def update(self, z, R=None, hx_args=()):
""" Update the CKF with the given measurements. On return,
self.x and self.P contain the new mean and covariance of the filter.
Parameters
----------
z : numpy.array of shape (dim_z)
measurement vector
R : numpy.array((dim_z, dim_z)), optional
Measurement noise. If provided, overrides self.R for
this function call.
hx_args : tuple, optional, default (,)
arguments to be passed into Hx function after the required state
variable.
"""
if z is None:
return
if not isinstance(hx_args, tuple):
hx_args = (hx_args, )
if R is None:
R = self.R
elif isscalar(R):
R = eye(self._dim_z) * R
for k in range(self._num_sigmas):
self.sigmas_h[k] = self.hx(self.sigmas_f[k], *hx_args)
# mean and covariance of prediction passed through unscented transform
#zp, Pz = UT(self.sigmas_h, self.Wm, self.Wc, R, self.z_mean, self.residual_z)
zp, Pz = ckf_transform(self.sigmas_h, R)
# compute cross variance of the state and the measurements
Pxz = zeros((self._dim_x, self._dim_z))
m = self._num_sigmas # literaure uses m for scaling factor
xf = self.x.flatten()
zpf = zp.flatten()
for k in range(m):
dx = self.sigmas_f[k] - xf
dz = self.sigmas_h[k] - zpf
Pxz += outer(dx, dz)
Pxz /= m
self.K = dot(Pxz, inv(Pz)) # Kalman gain
self.y = self.residual_z(z, zp) #residual
self.x = self.x + dot(self.K, self.y)
self.P = self.P - dot(self.K, Pz).dot(self.K.T)
if self.compute_log_likelihood:
self.log_likelihood = logpdf(x=self.y, cov=Pz)
self.likelihood = math.exp(self.log_likelihood)
if self.likelihood == 0:
self.likelihood = sys.float_info.min
def log_likelihood(self):
"""
log-likelihood of the last measurement.
"""
if self._log_likelihood is None:
self._log_likelihood = logpdf(x=self.y, cov=self.S)
return self._log_likelihood
def log_likelihood(self, z):
""" log likelihood of the measurement `z`. """
if z is None:
return math.log(sys.float_info.min)
else:
return logpdf(z, dot(self.H, self.x), self.S)
def update(self, z, R=None, UT=None, hx_args=()):
""" Update the UKF with the given measurements. On return,
self.x and self.P contain the new mean and covariance of the filter.
Parameters
----------
z : numpy.array of shape (dim_z)
measurement vector
R : numpy.array((dim_z, dim_z)), optional
Measurement noise. If provided, overrides self.R for
this function call.
UT : function(sigmas, Wm, Wc, noise_cov), optional
Optional function to compute the unscented transform for the sigma
points passed through hx. Typically the default function will
work - you can use x_mean_fn and z_mean_fn to alter the behavior
of the unscented transform.
hx_args : tuple, optional, default (,)
arguments to be passed into Hx function after the required state
variable.
"""
if z is None:
return
if not isinstance(hx_args, tuple):
hx_args = (hx_args,)
if UT is None:
UT = unscented_transform
if R is None:
R = self.R
elif isscalar(R):
R = eye(self._dim_z) * R
for i in range(self._num_sigmas):
self.sigmas_h[i] = self.hx(self.sigmas_f[i], *hx_args)
# mean and covariance of prediction passed through unscented transform
zp, Pz = UT(self.sigmas_h, self.Wm, self.Wc, R, self.z_mean, self.residual_z)
# compute cross variance of the state and the measurements
Pxz = zeros((self._dim_x, self._dim_z))
for i in range(self._num_sigmas):
dx = self.residual_x(self.sigmas_f[i], self.x)
dz = self.residual_z(self.sigmas_h[i], zp)
Pxz += self.Wc[i] * outer(dx, dz)
self.K = dot(Pxz, inv(Pz)) # Kalman gain
self.y = self.residual_z(z, zp) # residual
self.x = self.x + dot(self.K, self.y)
self.P = self.P - dot(self.K, Pz).dot(self.K.T)
self.log_likelihood = logpdf(self.y, np.zeros(len(self.y)), Pz)
def log_likelihood(self, z):
""" log likelihood of the measurement `z`. """
if z is None:
return math.log(sys.float_info.min)
else:
return logpdf(z, dot(self.H, self.x), self.S)
def update(self, z, R=None, UT=None, hx_args=()):
""" Update the UKF with the given measurements. On return,
self.x and self.P contain the new mean and covariance of the filter.
Parameters
----------
z : numpy.array of shape (dim_z)
measurement vector
R : numpy.array((dim_z, dim_z)), optional
Measurement noise. If provided, overrides self.R for
this function call.
UT : function(sigmas, Wm, Wc, noise_cov), optional
Optional function to compute the unscented transform for the sigma
points passed through hx. Typically the default function will
work - you can use x_mean_fn and z_mean_fn to alter the behavior
of the unscented transform.
hx_args : tuple, optional, default (,)
arguments to be passed into Hx function after the required state
variable.
