class Test_1x1(unittest.TestCase):
A = properties.ClassProperty('A')
def test_assign_0D(self):
self.A = properties.MatrixTemplate(1,1)
self.A = 1234
self.assertEqual(self.A.shape, (1,1))
self.assertEqual(self.A[0][0], 1234)
def test_assign_1D(self):
self.A = properties.MatrixTemplate(1,1)
# vector (1D) matrx
self.A = [3256]
self.assertEqual(self.A.shape, (1,1))
self.assertEqual(self.A[0][0], 3256)
def test_assign_2D(self):
self.A = properties.MatrixTemplate(1,1)
# matrix 1x1, 2D array-like
self.A = [[2394876]]
self.assertEqual(self.A.shape, (1,1))
self.assertEqual(self.A[0][0], 2394876)
def test_assign_3D(self):
self.A = properties.MatrixTemplate(1,1)
# 3D array-like
self.A = [[[32786]]]
self.assertEqual(self.A.shape, (1,1))
self.assertEqual(self.A[0][0], 32786)
class TestMatrix(unittest.TestCase):
A = properties.ClassProperty('A')
def test_invalid(self):
m,n = 3,5
self.A = properties.MatrixTemplate(m,n)
with self.assertRaises(ValueError):
# A 1D vector whose number of elements is the same as that of the matrix
self.A = np.zeros((m*n))
with self.assertRaises(ValueError):
# Transposed will not be matched automatically of both dimensions are greater than 1
self.A = np.zeros((n,m))
for shape in [(n,), (m,), (m,1), (1,m), (n,1), (1,n)]:
with self.assertRaises(ValueError):
self.A = np.zeros(shape)
self.A = np.zeros((m,n))
def test_multiplicative(self):
self.A = properties.MatrixTemplate(4,4,multiplicative=True)
self.A = 4
self.assertTrue(np.allclose(self.A, 4 * np.eye(4)))
with self.assertRaises(ValueError):
self.A = np.ones(4)
self.A = properties.MatrixTemplate(4,4,multiplicative=False)
with self.assertRaises(ValueError):
self.A = 4
with self.assertRaises(ValueError):
self.A = np.ones(4)
class TestZero(unittest.TestCase):
A = properties.ClassProperty('A')
F = properties.ClassProperty('F')
def test_zero_matrix(self):
for shape in [(1,4), (5, 1), (3,6), (7,3)]:
self.A = properties.MatrixTemplate(*shape)
self.A = 0
self.assertTrue(np.all(self.A == 0))
self.assertEqual(self.A.shape, shape)
def test_zero_matrix_function(self):
def zero():
return 0
for shape in [(1,4), (5, 1), (3,6), (7,3)]:
self.F = properties.MatrixFunctionTemplate(*shape)
self.F = zero
self.assertTrue(np.all(self.F() == 0))
self.assertEqual(self.F().shape, shape)
class AutomaticConversionOnAssignment(unittest.TestCase):
n = properties.ClassProperty('n');
def test_transformation(self):
class EnsureEvenNumber(properties.Template):
def __call__(self, v):
return v - (v % 2)
self.n = EnsureEvenNumber()
self.n = 20;
self.assertEqual(self.n, 20)
self.n = 11
self.assertEqual(self.n, 10)
class PropertyValidation(unittest.TestCase):
n = properties.ClassProperty('n')
def test_validation(self):
class ExpectEvenNumber(properties.Template):
def __call__(self, v):
assert (v % 2 == 0)
return v
self.n = ExpectEvenNumber()
self.n = 4
self.n = 8
with self.assertRaises(Exception):
self.n = 3
class TestVector(unittest.TestCase):
A = properties.ClassProperty('A')
def test_valid_multi_dimensional(self):
for n in range(2, 5):
x = list(range(n))
for shape in [(1,n), (n, 1)]:
self.A = properties.MatrixTemplate(*shape)
for d1 in range(3):
x = [xi for xi in x];
y = x
for d2 in range(3):
y = [y]
self.A = y
self.assertEqual(self.A.shape, shape);
def test_invalid_1D(self):
for n in range(2, 5):
for x in [list(range(1, n+2)), list(range(1, n))]:
for shape in [(1,n), (n, 1)]:
### CHANGE THE TEMPLATE HERE ###
self.A = properties.MatrixTemplate(*shape)
for d1 in range(3):
x = [xi for xi in x];
y = x
for d2 in range(3):
y = [y]
with self.assertRaises(ValueError):
self.A = y
def test_invalid_interpretations_from_2D(self):
m,n=3,5
for shape in [(m*n,1), (1,m*n)]:
self.A = properties.MatrixTemplate(*shape)
with self.assertRaises(ValueError):
# Number of elements of the whole matrix match the size of the vector
self.A = np.zeros((m,n))
with self.assertRaises(ValueError):
# Columns of the matrix match the vector shape, but not the matrix
self.A = np.zeros((m*n, 2))
with self.assertRaises(ValueError):
# Rows of the matrix match the vector shape, but not the matrix
self.A = np.zeros((2, m*n))
class ExtendedKalmanFilter(object):
""" Implements an extended Kalman filter (EKF). You are responsible for
setting the various state variables to reasonable values; the defaults
will not give you a functional filter.
