modules/math/statistics.test

# -*- tcl -*-
# statistics.test --
#    Test cases for the ::math::statistics package
#
# Note:
#    The tests assume tcltest 2.1, in order to compare
#    floating-point results

# -------------------------------------------------------------------------

source [file join \
	[file dirname [file dirname [file join [pwd] [info script]]]] \
	devtools testutilities.tcl]

testsNeedTcl     8.5;# statistics,linalg!
testsNeedTcltest 2.1

support {
    useLocal math.tcl math
    useLocal linalg.tcl math::linearalgebra
    useLocal optimize.tcl math::optimize
}
testing {
    useLocal statistics.tcl math::statistics
}

# -------------------------------------------------------------------------

set ::data_uniform  [list 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0]
set ::data_missing  [list 1.0 1.0 1.0 {} 1.0 {} {} 1.0 1.0 1.0 1.0 1.0 1.0]
set ::data_linear   [list 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0]
set ::data_empty    [list {} {} {}]
set ::data_missing2 [list 1.0 2.0 3.0 {} 4.0 5.0 6.0 7.0 8.0 9.0 10.0]

#
# Create and register (in that order!) custom matching procedures
#
proc matchTolerant { expected actual } {
   set match 1
   foreach a $actual e $expected {
      if { abs($e-$a)>0.0001*abs($e) &&
           abs($e-$a)>0.0001*abs($a)     } {
         set match 0
         break
      }
   }
   return $match
}
proc matchTolerant2 { expected actual } {
   set match 1
   foreach a $actual e $expected {
      if { abs($e-$a)>0.025*abs($e) &&
           abs($e-$a)>0.025*abs($a)     } {
         set match 0
         break
      }
   }
   return $match
}
proc matchAlmostZero { expected actual } {
   set match 1
   foreach a $actual {
      if { abs($a)>1.0e-6 } {
         set match 0
         break
      }
   }
   return $match
}
customMatch tolerant   matchTolerant
customMatch tolerant2  matchTolerant2
customMatch almostzero matchAlmostZero

#
# Test cases
#
test "BasicStats-1.0" "Basic statistics - uniform data" -match tolerant -body {
  set all_data [::math::statistics::BasicStats all $::data_uniform]
} -result [list 1.0 1.0 1.0 [llength $::data_uniform] 0.0 0.0 0.0 0.0]

test "BasicStats-1.1" "Basic statistics - empty data" -match glob -body {
  catch {
     set all_data [::math::statistics::BasicStats all $::data_empty]
  } msg
  set msg
} -result "Too*"

#
# Result must be the same as for 1.0! Hence ::data_empty and ::data_uniform
#
test "BasicStats-1.2" "Basic statistics - missing data" -match tolerant -body {
  set all_data [::math::statistics::BasicStats all $::data_missing]
} -result [list 1.0 1.0 1.0 [llength $::data_uniform] 0.0 0.0 0.0 0.0]

test "BasicStats-1.3" "Basic statistics - linear data - mean" -match tolerant -body {
  set value [::math::statistics::mean $::data_linear]
} -result 5.5

test "BasicStats-1.4" "Basic statistics - linear data - min" -match tolerant  -body {
  set value [::math::statistics::min $::data_linear]
} -result 1.0

test "BasicStats-1.5" "Basic statistics - linear data - max" -match tolerant  -body {
  set value [::math::statistics::max $::data_linear]
} -result 10.0

test "BasicStats-1.6" "Basic statistics - linear data - number" -match tolerant  -body {
  set value [::math::statistics::number $::data_linear]
} -result 10

test "BasicStats-1.7" "Basic statistics - missing data - number" -match tolerant  -body {
  set value [::math::statistics::number $::data_missing2]
} -result 10

test "BasicStats-1.8" "Basic statistics - missing data - stdev" -match almostzero -body {
  set value1 [::math::statistics::stdev  $::data_linear]
  set value2 [::math::statistics::stdev  $::data_missing2]
  expr {abs($value1-$value2)}
} -result 0.001 ;# Zero is impossible

test "BasicStats-1.9" "Basic statistics - missing data - var" -match almostzero -body {
  set value1 [::math::statistics::stdev  $::data_linear]
  set value2 [::math::statistics::var    $::data_missing2]
  expr {$value1*$value1-$value2}
} -result 0.001 ;# Zero is impossible

test "BasicStats-1.10" "Basic statistics - missing data - pstdev" -match almostzero -body {
  set value1 [::math::statistics::pstdev  $::data_linear]
  set value2 [::math::statistics::pstdev  $::data_missing2]
  expr {abs($value1-$value2)}
} -result 0.001 ;# Zero is impossible

test "BasicStats-1.11" "Basic statistics - missing data - pvar" -match almostzero -body {
  set value1 [::math::statistics::pstdev  $::data_linear]
  set value2 [::math::statistics::pvar    $::data_missing2]
  expr {$value1*$value1-$value2}
} -result 0.001 ;# Zero is impossible

#
# This test was added because the calculation of the standard deviation
# could fail with uniform data (the difference of two almost equal
# values became a small negative number)
#
# Further extension: more stable computation if the values are very
# close together. Due to this change the variance should be independent
# of the mean, however large (up to a point)
#
test "BasicStats-2.1" "Basic statistics - uniform data caused sqrt domain error" -body {
  set values [list]
  set count 0
  for { set i 0 } { $i < 20 } { incr i } {
     lappend values 0.6
     set value2 [::math::statistics::mean $values]
     incr count
  }
  set count
} -result 20 ;# We can finish the loop

test "BasicStats-2.2" "Basic statistics - large almost identical values" -match glob -body {
  catch {
     set data [list 100001 100002 100003 100004]
     set result_large [::math::statistics::BasicStats all $data]

     set data [list 1 2 3 4]
     set result_small [::math::statistics::BasicStats all $data]

     matchTolerant [lrange $result_small 3 end] [lrange $result_large 3 end]
  } msg
  set msg
} -result 1

#
# Histograms
#
test "Histogram-1.0" "Histogram - uniform data" -match glob -body {
  set values [::math::statistics::histogram {0 2} $::data_uniform]
} -result [list 0 [llength $::data_uniform] 0]

test "Histogram-1.1" "Histogram - missing data" -match glob -body {
  set values [::math::statistics::histogram {0 2} $::data_missing]
} -result [list 0 [::math::statistics::number $::data_missing] 0]

test "Histogram-1.2" "Histogram - linear data" -match glob -body {
  set values [::math::statistics::histogram {1.5 4.5 9.5} $::data_linear]
} -result {1 3 5 1}

test "Histogram-1.3" "Histogram - linear data 2" -match glob -body {
  set values [::math::statistics::histogram {1.5 2.5 10.5} $::data_linear]
} -result {1 1 8 0}

#
# Adding two dummy values should not influence the histogram (ticket 05d055c2f5)
#
test "Histogram-1.4" "Histogram - linear data 2 with weights" -match glob -body {
  set values [::math::statistics::histogram {1.5 2.5 10.5} [concat $::data_linear 0.0 0.0] \
      [concat [lrepeat [llength $::data_linear] 1] 0 0]]
} -result {1 1 8 0}

test "Histogram-1.5" "Histogram - linear data 2 with weights" -match glob -body {
  set values [::math::statistics::histogram {1.5 2.5} [concat $::data_linear 0.0 0.0] \
      [concat [lrepeat [llength $::data_linear] 1] 0 0]]
} -result {1 1 8}

#
# Alternative definition of the intervals (ticket 1502400fff)
# Note the difference in the expected bin sizes for the two
#
test "Histogram-2.1" "Histogram - alternative interval bounds" -match glob -body {
  set values [concat [::math::statistics::histogram-alt {5.0 7.0} $::data_linear] \
                     [::math::statistics::histogram     {5.0 7.0} $::data_linear]]
} -result {4 2 4 5 2 3}

#
# Quantiles
# Bug #1272910: related to rounding 0.5 - use different levels instead
#               because another bug was fixed, return to the original
#               levels again
#
test "Quantiles-1.0" "Quantiles - raw data" -match tolerant -body {
  set values [::math::statistics::quantiles $::data_linear {0.25 0.55 0.95}]
} -result {3.0 6.0 10.0}

test "Quantiles-1.1" "Quantiles - histogram" -match tolerant -body {
  set limits    {1.0 2.0 3.0 4.0}
  set data_hist {0 10 20 10 0}
  set values [::math::statistics::quantiles $limits $data_hist {0.25 0.5 0.9}]
} -result {2.0 2.5 3.6}

