def truncated_normal_ab_pdf(x, mu, sigma, a, b):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_AB_PDF evaluates the Truncated Normal PDF.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 24 January 2017
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real X, the argument of the PDF.
#
# Input, real MU, SIGMA, the mean and standard deviation of the
# parent Normal distribution.
#
# Input, real A, B, the lower and upper truncation limits.
#
# Output, real VALUE, the value of the PDF.
#
from normal_01_cdf import normal_01_cdf
from normal_01_pdf import normal_01_pdf
if (x < a):
value = 0.0
elif (x <= b):
alpha = (a - mu) / sigma
beta = (b - mu) / sigma
xi = (x - mu) / sigma
alpha_cdf = normal_01_cdf(alpha)
beta_cdf = normal_01_cdf(beta)
xi_pdf = normal_01_pdf(xi)
value = xi_pdf / (beta_cdf - alpha_cdf) / sigma
else:
value = 0.0
return value
def truncated_normal_ab_cdf_inv ( cdf, mu, sigma, a, b ):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_AB_CDF_INV inverts the truncated Normal CDF.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 09 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real CDF, the value of the CDF.
# 0.0 <= CDF <= 1.0.
#
# Input, real MU, SIGMA, the mean and standard deviation of the
# parent Normal distribution.
#
# Input, real A, B, the lower and upper truncation limits.
#
# Output, real X, the corresponding argument.
#
from normal_01_cdf import normal_01_cdf
from normal_01_cdf_inv import normal_01_cdf_inv
from sys import exit
if ( cdf < 0.0 or 1.0 < cdf ):
print ( '' )
print ( 'TRUNCATED_NORMAL_AB_CDF_INV - Fatal error!' )
print ( ' CDF < 0 or 1 < CDF.' )
exit ( 'TRUNCATED_NORMAL_AB_CDF_INV - Fatal error!' )
alpha = ( a - mu ) / sigma
beta = ( b - mu ) / sigma
alpha_cdf = normal_01_cdf ( alpha )
beta_cdf = normal_01_cdf ( beta )
xi_cdf = ( beta_cdf - alpha_cdf ) * cdf + alpha_cdf
xi = normal_01_cdf_inv ( xi_cdf )
x = mu + sigma * xi
return x
def truncated_normal_ab_sample ( mu, sigma, a, b, seed ): #*****************************************************************************80 # ## TRUNCATED_NORMAL_AB_SAMPLE samples the Truncated Normal distribution. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 08 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, real MU, SIGMA, the mean and standard deviation of the # parent Normal distribution. # # Input, real A, B, the lower and upper truncation limits. # # Input, integer SEED, a seed for the random number generator. # # Output, real X, a sample of the PDF. # # Output, integer SEED, a seed for the random number generator. # from normal_01_cdf import normal_01_cdf from normal_01_cdf_inv import normal_01_cdf_inv from r8_uniform_01 import r8_uniform_01 alpha = ( a - mu ) / sigma beta = ( b - mu ) / sigma alpha_cdf = normal_01_cdf ( alpha ) beta_cdf = normal_01_cdf ( beta ) u, seed = r8_uniform_01 ( seed ) xi_cdf = alpha_cdf + u * ( beta_cdf - alpha_cdf ) xi = normal_01_cdf_inv ( xi_cdf ) x = mu + sigma * xi return x, seed
def truncated_normal_ab_sample(mu, sigma, a, b, seed):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_AB_SAMPLE samples the Truncated Normal distribution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 08 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real MU, SIGMA, the mean and standard deviation of the
# parent Normal distribution.
#
# Input, real A, B, the lower and upper truncation limits.
#
# Input, integer SEED, a seed for the random number generator.
#
# Output, real X, a sample of the PDF.
#
# Output, integer SEED, a seed for the random number generator.
#
from normal_01_cdf import normal_01_cdf
from normal_01_cdf_inv import normal_01_cdf_inv
from r8_uniform_01 import r8_uniform_01
alpha = (a - mu) / sigma
beta = (b - mu) / sigma
alpha_cdf = normal_01_cdf(alpha)
beta_cdf = normal_01_cdf(beta)
u, seed = r8_uniform_01(seed)
xi_cdf = alpha_cdf + u * (beta_cdf - alpha_cdf)
xi = normal_01_cdf_inv(xi_cdf)
x = mu + sigma * xi
return x, seed
def normal_ms_cdf ( x, mu, sigma ): #*****************************************************************************80 # ## NORMAL_MS_CDF evaluates the CDF of the Normal MS distribution. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 05 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, real X, the argument of the CDF. # # Input, real MU, SIGMA, the parameters of the PDF. # 0.0 < SIGMA. # # Output, real VALUE, the value of the CDF. # from normal_01_cdf import normal_01_cdf y = ( x - mu ) / sigma value = normal_01_cdf ( y ) return value
def normal_ms_cdf(x, mu, sigma):
#*****************************************************************************80
#
## NORMAL_MS_CDF evaluates the CDF of the Normal MS distribution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 05 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real X, the argument of the CDF.
