#
#
# Nim's Runtime Library
# (c) Copyright 2015 Nim contributors
#
# See the file "copying.txt", included in this
# distribution, for details about the copyright.
#
## Statistical analysis framework for performing
## basic statistical analysis of data.
## The data is analysed in a single pass, when a data value
## is pushed to the ``RunningStat`` or ``RunningRegress`` objects
##
## ``RunningStat`` calculates for a single data set
## - n (data count)
## - min (smallest value)
## - max (largest value)
## - sum
## - mean
## - variance
## - varianceS (sample var)
## - standardDeviation
## - standardDeviationS (sample stddev)
## - skewness (the third statistical moment)
## - kurtosis (the fourth statistical moment)
##
## ``RunningRegress`` calculates for two sets of data
## - n
## - slope
## - intercept
## - correlation
##
## Procs have been provided to calculate statistics on arrays and sequences.
##
## However, if more than a single statistical calculation is required, it is more
## efficient to push the data once to the RunningStat object, and
## call the numerous statistical procs for the RunningStat object.
##
## .. code-block:: Nim
##
## var rs: RunningStat
## rs.push(MySeqOfData)
## rs.mean()
## rs.variance()
## rs.skewness()
## rs.kurtosis()
from math import FloatClass, sqrt, pow, round
{.push debugger: off.} # the user does not want to trace a part
# of the standard library!
{.push checks: off, line_dir: off, stack_trace: off.}
type
RunningStat* = object ## an accumulator for statistical data
n*: int ## number of pushed data
min*, max*, sum*: float ## self-explaining
mom1, mom2, mom3, mom4: float ## statistical moments, mom1 is mean
RunningRegress* = object ## an accumulator for regression calculations
n*: int ## number of pushed data
x_stats*: RunningStat ## stats for first set of data
y_stats*: RunningStat ## stats for second set of data
s_xy: float ## accumulated data for combined xy
# ----------- RunningStat --------------------------
proc clear*(s: var RunningStat) =
## reset `s`
s.n = 0
s.min = toBiggestFloat(int.high)
s.max = 0.0
s.sum = 0.0
s.mom1 = 0.0
s.mom2 = 0.0
s.mom3 = 0.0
s.mom4 = 0.0
proc push*(s: var RunningStat, x: float) =
## pushes a value `x` for processing
if s.n == 0: s.min = x
inc(s.n)
# See Knuth TAOCP vol 2, 3rd edition, page 232
if s.min > x: s.min = x
if s.max < x: s.max = x
s.sum += x
let n = toFloat(s.n)
let delta = x - s.mom1
let delta_n = delta / toFloat(s.n)
let delta_n2 = delta_n * delta_n
let term1 = delta * delta_n * toFloat(s.n - 1)
s.mom4 += term1 * delta_n2 * (n*n - 3*n + 3) +
6*delta_n2*s.mom2 - 4*delta_n*s.mom3
s.mom3 += term1 * delta_n * (n - 2) - 3*delta_n*s.mom2
s.mom2 += term1
s.mom1 += delta_n
proc push*(s: var RunningStat, x: int) =
## pushes a value `x` for processing.
##
## `x` is simply converted to ``float``
## and the other push operation is called.
s.push(toFloat(x))
proc push*(s: var RunningStat, x: openarray[float|int]) =
## pushes all values of `x` for processing.
##
## Int values of `x` are simply converted to ``float`` and
## the other push operation is called.
for val in x:
s.push(val)
proc mean*(s: RunningStat): float =
## computes the current mean of `s`
result = s.mom1
proc variance*(s: RunningStat): float =
## computes the current population variance of `s`
result = s.mom2 / toFloat(s.n)
proc varianceS*(s: RunningStat): float =
## computes the current sample variance of `s`
if s.n > 1: result = s.mom2 / toFloat(s.n - 1)
proc standardDeviation*(s: RunningStat): float =
## computes the current population standard deviation of `s`
result = sqrt(variance(s))
proc standardDeviationS*(s: RunningStat): float =
## computes the current sample standard deviation of `s`
result = sqrt(varianceS(s))
proc skewness*(s: RunningStat): float =
## computes the current population skewness of `s`
result = sqrt(toFloat(s.n)) * s.mom3 / pow(s.mom2, 1.5)
proc skewnessS*(s: RunningStat): float =
## computes the current sample skewness of `s`
let s2 = skewness(s)
result = sqrt(toFloat(s.n*(s.n-1)))*s2 / toFloat(s.n-2)
proc kurtosis*(s: RunningStat): float =
## computes the current population kurtosis of `s`
result = toFloat(s.n) * s.mom4 / (s.mom2 * s.mom2) - 3.0
proc kurtosisS*(s: RunningStat): float =
## computes the current sample kurtosis of `s`
result = toFloat(s.n-1) / toFloat((s.n-2)*(s.n-3)) *
(toFloat(s.n+1)*kurtosis(s) + 6)
