def show_x_error_chart(count):
""" displays x=123 with covariances showing error"""
plt.cla()
plt.gca().autoscale(tight=True)
cov = np.array([[0.03, 0], [0, 8]])
e = stats.covariance_ellipse(cov)
cov2 = np.array([[0.03, 0], [0, 4]])
e2 = stats.covariance_ellipse(cov2)
cov3 = np.array([[12, 11.95], [11.95, 12]])
e3 = stats.covariance_ellipse(cov3)
sigma = [1, 4, 9]
if count >= 1:
stats.plot_covariance_ellipse((0, 0), ellipse=e, variance=sigma)
if count == 2 or count == 3:
stats.plot_covariance_ellipse((5, 5), ellipse=e, variance=sigma)
if count == 3:
stats.plot_covariance_ellipse((5, 5),
ellipse=e3,
variance=sigma,
edgecolor='r')
if count == 4:
M1 = np.array([[5, 5]]).T
m4, cov4 = stats.multivariate_multiply(M1, cov2, M1, cov3)
e4 = stats.covariance_ellipse(cov4)
stats.plot_covariance_ellipse((5, 5),
ellipse=e,
variance=sigma,
alpha=0.25)
stats.plot_covariance_ellipse((5, 5),
ellipse=e3,
variance=sigma,
edgecolor='r',
alpha=0.25)
stats.plot_covariance_ellipse(m4[:, 0], ellipse=e4, variance=sigma)
plt.ylim((-9, 16))
#plt.ylim([0,11])
#plt.xticks(np.arange(1,4,1))
plt.xlabel("Position")
plt.ylabel("Velocity")
plt.show()
def show_x_error_chart(count):
""" displays x=123 with covariances showing error"""
plt.cla()
plt.gca().autoscale(tight=True)
cov = np.array([[0.03,0], [0,8]])
e = stats.covariance_ellipse (cov)
cov2 = np.array([[0.03,0], [0,4]])
e2 = stats.covariance_ellipse (cov2)
cov3 = np.array([[12,11.95], [11.95,12]])
e3 = stats.covariance_ellipse (cov3)
sigma=[1, 4, 9]
if count >= 1:
stats.plot_covariance_ellipse ((0,0), ellipse=e, variance=sigma)
if count == 2 or count == 3:
stats.plot_covariance_ellipse ((5,5), ellipse=e, variance=sigma)
if count == 3:
stats.plot_covariance_ellipse ((5,5), ellipse=e3, variance=sigma,
edgecolor='r')
if count == 4:
M1 = np.array([[5, 5]]).T
m4, cov4 = stats.multivariate_multiply(M1, cov2, M1, cov3)
e4 = stats.covariance_ellipse (cov4)
stats.plot_covariance_ellipse ((5,5), ellipse=e, variance=sigma,
alpha=0.25)
stats.plot_covariance_ellipse ((5,5), ellipse=e3, variance=sigma,
edgecolor='r', alpha=0.25)
stats.plot_covariance_ellipse (m4[:,0], ellipse=e4, variance=sigma)
plt.ylim((-9, 16))
#plt.ylim([0,11])
#plt.xticks(np.arange(1,4,1))
plt.xlabel("Position")
plt.ylabel("Velocity")
plt.show()
def do_plot_test():
import matplotlib.pyplot as plt
from numpy.random import multivariate_normal as mnormal
from filterpy.stats import covariance_ellipse, plot_covariance
p = np.array([[32, 15], [15., 40.]])
x, y = mnormal(mean=(0, 0), cov=p, size=5000).T
sd = 2
a, w, h = covariance_ellipse(p, sd)
print(np.degrees(a), w, h)
count = 0
color = []
for i in range(len(x)):
if _is_inside_ellipse(x[i], y[i], 0, 0, a, w, h):
color.append('b')
count += 1
else:
color.append('r')
plt.scatter(x, y, alpha=0.2, c=color)
plt.axis('equal')
plot_covariance(mean=(0., 0.),
cov=p,
std=[1, 2, 3],
alpha=0.3,
facecolor='none')
print(count / len(x))
def do_plot_test():
import matplotlib.pyplot as plt
from numpy.random import multivariate_normal as mnormal
from filterpy.stats import covariance_ellipse, plot_covariance
p = np.array([[32, 15], [15., 40.]])
