Module: Distribution::MathExtension::IncompleteGamma

Defined in:
lib/distribution/math_extension/incomplete_gamma.rb

Constant Summary collapse

NMAX =
5000
SMALL =
Float::EPSILON**3
PG21 =

PolyGamma[2,1]

-2.404113806319188570799476 # PolyGamma[2,1]

Class Method Summary collapse

Class Method Details

.a_greater_than_0(a, x, with_error = false) ⇒ Object

gamma_inc_a_gt_0


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 355

def a_greater_than_0(a, x, with_error = false)
  q       = q(a, x, with_error)
  q, q_err = q if with_error
  g       = Math.gamma(a)
  STDERR.puts("Warning: Don't know error for Math.gamma. Error will be incorrect") if with_error
  g_err   = Float::EPSILON
  result  = g * q
  error   = (g * q_err).abs + (g_err * q).abs if with_error
  with_error ? [result, error] : result
end

.continued_fraction(a, x, with_error = false) ⇒ Object

gamma_inc_CF


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 367

def continued_fraction(a, x, with_error = false)
  f = f_continued_fraction(a, x, with_error)
  f, f_error = f if with_error
  pre = Math.exp((a - 1.0) * Math.log(x) - x)
  STDERR.puts("Warning: Don't know error for Math.exp. Error will be incorrect") if with_error
  pre_error = Float::EPSILON
  result    = f * pre
  if with_error
    error     = (f_error * pre).abs + (f * pre_error) + (2.0 + a.abs) * Float::EPSILON * result.abs
    [result, error]
  else
    result
  end
end

.d(a, x, with_error = false) ⇒ Object

The dominant part, D(a,x) := x^a e^(-x) / Gamma(a+1) gamma_inc_D in GSL-1.9.


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 38

def d(a, x, with_error = false)
  error = nil
  if a < 10.0
    ln_a = Math.lgamma(a + 1.0).first
    lnr  = a * Math.log(x) - x - ln_a
    result = Math.exp(lnr)
    error = 2.0 * Float::EPSILON * (lnr.abs + 1.0) + result.abs if with_error
    with_error ? [result, error] : result
  else
    ln_term = ln_term_error = nil
    if x < 0.5 * a
      u       = x / a.to_f
      ln_u    = Math.log(u)
      ln_term = ln_u - u + 1.0
      ln_term_error = (ln_u.abs + u.abs + 1.0) * Float::EPSILON if with_error
    else
      mu      = (x - a) / a.to_f
      ln_term = Log.log_1plusx_minusx(mu, with_error)
      ln_term, ln_term_error = ln_term if with_error
    end
    gstar = Gammastar.evaluate(a, with_error)
    gstar, gstar_error = gstar if with_error
    term1 = Math.exp(a * ln_term) / Math.sqrt(2.0 * Math::PI * a)
    result = term1 / gstar
    error  = 2.0 * Float::EPSILON * ((a * ln_term).abs + 1.0) * result.abs + gstar_error / gstar.abs * result.abs if with_error
    with_error ? [result, error] : result
  end
end

.f_continued_fraction(a, x, with_error = false) ⇒ Object

gamma_inc_F_CF


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 231

def f_continued_fraction(a, x, with_error = false)
  hn = 1.0 # convergent
  cn = 1.0 / SMALL
  dn = 1.0
  n  = 2
  2.upto(NMAX - 1).each do |n|
    an = n.odd? ? 0.5 * (n - 1) / x : (0.5 * n - a) / x
    dn = 1.0 + an * dn
    dn = SMALL if dn.abs < SMALL
    cn = 1.0 + an / cn
    cn = SMALL if cn.abs < SMALL
    dn = 1.0 / dn
    delta = cn * dn
    hn *= delta
    break if (delta - 1.0).abs < Float::EPSILON
  end

  if n == NMAX
    STDERR.puts('Error: n reached NMAX in f continued fraction')
  else
    with_error ? [hn, 2.0 * Float::EPSILON * hn.abs + Float::EPSILON * (2.0 + 0.5 * n) * hn.abs] : hn
  end
end

