# UnscentedKalmanFilter¶

## Introduction and Overview¶

This implements the unscented Kalman filter.

Copyright 2015 Roger R Labbe Jr.

FilterPy library. http://github.com/rlabbe/filterpy

Supporting book at: https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

class filterpy.kalman.UnscentedKalmanFilter(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)[source]

Implements the Scaled Unscented Kalman filter (UKF) as defined by Simon Julier in , using the formulation provided by Wan and Merle in . 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) Measurement function. Converts state vector x into a measurement vector of shape (dim_z). fx : function(x,dt) function that returns the state x transformed by the state transistion 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  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. 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 += s * Wm[i] x += s * Wm[i] sum_sin += sin(s)*Wm[i] sum_cos += cos(s)*Wm[i] x = 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. def residual(a, b): y = a - b if y > np.pi: y -= 2*np.pi if y < -np.pi: y = 2*np.pi return y 

References

  Julier, Simon J. “The scaled unscented transformation,” American Control Converence, 2002, pp 4555-4559, vol 6.
  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.
  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).
  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
  Wan, Merle “The Unscented Kalman Filter,” chapter in Kalman Filtering and Neural Networks, John Wiley & Sons, Inc., 2001.
  R. Van der Merwe “Sigma-Point Kalman Filters for Probabilitic Inference in Dynamic State-Space Models” (Doctoral dissertation)

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, x])
>>>
>>> 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.

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 the last measurement. likelihood : float Computed from the log-likelihood. mahalanobis : float “ inv : function, default numpy.linalg.inv If you prefer another inverse function, such as the Moore-Penrose pseudo inverse, set it to that instead: kf.inv = np.linalg.pinv 
__init__(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)[source]

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.

predict(dt=None, UT=None, fx=None, **fx_args)[source]

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, **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).
update(z, R=None, UT=None, hx=None, **hx_args)[source]

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)
cross_variance(x, z, sigmas_f, sigmas_h)[source]

Compute cross variance of the state x and measurement z.

compute_process_sigmas(dt, fx=None, **fx_args)[source]

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.

batch_filter(zs, Rs=None, dts=None, UT=None, saver=None)[source]

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 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

# 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)

rts_smoother(Xs, Ps, Qs=None, dts=None, UT=None)[source]

Runs the Rauch-Tung-Striebal 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. x : numpy.ndarray smoothed means P : numpy.ndarray smoothed state covariances K : numpy.ndarray smoother gain at each step

Examples

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)

log_likelihood

log-likelihood of the last measurement.

likelihood

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

” Mahalanobis distance of measurement. E.g. 3 means measurement was 3 standard deviations away from the predicted value.

Returns: mahalanobis : float

class filterpy.kalman.MerweScaledSigmaPoints(n, alpha, beta, kappa, sqrt_method=None, subtract=None)[source]

Generates sigma points and weights according to Van der Merwe’s 2004 dissertation for the UnscentedKalmanFilter class.. It parametizes the sigma points using alpha, beta, kappa terms, and is the version seen in most publications.

Unless you know better, this should be your default choice.

Parameters: n : int Dimensionality of the state. 2n+1 weights will be generated. alpha : float Determins the spread of the sigma points around the mean. Usually a small positive value (1e-3) according to . beta : float Incorporates prior knowledge of the distribution of the mean. For Gaussian x beta=2 is optimal, according to . kappa : float, default=0.0 Secondary scaling parameter usually set to 0 according to , or to 3-n according to . sqrt_method : function(ndarray), default=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  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. subtract : callable (x, y), optional Function that computes the 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.

References

  R. Van der Merwe “Sigma-Point Kalman Filters for Probabilitic Inference in Dynamic State-Space Models” (Doctoral dissertation)

Examples

See my book Kalman and Bayesian Filters in Python https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

Attributes: Wm : np.array weight for each sigma point for the mean Wc : np.array weight for each sigma point for the covariance
__init__(n, alpha, beta, kappa, sqrt_method=None, subtract=None)[source]

x.__init__(…) initializes x; see help(type(x)) for signature

num_sigmas()[source]

Number of sigma points for each variable in the state x

sigma_points(x, P)[source]

Computes the sigma points for an unscented Kalman filter given the mean (x) and covariance(P) of the filter. Returns tuple of the sigma points and weights.