You will have to set the following attributes after constructing this
object for the filter to perform properly. Please note that there are
various checks in place to ensure that you have made everything the
'correct' size. However, it is possible to provide incorrectly sized
arrays such that the linear algebra can not perform an operation.
It can also fail silently - you can end up with matrices of a size that
allows the linear algebra to work, but are the wrong shape for the problem
you are trying to solve.
Parameters
----------
dim_x : int
Number of state variables for the Kalman filter. For example, if
you are tracking the position and velocity of an object in two
dimensions, dim_x would be 4.
This is used to set the default size of P, Q, and u
dim_z : int
Number of of measurement inputs. For example, if the sensor
provides you with position in (x,y), dim_z would be 2.
Attributes
----------
x : numpy.array(dim_x, 1)
State estimate vector
P : numpy.array(dim_x, dim_x)
Covariance matrix
x_prior : numpy.array(dim_x, 1)
Prior (predicted) state estimate. The *_prior and *_post attributes
are for convienence; they store the prior and posterior of the
current epoch. Read Only.
P_prior : numpy.array(dim_x, dim_x)
Prior (predicted) state covariance matrix. Read Only.
x_post : numpy.array(dim_x, 1)
Posterior (updated) state estimate. Read Only.
P_post : numpy.array(dim_x, dim_x)
Posterior (updated) state covariance matrix. Read Only.
R : numpy.array(dim_z, dim_z)
Measurement noise matrix
Q : numpy.array(dim_x, dim_x)
Process noise matrix
F : numpy.array()
State Transition matrix
H : numpy.array(dim_x, dim_x)
Measurement function
y : numpy.array
Residual of the update step. Read only.
K : numpy.array(dim_x, dim_z)
Kalman gain of the update step. Read only.
S : numpy.array
Systen uncertaintly projected to measurement space. Read only.
z : ndarray
Last measurement used in update(). Read only.
log_likelihood : float
log-likelihood of the last measurement. Read only.
likelihood : float
likelihood of last measurment. Read only.
Computed from the log-likelihood. The log-likelihood can be very
small, meaning a large negative value such as -28000. Taking the
exp() of that results in 0.0, which can break typical algorithms
which multiply by this value, so by default we always return a
number >= sys.float_info.min.
mahalanobis : float
mahalanobis distance of the innovation. E.g. 3 means measurement
was 3 standard deviations away from the predicted value.
Read only.
Examples
--------
See my book Kalman and Bayesian Filters in Python
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
"""
# overridable parameters guarded by checks
x = properties.ClassProperty('x')
P = properties.ClassProperty('P')
Q = properties.ClassProperty('Q')
B = properties.ClassProperty('B')
F = properties.ClassProperty('F')
R = properties.ClassProperty('R')
z = properties.ClassProperty('z')
def __init__(self, dim_x, dim_z, dim_u=0):
self.dim_x = dim_x
self.dim_z = dim_z
self.dim_u = dim_u
# Set templates that will hook the assignments and cast to
# the specific shape
self.x = properties.MatrixTemplate(dim_x, 1)
self.P = properties.MatrixTemplate(dim_x, dim_x)
self.Q = properties.MatrixTemplate(dim_x, dim_x)
self.B = properties.MatrixTemplate(dim_x, dim_u)
self.F = properties.MatrixTemplate(dim_x, dim_x)
self.R = properties.MatrixTemplate(dim_z, dim_z)
self.z = properties.MatrixTemplate(dim_z, 1)
self.x = zeros((dim_x, 1)) # state
self.P = eye(dim_x) # uncertainty covariance
self.B = 0 # control transition matrix
self.F = np.eye(dim_x) # state transition matrix
self.R = eye(dim_z) # state uncertainty
self.Q = eye(dim_x) # process uncertainty
self.y = zeros((dim_z, 1)) # residual
z = np.array([None]*self.dim_z)
self.z = reshape_z(z, self.dim_z, self.x.ndim)
# gain and residual are computed during the innovation step. We
# save them so that in case you want to inspect them for various
# purposes
self.K = np.zeros(self.x.shape) # kalman gain
self.y = zeros((dim_z, 1))
