def test_common_shape():
A = np.ones((4, 1))
B = 2
C = np.ones((1, 5)) * 5
assert_array_equal(common_shape(A, B, C), (4, 5))
assert_array_equal(common_shape(A, B, C, shape=(3, 4, 1)), (3, 4, 5))
A = np.ones((4, 1))
B = 2
C = np.ones((1, 5)) * 5
assert_array_equal(common_shape(A, B, C), (4, 5))
assert_array_equal(common_shape(A, B, C, shape=(3, 4, 1)), (3, 4, 5))
def test_common_shape():
A = np.ones((4, 1))
B = 2
C = np.ones((1, 5)) * 5
assert common_shape(A, B, C) == (4, 5)
assert common_shape(A, B, C, shape=(3, 4, 1)) == (3, 4, 5)
A = np.ones((4, 1))
B = 2
C = np.ones((1, 5)) * 5
assert common_shape(A, B, C) == (4, 5)
assert common_shape(A, B, C, shape=(3, 4, 1)) == (3, 4, 5)
def test_common_shape():
A = np.ones((4, 1))
B = 2
C = np.ones((1, 5)) * 5
assert_array_equal(common_shape(A, B, C), (4, 5))
assert_array_equal(common_shape(A, B, C, shape=(3, 4, 1)), (3, 4, 5))
A = np.ones((4, 1))
B = 2
C = np.ones((1, 5)) * 5
assert_array_equal(common_shape(A, B, C), (4, 5))
assert_array_equal(common_shape(A, B, C, shape=(3, 4, 1)), (3, 4, 5))
def cdfnorm2d(b1, b2, r):
'''
Returnc Bivariate Normal cumulative distribution function
Parameters
----------
b1, b2 : array-like
upper integration limits
r : real scalar
correlation coefficient (-1 <= r <= 1).
Returns
-------
bvn : ndarray
distribution function evaluated at b1, b2.
Notes
-----
CDFNORM2D computes bivariate normal probabilities, i.e., the probability
Prob(X1 <= B1 and X2 <= B2) with an absolute error less than 1e-15.
This function is based on the method described by Drezner, z and
G.O. Wesolowsky, (1989), with major modifications for double precision,
and for |r| close to 1.
Example
-------
>>> import wafo.gaussian as wg
>>> x = np.linspace(-5,5,20)
>>> [B1,B2] = np.meshgrid(x, x)
>>> r = 0.3;
>>> F = wg.cdfnorm2d(B1,B2,r)
surf(x,x,F)
See also
--------
cdfnorm
Reference
---------
Drezner, z and g.o. Wesolowsky, (1989),
"On the computation of the bivariate normal integral",
Journal of statist. comput. simul. 35, pp. 101-107,
'''
# Translated into Python
# Per A. Brodtkorb
#
# Original code
# by alan genz
# department of mathematics
# washington state university
# pullman, wa 99164-3113
# email : [email protected]
cshape = common_shape(b1, b2, r, shape=[1, ])
one = ones(cshape)
h, k, r = (-b1 * one).ravel(), (-b2 * one).ravel(), (r * one).ravel()
bvn = where(abs(r) > 1, nan, 0.0)
two = 2.e0
twopi = 6.283185307179586e0
hk = h * k
k0, = nonzero(abs(r) < 0.925e0)
if len(k0) > 0:
hs = (h[k0] ** 2 + k[k0] ** 2) / two
asr = arcsin(r[k0])
k1, = nonzero(r[k0] >= 0.75)
if len(k1) > 0:
k01 = k0[k1]
for i in range(10):
for sign in - 1, 1:
sn = sin(asr[k1] * (sign * _X20[i] + 1) / 2)
bvn[k01] = bvn[k01] + _W20[i] * \
exp((sn * hk[k01] - hs[k1]) / (1 - sn * sn))
k1, = nonzero((0.3 <= r[k0]) & (r[k0] < 0.75))
if len(k1) > 0:
k01 = k0[k1]
for i in range(6):
for sign in - 1, 1:
sn = sin(asr[k1] * (sign * _X12[i] + 1) / 2)
bvn[k01] = bvn[k01] + _W12[i] * \
exp((sn * hk[k01] - hs[k1]) / (1 - sn * sn))
k1, = nonzero(r[k0] < 0.3)
if len(k1) > 0:
k01 = k0[k1]
for i in range(3):
for sign in - 1, 1:
sn = sin(asr[k1] * (sign * _X6[i] + 1) / 2)
bvn[k01] = bvn[k01] + _W6[i] * \
exp((sn * hk[k01] - hs[k1]) / (1 - sn * sn))
bvn[k0] *= asr / (two * twopi)
bvn[k0] += fi(-h[k0]) * fi(-k[k0])
k1, = nonzero((0.925 <= abs(r)) & (abs(r) <= 1))
if len(k1) > 0:
k2, = nonzero(r[k1] < 0)
if len(k2) > 0:
k12 = k1[k2]
k[k12] = -k[k12]
hk[k12] = -hk[k12]
k3, = nonzero(abs(r[k1]) < 1)
if len(k3) > 0:
k13 = k1[k3]
a2 = (1 - r[k13]) * (1 + r[k13])
a = sqrt(a2)
b = abs(h[k13] - k[k13])
bs = b * b
c = (4.e0 - hk[k13]) / 8.e0
d = (12.e0 - hk[k13]) / 16.e0
asr = -(bs / a2 + hk[k13]) / 2.e0
k4, = nonzero(asr > -100.e0)
if len(k4) > 0:
bvn[k13[k4]] = (a[k4] * exp(asr[k4]) *
(1 - c[k4] * (bs[k4] - a2[k4]) *
(1 - d[k4] * bs[k4] / 5) / 3 +
c[k4] * d[k4] * a2[k4] ** 2 / 5))
k5, = nonzero(hk[k13] < 100.e0)
if len(k5) > 0:
# b = sqrt(bs);
k135 = k13[k5]
bvn[k135] = bvn[k135] - exp(-hk[k135] / 2) * sqrt(twopi) * fi(-b[k5] / a[k5]) * \
b[k5] * (1 - c[k5] * bs[k5] * (1 - d[k5] * bs[k5] / 5) / 3)
a /= two
for i in range(10):
for sign in - 1, 1:
xs = (a * (sign * _X20[i] + 1)) ** 2
rs = sqrt(1 - xs)
asr = -(bs / xs + hk[k13]) / 2
k6, = nonzero(asr > -100.e0)
if len(k6) > 0:
k136 = k13[k6]
bvn[k136] += (a[k6] * _W20[i] * exp(asr[k6]) *
(exp(-hk[k136] * (1 - rs[k6]) /
(2 * (1 + rs[k6]))) / rs[k6] -
(1 + c[k6] * xs[k6] *
(1 + d[k6] * xs[k6]))))
bvn[k3] = -bvn[k3] / twopi
k7, = nonzero(r[k1] > 0)
if len(k7):
k17 = k1[k7]
bvn[k17] += fi(-np.maximum(h[k17], k[k17]))
k8, = nonzero(r[k1] < 0)
if len(k8) > 0:
k18 = k1[k8]
bvn[k18] = -bvn[k18] + np.maximum(0, fi(-h[k18]) - fi(-k[k18]))
bvn.shape = cshape
return bvn
def cdfnorm2d(b1, b2, r):
'''
Returnc Bivariate Normal cumulative distribution function
Parameters
----------
b1, b2 : array-like
upper integration limits
r : real scalar
correlation coefficient (-1 <= r <= 1).
Returns
-------
bvn : ndarray
distribution function evaluated at b1, b2.
Notes
-----
CDFNORM2D computes bivariate normal probabilities, i.e., the probability
Prob(X1 <= B1 and X2 <= B2) with an absolute error less than 1e-15.
This function is based on the method described by Drezner, z and
G.O. Wesolowsky, (1989), with major modifications for double precision,
and for |r| close to 1.