"""
if z is None:
return
if not isinstance(hx_args, tuple):
hx_args = (hx_args, )
if UT is None:
UT = unscented_transform
if R is None:
R = self.R
elif isscalar(R):
R = eye(self._dim_z) * R
for i in range(self._num_sigmas):
self.sigmas_h[i] = self.hx(self.sigmas_f[i], *hx_args)
# mean and covariance of prediction passed through unscented transform
zp, Pz = UT(self.sigmas_h, self.Wm, self.Wc, R, self.z_mean,
self.residual_z)
# compute cross variance of the state and the measurements
Pxz = zeros((self._dim_x, self._dim_z))
for i in range(self._num_sigmas):
dx = self.residual_x(self.sigmas_f[i], self.x)
dz = self.residual_z(self.sigmas_h[i], zp)
Pxz += self.Wc[i] * outer(dx, dz)
self.K = dot(Pxz, inv(Pz)) # Kalman gain
self.y = self.residual_z(z, zp) # residual
self.x = self.x + dot(self.K, self.y)
self.P = self.P - dot(self.K, Pz).dot(self.K.T)
self.log_likelihood = logpdf(self.y, np.zeros(len(self.y)), Pz)
def update(self, z, R_inv=None):
"""
Add a new measurement (z) to the kalman filter. If z is None, nothing
is changed.
Parameters
----------
z : np.array
measurement for this update.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
"""
if z is None:
return
if R_inv is None:
R_inv = self.R_inv
elif np.isscalar(R_inv):
R_inv = eye(self.dim_z) * R_inv
# rename for readability and a tiny extra bit of speed
H = self.H
H_T = H.T
P_inv = self.P_inv
x = self.x
if self._no_information:
self.x = dot(P_inv, x) + dot(H_T, R_inv).dot(z)
self.P_inv = P_inv + dot(H_T, R_inv).dot(H)
self.log_likelihood = math.log(sys.float_info.min)
self.likelihood = sys.float_info.min
else:
# y = z - Hx
# error (residual) between measurement and prediction
self.y = z - dot(H, x)
# S = HPH' + R
# project system uncertainty into measurement space
self.S = P_inv + dot(H_T, R_inv).dot(H)
self.K = dot(self.inv(self.S), H_T).dot(R_inv)
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self.x = x + dot(self.K, self.y)
self.P_inv = P_inv + dot(H_T, R_inv).dot(H)
self.z = np.copy(reshape_z(z, self.dim_z, np.ndim(self.x)))
if self.compute_log_likelihood:
self.log_likelihood = logpdf(x=self.y, cov=self.S)
self.likelihood = math.exp(self.log_likelihood)
if self.likelihood == 0:
self.likelihood = sys.float_info.min
def update(self, z, R=None, H=None):
"""
Add a new measurement (z) to the Kalman filter. If z is None, nothing
is changed.
Parameters
----------
z : (dim_z, 1): array_like
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be convertible to a column vector.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
H : np.array, or None
Optionally provide H to override the measurement function for this
one call, otherwise self.H will be used.
"""
if z is None:
return
z = _reshape_z(z, self.dim_z, self.x.ndim)
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
if H is None:
H = self.H
# y = z - Hx
# error (residual) between measurement and prediction
self.y = z - dot(H, self.x)
# common subexpression for speed
PHT = dot(self.P, H.T)
# S = HPH' + R
# project system uncertainty into measurement space
self.S = dot(H, PHT) + R
# K = PH'inv(S)
# map system uncertainty into kalman gain
self.K = dot(PHT, self.inv(self.S))
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self.x = self.x + dot(self.K, self.y)
# P = (I-KH)P(I-KH)' + KRK'
I_KH = self._I - dot(self.K, H)
self.P = dot(dot(I_KH, self.P), I_KH.T) + dot(dot(self.K, R), self.K.T)
if self.compute_log_likelihood:
self.log_likelihood = logpdf(x=self.y, cov=self.S)
def test_logpdf():
assert 3.9 < exp(logpdf(1, 1, .01)) < 4.
assert 3.9 < exp(logpdf([1], [1], .01)) < 4.
assert 3.9 < exp(logpdf([[1]], [[1]], .01)) < 4.
logpdf([1., 2], [1.1, 2], cov=np.array([[1., 2], [2, 5]]), allow_singular=False)
logpdf([1., 2], [1.1, 2], cov=np.array([[1., 2], [2, 5]]), allow_singular=True)
def log_likelihood_of(kf, z):
"""
log likelihood of the measurement `z`. This should only be called
after a call to update(). Calling after predict() will yield an
incorrect result."""
if z is None:
return kf.log(sys.float_info.min)
return logpdf(z, kf.z, kf.S)
def test_logpdf2():
z = np.array([1., 2.])