self.S = np.zeros((dim_z, dim_z)) # system uncertainty
self.SI = np.zeros((dim_z, dim_z)) # inverse system uncertainty
# identity matrix. Do not alter this.
self._I = np.eye(dim_x)
self._log_likelihood = log(sys.float_info.min)
self._likelihood = sys.float_info.min
self._mahalanobis = None
# these will always be a copy of x,P after predict() is called
self.x_prior = self.x.copy()
self.P_prior = self.P.copy()
# these will always be a copy of x,P after update() is called
self.x_post = self.x.copy()
self.P_post = self.P.copy()
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.SI = linalg.inv(self.S)
self.K = dot(PHT, self.SI)
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)
# save measurement and posterior state
self.z = deepcopy(z)
self.x_post = self.x.copy()
self.P_post = self.P.copy()
# set to None to force recompute
self._log_likelihood = None
self._likelihood = None
self._mahalanobis = None
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`, posterior is not computed
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 z is None:
self.z = np.array([[None]*self.dim_z]).T
self.x_post = self.x.copy()
self.P_post = self.P.copy()
return
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.SI = linalg.inv(self.S)
self.K = PHT.dot(self.SI)
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)
# set to None to force recompute
self._log_likelihood = None
self._likelihood = None
self._mahalanobis = None
# save measurement and posterior state
self.z = deepcopy(z)
self.x_post = self.x.copy()
self.P_post = self.P.copy()
def predict_x(self, u=0):
"""
Predicts the next state of X. If you need to
compute the next state yourself, override this function. You would
need to do this, for example, if the usual Taylor expansion to
generate F is not providing accurate results for you.
"""
u = properties.as_matrix((self.dim_u, 1), u)
x = dot(self.F, self.x) + dot(self.B, u)
self.x = x
def predict(self, u=0):
"""
Predict next state (prior) using the Kalman filter state propagation
equations.
Parameters
----------
u : np.array
Optional control vector. If non-zero, it is multiplied by B
to create the control input into the system.
"""
self.predict_x(u)
self.P = dot(self.F, self.P).dot(self.F.T) + self.Q
# save prior
self.x_prior = np.copy(self.x)
self.P_prior = np.copy(self.P)
@property
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
@property
def likelihood(self):
"""
Computed from the log-likelihood. The log-likelihood can be very
small, meaning a large negative value such as -28000. Taking the
exp() of that results in 0.0, which can break typical algorithms
which multiply by this value, so by default we always return a
number >= sys.float_info.min.
"""
if self._likelihood is None:
self._likelihood = exp(self.log_likelihood)
if self._likelihood == 0:
self._likelihood = sys.float_info.min
return self._likelihood
@property
def mahalanobis(self):
"""
Mahalanobis distance of innovation. E.g. 3 means measurement
was 3 standard deviations away from the predicted value.
Returns
-------
mahalanobis : float
"""
if self._mahalanobis is None:
self._mahalanobis = sqrt(float(dot(dot(self.y.T, self.SI), self.y)))
return self._mahalanobis
def __repr__(self):
return '\n'.join([
'KalmanFilter object',
pretty_str('x', self.x),
pretty_str('P', self.P),
pretty_str('x_prior', self.x_prior),
pretty_str('P_prior', self.P_prior),
pretty_str('F', self.F),
pretty_str('Q', self.Q),
pretty_str('R', self.R),
pretty_str('K', self.K),
pretty_str('y', self.y),
pretty_str('S', self.S),
pretty_str('likelihood', self.likelihood),
pretty_str('log-likelihood', self.log_likelihood),
pretty_str('mahalanobis', self.mahalanobis)
])
class TestMatrixFunctionProperty(unittest.TestCase):
F = properties.ClassProperty('F')
def test_decorated_function(self):
for shape in [(3,5), (7,4)]:
@properties.MatrixFunction(*shape)
def f(x):
return np.zeros(x)
# check that a matrix was returned
self.assertEqual(f(shape).shape, shape)
with self.assertRaises(ValueError):
f(shape[::-1]) # Tell F to return a matrix with wrog shape
def test_decorated_function_assignment(self):
shape = (3,7)
@properties.MatrixFunction(*shape)
def f():
raise TypeError("This should not be called")
self.F = properties.MatrixFunctionTemplate(*shape)
self.F = f # matches the template
self.F = properties.MatrixFunctionTemplate(*shape[::-1])
with self.assertRaises(ValueError):
self.F = f # does not match the template
def test_undecorated_function_assignment(self):
shape = (3,7)
def f():
return 1
self.F = properties.MatrixFunctionTemplate(*shape)
# since f was not decorated it will accept expecting it
# to return matrices of the given shape
self.F = f
with self.assertRaises(ValueError):
self.F() # when f is called it will see an invalid value
def test_multiplicativify_of_function(self):
# Non multiplicative template
self.F = properties.MatrixFunctionTemplate(4,4,multiplicative=False)
def identity(x): return x
# Undecorated function
self.F = identity
with self.assertRaises(ValueError):
self.F(4)
self.assertTrue(np.allclose(self.F(4*np.eye(4)), 4*np.eye(4)))
# Non multiplicative template multiplicative function
self.F = properties.DecoratedMatrixFunction((4,4), identity, multiplicative=True)
self.assertTrue(np.allclose(self.F(4*np.eye(4)), 4*np.eye(4)))
# Multiplicative template
self.F = properties.MatrixFunctionTemplate(4,4,multiplicative=False)
# Multiplicative template undecorated function
self.F = identity
self.assertTrue(np.allclose(self.F(4*np.eye(4)), 4*np.eye(4)))
# Multiplicative template multiplicative function
self.F = properties.DecoratedMatrixFunction((4,4), identity, multiplicative=True)
self.assertTrue(np.allclose(self.F(4*np.eye(4)), 4*np.eye(4)))
# Multiplicative template non-multiplicative function
self.F = properties.DecoratedMatrixFunction((4,4), identity, multiplicative=False)
with self.assertRaises(ValueError):
self.F(6)
class Class:
prop1 = properties.ClassProperty('prop1')
prop2 = properties.ClassProperty('prop2')
class UnscentedKalmanFilter(object):
# pylint: disable=too-many-instance-attributes
# pylint: disable=invalid-name
r"""
Implements the Scaled Unscented Kalman filter (UKF) as defined by
Simon Julier in [1], using the formulation provided by Wan and Merle
in [2]. This filter scales the sigma points to avoid strong nonlinearities.
