> For the complete documentation index, see [llms.txt](https://yxy-adam.gitbook.io/ml-theory/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://yxy-adam.gitbook.io/ml-theory/kalman-filter.md). # Kalman Filter ### Model of Kalman Filter We assume the current state can be modeled as a Gaussian distribution $$ P(z\_t|z\_{t-1}) \sim \mathcal{N}(Az\_{t-1},Q) $$ We assume neural observations can also be modeled a Gaussian distribution $$ P(x\_t|z\_{t}) \sim \mathcal{N}(Cz\_{t},R) $$ We also assume abase case $$ P(z\_{1}) \sim \mathcal{N}(\Pi,V) $$ Thus the model parameters are: $$ \Theta={A,Q,\Pi,V,C,R} $$ ### Model Training We aim to maximize the joint likelihood of the state and observed date $$ \mathcal{D}={{x}^n, {z}^n}*{n=1}^N={{x\_1^n, \dots, x\_T^n}, {z\_1^n, \dots, z\_T^n}}*{n=1}^N $$ $$ \Theta^\* = \arg\max\_{\Theta} P({x}^n, {z}^n|\Theta)\\ \= \arg\max\_{\Theta} \prod\_{n=1}^N P(z^n\_1)\left( \prod\_{t=2}^T P(z^n\_t|z^n\_{t-1}) \right)\left( \prod\_{t=1}^T P(x^n\_t|z^n\_{t}) \right)\\ \= \arg\max\_{\Theta}\sum\_{n=1}^N \log P(z^n\_1) + \sum\_{t=2}^T \log P(z^n\_t|z^n\_{t-1}) + \prod\_{t=1}^T \log P(x^n\_t|z^n\_{t}) \\ \= \arg\max\_{\Theta} \sum\_{n=1}^N -\frac{1}{2} \log|V|-\frac{1}{2}(z^n\_1-\Pi)^{\top}V^{-1}(z^n\_1-\Pi) + \sum\_{t=2}^T \left(-\frac{1}{2} \log|Q|-\frac{1}{2}(z^n\_{t}-Az^n\_{t-1})^{\top}Q^{-1}(z^n\_{t}-Az^n\_{t-1})\right) + \sum\_{t=1}^T \left(-\frac{1}{2} \log|R|-\frac{1}{2}(x^n\_{t}-Cz^n\_{t})^{\top}R^{-1}(x^n\_{t}-Cz^n\_{t})\right) \\ \= \arg\min\_{\Theta} \sum\_{n=1}^N \log|V|+(z^n\_1-\Pi)^{\top}V^{-1}(z^n\_1-\Pi) + \sum\_{t=2}^T \left( \log|Q|+(z^n\_{t}-Az^n\_{t-1})^{\top}Q^{-1}(z^n\_{t}-Az^n\_{t-1})\right) + \sum\_{t=1}^T \left(\log|R|+\frac{1}{2}(x^n\_{t}-Cz^n\_{t})^{\top}R^{-1}(x^n\_{t}-Cz^n\_{t})\right) $$ Suppose $$ \mathcal{L}=\sum\_{n=1}^N \log|V|+(z^n\_1-\Pi)^{\top}V^{-1}(z^n\_1-\Pi) + \sum\_{t=2}^T \left( \log|Q|+(z^n\_{t}-Az^n\_{t-1})^{\top}Q^{-1}(z^n\_{t}-Az^n\_{t-1})\right) + \sum\_{t=1}^T \left(\log|R|+\frac{1}{2}(x^n\_{t}-Cz^n\_{t})^{\top}R^{-1}(x^n\_{t}-Cz^n\_{t})\right) $$ the minimize is achieved when the derivative vanishes $$ \nabla\_{\Pi} \mathcal{L} = \sum\_{n=1}^N -2 (z^n\_1-\Pi)^{\top}V^{-1} = 0 \to \Pi^\* = \frac{1}{N} \sum\_{n=1}^N