mean = np.array([1.1, 2])
cov = np.array([[1., 2], [2, 5]])
p = logpdf(z, mean, cov, allow_singular=False)
p2 = log_multivariate_normal_density(z, mean, cov)
print('p', p)
print('p2', p2)
print('p-p2', p - p2)
def _smooth_backstep(self, modal_probs_current, modal_probs__smoothed_next,
x_smooth_next, P_smooth_next, x_prior_next,
P_prior_next, x_post, P_post, likelihoods):
mixing_probs, _ = self._compute_mixing_probabilities_functional(
self.M, modal_probs_current)
mixing_probs, _ = self._compute_mixing_probabilities_functional(
M=mixing_probs, mu=modal_probs__smoothed_next)
[xs_smooth_mixed_n, Ps_smooth_mixed_n
] = self._compute_mixed_estimates(x_smooth_next, P_smooth_next,
mixing_probs)
xs_smooth_mixed_n = x_smooth_next
Ps_smooth_mixed_n = P_smooth_next
xs_smooth_model = []
Ps_smooth_model = []
for i, f in enumerate(self.filters):
[x_temp,
P_temp] = f._rts_backstep(x_post[i], P_post[i],
xs_smooth_mixed_n[i],
Ps_smooth_mixed_n[i], x_prior_next[i],
P_prior_next[i])
xs_smooth_model.append(x_temp)
Ps_smooth_model.append(P_temp)
likelihood_smooth = []
for j, f_j in enumerate(self.filters):
cumulation = 0.0
for i, f_i in enumerate(self.filters):
#Resizing by multiplying by lots of zeros slows the program down substantially
z = np.zeros(self.x.shape)
x_i = self.add(x_smooth_next[i], z)
x_j = self.add(x_prior_next[i], z)
P_j = self.add(P_prior_next[i], np.zeros(self.P.shape))
cumulation = cumulation + self.M[i, j] * logpdf(x_i, x_j, P_j)
#likelihood_smooth.append(likelihoods[j])
likelihood_smooth.append(cumulation)
mu_smooth = (likelihood_smooth * modal_probs_current)
mu_smooth = mu_smooth / mu_smooth.sum()
x_smooth_out = np.zeros(self.x.shape)
P_smooth_out = np.zeros(self.P.shape)
for i, (f, mu) in enumerate(zip(self.filters, mu_smooth)):
x_smooth_out = self.add(x_smooth_out, xs_smooth_model[i] * mu)
for i, (f, mu) in enumerate(zip(self.filters, mu_smooth)):
y = self.add(f.x, -self.x_smooth)
P_smooth_out = self.add(
P_smooth_out,
(self.add(mu * outer(y, y), mu * Ps_smooth_model[i])))
return x_smooth_out, P_smooth_out
def update(self, z, R=None, H=None):
if z is None:
return
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
# rename for readability and a tiny extra bit of speed
if H is None:
H = self.H
P = self.P
x = self.x
# handle special case: if z is in form [[z]] but x is not a column
# vector dimensions will not match
if x.ndim==1 and shape(z) == (1,1):
z = z[0]
if shape(z) == (): # is it scalar, e.g. z=3 or z=np.array(3)
z = np.asarray([z])
# y = z - Hx
# error (residual) between measurement and prediction
Hx = dot(H, x)
assert shape(Hx) == shape(z) or (shape(Hx) == (1,1) and shape(z) == (1,)), \
'shape of z should be {}, but it is {}'.format(
shape(Hx), shape(z))
self.y = z - Hx
# S = HPH' + R
# project system uncertainty into measurement space
self.S = dot3(H, P, H.T) + R
# K = PH'inv(S)
# map system uncertainty into kalman gain
self.K = dot3(P, H.T, linalg.inv(self.S))
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self.x = x + dot(self.K, self.y)
# P = (I-KH)P(I-KH)' + KRK'
I_KH = self.I - dot(self.K, H)
self.P = dot3(I_KH, P, I_KH.T) + dot3(self.K, R, self.K.T)
self.log_likelihood = logpdf(z, dot(H, x), self.S)
def update(self, z, R=None):
"""
Add a new measurement (z) to the kalman filter. If z is None, nothing
is changed.
Parameters
----------
z : np.array
measurement for this update.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
"""
if z is None:
return
if R is None:
R = self.R
elif np.isscalar(R):
R = eye(self.dim_z) * R
# y = z - Hx
# error (residual) between measurement and prediction
self.y = z - dot(self.H, self.x)
PHT = dot(self.P, self.H.T)
# S = HPH' + R
# project system uncertainty into measurement space
self.S = dot(self.H, PHT) + R
# K = PH'inv(S)
# map system uncertainty into kalman gain
self.K = PHT.dot(linalg.inv(self.S))
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self.x = self.x + dot(self.K, self.y)
# P = (I-KH)P(I-KH)' + KRK'
I_KH = self.I - dot(self.K, self.H)
self.P = dot(I_KH, self.P).dot(I_KH.T) + dot(self.K, R).dot(self.K.T)
if self.compute_log_likelihood:
self.log_likelihood = logpdf(x=self.y, cov=self.S)
def smooth(self):
"""
Performs IMM smoothing one step.
This should be performed after the update() is performed.