Parameters
----------
dim_x : int
Number of state variables for the filter. For example, if
you are tracking the position and velocity of an object in two
dimensions, dim_x would be 4.
dim_z : int
Number of of measurement inputs. For example, if the sensor
provides you with position in (x,y), dim_z would be 2.
This is for convience, so everything is sized correctly on
creation. If you are using multiple sensors the size of `z` can
change based on the sensor. Just provide the appropriate hx function
dt : float
Time between steps in seconds.
hx : function(x,**hx_args)
Measurement function. Converts state vector x into a measurement
vector of shape (dim_z).
fx : function(x,dt,**fx_args)
function that returns the state x transformed by the
state transition function. dt is the time step in seconds.
points : class
Class which computes the sigma points and weights for a UKF
algorithm. You can vary the UKF implementation by changing this
class. For example, MerweScaledSigmaPoints implements the alpha,
beta, kappa parameterization of Van der Merwe, and
JulierSigmaPoints implements Julier's original kappa
parameterization. See either of those for the required
signature of this class if you want to implement your own.
sqrt_fn : callable(ndarray), default=None (implies scipy.linalg.cholesky)
Defines how we compute the square root of a matrix, which has
no unique answer. Cholesky is the default choice due to its
speed. Typically your alternative choice will be
scipy.linalg.sqrtm. Different choices affect how the sigma points
are arranged relative to the eigenvectors of the covariance matrix.
Usually this will not matter to you; if so the default cholesky()
yields maximal performance. As of van der Merwe's dissertation of
2004 [6] this was not a well reseached area so I have no advice
to give you.
If your method returns a triangular matrix it must be upper
triangular. Do not use numpy.linalg.cholesky - for historical
reasons it returns a lower triangular matrix. The SciPy version
does the right thing as far as this class is concerned.
x_mean_fn : callable (sigma_points, weights), optional
Function that computes the mean of the provided sigma points
and weights. Use this if your state variable contains nonlinear
values such as angles which cannot be summed.
.. code-block:: Python
def state_mean(sigmas, Wm):
x = np.zeros(3)
sum_sin, sum_cos = 0., 0.
for i in range(len(sigmas)):
s = sigmas[i]
x[0] += s[0] * Wm[i]
x[1] += s[1] * Wm[i]
sum_sin += sin(s[2])*Wm[i]
sum_cos += cos(s[2])*Wm[i]
x[2] = atan2(sum_sin, sum_cos)
return x
z_mean_fn : callable (sigma_points, weights), optional
Same as x_mean_fn, except it is called for sigma points which
form the measurements after being passed through hx().
residual_x : callable (x, y), optional
residual_z : callable (x, y), optional
Function that computes the residual (difference) between x and y.
You will have to supply this if your state variable cannot support
subtraction, such as angles (359-1 degreees is 2, not 358). x and y
are state vectors, not scalars. One is for the state variable,
the other is for the measurement state.
.. code-block:: Python
def residual(a, b):
y = a[0] - b[0]
if y > np.pi:
y -= 2*np.pi
if y < -np.pi:
y += 2*np.pi
return y
state_add: callable (x, y), optional, default np.add
Function that subtracts two state vectors, returning a new
state vector. Used during update to compute `x + K@y`
You will have to supply this if your state variable does not
suport addition, such as it contains angles.
