Confusion related to Kalman filters density view - Cross Validated most recent 30 from stats.stackexchange.com 2019-08-20T01:31:22Z https://stats.stackexchange.com/feeds/question/46944 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/46944 3 Confusion related to Kalman filters density view user34790 https://stats.stackexchange.com/users/12329 2013-01-04T04:25:52Z 2013-09-02T08:15:59Z <p>I was reading this book related to Kalman filters and I didn't understand a couple of things. I have also attached the screenshot of the pages from the book where I had confusion.</p> <p>The book is Shumway, R.H. and Stoffer, D.S. Time Series Analysis and Its Applications: With R Examples</p> <p>From the first page, I didn't understand how this was derived</p> <p><img src="https://i.stack.imgur.com/wEHbF.png" alt="enter image description here"></p> <p>I mean how come we have $p(x_t|x_{t-1}) = f_w(x_t-\phi_{x_{t-1}})$ and has mean 0 and covariance Q. I mean it should be for the $w_t$ which has mean 0 and covariance Q not $p(x_t|x_{t-1})$</p> <p>From the second page, I didn't understand the following</p> <p><img src="https://i.stack.imgur.com/B0v7Q.png" alt="enter image description here"></p> <p>I mean obviously $p(\mu_t|Y_{t-1} =&gt; N(\mu_t^{t-1},P_t^{t-1}))$. But I didn't get how we get that from the integral</p> <p>From the third page, I didn't get how this was derived</p> <p><img src="https://i.stack.imgur.com/pKfGh.png" alt="enter image description here"></p> <p>I know I might be asking a lot, but I have spent several days to exactly grasp the derivation of the equations in Kalman filters. I would really appreciate if someone could help me. Thanks in advance</p> <p><img src="https://i.stack.imgur.com/Ru9wU.png" alt="enter image description here"></p> <p><img src="https://i.stack.imgur.com/whVb4.png" alt="enter image description here"></p> <p><img src="https://i.stack.imgur.com/A4diL.png" alt="enter image description here"></p> https://stats.stackexchange.com/questions/46944/-/46993#46993 1 Answer by F. Tusell for Confusion related to Kalman filters density view F. Tusell https://stats.stackexchange.com/users/892 2013-01-04T21:47:26Z 2013-01-04T21:47:26Z <p>Regarding your first question, $x_t|x_{t-1}$ has the same variance as $w_t$; if in the transition equation $x_t = \Phi x_{t-1} + w_t$ you hold $x_{t-1}$ fixed, the only source of variance in $x_t$ is just $w_t$.</p>