def plot_gaussians(xs, ps, x_range, y_range, N):
""" given a list of 2d states (x,y) and 2x2 covariance matrices, produce
a surface plot showing all of the gaussians"""
xs = np.asarray(xs)
x = np.linspace(x_range[0], x_range[1], N)
y = np.linspace(y_range[0], y_range[1], N)
xx, yy = np.meshgrid(x, y)
zv = np.zeros((N, N))
for mean, cov in zip(xs, ps):
zs = np.array([
multivariate_gaussian(np.array([i, j]), mean, cov)
for i, j in zip(np.ravel(xx), np.ravel(yy))
])
zv += zs.reshape(xx.shape)
ax = plt.figure().add_subplot(111, projection='3d')
ax.plot_surface(xx,
yy,
zv,
rstride=1,
cstride=1,
lw=.5,
edgecolors='#191919',
antialiased=True,
shade=True,
cmap=cm.autumn)
ax.view_init(elev=40., azim=230)
def run(self):
np.random.seed(13)
x = [2.5, 7.3]
mu = [2.0, 7.0]
P = [[8., 0], [0, 3.]]
mg = multivariate_gaussian(x, mu, P)
print(mg)
def test_multivariate_gaussian():
# test that we treat lists and arrays the same
mean= (0, 0)
cov=[[1, .5], [.5, 1]]
a = [[multivariate_gaussian((i, j), mean, cov)
for i in (-1, 0, 1)]
for j in (-1, 0, 1)]
b = [[multivariate_gaussian((i, j), mean, np.asarray(cov))
for i in (-1, 0, 1)]
for j in (-1, 0, 1)]
assert np.allclose(a, b)
a = [[multivariate_gaussian((i, j), np.asarray(mean), cov)
for i in (-1, 0, 1)]
for j in (-1, 0, 1)]
assert np.allclose(a, b)
def test_multivariate_gaussian():
# test that we treat lists and arrays the same
mean = (0, 0)
cov = [[1, .5], [.5, 1]]
a = [[multivariate_gaussian((i, j), mean, cov) for i in (-1, 0, 1)]
for j in (-1, 0, 1)]
b = [[
multivariate_gaussian((i, j), mean, np.asarray(cov))
for i in (-1, 0, 1)
] for j in (-1, 0, 1)]
assert np.allclose(a, b)
a = [[
multivariate_gaussian((i, j), np.asarray(mean), cov)
for i in (-1, 0, 1)
] for j in (-1, 0, 1)]
assert np.allclose(a, b)
def plot_3d_sampled_covariance(mean, cov):
""" plots a 2x2 covariance matrix positioned at mean. mean will be plotted
in x and y, and the probability in the z axis.
Parameters
----------
mean : 2x1 tuple-like object
mean for x and y coordinates. For example (2.3, 7.5)
cov : 2x2 nd.array
the covariance matrix
"""
# compute width and height of covariance ellipse so we can choose
# appropriate ranges for x and y
o,w,h = stats.covariance_ellipse(cov,3)
# rotate width and height to x,y axis
wx = abs(w*np.cos(o) + h*np.sin(o))*1.2
wy = abs(h*np.cos(o) - w*np.sin(o))*1.2
# ensure axis are of the same size so everything is plotted with the same
# scale
if wx > wy:
w = wx
else:
w = wy
minx = mean[0] - w
maxx = mean[0] + w
miny = mean[1] - w
maxy = mean[1] + w
count = 1000
x,y = multivariate_normal(mean=mean, cov=cov, size=count).T
xs = np.arange(minx, maxx, (maxx-minx)/40.)
ys = np.arange(miny, maxy, (maxy-miny)/40.)
xv, yv = np.meshgrid (xs, ys)
zs = np.array([100.* stats.multivariate_gaussian(np.array([xx,yy]),mean,cov) \
for xx,yy in zip(np.ravel(xv), np.ravel(yv))])
zv = zs.reshape(xv.shape)
ax = plt.figure().add_subplot(111, projection='3d')
ax.scatter(x,y, [0]*count, marker='.')
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.contour(xv, yv, zv, zdir='x', offset=minx-1, cmap=cm.autumn)
ax.contour(xv, yv, zv, zdir='y', offset=maxy, cmap=cm.BuGn)
def test_multivariate_gaussian():
# test that we treat lists and arrays the same
mean= (0, 0)
cov=[[1, .5], [.5, 1]]
a = [[multivariate_gaussian((i, j), mean, cov)
for i in (-1, 0, 1)]
for j in (-1, 0, 1)]
b = [[multivariate_gaussian((i, j), mean, np.asarray(cov))
for i in (-1, 0, 1)]
for j in (-1, 0, 1)]
assert np.allclose(a, b)
a = [[multivariate_gaussian((i, j), np.asarray(mean), cov)
for i in (-1, 0, 1)]
for j in (-1, 0, 1)]
assert np.allclose(a, b)
try:
multivariate_gaussian(1, 1, -1)
except:
pass
else:
assert False, "negative variances are meaningless"
# test that we get the same results as scipy.stats.multivariate_normal
xs = np.random.randn(1000)
mean = np.random.randn(1000)
var = np.random.random(1000) * 5
for x, m, v in zip(xs, mean, var):
assert abs(multivariate_gaussian(x, m, v) - scipy.stats.multivariate_normal(m, v).pdf(x)) < 1.e-12
def plot_gaussians(xs, ps, x_range, y_range, N):
""" given a list of 2d states (x,y) and 2x2 covariance matrices, produce
a surface plot showing all of the gaussians"""
xs = np.asarray(xs)
x = np.linspace (x_range[0], x_range[1], N)
y = np.linspace (y_range[0], y_range[1], N)
xx, yy = np.meshgrid(x, y)
zv = np.zeros((N, N))
for mean, cov in zip(xs, ps):
zs = np.array([multivariate_gaussian(np.array([i ,j]), mean, cov)
for i, j in zip(np.ravel(xx), np.ravel(yy))])
zv += zs.reshape(xx.shape)
ax = plt.figure().add_subplot(111, projection='3d')
ax.plot_surface(xx, yy, zv, rstride=1, cstride=1, lw=.5, edgecolors='#191919',
antialiased=True, shade=True, cmap=cm.autumn)
ax.view_init(elev=40., azim=230)
def plot_3d_sampled_covariance(mean, cov):
""" plots a 2x2 covariance matrix positioned at mean. mean will be plotted
in x and y, and the probability in the z axis.