Attributes
----------
x : numpy.array(dim_x)
state estimate vector
P : numpy.array(dim_x, dim_x)
covariance estimate matrix
x_prior : numpy.array(dim_x)
Prior (predicted) state estimate. The *_prior and *_post attributes
are for convienence; they store the prior and posterior of the
current epoch. Read Only.
P_prior : numpy.array(dim_x, dim_x)
Prior (predicted) state covariance matrix. Read Only.
x_post : numpy.array(dim_x)
Posterior (updated) state estimate. Read Only.
P_post : numpy.array(dim_x, dim_x)
Posterior (updated) state covariance matrix. Read Only.
z : ndarray
Last measurement used in update(). Read only.
R : numpy.array(dim_z, dim_z)
measurement noise matrix
Q : numpy.array(dim_x, dim_x)
process noise matrix
K : numpy.array
Kalman gain
y : numpy.array
innovation residual
log_likelihood : scalar
Log likelihood of last measurement update.
likelihood : float
likelihood of last measurment. Read only.
Computed from the log-likelihood. The log-likelihood can be very
small, meaning a large negative value such as -28000. Taking the
exp() of that results in 0.0, which can break typical algorithms
which multiply by this value, so by default we always return a
number >= sys.float_info.min.
mahalanobis : float
mahalanobis distance of the measurement. Read only.
inv : function, default numpy.linalg.inv
If you prefer another inverse function, such as the Moore-Penrose
pseudo inverse, set it to that instead:
.. code-block:: Python
kf.inv = np.linalg.pinv
Examples
--------
Simple example of a linear order 1 kinematic filter in 2D. There is no
need to use a UKF for this example, but it is easy to read.
>>> def fx(x, dt):
>>> # state transition function - predict next state based
>>> # on constant velocity model x = vt + x_0
>>> F = np.array([[1, dt, 0, 0],
>>> [0, 1, 0, 0],
>>> [0, 0, 1, dt],
>>> [0, 0, 0, 1]], dtype=float)
>>> return np.dot(F, x)
>>>
>>> def hx(x):
>>> # measurement function - convert state into a measurement
>>> # where measurements are [x_pos, y_pos]
>>> return np.array([x[0], x[2]])
>>>
>>> dt = 0.1
>>> # create sigma points to use in the filter. This is standard for Gaussian processes
>>> points = MerweScaledSigmaPoints(4, alpha=.1, beta=2., kappa=-1)
>>>
>>> kf = UnscentedKalmanFilter(dim_x=4, dim_z=2, dt=dt, fx=fx, hx=hx, points=points)
>>> kf.x = np.array([-1., 1., -1., 1]) # initial state
>>> kf.P *= 0.2 # initial uncertainty
>>> z_std = 0.1
>>> kf.R = np.diag([z_std**2, z_std**2]) # 1 standard
>>> kf.Q = Q_discrete_white_noise(dim=2, dt=dt, var=0.01**2, block_size=2)
>>>
>>> zs = [[i+randn()*z_std, i+randn()*z_std] for i in range(50)] # measurements
>>> for z in zs:
>>> kf.predict()
>>> kf.update(z)
>>> print(kf.x, 'log-likelihood', kf.log_likelihood)
For in depth explanations see my book Kalman and Bayesian Filters in Python
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
Also see the filterpy/kalman/tests subdirectory for test code that
may be illuminating.
References
----------
.. [1] Julier, Simon J. "The scaled unscented transformation,"
American Control Converence, 2002, pp 4555-4559, vol 6.
Online copy:
https://www.cs.unc.edu/~welch/kalman/media/pdf/ACC02-IEEE1357.PDF
.. [2] E. A. Wan and R. Van der Merwe, “The unscented Kalman filter for
nonlinear estimation,” in Proc. Symp. Adaptive Syst. Signal
Process., Commun. Contr., Lake Louise, AB, Canada, Oct. 2000.
Online Copy:
https://www.seas.harvard.edu/courses/cs281/papers/unscented.pdf
.. [3] S. Julier, J. Uhlmann, and H. Durrant-Whyte. "A new method for
the nonlinear transformation of means and covariances in filters
and estimators," IEEE Transactions on Automatic Control, 45(3),
pp. 477-482 (March 2000).
.. [4] E. A. Wan and R. Van der Merwe, “The Unscented Kalman filter for
Nonlinear Estimation,” in Proc. Symp. Adaptive Syst. Signal
Process., Commun. Contr., Lake Louise, AB, Canada, Oct. 2000.
https://www.seas.harvard.edu/courses/cs281/papers/unscented.pdf
.. [5] Wan, Merle "The Unscented Kalman Filter," chapter in *Kalman
Filtering and Neural Networks*, John Wiley & Sons, Inc., 2001.
