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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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Covariance matrix for a 2D state vector

I'm performing Optimal Interpolation (which in fact is a simplified Kalman filter with constant $\mathbf{K}$). My state variable is a 2D concentration field with a size of 370 x 400 on which I try to ...
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Custom Space State model using DLM in R

DLM package in R can model linear space state models of the form: I have a different category of equation which is also a linear polynomial equation of order 1 with constant coefficients. I would ...
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Parameters estimation using Kalman filter

I have a model like this: $$P_t=\alpha_t+gP_{t-1}+u_t$$ $$\alpha_t=\alpha_{t-1}+d+n_t$$ Where {$u_t$} and {$n_t$} are normal, iid with 0 mean but unknown variance. I want to estimate the parameters $...
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Updating the kalman filter and RTS with multiple measurements at each time step

I am currently working with kalman filter in area of target tracking. The sensor we are using gets me multiple measurements at each time step. Number of measurements is not fixed, meaning that in one ...
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31 views

Noise reduction with known noise distribution

I have a time signal with a known noise distribution parameters (gaussian, sd is known). I would like to estimate the true value statistically and in the best case obtain a confidence interval. As I ...
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Kalman filter-ish model, is this identifiable?

Time series of observations $y_t$. The proposed model is that there's unobservable scalar series $x_t$: $$x_t=\phi x_{t-1}+B_tu_t+w_t$$ where $u_t$ - vector predictirs and $w_t$ - noise. Then there's ...
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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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Time varying representation of Okun's law

I've estimated a dynamic linear model to capture time varying parameters in an Okun's law type of model: I set the starting values for the state vector all equal to zero and estimate the system ...
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Auto.arima differencing order test for the model with external regressors

Suppose that we have a regression model with ARIMA errors. We need to determine the appropriate level of differencing, so that the errors from the regression are actually stationary. In Hyndman's <...
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31 views

Observation Operator - Data Assimilation

In data assimilation, one assumes the existence of a observation operator $\mathcal{H}$ that maps the model-state vector $\bf{x_b}$ to $ \bf{y_b}$ (the model-equivalent of the observations $\bf{y_o}$) ...
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Fixed-Delay Kalman smoother with/without augmented measurements

There are several algorithms regarding fixed-lag Kalman smoothing. In most cases, an augmented state vector is defined in which the elements are the current and delays of the original state vector. ...
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38 views

Is a Kalman filter ever the optimal way to estimate a dynamic value given a full history of measurements?

I'm trying to get some intuition for Kalman filtering, and I conceived this toy example: Say that I have a sensor that tracks a moving 1-dimensional target. Say that the measurements from the sensor ...
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ARMA process forecasts and maximum likelihood parameters

I have some trouble understanding the forecasting/inference process of ARMA models. From Hamilton (which I am reading now), we can obtain forecasts at $Y$ from any linear process with r.v. values $X$...
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Trouble replicating the experiments in “On-line Novelty Detection Using the Kalman Filter and Extreme Value Theory”

I'm trying to replicate the online novelty detection algorithm from "On-line Novelty Detection Using the Kalman Filter and Extreme Value Theory" by Hyoung-joo Lee and Stephen J. Roberts. In the first ...
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Math questions in Kalman filter equation derivation

I am interested in data analysis. While my working data (actually it's shopping mall's daily sale) is accumlating, I wish to find some statistical laws underlying business phenomena. I left school for ...
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126 views

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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47 views

(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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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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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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60 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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69 views

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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44 views

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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118 views

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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36 views

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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Chi^2 Test: Alpha and it's Relation to Sigma

I'm using an Extended Kalman Filter and the $Chi^2$ test to test a part of it's residual during the update. Here I want to determine if the residual lays within 3 Sigma and reject outliers that are ...
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How to approach SSM models for time series forecasting in general?

I have worked on SSM model using KFAS package (https://cran.r-project.org/web/packages/KFAS/KFAS.pdf) in R. Package suggests me to use one of the Box_Jenkins method to implement SSM. So we convert ...
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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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Interpreting the standard deviation in this setting

I am having trouble relating the estimate of the standard deviation to the precision of the estimate of the $y$-variable. For example: $$\tilde{y}_t=a_{y,1}\,\tilde{y}_{t-1}+a_{y,2}\,\tilde{y}_{t-2}+\...
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Intuition behind definition of one-step-ahead prediction errors

So in this note, https://www.bankofengland.co.uk/-/media/boe/files/archive/discussion-paper/a-note-on-the-estimation-of-grach-m-models-using-the-kalman-filter the author defines the discrete system ...
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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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27 views

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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108 views

Implementation of kalman filter with inner ARIMA non seasonal model

I am trying to write an application which impute some missing values on one time series signal. I have done it similarly in R using ImputeTS package but now need to do it similarly in Java. I just ...
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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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121 views

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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168 views

Transaction and Observation matrices in Kalman Filtering for univariate data

I am working on Kalman Filtering and I have questions about observation and transition matrices. So, I have a variable and I want to clean the noise from my variable by using Kalman Filtering. What ...
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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 ...