1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
|
From d91eb9e61aa3b74240b79e3a33077cef179518d1 Mon Sep 17 00:00:00 2001
From: Matt Haberland <matt.haberland@gmail.com>
Date: Sun, 19 Oct 2025 07:41:20 -0700
Subject: [PATCH 1/5] ENH: scipy.stats: speed up bivariate normal cdf
---
scipy/stats/_qmvnt.py | 21 ++++
scipy/stats/_stats_pythran.py | 142 +++++++++++++++++++++++++
scipy/stats/tests/test_multivariate.py | 4 +-
3 files changed, 165 insertions(+), 2 deletions(-)
Index: scipy/scipy/stats/_qmvnt.py
===================================================================
--- scipy.orig/scipy/stats/_qmvnt.py 2025-10-27 00:20:27.016948021 +0100
+++ scipy/scipy/stats/_qmvnt.py 2025-10-27 00:20:27.006089715 +0100
@@ -32,11 +32,13 @@
# OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+import math
import numpy as np
from scipy.fft import fft, ifft
from scipy.special import ndtr as phi, ndtri as phinv
from scipy.stats._qmc import primes_from_2_to
+from scipy.stats._stats_pythran import _bvnu
from ._qmvnt_cy import _qmvn_inner, _qmvt_inner
@@ -178,6 +180,9 @@
prob = phi(high / covar**0.5) - phi(low / covar**0.5)
# More or less
est_error = 1e-15
+ elif n == 2:
+ prob = _bvn(low, high, covar)
+ est_error = 1e-15
else:
mi = min(limit, n * 1000)
prob = 0.0
@@ -452,3 +457,19 @@
t = x[slc1].copy()
x[slc1] = x[slc2].copy()
x[slc2] = t
+
+
+def _bvn(a, b, A):
+ # covariance matrix is written [[s1**2, rho*s1*s2], [rho*s1*s2, s2**2]]
+ # e.g. https://en.wikipedia.org/wiki/Multivariate_normal_distribution
+ # therefore, s12 = rho*s1*s2 -> rho = s12/(s1*s2)
+ s1 = math.sqrt(A[0, 0])
+ s2 = math.sqrt(A[1, 1])
+ s12 = A[0, 1]
+ r = s12 / (s1 * s2)
+ # the x and y coordinates seem to be normalized by the standard devs
+ xl, xu = a[0] / s1, b[0] / s1
+ yl, yu = a[1] / s2, b[1] / s2
+ p = _bvnu(xl, yl, r) - _bvnu(xu, yl, r) - _bvnu(xl, yu, r) + _bvnu(xu, yu, r)
+ p = max( 0., min( p, 1. ) )
+ return p
Index: scipy/scipy/stats/_stats_pythran.py
===================================================================
--- scipy.orig/scipy/stats/_stats_pythran.py 2025-10-27 00:20:27.016948021 +0100
+++ scipy/scipy/stats/_stats_pythran.py 2025-10-27 00:20:27.010392490 +0100
@@ -1,3 +1,4 @@
+import math
import numpy as np
@@ -209,3 +210,146 @@
cache[p] = np.cumsum(pmf) if tp=='cdf' else pmf
out[i] = cache[p][k[i]]
return out
+
+
+# function p = phid(z), p = erfc( -z/sqrt(2) )/2; % Normal cdf
+def phid(z):
+ return math.erfc(-z / math.sqrt(2)) / 2
+
+
+def np_dot(x, y):
+ return np.sum(x * y)
+
+
+# function p = bvnu( dh, dk, r )
+#pythran export _bvnu(float64, float64, float64)
+def _bvnu(dh, dk, r):
+ # Adapted from the MATLAB original implementation by Dr. Alan Genz;
+ # see license information in _qmvnt.py
+ math_inf, math_pi = np.inf, np.pi
+ # if dh == inf | dk == inf:p = 0;
+ if (dh == math_inf) or (dk == math_inf):
+ p = 0.
+ # elseif dh == -inf, if dk == -inf, p = 1; else p = phid(-dk); end
+ elif dh == -math_inf:
+ if dk == -math_inf:
+ p = 1.