.. [6] R. Van der Merwe "Sigma-Point Kalman Filters for Probabilitic
Inference in Dynamic State-Space Models" (Doctoral dissertation)
"""
# overridable parameters guarded by checks
x = properties.ClassProperty('x')
P = properties.ClassProperty('P')
Q = properties.ClassProperty('Q')
F = properties.ClassProperty('F')
H = properties.ClassProperty('H')
R = properties.ClassProperty('R')
M = properties.ClassProperty('M')
z = properties.ClassProperty('z')
def __init__(self, dim_x, dim_z, dt, hx, fx, points,
sqrt_fn=None, x_mean_fn=None, z_mean_fn=None,
residual_x=None,
residual_z=None,
state_add=None):
"""
Create a Kalman filter. You are responsible for setting the
various state variables to reasonable values; the defaults below will
not give you a functional filter.
"""
# Set templates that will hook the assignments and cast to
# the specific shape
self.x = properties.MatrixTemplate(dim_x, 1)
self.P = properties.MatrixTemplate(dim_x, dim_x)
self.Q = properties.MatrixTemplate(dim_x, dim_x)
self.F = properties.MatrixFunctionTemplate(dim_x, dim_x)
self.H = properties.MatrixFunctionTemplate(dim_z, dim_x)
self.R = properties.MatrixTemplate(dim_z, dim_z)
self.M = properties.MatrixTemplate(dim_x, dim_z)
self.z = properties.MatrixTemplate(dim_z, 1)
#pylint: disable=too-many-arguments
self.x = zeros(dim_x)
self.P = eye(dim_x)
self.x_prior = np.copy(self.x)
self.P_prior = np.copy(self.P)
self.Q = eye(dim_x)
self.R = eye(dim_z)
self._dim_x = dim_x
self._dim_z = dim_z
self.points_fn = points
self._dt = dt
self._num_sigmas = points.num_sigmas()
self.hx = hx
self.fx = fx
self.x_mean = x_mean_fn
self.z_mean = z_mean_fn
# Only computed only if requested via property
self._log_likelihood = log(sys.float_info.min)
self._likelihood = sys.float_info.min
self._mahalanobis = None
if sqrt_fn is None:
self.msqrt = cholesky
else:
self.msqrt = sqrt_fn
# weights for the means and covariances.
self.Wm, self.Wc = points.Wm, points.Wc
if residual_x is None:
self.residual_x = np.subtract
else:
self.residual_x = residual_x
if residual_z is None:
self.residual_z = np.subtract
else:
self.residual_z = residual_z
if state_add is None:
self.state_add = np.add
else:
self.state_add = state_add
# sigma points transformed through f(x) and h(x)
# variables for efficiency so we don't recreate every update
self.sigmas_f = zeros((self._num_sigmas, self._dim_x))
self.sigmas_h = zeros((self._num_sigmas, self._dim_z))
self.K = np.zeros((dim_x, dim_z)) # Kalman gain
self.y = np.zeros((dim_z)) # residual
self.z = np.array([[None]*dim_z]).T # measurement
self.S = np.zeros((dim_z, dim_z)) # system uncertainty
self.SI = np.zeros((dim_z, dim_z)) # inverse system uncertainty
self.inv = np.linalg.inv
# these will always be a copy of x,P after predict() is called
self.x_prior = self.x.copy()
self.P_prior = self.P.copy()
# these will always be a copy of x,P after update() is called
self.x_post = self.x.copy()
self.P_post = self.P.copy()
def predict(self, dt=None, UT=None, fx=None, **fx_args):
r"""
Performs the predict step of the UKF. On return, self.x and
self.P contain the predicted state (x) and covariance (P). '
Important: this MUST be called before update() is called for the first
time.
Parameters
----------
dt : double, optional
If specified, the time step to be used for this prediction.
self._dt is used if this is not provided.
fx : callable f(x, dt, **fx_args), optional
State transition function. If not provided, the default
function passed in during construction will be used.
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.
**fx_args : keyword arguments
optional keyword arguments to be passed into f(x).
"""
if dt is None:
dt = self._dt
if UT is None:
UT = unscented_transform
# calculate sigma points for given mean and covariance
self.compute_process_sigmas(dt, fx, **fx_args)
#and pass sigmas through the unscented transform to compute prior
self.x, self.P = UT(self.sigmas_f, self.Wm, self.Wc, self.Q,
self.x_mean, self.residual_x)
# update sigma points to reflect the new variance of the points
self.sigmas_f = self.points_fn.sigma_points(self.x, self.P)
# save prior
self.x_prior = np.copy(self.x)
self.P_prior = np.copy(self.P)
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 : callable h(x, **hx_args), optional
Measurement function. If not provided, the default
function passed in during construction will be used.