#
# Generate histogram limits
#

test "Limits-1.0" "Limits - based on mean/stdev" -match tolerant -body {
  set values [::math::statistics::mean-histogram-limits 1.0 1.0 4]
} -result {0.0 0.75 1.25 2.0}

test "Limits-1.1" "Limits - based on mean/stdev" -match tolerant -body {
  set values [::math::statistics::mean-histogram-limits 1.0 1.0 9]
} -result {-2.0 -1.0 0.0 0.75 1.0 1.25 2.0 3.0 4.0}

test "Limits-1.2" "Limits - based on mean/stdev" -match tolerant -body {
  set values [::math::statistics::mean-histogram-limits 0.0 1.0 11]
} -result {-3.0 -2.4 -1.8 -1.2 -0.6 0.0 0.6 1.2 1.8 2.4 3.0}

test "Limits-2.0" "Limits - based on min/max" -match tolerant -body {
  set values [::math::statistics::minmax-histogram-limits -2.0 2.0 9]
} -result {-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0}

test "Limits-2.1" "Limits - based on min/max" -match tolerant -body {
  set values [::math::statistics::minmax-histogram-limits -2.0 2.0 2]
} -result {-2.0 2.0}

#
# To do: design test cases for the following functions:
# - t-test-mean
# - estimate-mean-stdev
# - autocorr
# - crosscorr
# - linear-model
# - linear-residuals
# - pdf-*
# - cdf-*
# - random-*
# - histogram-*
#
# Crude test cases for Student's t test
#
test "Students-t-test-1.0" "Student's t - same sample" -match glob -body {
  set sample [::math::statistics::random-normal 0.0 1.0 40]
  set mean   0.0
  set stdev  1.0
  set confidence 0.95

  set result [::math::statistics::t-test-mean $sample $mean $stdev $confidence]
} -result 1

test "Students-t-test-1.1" "Student's t - different sample" -match glob -body {
  set sample [::math::statistics::random-normal 0.0 1.0 40]
  set mean   10.0
  set stdev   1.0
  set confidence 0.95

  set result [::math::statistics::t-test-mean $sample $mean $stdev $confidence]
} -result 0

test "Students-t-test-1.2" "Student's t - small sample" -match glob -body {
  set sample [::math::statistics::random-normal 0.0 1.0 2]
  set mean    2.0
  set stdev   1.0
  set confidence 0.90

  set result [::math::statistics::t-test-mean $sample $mean $stdev $confidence]
} -result 1

#
# Test private procedures
#
test "Cdf-toms322-1.0" "TOMS322 - erf(x)" -match tolerant2 -body {
  set result {}
  foreach z {4.417 3.891 3.291 2.576 2.241 1.960 1.645 1.150 0.674
             0.319 0.126 0.063 0.0125} {
     set prob [::math::statistics::Cdf-toms322 1 5000 [expr {$z*$z}]]
     lappend result [expr {1.0-$prob}]
  }
  set result
} -result {1.e-5 1.e-4 1.e-3 1.e-2 0.025 0.050 0.100 0.250 0.500
           0.750 0.900 0.950 0.990 }

test "Cdf-toms322-2.0" "TOMS322 - inverse erf(x)" -match tolerant2 -body {
  set result {}
  foreach p {0.5120 0.5948 0.7019 0.7996  0.8997  0.9505  0.9901  0.9980 } {
     set z [::math::statistics::Inverse-cdf-normal 0.0 1.0 $p]
     lappend result $z
  }
  set result
} -result    {0.03  0.24   0.53   0.84    1.28    1.65    2.33    2.88 }

#
# Correlation coefficients
#
test "Correlation-1.0" "Correlation - linear data" -match tolerant -body {
  set corr [::math::statistics::corr $::data_linear $::data_linear]
} -result 1.0
test "Correlation-1.1" "Correlation - linear/uniform" -match almostzero -body {
  set corr [::math::statistics::corr $::data_linear $::data_uniform]
} -result 0.0

#
# Test list procedures
#
proc matchListElements { expected actual } {
   if { [llength $expected] != [llength $actual] } {
      return 0
   } else {
      set match 1
      foreach a $actual e $expected {
         if { $a != $e } {
            set match 0
            break
         }
      }
   }
   return $match
}
customMatch matchList  matchListElements

set ::data_list {1 2 3 4 5 6 7 8 9 10}
set ::data_pairs {{1 2} {3 4} {5 6} {7 8} {9 10}}

test "Filter-1.0" "True filter" -match matchList -body {
   set data [::math::statistics::filter x $::data_list 1]
} -result $::data_list

test "Filter-1.1" "False filter" -match matchList -body {
   set data [::math::statistics::filter x $::data_list 0]
} -result {}

test "Filter-1.2" "Even filter" -match matchList -body {
   set data [::math::statistics::filter x $::data_list {$x%2==0}]
} -result {2 4 6 8 10}

test "Filter-2.1" "filter with parameter" -match matchList -body {
   set param 3.0
   set data [::math::statistics::filter x $::data_list {$x > $param}]
} -result {4 5 6 7 8 9 10}

test "Map-1.0" "Identity map" -match matchList -body {
   set data [::math::statistics::map x $::data_list {$x}]
} -result $::data_list

test "Map-1.1" "Is-even map" -match matchList -body {
   set data [::math::statistics::map x $::data_list {$x%2==0}]
} -result {0 1 0 1 0 1 0 1 0 1}

test "Map-1.2" "Double map" -match matchList -body {
   set data [::math::statistics::map x $::data_list {$x*2}]
} -result {2 4 6 8 10 12 14 16 18 20}

test "Map-2.1" "map with parameter" -match matchList -body {
   set param 3.0
   set data [::math::statistics::map x $::data_list {$x + $param}]
} -result {4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0}

test "Samplescount-1.0" "Single sublist" -match matchList -body {
   set data [::math::statistics::samplescount x [list $::data_list]]
} -result {10}

test "Samplescount-1.1" "List of singleton sublist" -match matchList -body {
   set data [::math::statistics::samplescount x $::data_list]
} -result {1 1 1 1 1 1 1 1 1 1}

test "Samplescount-1.2" "Pairs sublist" -match matchList -body {
   set data [::math::statistics::samplescount x $::data_pairs]
} -result {2 2 2 2 2}

test "Samplescount-1.3" "Select uneven sublist" -match matchList -body {
   set data [::math::statistics::samplescount x $::data_pairs {$x%2}]
} -result {1 1 1 1 1}

test "Samplescount-2.1" "Count with parameter" -match matchList -body {
   set param 3.0
   set data [::math::statistics::samplescount x $::data_pairs {$x>$param}]
} -result {0 1 2 2 2}

test "Median-1.1" "Median - odd number of data" -body {
   set data {1.0 3.0 2.0}
   set median [::math::statistics::median $data]
} -result 2.0

test "Median-1.2" "Median - even number of data" -body {
   set data {1.0 3.0 2.0 1.0}
   set median [::math::statistics::median $data]
} -result 1.5

test "Median-1.3" "Median - missing data" -body {
   set data {1.0 {} 3.0 2.0 1.0 {}}
   set median [::math::statistics::median $data]
} -result 1.5

test "test-2x2-1.0" "Test 2x2" -match tolerant -body {
   set data [::math::statistics::test-2x2 170 94 30 6]
} -result 5.1136364

test "test-xbar-1.0" "Test xbar procedure" -match exact -body {
    set data {}
    for { set i 0 } { $i < 500 } { incr i } {
        lappend data [expr {rand()}]
    }
    set limits  [::math::statistics::control-xbar $data]
    set newdata {1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 10.0 10.0 10.0 10.0}
    set result  [::math::statistics::test-xbar $limits $newdata]
} -result {0 2}

test "test-Rchart-1.0" "Test Rchart procedure" -match exact -body {
    set data {}
    for { set i 0 } { $i < 500 } { incr i } {
        lappend data [expr {rand()}]
    }
    set limits  [::math::statistics::control-Rchart $data]
    set newdata {0.0 1.0 2.0 1.0 0.4 0.5 0.6 0.5 10.0  0.0 10.0 10.0}
    set result  [::math::statistics::test-Rchart $limits $newdata]
} -result {0 2}