#
# Input, real MU, SIGMA, the parameters of the PDF.
# 0.0 < SIGMA.
#
# Output, real VALUE, the value of the CDF.
#
from normal_01_cdf import normal_01_cdf
y = (x - mu) / sigma
value = normal_01_cdf(y)
return value
def truncated_normal_b_cdf(x, mu, sigma, b):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_B_CDF evaluates the upper Truncated Normal CDF.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 24 January 2017
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real X, the argument of the CDF.
#
# Input, real MU, SIGMA, the mean and standard deviation of the
# parent Normal distribution.
#
# Input, real B, the upper truncation limit.
#
# Output, real VALUE, the value of the CDF.
#
from normal_01_cdf import normal_01_cdf
if (x <= b):
beta = (b - mu) / sigma
xi = (x - mu) / sigma
alpha_cdf = 0.0
beta_cdf = normal_01_cdf(beta)
xi_cdf = normal_01_cdf(xi)
value = (xi_cdf - alpha_cdf) / (beta_cdf - alpha_cdf)
else:
value = 1.0
return value
def truncated_normal_ab_variance ( mu, sigma, a, b ):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_AB_VARIANCE: variance of the Truncated Normal distribution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 08 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real MU, SIGMA, the parameters of the PDF.
#
# Input, real A, B, the lower and upper truncation limits.
#
# Output, real VALUE, the variance of the PDF.
#
from normal_01_cdf import normal_01_cdf
from normal_01_pdf import normal_01_pdf
alpha = ( a - mu ) / sigma
beta = ( b - mu ) / sigma
alpha_pdf = normal_01_pdf ( alpha )
beta_pdf = normal_01_pdf ( beta )
alpha_cdf = normal_01_cdf ( alpha )
beta_cdf = normal_01_cdf ( beta )
value = sigma * sigma * ( 1.0 \
+ ( alpha * alpha_pdf - beta * beta_pdf ) / ( beta_cdf - alpha_cdf ) \
- ( ( alpha_pdf - beta_pdf ) / ( beta_cdf - alpha_cdf ) ) ** 2 )
return value
def truncated_normal_ab_pdf ( x, mu, sigma, a, b ): #*****************************************************************************80 # ## TRUNCATED_NORMAL_AB_PDF evaluates the Truncated Normal PDF. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 09 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, real X, the argument of the PDF. # # Input, real MU, SIGMA, the mean and standard deviation of the # parent Normal distribution. # # Input, real A, B, the lower and upper truncation limits. # # Output, real VALUE, the value of the PDF. # from normal_01_cdf import normal_01_cdf from normal_01_pdf import normal_01_pdf alpha = ( a - mu ) / sigma beta = ( b - mu ) / sigma xi = ( x - mu ) / sigma alpha_cdf = normal_01_cdf ( alpha ) beta_cdf = normal_01_cdf ( beta ) xi_pdf = normal_01_pdf ( xi ) value = xi_pdf / ( beta_cdf - alpha_cdf ) / sigma return value
def truncated_normal_ab_variance(mu, sigma, a, b):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_AB_VARIANCE: variance of the Truncated Normal distribution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 08 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real MU, SIGMA, the parameters of the PDF.
