daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Added reference to Grewal2008 for observability.

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Roger Labbe 2015-12-13 10:53:45 -08:00
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06-Multivariate-Kalman-Filters.ipynb
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@ -2774,6 +2774,8 @@
"\n",
"We extended Gaussians into multiple dimensions to allow us to simultaneously handle multiple dimensions, both spacial and others (velocity, etc). This led us to use linear algebra. We made a key insight: hidden variables have the ability to significantly increase the accuracy of the filter. This counterintuitive result is because the hidden variables are correlated with the observed variables. Intuitively, I think of it as triangulating onto the solution - only a small number of states can be true for a given combination of a velocity Gaussian and position Gaussian, roughly described by their intersection. But of course this works for non-spatial problems as well. \n",
"\n",
"I gave an intuitive definition of *observability*. Observability was invented by Dr. Kalman for linear systems, and there is a fair amount of theory behind it. It answers the question of whether a system state can be determined by observing the system's output. For our problems this has been easy to determine, but more complicated systems may require rigorous analysis. Wikipedia's [Observability](https://en.wikipedia.org/wiki/Observability) article has an overview; if you need to learn the topic [[Grewal2008]](#References) is a good source. \n",
"\n",
"There is one important caveat about hidden variables. It is easy to construct a filter that produces estimates for hidden variables. I could write a filter that estimates the color of a tracked car. But there is no way to compute car color from positions, so the estimate for the color will be nonsense. The designer must verify that these variables are being estimated correctly. If you do not have a velocity sensor and yet are estimating velocity, you will need to test that the velocity estimates are correct.; do not trust that they are. For example, suppose the velocity has a periodic component to it - it looks like a sine wave. If your sample time is less than 2 times the frequency you will not be able to accurately estimate the velocity (due to Nyquist's Theorem). Imagine that the sample period is equal to the frequency of the velocity. The filter will report that the velocity is constant because each time it will be sampling the system at the same point on the sin wave. \n",
"\n",
"Initialization poses a particularly difficult problem for hidden variables. If you start with a bad initialization the filter can usually recover the observed variables, but may struggle and fail with the hidden one. Estimating hidden variables is a powerful tool, but a dangerous one. \n",
@ -2803,7 +2805,9 @@
"metadata": {},
"source": [
"- <A name=\"[wiki_article]\">[1]</A> 'Kalman Filters'. Wikipedia\n",
"https://en.wikipedia.org/wiki/Kalman_filter#Details"
"https://en.wikipedia.org/wiki/Kalman_filter#Details\n",
"\n",
"* **[Grewal2008]** Grewal, Mohinder S., Andrews, Angus P. *Kalman Filtering: Theory and Practice Using MATLAB*. Third Edition. John Wiley & Sons. 2008."
]
}
],