NAME¶

math::statistics - Basic statistical functions and procedures

SYNOPSIS¶

package require Tcl 8.5

package require math::statistics 1

::math::statistics::mean data

::math::statistics::min data

::math::statistics::max data

::math::statistics::number data

::math::statistics::stdev data

::math::statistics::var data

::math::statistics::pstdev data

::math::statistics::pvar data

::math::statistics::median data

::math::statistics::basic-stats data

::math::statistics::histogram limits values ?weights?

::math::statistics::histogram-alt limits values ?weights?

::math::statistics::corr data1 data2

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

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

::math::statistics::test-normal data significance

::math::statistics::lillieforsFit data

::math::statistics::test-Duckworth list1 list2 significance

::math::statistics::test-anova-F alpha args

::math::statistics::test-Tukey-range alpha args

::math::statistics::test-Dunnett alpha control args

::math::statistics::quantiles data confidence

::math::statistics::quantiles limits counts confidence

::math::statistics::autocorr data

::math::statistics::crosscorr data1 data2

::math::statistics::mean-histogram-limits mean stdev number

::math::statistics::minmax-histogram-limits min max number

::math::statistics::linear-model xdata ydata intercept

::math::statistics::linear-residuals xdata ydata intercept

::math::statistics::test-2x2 n11 n21 n12 n22

::math::statistics::print-2x2 n11 n21 n12 n22

::math::statistics::control-xbar data ?nsamples?

::math::statistics::control-Rchart data ?nsamples?

::math::statistics::test-xbar control data

::math::statistics::test-Rchart control data

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

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

::math::statistics::test-Levene groups

::math::statistics::test-Brown-Forsythe groups

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

::math::statistics::test-Wilcoxon sample_a sample_b

::math::statistics::spearman-rank sample_a sample_b

::math::statistics::spearman-rank-extended sample_a sample_b

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

::math::statistics::bootstrap data sampleSize ?numberSamples?

::math::statistics::wasserstein-distance prob1 prob2

::math::statistics::kl-divergence prob1 prob2

::math::statistics::logistic-model xdata ydata

::math::statistics::logistic-probability coeffs x

::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-triangular xmin xmax value

::math::statistics::pdf-symmetric-triangular 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::pdf-laplace location scale value

::math::statistics::pdf-kumaraswamy a b value

::math::statistics::pdf-negative-binomial r p 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-triangular xmin xmax value

::math::statistics::cdf-symmetric-triangular xmin xmax value

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

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

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

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

::math::statistics::cdf-weibull scale shape value

::math::statistics::cdf-gumbel location scale value

::math::statistics::cdf-pareto scale shape value

::math::statistics::cdf-cauchy location scale value

::math::statistics::cdf-F nf1 nf2 value

::math::statistics::cdf-laplace location scale value

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

::math::statistics::cdf-negative-binomial r p 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-triangular xmin xmax number

::math::statistics::random-symmetric-triangular 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::random-laplace location scale number

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

::math::statistics::random-negative-binomial r p 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::estimate-exponential values

::math::statistics::estimate-laplace values

::math::statistics::estimante-negative-binomial r values

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

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

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

::math::statistics::subdivide

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

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

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

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

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

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

DESCRIPTION¶

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

• Descriptive statistical parameters (mean, minimum, maximum, standard deviation)
• Estimates of the distribution in the form of histograms and quantiles
• Basic testing of hypotheses
• Probability and cumulative density functions

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

The purpose of this document is to describe the implemented procedures and provide some examples of their usage. As there is ample literature on the algorithms involved, we refer to relevant text books for more explanations. The package contains a fairly large number of public procedures. They can be distinguished in three sets: general procedures, procedures that deal with specific statistical distributions, list procedures to select or transform data and simple plotting procedures (these require Tk). Note: The data that need to be analyzed are always contained in a simple list. Missing values are represented as empty list elements. Note: With version 1.0.1 a mistake in the procs pdf-lognormal, cdf-lognormal and random-lognormal has been corrected. In previous versions the argument for the standard deviation was actually used as if it was the variance.

GENERAL PROCEDURES¶

The general statistical procedures are:

::math::statistics::mean data

Determine the mean value of the given list of data.