proc `+`*(a, b: RunningStat): RunningStat =
## combine two RunningStats.
##
## Useful if performing parallel analysis of data series
## and need to re-combine parallel result sets
result.clear()
result.n = a.n + b.n
let delta = b.mom1 - a.mom1
let delta2 = delta*delta
let delta3 = delta*delta2
let delta4 = delta2*delta2
let n = toFloat(result.n)
result.mom1 = (a.n.float*a.mom1 + b.n.float*b.mom1) / n
result.mom2 = a.mom2 + b.mom2 + delta2 * a.n.float * b.n.float / n
result.mom3 = a.mom3 + b.mom3 +
delta3 * a.n.float * b.n.float * (a.n.float - b.n.float)/(n*n);
result.mom3 += 3.0*delta * (a.n.float*b.mom2 - b.n.float*a.mom2) / n
result.mom4 = a.mom4 + b.mom4 +
delta4*a.n.float*b.n.float * toFloat(a.n*a.n - a.n*b.n + b.n*b.n) /
(n*n*n)
result.mom4 += 6.0*delta2 * (a.n.float*a.n.float*b.mom2 + b.n.float*b.n.float*a.mom2) /
(n*n) +
4.0*delta*(a.n.float*b.mom3 - b.n.float*a.mom3) / n
result.max = max(a.max, b.max)
result.min = min(a.min, b.min)
proc `+=`*(a: var RunningStat, b: RunningStat) {.inline.} =
## add a second RunningStats `b` to `a`
a = a + b
proc `$`*(a: RunningStat): string =
## produces a string representation of the ``RunningStat``. The exact
## format is currently unspecified and subject to change. Currently
## it contains:
##
## - the number of probes
## - min, max values
## - sum, mean and standard deviation.
result = "RunningStat(\n"
result.add " number of probes: " & $a.n & "\n"
result.add " max: " & $a.max & "\n"
result.add " min: " & $a.min & "\n"
result.add " sum: " & $a.sum & "\n"
result.add " mean: " & $a.mean & "\n"
result.add " std deviation: " & $a.standardDeviation & "\n"
result.add ")"
# ---------------------- standalone array/seq stats ---------------------
proc mean*[T](x: openArray[T]): float =
## computes the mean of `x`
var rs: RunningStat
rs.push(x)
result = rs.mean()
proc variance*[T](x: openArray[T]): float =
## computes the population variance of `x`
var rs: RunningStat
rs.push(x)
result = rs.variance()
proc varianceS*[T](x: openArray[T]): float =
## computes the sample variance of `x`
var rs: RunningStat
rs.push(x)
result = rs.varianceS()
proc standardDeviation*[T](x: openArray[T]): float =
## computes the population standardDeviation of `x`
var rs: RunningStat
rs.push(x)
result = rs.standardDeviation()
proc standardDeviationS*[T](x: openArray[T]): float =
## computes the sample standardDeviation of `x`
var rs: RunningStat
rs.push(x)
result = rs.standardDeviationS()
proc skewness*[T](x: openArray[T]): float =
## computes the population skewness of `x`
var rs: RunningStat
rs.push(x)
result = rs.skewness()
proc skewnessS*[T](x: openArray[T]): float =
## computes the sample skewness of `x`
var rs: RunningStat
rs.push(x)
result = rs.skewnessS()
proc kurtosis*[T](x: openArray[T]): float =
## computes the population kurtosis of `x`
var rs: RunningStat
rs.push(x)
result = rs.kurtosis()
proc kurtosisS*[T](x: openArray[T]): float =
## computes the sample kurtosis of `x`
var rs: RunningStat
rs.push(x)
result = rs.kurtosisS()
# ---------------------- Running Regression -----------------------------
proc clear*(r: var RunningRegress) =
## reset `r`
r.x_stats.clear()
r.y_stats.clear()
r.s_xy = 0.0
r.n = 0
proc push*(r: var RunningRegress, x, y: float) =
## pushes two values `x` and `y` for processing
r.s_xy += (r.x_stats.mean() - x)*(r.y_stats.mean() - y) *
toFloat(r.n) / toFloat(r.n + 1)
r.x_stats.push(x)
r.y_stats.push(y)
inc(r.n)
