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Questions tagged [kalman-filter]

The Kalman filter is an algorithm for estimating the mean vector and variance-covariance matrix of the unknown state in a state space model.

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Bayes rule and terms with expectation

I am reading the following paper in economics; link On page 495, authors give an expression with Bayes rule. As an example, say that there is a random variable $\beta$ which can be either $\beta_L$ or ...
optimal control's user avatar
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Intuitive explanation of "Information Filter" formation of Kalman filter

Can someone intuitively explain this "Information Filter" formation quoted from wikipedia ? In particular I struggle to understand why $\mathbf{I}_k = \mathbf{H}_k^\textsf{T} \mathbf{R}_k^{-...
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kalman filter equations when I have position and velocity but force is unknown

I have a car for which I measure position and velocity using GPS receiver, but I do not know forces, which change the car velocity. I wonder how to build equations for Kalman filter. For measurements ...
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Covariance inversion for Gaussian process

Background Let $x=f(u_x)\in\mathbb{R}$ and let $y=[f(u_y^1)\cdots f(u_y^{N})]\in\mathbb{R}^N$ for some function $f:u \in \mathbb{R}\mapsto \mathbb{R}$. Given $y$, $u_x$, $u_{y}^1,\dots, u_{y}^{N}$, I ...
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How to Interpret and Address Non-Gaussian Measurement Post-Fit Residuals of a Kalman Filter?

I'm using a Kalman filter, optimized via hyperparameter search for both the feature matrix (about 300 variables post my most recent hyperparameter grid search), a ridge regression-based observation ...
Captain Ahab's user avatar
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Best way to approach sensor fusion

I'm fusing acceleration data from an accelerometer and the derived acceleration from a distance sensor to learn about sensor fusion. The derived acceleration (2nd derivative) from the distance sensor ...
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Computing the Kalman filter measurement covariance matrix directly from measurement data?

I'm looking into methods for estimating the parameters of a Kalman Filter, in particular the covariance matrices. I have seen it suggested a couple times (for example here) that the measurement ...
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Looking for books and/or other resources for Multivariate Statistics for Optimal Estimation

I was trying to familiarize myself with state estimation theory by going through Optimal State Estimation and I realized I don't have the required background in Multiple Random Variables and ...
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Is reinforcement learning conceptually equivalent to time-series with a latent dependent variable?

In reinforcement learning, there is a state $s_t$, an action $a_t$, and a policy $\pi(a|s)$ that maps states to the Probability Distribution Function (PDF) of actions. The goal is to choose the ...
Colin T Bowers's user avatar
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Kalman Filtering and Smoothing - the effect of adding a new set of observed values to latent state

For such a latent space model $$z_t = A z_{t-1} + w_t$$ $$x_t = C z_t + v_t$$ where the noise is being drawn $w_t \sim N(0, Q)$ and $v_t \sim N(0, R)$ I have the derivation of the forward and backward ...
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Kalman filter + MCMC vs pure MCMC for bayesian dynamic linear(state space) models

I have heard that you can use both kalman filter integrated with MCMC to estimate bayesian state space models, i have also heard that you can use pure MCMC to estimate bayesian state space models. Why ...
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Computing the Jacobian in the Extended Kalman Filter with Non-additive noise

I have the following problem. I have the following Kalman filter: $ \boldsymbol{x}_k=\boldsymbol{x}_{k-1} + \boldsymbol{w}_k$ $ \boldsymbol{y}_k=h(\boldsymbol{x}_{k}, \boldsymbol{v}_k)$ where $\...
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regression model with arma errors: forecasting the residuals

Suppose I estimate the following model: $$ y_t = \beta_0 + \beta_1 x_t + \eta_t $$ where $\eta_t$ is an AR(1) model, say. I can do that with forecast::Arima() as ...
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What is the "process noise" (Q) parameter to a Kalman filter?

I'm teaching myself about Kalman filters; I've studied a number of videos and articles and have enough of an understanding now that I can effectively construct and use them. However, most of what I've ...
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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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How can mahalanobis and chi2-test be used to determine of an observation is acceptable?

Assume that you have a model $$\dot x = Ax + Bu$$ $$y = Cx$$ And this model is SISO. Single input and single output. You got the mission to determine of an observation is acceptable for the kalman ...
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Assessing probability that one set of measurements extends the other with Kalman smoother

I have two sets of N-dimensional measurements following each other with a certain time gap in between. Let's name those sets $A$ and $B$, respectively. All observations have constant Gaussian white ...
emerte's user avatar
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Exact diffuse initialization of the Kalman Filter: what does the design matrix look like?