Parameters
----------
mean : 2x1 tuple-like object
mean for x and y coordinates. For example (2.3, 7.5)
cov : 2x2 nd.array
the covariance matrix
"""
# compute width and height of covariance ellipse so we can choose
# appropriate ranges for x and y
o, w, h = stats.covariance_ellipse(cov, 3)
# rotate width and height to x,y axis
wx = abs(w * np.cos(o) + h * np.sin(o)) * 1.2
wy = abs(h * np.cos(o) - w * np.sin(o)) * 1.2
# ensure axis are of the same size so everything is plotted with the same
# scale
if wx > wy:
w = wx
else:
w = wy
minx = mean[0] - w
maxx = mean[0] + w
miny = mean[1] - w
maxy = mean[1] + w
count = 1000
x, y = multivariate_normal(mean=mean, cov=cov, size=count).T
xs = np.arange(minx, maxx, (maxx - minx) / 40.)
ys = np.arange(miny, maxy, (maxy - miny) / 40.)
xv, yv = np.meshgrid(xs, ys)
zs = np.array([100.* stats.multivariate_gaussian(np.array([xx,yy]),mean,cov) \
for xx,yy in zip(np.ravel(xv), np.ravel(yv))])
zv = zs.reshape(xv.shape)
ax = plt.gcf().add_subplot(111, projection='3d')
ax.scatter(x, y, [0] * count, marker='.')
ax.set_xlabel('X')
ax.set_ylabel('Y')
x = mean[0]
zs = np.array([
100. * stats.multivariate_gaussian(np.array([x, y]), mean, cov)
for _, y in zip(np.ravel(xv), np.ravel(yv))
])
zv = zs.reshape(xv.shape)
ax.contour(xv, yv, zv, zdir='x', offset=minx - 1, cmap=cm.binary)
y = mean[1]
zs = np.array([
100. * stats.multivariate_gaussian(np.array([x, y]), mean, cov)
for x, _ in zip(np.ravel(xv), np.ravel(yv))
])
zv = zs.reshape(xv.shape)
ax.contour(xv, yv, zv, zdir='y', offset=maxy, cmap=cm.binary)
def plot_3d_covariance(mean, cov):
""" plots a 2x2 covariance matrix positioned at mean. mean will be plotted
in x and y, and the probability in the z axis.
Parameters
----------
mean : 2x1 tuple-like object
mean for x and y coordinates. For example (2.3, 7.5)
cov : 2x2 nd.array
the covariance matrix
"""
# compute width and height of covariance ellipse so we can choose
# appropriate ranges for x and y
o,w,h = stats.covariance_ellipse(cov,3)
# rotate width and height to x,y axis
wx = abs(w*np.cos(o) + h*np.sin(o))*1.2
wy = abs(h*np.cos(o) - w*np.sin(o))*1.2
# ensure axis are of the same size so everything is plotted with the same
# scale
if wx > wy:
w = wx
else:
w = wy
minx = mean[0] - w
maxx = mean[0] + w
miny = mean[1] - w
maxy = mean[1] + w
xs = np.arange(minx, maxx, (maxx-minx)/40.)
ys = np.arange(miny, maxy, (maxy-miny)/40.)
xv, yv = np.meshgrid(xs, ys)
zs = np.array([100.* stats.multivariate_gaussian(np.array([x,y]),mean,cov) \
for x, y in zip(np.ravel(xv), np.ravel(yv))])
zv = zs.reshape(xv.shape)
maxz = np.max(zs)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
#ax = plt.gca(projection='3d')
ax.plot_surface(xv, yv, zv, rstride=1, cstride=1, cmap=cm.autumn)
ax.set_xlabel('X')
ax.set_ylabel('Y')
# For unknown reasons this started failing in Jupyter notebook when
# using `%matplotlib inline` magic. Still works fine in IPython or when
# `%matplotlib notebook` magic is used.
x = mean[0]
zs = np.array([100.* stats.multivariate_gaussian(np.array([x, y]),mean,cov)
for _, y in zip(np.ravel(xv), np.ravel(yv))])
zv = zs.reshape(xv.shape)
try:
pass
#ax.contour(xv, yv, zv, zdir='x', offset=minx-1, cmap=cm.binary)
except:
pass
y = mean[1]
zs = np.array([100.* stats.multivariate_gaussian(np.array([x, y]),mean,cov)
for x, _ in zip(np.ravel(xv), np.ravel(yv))])
zv = zs.reshape(xv.shape)
try:
pass
#ax.contour(xv, yv, zv, zdir='y', offset=maxy, cmap=cm.binary)
except:
pass