proc push*(r: var RunningRegress, x, y: int) {.inline.} =
## pushes two values `x` and `y` for processing.
##
## `x` and `y` are converted to ``float``
## and the other push operation is called.
r.push(toFloat(x), toFloat(y))
proc push*(r: var RunningRegress, x, y: openarray[float|int]) =
## pushes two sets of values `x` and `y` for processing.
assert(x.len == y.len)
for i in 0..<x.len:
r.push(x[i], y[i])
proc slope*(r: RunningRegress): float =
## computes the current slope of `r`
let s_xx = r.x_stats.varianceS()*toFloat(r.n - 1)
result = r.s_xy / s_xx
proc intercept*(r: RunningRegress): float =
## computes the current intercept of `r`
result = r.y_stats.mean() - r.slope()*r.x_stats.mean()
proc correlation*(r: RunningRegress): float =
## computes the current correlation of the two data
## sets pushed into `r`
let t = r.x_stats.standardDeviation() * r.y_stats.standardDeviation()
result = r.s_xy / (toFloat(r.n) * t)
proc `+`*(a, b: RunningRegress): RunningRegress =
## combine two `RunningRegress` objects.
##
## Useful if performing parallel analysis of data series
## and need to re-combine parallel result sets
result.clear()
result.x_stats = a.x_stats + b.x_stats
result.y_stats = a.y_stats + b.y_stats
result.n = a.n + b.n
let delta_x = b.x_stats.mean() - a.x_stats.mean()
let delta_y = b.y_stats.mean() - a.y_stats.mean()
result.s_xy = a.s_xy + b.s_xy +
toFloat(a.n*b.n)*delta_x*delta_y/toFloat(result.n)
proc `+=`*(a: var RunningRegress, b: RunningRegress) =
## add RunningRegress `b` to `a`
a = a + b
{.pop.}
{.pop.}
runnableExamples:
static:
block:
var statistics: RunningStat ## Must be "var"
statistics.push(@[1.0, 2.0, 1.0, 4.0, 1.0, 4.0, 1.0, 2.0])
doAssert statistics.n == 8
template `===`(a, b: float): bool = (abs(a - b) < 1e-9)
doAssert statistics.mean() === 2.0
doAssert statistics.variance() === 1.5
doAssert statistics.varianceS() === 1.714285714285715
doAssert statistics.skewness() === 0.8164965809277261
doAssert statistics.skewnessS() === 1.018350154434631
doAssert statistics.kurtosis() === -1.0
doAssert statistics.kurtosisS() === -0.7000000000000008
when isMainModule:
proc clean(x: float): float =
result = round(1.0e8*x).float * 1.0e-8
var rs: RunningStat
rs.push(@[1.0, 2.0, 1.0, 4.0, 1.0, 4.0, 1.0, 2.0])
doAssert(rs.n == 8)
doAssert(clean(rs.mean) == 2.0)
doAssert(clean(rs.variance()) == 1.5)
doAssert(clean(rs.varianceS()) == 1.71428571)
doAssert(clean(rs.skewness()) == 0.81649658)
doAssert(clean(rs.skewnessS()) == 1.01835015)
doAssert(clean(rs.kurtosis()) == -1.0)
doAssert(clean(rs.kurtosisS()) == -0.7000000000000001)
var rs1, rs2: RunningStat
rs1.push(@[1.0, 2.0, 1.0, 4.0])
rs2.push(@[1.0, 4.0, 1.0, 2.0])
let rs3 = rs1 + rs2
doAssert(clean(rs3.mom2) == clean(rs.mom2))
doAssert(clean(rs3.mom3) == clean(rs.mom3))
doAssert(clean(rs3.mom4) == clean(rs.mom4))
rs1 += rs2
doAssert(clean(rs1.mom2) == clean(rs.mom2))
doAssert(clean(rs1.mom3) == clean(rs.mom3))
doAssert(clean(rs1.mom4) == clean(rs.mom4))
rs1.clear()
rs1.push(@[1.0, 2.2, 1.4, 4.9])
doAssert(rs1.sum == 9.5)
doAssert(rs1.mean() == 2.375)
when not defined(cpu32):
# XXX For some reason on 32bit CPUs these results differ
var rr: RunningRegress
rr.push(@[0.0, 1.0, 2.8, 3.0, 4.0], @[0.0, 1.0, 2.3, 3.0, 4.0])
doAssert(rr.slope() == 0.9695585996955861)
doAssert(rr.intercept() == -0.03424657534246611)
doAssert(rr.correlation() == 0.9905100362239381)
var rr1, rr2: RunningRegress
rr1.push(@[0.0, 1.0], @[0.0, 1.0])
rr2.push(@[2.8, 3.0, 4.0], @[2.3, 3.0, 4.0])
let rr3 = rr1 + rr2
doAssert(rr3.correlation() == rr.correlation())
doAssert(clean(rr3.slope()) == clean(rr.slope()))
doAssert(clean(rr3.intercept()) == clean(rr.intercept()))