I am using Python (statsmodels) to create a dynamic factor model on which I apply the Kalman filter. Thanks to earlier questions on this forum, I landed upon using exact diffuse initialization. My ...
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Kalman Filter Terminology: Prediction vs Estimation

The linear dynamical system model underlying the Kalman filter technique involves a random process $(x_{0}, v_{1}, w_{1}, x_{1}, z_{1}, v_{2}, w_{2}, x_{2}, z_{2}, \ldots)$ where $x_{k}$ represents ...
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Updating Dynamic Factor Model within a time-period

I have the following question. Assume that we have the standard Dynamic Factor Model: $$ X_{i,q} = \beta_i F_q + \epsilon_{i,q}, \qquad \epsilon_{i,q} \sim \mathcal{N}(0, \sigma_i^2), $$ and $$ F_q = \...
Borys Koval's user avatar
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Some Problems in Auxiliary Particle Filter

recently I am studying PF. And I am stuck in APF for a few days, though I derived many times. Here is my question: I followed the framework of this paper. The APF is defined in Algorithm 1: The ...
stander Qiu's user avatar
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Kalman filter) Observation matrix of measurement equation and what is a good signal?

I am trying to use a Kalman filter, but my data are somewhat deviating from the assumptions. The noises in my measurement equation are not normally distributed. First of all, they are not zero-mean. ...
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How to handle exact diffuse initialization of a Kalman filter?

This is partially a coding question so I hope I'm on the right platform for this. I am fitting a dynamic factor model using the state space framework. I don't know the initial distribution of the ...
eork's user avatar
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Inferences with Filtered and Smoothed state estimates from a tracking problem

I am working on a synthetic tracking problem in a two-dimensional space, where the start and end positions are known and noisy measurements of the state variables are given at discrete time steps. As ...
Carlo Berger's user avatar
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How to treat non-normal measurement noise in a Kalman filter? Plus, how to treat non-zero mean of noise in Kalman filter?

Typically in textbooks, it is assumed that measurement noise $\nu_{t}$ is normally distributed. Suppose that $S_{t}$ is a signal. Then a measurement equation is $$ S_{t} = x_{t} +\nu_{t} $$ where $$ \...
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Why do popular ML and statistical packages simply ignore classical estimation and detection algorithms for statistical signal processing? [closed]

For those who had a hard time to study and understand classical estimation and detection algorithms, and unfortunately realized that these algorithms are simply ignored by many packages that have the ...
Rubem Pacelli's user avatar
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Proving consistent/inconsistency of a fusion of KF estimates

I have a distributed fusion scenario with a single target where two sensor nodes $i,j$ estimate the true state $\mathbf{x}$ using a local Kalman filter. The (linear, Gaussian) measurement errors of ...
Nikhil Sharma's user avatar
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Smoothing of GPS tracks - remove noise and stop-go clusters

I know there are several posts about this, but I could not find exactly what I need. I have GPS track data (from an underwater vehicle) for short intervals of 1 second (time-stamps on data). The data ...
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Instantaneous propagation of process covariance matrix modification's effect on state

I am trying to build a zero-delay kalman filter which udates its process noise covariance matrix $Q_k$ depending on the value of the residues $z_k - H\cdot x_k$. My problem is, I have adopted a 1D ...
ArraysShouldStartAt1's user avatar
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Inferring a random walk from noisy "images"

I'm interested in the following inference / filtering problem in a hidden Markov model setting. Suppose we have a simple random walk $x_t\in\mathbb{Z}$ and observations are "images" ...
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Negative value of likelihood function in Kalman filter

I use kalman filter algorithm, where I minimize the value of likelihood function. But after some iteration I got negative value of likelihood function. Is that a problem?
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Comparison of two models with different number of parameters

I want to compare two models, which has different number of parameters. The first model is Arbitrage free Nelson-Siegel model, which has the following equation: $y_{t}(\tau )=X_{1,t}+X_{2,t}(\frac{1-e^...
Shelley's user avatar
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Do we need to propagate state covariance matrix 'P' during missing observations in the Extended Kalman Filter?