#
# Testing for normal distribution
#
test "Testnormal-1.0" "Determine normality statistic for birth weight data" -match tolerant -body {
    ::math::statistics::lillieforsFit {72 112 111 107 119  92 126  80 81 84 115
                                       118 128 128 123 116 125 126 122 126 127 86
                                       142 132  87 123 133 106 103 118 114 94}
} -result 0.82827415657

test "Testnormal-1.0" "Test birthweight data for normality - 20% significance" -match exact -body {
    ::math::statistics::test-normal {72 112 111 107 119  92 126  80 81 84 115
                                     118 128 128 123 116 125 126 122 126 127 86
                                     142 132  87 123 133 106 103 118 114 94} 0.20
} -result 1

test "Testnormal-1.0" "Test birthweight data for normality - 5% significance" -match exact -body {
    ::math::statistics::test-normal {72 112 111 107 119  92 126  80 81 84 115
                                     118 128 128 123 116 125 126 122 126 127 86
                                     142 132  87 123 133 106 103 118 114 94} 0.05
} -result 0

test "Test-Duckworth-1.0" "Test Tukey-Duckworth - 5% significance" -match exact -body {
    set list1 {10 2 3 4 6}
    set list2 {12 3 4 6}

    ::math::statistics::test-Duckworth $list1 $list2 0.05
} -result 1

test "Test-Duckworth-1.1" "Test Tukey-Duckworth - symmetry" -match exact -body {
    set list1 {1 2 3 4 5 6 7 8 9 10}
    set list2 {6 7 8 9 10 11 12 13 14 15 16 17}

    set result [list [::math::statistics::test-Duckworth $list1 $list2 0.05] \
                     [::math::statistics::test-Duckworth $list2 $list1 0.05]]
} -result {0 0}

test "Test-Duckworth-1.2" "Test Tukey-Duckworth - applicability" -match exact -body {
    set list1 {2 3 4 6 20}
    set list2 {12 3 4 6}

    ::math::statistics::test-Duckworth $list1 $list2 0.05
} -result -1

#
# Testing multivariate linear regression
#
# Provide some data
test "Testmultivar-1.0" "Ordinary multivariate regression - three independent variables" \
        -match tolerant -body {
    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]

    # Flatten the result (so that we can use the tolerant comparison method)
    eval concat [eval concat $results]
} -result {0.887239767929 0.830859651893
3.33854942057 -1.58346976987 0.0362328113288 32.571621244
1.03305463908 0.237943867401 0.234143883673 19.4700016828
0.810755783819 5.86634305732
-2.16569743834 -1.00124210139 -0.536696631937 0.609162254594
-15.0697565684 80.2129990564}


test "Testmultivar-1.1" "Ordinary multivariate regression - zero variation" -body {
    set results [::math::statistics::mv-ols {{0 25125 128} {0 23224 64} {0 37903 512} {0 21263 32}
                                             {0 22053 64} {0 25745 256} {0 25745 256} {0 21557 32}
                                             {0 24935 128} {0 22904 64} {0 21422 32} {0 21947 32}
                                             {0 33244 512} {0 33244 512} {0 30060 512} {0 29691 256}
                                             {0 30439 256} {0 23724 128} {0 22541 64} {0 23640 128}
                                             {0 21422 32} {0 23640 128} {0 22249 64} {0 28247 512}
                                             {0 23333 32} {0 29841 256} {0 23959 128} {0 30819 512}
                                             {0 26333 256} {0 22145 32} {0 23863 128} {0 20772 32}
                                             {0 28511 512} {0 22425 64} {0 21598 32} {0 26335 256}
                                             {0 23816 128} {0 21157 32} {0 20973 32} {0 20973 32}
                                             {0 35125 512} {0 20679 32} {0 21241 64} {0 25297 256}
                                             {0 22301 32} {0 22007 32} {0 33351 512} {0 24115 128}
                                             {0 24115 128} {0 22301 32} {0 22797 64} {0 22593 64}
                                             {0 26439 256} {0 21255 32} {0 22645 32} {0 23447 128}
                                             {0 24205 64} {0 25051 128} {0 21007 32} {0 28237 256}
                                             {0 25546 128} {0 25669 256} {0 25669 256} {0 25669 256}
                                             {0 21977 64} {0 21977 64} {0 26187 128} {0 38360 512}
                                             {0 31846 256} {0 28349 256} {0 26450 128}}]
    set r2 [lindex $results 0]
} -result 1.0

#
# pdf/cdf tests - transformed from the contributions by Eric K. Benedict
#                 Cf. the examples.
#
# Note: cases with integer numbers test if divisions are done in floating-point or not
#

test "uniform-distribution-1.0" "Test pdf-uniform" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-uniform   0   10  5] \
        [::math::statistics::pdf-uniform   0.0 1.0 0.5] \
        [::math::statistics::pdf-uniform -10.0 1.0 -4.5] \
        [::math::statistics::pdf-uniform  -2.0 2.0  1.0]]
} -result {0.1 1.0 0.0909090909 0.25}

test "uniform-distribution-1.1" "Test cdf-uniform" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-uniform   0   10  5] \
        [::math::statistics::cdf-uniform   0.0 1.0 0.5] \
        [::math::statistics::cdf-uniform -10.0 1.0 -4.5] \
        [::math::statistics::cdf-uniform  -2.0 2.0  1.0]]
} -result {0.5 0.5 0.5 0.75}

test "triangular-distribution-1.0" "Test pdf-triangular" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-triangular   0   1.0 0.5]  \
        [::math::statistics::pdf-triangular   1.0 0.0 0.5]  \
        [::math::statistics::pdf-triangular   0.0 1.0 0.25] \
        [::math::statistics::pdf-triangular   1.0 0.0 0.25] \
        [::math::statistics::pdf-triangular   0.0 2.0 0.0]  \
        [::math::statistics::pdf-triangular   0.0 2.0 1.0]  \
        [::math::statistics::pdf-triangular   0.0 2.0 2.0]]
} -result {1.0 1.0 1.5 0.5 2.0 1.0 0.0}
test "triangular-distribution-1.1" "Test cdf-triangular" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-triangular   0   1.0  0.5] \
        [::math::statistics::cdf-triangular   1.0 0.0  0.5] \
        [::math::statistics::cdf-triangular   0.0 1.0  0.25] \
        [::math::statistics::cdf-triangular   1.0 0.0  0.25] \
        [::math::statistics::cdf-triangular   0.0 2.0  0.0] \
        [::math::statistics::cdf-triangular   0.0 2.0  1.0] \
        [::math::statistics::cdf-triangular   0.0 2.0  2.0]]
} -result {0.75 0.25 0.4375 0.0625 0.0 0.75 1.0}

test "triangular-symmetric-distribution-1.0" "Test pdf-symmetric-triangular" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-symmetric-triangular 0.0 2.0 -0.5]  \
        [::math::statistics::pdf-symmetric-triangular 0.0 2.0  0.0]  \
        [::math::statistics::pdf-symmetric-triangular 0.0 2.0  0.5]  \
        [::math::statistics::pdf-symmetric-triangular 0.0 2.0  1.0]  \
        [::math::statistics::pdf-symmetric-triangular 0.0 2.0  1.5]  \
        [::math::statistics::pdf-symmetric-triangular 0.0 2.0  2.0]  \
        [::math::statistics::pdf-symmetric-triangular 0.0 2.0  2.5]]
} -result {0.0 0.0 0.5 1.0 0.5 0.0 0.0}

test "triangular-symmetric-distribution-1.1" "Test cdf-symmetric-triangular" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-symmetric-triangular 0.0 2.0 -0.5]  \
        [::math::statistics::cdf-symmetric-triangular 0.0 2.0  0.0]  \
        [::math::statistics::cdf-symmetric-triangular 0.0 2.0  0.5]  \
        [::math::statistics::cdf-symmetric-triangular 0.0 2.0  1.0]  \
        [::math::statistics::cdf-symmetric-triangular 0.0 2.0  1.5]  \
        [::math::statistics::cdf-symmetric-triangular 0.0 2.0  2.0]  \
        [::math::statistics::cdf-symmetric-triangular 0.0 2.0  2.5]]
} -result {0.0 0.0 0.125 0.5 0.875 1.0 1.0}

test "exponential-distribution-1.0" "Test pdf-exponential" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-exponential 2   1] \
        [::math::statistics::pdf-exponential 1.0 1.0] \
        [::math::statistics::pdf-exponential 2.0 2.0] \
        [::math::statistics::pdf-exponential 2.0 1.0]]
} -result {0.3032653298563167 0.36787944117144233 0.18393972058572117 0.3032653298563167}

test "exponential-distribution-1.1" "Test cdf-exponential" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-exponential 2   1] \
        [::math::statistics::cdf-exponential 1.0 1.0] \
        [::math::statistics::cdf-exponential 2.0 2.0] \
        [::math::statistics::cdf-exponential 2.0 1.0]]
} -result {0.3934693402873666 0.6321205588285577 0.6321205588285577 0.3934693402873666}

