Least Squares Module

The leastsq module contains least squares filter implementations.

leastsq

Copyright 2015 Roger R Labbe Jr.

FilterPy library. https://github.com/GeorgePearse/bayesian_filters

Documentation at: https://georgepearse.github.io/bayesian_filters

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.

__all__ = ['least_squares'] module-attribute

LeastSquaresFilter

Bases: object

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.

Parameters:

Name	Type	Description	Default
dt	float	time step per update	required
order	int	order of filter 0..2	required
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.	0.0

Attributes:

Name	Type	Description
n	int	step in the recursion. 0 prior to first call, 1 after the first call, etc.
K	array	Gains for the filter. K[0] for all orders, K[1] for orders 0 and 1, and K[2] for order 2
x	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).

Examples:

.. code-block:: Python

from bayesian_filters.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))

References

.. [1] Zarchan and Musoff. "Fundamentals of Kalman Filtering: A Practical Approach." Third Edition. AIAA, 2009.

errors()

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

Returns:

Name	Type	Description
error	np.array size 1xorder+1
std	np.array size 1xorder+1

reset()

reset filter back to state at time of construction

update(z)

Update filter with new measurement z

Returns:

Name	Type	Description
x	array	estimate for this time step (same as self.x)