# 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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### Initial conditions of differentiated Kalman filter for MLE

I want some help about the initial conditions for the derivative of a Kalman filter. (Differentiating the filtering equations necessary for the calculation of the gradient of the log-likelihood ...
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### Univariate Kalman filtering with factor in state-equation

I have a simple Kalman problem: how does one estimate the following local level univariate state-space model, but with some driving factor: ...
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### Kalman filter equation derivation

I'm studying the Kalman Filter for tracking and smoothing. Even if I have understood the Bayesian filter concept, and I can efficiently use some of Kalman Filter implementation I'm stucked on ...
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### Linearisation of Kalman filter

Assume we have the following state-space model: $$z_{k} = \theta_{k} z_{k-1} + v_{k}\\ \theta_{k} = \phi \theta_{k-1} + w_{k},$$ where $v_{k}$ and $w_{k}$ are independent and normal for all $k$. The ...
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### Kalman filter: asymptotic of state estimate

Assume we have a linear state-space model: $$z_{k} = Hx_{k} + v_{k}\\ x_{k} = F x_{k-1} + Bu + w_{k},$$ where $u$ is some control variable (constant intercept is the simplest case). Kalman filter, ...
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### RegARMA in state space representation

I am attempting to fit a state space regression model of the form: $Y_{t} = i^* + \beta_{1}Y_{t-1} + \beta_{2}X_{t} + \epsilon_{1,t}$ $i^* = i^*_{t-1} + \epsilon_{2,t}$ How could I represent the ...
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### Finite grid approximation to the Bayesian filtering problem

I need some hints for solving Ecercise 4.4 from Bayesian Filtering & Smoothing by Simo Särkkä: Select a finite interval in the state space, say, $x \in [-10, 10]$ and discretize it evenly to N ...
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### covariances in Kalman Filter

I am confused with the Kalman filter. Could you, please, explain the solution here https://stackoverflow.com/questions/46198246/em-algorithm-with-pykalman/58560992#58560992 In the simulations ...
28 views

### Kalman EM estimation of observation variance

Let us consider a simple AR(1) process: $$y_{t} = \mu + \beta y_{t-1} + \varepsilon_{t},$$ with $t = 0, \dots, N$. Assume that the parameters $\mu$ and $\beta$ slowly change in time and let's ...
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### What are the differences between Bayesian filters and adaptive filters?

I am learning about state estimation and I am having difficulty understanding the difference between Bayesian filters such as Kalman filter and particle filters compared to adaptive filters. According ...
786 views

### Trouble training LSTM for sequence to sequence learning of sensor time series

I'm experimenting with using RNNs/LSTMs in place of a Kalman Filter (KF) for sensor fusion. I'm struggling to make much progress, and would appreciate some feedback/advice. I have several multi-...
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### MLE vs Expectation Maximization to estimate time-changing parameters in state space model

Suppose I have a generic model in state-space form described as $$x_{t+1}=\phi_{t} x_{t}+w_{t+1}\epsilon_{t+1}$$ $$y_{t}=H_{t}x_{t}+v_{t}e_{t+1}$$ where both $e_{t+1}$ and $\epsilon_{t+1}$ are iid ...
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### Kalman Filter with heteroscedastic Q (covariance of the transition noise)

I am looking at a generic derivation of the Kalman Filter (like this but you can take any). And I was wondering, checking all the derivation, why are we forced to assume that the covariance matrix Q ...
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### In a LGSSM how do we know that the prediction distribution is Gaussian?

I am trying to follow lecture notes regarding the Kalman Filter from a course taught at Stanford. The lecture notes can be found here. The linear Gaussian state space model (LGSSM) is introduced as ...
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### What is behind “forecast” in Eviews?

I have been trying to use state space models in order to represent some gestural data. Until now I have been using Eviews to to do all the dynamic forecasting part, so I was curious what is behind ...
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### R- Auto Arima with Kalman Filter

In Python auto arima, it is clearly stated that when you set the method parameter as "ml" (maximum likelihood), residuals are obtained via the Kalman Filter. What is its equivalent in R?
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### For regression with time varying parameters, SGD or Kalman filter?

What is the advantage of kalman filters as an online update mechanism instead of the stochastic gradient descent?
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I'm interested doing a dynamic factor model (DLM) similar to Doz, Giannone and Reichlin (2011) and Giannone, Reichlin and Small (2008). Moreover, I'm trying doing macroeconomic nowcasting model. In ...
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### How to plot results from Kalman filter

I am interested in representing the performance/consistency of my Kalman filter in a single plot. I would like to compare the norm of the estimate error against 3$\sigma$ error. I would also like the ...
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### How does one apply Kalman smoothing with irregular time steps?

I would like to apply Kalman smoothing to a series of data sampled at irregular time points. There is a claim on Stack Exchange that "For irregular spaced time series it's easy to construct a Kalman ...
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I see some state space models specify their innovation process as log innovations and some squaring the term. For example, the examples in the R package DLM favours the use of log innovations when ...
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### Tracking and data association with Kalman filters

I am trying to solve tracking problem. At certain points in time I receive object location and I should make decision whether received object location belongs to existing track or not. If not, I ...
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### Multi-target Tracking: calculate the association gate from Kalman filter

I'm trying to implement a multi target tracking with Kalman filter. Each object has an instance of Kalman Filter. The true position of the objects $(x,Y)$ are the corrected state out of the KF after ...
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### Multi-Target Tracking Filters

I am trying to solve a multi-target tracking problem, which is in some parts different to some filters I have already researched such as the PHD filter. I am asking for advise which filters to start ...
Consider a State Space Model, where the observed process is $Y_t$ $$Y_t = B F_t + \epsilon_t \\ F_t = \Phi F_{t-1} + \nu_t$$ where the error terms are white noise. Later on, I want to compute the ...