Works with both scalar and array inputs: sigma_points (5, 9, 2) # mean 5, covariance 9 sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I

Parameters: x : An array-like object of the means of length n Can be a scalar if 1D. examples: 1, [1,2], np.array([1,2]) P : scalar, or np.array Covariance of the filter. If scalar, is treated as eye(n)*P. sigmas : np.array, of size (n, 2n+1) Two dimensional array of sigma points. Each column contains all of the sigmas for one dimension in the problem space. Ordered by Xi_0, Xi_{1..n}, Xi_{n+1..2n}

class filterpy.kalman.JulierSigmaPoints(n, kappa=0.0, sqrt_method=None, subtract=None)[source]

Generates sigma points and weights according to Simon J. Julier and Jeffery K. Uhlmann’s original paper. It parametizes the sigma points using kappa.

Parameters: n : int Dimensionality of the state. 2n+1 weights will be generated. kappa : float, default=0. Scaling factor that can reduce high order errors. kappa=0 gives the standard unscented filter. According to [Julier], if you set kappa to 3-dim_x for a Gaussian x you will minimize the fourth order errors in x and P. sqrt_method : function(ndarray), default=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  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. subtract : callable (x, y), optional Function that computes the 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

References

  Julier, Simon J.; Uhlmann, Jeffrey “A New Extension of the Kalman Filter to Nonlinear Systems”. Proc. SPIE 3068, Signal Processing, Sensor Fusion, and Target Recognition VI, 182 (July 28, 1997)
Attributes: Wm : np.array weight for each sigma point for the mean Wc : np.array weight for each sigma point for the covariance
__init__(n, kappa=0.0, sqrt_method=None, subtract=None)[source]

x.__init__(…) initializes x; see help(type(x)) for signature

num_sigmas()[source]

Number of sigma points for each variable in the state x

sigma_points(x, P)[source]

Computes the sigma points for an unscented Kalman filter given the mean (x) and covariance(P) of the filter. kappa is an arbitrary constant. Returns sigma points.

Works with both scalar and array inputs: sigma_points (5, 9, 2) # mean 5, covariance 9 sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I

Parameters: x : array-like object of the means of length n Can be a scalar if 1D. examples: 1, [1,2], np.array([1,2]) P : scalar, or np.array Covariance of the filter. If scalar, is treated as eye(n)*P. kappa : float Scaling factor. sigmas : np.array, of size (n, 2n+1) 2D array of sigma points $$\chi$$. Each column contains all of the sigmas for one dimension in the problem space. They are ordered as: \begin{eqnarray} \chi = &x \\ \chi[1..n] = &x + [\sqrt{(n+\kappa)P}]_k \\ \chi[n+1..2n] = &x - [\sqrt{(n+\kappa)P}]_k \end{eqnarray}

class filterpy.kalman.SimplexSigmaPoints(n, alpha=1, sqrt_method=None, subtract=None)[source]

Generates sigma points and weights according to the simplex method presented in .

Parameters: n : int Dimensionality of the state. n+1 weights will be generated. sqrt_method : function(ndarray), default=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 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. subtract : callable (x, y), optional Function that computes the 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.

References

  Phillippe Moireau and Dominique Chapelle “Reduced-Order Unscented Kalman Filtering with Application to Parameter Identification in Large-Dimensional Systems” DOI: 10.1051/cocv/2010006
Attributes: Wm : np.array weight for each sigma point for the mean Wc : np.array weight for each sigma point for the covariance
__init__(n, alpha=1, sqrt_method=None, subtract=None)[source]

x.__init__(…) initializes x; see help(type(x)) for signature

num_sigmas()[source]

Number of sigma points for each variable in the state x

sigma_points(x, P)[source]

Computes the implex sigma points for an unscented Kalman filter given the mean (x) and covariance(P) of the filter. Returns tuple of the sigma points and weights.

Works with both scalar and array inputs: sigma_points (5, 9, 2) # mean 5, covariance 9 sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I

Parameters: x : An array-like object of the means of length n Can be a scalar if 1D. examples: 1, [1,2], np.array([1,2]) P : scalar, or np.array Covariance of the filter. If scalar, is treated as eye(n)*P. sigmas : np.array, of size (n, n+1) Two dimensional array of sigma points. Each column contains all of the sigmas for one dimension in the problem space. Ordered by Xi_0, Xi_{1..n}