z^n\_1\\ \nabla\_{V} \mathcal{L} = \sum\_{n=1}^N (V^{-1})^{\top} + (z^n\_1-\Pi)(z^n\_1-\Pi)^{\top} = 0 \to V^\* = \frac{1}{N}\sum\_{n=1}^N (z^n\_1-\Pi^*)(z^n\_1-\Pi^*)^{\top} \\ \nabla\_{A} \mathcal{L} = \sum\_{n=1}^N \sum\_{t=2}^T -2Q^{-1}(z^n\_{t}-Az^n\_{t-1})(z^n\_{t-1})^{\top} = 0 \to A^\* = \left(\sum\_{n=1}^N \sum\_{t=2}^T z^n\_{t}(z^n\_{t-1})^{\top} \right)\left(\sum\_{n=1}^N \sum\_{t=2}^T z^n\_{t-1}(z^n\_{t-1})^{\top} \right)^{-1}\\ \nabla\_{Q} \mathcal{L} = \sum\_{n=1}^N \sum\_{t=2}^T (Q^{-1})^{\top} + (z^n\_{t}-Az^n\_{t-1})(z^n\_{t}-Az^n\_{t-1})^{\top} = 0 \to Q^\* = \frac{1}{N(T-1)} \sum\_{n=1}^N \sum\_{t=2}^T (z^n\_{t}-A^*z^n\_{t-1})(z^n\_{t}-A^*z^n\_{t-1})^{\top}\\ \nabla\_{C} \mathcal{L} = \sum\_{n=1}^N \sum\_{t=1}^T -2R^{-1}(x^n\_{t}-Cz^n\_{t})(z^n\_{t})^{\top} = 0 \to C^* =\left(\sum\_{n=1}^N \sum\_{t=1}^T x^n\_{t}(z^n\_{t})^{\top} \right)\left(\sum\_{n=1}^N \sum\_{t=1}^T z^n\_{t}(z^n\_{t})^{\top} \right)^{-1}\\ \nabla\_{R} \mathcal{L} = \sum\_{n=1}^N \sum\_{t=1}^T (R^{-1})^{\top} + (x^n\_{t}-Cz^n\_{t})(x^n\_{t}-Cz^n\_{t})^{\top} = 0 \to R^* = \frac{1}{NT} \sum\_{n=1}^N \sum\_{t=1}^T (x^n\_{t}-C^\*z^n\_{t})(x^n\_{t}-C^\*z^n\_{t})^{\top} $$ #### Testing the Model In the testing phase, we aim to compute$$P(z\_t|x\_1,\dots,x\_t)$$. At each time step, we apply two sub-steps, a one-step prediction and then a measurement update. * One step Prediction $$ P(z\_t|x\_1,\dots,x\_{t-1}) = \int P(z\_t|z\_{t-1}) P(z\_{t-1}|x\_1,\dots,x\_{t-1}) dz\_{t-1} $$ * Measurement update $$ P(z\_t|x\_1,\dots,x\_{t}) = \frac{P(x\_t|z\_t)P(z\_t|x\_1,\dots,x\_{t-1})}{P(x\_t|x\_{t-1})} $$ We’ll use the following notation for mean and convenience: $$ \mu\_t^t = E\[z\_t|x\_1,\dots,x\_{t}]\\ \Sigma\_t^t = cov\[z\_t|x\_1,\dots,x\_{t}] $$ **One step Prediction** We assume that we have the mean $$\mu\_{t-1}^{t-1}$$*and covariance*$$\Sigma\_{t-1}^{t-1}$$ from the previous iteration. We need to compute $$\mu\_{t}^{t-1}$$*and* $$\Sigma\_{t}^{t-1}$$. Because$$z\_t = Az\_{t-1} + \gamma, \gamma \in \mathcal{N}(0, Q)$$ $$ \mu\_{t}^{t-1} = E\[z\_t|x\_1,\dots,x\_{t-1}] \\ \= E\[ Az\_{t-1} + \gamma|x\_1,\dots,x\_{t-1}]\\ \= AE\[ z\_{t-1}|x\_1,\dots,x\_{t-1}] + E\[\gamma|x\_1,\dots,x\_{t-1}]\\ \= A\mu\_{t-1}^{t-1} $$ $$ \Sigma\_{t}^{t-1} = cov\[z\_t|x\_1,\dots,x\_{t-1}] \\ \= cov\[ Az\_{t-1} + \gamma|x\_1,\dots,x\_{t-1}]\\ \= E\[(Az\_{t-1} + \gamma - \mu\_{t}^{t-1})(Az\_{t-1} + \gamma-\mu\_{t}^{t-1})^{\top}|x\_1,\dots,x\_{t-1}] \\ \= E\[(Az\_{t-1} - \mu\_{t}^{t-1})(Az\_{t-1} -\mu\_{t}^{t-1})^{\top}|x\_1,\dots,x\_{t-1}] + E\[\gamma \gamma^{\top}|x\_1,\dots,x\_{t-1}]\\ \= E\[(Az\_{t-1} - A\mu\_{t-1}^{t-1})(Az\_{t-1} -A\mu\_{t-1}^{t-1})^{\top}|x\_1,\dots,x\_{t-1}] + Q\\ \= A E\[(z\_{t-1} - \mu\_{t-1}^{t-1})(z\_{t-1} -\mu\_{t-1}^{t-1})^{\top}|x\_1,\dots,x\_{t-1}] A^\top + Q\\ \= A \Sigma\_{t-1}^{t-1} A^\top + Q $$ **Measurement update** We take advantage of the property of the Gaussian distributions If$$x=\begin{bmatrix}x\_a\ x\_b\end{bmatrix}\sim \mathcal{N}\left(\begin{bmatrix} \mu\_a \ \mu\_b\end{bmatrix}, \begin{bmatrix}\Sigma\_{aa}&\Sigma\_{ab}\ \Sigma\_{ba}&\Sigma\_{bb}\end{bmatrix}\right)$$, then $$P(x\_a|x\_b)$$ is Gaussian with $$ E(x\_a|x\_b) = \mu\_a + \Sigma\_{ab}\Sigma\_{bb}^{-1}(x\_b-\mu\_b)\\ cov(x\_a|x\_b) = \Sigma\_{aa} + \Sigma\_{ab}\Sigma\_{bb}^{-1}\Sigma\_{ba} $$ Because$$x\_t = Cz\_{t} + \sigma, \sigma \in \mathcal{N}(0, R)$$, Then$$\mathbb{E}\[x\_t|x\_1,\dots,x\_{t-1}]=E\[Cz\_{t} + \sigma|x\_1,\dots,x\_{t-1}]=C \mu\_{t}^{t-1}$$*. Similarly* $$cov\[x\_t|x\_1,\dots,x\_{t-1}]=C \Sigma\_{t}^{t-1}C^{\top} + R$$ $$ cov\[z\_t|x\_1,\dots,x\_{t-1}] = C\Sigma\_{t}^{t-1} $$ Now we have the joint distribution $$P(\begin{bmatrix}x\_t\ z\_t\end{bmatrix}|x\_1,\dots,x\_{t-1})\sim \mathcal{N}\left(\begin{bmatrix}C \mu\_t^{t-1}\mu\_t^{t-1}\end{bmatrix}, \begin{bmatrix}C \Sigma\_{t}^{t-1}C^{\top} + R \&C\Sigma\_{t}^{t-1}\ \Sigma{t}^{t-1}C^{\top}&\Sigma\_{t}^{t-1}\end{bmatrix}\right)$$ We can write the measurement updates with the Kalman gain $$ \mu\_t^t = \mu\_{t}^{t-1} + K\_t (x\_t-C \mu\_t^{t-1})\\ \Sigma\_t^t = \Sigma\_{t}^{t-1} - K\_t C \Sigma\_{t}^{t-1} $$ where $$K\_t = \Sigma\_t^{t-1}C^{\top}(C\Sigma\_{t}^{t-1}C^{\top}+R)^{-1}$$ ### Reference * Matrix cookbook * Maximum likelihood for Multivariant Gaussian --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter: ``` GET https://yxy-adam.gitbook.io/ml-theory/kalman-filter.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.