test "laplace-distribution-1.0" "Test pdf-laplace" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-laplace  1   1   1] \
        [::math::statistics::pdf-laplace  1.0 1.0 2] \
        [::math::statistics::pdf-laplace -1.0 2.0 0] \
        [::math::statistics::pdf-laplace  1.5 2.5 3]]
} -result {0.5 0.18393972058572117 0.15163266492815836 0.1097623272188053}

test "laplace-distribution-1.1" "Test cdf-laplace" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-laplace  1   1   1] \
        [::math::statistics::cdf-laplace  1.0 1.0 2] \
        [::math::statistics::cdf-laplace -1.0 2.0 0] \
        [::math::statistics::cdf-laplace  1.5 2.5 3]]
} -result {0.5 0.8160602794142788 0.6967346701436833 0.7255941819529867}

test "kumaraswamy-distribution-1.0" "Test pdf-kumaraswamy" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-kumaraswamy  1   1   0.5] \
        [::math::statistics::pdf-kumaraswamy  0.5 0.5 0.5] \
        [::math::statistics::pdf-kumaraswamy  1.0 2.0 0] \
        [::math::statistics::pdf-kumaraswamy  1.0 2.0 0.5]]
} -result {1.0 0.6532814824381884 2.0 1.0}

test "kumaraswamy-distribution-1.1" "Test cdf-kumaraswamy" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-kumaraswamy  1   1   0.5] \
        [::math::statistics::cdf-kumaraswamy  0.5 0.5 0.5] \
        [::math::statistics::cdf-kumaraswamy  1.0 2.0 0] \
        [::math::statistics::cdf-kumaraswamy  1.0 2.0 0.5]]
} -result {0.5 0.4588038998538031 0.0 0.75}

# Non-trivial examples from Wikipedia
test "negative-binomial-distribution-1.0" "Test pdf-negative-binomial" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-negative-binomial 1 0.3 0] \
        [::math::statistics::pdf-negative-binomial 1 0.9 0] \
        [::math::statistics::pdf-negative-binomial 5 0.4 0] \
        [::math::statistics::pdf-negative-binomial 5 0.4 1] \
        [::math::statistics::pdf-negative-binomial 5 0.4 2] \
        [::math::statistics::pdf-negative-binomial 5 0.4 3] \
        [::math::statistics::pdf-negative-binomial 5 0.4 5]]
} -result {0.7 0.1 0.07776 0.15552 0.186624 0.1741824 0.10032906}

test "negative-binomial-distribution-1.1" "Test cdf-negative-binomial" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-negative-binomial 5 0.4 3] \
        [::math::statistics::cdf-negative-binomial 1 0.9 0] \
        [::math::statistics::cdf-negative-binomial 1 0.9 1] \
        [::math::statistics::cdf-negative-binomial 1 0.9 2]]
} -result {0.59408 0.1 0.19 0.271}

test "normal-distribution-1.0" "Test pdf-normal" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-normal  0   1   1] \
        [::math::statistics::pdf-normal  0.0 1.0 1.0] \
        [::math::statistics::pdf-normal  2.0 2.0 4.0] \
        [::math::statistics::pdf-normal -2.0 2.0 0.0] \
        [::math::statistics::pdf-normal  2.0 2.0 3.0]]
} -result {0.24197072451914337 0.24197072451914337 0.12098536225957168 0.12098536225957168 0.17603266338214976}

test "normal-distribution-1.1" "Test cdf-normal" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-normal  0   1   1] \
        [::math::statistics::cdf-normal  0.0 1.0 1.0] \
        [::math::statistics::cdf-normal  2.0 2.0 4.0] \
        [::math::statistics::cdf-normal -2.0 2.0 0.0] \
        [::math::statistics::cdf-normal  2.0 2.0 3.0]]
} -result {0.8413205502059895 0.8413205502059895 0.8413205502059895 0.8413205502059895 0.691451459572962}

#
# The values of the mean and the standard deviation are reconstructed from mu and sigma.
# If you use mu and sigma in pdf-normal/cdf-normal, the same numbers should arise
#
test "lognormal-distribution-1.0" "Test pdf-lognormal" -match tolerant -body {
    foreach {mu sigma mean stdev} {0.0 1.0 mean1 stdev1 2.0 2.0 mean2 stdev2 -2.0 2.0 mean3 stdev3} {
        set m      [expr {exp($mu + $sigma*$sigma/2.0)}]
        set $mean  $m
        set $stdev [expr {sqrt(exp($sigma*$sigma) - 1.0) * $m}]
    }

    set x [list \
        [::math::statistics::pdf-lognormal  $mean1 $stdev1 [expr {exp(1.0)}]] \
        [::math::statistics::pdf-lognormal  $mean2 $stdev2 [expr {exp(4.0)}]] \
        [::math::statistics::pdf-lognormal  $mean3 $stdev3 [expr {exp(0.0)}]] \
        [::math::statistics::pdf-lognormal  $mean2 $stdev2 [expr {exp(3.0)}]]]
} -result {0.24197072451914337 0.12098536225957168 0.12098536225957168 0.17603266338214976}

test "lognormal-distribution-1.1" "Test cdf-lognormal" -match tolerant -body {
    foreach {mu sigma mean stdev} {0.0 1.0 mean1 stdev1 2.0 2.0 mean2 stdev2 -2.0 2.0 mean3 stdev3} {
        set m      [expr {exp($mu + $sigma*$sigma/2.0)}]
        set $mean  $m
        set $stdev [expr {sqrt(exp($sigma*$sigma) - 1.0) * $m}]
    }

    set x [list \
        [::math::statistics::cdf-lognormal  $mean1 $stdev1 [expr {exp(1.0)}]] \
        [::math::statistics::cdf-lognormal  $mean2 $stdev2 [expr {exp(4.0)}]] \
        [::math::statistics::cdf-lognormal  $mean3 $stdev3 [expr {exp(0.0)}]] \
        [::math::statistics::cdf-lognormal  $mean2 $stdev2 [expr {exp(3.0)}]]]
} -result {0.8413205502059895 0.8413205502059895 0.8413205502059895 0.691451459572962}

test "gamma-distribution-1.0" "Test pdf-gamma" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-gamma 1.5 2.7 3.0] \
        [::math::statistics::pdf-gamma 7.5 0.2 30.0] \
        [::math::statistics::pdf-gamma 15.0 1.2 2.0]]
} -result {0.00263194027271168 0.0302770403110644 2.62677891379834e-07}

test "gamma-distribution-1.1" "Test cdf-gamma" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-gamma 1.9 0.45 2.5] \
        [::math::statistics::cdf-gamma 45.0 2.2 32.7]]
} -result {0.340299345090375 0.999731419881902}

test "poisson-distribution-1.0" "Test pdf-poisson" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-poisson 100 130] \
        [::math::statistics::pdf-poisson 27.2 37] \
        [::math::statistics::pdf-poisson 7.3 11.2]]
} -result {0.000575252683815462 0.0134122817590761 0.0530940708960824}

test "poisson-distribution-1.1" "Test cdf-poisson" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-poisson 4 7] \
        [::math::statistics::cdf-poisson 80 70] \
        [::math::statistics::cdf-poisson 4.9 6.2]]
} -result {0.948866384207153 0.14338996716003 0.77665467292263}

test "chisquare-distribution-1.0" "Test pdf-chisquare" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-chisquare 3 1.75]  \
        [::math::statistics::pdf-chisquare 10 2.9]  \
        [::math::statistics::pdf-chisquare 4 17.45] \
        [::math::statistics::pdf-chisquare 2.5 1.8]]
} -result {0.219999360547348 0.0216024880121444 0.000708787557977144 0.218446210041615}

test "chisquare-distribution-1.1" "Test cdf-chisquare" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-chisquare 2 3.5]   \
        [::math::statistics::cdf-chisquare 5 2.2]   \
        [::math::statistics::cdf-chisquare 5 100]   \
        [::math::statistics::cdf-chisquare 3.9 4.2] \
        [::math::statistics::cdf-chisquare 1  2.0]  \
        [::math::statistics::cdf-chisquare 3 -2.0]]
} -result {0.826226056549555 0.179164030785504 1.0 0.634682741547709 0.842700792949715 0.0}

test "students-t-distribution-1.0" "Test pdf-students-t" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-students-t 1 0.1]  \
        [::math::statistics::pdf-students-t 0.5 0.1]  \
        [::math::statistics::pdf-students-t 4 3.2]  \
        [::math::statistics::pdf-students-t 3 2.0]  \
        [::math::statistics::pdf-students-t 3 7.5]]
} -result {0.315158303152268 0.265700672177405 0.0156821741652879 0.0675096606638929 0.000942291548015668}