::math::statistics::min data

Determine the minimum value of the given list of data.

::math::statistics::max data

Determine the maximum value of the given list of data.

::math::statistics::number data

Determine the number of non-missing data in the given list

::math::statistics::stdev data

Determine the sample standard deviation of the data in the given list

::math::statistics::var data

Determine the sample variance of the data in the given list

::math::statistics::pstdev data

Determine the population standard deviation of the data in the given list

::math::statistics::pvar data

Determine the population variance of the data in the given list

::math::statistics::median data

Determine the median of the data in the given list (Note that this requires sorting the data, which may be a costly operation)

::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.)

::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.

::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.

::math::statistics::corr data1 data2

Determine the correlation coefficient between two sets 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)

::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.

::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.

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.

::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.

::math::statistics::test-anova-F alpha args

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

    test-anova-F 0.05 $A $B $C


    #


    # Or equivalently:


    #


    test-anova-F 0.05 [list $A $B $C]

::math::statistics::test-Tukey-range alpha args

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

::math::statistics::test-Dunnett alpha control args

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

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

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

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

::math::statistics::quantiles data confidence

Return the quantiles for a given set of data

::math::statistics::quantiles limits counts confidence

Return the quantiles based on histogram information (alternative to the call with two arguments)

::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

::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.

::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.

::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.

::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

::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.

::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.

::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.

::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.

::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.

::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.

::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.

::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.

::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.

::math::statistics::test-Levene groups

Compute the Levene statistic to determine if groups of data have the same variance (are homoscadastic) or not. The data are organised in groups. This version uses the mean of the data as the measure to determine the deviations. The statistic is equivalent to an F statistic with degrees of freedom k-1 and N-k, k being the number of groups and N the total number of data.

::math::statistics::test-Brown-Forsythe groups

Compute the Brown-Forsythe statistic to determine if groups of data have the same variance (are homoscadastic) or not. Like the Levene test, but this version uses the median of the 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.

::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.

::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.

::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.

::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.

::math::statistics::bootstrap data sampleSize ?numberSamples?

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

::math::statistics::wasserstein-distance prob1 prob2

Compute the Wasserstein distance or earth mover's distance for two equidstantly spaced histograms or probability densities. The histograms need not to be normalised to sum to one, but they must have the same number of entries.

Note: the histograms are assumed to be based on the same equidistant intervals. As the bounds are not passed, the value is expressed in the length of the intervals.

::math::statistics::kl-divergence prob1 prob2

Compute the Kullback-Leibler (KL) divergence for two equidstantly spaced histograms or probability densities. The histograms need not to be normalised to sum to one, but they must have the same number of entries.

Note: the histograms are assumed to be based on the same equidistant intervals. As the bounds are not passed, the value is expressed in the length of the intervals.

Note also that the KL divergence is not symmetric and that the second histogram should not contain zeroes in places where the first histogram has non-zero values.

::math::statistics::logistic-model xdata ydata

Estimate the coefficients of the logistic model that fits the data best. The data consist of independent x-values and the outcome 0 or 1 for each of the x-values. The result can be used to estimate the probability that a certain x-value gives 1.

::math::statistics::logistic-probability coeffs x

Calculate the probability of success for the value x given the coefficients of the logistic model.

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.

::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:

Arguments:

::math::statistics::mv-ols values

Carries out an ordinary least squares linear regression for the data points provided.

This procedure simply calls ::mvlinreg::wls with the weights set to 1.0, and returns the same information.

Example of the use:

# Store the value of the unicode value for the "+/-" character
set pm "\u00B1"
# Provide some data
set data {{  -.67  14.18  60.03 -7.5  }


