Copyright 2015 Roger R Labbe Jr.

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

Documentation at: https://filterpy.readthedocs.org

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

This is licensed under an MIT license. See the readme.MD file for more information.

class filterpy.leastsq.LeastSquaresFilter(dt, order, noise_sigma=0.0)[source]

Implements a Least Squares recursive filter. Formulation is per Zarchan [1].

Filter may be of order 0 to 2. Order 0 assumes the value being tracked is a constant, order 1 assumes that it moves in a line, and order 2 assumes that it is tracking a second order polynomial.

dt : float

time step per update

order : int

order of filter 0..2

noise_sigma : float

sigma (std dev) in x. This allows us to calculate the error of the filter, it does not influence the filter output.


[1](1, 2) Zarchan and Musoff. “Fundamentals of Kalman Filtering: A Practical Approach.” Third Edition. AIAA, 2009.


from filterpy.leastsq import LeastSquaresFilter

lsq = LeastSquaresFilter(dt=0.1, order=1, noise_sigma=2.3)

while True:
    z = sensor_reading()  # get a measurement
    x = lsq.update(z)     # get the filtered estimate.
    print('error: {}, velocity error: {}'.format(
          lsq.error, lsq.derror))
n : int

step in the recursion. 0 prior to first call, 1 after the first call, etc.

K : np.array

Gains for the filter. K[0] for all orders, K[1] for orders 0 and 1, and K[2] for order 2

x: np.array (order + 1, 1)

estimate(s) of the output. It is a vector containing the estimate x and the derivatives of x: [x x’ x’‘].T. It contains as many derivatives as the order allows. That is, a zero order filter has no derivatives, a first order has one derivative, and a second order has two.

y : float

residual (difference between measurement projection of previous estimate to current time).

__init__(dt, order, noise_sigma=0.0)[source]

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


reset filter back to state at time of construction


Update filter with new measurement z

x : np.array

estimate for this time step (same as self.x)


Computes and returns the error and standard deviation of the filter at this time step.

error : np.array size 1xorder+1
std : np.array size 1xorder+1