Module: Math

Defined in:
lib/standard/facets/math/ec.rb,
lib/standard/facets/math/abs.rb,
lib/standard/facets/math/amd.rb,
lib/standard/facets/math/cdf.rb,
lib/standard/facets/math/cot.rb,
lib/standard/facets/math/csc.rb,
lib/standard/facets/math/gcd.rb,
lib/standard/facets/math/lcm.rb,
lib/standard/facets/math/min.rb,
lib/standard/facets/math/pow.rb,
lib/standard/facets/math/rmd.rb,
lib/standard/facets/math/sec.rb,
lib/standard/facets/math/sqr.rb,
lib/standard/facets/math/std.rb,
lib/standard/facets/math/sum.rb,
lib/standard/facets/math/tau.rb,
lib/standard/facets/math/acot.rb,
lib/standard/facets/math/acsc.rb,
lib/standard/facets/math/asec.rb,
lib/standard/facets/math/beta.rb,
lib/standard/facets/math/ceil.rb,
lib/standard/facets/math/coth.rb,
lib/standard/facets/math/csch.rb,
lib/standard/facets/math/exp2.rb,
lib/standard/facets/math/log2.rb,
lib/standard/facets/math/mean.rb,
lib/standard/facets/math/root.rb,
lib/standard/facets/math/sech.rb,
lib/standard/facets/math/sign.rb,
lib/standard/facets/math/sinc.rb,
lib/standard/facets/math/acoth.rb,
lib/standard/facets/math/acsch.rb,
lib/standard/facets/math/asech.rb,
lib/standard/facets/math/delta.rb,
lib/standard/facets/math/exp10.rb,
lib/standard/facets/math/floor.rb,
lib/standard/facets/math/round.rb,
lib/standard/facets/math/median.rb,
lib/standard/facets/math/tgamma.rb,
lib/standard/facets/math/epsilon.rb,
lib/standard/facets/math/lngamma.rb,
lib/standard/facets/math/sqsolve.rb,
lib/standard/facets/math/distance.rb,
lib/standard/facets/math/linsolve.rb,
lib/standard/facets/math/variance.rb,
lib/standard/facets/math/factorial.rb,
lib/standard/facets/math/percentile.rb,
lib/standard/facets/math/theil_index.rb,
lib/standard/facets/math/approx_equal.rb,
lib/standard/facets/math/kldivergence.rb,
lib/standard/facets/math/summed_sqdevs.rb,
lib/standard/facets/math/atkinson_index.rb,
lib/standard/facets/math/gini_coefficient.rb

Constant Summary collapse

EC =

Euler’s constant.

0.577_215_664_901_532_861
TAU =

See tauday.com/tau-manifesto

2 * PI
INVERSE_LN_2 = 
1.0 / ::Math.log(2.0)
FACTORIALS =

First 16 factorials.

[
  1,
  1,
  2,
  6,
  24,
  120,
  720,
  5_040,
  40_320,
  362_880,
  3_628_800,
  39_916_800,
  479_001_600,
  6_227_020_800,
  87_178_291_200,
  1_307_674_368_000
]
EPSILON = 
0.000000001

Class Method Summary collapse

Instance Method Summary collapse

Class Method Details

.abs(x) ⇒ Object

Absolute value of x.



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# File 'lib/standard/facets/math/abs.rb', line 4

def self.abs(x)
  x.abs
end

.acosec(x) ⇒ Object

Arcus cosecans of ‘x`.



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# File 'lib/standard/facets/math/acsc.rb', line 9

def self.acosec(x)
  asin(1.0 / x)
end

.acot(x) ⇒ Object

Arcus cotangens of x



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# File 'lib/standard/facets/math/acot.rb', line 4

def self.acot(x)
  (PI * 0.5) - atan(x)
end

.acoth(x) ⇒ Object

Area cotangens hyperbolicus of x



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# File 'lib/standard/facets/math/acoth.rb', line 4

def self.acoth(x)
  0.5 * log((x + 1.0) / (x - 1.0))
end

.acsc(x) ⇒ Object

Arcus cosecans of ‘x`.



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# File 'lib/standard/facets/math/acsc.rb', line 4

def self.acsc(x)
  asin(1.0 / x)
end

.amd(array) ⇒ Object Also known as: absolute_mean_difference

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



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# File 'lib/standard/facets/math/amd.rb', line 8

def self.amd(array)
 rmd(array) * mean(array)
end

.approx_equal(a, b, epsilon = EPSILON) ⇒ Object

Approximately equal.

