Files

Math

Constants

EPSILON
FACTORIALS

First 16 factorials.

INVERSE_LN_2

Public Class Methods

abs(x) click to toggle source

Absolute value of x.

# File lib/facets/math/abs.rb, line 4
def self.abs(x)
  x.abs
end
absolute_mean_difference(array) click to toggle source
Alias for: amd
acot(x) click to toggle source

Arcus cotangens of x

# File lib/facets/math/acot.rb, line 4
def self.acot(x)
  (PI * 0.5) - atan(x)
end
acoth(x) click to toggle source

Area cotangens hyperbolicus of x

# File lib/facets/math/acoth.rb, line 4
def self.acoth(x)
  0.5 * log((x + 1.0) / (x - 1.0))
end
acsc(x) click to toggle source

Arcus cosecans of x

# File lib/facets/math/acsc.rb, line 4
def self.acsc(x)
  asin(1.0 / x)
end
amd(array) click to toggle source

The average absolute difference of two independent values drawn from the sample. Equal to the RMD * mean.

# File lib/facets/math/amd.rb, line 8
def self.amd(array)
        rmd(array) * mean(array)
end
Also aliased as: absolute_mean_difference
approx_equal(a, b, epsilon=EPSILON) click to toggle source

Approximately equal.

TODO: Use core extension Numeric#approx? instead (?)

# File lib/facets/math/approx_equal.rb, line 9
def self.approx_equal(a, b, epsilon=EPSILON)
        c = a - b
        c *= -1.0 if c < 0
  c < epsilon
end
asec(x) click to toggle source

Arcus secans of x

# File lib/facets/math/asec.rb, line 4
def self.asec(x)
  acos(1.0 / x)
end
atkinson_index(array) click to toggle source

Closely related to the Theil index and easily expressible in terms of it.

AI = 1-e^{theil_index}

en.wikipedia.org/wiki/Atkinson_index

# File lib/facets/math/atkinson_index.rb, line 11
def self.atkinson_index(array)
  t = theil_index(array)
  (t < 0) ? -1 : 1-Math::E**(-t)
end
beta(x, y) click to toggle source

Beta function of x and y - beta(x, y) = tgamma(x) * tgamma(y) / tgamma(x + y)

# File lib/facets/math/beta.rb, line 5
def self.beta(x, y)
  exp(lgamma(x) + lgamma(y) - lgamma(x+y))
end
cdf(array, normalised=1.0) click to toggle source

Returns the Cumulative Density Function of this sample (normalised to a fraction of 1.0).

# File lib/facets/math/cdf.rb, line 5
def self.cdf(array, normalised=1.0)
  s = sum(array).to_f
  array.sort.inject([0.0]) { |c,d| c << c[-1] + normalised*d.to_f/s }
end
ceil(x) click to toggle source

Smallest integer not smaller than x.

# File lib/facets/math/ceil.rb, line 4
def self.ceil(x)
  x.ceil
end
cot(x) click to toggle source

Cotangens of x

# File lib/facets/math/cot.rb, line 4
def self.cot(x)
  tan((PI * 0.5) - x)
end
coth(x) click to toggle source

Cotangens hyperbolicus of x

# File lib/facets/math/coth.rb, line 4
def self.coth(x)
  1.0 / tanh(x)
end
csc(x) click to toggle source

Cosecans of x

# File lib/facets/math/csc.rb, line 4
def self.csc(x)
  1.0 / sin(x)
end
csch(x) click to toggle source

Cosecans hyperbolicus of x

# File lib/facets/math/csch.rb, line 4
def self.csch(x)
  1.0 / sinh(x)
end
delta(i, j) click to toggle source

Kronecker symbol of i and j. Returns 1 if i and j are equal, 0 otherwise.