          { 36.97  15.52  34.24 14.61 }


          {-29.57  21.85  83.36 -7.   }


          {-16.9   11.79  51.67 -6.56 }


          { 14.09  16.24  36.97 -12.84}


          { 31.52  20.93  45.99 -25.4 }


          { 24.05  20.69  50.27  17.27}


          { 22.23  16.91  45.07  -4.3 }


          { 40.79  20.49  38.92  -.73 }


          {-10.35  17.24  58.77  18.78}}
# Call the ols routine
set results [::math::statistics::mv-ols $data]
# Pretty-print the results
puts "R-squared: [lindex $results 0]"
puts "Adj R-squared: [lindex $results 1]"
puts "Coefficients $pm s.e. -- \[95% confidence interval\]:"
foreach val [lindex $results 2] se [lindex $results 3] bounds [lindex $results 4] {


    set lb [lindex $bounds 0]


    set ub [lindex $bounds 1]


    puts "   $val $pm $se -- \[$lb to $ub\]"
}

STATISTICAL DISTRIBUTIONS¶

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

• The normal or Gaussian distribution as well as the log-normal distribution
• The uniform distribution - equal probability for all data within a given interval
• The exponential distribution - useful as a model for certain extreme-value distributions.
• The gamma distribution - based on the incomplete Gamma integral
• The beta distribution
• The chi-square distribution
• The student's T distribution
• The Poisson distribution
• The Pareto distribution
• The Gumbel distribution
• The Weibull distribution
• The Cauchy distribution
• The F distribution (only the cumulative density function)
• PM - binomial.

In principle for each distribution one has procedures for:

• The probability density (pdf-*)
• The cumulative density (cdf-*)
• Quantiles for the given distribution (quantiles-*)
• Histograms for the given distribution (histogram-*)
• List of random values with the given distribution (random-*)

The following procedures have been implemented:

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

Return the probability of a given value for a normal distribution with given mean and standard deviation.

::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.

::math::statistics::pdf-exponential mean value

Return the probability of a given value for an exponential distribution with given mean.

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

Return the probability of a given value for a uniform distribution with given extremes.

::math::statistics::pdf-triangular xmin xmax value

Return the probability of a given value for a triangular distribution with given extremes. If the argument min is lower than the argument max, then smaller values have higher probability and vice versa. In the first case the probability density function is of the form f(x) = 2(1-x) and the other case it is of the form f(x) = 2x.

::math::statistics::pdf-symmetric-triangular xmin xmax value

Return the probability of a given value for a symmetric triangular distribution with given extremes.

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

Return the probability of a given value for a Gamma distribution with given shape and rate parameters

::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)

::math::statistics::pdf-chisquare df value

Return the probability of a given value for a chi square distribution with given degrees of freedom

::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

::math::statistics::pdf-gamma a b value

Return the probability of a given value for a Gamma distribution with given shape and rate parameters

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

Return the probability of a given value for a Beta distribution with given shape parameters

::math::statistics::pdf-weibull scale shape value

Return the probability of a given value for a Weibull distribution with given scale and shape parameters

::math::statistics::pdf-gumbel location scale value

Return the probability of a given value for a Gumbel distribution with given location and shape parameters

::math::statistics::pdf-pareto scale shape value

Return the probability of a given value for a Pareto distribution with given scale and shape parameters

::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.

::math::statistics::pdf-laplace location scale value

Return the probability of a given value for a Laplace distribution with given location and shape parameters. The Laplace distribution consists of two exponential functions, is peaked and has heavier tails than the normal distribution.

::math::statistics::pdf-kumaraswamy a b value

Return the probability of a given value for a Kumaraswamy distribution with given parameters a and b. The Kumaraswamy distribution is related to the Beta distribution, but has a tractable cumulative distribution function.

::math::statistics::pdf-negative-binomial r p value

Return the probability of a given value for a negative binomial distribution with an allowed number of failures and the probability of success.

::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.

::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.

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

Return the cumulative probability of a given value for an exponential distribution with given mean.

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

Return the cumulative probability of a given value for a uniform distribution with given extremes.

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

Return the cumulative probability of a given value for a triangular distribution with given extremes. If xmin < xmax, then lower values have a higher probability and vice versa, see also pdf-triangular

::math::statistics::cdf-symmetric-triangular xmin xmax value

Return the cumulative probability of a given value for a symmetric triangular distribution with given extremes.

::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.

::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.

::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).

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

Return the cumulative probability of a given value for a Beta distribution with given shape parameters

::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.

::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.

::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

::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.

::math::statistics::cdf-F nf1 nf2 value

Return the cumulative probability of a given value for an F distribution with nf1 and nf2 degrees of freedom.

