/usr/lib/python2.7/dist-packages/dipy/core/sphere_stats.py is in python-dipy 0.10.1-1.
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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 | """ Statistics on spheres
"""
from __future__ import division, print_function, absolute_import
import numpy as np
import dipy.core.geometry as geometry
from itertools import permutations
def random_uniform_on_sphere(n=1, coords='xyz'):
r'''Random unit vectors from a uniform distribution on the sphere.
Parameters
-----------
n : int
Number of random vectors
coords : {'xyz', 'radians', 'degrees'}
'xyz' for cartesian form
'radians' for spherical form in rads
'degrees' for spherical form in degrees
Notes
------
The uniform distribution on the sphere, parameterized by spherical
coordinates $(\theta, \phi)$, should verify $\phi\sim U[0,2\pi]$, while
$z=\cos(\theta)\sim U[-1,1]$.
References
-----------
.. [1] http://mathworld.wolfram.com/SpherePointPicking.html.
Returns
--------
X : array, shape (n,3) if coords='xyz' or shape (n,2) otherwise
Uniformly distributed vectors on the unit sphere.
Examples
---------
>>> from dipy.core.sphere_stats import random_uniform_on_sphere
>>> X = random_uniform_on_sphere(4, 'radians')
>>> X.shape
(4, 2)
>>> X = random_uniform_on_sphere(4, 'xyz')
>>> X.shape
(4, 3)
'''
z = np.random.uniform(-1, 1, n)
theta = np.arccos(z)
phi = np.random.uniform(0, 2*np.pi, n)
if coords == 'xyz':
r = np.ones(n)
return np.vstack(geometry.sphere2cart(r, theta, phi)).T
angles = np.vstack((theta, phi)).T
if coords == 'radians':
return angles
if coords == 'degrees':
return np.rad2deg(angles)
def eigenstats(points, alpha=0.05):
r'''Principal direction and confidence ellipse
Implements equations in section 6.3.1(ii) of Fisher, Lewis and
Embleton, supplemented by equations in section 3.2.5.
Parameters
----------
points : arraey_like (N,3)
array of points on the sphere of radius 1 in $\mathbb{R}^3$
alpha : real or None
1 minus the coverage for the confidence ellipsoid, e.g. 0.05 for 95% coverage.
Returns
-------
centre : vector (3,)
centre of ellipsoid
b1 : vector (2,)
lengths of semi-axes of ellipsoid
'''
n = points.shape[0]
# the number of points
rad2deg = 180/np.pi
# scale angles from radians to degrees
# there is a problem with averaging and axis data.
'''
centroid = np.sum(points, axis=0)/n
normed_centroid = geometry.normalized_vector(centroid)
x,y,z = normed_centroid
#coordinates of normed centroid
polar_centroid = np.array(geometry.cart2sphere(x,y,z))*rad2deg
'''
cross = np.dot(points.T,points)/n
# cross-covariance of points
evals, evecs = np.linalg.eigh(cross)
# eigen decomposition assuming that cross is symmetric
order = np.argsort(evals)
# eigenvalues don't necessarily come in an particular order?
tau = evals[order]
# the ordered eigenvalues
h = evecs[:,order]
# the eigenvectors in corresponding order
h[:,2] = h[:,2]*np.sign(h[2,2])
# map the first principal direction into upper hemisphere
centre = np.array(geometry.cart2sphere(*h[:,2]))[1:]*rad2deg
# the spherical coordinates of the first principal direction
e = np.zeros((2,2))
p0 = np.dot(points,h[:,0])
p1 = np.dot(points,h[:,1])
p2 = np.dot(points,h[:,2])
# the principal coordinates of the points
e[0,0] = np.sum((p0**2)*(p2**2))/(n*(tau[0]-tau[2])**2)
e[1,1] = np.sum((p1**2)*(p2**2))/(n*(tau[1]-tau[2])**2)
e[0,1] = np.sum((p0*p1*(p2**2))/(n*(tau[0]-tau[2])*(tau[1]-tau[2])))
e[1,0] = e[0,1]
# e is a 2x2 helper matrix
b1 = np.array([np.NaN,np.NaN])
d = -2*np.log(alpha)/n
s,w = np.linalg.eig(e)
g = np.sqrt(d*s)
b1= np.arcsin(g)*rad2deg
# b1 are the estimated 100*(1-alpha)% confidence ellipsoid semi-axes
# in degrees
return centre, b1
'''
# b2 is equivalent to b1 above
# try to invert e and calculate vector b the standard errors of
# centre - these are forced to a mixture of NaN and/or 0 in singular cases
b2 = np.array([np.NaN,np.NaN])
if np.abs(np.linalg.det(e)) < 10**-20:
b2 = np.array([0,np.NaN])
else:
try:
f = np.linalg.inv(e)
except np.linalg.LigAlgError:
b2 = np.array([np.NaN, np.NaN])
else:
t, y = np.linalg.eig(f)
d = -2*np.log(alpha)/n
g = np.sqrt(d/t)
b2= np.arcsin(g)*rad2deg
'''
def compare_orientation_sets(S,T):
r'''Computes the mean cosine distance of the best match between
points of two sets of vectors S and T (angular similarity)
Parameters
-----------
S : array, shape (m,d)
First set of vectors.