# File lib/facets/math/delta.rb, line 5
def self.delta(i, j)
  return Integer(i) == Integer(j) ? 1 : 0
end
epsilon(i, j, k) click to toggle source

Levi-Civita symbol of i, j, and k - 1 if (i, j, k) is (1, 2, 3), (2, 3, 1), or (3, 1, 2), -1 if it is (1, 3, 2), (2, 1, 3), or (3, 2, 1), 0 as long as i, j, and k are all elements of {1, 2, 3}, otherwise returns nil.

# File lib/facets/math/epsilon.rb, line 7
def self.epsilon(i, j, k)
  i = Integer(i)
  return nil if i < 1 or i > 3
  j = Integer(j)
  return nil if j < 1 or j > 3
  k = Integer(k)
  return nil if k < 1 or k > 3
  case i * 16 + j * 4 + k
    when 27, 45, 54 then return  1
    when 30, 39, 57 then return -1
  end
  0
end
exp10(x) click to toggle source

10 to the power x

# File lib/facets/math/exp10.rb, line 4
def self.exp10(x)
  10.0 ** x
end
exp2(x) click to toggle source

2 to the power x

# File lib/facets/math/exp2.rb, line 4
def self.exp2(x)
  2.0 ** x
end
factorial(n) click to toggle source

1 * 2 * … * n, nil for negative numbers

# File lib/facets/math/factorial.rb, line 24
def self.factorial(n)
  n = Integer(n)
  if n < 0
    nil
  elsif FACTORIALS.length > n
    FACTORIALS[n]
  else
    h = FACTORIALS.last
    (FACTORIALS.length .. n).each { |i| FACTORIALS.push h *= i }
    h
  end
end
floor(x) click to toggle source

Largest integer not larger than x.

# File lib/facets/math/floor.rb, line 4
def self.floor(x)
  x.floor
end
gamma(x) click to toggle source
# File lib/facets/math/gamma.rb, line 7
def self.gamma(x)
  exp(lgamma(x))
end
gcd(m, n) click to toggle source

Greatest common divisor of m and n, nil for non-positive numbers - gcd is computed by means of the Euclidian algorithm.

# File lib/facets/math/gcd.rb, line 5
def self.gcd(m, n)
  m = Integer(m)
  n = Integer(n)
  if m <= 0 || n <= 0
    return nil
  end
  loop {
    if m < n
      m, n = n, m
    end
    if (l = m % n) == 0
      break
    end
    m = l
  }
  n
end
gini_coefficient(array) click to toggle source

Calculates the Gini Coefficient (a measure of inequality of a distribution based on the area between the Lorenz curve and the uniform curve).

en.wikipedia.org/wiki/Gini_coefficient

This is a slightly cleaner way of calculating the Gini Coefficient then the previous implementationj.

GC = \frac{\sum_{i=1}^N (2i-N-1)x_i}{N^2-\bar{x}}
# File lib/facets/math/gini_coefficient.rb, line 15
def self.gini_coefficient(array)
  return -1 if size <= 0 or any? { |x| x < 0 }
  return 0 if size < 2 or all? { |x| approx_equal(x,0) }
  s = 0
  sort.each_with_index { |li,i| s += (2*i+1-size)*li }
  s.to_f/(size**2*mean).to_f
end
kldivergence(array, q) click to toggle source

The Kullback-Leibler divergence from this array to that of q.

NB: You will possibly want to sort both P and Q before calling this depending on what you’re actually trying to measure.

en.wikipedia.org/wiki/Kullback-Leibler_divergence

# File lib/facets/math/kldivergence.rb, line 10
def self.kldivergence(array, q)
  fail "Buggy."
  fail "Cannot compare differently sized arrays." unless size = q.size
  kld = 0
  each_with_index { |pi,i| kld += pi*Math::log(pi.to_f/q[i].to_f) }
  kld
end
lcm(m, n) click to toggle source

Least common multiple of m and n, computed by multiplying m and n and dividing the product by the gcd of m and n, nil for non-positive numbers.

