Standard error of the intercept A

                     o      Significance level of A

                     o      Standard error of the slope B

                     o      Significance level of B


       ::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)


       ::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.


       ::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.


       ::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


       ::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


       ::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


       ::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


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.


       ::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:

              o      The r-squared statistic

              o      The adjusted r-squared statistic

              o      A list containing the estimated coefficients b1, ... bN,
                     b0 (The constant b0 comes last in the list.)

              o      A list containing the standard errors of the coefficients

              o      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:

       o      The normal or Gaussian distribution

       o      The uniform distribution - equal probability for all data within
              a given interval

       o      The exponential distribution - useful as a model for certain
              extreme-value distributions.

       o      The gamma distribution - based on the incomplete Gamma integral

       o      The chi-square distribution

       o      The student's T distribution

       o      The Poisson distribution

       o      PM - binomial,F.

       In principle for each distribution one has procedures for:

       o      The probability density (pdf-*)

       o      The cumulative density (cdf-*)

       o      Quantiles for the given distribution (quantiles-*)

       o      Histograms for the given distribution (histogram-*)

       o      List of random values with the given distribution (random-*)

       The following procedures have been implemented:

       ::math::statistics::pdf-normal mean stdev value
              Return the probability of a given value for a normal
              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::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


       ::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


       ::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


       ::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


       ::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


       ::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


       ::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
                     - First shape parameter

              float value
                     - Value for which the probability is required


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

              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::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


       ::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


       ::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


       ::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


       ::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


       ::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
                     - First shape parameter

              float value
                     - Value for which the probability is required


       ::math::statistics::random-normal mean stdev number
              Return a list of "number" random values satisfying a normal
              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::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


       ::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


       ::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


       ::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


       ::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


       ::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


       ::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


       ::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)


       ::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)


       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


       ::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


       ::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


       ::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

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


       ::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)


       ::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)


       ::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)


       ::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)


       ::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)


THINGS TO DO
       The following procedures are yet to be implemented:

       o      F-test-stdev

       o      interval-mean-stdev

       o      histogram-normal

       o      histogram-exponential

       o      test-histogram

       o      test-corr

       o      quantiles-*

       o      fourier-coeffs

       o      fourier-residuals

       o      onepar-function-fit

       o      onepar-function-residuals

       o      plot-linear-model

       o      subdivide

EXAMPLES
       The code below is a small example of how you can examine a set of data:

              # Simple example:
              # - Generate data (as a cheap way of getting some)
              # - Perform statistical analysis to describe the data
              #
              package require math::statistics

              #
              # Two auxiliary procs
              #
              proc pause {time} {
                 set wait 0
                 after [expr {$time*1000}] {set ::wait 1}
                 vwait wait
              }

              proc print-histogram {counts limits} {
                 foreach count $counts limit $limits {
                    if { $limit != {} } {
                       puts [format "<%12.4g\t%d" $limit $count]
                       set 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"
              ::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

              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"

              puts "Quantiles: 0.1, 0.2, 0.5, 0.8, 0.9"
              puts "First:  [::math::statistics::quantiles $data1 {0.1 0.2 0.5 0.8 0.9}]"
              puts "Second: [::math::statistics::quantiles $data2 {0.1 0.2 0.5 0.8 0.9}]"


       If you run this example, then the following should be clear:

       o      There is a strong correlation between two time series, as
              displayed by the raw data and especially by the correlation
              functions.

       o      Both time series show a significant periodic component

       o      The histograms are not very useful in identifying the nature of
              the time series - they do not show the periodic nature.

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 SF Trackers
       [http://sourceforge.net/tracker/?group_id=12883].  Please also report
       any ideas for enhancements you may have for either package and/or
       documentation.

KEYWORDS
       data analysis, mathematics, statistics

CATEGORY
       Mathematics



math                                  0.5                  math::statistics(n)