.p(a, x, with_error = false) ⇒ Object

The incomplete gamma function. gsl_sf_gamma_inc_P_e


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 109

def p(a, x, with_error = false)
  fail(ArgumentError, 'Range Error: a must be positive, x must be non-negative') if a <= 0.0 || x < 0.0
  if x == 0.0
    return with_error ? [0.0, 0.0] : 0.0
  elsif x < 20.0 || x < 0.5 * a
    return p_series(a, x, with_error)
  elsif a > 1e6 && (x - a) * (x - a) < a
    return q_asymptotic_uniform_complement a, x, with_error
  elsif a <= x
    if a > 0.2 * x
      return q_continued_fraction_complement(a, x, with_error)
    else
      return q_large_x_complement(a, x, with_error)
    end
  elsif (x - a) * (x - a) < a
    return q_asymptotic_uniform_complement a, x, with_error
  else
    return p_series(a, x, with_error)
  end
end

.p_series(a, x, with_error = false) ⇒ Object

gamma_inc_P_series


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 68

def p_series(a, x, with_error = false)
  d = d(a, x, with_error)
  d, d_err = d if with_error
  sum      = 1.0
  term     = 1.0
  n        = 1
  1.upto(NMAX - 1) do |n|
    term *= x / (a + n).to_f
    sum += term
    break if (term / sum).abs < Float::EPSILON
  end

  result   = d * sum

  if n == NMAX
    STDERR.puts('Error: n reached NMAX in p series')
  else
    return with_error ? [result, d_err * sum.abs + (1.0 + n) * Float::EPSILON * result.abs] : result
  end
end

.q(a, x, with_error = false) ⇒ Object

gamma_inc_Q_e


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 131

def q(a, x, with_error = false)
  fail(ArgumentError, 'Range Error: a and x must be non-negative') if a < 0.0 || x < 0.0
  if x == 0.0
    return with_error ? [1.0, 0.0] : 1.0
  elsif a == 0.0
    return with_error ? [0.0, 0.0] : 0.0
  elsif x <= 0.5 * a
    # If series is quick, do that.
    p = p_series(a, x, with_error)
    p, p_err = p if with_error
    result  = 1.0 - p
    return with_error ? [result, p_err + 2.0 * Float::EPSILON * result.abs] : result
  elsif a >= 1.0e+06 && (x - a) * (x - a) < a # difficult asymptotic regime, only way to do this region
    return q_asymptotic_uniform(a, x, with_error)
  elsif a < 0.2 && x < 5.0
    return q_series(a, x, with_error)
  elsif a <= x
    return x <= 1.0e+06 ? q_continued_fraction(a, x, with_error) : q_large_x(a, x, with_error)
  else
    if x > a - Math.sqrt(a)
      return q_continued_fraction(a, x, with_error)
    else
      p = p_series(a, x, with_error)
      p, p_err = p if with_error
      result = 1.0 - p
      return with_error ? [result, p_err + 2.0 * Float::EPSILON * result.abs] : result
    end
  end
end

.q_asymptotic_uniform(a, x, with_error = false) ⇒ Object

Uniform asymptotic for x near a, a and x large gamma_inc_Q_asymp_unif


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 201

def q_asymptotic_uniform(a, x, with_error = false)
  rta = Math.sqrt(a)
  eps = (x - a).quo(a)

  ln_term = Log.log_1plusx_minusx(eps, with_error)
  ln_term, ln_term_err = ln_term if with_error

  eta     = (eps >= 0 ? 1 : -1) * Math.sqrt(-2 * ln_term)

  erfc    = Math.erfc_e(eta * rta / SQRT2, with_error)
  erfc, erfc_err = erfc if with_error

  c0 = c1 = nil
  if eps.abs < ROOT5_FLOAT_EPSILON
    c0 = -1.quo(3) + eps * (1.quo(12) - eps * (23.quo(540) - eps * (353.quo(12_960) - eps * 589.quo(30_240))))
    c1 = -1.quo(540) - eps.quo(288)
  else
    rt_term = Math.sqrt(-2 * ln_term.quo(eps * eps))
    lam     = x.quo(a)
    c0      = (1 - 1 / rt_term) / eps
    c1      = -(eta**3 * (lam * lam + 10 * lam + 1) - 12 * eps**3).quo(12 * eta**3 * eps**3)
  end

  r = Math.exp(-0.5 * a * eta * eta) / (SQRT2 * SQRTPI * rta) * (c0 + c1.quo(a))

  result = 0.5 * erfc + r
  with_error ? [result, Float::EPSILON + (r * 0.5 * a * eta * eta).abs + 0.5 * erfc_err + 2.0 * Float::EPSILON + result.abs] : result
end

.q_asymptotic_uniform_complement(a, x, with_error = false) ⇒ Object

This function does not exist in GSL, but is nonetheless GSL code. It's for calculating two specific ranges of p.