test "beta-distribution-1.0" "Test pdf-beta" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-beta 1.3 2.4 0.2] \
        [::math::statistics::pdf-beta 1 1 0.5] \
        [::math::statistics::pdf-beta 3.7 0.9 0.0] \
        [::math::statistics::pdf-beta 1.8 4.2 1.0] \
        [::math::statistics::pdf-beta 320 400 0.4] \
        [::math::statistics::pdf-beta 500   1 0.2] \
        [::math::statistics::pdf-beta 1000 1000 0.50]]
} -result {1.68903180472449 1.0 0.0 0.0 1.18192376783860 0.0 35.6780222917086}

test "beta-distribution-1.1" "Test cdf-beta" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-beta 2.1 3.0 0.2] \
        [::math::statistics::cdf-beta 4.2 17.3 0.5] \
        [::math::statistics::cdf-beta 500 375 0.7] \
        [::math::statistics::cdf-beta 250 760 0.2] \
        [::math::statistics::cdf-beta 43.2 19.7 0.6] \
        [::math::statistics::cdf-beta 500 640 0.3] \
        [::math::statistics::cdf-beta 400 640 0.3] \
        [::math::statistics::cdf-beta 0.1 30 0.1] \
        [::math::statistics::cdf-beta 0.01 0.03 0.9] \
        [::math::statistics::cdf-beta 2 3 0.9999] \
        [::math::statistics::cdf-beta 249.9999 759.99999 0.2] \
        [::math::statistics::cdf-beta 1000 1000 0.4] \
        [::math::statistics::cdf-beta 1000 1000 0.499] \
        [::math::statistics::cdf-beta 1000 1000 0.5] \
        [::math::statistics::cdf-beta 1000 1000 0.7] \
        [::math::statistics::cdf-beta 2 3 0.6]]
} -result {0.16220409275804 0.998630771123192 1.0 0.000125234318666948 0.0728881294218269
           2.99872547567313e-23 3.07056696205524e-09 0.998641008671625 0.765865005703006
           0.999999999996 0.000125237075575121 8.23161135486914e-20 0.464369443974288
           0.5 1.0 0.8208}

#
# TODO: chose the tests with _integer_ arguments more carefully
#
test "gumbel-distribution-1.0" "Test pdf-gumbel" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-gumbel 1.0 1.0 0.0] \
        [::math::statistics::pdf-gumbel 1.0 1.0 0.1] \
        [::math::statistics::pdf-gumbel 1.0 1.0 0.2] \
        [::math::statistics::pdf-gumbel 1.0 1.0 1.0] \
        [::math::statistics::pdf-gumbel 1.0 1.0 2.0] \
        [::math::statistics::pdf-gumbel 1.0 1.0 5.0] \
        [::math::statistics::pdf-gumbel 0.1 2.0 0.0] \
        [::math::statistics::pdf-gumbel 0.1 2.0 1.0] \
        [::math::statistics::pdf-gumbel 0.1 2.0 2.0] \
        [::math::statistics::pdf-gumbel 0.1 2.0 5.0] \
        [::math::statistics::pdf-gumbel 1   1   5  ] ]
} -result {0.179374 0.210219 0.240378 0.367879 0.254646 0.017983 0.183706 0.168507 0.131350 0.039580 0.017983}

test "gumbel-distribution-1.1" "Test cdf-gumbel" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-gumbel 1.0 1.0 0.0] \
        [::math::statistics::cdf-gumbel 1.0 1.0 0.2] \
        [::math::statistics::cdf-gumbel 1.0 1.0 1.0] \
        [::math::statistics::cdf-gumbel 1.0 1.0 2.0] \
        [::math::statistics::cdf-gumbel 0.1 2.0 0.0] \
        [::math::statistics::cdf-gumbel 0.1 2.0 1.0] \
        [::math::statistics::cdf-gumbel 0.1 2.0 2.0] \
        [::math::statistics::cdf-gumbel 1   1   2  ] ]
} -result {0.065988 0.108009 0.367879 0.692201 0.349493 0.528544 0.679266 0.692201}

test "weibull-distribution-1.0" "Test pdf-weibull" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-weibull 1.0 1.0 -1.0] \
        [::math::statistics::pdf-weibull 1.0 1.0 0.0] \
        [::math::statistics::pdf-weibull 1.0 1.0 0.1] \
        [::math::statistics::pdf-weibull 1.0 1.0 0.2] \
        [::math::statistics::pdf-weibull 1.0 1.0 1.0] \
        [::math::statistics::pdf-weibull 1.0 1.0 2.0] \
        [::math::statistics::pdf-weibull 1.0 1.0 5.0] \
        [::math::statistics::pdf-weibull 2.0 2.0 0.0] \
        [::math::statistics::pdf-weibull 2.0 2.0 1.0] \
        [::math::statistics::pdf-weibull 2.0 2.0 2.0] \
        [::math::statistics::pdf-weibull 2.0 2.0 5.0] ]
} -result {0 1.0 0.904837 0.818730 0.367879 0.135335 0.006738 0 0.389400 0.367879 0.004826}

test "weibull-distribution-1.1" "Test cdf-weibull" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-weibull 1.0 1.0 -1.0] \
        [::math::statistics::cdf-weibull 1.0 1.0 0.0] \
        [::math::statistics::cdf-weibull 1.0 1.0 0.2] \
        [::math::statistics::cdf-weibull 1.0 1.0 1.0] \
        [::math::statistics::cdf-weibull 1.0 1.0 2.0] \
        [::math::statistics::cdf-weibull 2.0 2.0 0.0] \
        [::math::statistics::cdf-weibull 2.0 2.0 1.0] \
        [::math::statistics::cdf-weibull 2.0 2.0 2.0] \
        [::math::statistics::cdf-weibull 2   2   2  ] ]
} -result {0 0 0.181269 0.632106 0.864665 0 0.221199 0.632121 0.632121}

test "pareto-distribution-1.0" "Test pdf-pareto" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-pareto 1.0 1.0 0.0] \
        [::math::statistics::pdf-pareto 1.0 1.0 1.1] \
        [::math::statistics::pdf-pareto 1.0 1.0 1.2] \
        [::math::statistics::pdf-pareto 1.0 1.0 2.0] \
        [::math::statistics::pdf-pareto 1.0 1.0 3.0] \
        [::math::statistics::pdf-pareto 1.0 1.0 5.0] \
        [::math::statistics::pdf-pareto 2.0 2.0 2.1] \
        [::math::statistics::pdf-pareto 2.0 2.0 3.0] \
        [::math::statistics::pdf-pareto 2.0 2.0 5.0] \
        [::math::statistics::pdf-pareto 2.0 2.0 10.0] ]
} -result {0 0.826446 0.694444 0.25 0.111111 0.04 0.863838 0.296296 0.064 0.008}

test "pareto-distribution-1.1" "Test cdf-pareto" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-pareto 1.0 1.0 0.0] \
        [::math::statistics::cdf-pareto 1.0 1.0 1.1] \
        [::math::statistics::cdf-pareto 1.0 1.0 1.2] \
        [::math::statistics::cdf-pareto 1.0 1.0 2.0] \
        [::math::statistics::cdf-pareto 1.0 1.0 3.0] \
        [::math::statistics::cdf-pareto 2.0 2.0 2.1] \
        [::math::statistics::cdf-pareto 2.0 2.0 3.0] \
        [::math::statistics::cdf-pareto 2.0 2.0 5.0] \
        [::math::statistics::cdf-pareto 2   2   3  ] ]
} -result {0 0.090909 0.1666667 0.5 0.666667 0.092971 0.555556 0.84 0.555556}

test "cauchy-distribution-1.0" "Test pdf-cauchy" -match tolerant -body {
    set x [list \
        [::math::statistics::pdf-cauchy 1.0 1.0 0.0] \
        [::math::statistics::pdf-cauchy 2.0 1.0 1.0] \
        [::math::statistics::pdf-cauchy 1.0 2.0 2.0] \
        [::math::statistics::pdf-cauchy 2.0 2.0 2.0] ]
} -result {0.1591555 0.1591555 0.1273240 0.1591550}

test "cauchy-distribution-1.1" "Test cdf-cauchy" -match tolerant -body {
    set x [list \
        [::math::statistics::cdf-cauchy 1.0 1.0 0.0] \
        [::math::statistics::cdf-cauchy 2.0 1.0 1.0] \
        [::math::statistics::cdf-cauchy 1.0 2.0 2.0] \
        [::math::statistics::cdf-cauchy 2.0 2.0 2.0] ]
} -result {0.25 0.25 0.6475836 0.5}

test "F-distribution-1.1" "Test cdf-F" -match tolerant -body {
    # Note: returned values are 1-0.05, 1-0.10 and 1-0.01
    set x [list \
        [::math::statistics::cdf-F  3  3 9.277] \
        [::math::statistics::cdf-F  1 30 4.171] \
        [::math::statistics::cdf-F  7  8 3.500] \
        [::math::statistics::cdf-F  2 10 2.924] \
        [::math::statistics::cdf-F 17  2 9.433] \
        [::math::statistics::cdf-F  5  6 8.746] \
        [::math::statistics::cdf-F  8 11 4.744] ]
} -result {0.95 0.95 0.95 0.90 0.90 0.99 0.99}

test "empirical-distribution-1.0" "Test empirical-distribution" -match tolerant -body {
    set x {10 4 3 2 5 6 7}
    set distribution [::math::statistics::empirical-distribution $x]
} -result {2 0.086207 3 0.224138 4 0.36207 5 0.5 6 0.637910 7 0.775862 10 0.913793}