+ else:
+ p = phid(-dk)
+ # elseif dk == -inf, p = phid(-dh);
+ elif dk == -math_inf:
+ p = phid(-dh)
+ # elseif r == 0, p = phid(-dh)*phid(-dk);
+ elif r == 0:
+ p = phid(-dh) * phid(-dk)
+ # else, tp = 2*pi; h = dh; k = dk; hk = h*k; bvn = 0;
+ else:
+ tp = 2*math_pi
+ h = dh
+ k = dk
+ hk = h*k
+ bvn = 0.
+ # if abs(r) < 0.3 % Gauss Legendre points and weights, n = 6
+ # w(1:3) = [0.1713244923791705 0.3607615730481384 0.4679139345726904];
+ # x(1:3) = [0.9324695142031522 0.6612093864662647 0.2386191860831970];
+ if abs(r) < 0.3:
+ w = [0.1713244923791705, 0.3607615730481384, 0.4679139345726904]
+ x = [0.9324695142031522, 0.6612093864662647, 0.2386191860831970]
+ # elseif abs(r) < 0.75 % Gauss Legendre points and weights, n = 12
+ # w(1:3) = [.04717533638651177 0.1069393259953183 0.1600783285433464];
+ # w(4:6) = [0.2031674267230659 0.2334925365383547 0.2491470458134029];
+ # x(1:3) = [0.9815606342467191 0.9041172563704750 0.7699026741943050];
+ # x(4:6) = [0.5873179542866171 0.3678314989981802 0.1252334085114692];
+ elif abs(r) < 0.75:
+ w = [.04717533638651177, 0.1069393259953183, 0.1600783285433464,
+ 0.2031674267230659, 0.2334925365383547, 0.2491470458134029]
+ x = [0.9815606342467191, 0.9041172563704750, 0.7699026741943050,
+ 0.5873179542866171, 0.3678314989981802, 0.1252334085114692]
+ # else, % Gauss Legendre points and weights, n = 20
+ # w(1:3) = [.01761400713915212 .04060142980038694 .06267204833410906];
+ # w(4:6) = [.08327674157670475 0.1019301198172404 0.1181945319615184];
+ # w(7:9) = [0.1316886384491766 0.1420961093183821 0.1491729864726037];
+ # w(10) = 0.1527533871307259;
+ # x(1:3) = [0.9931285991850949 0.9639719272779138 0.9122344282513259];
+ # x(4:6) = [0.8391169718222188 0.7463319064601508 0.6360536807265150];
+ # x(7:9) = [0.5108670019508271 0.3737060887154196 0.2277858511416451];
+ # x(10) = 0.07652652113349733;
+ else:
+ w = [.01761400713915212, .04060142980038694, .06267204833410906,
+ .08327674157670475, 0.1019301198172404, 0.1181945319615184,
+ 0.1316886384491766, 0.1420961093183821, 0.1491729864726037,
+ 0.1527533871307259]
+ x = [0.9931285991850949, 0.9639719272779138, 0.9122344282513259,
+ 0.8391169718222188, 0.7463319064601508, 0.6360536807265150,
+ 0.5108670019508271, 0.3737060887154196, 0.2277858511416451,
+ 0.07652652113349733]
+ # end, w = [w w]; x = [1-x 1+x];
+ w, x = np.asarray(w), np.asarray(x)
+ w = np.concatenate((w, w))
+ x = np.concatenate((1-x, 1+x))
+ # if abs(r) < 0.925, hs = ( h*h + k*k )/2; asr = asin(r)/2;
+ if abs(r) < 0.925:
+ hs = ( h*h + k*k )/2
+ asr = math.asin(r)/2
+ # sn = sin(asr*x); bvn = exp((sn*hk-hs)./(1-sn.^2))*w';
+ sn = np.sin(asr*x)
+ bvn = np_dot(np.exp((sn*hk-hs) / (1-sn**2)), w)
+ # bvn = bvn*asr/tp + phid(-h)*phid(-k);
+ bvn = bvn*asr/tp + phid(-h)*phid(-k)
+ # else, if r < 0, k = -k; hk = -hk; end
+ else:
