Kalman Filter textbook using Ipython Notebook. This book takes a minimally mathematical approach, focusing on building intuition and experience, not formal proofs. Includes Kalman filters, Extended Kalman filters, unscented filters, and more. Includes exercises with solutions.
import numpy as np
from scipy.stats import multivariate_normal
from numpy import dot, log, exp
import scipy.linalg as la
def gaus_pdf(X, M, S):
DX = (X-M)[0,0]
E = 0.5*np.dot(DX.T, (S/DX))
d = M.shape[0]
E = E + 0.5 * d * log(2*np.pi) + 0.5 * log(la.det(S));
P = exp(-E)
return P
def kf_liklihood(x, P, z, H, R):
IM = np.dot(H, x)
S = np.dot(H, P).dot(H.T) + R
print(gaus_pdf(z, IM, S))
print(multivariate_normal.pdf(z, mean=IM, cov=S))
return multivariate_normal.pdf(z, mean=IM, cov=S)from scipy.stats import multivariate_normal
def likelihood(x, P, z, H, R):
IM = np.dot(H, x)
S = np.dot(H, P).dot(H.T) + R
return multivariate_normal.pdf(z, mean=IM, cov=S)
When I install the packge filterpy，I got the following error:
C:\WINDOWS\system32>conda install --channel https://conda.anaconda.org/phios filterpy
Fetching package metadata: ......
Solving package specifications: ..........
Error: Unsatisfiable package specifications.
Generating hint:
[ COMPLETE ]|##################################################| 100%
Hint: the following packages conflict with each other:
- filterpy
- python 3.5*
Use 'conda info filterpy' etc. to see the dependencies for each package.
Note that the following features are enabled:
- vc14
I want to know wether the filterpy can be installed with python3.5 and how can solute this problems.Thanks.
pip install filterpy instead; that should work. Recently I tried to get filterpy working with conda; conda install filterpy might work for windows, but I have reports it doesn't work for mac and linux yet. I need to put more effort on this.
raise ValueError('object arrays are not supported')
ValueError: object arrays are not supported
Although I did not read all chapters yet, it came to my understanding that outliers (measurements containing no information) are pretty bad for a 'standard' Kalman filter. By just searching trough the pdf version I didn't find any mention of outliers. So I was wondering if you could point me to any resources that introduce outlier handling within kalman filtering? Of course I googled but I cannot really judge the quality for a beginner of all the different results I get back...
It is clear, didactic and well-documented.
I had to build my very first Kalman filter in a quite complex configuration (7 state variables, strong non-linearities and very low signal to noise ratio).
Your book has brought me tremendous help in doing that - although, as a Scilab user, I have found the recourse to Python more troublesome than helpful -.
It is rare to find such a thorough, simple, user-oriented while scientifically sound presentation of the Kalman filter.
I think you are a born pedagogue.
Lots of thanks.
@rlabbe I just started working through the Jupyter notebooks. Your intuitive approach to the subject matter is very refreshing. I do have a question/observation about the material in Chapter 3... I find the discussion of the product vs sum of "Gaussians" a bit confusing. It seems that you are discussing the sum of Gaussian random "variables" and the product of Gaussian probability "distributions". The sum of two independent Gaussian random variables is also Gaussian-distributed. The product of two Gaussian random variables is not, in general, Gaussian-distributed.
Having now made it through Chapter 4... I think the source of the confusion is that, in both cases, we are really talking about operations on Gaussian "distributions" rather than random "variables" . The mathematical operation involved in the "prediction" step is really a convolution, rather than a sum, of Gaussian "distributions", which can be shown to be a Gaussian "distribution" with mean and variance as described in Chapter 3. At least, that's what I think after reading Chapter 4... Looking forward to further enlightenment in the upcoming chapters... :-)
1 reply
In chapter 6: Multivariate Kalman Filters, in the part
what if 𝑥=𝑥˙Δ𝑡 ? (set F00 to 0, the rest at defaults)
leads to the following plot of :
But I can't get the idea behind it, I mean how is the variance for x (position) reduced to minimum ? On using 𝑥0 = 0, the resulting matrix for P is:
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Hi @rlabbe I came across the part "Stable Computation of the Posterior Covariance" and couldn't understand how the Joseph-equation-derived-covariance mitigates a non-optimal K (Kalman gain) ? I see that to be possible only when a different method is used to compute K.
Can you please help ?
why the the gain_rate equation given by "gain_rate = gain_rate + gain_scale (residual/time_step)", and not "gain_rate = gain_rate + gain_scale ((residual-gain_rate)/time_step)", here 1 is our predicted gain rate. Shouldn't the residual give us the measured gain_rate (from sensor) and we already have our previous gain_rate. Using both of these , we update our next gain_rate.
please clarify