#
# Crude tests for the random number generators
# Mainly to verify that there are no obvious errors
#
# To verify that the values are scaled properly, use a fixed seed
#
set ::rseed 1000000

test "random-numbers-1.0" "Test random-uniform" -body {
    expr {srand($::rseed)}

    set rnumbers [::math::statistics::random-uniform 0 10 100]

    set inrange 1
    foreach r $rnumbers {
        if { $r < 0.0 || $r > 10.0 } {
            set inrange 0
            break
        }
    }

    expr {srand($::rseed)}
    set scaled    1
    set rnumbers2 [::math::statistics::random-uniform 0 20 100]
    foreach r1 $rnumbers r2 $rnumbers2 {
        set scale [expr {$r2 / $r1}]
        if { abs($scale - 2.0) > 0.00001 } {
            set scaled 0
        }
    }
    expr {srand($::rseed)}
    set shifted   1
    set rnumbers3 [::math::statistics::random-uniform 10 20 100]
    foreach r1 $rnumbers r3 $rnumbers3 {
        set shift [expr {$r3 - $r1}]
        if { abs($shift - 10.0) > 0.00001 } {
            set shifted 0
        }
    }

    set result [list $inrange [llength $rnumbers] $scaled $shifted]
} -result {1 100 1 1}

test "random-numbers-1.1" "Test random-exponential" -body {
    expr {srand($::rseed)}
    set rnumbers [::math::statistics::random-exponential 1 100]

    set inrange 1
    foreach r $rnumbers {
        if { $r < 0.0 } {
            set inrange 0
            break
        }
    }

    expr {srand($::rseed)}
    set scaled    1
    set rnumbers2 [::math::statistics::random-exponential 2 100]
    foreach r1 $rnumbers r2 $rnumbers2 {
        set scale [expr {$r2 / $r1}]
        if { abs($scale - 2.0) > 0.00001 } {
            set scaled 0
        }
    }
    set result [list $inrange [llength $rnumbers] $scaled]
} -result {1 100 1}

test "random-numbers-1.2" "Test random-normal" -body {
    set rnumbers [::math::statistics::random-normal 0 1 100]
    set result [llength $rnumbers]
} -result 100

test "random-numbers-1.3" "Test random-gamma" -body {
    set rnumbers [::math::statistics::random-gamma 1.5 2.7 100]
    set result [llength $rnumbers]
} -result 100

test "random-numbers-1.4" "Test random-poisson" -body {
    set rnumbers [::math::statistics::random-poisson 2.5 100]
    set result [llength $rnumbers]
} -result 100

test "random-numbers-1.5" "Test random-chisquare" -body {
    set rnumbers [::math::statistics::random-chisquare 3 100]
    set result [llength $rnumbers]
} -result 100

test "random-numbers-1.6" "Test random-students-t" -body {
    set rnumbers [::math::statistics::random-students-t 3 100]
    set result [llength $rnumbers]
} -result 100

test "random-numbers-1.7" "Test random-beta" -body {
    set rnumbers [::math::statistics::random-beta 1.3 2.4 100]
    set result 1
    foreach r $rnumbers {
        if { $r < 0.0 || $r > 1.0 } {
            result 0
            break
        }
    }
    lappend result [llength $rnumbers]
} -result {1 100}

test "random-numbers-1.8" "Test random-gumbel" -body {
    set rnumbers [::math::statistics::random-gumbel 1.0 3.0 100]
    set result [llength $rnumbers]
} -result 100

test "random-numbers-1.9" "Test random-weibull" -body {
    set rnumbers [::math::statistics::random-weibull 1.0 3.0 100]
    set result [llength $rnumbers]
} -result 100

test "random-numbers-1.10" "Test random-pareto" -body {
    set rnumbers [::math::statistics::random-pareto 1.0 3.0 100]
    set result 1
    foreach r $rnumbers {
        if { $r < 1.0 } {
            result 0
            break
        }
    }
    lappend result [llength $rnumbers]
} -result {1 100}

test "random-numbers-1.11" "Test random-lognormal" -body {
    set rnumbers [::math::statistics::random-lognormal 1 1 100]
    set result [llength $rnumbers]
} -result 100

test "random-numbers-1.11" "Test random-cauchy" -body {
    set rnumbers [::math::statistics::random-cauchy 0 1 100]
    set result [llength $rnumbers]
} -result 100

test "random-numbers-1.12" "Test random-triangular" -body {
    set result 1
    set rnumbers [::math::statistics::random-triangular -10 10 100]
    # Check the scaling
    foreach r $rnumbers {
        if { $r < -10.0 || $r > 10.0 } {
            set result 0
            break
        }
    }

    # Also the alternative triangle
    set rnumbers [::math::statistics::random-triangular 10 -10 100]
    # Check the scaling
    foreach r $rnumbers {
        if { $r < -10.0 || $r > 10.0 } {
            set result 0
            break
        }
    }

    # The symmetric triangle
    set rnumbers [::math::statistics::random-symmetric-triangular 10 -10 100]
    # Check the scaling
    foreach r $rnumbers {
        if { $r < -10.0 || $r > 10.0 } {
            set result 0
            break
        }
    }

    set result
} -result 1

test "random-numbers-2.1" "Test random/estimate-pareto" -match tolerant -body {
    expr {srand($::rseed)}
    set rnumbers [::math::statistics::random-pareto 1.0 3.0 100]
    set result   [::math::statistics::estimate-pareto $rnumbers]
} -result {1.000519 3.668162 0.3668162}

test "random-numbers-2.2" "Test random/estimate-exponential" -match tolerant -body {
    expr {srand($::rseed)}
    set rnumbers [::math::statistics::random-exponential 2.0 1000]
    set result   [::math::statistics::estimate-exponential $rnumbers]
} -result {1.9976327295438587 0.06317069353857727}

test "random-numbers-2.3" "Test random/estimate-laplace" -match tolerant -body {
    expr {srand($::rseed)}
    set rnumbers [::math::statistics::random-laplace 1.0 1.0 1000]
    set result   [::math::statistics::estimate-laplace $rnumbers]
} -result {1.0106101707362272 0.9319576942526239}

# Very simple test - just check that the numbers lie between 0 and 1
test "random-numbers-2.4" "Test random-kumaraswamy" -body {
    expr {srand($::rseed)}
    set result   1

    foreach {a b} {0.5 0.5  1.0 1.0  0.5 1.0  1.0 0.5  2.0 0.1  0.1 2.0} {
        set rnumbers [::math::statistics::random-kumaraswamy $a $b 1000]
        foreach r $rnumbers {
            if { $r < 0.0 || $r > 1.0 } {
                set result 0
                break
            }
        }
    }
    set result
} -result 1

# Simple test, exact matching
test "random-numbers-2.5" "Test random-negative-binomial" -body {
    expr {srand($::rseed)}
    set rnumbers [::math::statistics::random-negative-binomial 1 0.3 10]
} -result {0 0 3 0 0 0 1 0 1 1}

test "random-numbers-2.6" "Test random/estimate-negative-binomial" -match tolerant -body {
    expr {srand($::rseed)}
    set r        3
    set rnumbers [::math::statistics::random-negative-binomial $r 0.5 1000]
    set pvalue   [::math::statistics::estimate-negative-binomial $r $rnumbers]
} -result 0.50190935

test "kruskal-wallis-1.0" "Test analysis Kruskal-Wallis" -match tolerant -body {
    ::math::statistics::analyse-Kruskal-Wallis {6.4 6.8 7.2 8.3 8.4 9.1 9.4 9.7} {2.5 3.7 4.9 5.4 5.9 8.1 8.2} {1.3 4.1 4.9 5.2 5.5 8.2}
} -result {9.83627087199 0.00731275323967}
test "kruskal-wallis-1.1" "Test test Kruskal-Wallis" -match tolerant -body {
    ::math::statistics::test-Kruskal-Wallis 0.95 {6.4 6.8 7.2 8.3 8.4 9.1 9.4 9.7} {2.5 3.7 4.9 5.4 5.9 8.1 8.2} {1.3 4.1 4.9 5.2 5.5 8.2}
} -result 1

