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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::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::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::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.

## 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 ?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::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

::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 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::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.

::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::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)

::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 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:

    - (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::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

::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

::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

::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

::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

::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)

## 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:

  - 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

- 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::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-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

::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-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

::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

::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

::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

::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

::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

::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-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

::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

    - Second shape parameter

  float value

    - Value for which the probability is required

::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

::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

::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

::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

::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

::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-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

::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-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

::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::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

::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

::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

::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

::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)

::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

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::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:

- 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.

## Keywords

data analysis, mathematics, statistics

## Category

Mathematics