# File lib/facets/math/lcm.rb, line 6
def self.lcm(m, n)
  m = Integer(m)
  n = Integer(n)
  if m <= 0 || n <= 0
    return nil
  end
  m / gcd(m, n) * n
end
lgamma(x) click to toggle source

Logarithmus naturalis of gamma function of x.

@author Josef Schugt

# File lib/facets/math/lgamma.rb, line 8
def self.lgamma(x)
  h  = x + 5.5
  h -= (x + 0.5) * log(h)
  sum  =  1.000_000_000_190_015
  sum += 76.180_091_729_471_46           / (x + 1.0)
  sum -= 86.505_320_329_416_77           / (x + 2.0)
  sum += 24.014_098_240_830_91           / (x + 3.0)
  sum -=  1.231_739_572_450_155          / (x + 4.0)
  sum +=  0.120_865_097_386_617_9e-2 / (x + 5.0)
  sum -=  0.539_523_938_495_3e-5     / (x + 6.0)
  -h + log(2.506_628_274_631_000_5 * sum / x)
end
linsolve(a, b, c = 0.0) click to toggle source

Returns real solution(s) of +a+x + b = c or nil if no or an infinite number of solutions exist. If c is missing it is assumed to be 0.

@author Josef Schugt

# File lib/facets/math/linsolve.rb, line 8
def self.linsolve(a, b, c = 0.0)
  a == 0 ? nil : (c - b) / a
end
log2(x) click to toggle source

Logarithmus dualis of x.

# File lib/facets/math/log2.rb, line 8
def self.log2(x)
  Math.log(x) * INVERSE_LN_2
end
max(array, block) click to toggle source
# File lib/facets/math/min.rb, line 20
def self.max(array, block)
  if block_given?
    if max = find{|i| i}
      max = yield(max)
      each{|i|
        j = yield(i)
        max = j if max < j
      }
      max
    end
  else
    array.max
  end
end
mean(array, &blk) click to toggle source

Mean average.

# File lib/facets/math/mean.rb, line 6
def self.mean(array, &blk)
  s = array.size
  return 0.0 if s == 0
  sum(array, &blk) / s
end
Also aliased as: mean_average
mean_average(array, &blk) click to toggle source
Alias for: mean
median(array) click to toggle source

Returns the numerical median for the an array of values; or nil if array is empty.

# File lib/facets/math/median.rb, line 8
def self.median(array)
  percentile(array, 50)
end
min(array, &block) click to toggle source
# File lib/facets/math/min.rb, line 4
def self.min(array, &block)
  if block_given?
    if min = array.find{ |i| i }
      min = yield(min)
      array.each do |i|
        j = yield(i)
        min = j if min > j
      end
      min
    end
  else
    array.min
  end
end
percentile(array, pcnt) click to toggle source

Returns the percentile value for percentile pcnt; nil if array is empty.

pcnt should be expressed as an integer, e.g. `percentile(90)` returns the 90th percentile of the array.

Algorithm from NIST

NOTE: This is not a common core extension and is not loaded automatically when using require 'facets'.

CREDIT: Ben Koski

@non-core

require 'facets/array/precentile'
# File lib/facets/math/percentile.rb, line 18
def self.percentile(array, pcnt)
  sorted_array = array.sort

  return nil if array.length == 0

  rank  = (pcnt.to_f / 100) * (array.length + 1)
  whole = rank.truncate
 
  # if has fractional part
  if whole != rank
    s0 = sorted_array[whole - 1]
    s1 = sorted_array[whole]

    f = (rank - rank.truncate).abs

    return (f * (s1 - s0)) + s0
  else
    return sorted_array[whole - 1]
  end
end
pow(x, y) click to toggle source

x to the power y

# File lib/facets/math/pow.rb, line 4
def self.pow(x, y)
  x ** y
end
pstd(array, &block) click to toggle source

Standard deviation of a population.

# File lib/facets/math/std.rb, line 17
def self.pstd(array, &block)
  Math::sqrt(pvariance(array, &block))
end
pvariance(array) click to toggle source

Variance of a population. Variance of 0 or 1 elements is 0.0.