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



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# File 'lib/standard/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) ⇒ Object

Arcus secans of x



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# File 'lib/standard/facets/math/asec.rb', line 4

def self.asec(x)
  acos(1.0 / x)
end

.atkinson_index(array) ⇒ Object

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

AI = 1-e^theil_index

en.wikipedia.org/wiki/Atkinson_index



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# File 'lib/standard/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) ⇒ Object

Beta function of ‘x` and `y`.

beta(x, y) = tgamma(x) * tgamma(y) / tgamma(x + y)



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# File 'lib/standard/facets/math/beta.rb', line 9

def self.beta(x, y)
  #exp(lgamma(x).first + lgamma(y).first - lgamma(x+y).first)
  tgamma(x) * tgamma(y) / tgamma(x + y)
end

.cdf(array, normalised = 1.0) ⇒ Object

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



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# File 'lib/standard/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) ⇒ Object

Smallest integer not smaller than x.



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# File 'lib/standard/facets/math/ceil.rb', line 4

def self.ceil(x)
  x.ceil
end

.cosec(x) ⇒ Object

Cosecans of ‘x`.



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# File 'lib/standard/facets/math/csc.rb', line 9

def self.cosec(x)
  1.0 / sin(x)
end

.cosech(x) ⇒ Object

Cosecans hyperbolicus of ‘x`.



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# File 'lib/standard/facets/math/csch.rb', line 9

def self.cosech(x)
  1.0 / sinh(x)
end

.cot(x) ⇒ Object

Cotangens of x



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# File 'lib/standard/facets/math/cot.rb', line 4

def self.cot(x)
  tan((PI * 0.5) - x)
end

.coth(x) ⇒ Object

Cotangens hyperbolicus of x



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# File 'lib/standard/facets/math/coth.rb', line 4

def self.coth(x)
  1.0 / tanh(x)
end

.csc(x) ⇒ Object

Cosecans of ‘x`.



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# File 'lib/standard/facets/math/csc.rb', line 4

def self.csc(x)
  1.0 / sin(x)
end

.csch(x) ⇒ Object

Cosecans hyperbolicus of ‘x`.



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# File 'lib/standard/facets/math/csch.rb', line 4

def self.csch(x)
  1.0 / sinh(x)
end

.delta(i, j) ⇒ Object

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



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# File 'lib/standard/facets/math/delta.rb', line 5

def self.delta(i, j)
  return Integer(i) == Integer(j) ? 1 : 0
end

.distance(p, q) ⇒ Object

Calculates the Euclidean Distance between points p and q.

‘p`, `q` is assumed to described coordinates in N-dimensions, e. g.:

Math.distance([1, 1], [2, 2])          # 2D coordinates
Math.distance([1, 1, 1], [2, 2, 2])    # 3D coordinates

If N is 1, then ‘::distance` may also be invoked like so:

Math.distance(1, 1)



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# File 'lib/standard/facets/math/distance.rb', line 14

def self.distance(p, q)
  p, q = [p].flatten, [q].flatten
  sqrt(p.zip(q).inject(0){ |sum, coord| sum + (coord.first - coord.last)**2 })
end

.epsilon(i, j, k) ⇒ Object

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 2, 3, otherwise returns nil.



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# File 'lib/standard/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) ⇒ Object

10 to the power x



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# File 'lib/standard/facets/math/exp10.rb', line 4

def self.exp10(x)
  10.0 ** x
end

.exp2(x) ⇒ Object

2 to the power x



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# File 'lib/standard/facets/math/exp2.rb', line 4

def self.exp2(x)
  2.0 ** x
end

.factorial(n) ⇒ Object

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



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# File 'lib/standard/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) ⇒ Object

Largest integer not larger than x.



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# File 'lib/standard/facets/math/floor.rb', line 4

def self.floor(x)
  x.floor
end

.gcd(m, n) ⇒ Object

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



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# File 'lib/standard/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) ⇒ Object

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}}



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# File 'lib/standard/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) ⇒ Object

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



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# File 'lib/standard/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) ⇒ Object

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.



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# File 'lib/standard/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

.linsolve(a, b, c = 0.0) ⇒ Object

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



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# File 'lib/standard/facets/math/linsolve.rb', line 8

def self.linsolve(a, b, c = 0.0)
  a == 0 ? nil : (c - b) / a
end

.ln_gamma(x) ⇒ Object

Old name used by Extmath library.



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# File 'lib/standard/facets/math/lngamma.rb', line 13

def self.ln_gamma(x)
  lgamma(x).first
end

.lngamma(x) ⇒ Object

Logarithmus naturalis of gamma function of ‘x`.

Notice the use of ‘ln` prefix to differentiate from Ruby’s built-in ‘#lgamma` function which returns an Array.



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# File 'lib/standard/facets/math/lngamma.rb', line 8

def self.lngamma(x)
  lgamma(x).first
end

.log2(x) ⇒ Object

Logarithmus dualis of x.



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# File 'lib/standard/facets/math/log2.rb', line 8

def self.log2(x)
  Math.log(x) * INVERSE_LN_2
end

.max(array, block) ⇒ Object 



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# File 'lib/standard/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) ⇒ Object Also known as: mean_average

Mean average.



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# File 'lib/standard/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

.median(array) ⇒ Object

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



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# File 'lib/standard/facets/math/median.rb', line 8

def self.median(array)
  percentile(array, 50)
end

.min(array, &block) ⇒ Object 



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# File 'lib/standard/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) ⇒ Object

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'



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# File 'lib/standard/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) ⇒ Object

‘x` to the power `y`.



