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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Kalman Filter vs. Regression

I'm an economics undergraduate with a fundamental understanding of regression and some experience with machine learning models (e.g. regression trees, boosting). To my knowledge, Kalman Filter is ...
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(Online) intuitive explanation of state space models

I have a similar question to the one in the link below: Intuitive explanation of state space models In the link they recommend the book by Commandeur and Koopman. I have this book already. I was ...
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How to create the initial ensemble samples for EnKF

As we know, for the ensemble Kalman filter (EnKF), we need to create a set of samples in the beginning and then to run the predict and analysis step. But for now I have a question of how to create the ...
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664 views

Examples of state space models where the filtering problem can be solved analytically

Background A discrete-time, Markovian state space model takes the form \begin{align} \mathbf{y}_t&\sim p(\mathbf{y}_t\,|\,\mathbf{s}_t,\,\boldsymbol{\theta})\\ \mathbf{s}_t&\sim p(\mathbf{s}...
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How can I not show the initialization of the estimation in the Extended Kalman Filter?

I'm making estimates through the Extended Kalman Filter and I have a problem related to the vertical axis of my figure, it's too big, so I can not see population dynamics. However, I wish it did not ...
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Tracking Moving Objects with Kalman Filters— Over-fitting over time?

I've been learning about Kalman Filters, and the classic example given is tracking an object via radar/gps. My issue here is that each time you get a new data point, you update the error in the ...
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152 views

How can one use Kalman filtering to estimate stochastic volatility models?

Assume that we have returns modelled by a stochastic volatility model with parameters that are unknown. Say we want to estimate the parameters with Quasi-Maximum Likelihood estimation and the ...
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Are there any R code examples for estimating the state space vector in this case?

I couldn't make sure Whether the model I'm using is a local level model with multiplicative components (state vector $\times$ regressor vector) or a linear gaussian state-space model. And couldn't ...
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111 views

How to sample an unobserved Markov process using the forward-backward algorithm?

The setup Let $X = (x_1, \ldots, x_T)$ denote a state variable that follows a Markov process, where $x_t \in S$. The transition distribution is denoted by \begin{equation} p(x_{t}|x_{t-1}) . \end{...
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How to interpret log-likelihood score as compared to mse

Say one has a linear dynamic system as follows: $x_k = Fx_{k-1} + v_k$ $y_k = Hx_{k-1} + w_k$ with $v \sim (0, Q)$ and $w \sim (0, R)$. I am estimating $(x)_k$ using a normal Kalman Filter and ...
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State-space model with contemporaneous effects

I have the following system of equations: $$ \begin{align} y_t^{(1)}&=y_t^{(2)}-x_t+\epsilon_t\\ y_t^{(2)}&=x_t+\nu_t\\ x_t&=\alpha x_{t-1}+u_t \end{align} $$ where $y_t^{(1)}, y_t^{(2)}$ ...
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Probability of a measurement with uncertainty covariance being generated by a normal distribution

I have the following situation: A set of Kalman filters with the same model, each with its own current estimated state and state covariance. A measurement with a covariance matrix expressing its ...
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Proper Imputation and bias-correction on degrading signal with Kalman Filtering?

A signal degrades in its quality. Some signals are far more robust to degradation while others are not. We will simulate degradation by randomly removing values from a function and then applying ...
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State Space Model Form for Equations

I have a set of equations which I have to write in state space model form but unfortunately I'm having a bit of difficulty doing so. They are given as: $y_{t} = x_{t} + z_{t}$ $x_{t} = x_{t-1} + w_{...
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Deriving a filter like a Kalman filter from a non-Gaussian state space model

Assume we specify a state space model as $$Y_t = a X_t + W_t$$ and $$X_{t+1} = b X_t + V_t$$ where $b,a \in R$, $E[W_t] = E[V_t] = 0 \quad \forall{t }$ and $W_t $ and $V_t$ are indipendent for ...
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Student doubts about maximmization

I am an economics student and I am having doubts about optimization. For example, at some point in my course I will estimate a state space model via kalman filter and I will need to find parameters ...
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Using Kalman Filters with different dimensionality in an Interacting Multiple Model Algorithm

I am currently reading a lot about Kalman Filtering and am especially interested in the IMM - Interactive Multiple Model Algorithm. In the literature (e.g. here), IMM is used for Kalman Filters with ...
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Why is Qk not included in the cost function that is optimised by the Kalman filter?

Assume the following linear discrete system: $x_k = Fx_{k-1} + w_{k-1}$ where $w_{k} \sim N(0, Q)$ $y_k = Hx_k + v_{k}$ where $v_{k} \sim N(0, R)$ One way to prove that the Kalman filter is optimal ...
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What is the difference between Noise, error and residuals?

I was reading about Kalman filter. http://web.mit.edu/kirtley/kirtley/binlustuff/literature/control/Kalman%20filter.pdf They talk about additive noise and error. I need to understand difference ...
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Does the MLE-Kalman prediction maximize the likelyhood of the prediction?

The question is the following. Say I have observations of a Gaussian stochastic process ($\{x_i\}_{i=1}^n$) for which is convenient to use the state space formalism (and Kalman recursions) to describe ...
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Which is the random variable in a Kalman filter?