#
# Input, real A, B, the lower and upper truncation limits.
#
# Output, real VALUE, the variance of the PDF.
#
from normal_01_cdf import normal_01_cdf
from normal_01_pdf import normal_01_pdf
alpha = (a - mu) / sigma
beta = (b - mu) / sigma
alpha_pdf = normal_01_pdf(alpha)
beta_pdf = normal_01_pdf(beta)
alpha_cdf = normal_01_cdf(alpha)
beta_cdf = normal_01_cdf(beta)
value = sigma * sigma * ( 1.0 \
+ ( alpha * alpha_pdf - beta * beta_pdf ) / ( beta_cdf - alpha_cdf ) \
- ( ( alpha_pdf - beta_pdf ) / ( beta_cdf - alpha_cdf ) ) ** 2 )
return value
def truncated_normal_a_cdf ( x, mu, sigma, a ): #*****************************************************************************80 # ## TRUNCATED_NORMAL_A_CDF evaluates the lower Truncated Normal CDF. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 08 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, real X, the argument of the CDF. # # Input, real MU, SIGMA, the mean and standard deviation of the # parent Normal distribution. # # Input, real A, the lower truncation limit. # # Output, real VALUE, the value of the CDF. # from normal_01_cdf import normal_01_cdf alpha = ( a - mu ) / sigma xi = ( x - mu ) / sigma alpha_cdf = normal_01_cdf ( alpha ) beta_cdf = 1.0 xi_cdf = normal_01_cdf ( xi ) value = ( xi_cdf - alpha_cdf ) / ( beta_cdf - alpha_cdf ) return value
def truncated_normal_b_cdf ( x, mu, sigma, b ): #*****************************************************************************80 # ## TRUNCATED_NORMAL_B_CDF evaluates the upper Truncated Normal CDF. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 08 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, real X, the argument of the CDF. # # Input, real MU, SIGMA, the mean and standard deviation of the # parent Normal distribution. # # Input, real B, the upper truncation limit. # # Output, real VALUE, the value of the CDF. # from normal_01_cdf import normal_01_cdf beta = ( b - mu ) / sigma xi = ( x - mu ) / sigma alpha_cdf = 0.0 beta_cdf = normal_01_cdf ( beta ) xi_cdf = normal_01_cdf ( xi ) value = ( xi_cdf - alpha_cdf ) / ( beta_cdf - alpha_cdf ) return value
def truncated_normal_a_cdf_inv ( cdf, mu, sigma, a ):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_A_CDF_INV inverts the lower truncated Normal CDF.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 09 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real CDF, the value of the CDF.
# 0.0 <= CDF <= 1.0.
#
# Input, real MU, SIGMA, the mean and standard deviation of the
# parent Normal distribution.
#
# Input, real A, the lower truncation limit.
#
# Output, real X, the corresponding argument.
#
from normal_01_cdf import normal_01_cdf
from normal_01_cdf_inv import normal_01_cdf_inv
from sys import exit
if ( cdf < 0.0 or 1.0 < cdf ):
print ''
print 'TRUNCATED_NORMAL_A_CDF_INV - Fatal error!'
print ' CDF < 0 or 1 < CDF.'
exit ( 'TRUNCATED_NORMAL_A_CDF_INV - Fatal error!' )
alpha = ( a - mu ) / sigma
alpha_cdf = normal_01_cdf ( alpha )
beta_cdf = 1.0
xi_cdf = ( beta_cdf - alpha_cdf ) * cdf + alpha_cdf
xi = normal_01_cdf_inv ( xi_cdf )
x = mu + sigma * xi
return x
def truncated_normal_a_mean(mu, sigma, a):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_A_MEAN returns the mean of the lower Truncated Normal distribution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 08 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real MU, SIGMA, the parameters of the parent Normal Distribution.
#
# Input, real A, the lower truncation limit.
#
# Output, real VALUE, the mean of the PDF.
#
from normal_01_cdf import normal_01_cdf
from normal_01_pdf import normal_01_pdf
alpha = (a - mu) / sigma
alpha_cdf = normal_01_cdf(alpha)
beta_cdf = 1.0
alpha_pdf = normal_01_pdf(alpha)
beta_pdf = 0.0
value = mu + sigma * (alpha_pdf - beta_pdf) / (beta_cdf - alpha_cdf)
return value
def truncated_normal_b_mean ( mu, sigma, b ): #*****************************************************************************80 # ## TRUNCATED_NORMAL_B_MEAN returns the mean of the upper Truncated Normal distribution. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 08 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, real MU, SIGMA, the parameters of the parent Normal Distribution. # # Input, real B, the upper truncation limit. # # Output, real VALUE, the mean of the PDF. # from normal_01_cdf import normal_01_cdf from normal_01_pdf import normal_01_pdf beta = ( b - mu ) / sigma alpha_cdf = 0.0 beta_cdf = normal_01_cdf ( beta ) alpha_pdf = 0.0 beta_pdf = normal_01_pdf ( beta ) value = mu + sigma * ( alpha_pdf - beta_pdf ) / ( beta_cdf - alpha_cdf ) return value
def truncated_normal_b_moment(order, mu, sigma, b):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_B_MOMENT: a moment of upper truncated Normal distribution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 09 March 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# Phoebus Dhrymes,
# Moments of Truncated Normal Distributions,
# May 2005.
#
# Parameters:
#
# Input, integer ORDER, the order of the moment.
# 0 <= ORDER.
#
# Input, real MU, SIGMA, the mean and standard deviation of the
# parent Normal distribution.
# 0 < S.
#
# Input, real B, the upper truncation limit.
#
# Output, real VALUE, the moment of the PDF.
#
from normal_01_cdf import normal_01_cdf
from normal_01_pdf import normal_01_pdf
from r8_choose import r8_choose
from sys import exit
if (order < 0):
print ''
print 'TRUNCATED_NORMAL_B_MOMENT - Fatal error!'