**hx_args : keyword argument
arguments to be passed into h(x) after x -> h(x, **hx_args)
"""
if z is None:
self.z = np.array([[None]*self._dim_z]).T
self.x_post = self.x.copy()
self.P_post = self.P.copy()
return
else:
z = properties.as_matrix((self._dim_z, 1), z, 'z')
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
else:
R = properties.as_matrix((self._dim_z,)*2, R, '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 = [hx(s, **hx_args) for s in self.sigmas_f]
sigmas_h = np.reshape(sigmas_h, [-1, self._dim_z, 1])
# mean and covariance of prediction passed through unscented transform
zp, self.S = UT(self.sigmas_h, self.Wm, self.Wc, R, self.z_mean, self.residual_z)
self.SI = self.inv(self.S)
# 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.SI) # Kalman gain
self.y = self.residual_z(z, zp) # residual
# update Gaussian state estimate (x, P)
self.x = self.state_add(self.x, dot(self.K, self.y))
self.P = self.P - dot(self.K, dot(self.S, self.K.T))
# save measurement and posterior state
self.z = deepcopy(z)
self.x_post = self.x.copy()
self.P_post = self.P.copy()
# set to None to force recompute
self._log_likelihood = None
self._likelihood = None
self._mahalanobis = None
def cross_variance(self, x, z, sigmas_f, sigmas_h):
"""
Compute cross variance of the state `x` and measurement `z`.
"""
Pxz = zeros((sigmas_f.shape[1], sigmas_h.shape[1]))
N = sigmas_f.shape[0]
x = np.reshape(x, (-1,))
z = np.reshape(z, (-1,))
for i in range(N):
dx = self.residual_x(sigmas_f[i], x)
dz = self.residual_z(sigmas_h[i], z)
Pxz += self.Wc[i] * outer(dx, dz)
return Pxz
def compute_process_sigmas(self, dt, fx=None, **fx_args):
"""
computes the values of sigmas_f. Normally a user would not call
this, but it is useful if you need to call update more than once
between calls to predict (to update for multiple simultaneous
measurements), so the sigmas correctly reflect the updated state
x, P.
"""
if fx is None:
fx = self.fx
# calculate sigma points for given mean and covariance
sigmas = self.points_fn.sigma_points(self.x, self.P)
for i, s in enumerate(sigmas):
self.sigmas_f[i] = np.squeeze(fx(s, dt, **fx_args))
def batch_filter(self, zs, Rs=None, dts=None, UT=None, saver=None):
"""
Performs the UKF filter over the list of measurement in `zs`.
Parameters
----------
zs : list-like
list of measurements at each time step `self._dt` Missing
measurements must be represented by 'None'.
Rs : None, np.array or list-like, default=None
optional list of values to use for the measurement error
covariance R.
If Rs is None then self.R is used for all epochs.
If it is a list of matrices or a 3D array where
len(Rs) == len(zs), then it is treated as a list of R values, one
per epoch. This allows you to have varying R per epoch.
dts : None, scalar or list-like, default=None
optional value or list of delta time to be passed into predict.
If dtss is None then self.dt is used for all epochs.
If it is a list where len(dts) == len(zs), then it is treated as a
list of dt values, one per epoch. This allows you to have varying
epoch durations.
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.
saver : filterpy.common.Saver, optional
filterpy.common.Saver object. If provided, saver.save() will be
called after every epoch
Returns
-------
means: ndarray((n,dim_x,1))
array of the state for each time step after the update. Each entry
is an np.array. In other words `means[k,:]` is the state at step
`k`.
covariance: ndarray((n,dim_x,dim_x))
array of the covariances for each time step after the update.
In other words `covariance[k,:,:]` is the covariance at step `k`.
Examples
--------
.. code-block:: Python
# this example demonstrates tracking a measurement where the time
# between measurement varies, as stored in dts The output is then smoothed
# with an RTS smoother.
zs = [t + random.randn()*4 for t in range (40)]
(mu, cov, _, _) = ukf.batch_filter(zs, dts=dts)
(xs, Ps, Ks) = ukf.rts_smoother(mu, cov)
"""
#pylint: disable=too-many-arguments
try:
z = zs[0]
except TypeError:
raise TypeError('zs must be list-like')
if self._dim_z == 1:
if not(isscalar(z) or (z.ndim == 1 and len(z) == 1)):
raise TypeError('zs must be a list of scalars or 1D, 1 element arrays')
else:
if len(z) != self._dim_z:
raise TypeError(
'each element in zs must be a 1D array of length {}'.format(self._dim_z))
z_n = np.size(zs, 0)
if Rs is None:
Rs = [self.R] * z_n
if dts is None:
dts = [self._dt] * z_n
# mean estimates from Kalman Filter
if self.x.ndim == 1:
means = zeros((z_n, self._dim_x))
else:
means = zeros((z_n, self._dim_x, 1))
# state covariances from Kalman Filter
covariances = zeros((z_n, self._dim_x, self._dim_x))
for i, (z, r, dt) in enumerate(zip(zs, Rs, dts)):
self.predict(dt=dt, UT=UT)
self.update(z, r, UT=UT)
means[i, :] = self.x
covariances[i, :, :] = self.P
if saver is not None:
saver.save()
return (means, covariances)
def rts_smoother(self, Xs, Ps, Qs=None, dts=None, UT=None):
"""
Runs the Rauch-Tung-Striebel Kalman smoother on a set of
means and covariances computed by the UKF. The usual input
would come from the output of `batch_filter()`.