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 90

def q_asymptotic_uniform_complement(a, x, with_error = false)
  q = q_asymptotic_uniform(a, x, with_error)
  q, q_err = q if with_error
  result = 1.0 - q
  with_error ? [result, q_err + 2.0 * Float::EPSILON * result.abs] : result
end

.q_continued_fraction(a, x, with_error = false) ⇒ Object

gamma_inc_Q_CF


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 162

def q_continued_fraction(a, x, with_error = false)
  d = d(a, x, with_error)
  f = f_continued_fraction(a, x, with_error)

  if with_error
    [d.first * (a / x).to_f * f.first, d.last * ((a / x).to_f * f.first).abs + (d.first * a / x * f.last).abs]
  else
    d * (a / x).to_f * f
  end
end

.q_continued_fraction_complement(a, x, with_error = false) ⇒ Object


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 97

def q_continued_fraction_complement(a, x, with_error = false)
  q = q_continued_fraction(a, x, with_error)
  with_error ? [1.0 - q.first, q.last + 2.0 * Float::EPSILON * (1.0 - q.first).abs] : 1.0 - q
end

.q_large_x(a, x, with_error = false) ⇒ Object

gamma_inc_Q_large_x in GSL-1.9


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 174

def q_large_x(a, x, with_error = false)
  d = d(a, x, with_error)
  d, d_err = d if with_error
  sum  = 1.0
  term = 1.0
  last = 1.0
  n    = 1
  1.upto(NMAX - 1).each do |n|
    term *= (a - n) / x
    break if (term / last).abs > 1.0
    break if (term / sum).abs < Float::EPSILON
    sum += term
    last  = term
  end

  result = d * (a / x) * sum
  error  = d_err * (a / x).abs * sum if with_error

  if n == NMAX
    STDERR.puts('Error: n reached NMAX in q_large_x')
  else
    return with_error ? [result, error] : result
  end
end

.q_large_x_complement(a, x, with_error = false) ⇒ Object


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 102

def q_large_x_complement(a, x, with_error = false)
  q = q_large_x(a, x, with_error)
  with_error ? [1.0 - q.first, q.last + 2.0 * Float::EPSILON * (1.0 - q.first).abs] : 1.0 - q
end

.q_series(a, x, with_error = false) ⇒ Object


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 255

def q_series(a, x, with_error = false)
  term1 = nil
  sum   = nil
  term2 = nil
  begin
    lnx  = Math.log(x)
    el   = EULER + lnx
    c1   = -el
    c2   = Math::PI * Math::PI / 12.0 - 0.5 * el * el
    c3   = el * (Math::PI * Math::PI / 12.0 - el * el / 6.0) + PG21 / 6.0
    c4   = -0.04166666666666666667 *
           (-1.758243446661483480 + lnx) *
           (-0.764428657272716373 + lnx) *
           (0.723980571623507657 + lnx) *
           (4.107554191916823640 + lnx)
    c5 = -0.0083333333333333333 *
         (-2.06563396085715900 + lnx) *
         (-1.28459889470864700 + lnx) *
         (-0.27583535756454143 + lnx) *
         (1.33677371336239618 + lnx) *
         (5.17537282427561550 + lnx)
    c6 = -0.0013888888888888889 *
         (-2.30814336454783200 + lnx) *
         (-1.65846557706987300 + lnx) *
         (-0.88768082560020400 + lnx) *
         (0.17043847751371778 + lnx) *
         (1.92135970115863890 + lnx) *
         (6.22578557795474900 + lnx)
    c7 = -0.00019841269841269841
    (-2.5078657901291800 + lnx) *
      (-1.9478900888958200 + lnx) *
      (-1.3194837322612730 + lnx) *
      (-0.5281322700249279 + lnx) *
      (0.5913834939078759 + lnx) *
      (2.4876819633378140 + lnx) *
      (7.2648160783762400 + lnx)
    c8 = -0.00002480158730158730 *
         (-2.677341544966400 + lnx) *
         (-2.182810448271700 + lnx) *
         (-1.649350342277400 + lnx) *
         (-1.014099048290790 + lnx) *
         (-0.191366955370652 + lnx) *
         (0.995403817918724 + lnx) *
         (3.041323283529310 + lnx) *
         (8.295966556941250 + lnx) *
         c9 = -2.75573192239859e-6 *
              (-2.8243487670469080 + lnx) *
              (-2.3798494322701120 + lnx) *
              (-1.9143674728689960 + lnx) *
              (-1.3814529102920370 + lnx) *
              (-0.7294312810261694 + lnx) *
              (0.1299079285269565 + lnx) *
              (1.3873333251885240 + lnx) *
              (3.5857258865210760 + lnx) *
              (9.3214237073814600 + lnx) *
              c10 = -2.75573192239859e-7 *
                    (-2.9540329644556910 + lnx) *
                    (-2.5491366926991850 + lnx) *
                    (-2.1348279229279880 + lnx) *
                    (-1.6741881076349450 + lnx) *
                    (-1.1325949616098420 + lnx) *
                    (-0.4590034650618494 + lnx) *
                    (0.4399352987435699 + lnx) *
                    (1.7702236517651670 + lnx) *
                    (4.1231539047474080 + lnx) *
                    (10.342627908148680 + lnx)
    term1 = a * (c1 + a * (c2 + a * (c3 + a * (c4 + a * (c5 + a * (c6 + a * (c7 + a * (c8 + a * (c9 + a * c10)))))))))
  end