"""
x_smooths, P_smooths, likelihood_smooth = [], [], []
mu_smooth = self.mu.copy()
for i, f in enumerate(self.filters):
# propagate using the mixed state estimate and covariance
Smoothing_gain = dot(f.P_post.dot(f.F.T), np.linalg.inv(f.P_prior))
x_smooths.append(f.x_post +
Smoothing_gain.dot(self.x_mixed[i] - f.x_prior))
P_smooths.append(f.P_post - Smoothing_gain.dot(
self.P_mixed[i] - f.P_prior).dot(Smoothing_gain.T))
for j, f_j in enumerate(self.filters):
cumulation = 0.0
for i, f_i in enumerate(self.filters):
#Resizing by multiplying by lots of zeros slows the program down substantially
z = np.zeros(self.x.shape)
x_i = self.add(x_smooths[i], z)
x_j = self.add(f_i.x_prior, z)
P_j = self.add(f_i.P_prior, np.zeros(self.P.shape))
cumulation = cumulation + self.M[i, j] * logpdf(x_i, x_j, P_j)
likelihood_smooth.append(cumulation)
likelihood_smooth.clear()
for j, f_j in enumerate(self.filters):
likelihood_smooth.append(f_j.likelihood)
mu_smooth = (likelihood_smooth * mu_smooth)
mu_smooth = mu_smooth / mu_smooth.sum()
self.x_smooth.fill(0)
for i, (f, mu) in enumerate(zip(self.filters, mu_smooth)):
self.x_smooth = self.add(self.x_smooth, x_smooths[i] * mu)
self.P_smooth.fill(0)
for i, (f, mu) in enumerate(zip(self.filters, mu_smooth)):
y = self.add(f.x, -self.x_smooth)
self.P_smooth = self.add(
self.P_smooth, (self.add(mu * outer(y, y), mu * P_smooths[i])))
def update_correlated(self, z, R=None, H=None):
if z is None:
return
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
# rename for readability and a tiny extra bit of speed
if H is None:
H = self.H
x = self.x
P = self.P
M = self.M
# handle special case: if z is in form [[z]] but x is not a column
# vector dimensions will not match
if x.ndim==1 and shape(z) == (1,1):
z = z[0]
if shape(z) == (): # is it scalar, e.g. z=3 or z=np.array(3)
z = np.asarray([z])
# y = z - Hx
# error (residual) between measurement and prediction
self.y = z - dot(H, x)
# project system uncertainty into measurement space
self.S = dot3(H, P, H.T) + dot(H, M) + dot(M.T, H.T) + R
# K = PH'inv(S)
# map system uncertainty into kalman gain
self.K = dot(dot(P, H.T) + M, linalg.inv(self.S))
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self.x = x + dot(self.K, self.y)
self.P = P - dot(self.K, dot(H, P) + M.T)
# compute log likelihood
self.log_likelihood = logpdf(z, dot(H, x), self.S)
def update(self, z, R=None, UT=None, hx_args=()):
if z is None:
return
if not isinstance(hx_args, tuple):
hx_args = (hx_args,)
if UT is None:
UT = unscented_transform
if R is None:
R = self.R
elif isscalar(R):
R = eye(self._dim_z) * R
for i in range(self._num_sigmas):
self.sigmas_h[i] = self.hx(self.sigmas_f[i], *hx_args)
# mean and covariance of prediction passed through unscented transform
zp, Pz = UT(self.sigmas_h, self.Wm, self.Wc, R, self.z_mean, self.residual_z)
# compute cross variance of the state and the measurements
Pxz = zeros((self._dim_x, self._dim_z))
for i in range(self._num_sigmas):
dx = self.residual_x(self.sigmas_f[i], self.x)
dz = self.residual_z(self.sigmas_h[i], zp)
Pxz += self.Wc[i] * outer(dx, dz)
self.K = dot(Pxz, inv(Pz)) # Kalman gain
self.y = self.residual_z(z, zp) # residual
self.x = self.x + dot(self.K, self.y)
self.P = self.P - dot3(self.K, Pz, self.K.T)
self.log_likelihood = logpdf(self.y, np.zeros(len(self.y)), Pz)
# Store transformed state uncertainty but without added sensor noise
self.Pz = Pz - R
def find_likelihood(self, z, R=None, H=None):
if np.all(np.isnan(z)):
x_mean_velocity = np.mean(self.velocity_x[-5:])
y_mean_velocity = np.mean(self.velocity_y[-5:])
z_noise = np.random.normal(loc=0, scale=0, size=2)
z = list(
self.x[0:2] - [self.velocity_x[-1], self.velocity_y[-1]] + [x_mean_velocity, y_mean_velocity] + z_noise)
z = reshape_z(z, self.dim_z, self.x.ndim)
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
if H is None:
H = self.H
y = z - dot(H, self.x)
PHT = dot(self.P, H.T)
S = dot(H, PHT) + R
_log_likelihood = logpdf(x=y, cov=S)
_likelihood = exp(_log_likelihood)
if _likelihood == 0:
_likelihood = sys.float_info.min
return _likelihood
def update(self,
z,
HJacobian,
Hx,
R=None,
args=(),
hx_args=(),
residual=np.subtract):
""" Performs the update innovation of the extended Kalman filter.
Parameters
----------
z : np.array
measurement for this step.
If `None`, only predict step is perfomed.
HJacobian : function
function which computes the Jacobian of the H matrix (measurement
function). Takes state variable (self.x) as input, returns H.