Parameters
----------
Xs : numpy.array
array of the means (state variable x) of the output of a Kalman
filter.
Ps : numpy.array
array of the covariances of the output of a kalman filter.
Qs: list-like collection of numpy.array, optional
Process noise of the Kalman filter at each time step. Optional,
if not provided the filter's self.Q will be used
dt : optional, float or array-like of float
If provided, specifies the time step of each step of the filter.
If float, then the same time step is used for all steps. If
an array, then each element k contains the time at step k.
Units are seconds.
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.
Returns
-------
x : numpy.ndarray
smoothed means
P : numpy.ndarray
smoothed state covariances
K : numpy.ndarray
smoother gain at each step
Examples
--------
.. code-block:: Python
zs = [t + random.randn()*4 for t in range (40)]
(mu, cov, _, _) = kalman.batch_filter(zs)
(x, P, K) = rts_smoother(mu, cov, fk.F, fk.Q)
"""
#pylint: disable=too-many-locals, too-many-arguments
if len(Xs) != len(Ps):
raise ValueError('Xs and Ps must have the same length')
n, dim_x = Xs.shape
if dts is None:
dts = [self._dt] * n
elif isscalar(dts):
dts = [dts] * n
if Qs is None:
Qs = [self.Q] * n
if UT is None:
UT = unscented_transform
# smoother gain
Ks = zeros((n, dim_x, dim_x))
num_sigmas = self._num_sigmas
xs, ps = Xs.copy(), Ps.copy()
sigmas_f = zeros((num_sigmas, dim_x))
for k in reversed(range(n-1)):
# create sigma points from state estimate, pass through state func
sigmas = self.points_fn.sigma_points(xs[k], ps[k])
for i in range(num_sigmas):
sigmas_f[i] = self.fx(sigmas[i], dts[k])
xb, Pb = UT(
sigmas_f, self.Wm, self.Wc, self.Q,
self.x_mean, self.residual_x)
# compute cross variance
Pxb = 0
for i in range(num_sigmas):
y = self.residual_x(sigmas_f[i], xb)
z = self.residual_x(sigmas[i], Xs[k])
Pxb += self.Wc[i] * outer(z, y)
# compute gain
K = dot(Pxb, self.inv(Pb))
# update the smoothed estimates
xs[k] += dot(K, self.residual_x(xs[k+1], xb))
ps[k] += dot(K, ps[k+1] - Pb).dot(K.T)
Ks[k] = K
return (xs, ps, Ks)
@property
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
@property
def likelihood(self):
"""
Computed from the log-likelihood. The log-likelihood can be very
small, meaning a large negative value such as -28000. Taking the
exp() of that results in 0.0, which can break typical algorithms
which multiply by this value, so by default we always return a
number >= sys.float_info.min.
"""
if self._likelihood is None:
self._likelihood = exp(self.log_likelihood)
if self._likelihood == 0:
self._likelihood = sys.float_info.min
return self._likelihood
@property
def mahalanobis(self):
""""
Mahalanobis distance of measurement. E.g. 3 means measurement
was 3 standard deviations away from the predicted value.
Returns
-------
mahalanobis : float
"""
if self._mahalanobis is None:
self._mahalanobis = sqrt(float(dot(dot(self.y.T, self.SI), self.y)))
return self._mahalanobis
def __repr__(self):
return '\n'.join([
'UnscentedKalmanFilter object',
pretty_str('x', self.x),
pretty_str('P', self.P),
pretty_str('x_prior', self.x_prior),
pretty_str('P_prior', self.P_prior),
pretty_str('Q', self.Q),
pretty_str('R', self.R),
pretty_str('S', self.S),
pretty_str('K', self.K),
pretty_str('y', self.y),
pretty_str('log-likelihood', self.log_likelihood),
pretty_str('likelihood', self.likelihood),
pretty_str('mahalanobis', self.mahalanobis),
pretty_str('sigmas_f', self.sigmas_f),
pretty_str('h', self.sigmas_h),
pretty_str('Wm', self.Wm),
pretty_str('Wc', self.Wc),
pretty_str('residual_x', self.residual_x),
pretty_str('residual_z', self.residual_z),
pretty_str('msqrt', self.msqrt),
pretty_str('hx', self.hx),
pretty_str('fx', self.fx),
pretty_str('x_mean', self.x_mean),
pretty_str('z_mean', self.z_mean)
])
