path: root/tcllib/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     8.4;# statistics,linalg!
testsNeedTcltest 2.1

support {
    useLocal math.tcl math
    useLocal linalg.tcl math::linearalgebra
}
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}

#
# 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 "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 "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}

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 $stdev [expr {(exp($sigma*$sigma) - 1.0) * $m*$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 $stdev [expr {(exp($sigma*$sigma) - 1.0) * $m*$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 "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-2.1" "Test 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 "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

# 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

# End of test cases
testsuiteCleanup