Basically as the title says: in a scenario, we have missing observations where the entire state vector is unknown for consecutive time steps. Do we just run through the prediction section of the ...
user383687's user avatar
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Method of evaluating the feature map of a polynomial kernel feature mapping

I'm attempting to implement an adaptive kernel Kalman filter following this paper https://arxiv.org/abs/2203.08300, but I'm struggling to find a method of evaluating the feature mapping for a ...
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State Space with Space Lags in R (dlm, MARSS or anything else)

** Edited to reflect on some first comments ** I am trying to estimate a state space model which does a kind of disaggregation. In particular, I am interested in estimating high-frequency unobserved ...
Carl Dhreiner's user avatar
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How does maximum likelihood estimation from the Kalman filter work?

My understanding is Step 1: You would run through the Kalman filter equations with initial parameter values. Step 2: After you run through the Kalman filter equations, you will have innovations ...
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Can you create Kalman filter (or a recurssive state estimator) with Beta and Binomial distributions?

I have to infer the probability of a system failing from observations. Since probabilities are bounded between 0 and 1, they are sometimes modeled using Beta distribution. While the traditional Kalman ...
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The mean of Gaussian distribution subject to another Gaussian distribution, how to derive it?

I got a fomula $$N(y;cx,R)N(x;\bar{x},\Sigma)=N(y;c\bar{x},S)N(x;g,F)$$ where: \begin{align}S&=c\Sigma c^T+R\\g &=\bar{x}+\Sigma c^Ts^{-1}(y-c\bar{x})\\F &= \Sigma - \Sigma c^T s^{-1} c \...
night3759's user avatar
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How can I use Kalman Filter formulation for inferring the probability of system failure?

I want to use the Kalman filter to sense the state of a machine. The machine could be either working or damaged. I'm trying to infer the probability of the machine working, given observations from ...
PPR's user avatar
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Kalman Filter Propagation using two previous time steps

Kalman Filters propagate using a single previous state estimate $\hat{x}$ with covariance $P$. Is there a formal way of propagating using two previous state estimates with different associated ...
esatemporis's user avatar
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What is the difference between these two observational update models in an information filter?

The Information Filter is defined as the mathematical inverse of the Kalman filter. As defined in this Wikipedia article, the observation update of the Information Matrix is defined as $$y_{k|k} = y_{...
Prometheus2508's user avatar
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Parameter estimation of state-space models with hidden variables

I have a time-series analysis problem, that I am having trouble finding a suitable regression technique for. I have a coupled linear three dimensional system \begin{align*} X_{t} & =\left(1+J\...
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Is there some standard way to diagnose a structural time series model (also called simple unobserved components model)?

I am dealing with a structural time series model (also called a simple unobserved components model), and I wonder if there is some standard way to diagnose this sort of models. In most reference books ...
zyy's user avatar
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Is there a probabilistic or bayesian interpretation of the kalman filter gain?

The Kalman filter makes sense to me as the repeated application of Bayes' theorem - if you correctly propagate the gaussian prior at each step and then update on new observation, you get a gaussian ...
shimao's user avatar
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How to apply Kalman filter to improve my model?

I'm trying to build a model that predicts humidity W This is the data I'm working with : ...
wageeh's user avatar
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Kalman forecast of AR(1)

I'm trying to work out the details of the proof of the following statement: Suppose $\xi_t = \rho \xi_{t-1} + \epsilon_t$ is an AR(1) process. Using Kalman filter, one can prove that $\mathbb{E}_t\{\...
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What is the scope of application of Kalman filter?

Recently I learned some basics about Kalman Filter 1D As I know, Kalman Filter is useful in Telecommunication and GPS positioning. My estimation goal is to measure the reliability of the Circuit using ...
less force's user avatar
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Approximating a 1-d Kalman Filter with non-Gaussian Observation Noise

I'm looking for a Bayesian filter where observations are generated according to $s_t = \gamma s_{t-1} + w_p$ and $w_p \sim Normal(0, \sigma_p^2)$. Both $\gamma$ and the variance of the process noise $\...
ratatosk's user avatar
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Is the state covariance matrix (or estimate uncertainty) in a Kalman filter in 1D equal to the variance of the current normal distribution?

I'm trying to use a Kalman-filter for some kind of anomaly detection. But I think that maybe I have misunderstood something fundamental about the filter. I'm following this "guide". I'm ...
afd's user avatar
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(Co)variance interpretation in Kalman filter

Let's say I have a device which uses Kalman filter to fuse sensor data and produce an optimal estimate of the system parameters. As it should, it also estimates parameter covariance matrices at each ...
Ilya G.'s user avatar

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