#! /usr/bin/env python
#
def burr_cdf ( x, a, b, c, d ):

#*****************************************************************************80
#
## BURR_CDF evaluates the Burr CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    08 August 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real X, the argument of the CDF.
#
#    Input, real A, B, C, D, the parameters of the PDF.
#    0 < B,
#    0 < C.
#
#    Output, real CDF, the value of the CDF.
#
  if ( x <= a ):

    cdf = 0.0

  else:

    y = ( x - a ) / b

    cdf = 1.0 - 1.0 / ( 1.0 + y ** c  ) ** d

  return cdf

def burr_cdf_inv ( cdf, a, b, c, d ):

#*****************************************************************************80
#
## BURR_CDF_INV inverts the Burr CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    08 August 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real CDF, the value of the CDF.
#    0.0 <= CDF <= 1.0.
#
#    Input, real A, B, C, D, the parameters of the PDF.
#    0 < B,
#    0 < C.
#
#    Output, real X, the corresponding argument.
#
  from sys import exit

  if ( cdf < 0.0 or 1.0 < cdf ):
    print ( '' )
    print ( 'BURR_CDF_INV - Fatal error!' )
    print ( '  CDF < 0 or 1 < CDF.' )
    exit ( 'BURR_CDF_INV - Fatal error!' )

  y = ( ( 1.0 / ( 1.0 - cdf ) ) ** ( 1.0 / d ) - 1.0 ) ** ( 1.0 / c )

  x = a + b * y

  return x

def burr_cdf_test ( ):

#*****************************************************************************80
#
## BURR_CDF_TEST tests BURR_CDF, BURR_CDF_INV, BURR_PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    14 March 2016
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'BURR_CDF_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  BURR_CDF evaluates the Burr CDF' )
  print ( '  BURR_CDF_INV inverts the Burr CDF.' )
  print ( '  BURR_PDF evaluates the Burr PDF' )

  a = 1.0
  b = 2.0
  c = 3.0
  d = 2.0

  check = burr_check ( a, b, c, d )

  if ( not check ):
    print ( '' )
    print ( 'BURR_CDF_TEST - Fatal error!' )
    print ( '  The parameters are not legal.' )
    return

  print ( '' )
  print ( '  PDF parameter A = %14g' % ( a ) )
  print ( '  PDF parameter B = %14g' % ( b ) )
  print ( '  PDF parameter C = %14g' % ( c ) )
  print ( '  PDF parameter D = %14g' % ( d ) )

  seed = 123456789

  print ( '' )
  print ( '       X            PDF           CDF            CDF_INV' )
  print ( '' )

  for i in range ( 0, 10 ):

    x, seed = burr_sample ( a, b, c, d, seed )

    pdf = burr_pdf ( x, a, b, c, d )

    cdf = burr_cdf ( x, a, b, c, d )

    x2 = burr_cdf_inv ( cdf, a, b, c, d )

    print ( ' %14g  %14g  %14g  %14g' % ( x, pdf, cdf, x2 ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'BURR_DF_TEST' )
  print ( '  Normal end of execution.' )
  return

def burr_check ( a, b, c, d ):

#*****************************************************************************80
#
## BURR_CHECK checks the parameters of the Burr CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    14 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, B, C, D, the parameters of the PDF.
#    0 < B,
#    0 < C.
#
#    Output, logical CHECK, is TRUE if the parameters are legal.
#
  check = True

  if ( b <= 0.0 ):
    print ( '' )
    print ( 'BURR_CHECK - Fatal error!' )
    print ( '  B <= 0.' )
    check = False

  if ( c <= 0 ):
    print ( '' )
    print ( 'BURR_CHECK - Fatal error!' )
    print ( '  C <= 0.' )
    check = False

  return check

def burr_mean ( a, b, c, d ):

#*****************************************************************************80
#
## BURR_MEAN returns the mean of the Burr PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    08 August 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, B, C, D, the parameters of the PDF.
#    0 < B,
#    0 < C.
#
#    Output, real MEAN, the mean of the PDF.
#
  from r8_gamma import r8_gamma

  ymean = d * r8_gamma ( d       - 1.0 / c ) \
            * r8_gamma (     1.0 + 1.0 / c ) \
            / r8_gamma ( d + 1.0           )

  mean = a + b * ymean

  return mean

def burr_pdf ( x, a, b, c, d ):