# Data from Statistical methods in Engineering and Quality Assurance by Peter W.M. John
test "wilcoxon-1.0" "Test test Wilcoxon" -match tolerant -body {
    ::math::statistics::test-Wilcoxon {71.1 68.3 74.8 72.1 71.2 70.4 73.6 66.3 72.7 74.1 70.1 68.5} \
                                      {73.3 70.9 74.6 72.1 72.8 74.2 74.7 69.2 75.5 75.8 70.0 72.1}
} -result -1.67431578065

# Very simple tests, merely to show that the procedure "works"
# (No numerical example available)
test "levene-brown-forsythe-1.0" "Test test Levene/Brown-Forsythe" -match tolerant -body {
    set values {{1 2 3} {2 3 4} {5 6 7}}
    ::math::statistics::test-Levene $values
} -result 0.0

test "levene-brown-forsythe-1.1" "Test test Levene/Brown-Forsythe" -match tolerant -body {
    set values {{1 2 3} {2 3 4} {5 6 7}}
    ::math::statistics::test-Brown-Forsythe $values
} -result 0.0

test "levene-brown-forsythe-1.2" "Test test Levene/Brown-Forsythe" -match tolerant -body {
    set values {{1 2 2} {2 3 4} {5 6 7.4}}
    ::math::statistics::test-Levene $values
} -result 0.47937131630648294

test "levene-brown-forsythe-1.3" "Test test Levene/Brown-Forsythe" -match tolerant -body {
    set values {{1 2 2} {2 3 4} {5 6 7.4}}
    ::math::statistics::test-Brown-Forsythe $values
} -result 0.4382022471910113

# Data taken from https://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm
# Note: the webpage talks of the Levene test, but is actually the Brown-Forsythe
test "levene-brown-forsythe-1.4" "Test test Levene/Brown-Forsythe" -match tolerant -body {
    set values {{ 1.006 0.996 0.998 1.000 0.992 0.993 1.002 0.999 0.994 1.000 }
                { 0.998 1.006 1.000 1.002 0.997 0.998 0.996 1.000 1.006 0.988 }
                { 0.991 0.987 0.997 0.999 0.995 0.994 1.000 0.999 0.996 0.996 }
                { 1.005 1.002 0.994 1.000 0.995 0.994 0.998 0.996 1.002 0.996 }
                { 0.998 0.998 0.982 0.990 1.002 0.984 0.996 0.993 0.980 0.996 }
                { 1.009 1.013 1.009 0.997 0.988 1.002 0.995 0.998 0.981 0.996 }
                { 0.990 1.004 0.996 1.001 0.998 1.000 1.018 1.010 0.996 1.002 }
                { 0.998 1.000 1.006 1.000 1.002 0.996 0.998 0.996 1.002 1.006 }
                { 1.002 0.998 0.996 0.995 0.996 1.004 1.004 0.998 0.999 0.991 }
               { 0.991 0.995 0.984 0.994 0.997 0.997 0.991 0.998 1.004 0.997 } }
    ::math::statistics::test-Brown-Forsythe $values
} -result 1.705910

# Data from the Wikipedia page on Spearman's rank correlation coefficient
test "spearman-rank-1.0" "Test Spearman rank correlation" -match tolerant -body {
    ::math::statistics::spearman-rank {106  86 100 101  99 103  97 113 112 110} \
                                      {  7   0  27  50  28  29  20  12   6  17}
} -result -0.175757575758

test "spearman-rank-extended-1.0" "Test extended Spearman rank correlation procedure" -match tolerant -body {
    ::math::statistics::spearman-rank-extended {106  86 100 101  99 103  97 113 112 110} \
                                               {  7   0  27  50  28  29  20  12   6  17}
} -result {-0.175757575758 10 -0.456397284}

#
# Note: for the uniform and the logistic kernel the sum deviates more from 1 than for the others.
# For the logistic kernel this is because the density function is very widespread. For the
# uniform kernel the reason is not quite clear. Hence the margin per kernel.
#
test "kernel-density-1.0" "Test various kernel functions" -body {
    set data {1 2 3 4 5 6 7 8 9 10}

    set roughlyOne {}

    foreach kernel {gaussian uniform triangular epanechnikov biweight cosine logistic} \
            margin {0.01     0.02    0.01       0.01         0.01     0.01   0.05    } {
        set result [::math::statistics::kernel-density $data -kernel $kernel]

        set sum 0.0
        set xbegin [lindex $result 2 0]
        set xend   [lindex $result 2 1]
        set number [llength [lindex $result 0]]
        set dx     [expr {($xend-$xbegin) / $number}]

        #
        # Integral should be roughly one
        #
        set sum 0.0
        foreach v [lindex $result 1] {
            set sum [expr {$sum + $dx * $v}]
        }

        lappend roughlyOne [expr {abs($sum-1.0) < $margin}]
    }

    return $roughlyOne
} -result {1 1 1 1 1 1 1}

test "kernel-density-1.1" "Test various options - just that they have effect" -body {
    set subResults {}

    set data {1 2 3 4 5 6 7 8 9 10}

    set result [::math::statistics::kernel-density $data -number 20]
    lappend subResults [llength [lindex $result 0]]  ;# Number of bins
    lappend subResults [llength [lindex $result 1]]  ;# Number of density values

    set result [::math::statistics::kernel-density $data -interval {0 20}]
    lappend subResults [lindex $result 2 0]          ;# Beginning of interval
    lappend subResults [lindex $result 2 1]          ;# End of interval
    lappend subResults [expr {[lindex $result 0 0]   > [lindex $result 2 0]}] ;# First bin -- beginning of interval
    lappend subResults [expr {[lindex $result 0 0]   < [lindex $result 2 1]}] ;# First bin -- end of interval
    lappend subResults [expr {[lindex $result 0 end] > [lindex $result 2 0]}] ;# Last bin -- beginning of interval
    lappend subResults [expr {[lindex $result 0 end] < [lindex $result 2 1]}] ;# Last bin -- end of interval

    set result [::math::statistics::kernel-density $data -bandwidth 2]
    lappend subResults [lindex $result 2 end]        ;# Bandwidth

    return $subResults
} -result {20 20 0 20 1 1 1 1 2}

test "kernel-density-1.2" "Dealing with missing values" -body {
    set subResults {}

    set data {1 2 3 4 {} 6 7 8 9 10}

    set result [::math::statistics::kernel-density $data]

    set sum 0.0
    set xbegin [lindex $result 2 0]
    set xend   [lindex $result 2 1]
    set number [llength [lindex $result 0]]
    set dx     [expr {($xend-$xbegin) / $number}]

    #
    # Integral should be roughly one
    #
    set sum 0.0
    foreach v [lindex $result 1] {
        set sum [expr {$sum + $dx * $v}]
    }

    return [expr {abs($sum-1.0) < 0.01}]
} -result 1

test "anova-one-way-1.1" "ANOVA test from Wikipedia" -body {
    ::math::statistics::test-anova-F 0.05 {6 8 4 5 3 4} {8 12 9 11 6 8} {13 9 11 8 7 12}
} -result 0

test "anova-one-way-1.2" "ANOVA test from Wikipedia - using nested list" -body {
    ::math::statistics::test-anova-F 0.05 {{6 8 4 5 3 4} {8 12 9 11 6 8} {13 9 11 8 7 12}}
} -result 0

#
# Data from http://www.itl.nist.gov/div898/handbook/prc/section4/prc436.htm#example1
# See also http://www.itl.nist.gov/div898/handbook/prc/section4/prc471.htm
# Caveat: the calculation produces slightly different confidence intervals. I checked whether
# I got the calculation of the pooled variance right against the example appearing on Wikipedia
# (https://en.wikipedia.org/wiki/Pooled_variance) and that seems okay.
# No idea where the numerical difference is coming from.
#
test "Tukey-range-test-1.1" "Tukey range test" -body {
    set data {
        Group 1        {6.9     5.4     5.8     4.6     4.0}
        Group 2        {8.3     6.8     7.8     9.2     6.5}
        Group 3        {8.0     10.5    8.1     6.9     9.3}
        Group 4        {5.8     3.8     6.1     5.6     6.2}
    }

    set groupData {}
    foreach {dummy label d} $data {
        lappend groupData $d
    }

    set tukeyRange [::math::statistics::test-Tukey-range 0.05 $groupData]

    set indications {}
    foreach t $tukeyRange {
        lappend indications [lindex $t 2]
    }
    set indications
} -result {1 1 0 0 0 1}

