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POExplain process noise terminology in Kalman Filter
 

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  1. POExplain process noise terminology in Kalman Filter
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    <p>I am just learning Kalman filter. In the Kalman Filter terminology, I am having some difficulty with process noise. Process noise seems to be ignored in many concrete examples (most focused on measurement noise). If someone can point me to some introductory level link that described process noise well with examples, that’d be great.</p> <p>Let’s use a concrete scalar example for my question, given:</p> <pre><code>x_j = a x_j-1 + b u_j + w_j </code></pre> <p>Let’s say <code>x_j</code> models the temperature within a fridge with time. It is 5 degrees and should stay that way, so we model with <code>a = 1</code>. If at some point <code>t = 100</code>, the temperature of the fridge becomes 7 degrees (ie. hot day, poor insulation), then I believe the process noise at this point is 2 degrees. So our state variable <code>x_100 = 7</code> degrees, and this is the true value of the system.</p> <p><strong>Question 1:</strong> </p> <p>If I then paraphrase the phrase I often see for describing Kalman filter, “we filter the signal x so that the effects of the noise w are minimized “, <a href="http://www.swarthmore.edu/NatSci/echeeve1/Ref/Kalman/ScalarKalman.html" rel="nofollow noreferrer">http://www.swarthmore.edu/NatSci/echeeve1/Ref/Kalman/ScalarKalman.html</a> if we minimize the effects of the 2 degrees, are we trying to get rid of the 2 degree difference? But the true state at is <code>x_100 == 7</code> degrees. What are we doing to the process noise w exactly when we Kalmen filter?</p> <p><strong>Question 2:</strong></p> <p>The process noise has a variance of <code>Q</code>. In the simple fridge example, it seems easy to model because you know the underlying true state is 5 degrees and you can take <code>Q</code> as the deviation from that state. But if the true underlying state is fluctuating with time, when you model, what part of this would be considered state fluctuation vs. “process noise”. And how do we go about determining a good <code>Q</code> (again example would be nice)? </p> <p>I have found that as <code>Q</code> is always added to the covariance prediction no matter which time step you are at, (see Covariance prediction formula from <a href="http://greg.czerniak.info/guides/kalman1/" rel="nofollow noreferrer">http://greg.czerniak.info/guides/kalman1/</a>) that if you select an overly large <code>Q</code>, then it doesn’t seem like the Kalman filter would be well-behaved.</p> <p>Thanks.</p> <p><strong>EDIT1 My Interpretation</strong></p> <p>My interpretation of the term process noise is the difference between the actual state of the system and the state modeled from the state transition matrix (ie. <code>a * x_j-1</code>). And what Kalman filter tries to do, is to bring the prediction closer to the actual state. In that sense, it actually partially "incorporate" the process noise into the prediction through the residual feedback mechanism, rather than "eliminate" it, so that it can predict the actual state better. I have not read such an explanation anywhere in my search, and I would appreciate anyone commenting on this view.</p>
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