::math::statistics::cdf-laplace location scale value

Return the cumulative probability of a given value for a Laplace distribution with given location and shape parameters. The Laplace distribution consists of two exponential functions, is peaked and has heavier tails than the normal distribution.

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

Return the cumulative probability of a given value for a Kumaraswamy distribution with given parameters a and b. The Kumaraswamy distribution is related to the Beta distribution, but has a tractable cumulative distribution function.

::math::statistics::cdf-negative-binomial r p value

Return the cumulative probability of a given value for a negative binomial distribution with an allowed number of failures and the probability of success.

::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)

::math::statistics::random-normal mean stdev number

Return a list of "number" random values satisfying a normal distribution with given mean and standard deviation.

::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.

::math::statistics::random-exponential mean number

Return a list of "number" random values satisfying an exponential distribution with given mean.

::math::statistics::random-uniform xmin xmax number

Return a list of "number" random values satisfying a uniform distribution with given extremes.

::math::statistics::random-triangular xmin xmax number

Return a list of "number" random values satisfying a triangular distribution with given extremes. If xmin < xmax, then lower values have a higher probability and vice versa (see also pdf-triangular.

::math::statistics::random-symmetric-triangular xmin xmax number

Return a list of "number" random values satisfying a symmetric triangular distribution with given extremes.

::math::statistics::random-gamma alpha beta number

Return a list of "number" random values satisfying a Gamma distribution with given shape and rate parameters.

::math::statistics::random-poisson mu number

Return a list of "number" random values satisfying a Poisson distribution with given mean.

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

Return a list of "number" random values satisfying a chi square distribution with given degrees of freedom.

::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.

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

Return a list of "number" random values satisfying a Beta distribution with given shape parameters.

::math::statistics::random-weibull scale shape number

Return a list of "number" random values satisfying a Weibull distribution with given scale and shape parameters.

::math::statistics::random-gumbel location scale number

Return a list of "number" random values satisfying a Gumbel distribution with given location and scale parameters.

::math::statistics::random-pareto scale shape number

Return a list of "number" random values satisfying a Pareto distribution with given scale and shape parameters.

::math::statistics::random-cauchy location scale number

Return a list of "number" random values satisfying a Cauchy distribution with given location and scale parameters.

::math::statistics::random-laplace location scale number

Return a list of "number" random values satisfying a Laplace distribution with given location and shape parameters. The Laplace distribution consists of two exponential functions, is peaked and has heavier tails than the normal distribution.

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

Return a list of "number" random values satisying a Kumaraswamy distribution with given parameters a and b. The Kumaraswamy distribution is related to the Beta distribution, but has a tractable cumulative distribution function.

::math::statistics::random-negative-binomial r p number

Return a list of "number" random values satisying a negative binomial distribution.

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

Return the expected histogram for a uniform distribution.

::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

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

Evaluate the incomplete Beta integral

::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.

::math::statistics::estimate-exponential values

Estimate the parameter for the exponential distribution that comes closest to the given values. Returns an estimate of the one parameter and of the standard error.

::math::statistics::estimate-laplace values

Estimate the parameters for the Laplace distribution that comes closest to the given values. Returns an estimate of respectively the location and scale parameters, based on maximum likelihood.

::math::statistics::estimante-negative-binomial r values

Estimate the probability of success for the negative binomial distribution that comes closest to the given values. The allowed number of failures must be given.

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).

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

Return a list consisting of the data that are transformed via the expression.

::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.

::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.

::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.

::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.

::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.

::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.

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

Create a simple histogram in the given canvas

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}]"

• There is a strong correlation between two time series, as displayed by the raw data and especially by the correlation functions.
• Both time series show a significant periodic component
• The histograms are not very useful in identifying the nature of the time series - they do not show the periodic nature.

BUGS, IDEAS, FEEDBACK¶

This document, and the package it describes, will undoubtedly contain bugs and other problems. Please report such in the category math :: statistics of the Tcllib Trackers [http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas for enhancements you may have for either package and/or documentation.

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

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

KEYWORDS¶

data analysis, mathematics, statistics

CATEGORY¶

Mathematics