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# File 'lib/standard/facets/math/pow.rb', line 4

def self.pow(x, y)
  x ** y
end

.pstd(array, &block) ⇒ Object

Standard deviation of a population.



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# File 'lib/standard/facets/math/std.rb', line 17

def self.pstd(array, &block)
  Math::sqrt(pvariance(array, &block))
end

.pvariance(array) ⇒ Object

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



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# File 'lib/standard/facets/math/variance.rb', line 26

def self.pvariance(array)
  return 0.0 if array.size < 2
  summed_sqdevs(array) / array.size
end

.pwr(x, y) ⇒ Object

‘x` to the power `y`.



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# File 'lib/standard/facets/math/pow.rb', line 9

def self.pwr(x, y)
  x ** y
end

.rmd(array) ⇒ Object Also known as: relative_mean_difference

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



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# File 'lib/standard/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

.root(x, y) ⇒ Object

The ‘y` root of `x`.



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# File 'lib/standard/facets/math/root.rb', line 4

def self.root(x, y)
  x ** (1.0 / y)
end

.round(x) ⇒ Object

Round number to an integer.



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# File 'lib/standard/facets/math/round.rb', line 5

def self.round(x)
  x.round
end

.sec(x) ⇒ Object

Secans of x.



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# File 'lib/standard/facets/math/sec.rb', line 4

def self.sec(x)
  1.0 / cos(x)
end

.sech(x) ⇒ Object

Secans hyperbolicus of x



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# File 'lib/standard/facets/math/sech.rb', line 4

def self.sech(x)
  1.0 / cosh(x)
end

.sgn(x, zero = 0.0) ⇒ Object

Same as ‘Math.sign`.



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# File 'lib/standard/facets/math/sign.rb', line 10

def self.sgn(x, zero=0.0)
  (x > 0.0) ? 1.0 : ((x < 0.0) ? -1.0 : zero)
end

.sign(x, zero = 0.0) ⇒ Object

Sign of ‘x`. This function returns `-1.0` if `x` is negative, `+1.0` if `x` is positive `x`, and `0.0` if `x = 0`.



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# File 'lib/standard/facets/math/sign.rb', line 5

def self.sign(x, zero=0.0)
  (x > 0.0) ? 1.0 : ((x < 0.0) ? -1.0 : zero)
end

.sinc(x) ⇒ Object

Sinc function of x.



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# File 'lib/standard/facets/math/sinc.rb', line 4

def self.sinc(x)
  (x == 0.0) ? 1.0 : sin(x) / x
end

.sqr(x) ⇒ Object

Square of number.



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# File 'lib/standard/facets/math/sqr.rb', line 4

def self.sqr(x)
  x * x
end

.sqsolve(a, b, c, d = 0.0) ⇒ Object

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



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# File 'lib/standard/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

.std(array, &block) ⇒ Object Also known as: standard_deviation

Standard deviation of a sample.



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# File 'lib/standard/facets/math/std.rb', line 7

def self.std(array, &block)
  sqrt(variance(array, &block))
end

.stderr(array) ⇒ Object

Calculates the standard error of a sample.



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# File 'lib/standard/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) ⇒ Object

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



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# File 'lib/standard/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) ⇒ Object

The sum of the squared deviations from the mean.



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# File 'lib/standard/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) ⇒ Object

Exp of LGamma.



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# File 'lib/standard/facets/math/tgamma.rb', line 6

def self.tgamma(x)
  exp(lngamma(x)) #exp(log(gamma(x).abs)
end

.theil_index(array) ⇒ Object

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

TI = sum_i=1^N fracx_isum_{j=1^N x_j} ln fracx_ibar{x}

http://en.wikipedia.org/wiki/Theil_index



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# File 'lib/standard/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

.unit_step(x, zero = 1.0) ⇒ Object

The *Heaviside step function*, also called the the *unit step function*. This functions works like ‘Math.sign` but by default returns `1.0` for zero.



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# File 'lib/standard/facets/math/sign.rb', line 16

def self.unit_step(x, zero=1.0)
  (x > 0.0) ? 1.0 : ((x < 0.0) ? -1.0 : zero)
end

.variance(array, &block) ⇒ Object 



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# File 'lib/standard/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) ⇒ Object

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

TODO: Same as #variance? Then choose one.



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# File 'lib/standard/facets/math/variance.rb', line 19

def self.variance2(array)
  return 0.0 if array.size < 2
  summed_sqdevs(array) / (array.size - 1)
end

Instance Method Details

#acsch(x) ⇒ Object

Area cosecans hyperbolicus of x



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# File 'lib/standard/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) ⇒ Object

Area secans hyperbolicus of x



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# File 'lib/standard/facets/math/asech.rb', line 4

def asech(x)
  log((1.0 + sqrt(1.0 - x * x)) / x)
end