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# stat_kernel.tcl --
#
# Part of the statistics package for basic statistical analysis
# Based on http://en.wikipedia.org/wiki/Kernel_(statistics) and
# http://en.wikipedia.org/wiki/Kernel_density_estimation
#
# version 0.1: initial implementation, january 2014
# kernel-density --
# Estimate the probability density using the kernel density
# estimation method
#
# Arguments:
# data List of univariate data
# args List of options in the form of keyword-value pairs:
# -weights weights: per data point the weight
# -bandwidth value: bandwidth to be used for the estimation
# -number value: number of bins to be returned
# -interval {begin end}: begin and end of the interval for
# which the density is returned
# -kernel function: kernel to be used (gaussian, cosine,
# epanechnikov, uniform, triangular, biweight,
# logistic)
# For all options more or less sensible defaults are
# provided.
#
# Result:
# A list of the bin centres, a list of the corresponding density
# estimates and a list containing several computational parameters:
# begin and end of the interval, mean, standard deviation and bandwidth
#
# Note:
# The conditions for the kernel function are fairly weak:
# - It should integrate to 1
# - It should be symmetric around 0
#
# As for the implementation in Tcl: it should be reachable in the
# ::math::statistics namespace. As a consequence, you can define
# your own kernel function too. Hence there is no check.
#
proc ::math::statistics::kernel-density {data args} {
#
# Determine the basic statistics
#
set basicStats [BasicStats all $data]
set mean [lindex $basicStats 0]
set ndata [lindex $basicStats 3]
set stdev [lindex $basicStats 4]
if { $ndata < 1 } {
return -code error -errorcode ARG -errorinfo "Too few actual data"
}
#
# Get the options (providing defaults as needed)
#
set opt(-weights) {}
set opt(-number) 100
set opt(-kernel) gaussian
#
# The default bandwidth is set via a simple expression, which
# is supposed to be optimal for the Gaussian kernel.
# Perhaps a more sophisticated method should be provided as well
#
set opt(-bandwidth) [expr {1.06 * $stdev / pow($ndata,0.2)}]
#
# The default interval is derived from the mean and the
# standard deviation
#
set opt(-interval) [list [expr {$mean - 3.0 * $stdev}] [expr {$mean + 3.0 * $stdev}]]
#
# Retrieve the given options from $args
#
if { [llength $args] % 2 != 0 } {
return -code error -errorcode ARG -errorinfo "The options must all have a value"
}
array set opt $args
#
# Elementary checks
#
if { $opt(-bandwidth) <= 0.0 } {
return -code error -errorcode ARG -errorinfo "The bandwidth must be positive: $opt(-bandwidth)"
}
if { $opt(-number) <= 0.0 } {
return -code error -errorcode ARG -errorinfo "The number of bins must be positive: $opt(-number)"
}
if { [lindex $opt(-interval) 0] == [lindex $opt(-interval) 1] } {
return -code error -errorcode ARG -errorinfo "The interval has length zero: $opt(-interval)"
}
if { [llength [info proc $opt(-kernel)]] == 0 } {
return -code error -errorcode ARG -errorinfo "Unknown kernel function: $opt(-kernel)"
}
#
# Construct the weights
#
if { [llength $opt(-weights)] > 0 } {
if { [llength $data] != [llength $opt(-weights)] } {
return -code error -errorcode ARG -errorinfo "The list of weights must match the data"
}
set sum 0.0
foreach d $data w $opt(-weights) {
if { $d != {} } {
set sum [expr {$sum + $w}]
}
}
set scale [expr {1.0/$sum/$ndata}]
set weight {}
foreach w $opt(-weights) {
if { $d != {} } {
lappend weight [expr {$w / $scale}]
} else {
lappend weight {}
}
}
} else {
set weight [lrepeat [llength $data] [expr {1.0/$ndata}]] ;# Note: missing values have weight zero
}
#
# Construct the centres of the bins
#
set xbegin [lindex $opt(-interval) 0]
set xend [lindex $opt(-interval) 1]
set dx [expr {($xend - $xbegin) / double($opt(-number))}]
set xb [expr {$xbegin + 0.5 * $dx}]
set xvalue {}
for {set i 0} {$i < $opt(-number)} {incr i} {
lappend xvalue [expr {$xb + $i * $dx}]
}
#
# Construct the density function
#
set density {}
set scale [expr {1.0/$opt(-bandwidth)}]
foreach x $xvalue {
set sum 0.0
foreach d $data w $weight {
if { $d != {} } {
set kvalue [$opt(-kernel) [expr {$scale * ($x-$d)}]]
set sum [expr {$sum + $w * $kvalue}]
}
}
lappend density [expr {$sum * $scale}]
}
#
# Return the result
#
return [list $xvalue $density [list $xbegin $xend $mean $stdev $opt(-bandwidth)]]
}
# gaussian, uniform, triangular, epanechnikov, biweight, cosine, logistic --
# The Gaussian kernel
#
# Arguments:
# x (Scaled) argument
#
# Result:
# Value of the kernel
#
# Note:
# The standard deviation is 1.
#
proc ::math::statistics::gaussian {x} {
return [expr {exp(-0.5*$x*$x) / sqrt(2.0*acos(-1.0))}]
}
proc ::math::statistics::uniform {x} {
if { abs($x) <= 1.0 } {
return 0.5
} else {
return 0.0
}
}
proc ::math::statistics::triangular {x} {
if { abs($x) < 1.0 } {
return [expr {1.0 - abs($x)}]
} else {
return 0.0
}
}
proc ::math::statistics::epanechnikov {x} {
if { abs($x) < 1.0 } {
return [expr {0.75 * (1.0 - abs($x)*abs($x))}]
} else {
return 0.0
}
}
proc ::math::statistics::biweight {x} {
if { abs($x) < 1.0 } {
return [expr {0.9375 * pow((1.0 - abs($x)*abs($x)),2)}]
} else {
return 0.0
}
}
proc ::math::statistics::cosine {x} {
if { abs($x) < 1.0 } {
return [expr {0.25 * acos(-1.0) * cos(0.5 * acos(-1.0) * $x)}]
} else {
return 0.0
}
}
proc ::math::statistics::logistic {x} {
return [expr {1.0 / (exp($x) + 2.0 + exp(-$x))}]
}
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