print ' ORDER < 0.'
exit('TRUNCATED_NORMAL_B_MOMENT - Fatal error!')
if (sigma <= 0.0):
print ''
print 'TRUNCATED_NORMAL_B_MOMENT - Fatal error!'
print ' SIGMA <= 0.0.'
exit('TRUNCATED_NORMAL_B_MOMENT - Fatal error!')
b_h = (b - mu) / sigma
b_pdf = normal_01_pdf(b_h)
b_cdf = normal_01_cdf(b_h)
if (b_cdf == 0.0):
print ''
print 'TRUNCATED_NORMAL_B_MOMENT - Fatal error!'
print ' PDF/CDF ratio fails, because B_CDF too small.'
print ' B_PDF = %g' % (b_pdf)
print ' B_CDF = %g' % (b_cdf)
exit('TRUNCATED_NORMAL_B_MOMENT - Fatal error!')
f = b_pdf / b_cdf
value = 0.0
irm2 = 0.0
irm1 = 0.0
for r in range(0, order + 1):
if (r == 0):
ir = 1.0
elif (r == 1):
ir = -f
else:
ir = -b_h**(r - 1) * f + (r - 1) * irm2
value = value + r8_choose ( order, r ) \
* mu ** ( order - r ) \
* sigma ** r * ir
irm2 = irm1
irm1 = ir
return value
def truncated_normal_ab_moment ( order, mu, sigma, a, b ):
#*****************************************************************************80
#
## TRUNCATED_NORMAL_AB_MOMENT: a moment of the truncated Normal distribution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 09 March 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# Phoebus Dhrymes,
# Moments of Truncated Normal Distributions,
# May 2005.
#
# Parameters:
#
# Input, integer ORDER, the order of the moment.
# 0 <= ORDER.
#
# Input, real MU, SIGMA, the mean and standard deviation of the
# parent Normal distribution.
# 0 < S.
#
# Input, real A, B, the lower and upper truncation limits.
# A < B.
#
# Output, real VALUE, the moment of the PDF.
#
from normal_01_cdf import normal_01_cdf
from normal_01_pdf import normal_01_pdf
from r8_choose import r8_choose
from sys import exit
if ( order < 0 ):
print ''
print 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!'
print ' ORDER < 0.'
exit ( 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!' )
if ( sigma <= 0.0 ):
print ''
print 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!'
print ' SIGMA <= 0.0.'
exit ( 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!' )
if ( b <= a ):
print ''
print 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!'
print ' B <= A.'
exit ( 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!' )
a_h = ( a - mu ) / sigma
a_pdf = normal_01_pdf ( a_h )
a_cdf = normal_01_cdf ( a_h )
if ( a_cdf == 0.0 ):
print ''
print 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!'
print ' PDF/CDF ratio fails, because A_CDF is too small.'
print ' A_PDF = %g' % ( a_pdf )
print ' A_CDF = %g' % ( a_cdf )
exit ( 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!' )
b_h = ( b - mu ) / sigma
b_pdf = normal_01_pdf ( b_h )
b_cdf = normal_01_cdf ( b_h )
if ( b_cdf == 0.0 ):
print ''
print 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!'
print ' PDF/CDF ratio fails, because B_CDF too small.'
print ' B_PDF = %g' % ( b_pdf )
print ' B_CDF = %g' % ( b_cdf )
exit ( 'TRUNCATED_NORMAL_AB_MOMENT - Fatal error!' )
value = 0.0
irm2 = 0.0
irm1 = 0.0
for r in range ( 0, order + 1 ):
if ( r == 0 ):
ir = 1.0
elif ( r == 1 ):
ir = - ( b_pdf - a_pdf ) / ( b_cdf - a_cdf )
else:
ir = ( r - 1 ) * irm2 \
- ( b_h ** ( r - 1 ) * b_pdf - a_h ** ( r - 1 ) * a_pdf ) \
/ ( b_cdf - a_cdf )
value = value + r8_choose ( order, r ) \
* mu ** ( order - r ) \
* sigma ** r * ir
irm2 = irm1
irm1 = ir
return value