  n   = 1
  begin
    t   = 1.0
    sum = 1.0
    1.upto(NMAX - 1).each do |n|
      t *= -x / (n + 1.0)
      sum += (a + 1.0) / (a + n + 1.0) * t
      break if (t / sum).abs < Float::EPSILON
    end
  end

  if n == NMAX
    STDERR.puts('Error: n reached NMAX in q_series')
  else
    term2 = (1.0 - term1) * a / (a + 1.0) * x * sum
    result = term1 + term2
    with_error ? [result, Float::EPSILON * term1.abs + 2.0 * term2.abs + 2.0 * Float::EPSILON * result.abs] : result
  end
end

.series(a, x, with_error = false) ⇒ Object

gamma_inc_series


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 345

def series(a, x, with_error = false)
  q = q_series(a, x, with_error)
  g = Math.gamma(a)
  STDERR.puts("Warning: Don't know error for Math.gamma. Error will be incorrect") if with_error
  # When we get the error from Gamma, switch the comment on the next to lines
  # with_error ? [q.first*g.first, (q.first*g.last).abs + (q.last*g.first).abs + 2.0*Float::EPSILON*(q.first*g.first).abs] : q*g
  with_error ? [q.first * g, (q.first * Float::EPSILON).abs + (q.last * g.first).abs + 2.0 * Float::EPSILON(q.first * g).abs] : q * g
end

.unnormalized(a, x, with_error = false) ⇒ Object

Unnormalized incomplete gamma function. gsl_sf_gamma_inc_e


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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 384

def unnormalized(a, x, with_error = false)
  fail(ArgumentError, 'x cannot be negative') if x < 0.0

  if x == 0.0
    result  = Math.gamma(a.to_f)
    STDERR.puts("Warning: Don't know error for Math.gamma. Error will be incorrect") if with_error
    return with_error ? [result, Float::EPSILON] : result
  elsif a == 0.0
    return ExponentialIntegral.first_order(x.to_f, with_error)
  elsif a > 0.0
    return a_greater_than_0(a.to_f, x.to_f, with_error)
  elsif x > 0.25
    # continued fraction seems to fail for x too small
    return continued_fraction(a.to_f, x.to_f, with_error)
  elsif a.abs < 0.5
    return series(a.to_f, x.to_f, with_error)
  else
    fa = a.floor.to_f
    da = a - fa
    g_da = da > 0.0 ? a_greater_than_0(da, x.to_f, with_error) : ExponentialIntegral.first_order(x.to_f, with_error)
    g_da, g_da_err = g_da if with_error
    alpha = da
    gax = g_da

    # Gamma(alpha-1,x) = 1/(alpha-1) (Gamma(a,x) - x^(alpha-1) e^-x)
    begin
      shift  = Math.exp(-x + (alpha - 1.0) * Math.log(x))
      gax    = (gax - shift) / (alpha - 1.0)
      alpha -= 1.0
    end while alpha > a

    result = gax
    return with_error ? [result, 2.0 * (1.0 + a.abs) * Float::EPSILON * gax.abs] : result
  end
end