# Questions tagged [kalman-filter]

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### Can addition of noise to dynamical system reduce estimation errors

I am using Kalman filter to estimate the states of a stochastic dynamical system which has very very small noise( consider zero ). The filter is not aware that the noise is zero. Implementation of KF ...
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### Are Kalman Filter suited for parameter estimation in time-varying black-box models?

I want to solve a computer model calibration problem, i.e. to inverse estimate parameters from a physics simulation such that the observed physic process matches the output from the simulation. I came ...
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### General state space form for ARIMA(p,d,q)(P,D,Q)

I cannot for the life of me find a resource which gives the general state space form for an ARIMA(p,d,q)(P,D,Q) model. I am reading Time Series Analysis by State Space Methods, by Durbin & Koopman,...
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### Minimum residual error in estimation of deterministic system using Kalman filter

Let the process equation for a state vector $\mathbf{x}_t$ at time $t$ be: $$\bf{x}_{t+1} = \bf{f}\left(\bf{x}_t\right)$$ where, $\mathbf{f}\left(.\right)$ is a nonlinear ...
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### What motivates the linear form of the Kalman Filter?

The derivations of the Kalman Filter I've read express the state update estimate as $\hat{x}_t = F\hat{x}_{t-1} + K\tilde{y}$ (using Wikipedia notation). The derviations will go on to derive K by ...
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### Spectral radius of $I-KH$, where $K$ is a Kalman gain

I have noticed numerically that the spectral radius of $I-KH$, where $K$ is a Kalman gain, is less than or equal to 1. In other words, for some symmetric positive definite matrices $R$ and $C$, and ...
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### Matrix Valued Inner Product

I was studying Kalman Filters lately and came across an interesting concept. A textbook explained that if you have two random vectors X in $R^n$ in $R^m$ then you can define a sort of "inner ...
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### Quaternion process noise in EKF

I'm learning about Extended Kalman Filters for position and orientation estimation and I found the documentation for the OpenIMU project to be a good resource because it shows a step-by-step ...
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### How critical is the estimation of the covariance matrices in Kalman-Filter

Considering the follow basic Kalman Filter, following the Wikipedia convention $$\begin{split} x_k &= F_kx_{k-1} + B_k u_k +w_k\\ y_k &= H_kx_k + v_k \end{split}$$ ...
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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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### would Kalman Filter capture consistent drift in observation model?

I am studying Kalman filter and I wonder how it handles the case of consistent bias in observation model? Let's take this example from wikipedia: https://en.wikipedia.org/wiki/Kalman_filter#...
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### Extended Kalman filter with Non-additive noise

I have the following problem. I have the following Kalman filter: \begin{split} \boldsymbol{x}_k=\boldsymbol{x}_{k-1} + \boldsymbol{w}_k \\ \boldsymbol{y}_k=h(\boldsymbol{x}_{k}, \...
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### Using Bayesian statistics in time series forecasting

I would like to forecast demand count time series of taxi fleets at different locations on the map at different points in time. I.e. multivariate demand Time series forecasting. Given hierarchinal ...
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### Russell & Norvig: Connecting models of probabilistic reasoning to Stochastic Differential Equations

Artificial Intelligence: A Modern Approach, 4th Global ed. by Stuart Russell and Peter Norvig contains the following footnote on page 480 of chapter 14: Uncertainty over continuous time can be ...
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### Continuous-time & Discrete-time kalman filter(KF)

I seek the general difference between continuous time and discrete time Kalman filter(KF). I want to design the continuous-time transition model &discrete-time measurement model KF and discrete-...
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### Extending Kalman filters with discrete (catergorical) state variable?