Hx : function
function which takes as input the state variable (self.x) along
with the optional arguments in hx_args, and returns the measurement
that would correspond to that state.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
args : tuple, optional, default (,)
arguments to be passed into HJacobian after the required state
variable. for robot localization you might need to pass in
information about the map and time of day, so you might have
`args=(map_data, time)`, where the signature of HCacobian will
be `def HJacobian(x, map, t)`
hx_args : tuple, optional, default (,)
arguments to be passed into Hx function after the required state
variable.
residual : function (z, z2), optional
Optional function that computes the residual (difference) between
the two measurement vectors. If you do not provide this, then the
built in minus operator will be used. You will normally want to use
the built in unless your residual computation is nonlinear (for
example, if they are angles)
"""
if not isinstance(args, tuple):
args = (args, )
if not isinstance(hx_args, tuple):
hx_args = (hx_args, )
if R is None:
R = self.R
elif np.isscalar(R):
R = eye(self.dim_z) * R
if np.isscalar(z) and self.dim_z == 1:
z = np.asarray([z], float)
H = HJacobian(self.x, *args)
PHT = dot(self.P, H.T)
self.S = dot(H, PHT) + R
self.K = PHT.dot(linalg.inv(self.S))
hx = Hx(self.x, *hx_args)
self.y = residual(z, hx)
self.x = self.x + dot(self.K, self.y)
# P = (I-KH)P(I-KH)' + KRK' is more numerically stable
# and works for non-optimal K vs the equation
# P = (I-KH)P usually seen in the literature.
I_KH = self._I - dot(self.K, H)
self.P = dot(I_KH, self.P).dot(I_KH.T) + dot(self.K, R).dot(self.K.T)
if self.compute_log_likelihood:
self.log_likelihood = logpdf(x=self.y, cov=self.S)
self.likelihood = math.exp(self.log_likelihood)
if self.likelihood == 0:
self.likelihood = sys.float_info.min
def predict_update(self, z, HJacobian, Hx, args=(), hx_args=(), u=0):
""" Performs the predict/update innovation of the extended Kalman
filter.
Parameters
----------
z : np.array
measurement for this step.
If `None`, only predict step is perfomed.
HJacobian : function
function which computes the Jacobian of the H matrix (measurement
function). Takes state variable (self.x) as input, along with the
optional arguments in args, and returns H.
Hx : function
function which takes as input the state variable (self.x) along
with the optional arguments in hx_args, and returns the measurement
that would correspond to that state.
args : tuple, optional, default (,)
arguments to be passed into HJacobian after the required state
variable.
hx_args : tuple, optional, default (,)
arguments to be passed into Hx after the required state
variable.
u : np.array or scalar
optional control vector input to the filter.
"""
#pylint: disable=too-many-locals
if not isinstance(args, tuple):
args = (args, )
if not isinstance(hx_args, tuple):
hx_args = (hx_args, )
if np.isscalar(z) and self.dim_z == 1:
z = np.asarray([z], float)
F = self.F
B = self.B
P = self.P
Q = self.Q
R = self.R
x = self.x
H = HJacobian(x, *args)
# predict step
x = dot(F, x) + dot(B, u)
P = dot(F, P).dot(F.T) + Q
# save prior
self.x_prior = np.copy(self.x)
self.P_prior = np.copy(self.P)
# update step
PHT = dot(P, H.T)
self.S = dot(H, PHT) + R
self.K = dot(PHT, linalg.inv(self.S))
self.y = z - Hx(x, *hx_args)
self.x = x + dot(self.K, self.y)
I_KH = self._I - dot(self.K, H)
self.P = dot(I_KH, P).dot(I_KH.T) + dot(self.K, R).dot(self.K.T)
if self.compute_log_likelihood:
self.log_likelihood = logpdf(x=self.y, cov=self.S)
self.likelihood = math.exp(self.log_likelihood)
if self.likelihood == 0:
self.likelihood = sys.float_info.min
def update(x, P, z, R, H=None, return_all=False):
"""
Add a new measurement (z) to the Kalman filter. If z is None, nothing
is changed.
This can handle either the multidimensional or unidimensional case. If
all parameters are floats instead of arrays the filter will still work,
and return floats for x, P as the result.
update(1, 2, 1, 1, 1) # univariate
update(x, P, 1
Parameters
----------
x : numpy.array(dim_x, 1), or float
State estimate vector
P : numpy.array(dim_x, dim_x), or float
Covariance matrix
z : numpy.array(dim_z, 1), or float
measurement for this update.
R : numpy.array(dim_z, dim_z), or float
Measurement noise matrix
H : numpy.array(dim_x, dim_x), or float, optional
Measurement function. If not provided, a value of 1 is assumed.
return_all : bool, default False
If true, y, K, S, and log_likelihood are returned, otherwise
only x and P are returned.
Returns
-------
x : numpy.array
Posterior state estimate vector
P : numpy.array
Posterior covariance matrix
y : numpy.array or scalar
Residua. Difference between measurement and state in measurement space
K : numpy.array
Kalman gain
S : numpy.array
System uncertainty in measurement space
log_likelihood : float
log likelihood of the measurement
"""
if z is None:
if return_all:
return x, P, None, None, None, None
else:
return x, P
if H is None:
H = np.array([1])
if np.isscalar(H):
H = np.array([H])
if not np.isscalar(x):
# handle special case: if z is in form [[z]] but x is not a column
# vector dimensions will not match
if x.ndim == 1 and shape(z) == (1, 1):
z = z[0]
if shape(z) == (): # is it scalar, e.g. z=3 or z=np.array(3)
z = np.asarray([z])
# error (residual) between measurement and prediction
y = z - dot(H, x)
# project system uncertainty into measurement space
S = dot3(H, P, H.T) + R
# map system uncertainty into kalman gain
try:
K = dot3(P, H.T, linalg.inv(S))
except:
K = dot3(P, H.T, 1 / S)
# predict new x with residual scaled by the kalman gain
x = x + dot(K, y)
# P = (I-KH)P(I-KH)' + KRK'
KH = dot(K, H)
try:
I_KH = np.eye(KH.shape[0]) - KH
except:
I_KH = np.array(1 - KH)
P = dot3(I_KH, P, I_KH.T) + dot3(K, R, K.T)
if return_all:
# compute log likelihood
log_likelihood = logpdf(z, dot(H, x), S)
return x, P, y, K, S, log_likelihood
else:
return x, P
def update_correlated(self, z, R=None, H=None):
""" Add a new measurement (z) to the Kalman filter assuming that
process noise and measurement noise are correlated as defined in
the `self.M` matrix.