# File lib/facets/math/variance.rb, line 26
def self.pvariance(array)
  return 0.0 if array.size < 2
  summed_sqdevs(array) / array.size
end
relative_mean_difference(array) click to toggle source
Alias for: rmd
rmd(array) click to toggle source

Calculates the relative mean difference of this sample. Makes use of the fact that the Gini Coefficient is half the RMD.

# File lib/facets/math/rmd.rb, line 7
def self.rmd(array)
  return 0.0 if approx_equal(mean(array), 0.0)
  gini_coefficient(array) * 2
end
Also aliased as: relative_mean_difference
root(x, y) click to toggle source

The y root of x.

# File lib/facets/math/root.rb, line 4
def self.root(x, y)
  x ** (1.0 / y)
end
round(x) click to toggle source

Round number to an integer.

# File lib/facets/math/round.rb, line 5
def self.round(x)
  x.round
end
sec(x) click to toggle source

Secans of x.

# File lib/facets/math/sec.rb, line 4
def self.sec(x)
  1.0 / cos(x)
end
sech(x) click to toggle source

Secans hyperbolicus of x

# File lib/facets/math/sech.rb, line 4
def self.sech(x)
  1.0 / cosh(x)
end
sign(x) click to toggle source

Sign of x. Returns -1 for negative x, +1 for positive x and zero for x = 0

# File lib/facets/math/sign.rb, line 5
def self.sign(x)
  (x > 0.0) ? 1.0 : ((x < 0.0) ? -1.0 : 0.0)
end
sinc(x) click to toggle source

Sinc function of x.

# File lib/facets/math/sinc.rb, line 4
def self.sinc(x)
  (x == 0.0) ? 1.0 : sin(x) / x
end
sqr(x) click to toggle source

Square of number.

# File lib/facets/math/sqr.rb, line 4
def self.sqr(x)
  x * x
end
sqsolve(a, b, c, d = 0.0) click to toggle source

Returns array of real solution of ax**2 + bx + c = d or nil if no or an infinite number of solutions exist. If d is missing it is assumed to be 0.

In order to solve ax**2 + bx + c = d sqsolve identifies several cases:

  • a == 0: The equation to be solved is the linear equation bx + c = d. sqsolve> delegates the computation to linsolve>. If it results in nil, nil is returned (not [nil]!). Otherwise a one-element array containing result of linsolve is returned.

  • a != 0:

    The equation to be solved actually is a second order one.
    * <code>c == d</code>
      The equation to be solved is <code>ax**2 + bx = 0</code>. One solution of this equation obviously is
      <code>x = 0</code>, the second one solves <code>ax + b = 0</code>. The solution of the latter is
      delegated to +linsolve+. An array containing both results in ascending order is returned.
    * <code>c != d</code>
      The equation cannot be separated into <code>x</code> times some factor.
      * <code>b == 0</code>
        The equation to be solved is <code>ax**2 + c = d</code>. This can be written as the linear equation
        <code>ay + c = d</code> with <code>y = x ** 2</code>. The solution of the linear equation is delegated
        to +linsolve+. If the returned value for +y+ is +nil+, that becomes the overall return value.
        Otherwise an array containing the negative and positive squareroot of +y+ is returned
      * <code>b != 0 </code>
        The equation cannot be reduced to simpler cases. We now first have to compute what is called the
        discriminant <code>x = b**2 + 4a(d - c)</code> (that's what we need to compute the square root of).
        If the descriminant is negative no real solution exists and <code>nil</code> is returned. The ternary
        operator checking whether <code>b</code> is negative does ensure better numerical stability --only one
        of the two solutions is computed using the widely know formula for solving second order equations.
        The second one is computed from the fact that the product of both solutions is <code>(c - d) / a</code>.
        Take a look at a book on numerical mathematics if you don't understand why this should be done.