I'm new to Kalman filters. I have a use case similar to the one-dimensional train example. But I have railroad track with switches and mergers. So it's a non-trivial topology. I would like to model ...
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### Numerical integration of a Lyapunov differential equation in a kalman filter

I'm struggling understanding the following section of the MSCKF paper. This is regarding the covariance matrix propagation of the kalman filter using a continuous-time model. $P_{II_{k+1|k}}$ is ...
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### Koopman's Kalman Filter Derivation

i am currently doing my undergraduate thesis which uses State Space Model and Kalman Filter as my main model. The State Space Model and Kalman Filter is based on the one used in "Time Series ...
1 vote
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### Calculations of the error quaternion from the MSCKF paper

In the MSCKF paper proposed for visual-inertial odometry, the authors use an error-state extended Kalman filter. When describing the error quaternion, they state that: for the quaternion a different ...
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### How does autocorrelated noise models converge to white noise models when considering kinematic tracking models?

In the article: A jerk model for tracking highly maneuvering targets see http://eprints.iisc.ac.in/2710/ The jerk model is explained. What happens with the process noise matrix Q if the limit is taken ...
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### Kalman filter specification in Python

I'm learning about state space models and I've written a simple code to apply a Kalman filter. I want to apply it to the following time-dependent model: \begin{align*} x_{t} &=\Phi_tx_{t-1}+ \...
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### Convergence of the conditional expectation in Kalman-Bucy filter for small noise

I am reading different papers on parameter estimation in the Kalman-Bucy filter scheme for small noise ("ON FREQUENCY ESTIMATION FOR PARTIALLY OBSERVED SYSTEM WITH SMALL NOISES IN STATE AND ...
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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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### Why should a part of the updated posterior variance using kalman filter have a negative sign?

enter image description here Hi, I am learning the Kalman filter and am really confused at the estimated posterior variance. The conditional posterior variance has a negative term, but to me, it is ...
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### Quaternion vs Axis Angle parameterization in state estimation.

Euler angles are known to have gimbal lock so quaternions are sometimes used instead for state estimation problems. I'm curious, why isn't the axis angle representation used and how does that compare ...
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### Process Covariance Matrix in a Kalman Filter

I have a question with regards to the Kalman Filter Covariance matrix. Specifically the differences the Process noise Covariance matrix (Q) and the process noise(W). I am attempting to implement a ...
1 vote
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### What is the correct understanding of Kalman process noise with respect to ground truth?

I'm confused between two different ways of looking at what Kalman filtering does exactly. I'm trying to simulate a particle going from (-3,0) to (3,0) with a constant velocity and some noise (e.g. the ...
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### How can I estimate the variance of a dependent variable from a random variable with nonlinear relationship?

I am working on a Kalman Filter and I've added a new state variable that is observable via one of the measurements via nonlinear relationship. My sensor reads $y \in \mathbb{R}$ and I am assuming that ...
1 vote
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### What is the filtered probability space used to study linear SDEs with constant coeficients?

Context: I am currently working with Kalman-like filters. As a result I deal with linear Stochastic Differential Equations (SDEs) with constant coefficients such as: $$dx(t) = Ax(t)dt + Bdw(t)$$ ...
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### How to verify if the kalman gain matrix K is working properly?

If I have a state space model. $$x(k + 1) = Ax(k) + Bu(k)$$ $$y(k) = Cx(k) + Du(k)$$ And a kalman gain matrix $K$. Then, how do I know if the kalman gain matrix $K$ is properly designed for my state ...
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### How to show that a set is an invariant set for my system?

I have this type of systems: $\;x(k+1) = -0.5x(k)$ and I have this set: $\;\big\{x\;|\;x^4\leqslant5\big\}\;.$ How can I show that this set is an invariant set ? I would like to understand it ...
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### Kalman Filtering the Vasicek Model, are there different Kalman Filters for the same application? [closed]

While studying parameter estimation in affine term structure models, I stumbeled across two papers. Affine Term-Structure Models: Theory and Implementation by David Bolder (2001) https://www....
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### Kalman filter, intuitively

I am currently working my way through the Kalman filter equations. Be warned, I think do have a solid understanding of math, yet I am just an engineer. So first of all there is this excellent website ...
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### How to derive an optimal, continuous-time linear quadratic estimator from a Luenberger state observer?

How does one derive an optimal, continuous-time linear quadratic estimator from a Luenberger state observer? I am aware of a Kalman filter, but I would like to see a derivation of an observer without ...
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