+ if r < 0:
+ k = -k
+ hk = -hk
+ # if abs(r) < 1, as = 1-r^2; a = sqrt(as); bs = (h-k)^2;
+ if abs(r) < 1:
+ as_ = 1-r**2
+ a = math.sqrt(as_)
+ bs = (h-k)**2
+ # asr = -( bs/as + hk )/2; c = (4-hk)/8 ; d = (12-hk)/80;
+ asr = -( bs/as_ + hk )/2
+ c = (4-hk)/8
+ d = (12-hk)/80
+ # if asr > -100, bvn = a*exp(asr)*(1-c*(bs-as)*(1-d*bs)/3+c*d*as^2); end
+ if asr > -100:
+ bvn = a*math.exp(asr)*(1-c*(bs-as_)*(1-d*bs)/3+c*d*as_**2)
+ # if hk > -100, b = sqrt(bs); sp = sqrt(tp)*phid(-b/a);
+ if hk > -100:
+ b = math.sqrt(bs)
+ sp = math.sqrt(tp)*phid(-b/a)
+ # bvn = bvn - exp(-hk/2)*sp*b*( 1 - c*bs*(1-d*bs)/3 );
+ bvn = bvn - math.exp(-hk/2)*sp*b*( 1 - c*bs*(1-d*bs)/3 )
+
+ # end, a = a/2; xs = (a*x).^2; asr = -( bs./xs + hk )/2;
+ a = a/2
+ xs = (a*x)**2
+ asr = -( bs / xs + hk )/2
+ # ix = find( asr > -100 ); xs = xs(ix); sp = ( 1 + c*xs.*(1+5*d*xs) );
+ ix = asr > -100
+ xs = xs[ix]
+ sp = 1 + c*xs * (1+5*d*xs)
+ # rs = sqrt(1-xs); ep = exp( -(hk/2)*xs./(1+rs).^2 )./rs;
+ rs = np.sqrt(1-xs)
+ ep = np.exp( -(hk/2)*xs / (1+rs)**2 )/rs
+ # bvn = ( a*( (exp(asr(ix)).*(sp-ep))*w(ix)' ) - bvn )/tp;
+ bvn = ( a*np_dot( (np.exp(asr[ix]) * (sp-ep)), w[ix] ) - bvn )/tp
+ # end
+ # if r > 0, bvn = bvn + phid( -max( h, k ) );
+ if r > 0:
+ bvn = bvn + phid( -max( h, k ) )
+ # elseif h >= k, bvn = -bvn;
+ elif h >= k:
+ bvn = -bvn
+ # else, if h < 0, L = phid(k)-phid(h); else, L = phid(-h)-phid(-k); end
+ else:
+ if h < 0:
+ L = phid(k)-phid(h)
+ else:
+ L = phid(-h)-phid(-k)
+ # bvn = L - bvn;
+ bvn = L - bvn
+ # end
+ # end, p = max( 0, min( 1, bvn ) );
+ p = max( 0, min( 1, bvn ) )
+ # end
+ return p
Index: scipy/scipy/stats/tests/test_multivariate.py
===================================================================
--- scipy.orig/scipy/stats/tests/test_multivariate.py 2025-10-27 00:20:27.016948021 +0100
+++ scipy/scipy/stats/tests/test_multivariate.py 2025-10-27 00:20:27.008391611 +0100
@@ -1039,8 +1039,8 @@
assert_allclose(cdf, cdf[0]*expected_signs)
@pytest.mark.slow
- def test_cdf_vs_cubature(self):
- ndim = 3
+ @pytest.mark.parametrize("ndim", [2, 3])
+ def test_cdf_vs_cubature(self, ndim):
rng = np.random.default_rng(123)
a = rng.uniform(size=(ndim, ndim))
cov = a.T @ a
Index: scipy/scipy/stats/meson.build
===================================================================
--- scipy.orig/scipy/stats/meson.build 2025-10-27 00:20:27.016948021 +0100
+++ scipy/scipy/stats/meson.build 2025-10-27 00:20:27.011288415 +0100
@@ -77,7 +77,8 @@
if use_pythran
py3.extension_module('_stats_pythran',
pythran_gen.process('_stats_pythran.py'),
- cpp_args: cpp_args_pythran,
+ # `_cpp_Wno_unused_local_typedefs` can be removed when we require pythran >= 0.19.0
+ cpp_args: [cpp_args_pythran, _cpp_Wno_unused_local_typedefs],
dependencies: [pythran_dep, np_dep],
link_args: version_link_args,
install: true,
|
