def _tmeancost(mu, sigma, bound):
''' additive constant for the mean of the truncated normal '''
l, u = bound
l = (l - mu) / sigma
u = (u - mu) / sigma
n = norm_pdf(u) - norm_pdf(l)
d = norm_cdf(u) - norm_cdf(l)
c = sigma * n / d
return c
def tnorm_pdf(x, mu, sigma, bound):
''' truncated normal density function '''
x = np.asarray(x)
x = (x - mu) / sigma
u = (bound[1] - mu) / sigma
l = (bound[0] - mu) / sigma
c = norm_cdf(u) - norm_cdf(l)
d = norm_pdf(x) / (c * sigma)
return np.where((x >= l) & (x <= u), d, 0.)
def tnorm_cdf(x, mu, sigma, bound):
''' truncated normal distribution function '''
x = np.asarray(x)
x = (x - mu) / sigma
u = (bound[1] - mu) / sigma
l = (bound[0] - mu) / sigma
c = norm_cdf(u) - norm_cdf(l)
p = (norm_cdf(x) - norm_cdf(l)) / c
p[x < l] = 0
p[x > u] = 1
return p
def _tvarcost(mu, sigma, bound):
''' multiplicative factor for the variance of the truncated normal '''
l,u = bound
l = (l - mu) / sigma
u = (u - mu) / sigma
# as x --> +/- Inf, x * f(x) --> 0 for the gaussian density f, but 0. *
# Inf would normally give NaN
n1_1 = 0. if np.isinf(u) else u * norm_pdf(u)
n1_2 = 0. if np.isinf(l) else l * norm_pdf(l)
n1 = n1_1 - n1_2
n2 = norm_pdf(u) - norm_pdf(l)
d = norm_cdf(u) - norm_cdf(l)
c = 1 + n1 / d - (n2 /d )**2
assert c > 0, "c = %g " % c
return c