Questions tagged [stochastic-filtering]

For questions regarding the problem of determining the state of a system from an incomplete and potentially noisy set of observations.

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Example of Kalman-filter

I am trying to understand Example 2 in the original article of Kalman. I would like to use the notion of Theorem 2.5 in my lecture notes to determine the Kalman equations. Moreover, the example shows ...
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Innovation error covariance decreasing but state error covariance inceasing

I am trying to implement Kalman Filter to estimate some random variables. I see that for the system I am using, the innovation error is zero for all times and the innovation error covariance matrix is ...
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Error propagation on SO(3).

I have seen Quaternions, Euler angles and I believe I even saw Axis-Angle forms of parametrization for SO(3). When implementing a filter(EKF, UKF, etc) which parametrization is best? I read something ...
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Deriving variance from the expected deviation from the mean of a normal distribution

I know the expected absolute deviation from the mean of a normal distribution $E[|X-\mu_x|]$. From this I want to derive the variance $\sigma^2$ of said distribution. This is done to tune a filter of $...
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Is HMM Smoothing more computationally complex than HMM filtering?

Given a sequence of observations $y_1, y_2, \dots, y_n$, of some lates sequence of Markovian states $x_1, x_2, \dots, x_n$: HMM filtering computes $p(x_k|y_1,\dots,y_k) \\$ HMM smoothing computes $p(...
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Filtering noise of unknown variance

Say that there is an unknown state $\theta\in \{0,1\}$ with prior $p_t$ (that state is 1) and we observe a "discrete" realization of the signal $$ dY_t = \sigma_\theta dW_t$$ meaning that we ...
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Optimal filtering with changing variance

I have seen on some books (e.g. Lipster and Shiryayev (1977)) some concepts from optimal filtering. One idea is that, given a hidden state $\theta$ (say time invariant and taking finitely many values) ...
ensue's user avatar
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Moving averages: a field of golden crosses?

I recently learned about moving averages, and the use of the intersection of a "slow" moving average with a "fast" moving average to predict significant long term-shifts in ...
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Estimation of unobserved variable

I am applying Kalman Filter to estimate the states of a particle moving along x direction. The states of the particle at any discrete time $t = k$ are $\left[x_k \hspace{3pt} \dot{x}_k \hspace{3pt} \...
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Derivation of the Kalman filter prediction step

I've been working through Murphy's Machine Learning A Probabilistic Perspective and have had a slight issue with the section on the Kalman Filter. As a setup we're assuming a linear-Gaussian state ...
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Special name for this Ito's process?

I came across with this formula coming from the optimal filtering of stochastic process. $$dX_{t} = X_{t} (1-X_{t}) dt + X_{t} (1-X_{t}) \sigma dZ_{t}$$ where $\sigma$ denotes the variance of ...
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Filtration of Brownian motion with drift

I have been self-studying Liptser and Shiryaev (2001), and came up with a little twist on this example related to the optimal filtering of Brownian motion (Page 371, example 1). There is a Brownian ...
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Kalman filtering: Processing all measurements together vs processing them sequentially

If I have $m$ measurements to estimate an $n $ dimensional state vector, and I am using Kalman filter to do the filtering, then: Should I put all the $m$ measurements together in the measurement ...
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What is the distribution of a Stochastic Process after passing through a convolutional filter?

Let $X(t)$ be a continuous-time, mean-zero, wide-sense stationary (covariance stationary) stochastic process, and $p(t)$ be an integrable function. Is there a general formula for the distribution ...
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Prove the Markovness from independent variables

Let $\mathbf{w}$ and $\mathbf{v}$ be two independent random variables. Consider the following equations: \begin{align} \mathbf{x} &= f(\mathbf{y}, \mathbf{w}),\\ \mathbf{z} &= g(\mathbf{y}, \...
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Confusion regarding usage of Mahalanobis distance for update rejection in Kalman filtering

I recently came across some material that discussed a method for performing update rejection in Kalman filters when bad measurements are received. [Paper 1] [Paper 2: see Section III(E)] This method ...
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For a random variable $X$, how can we characterize the events that unambigously describe possible properties of the outcome of $X$?

Let $(\Omega, \mathcal{A}, P)$ be a probability space, $(\hat{\Omega}, \hat{\mathcal{A}})$ be a measurable space and $X : \Omega \rightarrow \hat{\Omega}$ be a random variable. We say that an event $...
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Difference of Stochastic Filtering and Stochastic Smoothin

I know the stochastic filtering problem estimates the dynamics of the density $\pi_t(\phi)$ of the random variable $$ \mathbb{E}\left[ \phi(X_t) \mid \mathfrak{F}_t^Y \right]^o, $$ where $X_t$ is the ...
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