If z is None, nothing is changed.
Parameters
----------
z : np.array
measurement for this update.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
H : np.array, or None
Optionally provide H to override the measurement function for this
one call, otherwise self.H will be used.
"""
if z is None:
return
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
# rename for readability and a tiny extra bit of speed
if H is None:
H = self.H
x = self._x
P = self._P
M = self.M
# handle special case: if z is in form [[z]] but x is not a column
# vector dimensions will not match
if x.ndim == 1 and shape(z) == (1, 1):
z = z[0]
if shape(z) == (): # is it scalar, e.g. z=3 or z=np.array(3)
z = np.asarray([z])
# y = z - Hx
# error (residual) between measurement and prediction
self._y = z - dot(H, x)
# project system uncertainty into measurement space
S = dot3(H, P, H.T) + dot(H, M) + dot(M.T, H.T) + R
# K = PH'inv(S)
# map system uncertainty into kalman gain
K = dot(dot(P, H.T) + M, linalg.inv(S))
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self._x = x + dot(K, self._y)
self._P = P - dot(K, dot(H, P) + M.T)
self._S = S
self._K = K
# compute log likelihood
self.log_likelihood = logpdf(z, dot(H, x), S)
def update(self, z, R=None, H=None):
"""
Add a new measurement (z) to the Kalman filter. If z is None, nothing
is changed.
Parameters
----------
z : np.array
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be a column vector.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
H : np.array, or None
Optionally provide H to override the measurement function for this
one call, otherwise self.H will be used.
"""
if z is None:
return
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
# rename for readability and a tiny extra bit of speed
if H is None:
H = self.H
P = self._P
x = self._x
# handle special case: if z is in form [[z]] but x is not a column
# vector dimensions will not match
if x.ndim == 1 and shape(z) == (1, 1):
z = z[0]
if shape(z) == (): # is it scalar, e.g. z=3 or z=np.array(3)
z = np.asarray([z])
# y = z - Hx
# error (residual) between measurement and prediction
Hx = dot(H, x)
assert shape(Hx) == shape(z) or (shape(Hx) == (1,1) and shape(z) == (1,)), \
'shape of z should be {}, but it is {}'.format(
shape(Hx), shape(z))
self._y = z - Hx
# S = HPH' + R
# project system uncertainty into measurement space
S = dot3(H, P, H.T) + R
# K = PH'inv(S)
# map system uncertainty into kalman gain
K = dot3(P, H.T, linalg.inv(S))
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self._x = x + dot(K, self._y)
# P = (I-KH)P(I-KH)' + KRK'
I_KH = self._I - dot(K, H)
self._P = dot3(I_KH, P, I_KH.T) + dot3(K, R, K.T)
self._S = S
self._K = K
self.log_likelihood = logpdf(z, dot(H, x), S)
def log_likelihood(self):
if self._log_likelihood is None:
self._log_likelihood = logpdf(x=self.y, cov=self.S)
return self._log_likelihood
def update(self, z, R=None, UT=None, hx=None, **hx_args):
"""
Update the UKF with the given measurements. On return,
self.x and self.P contain the new mean and covariance of the filter.
Parameters
----------
z : numpy.array of shape (dim_z)
measurement vector
R : numpy.array((dim_z, dim_z)), optional
Measurement noise. If provided, overrides self.R for
this function call.
UT : function(sigmas, Wm, Wc, noise_cov), optional
Optional function to compute the unscented transform for the sigma
points passed through hx. Typically the default function will
work - you can use x_mean_fn and z_mean_fn to alter the behavior
of the unscented transform.
**hx_args : keyword argument
arguments to be passed into h(x) after x -> h(x, **hx_args)
"""
if z is None:
return
if hx is None:
hx = self.hx
if UT is None:
UT = unscented_transform
if R is None:
R = self.R
elif isscalar(R):
R = eye(self._dim_z) * R
# pass prior sigmas through h(x) to get measurement sigmas
# the shape of sigmas_h will vary if the shape of z varies, so
# recreate each time
sigmas_h = []
for s in self.sigmas_f:
sigmas_h.append(hx(s, **hx_args))
self.sigmas_h = np.atleast_2d(sigmas_h)
# mean and covariance of prediction passed through unscented transform
zp, Pz = UT(self.sigmas_h, self.Wm, self.Wc, R, self.z_mean,
self.residual_z)
# compute cross variance of the state and the measurements
Pxz = self.cross_variance(self.x, zp, self.sigmas_f, self.sigmas_h)
self.K = dot(Pxz, self.inv(Pz)) # Kalman gain
self.y = self.residual_z(z, zp) # residual
# update Gaussian state estimate (x, P)
self.x = self.x + dot(self.K, self.y)
self.P = self.P - dot(self.K, dot(Pz, self.K.T))
if self.compute_log_likelihood:
self.log_likelihood = logpdf(x=self.y, cov=Pz)
self.likelihood = math.exp(self.log_likelihood)
if self.likelihood == 0:
self.likelihood = sys.float_info.min
def dist(self, observation):
# negative log likelihood (so that smaller is better)
kf = self._kf
residual = observation.centroid - self.mean()
S = np.dot(kf.H, np.dot(kf.P, kf.H.T)) + observation.covariance
return -logpdf(x=residual, cov=S)
def update(x, P, z, R, H=None, return_all=False):
"""
Add a new measurement (z) to the Kalman filter. If z is None, nothing
is changed.