Example
-------
>>> import wafo.gaussian as wg
>>> x = np.linspace(-5,5,20)
>>> [B1,B2] = np.meshgrid(x, x)
>>> r = 0.3;
>>> F = wg.cdfnorm2d(B1,B2,r)
surf(x,x,F)
See also
--------
cdfnorm
Reference
---------
Drezner, z and g.o. Wesolowsky, (1989),
"On the computation of the bivariate normal integral",
Journal of statist. comput. simul. 35, pp. 101-107,
'''
# Translated into Python
# Per A. Brodtkorb
#
# Original code
# by alan genz
# department of mathematics
# washington state university
# pullman, wa 99164-3113
# email : [email protected]
cshape = common_shape(b1, b2, r, shape=[
1,
])
one = ones(cshape)
h, k, r = (-b1 * one).ravel(), (-b2 * one).ravel(), (r * one).ravel()
bvn = where(abs(r) > 1, nan, 0.0)
two = 2.e0
twopi = 6.283185307179586e0
hk = h * k
k0, = nonzero(abs(r) < 0.925e0)
if len(k0) > 0:
hs = (h[k0]**2 + k[k0]**2) / two
asr = arcsin(r[k0])
k1, = nonzero(r[k0] >= 0.75)
if len(k1) > 0:
k01 = k0[k1]
for i in range(10):
for sign in -1, 1:
sn = sin(asr[k1] * (sign * _X20[i] + 1) / 2)
bvn[k01] = bvn[k01] + _W20[i] * \
exp((sn * hk[k01] - hs[k1]) / (1 - sn * sn))
k1, = nonzero((0.3 <= r[k0]) & (r[k0] < 0.75))
if len(k1) > 0:
k01 = k0[k1]
for i in range(6):
for sign in -1, 1:
sn = sin(asr[k1] * (sign * _X12[i] + 1) / 2)
bvn[k01] = bvn[k01] + _W12[i] * \
exp((sn * hk[k01] - hs[k1]) / (1 - sn * sn))
k1, = nonzero(r[k0] < 0.3)
if len(k1) > 0:
k01 = k0[k1]
for i in range(3):
for sign in -1, 1:
sn = sin(asr[k1] * (sign * _X6[i] + 1) / 2)
bvn[k01] = bvn[k01] + _W6[i] * \
exp((sn * hk[k01] - hs[k1]) / (1 - sn * sn))
bvn[k0] *= asr / (two * twopi)
bvn[k0] += fi(-h[k0]) * fi(-k[k0])
k1, = nonzero((0.925 <= abs(r)) & (abs(r) <= 1))
if len(k1) > 0:
k2, = nonzero(r[k1] < 0)
if len(k2) > 0:
k12 = k1[k2]
k[k12] = -k[k12]
hk[k12] = -hk[k12]
k3, = nonzero(abs(r[k1]) < 1)
if len(k3) > 0:
k13 = k1[k3]
a2 = (1 - r[k13]) * (1 + r[k13])
a = sqrt(a2)
b = abs(h[k13] - k[k13])
bs = b * b
c = (4.e0 - hk[k13]) / 8.e0
d = (12.e0 - hk[k13]) / 16.e0
asr = -(bs / a2 + hk[k13]) / 2.e0
k4, = nonzero(asr > -100.e0)
if len(k4) > 0:
bvn[k13[k4]] = (a[k4] * exp(asr[k4]) *
(1 - c[k4] * (bs[k4] - a2[k4]) *
(1 - d[k4] * bs[k4] / 5) / 3 +
c[k4] * d[k4] * a2[k4]**2 / 5))
k5, = nonzero(hk[k13] < 100.e0)
if len(k5) > 0:
# b = sqrt(bs);
k135 = k13[k5]
bvn[k135] = bvn[k135] - exp(-hk[k135] / 2) * sqrt(twopi) * fi(-b[k5] / a[k5]) * \
b[k5] * (1 - c[k5] * bs[k5] * (1 - d[k5] * bs[k5] / 5) / 3)
a /= two
for i in range(10):
for sign in -1, 1:
xs = (a * (sign * _X20[i] + 1))**2
rs = sqrt(1 - xs)
asr = -(bs / xs + hk[k13]) / 2
k6, = nonzero(asr > -100.e0)
if len(k6) > 0:
k136 = k13[k6]
bvn[k136] += (a[k6] * _W20[i] * exp(asr[k6]) *
(exp(-hk[k136] * (1 - rs[k6]) /
(2 * (1 + rs[k6]))) / rs[k6] -
(1 + c[k6] * xs[k6] *
(1 + d[k6] * xs[k6]))))
bvn[k3] = -bvn[k3] / twopi
k7, = nonzero(r[k1] > 0)
if len(k7):
k17 = k1[k7]
bvn[k17] += fi(-np.maximum(h[k17], k[k17]))
k8, = nonzero(r[k1] < 0)
if len(k8) > 0:
k18 = k1[k8]
bvn[k18] = -bvn[k18] + np.maximum(0, fi(-h[k18]) - fi(-k[k18]))
bvn.shape = cshape
return bvn