When estimating a hidden state $x$ with a Kalman filter, there is the posterior and prior estimate. There are also covariances associated with those estimates. Some authors call these the covariances ...
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Extended kalman filter vs online passive-aggressive

I was wondering, what are the advantages and disadvantages of extended Kalman filter and online passive-aggressive algorithm when we use them to train our networks. I have RBF neural network and I'm ...
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Bayes filter with delayed measurements

I have some straight and curve pieces with numbers, they are used to build tracks (of $5$ lanes) for my cars (figure $1$), I can send commands to the cars using an SDK on the Raspberry (set the speed ...
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best statistical approach to study the time evolution of clustering in a data set

I am using a stochastic method for the clustering of a data set. The number of clusters that this approach returns, can differ in each iteration. On the other hand, I would like to study the evolution ...
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traditional state-space models and LSTMs

I am trying to understand the nature of LSTMs in relation to intuitions from traditional state-space models (e.g., Kalman filtering). The code below aims to simulate a simple univariate linear state-...
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Initialization of the Kalman Filter

I would like to try out different initialization procedures of the Kalman Filter in order to see if it effects the estimation paths of the state variables. One way of initializing is to use the first ...
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unscented kalman filter for non-linear state-space

I intend to use unscented kalman filter to estimate a non-linear state -space problem. latent factor $X_t$ in the formulation has usual VAR(1) specification $$X_t = \phi X_{t-1} +\epsilon_t$$ ...
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Negative variances in Kalman smoother (FFBS)

I have implemented the forward-filtering-backwards-sampling (ffbs) algorithm. It consists of kalman filtering forward in time (to obtain mean and sigma). Then it uses these values and the Kalman ...
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Is a Kalman Filter applicable for irregular, infrequent measurement?

I have taken on a project previously approached by someone else, looking at sensor data. Each sensor produces about three days of data (sampling about once a second), and each day a calibration is ...
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51 views

MLE derivation of the Recursive Least Squares estimator

I think I'm able to derive the RLS estimate using simple properties of the likelihood/score function, assuming standard normal errors. If the model is $$Y_t = X_t\beta + W_t$$ then the likelihood ...
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567 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 does one approximate $\mu$ and $\sigma$ in an arithmetic Brownian motion using a Kalman filter?

My concern arises from the fact that in the following system: $x_k = (\mu, \sigma)^T = x_{k-1}$ $Y_k = Y_{k-1} + \mu + \sigma Z_k \quad Z_k \sim N(0,1)$ that I cannot separate the states I want to ...
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811 views

Kalman Filter prediction using different time step

Typically Kalman Filter or any other time series forecasting methods use a single step prediction - update step. For eg: Let us say I have sensor data collected at every 1ms. Let z denote ...
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Including features in a Dirichlet model with Markov dynamics

I have a fantasizing about a model here, so please keep in mind that this is not even half-baked: I have categorical time series data $y_t\sim\text{Cat}(y\ |\ \lambda_t)$ with a hidden variable $\...
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kalman filter in R when restoring missing values

I did not know, where it would be more correct to ask my question, on CrossValidate or on stakoverflow, but decided here. If I'm wrong, just let me know, I'll delete the post and create it on a ...
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439 views

EM Algorithm seems to work, but Q is not monotonic. Possible reasons?

I have implemented Expectation maximization to fit some of the parameters of a linear Gaussian state space model using Kalman filtering / smoothing. The model is: $x(t) = Ax(t - 1) + w(t); w(t) \sim ...
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Evaluation of Jacobian for Extended Kalman Filter

For the non-additive noise case, \begin{equation} x_k = f(x_{k-1}, u_{k-1}, \xi_{k-1}) \\ y_k = h(x_k, \nu_k) \end{equation} the EKF takes into account the jacobian wrt to the noise terms $ L_{k-1} =...
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Search for tracking techniques

I have an image with scatter points. Check the following figures. We can see a line and a sin function in the images, which are corrupted by noises. The tracks of the straight line and the sin ...
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ARIMA in state space and Kalman filter for predicted values [closed]

Given the coefficients of an arima Model arimaM <- arima(y, order = c(1,0,2), transform.pars = FALSE, fixed = c(0.5,2,1.5,NA)) how can I compute the one step ...
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105 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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410 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 ...
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prior for initial values of Kalman Filter

I'm studying Carter and Kohn's (1994) implementation of the Gibbs sampler for Bayesian analysis of state space models. In their paper, they assume the starting value, call it $\beta_0$, of the state ...
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Kalman filter Welch and Bishop

I am trying to understand Kalman filter from a highly recommended pdf by Welch and Bishop https://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf. I am highly confused with one terminilogy x_k. There ...
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How to use Kalman filter in regression?

I read that Kalman filter can be applied to perform regression with a dynamic beta, calculated on the fly. Can someone please break this down for me, with some simple example of single-variable ...
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ARIMA and SARIMA state space form

I need to write down a program that place ARIMA(p,d,q) and SARIMA models in state space form, however I cannot figure out the composition of the system matrices. In the book of Koopman (pag. 54) the ...
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Estimate standard deviation of random-walk using Kalman filter

I'm new to Kalman filters so this might be a stupid question. I created a Kalman filter that takes in time series observations and estimates the mean of that time series. This is simply modeling a ...
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Number of the samples for an Ensemble Kalman Filter EnKF

Can someone lead me to some references related to how to choose the samples number for the ensemble Kalman filter EnKF.
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The difference between systems with and without direct feedthrough

Generally, in nonlinear state estimation the state space model is defined by the following pair of difference equations in discrete-time: \begin{equation} \begin{aligned} x_k & = f(x_{k-1},u_{k-1}...
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Can Kalman Filtering be done hierarchically - estimated from multiple time series with the same parameters?

I have a large number of of noisy time series recordings (trials), for which I wish to estimate the state transition model underlying them using the Kalman filter. The process generating the time ...
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DLM implementation of the mean reverting model

I am trying to use DLM package in R to estimate a state space repersentation of the term structure model, where observation and state equation are as follows $y(t )= F* x_t +e_t$ $x_t- \mu = G* (x_{...