@author Josef Schugt

# File lib/facets/math/sqsolve.rb, line 37
def self.sqsolve(a, b, c, d = 0.0)
  if a == 0.0
    x = linsolve(b, c, d)
    return x.nil? ? nil: [ linsolve(b, c, d) ]
  else
    return [0.0, linsolve(a, b)].sort if c == d
    if b == 0.0
      x = linsolve(a, c, d)
      x < 0.0 ? nil : [-Math.sqrt(x), Math.sqrt(x)]
    else
      x = b * b + 4.0 * a * (d - c)
      return nil if x < 0.0
      x = b < 0 ? b - Math.sqrt(x) : b + Math.sqrt(x)
      [-0.5 * x / a, 2.0 * (d - c) / x].sort
    end
  end
end
standard_deviation(array, &block) click to toggle source
Alias for: std
std(array, &block) click to toggle source

Standard deviation of a sample.

# File lib/facets/math/std.rb, line 7
def self.std(array, &block)
  sqrt(variance(array, &block))
end
Also aliased as: standard_deviation
stderr(array) click to toggle source

Calculates the standard error of a sample.

# File lib/facets/math/std.rb, line 22
def self.stderr(array)
  return 0.0 if array.size < 2
  std(array) / sqrt(array.size)
end
sum(array) click to toggle source

Returns sum. When a block is given, summation is taken over the each result of block evaluation.

# File lib/facets/math/sum.rb, line 6
def self.sum(array) #:yield:
  sum = 0.0
  if block_given?
    array.each{|i| sum += yield(i)}
  else
    array.each{|i| sum += i}
  end
  sum
end
summed_sqdevs(array) click to toggle source

The sum of the squared deviations from the mean.

# File lib/facets/math/summed_sqdevs.rb, line 8
def self.summed_sqdevs(array)
  return 0 if array.size < 2
  m = mean(array)
  sum(array.map{ |x| (x - m) ** 2 })
end
tgamma(x) click to toggle source

Exp of LGamma.

# File lib/facets/math/tgamma.rb, line 6
def self.tgamma(x)
  exp(lgamma(x))
end
theil_index(array) click to toggle source

Calculates the Theil index (a statistic used to measure economic inequality).

TI = sum_{i=1}^N frac{x_i}{sum_{j=1}^N x_j} ln frac{x_i}{bar{x}}

http://en.wikipedia.org/wiki/Theil_index
# File lib/facets/math/theil_index.rb, line 14
def self.theil_index(array)
  return -1 if array.size <= 0 or any? { |x| x < 0 }
  return  0 if array.size <  2 or all? { |x| approx_equal(x, 0) }
  m = mean(array)
  s = sum(array).to_f
  inject(0) do |theil, xi|
          theil + ((xi > 0) ? (log(xi.to_f/m) * xi.to_f/s) : 0.0)
  end
end
variance(array, &block) click to toggle source
# File lib/facets/math/variance.rb, line 6
def self.variance(array, &block)
  sum2 = if block_given?
    sum(array){ |i| j = block[i]; j*j }
  else
    sum(array){ |i| i**2 }
  end
  sum2/array.size - mean(array, &block)**2
end
variance2(array) click to toggle source

Variance of the sample. Variance of 0 or 1 elements is 0.0.

TODO: Same as variance? Then choose one.

# File lib/facets/math/variance.rb, line 19
def self.variance2(array)
  return 0.0 if array.size < 2
  summed_sqdevs(array) / (array.size - 1)
end

Public Instance Methods

acsch(x) click to toggle source

Area cosecans hyperbolicus of x

# File lib/facets/math/acsch.rb, line 4
def acsch(x)
  ::Math.log(1.0 / x + Math.sqrt(1.0 + 1.0 / (x * x)))
end
asech(x) click to toggle source

Area secans hyperbolicus of x

# File lib/facets/math/asech.rb, line 4
def asech(x)
  log((1.0 + sqrt(1.0 - x * x)) / x)
end

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