This can handle either the multidimensional or unidimensional case. If
all parameters are floats instead of arrays the filter will still work,
and return floats for x, P as the result.
update(1, 2, 1, 1, 1) # univariate
update(x, P, 1
Parameters
----------
x : numpy.array(dim_x, 1), or float
State estimate vector
P : numpy.array(dim_x, dim_x), or float
Covariance matrix
z : (dim_z, 1): array_like
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be convertible to a column vector.
R : numpy.array(dim_z, dim_z), or float
Measurement noise matrix
H : numpy.array(dim_x, dim_x), or float, optional
Measurement function. If not provided, a value of 1 is assumed.
return_all : bool, default False
If true, y, K, S, and log_likelihood are returned, otherwise
only x and P are returned.
Returns
-------
x : numpy.array
Posterior state estimate vector
P : numpy.array
Posterior covariance matrix
y : numpy.array or scalar
Residua. Difference between measurement and state in measurement space
K : numpy.array
Kalman gain
S : numpy.array
System uncertainty in measurement space
log_likelihood : float
log likelihood of the measurement
"""
if z is None:
if return_all:
return x, P, None, None, None, None
else:
return x, P
if H is None:
H = np.array([1])
if np.isscalar(H):
H = np.array([H])
Hx = np.atleast_1d(dot(H, x))
z = _reshape_z(z, Hx.shape[0], x.ndim)
# error (residual) between measurement and prediction
y = z - Hx
# project system uncertainty into measurement space
S = dot(dot(H, P), H.T) + R
# map system uncertainty into kalman gain
try:
K = dot(dot(P, H.T), linalg.inv(S))
except:
K = dot(dot(P, H.T), 1./S)
# predict new x with residual scaled by the kalman gain
x = x + dot(K, y)
# P = (I-KH)P(I-KH)' + KRK'
KH = dot(K, H)
try:
I_KH = np.eye(KH.shape[0]) - KH
except:
I_KH = np.array([1 - KH])
P = dot(dot(I_KH, P), I_KH.T) + dot(dot(K, R), K.T)
if return_all:
# compute log likelihood
log_likelihood = logpdf(z, dot(H, x), S)
return x, P, y, K, S, log_likelihood
else:
return x, P
def update(x, P, z, R, H=None, return_all=False):
"""
Add a new measurement (z) to the Kalman filter. If z is None, nothing
is changed.
This can handle either the multidimensional or unidimensional case. If
all parameters are floats instead of arrays the filter will still work,
and return floats for x, P as the result.
update(1, 2, 1, 1, 1) # univariate
update(x, P, 1
Parameters
----------
x : numpy.array(dim_x, 1), or float
State estimate vector
P : numpy.array(dim_x, dim_x), or float
Covariance matrix
z : numpy.array(dim_z, 1), or float
measurement for this update.
R : numpy.array(dim_z, dim_z), or float
Measurement noise matrix
H : numpy.array(dim_x, dim_x), or float, optional
Measurement function. If not provided, a value of 1 is assumed.
return_all : bool, default False
If true, y, K, S, and log_likelihood are returned, otherwise
only x and P are returned.
Returns
-------
x : numpy.array
Posterior state estimate vector
P : numpy.array
Posterior covariance matrix
y : numpy.array or scalar
Residua. Difference between measurement and state in measurement space
K : numpy.array
Kalman gain
S : numpy.array
System uncertainty in measurement space
log_likelihood : float
log likelihood of the measurement
"""
if z is None:
if return_all:
return x, P, None, None, None, None
else:
return x, P
if H is None:
H = np.array([1])
if np.isscalar(H):
H = np.array([H])
if not np.isscalar(x):
# handle special case: if z is in form [[z]] but x is not a column
# vector dimensions will not match
if x.ndim==1 and shape(z) == (1,1):
z = z[0]
if shape(z) == (): # is it scalar, e.g. z=3 or z=np.array(3)
z = np.asarray([z])
# error (residual) between measurement and prediction
y = z - dot(H, x)
# project system uncertainty into measurement space
S = dot3(H, P, H.T) + R
# map system uncertainty into kalman gain
try:
K = dot3(P, H.T, linalg.inv(S))
except:
K = dot3(P, H.T, 1/S)
# predict new x with residual scaled by the kalman gain
x = x + dot(K, y)
# P = (I-KH)P(I-KH)' + KRK'
KH = dot(K, H)
try:
I_KH = np.eye(KH.shape[0]) - KH
except:
I_KH = np.array(1 - KH)
P = dot3(I_KH, P, I_KH.T) + dot3(K, R, K.T)
if return_all:
# compute log likelihood
log_likelihood = logpdf(z, dot(H, x), S)
return x, P, y, K, S, log_likelihood
else:
return x, P
def update_correlated(self, z, R=None, H=None):
""" Add a new measurement (z) to the Kalman filter assuming that
process noise and measurement noise are correlated as defined in
the `self.M` matrix.
