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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158 views

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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135 views

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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5answers
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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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70 views

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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1answer
48 views

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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67 views

Kalman filter for AR(1) plus noise

I am working the following AR(1) plus noise state-space model $$ z_{t} = x_{t} + v_{t}\\ x_{t} = \phi x_{t-1} + c + w_{t} $$ Therefore, the transition matrix is $[\phi]$, the observation matrix is $[1]...
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1answer
146 views

Are Kalman Filter recursions valid when the state noise has a singular covariance matrix?

Consider a Linear Gaussian State-Space Model where the states are denoted by $X_t$ and observations are denoted by $Y_t$: \begin{align} X_t &= A X_{t-1} + \epsilon_t, &&\epsilon_t \sim \...
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29 views

Exponential moving average before computing std

In which cases it would make sense to use exponentially weighted moving average (EWMA) before, for example, computing sample variance or other statistical analysis? Could you give an example when one ...
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18 views

Square root cubature Kalman filter stability

*Square root cubature Kalman filter gain $\mathbf{K}_k$ directly without the need for a matrix inversion: Efficient least-squares: The least-squares method is used to compute the Kalman filter gain. ...
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16 views

Kalman Filter and Monte Carlo

What is the consequence on the uncertainty of our estimate when applying the standard kalman filter to nonlinear systems? If we are unaware of the functional form of these non linear systems how do ...
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13 views

how to describe the process noise correctly

A state space represenation is given the following way x(k+1)=Ax+[1 2;3 4][u;w] z(k)=Cx+v w is a white noise. How can I bring the system in the form ...
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2answers
151 views

Is there a difference between recursive parameter estimates and time-varying parameters?

As the title indicates, is there a difference between recursive parameter estimates and time-varying parameters. I ask this in the context of time-series. For example, recursive parameter estimates ...
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An approach to missing values in time series: easy, universal (and wrong?)

Suppose I have time series data and am fitting some appropriate model to the dependent variable, with or without independent variables, but definitely with some temporal structure that I wish to ...
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R code for time series forecasting using Kalman filter

Does anybody have a good example for Time Series Forecasting/smoothing using Kalman Filter in R?
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89 views

AR(1) model with autoregressive intercept

Let us consider the following model: $$ y_{t} = c_{t} + \alpha y_{t-1} + v_{t} \\ c_{t+1} = c_{0} + \beta c_{t} + w_{t} $$ where $v_{t} \in \mathcal{N}(0, \sigma^{2}_{v})$ and $w_{t} \in \mathcal{N}(...
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1answer
33 views

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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1answer
51 views

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 ...
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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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1answer
54 views

Stationary Kalman Filter

Ecercise 4.5 from Bayesian Filtering & Smoothing by Simo Särkkä: Derive the stationary Kalman filter for the Gaussian random walk model. That is, compute the limiting Kalman filter gain when $...
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Kalman filter multi-step prediction

I have historical data for five correlated time series, denoted by $\{\mathbf{x}_t\}_{t=-m}^0$, where $$\mathbf{x}_t\equiv[x_{1t}, x_{2t}, x_{3t}, x_{4t}, x_{5t}]^\intercal.$$ I already have ...
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1answer
593 views

Bayesian Filtering for linear but non-Gaussian estimation problems

It seems that most optimal estimation literature is divided into either linear Gaussian problems, for which you use Kalman Filter, or non linear and non Gaussian problems for which you use EKF, UKF or ...
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How to estimate and determine the confidence level of the presence of an object with respect to another object?

I'm working on a university project to estimate the confidence (And therefore, the change in confidence) of the presence an area of given dimensions with respect to positions obtained from a moving ...
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1answer
469 views

Kalman Filter Vs Recursive Least Squares

Does the Kalman Filter boil down to Recursive (i.e., incremental) Least Squares if the state is constant? I expect it does but I am not sure. Assume that all simplifying assumptions hold (i.e, ...
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Negative Hessian matrix in R optim [closed]

I used the optim() function in R to find the min log likelihood, however the diagonal elements of the inverse of Hessian matrix turned out to be negative. ...
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108 views

RNN vs Kalman filter : learning the underlying dynamics?

Being recently interested in Kalman filters and Recurrent neural networks, it appears to me that the two are closely related, yet I can't find relevant enough litterature : In a Kalman filter, the ...
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25 views

zero-lag filter: size of negative part of filter weights: when in-phase with sinusoid?

This question is about negative weights in causal filters and their effect on the lag, or "synchronization" with a sinusoidal signal. There are a few types of moving averages that use negative ...
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2answers
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Can we use bootstrap samples that are smaller than original sample?

I want to use bootstrapping to estimate confidence intervals for estimated parameters from a panel dataset with N=250 firms and T=50 month. The estimation of parameters is computationally expensive (...
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1answer
72 views

Retrodiction / Specific filter to obtain initial state

Problem I have a system that is measured at regular intervals. The state of the system at those times is given by the vector $\vec x=(x_0, x_1, x_2,\cdots)$. In between each measurement, a random ...
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1answer
22 views

Examples of Real Applications for Time-series with Continuous-valued Targets and Continuous-valued Observations

Suppose that we are interested in estimating continuous-valued targets $y_t$ from continuous-valued observations $x_t$ over discrete time steps $t = \{1,2,3,\dots,T\}$. Could you give me some ...
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1answer
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Unscented Kalman filter-negative covariance matrix

I have recently started working on the unscented Kalman filter. I coded the numerically stable version (i.e., square root Kalman filter) and used MATLAB for implementing. In the final update step, ...
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Outliers Kalman Filtering

This might not be the right place to ask this questions, but I figured it's more of a machine learning question. I am also asking on the pyro forum for brevity. I'm working with the simple extended ...
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1answer
42 views

Clarification on Akaike's IC (AIC) and BIC for Expectation Maximization with time-changing parameters

I apologize in advance for the trivial question, but I need a clarification on the following issue. Suppose I have a generic model in state-space form described as $$x_{t+1}=\phi_{t} x_{t}+w_{t+1}$$ $...
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1answer
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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 ...
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1answer
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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20 views

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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2answers
54 views

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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1answer
53 views

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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22 views

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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757 views

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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Dynamic factor model (DFM) with R, please help

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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1answer
3k views

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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Log innovation vs squared

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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1answer
1k views

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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2answers
2k views

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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1answer
82 views

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 ...
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1answer
41 views

How do I model the chaotic behaviour(like the sequence from Lorenz attractor) in a stochastic sense?

Recently, I encountered a difficulty of prediction Lorenz attractor by using a GRU. (See the code from here.) I think that it's inevitable since the original system, i.e. Lorenz equation, is too ...
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1answer
459 views

State space models: Advantage of Stationary State Vector?

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 ...