x, y = mnormal(mean=(0, 0), cov=p, size=5000).T
sd = 2
a, w, h = covariance_ellipse(p, sd)
print(np.degrees(a), w, h)
count = 0
color = []
for i in range(len(x)):
if _is_inside_ellipse(x[i], y[i], 0, 0, a, w, h):
color.append('b')
count += 1
else:
color.append('r')
plt.scatter(x, y, alpha=0.2, c=color)
plt.axis('equal')
plot_covariance(mean=(0., 0.),
cov=p,
std=[1,2,3],
alpha=0.3,
facecolor='none')
print(count / len(x))
def show_x_with_unobserved():
""" shows x=1,2,3 with velocity superimposed on top """
# plot velocity
sigma=[0.5,1.,1.5,2]
cov = np.array([[1,1],[1,1.1]])
stats.plot_covariance_ellipse ((2,2), cov=cov, variance=sigma, axis_equal=False)
# plot positions
cov = np.array([[0.003,0], [0,12]])
sigma=[0.5,1.,1.5,2]
e = stats.covariance_ellipse (cov)
stats.plot_covariance_ellipse ((1,1), ellipse=e, variance=sigma, axis_equal=False)
stats.plot_covariance_ellipse ((2,1), ellipse=e, variance=sigma, axis_equal=False)
stats.plot_covariance_ellipse ((3,1), ellipse=e, variance=sigma, axis_equal=False)
# plot intersection cirle
isct = Ellipse(xy=(2,2), width=.2, height=1.2, edgecolor='r', fc='None', lw=4)
plt.gca().add_artist(isct)
plt.ylim([0,11])
plt.xlim([0,4])
plt.xticks(np.arange(1,4,1))
plt.xlabel("Position")
plt.ylabel("Time")
plt.show()
def show_x_with_unobserved():
""" shows x=1,2,3 with velocity superimposed on top """
# plot velocity
sigma=[0.5,1.,1.5,2]
cov = np.array([[1,1],[1,1.1]])
stats.plot_covariance_ellipse ((2,2), cov=cov, variance=sigma, axis_equal=False)
# plot positions
cov = np.array([[0.003,0], [0,12]])
sigma=[0.5,1.,1.5,2]
e = stats.covariance_ellipse (cov)
stats.plot_covariance_ellipse ((1,1), ellipse=e, variance=sigma, axis_equal=False)
stats.plot_covariance_ellipse ((2,1), ellipse=e, variance=sigma, axis_equal=False)
stats.plot_covariance_ellipse ((3,1), ellipse=e, variance=sigma, axis_equal=False)
# plot intersection cirle
isct = Ellipse(xy=(2,2), width=.2, height=1.2, edgecolor='r', fc='None', lw=4)
plt.gca().add_artist(isct)
plt.ylim([0,11])
plt.xlim([0,4])
plt.xticks(np.arange(1,4,1))
plt.xlabel("Position")
plt.ylabel("Time")
plt.show()
def plot_covariance(
mean, cov=None, variance=1.0, std=None, interval=None,
ellipse=None, title=None, axis_equal=False,
show_semiaxis=False, show_center=True,
facecolor=None, edgecolor=None,
fc='none', ec='#004080',
alpha=1.0, xlim=None, ylim=None,
ls='solid',ax=None):
"filterpy.stats covariance ellipse plot function with added parameter for custom axis"
from matplotlib.patches import Ellipse
import matplotlib.pyplot as plt
if cov is not None and ellipse is not None:
raise ValueError('You cannot specify both cov and ellipse')
if cov is None and ellipse is None:
raise ValueError('Specify one of cov or ellipse')
if facecolor is None:
facecolor = fc
if edgecolor is None:
edgecolor = ec
if cov is not None:
ellipse = covariance_ellipse(cov)
if axis_equal:
plt.axis('equal')
if title is not None:
plt.title(title)
angle = np.degrees(ellipse[0])
width = ellipse[1] * 2.
height = ellipse[2] * 2.
std = _std_tuple_of(variance, std, interval)
for sd in std:
e = Ellipse(xy=mean, width=sd*width, height=sd*height, angle=angle,
facecolor=facecolor,
edgecolor=edgecolor,
alpha=alpha,
lw=2, ls=ls)
ax.add_patch(e)
x, y = mean
if show_center:
ax.scatter(x, y, marker='+', color=edgecolor)
if show_semiaxis:
a = ellipse[0]
h, w = height/4, width/4
ax.plot([x, x+ h*cos(a+np.pi/2)], [y, y + h*sin(a+np.pi/2)])
ax.plot([x, x+ w*cos(a)], [y, y + w*sin(a)])
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 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
def show_x_error_chart(count):
""" displays x=123 with covariances showing error"""
plt.cla()
plt.gca().autoscale(tight=True)
cov = np.array([[0.1, 0], [0, 8]])
pos_ellipse = stats.covariance_ellipse(cov)
cov2 = np.array([[0.1, 0], [0, 4]])
cov3 = np.array([[12, 11.95], [11.95, 12]])
vel_ellipse = stats.covariance_ellipse(cov3)
sigma = [1, 4, 9]
if count < 4:
stats.plot_covariance_ellipse((0, 0),
ellipse=pos_ellipse,
variance=sigma)
if count == 2:
stats.plot_covariance_ellipse((0, 0),
ellipse=vel_ellipse,
variance=sigma,
edgecolor='r')
if count == 3:
stats.plot_covariance_ellipse((5, 5),
ellipse=pos_ellipse,
variance=sigma)
stats.plot_covariance_ellipse((0, 0),
ellipse=vel_ellipse,
variance=sigma,
edgecolor='r')
if count == 4:
M0 = np.array([[0, 0]]).T
M1 = np.array([[5, 5]]).T
_, cov4 = stats.multivariate_multiply(M0, cov2, M1, cov3)
e4 = stats.covariance_ellipse(cov4)
stats.plot_covariance_ellipse((0, 0),
ellipse=pos_ellipse,
variance=sigma,
alpha=0.25)
stats.plot_covariance_ellipse((5, 5),
ellipse=pos_ellipse,
variance=sigma,
alpha=0.25)
stats.plot_covariance_ellipse((0, 0),
ellipse=vel_ellipse,
variance=sigma,
edgecolor='r',
alpha=0.25)
stats.plot_covariance_ellipse((5, 5), ellipse=e4, variance=sigma)
plt.ylim((-9, 16))
plt.xlabel("Position")
plt.ylabel("Velocity")
plt.axis('equal')
plt.show()