If z is None, nothing is changed.
Parameters
----------
z : np.array
measurement for this update.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
H : np.array, or None
Optionally provide H to override the measurement function for this
one call, otherwise self.H will be used.
"""
if z is None:
return
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
# rename for readability and a tiny extra bit of speed
if H is None:
H = self.H
x = self._x
P = self._P
M = self.M
# handle special case: if z is in form [[z]] but x is not a column
# vector dimensions will not match
if x.ndim==1 and shape(z) == (1,1):
z = z[0]
if shape(z) == (): # is it scalar, e.g. z=3 or z=np.array(3)
z = np.asarray([z])
# y = z - Hx
# error (residual) between measurement and prediction
self._y = z - dot(H, x)
# project system uncertainty into measurement space
S = dot3(H, P, H.T) + dot(H, M) + dot(M.T, H.T) + R
# K = PH'inv(S)
# map system uncertainty into kalman gain
K = dot(dot(P, H.T) + M, linalg.inv(S))
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self._x = x + dot(K, self._y)
self._P = P - dot(K, dot(H, P) + M.T)
self._S = S
self._K = K
# compute log likelihood
self.log_likelihood = logpdf(z, dot(H, x), S)
def update(self, z, R=None, H=None):
"""
Add a new measurement (z) to the Kalman filter. If z is None, nothing
is changed.
Parameters
----------
z : np.array
measurement for this update. z can be a scalar if dim_z is 1,
otherwise it must be a column vector.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
H : np.array, or None
Optionally provide H to override the measurement function for this
one call, otherwise self.H will be used.
"""
if z is None:
return
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
# rename for readability and a tiny extra bit of speed
if H is None:
H = self.H
P = self._P
x = self._x
# handle special case: if z is in form [[z]] but x is not a column
# vector dimensions will not match
if x.ndim==1 and shape(z) == (1,1):
z = z[0]
if shape(z) == (): # is it scalar, e.g. z=3 or z=np.array(3)
z = np.asarray([z])
# y = z - Hx
# error (residual) between measurement and prediction
Hx = dot(H, x)
assert shape(Hx) == shape(z) or (shape(Hx) == (1,1) and shape(z) == (1,)), \
'shape of z should be {}, but it is {}'.format(
shape(Hx), shape(z))
self._y = z - Hx
# S = HPH' + R
# project system uncertainty into measurement space
S = dot3(H, P, H.T) + R
# K = PH'inv(S)
# map system uncertainty into kalman gain
K = dot3(P, H.T, linalg.inv(S))
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self._x = x + dot(K, self._y)
# P = (I-KH)P(I-KH)' + KRK'
I_KH = self._I - dot(K, H)
self._P = dot3(I_KH, P, I_KH.T) + dot3(K, R, K.T)
self._S = S
self._K = K
self.log_likelihood = logpdf(z, dot(H, x), S)
def update(self, z, R=None, UT=None, hx_args=()):
"""
Update the UKF with the given measurements. On return,
self.x and self.P contain the new mean and covariance of the filter.
Parameters
----------
z : numpy.array of shape (dim_z)
measurement vector
R : numpy.array((dim_z, dim_z)), optional
Measurement noise. If provided, overrides self.R for
this function call.
UT : function(sigmas, Wm, Wc, noise_cov), optional
Optional function to compute the unscented transform for the sigma
points passed through hx. Typically the default function will
work - you can use x_mean_fn and z_mean_fn to alter the behavior
of the unscented transform.
hx_args : tuple, optional, default (,)
arguments to be passed into Hx function after the required state
variable.
"""
if z is None:
return
if not isinstance(hx_args, tuple):
hx_args = (hx_args, )
if UT is None:
UT = unscented_transform
if R is None:
R = self.R
elif isscalar(R):
R = eye(self._dim_z) * R
for i, s in enumerate(self.sigmas_f):
self.sigmas_h[i] = self.hx(s, *hx_args)
# mean and covariance of prediction passed through unscented transform
zp, Pz = UT(self.sigmas_h, self.Wm, self.Wc, R, self.z_mean,
self.residual_z)
# compute cross variance of the state and the measurements
Pxz = self.cross_variance(self.x, zp, self.sigmas_f, self.sigmas_h)
self.K = dot(Pxz, self.inv(Pz)) # Kalman gain
self.y = self.residual_z(z, zp) # residual
# update Gaussian state estimate (x, P)
self.x = self.x + dot(self.K, self.y)
self.P = self.P - dot(self.K, dot(Pz, self.K.T))
if self.compute_log_likelihood:
self.log_likelihood = logpdf(x=self.y, cov=Pz)
self.likelihood = math.exp(self.log_likelihood)
if self.likelihood == 0:
self.likelihood = sys.float_info.min