#*****************************************************************************80
#
## BURR_PDF evaluates the Burr PDF.
#
#  Discussion:
#
#    Y = ( X - A ) / B;
#
#    PDF(X)(A,B,C,D) = ( C * D / B ) * Y ^ ( C - 1 ) / ( 1 + Y ^ C ) ^ ( D + 1 )
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    08 August 2016
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    M E Johnson,
#    Multivariate Statistical Simulation,
#    Wiley, New York, 1987.
#
#  Parameters:
#
#    Input, real X, the argument of the PDF.
#    A <= X
#
#    Input, real A, B, C, D, the parameters of the PDF.
#    0 < B,
#    0 < C.
#
#    Output, real PDF, the value of the PDF.
#
  if ( x <= a ):

    pdf = 0.0

  else:

    y = ( x - a ) / b

    pdf = ( c * d / b ) * y ** ( c - 1.0 ) / ( 1.0 + y ** c ) ** ( d + 1.0 )

  return pdf

def burr_sample ( a, b, c, d, seed ):

#*****************************************************************************80
#
## BURR_SAMPLE samples the Burr PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    14 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, B, C, D, the parameters of the PDF.
#    0 < B,
#    0 < C.
#
#    Input, integer SEED, a seed for the random number generator.
#
#    Output, real X, a sample of the PDF.
#
#    Output, integer SEED, an updated seed for the random number generator.
#
  from r8_uniform_01 import r8_uniform_01

  cdf, seed = r8_uniform_01 ( seed )

  x = burr_cdf_inv ( cdf, a, b, c, d )

  return x, seed

def burr_sample_test ( ):

#*****************************************************************************80
#
## BURR_SAMPLE_TEST tests BURR_MEAN, BURR_VARIANCE, BURR_SAMPLE
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    14 March 2016
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform
  from r8vec_max import r8vec_max
  from r8vec_mean import r8vec_mean
  from r8vec_min import r8vec_min
  from r8vec_variance import r8vec_variance

  nsample = 1000
  seed = 123456789

  print ( '' )
  print ( 'BURR_SAMPLE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  BURR_MEAN computes the Burr mean' )
  print ( '  BURR_VARIANCE computes the Burr variance' )
  print ( '  BURR_SAMPLE Burr samples the distribution' )

  a = 1.0
  b = 2.0
  c = 3.0
  d = 2.0

  check = burr_check ( a, b, c, d )

  if ( not check ):
    print ( '' )
    print ( 'BURR_SAMPLE_TEST - Fatal error!' )
    print ( '  The parameters are not legal.' )
    return

  mean = burr_mean ( a, b, c, d )
  variance = burr_variance ( a, b, c, d )

  print ( '' )
  print ( '  PDF parameter A = %14g' % ( a ) )
  print ( '  PDF parameter B = %14g' % ( b ) )
  print ( '  PDF parameter C = %14g' % ( c ) )
  print ( '  PDF parameter D = %14g' % ( d ) )
  print ( '  PDF mean =        %14g' % ( mean ) )
  print ( '  PDF variance =    %14g' % ( variance ) )

  x = np.zeros ( nsample )

  for i in range ( 0, nsample ):
    x[i], seed = burr_sample ( a, b, c, d, seed )

  mean = r8vec_mean ( nsample, x )
  variance = r8vec_variance ( nsample, x )
  xmax = r8vec_max ( nsample, x )
  xmin = r8vec_min ( nsample, x )

  print ( '' )
  print ( '  Sample size =     %6d' % ( nsample ) )
  print ( '  Sample mean =     %14f' % ( mean ) )
  print ( '  Sample variance = %14f' % ( variance ) )
  print ( '  Sample maximum =  %14f' % ( xmax ) )
  print ( '  Sample minimum =  %14f' % ( xmin ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'BURR_SAMPLE_TEST' )
  print ( '  Normal end of execution.' )
  return

def burr_variance ( a, b, c, d ):

#*****************************************************************************80
#
## BURR_VARIANCE returns the variance of the Burr PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    08 August 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, B, C, D, the parameters of the PDF.
#    0 < B,
#    0 < C.
#
#    Output, real VARIANCE, the variance of the PDF.
#
  from r8_gamma import r8_gamma
  from r8_huge import r8_huge

  if ( c <= 2.0 ):

    print ( '' )
    print ( 'BURR_VARIANCE - Warning!' )
    print ( '  Variance undefined for C <= 2.' )
    variance = r8_huge ( )

  else:

    mu1 = b     * d * r8_gamma ( ( c * d     - 1.0 ) / c ) \
                    * r8_gamma ( (         c + 1.0 ) / c ) \
                    / r8_gamma ( ( c * d + c       ) / c )

    mu2 = b * b * d * r8_gamma ( ( c * d     - 2.0 ) / c ) \
                    * r8_gamma ( (         c + 2.0 ) / c ) \
                    / r8_gamma ( ( c * d + c       ) / c )

    variance = - mu1 * mu1 + mu2

  return variance

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( ) 
  burr_cdf_test ( )
  burr_sample_test ( )
  timestamp ( )