T : array, shape (n,d)
Second set of vectors.
Returns
--------
max_mean_cosine : float
Maximum mean cosine distance.
Examples
---------
>>> from dipy.core.sphere_stats import compare_orientation_sets
>>> S=np.array([[1,0,0],[0,1,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,0,1]])
>>> compare_orientation_sets(S,T)
1.0
>>> T=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> S=np.array([[1,0,0],[0,0,1]])
>>> compare_orientation_sets(S,T)
1.0
>>> from dipy.core.sphere_stats import compare_orientation_sets
>>> S=np.array([[-1,0,0],[0,1,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,0,-1]])
>>> compare_orientation_sets(S,T)
1.0
'''
m = len(S)
n = len(T)
if m < n:
A = S.copy()
a = m
S = T
T = A
m = n
n = a
v = [np.sum([np.abs(np.dot(p[i],T[i])) for i in range(n)]) for p in permutations(S,n)]
return np.max(v)/np.float(n)
#return np.max(v)*np.float(n)/np.float(m)
def angular_similarity(S,T):
r'''Computes the cosine distance of the best match between
points of two sets of vectors S and T
Parameters
-----------
S : array, shape (m,d)
T : array, shape (n,d)
Returns
--------
max_cosine_distance:float
Examples
---------
>>> import numpy as np
>>> from dipy.core.sphere_stats import angular_similarity
>>> S=np.array([[1,0,0],[0,1,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,0,1]])
>>> angular_similarity(S,T)
2.0
>>> T=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> S=np.array([[1,0,0],[0,0,1]])
>>> angular_similarity(S,T)
2.0
>>> S=np.array([[-1,0,0],[0,1,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,0,-1]])
>>> angular_similarity(S,T)
2.0
>>> T=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> S=np.array([[1,0,0],[0,1,0],[0,0,1]])
>>> angular_similarity(S,T)
3.0
>>> S=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,np.sqrt(2)/2.,np.sqrt(2)/2.],[0,0,1]])
>>> angular_similarity(S,T)
2.7071067811865475
>>> S=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> T=np.array([[1,0,0]])
>>> angular_similarity(S,T)
1.0
>>> S=np.array([[0,1,0],[1,0,0]])
>>> T=np.array([[0,0,1]])
>>> angular_similarity(S,T)
0.0
>>> S=np.array([[0,1,0],[1,0,0]])
>>> T=np.array([[0,np.sqrt(2)/2.,np.sqrt(2)/2.]])
Now we use ``print`` to reduce the precision of of the printed output
(so the doctests don't detect unimportant differences)
>>> print('%.12f' % angular_similarity(S,T))
0.707106781187
>>> S=np.array([[0,1,0]])
>>> T=np.array([[0,np.sqrt(2)/2.,np.sqrt(2)/2.]])
>>> print('%.12f' % angular_similarity(S,T))
0.707106781187
>>> S=np.array([[0,1,0],[0,0,1]])
>>> T=np.array([[0,np.sqrt(2)/2.,np.sqrt(2)/2.]])
>>> print('%.12f' % angular_similarity(S,T))
0.707106781187
'''
m = len(S)
n = len(T)
if m < n:
A = S.copy()
a = m
S = T
T = A
m = n
n = a
"""
v=[]
for p in permutations(S,n):
angles=[]
for i in range(n):
angles.append(np.abs(np.dot(p[i],T[i])))
v.append(np.sum(angles))
print(v)
"""
v = [np.sum([np.abs(np.dot(p[i],T[i])) for i in range(n)]) for p in permutations(S,n)]
return np.float(np.max(v))#*np.float(n)/np.float(m)
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