#
# Data from https://en.wikipedia.org/wiki/Dunnett's_test
# Note that the Wikipedia uses t = 2.94, whereas it should have been 2.88
#
test "Dunnet-test-1.1" "Dunnett test" -body {
    set control {55 47 48}
    set data    {{55 64 64} {55 49 52} {50 44 41}}

    set dunnett [::math::statistics::test-Dunnett 0.05 $control $data]

    set indications {}
    foreach t $dunnett {
        lappend indications [lindex $t 0]
    }
    set indications
} -result {1 0 0}

#
# Bootstrap method to create new samples
#
test "Bootstrap-1.1" "Bootstrap - construct a single sample" -body {
    set data {1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20}

    set sample [::math::statistics::bootstrap $data 10]
    #
    # Check the sample
    #
    set numberOkay 0
    foreach value $sample {
        incr numberOkay [expr {$value in $data}]
    }

    set result [list [llength $sample] $numberOkay]
} -result {10 10}

test "Bootstrap-1.2" "Bootstrap - variation in a single sample" -body {
    set data 0
    for {set i 0} {$i < 100} {incr i} {
        lappend data $i
    }

    set sample [::math::statistics::bootstrap $data 10]
    #
    # It is improbable that all values are the same
    #
    set numberInHalf1 0
    set numberInHalf2 0
    set halfData1 [lrange $data  0 49]
    set halfData2 [lrange $data 50 99]
    foreach value $sample {
        incr numberInHalf1 [expr {$value in $halfData1}]
        incr numberInHalf2 [expr {$value in $halfData2}]
    }

    set result [list [expr {$numberInHalf1 > 0}] [expr {$numberInHalf2 > 0}]]
} -result {1 1}

test "Bootstrap-1.3" "Bootstrap - list of samples" -body {
    set data 0
    for {set i 0} {$i < 100} {incr i} {
        lappend data $i
    }

    set sampleSize    10
    set numberSamples 20
    set samples [::math::statistics::bootstrap $data $sampleSize $numberSamples]

    set result 1

    if { [llength $samples] != $numberSamples } {
        set result 0
    }

    foreach sample $samples {
        if { [llength $sample] != $sampleSize } {
            set result 0
        }
    }

    set result
} -result 1


#
# Tests for Wasserstein distance and KL divergence
#
# Tests for error conditions
#
test "Wasserstein-1.1" "Error: lengths differ" -body {
    set distance [::math::statistics::wasserstein-distance {0 1} {1 0 0 0}]
} -returnCodes 1 -result {Lengths of the probability histograms must be the same}

#
# Check the symmetry for arbitrary histograms
#
test "Wasserstein-1.2" "Test symmetry" -body {
    set okay 1
    for {set i 0} {$i < 10} {incr i} {
        set histogram1 {}
        set histogram2 {}
        for {set j 0} {$j < 3*($i+1)} {incr j} {
            lappend histogram1 [expr {rand()}]
            lappend histogram2 [expr {rand()}]
        }

        set distance1 [::math::statistics::wasserstein-distance $histogram1 $histogram2]
        set distance2 [::math::statistics::wasserstein-distance $histogram2 $histogram1]

        if { abs($distance1-$distance2) > 1.0e-6 } {
            set okay 0
        }
    }

    return $okay

} -result 1

#
# Check the non-negativity for arbitrary histograms
#
test "Wasserstein-1.3" "Test non-negativity" -body {
    set okay 1
    for {set i 0} {$i < 10} {incr i} {
        set histogram1 {}
        set histogram2 {}
        for {set j 0} {$j < 3*($i+1)} {incr j} {
            lappend histogram1 [expr {rand()}]
            lappend histogram2 [expr {rand()}]
        }

        set distance1 [::math::statistics::wasserstein-distance $histogram1 $histogram2]

        if { $distance1 < 0.0 } {
            set okay 0
        }
    }

    return $okay

} -result 1

#
# Check the non-normalised histograms
#
test "Wasserstein-1.4" "Test non-normalised histograms" -match tolerant -body {
    return [list [::math::statistics::wasserstein-distance {2 0 0} {0 0 0.5}]  \
                 [::math::statistics::wasserstein-distance {1 0 0} {0 0 0.25}] ]
} -result {2.0 2.0}

#
# Check arbitrarily extended histograms
#
test "Wasserstein-1.5" "Test extended histograms" -match tolerant -body {
    return [list [::math::statistics::wasserstein-distance {1 0 0}     {0 0 1}]     \
                 [::math::statistics::wasserstein-distance {0 0 1 0 0} {0 0 0 0 1}] \
                 [::math::statistics::wasserstein-distance {1 0 0 0 0} {0 0 1 0 0}] ]
} -result {2.0 2.0 2.0}

#
# Check numerical results
#
test "Wasserstein-1.6" "Test numerical results" -match tolerant -body {
    return [list [::math::statistics::wasserstein-distance {1 0 0}     {0 0.5 0.5}]             \
                 [::math::statistics::wasserstein-distance {1 0 0 0 0} {0 0 0 0.5 0.5}]         \
                 [::math::statistics::wasserstein-distance {1 0 0 0 0} {0 0.25 0.25 0.25 0.25}] ]
} -result {1.5 3.5 2.5}

#
# Tests for error conditions
#
test "KL-divergence-1.1" "Error: lengths differ" -body {
    set distance [::math::statistics::kl-divergence {0 1} {1 0 0 0}]
} -returnCodes 1 -result {Lengths of the two probability histograms must be the same}

test "KL-divergence-1.2" "Error: unmatched zeroes" -body {
    set distance [::math::statistics::kl-divergence {0.3 0.3 0.4} {1 0 0}]
} -returnCodes 1 -result {Second probability histogram contains unmatched zeroes}


test "KL-divergence-1.3" "Matched zeroes should be accepted" -match tolerant -body {
    set distance [::math::statistics::kl-divergence {0.3 0.3 0.4 0} {0.7 0.2 0.1 0}]
} -returnCodes 0 -result 0.42196792

#
# Tests for equal histograms (not all normalised)
#
test "KL-divergence-1.4" "Equal histograms give zero divergence" -body {
    set distance [::math::statistics::kl-divergence {0.3 0.3 0.4} {0.3 0.3 0.4}]
} -result 0.0

test "KL-divergence-1.5" "Non-normalised but equal histograms give zero divergence" -body {
    set distance [::math::statistics::kl-divergence {0.3 0.3 0.4} {0.6 0.6 0.8}]
} -result 0.0

#
# Numerical tests - note: the expected values were taken from the implementation
#                         No independent source found
#
test "KL-divergence-1.6" "Shifted histograms" -match tolerant -body {
    set distance [::math::statistics::kl-divergence {1.0e-8 0.3 0.3 0.3 0.1} {0.3 0.3 0.3 0.1 1.0e-8}]
} -result 1.9413931

test "KL-divergence-1.7" "Arbitrary histograms" -match tolerant -body {
    set distance [::math::statistics::kl-divergence {0.4 0.2 0.3 0.1} {0.1 0.1 0.7 0.1}]
} -result 0.4389578


#
# Tests for logistic regression
#
set xdata {0.50 0.75 1.00 1.25 1.50 1.75 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 4.00 4.25 4.50 4.75 5.00 5.50}
set ydata {0    0    0    0    0    0    1    0    1    0    1    0    1    0    1    1    1    1    1    1   }

test "Logit-regression-1.0" "Logistic regression - coefficients" -match tolerant -body {
    set coeffs [::math::statistics::logistic-model $xdata $ydata]
} -result {-4.07679035286303 1.5045982920571572}

test "Logit-regression-1.1" "Logistic regression - probabilities" -match tolerant -body {
    set coeffs [::math::statistics::logistic-model $xdata $ydata]

    set probabilities {}
    foreach x {1 2 3 4 5} {
        lappend probabilities [::math::statistics::logistic-probability $coeffs $x]
    }

    return $probabilities

} -result {0.07094967663389527 0.2558609520815849 0.6075450376084848 0.8745281239093334 0.9691176473773739}


# End of test cases
testsuiteCleanup