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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622 views

Estimation of ARMA: state space vs. alternatives

I am interested in estimation of ARMA models. I understand that a popular approach is to write the model down in the state space form and then maximize the likelihood of the model using some ...
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1answer
4k views

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, ...
5
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1answer
92 views

Is there something analogous to a Kalman filter for estimating continuous variables that are supported over bounded intervals?

Suppose that I have a robot which is somewhere in a $100 \times 100$ arena. A Kalman filter could be used to estimate its position from noisy measurements. The estimate produced from the Kalman ...
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3k views

Possible causes for the state noise variance to become negative in a Kalman Filter?

I am having some trouble debugging an application of a linear discreet Kalman Filter. From time to time, I find that there are diagonal elements of the covariance matrix that become negative. This is ...
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400 views

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-...
4
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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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290 views

Relation between Wiener and Kalman filtering

What is the relationship from an historical point of view between Kalman and Wiener filtering? Can the first be logically seen a consequence of the latter?
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447 views

What state-space representation of VARMA is commonly used for fitting

What state-space representation of VARMA is commonly used for fitting? Is Kalman filter + MLE approach used for fitting VARMA model as a common practice? Does the choice of which state-space ...
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268 views

Better prediction models with polling data?

I've been working on a project on measuring polls' accuracy in complex contexts (more than two candidates) where there are a small number of inaccurate polling data points. I thought it would be the ...
3
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107 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 ...
3
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218 views

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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241 views

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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476 views

Implementing ARMA Log Likelihood with the Kalman Filter Algorithm

A popular algorithm to determine the (complex) log likelihood function of an ARMA(p,q) process involves generating it through the use of a state-space model and the Kalman Filter. I started reading ...
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185 views

How to add stochastic drift in dynamic linear model?

As I'm not able to comment (yet), my question follows the one raised by @mzuba here I would like to use the DLM R package to model the local linear trend model, which unlike mzuba specified, has a ...
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222 views

Unscented Transform of non-Gaussian data

I'm interested in the unscented transformation, but have a slightly unusual problem. The typical unscented transformation has several parameters which are optionally tuned by the researcher, but, let'...
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1k views

Kalman Filter to correct model simulation bias

I am working with a large scale deterministic model, which attempts to simulate CO2 emissions in different regions. When compared to historic data, the model output suffers from systematic biases. ...
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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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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
145 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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50 views

Continuous-time Kalman filter with no observation/measurement noise

The continuous-time (linear) state space model can be written \begin{align*} \text{d}\mathbf{x}_t &= \mathbf{F} \,\mathbf{x}_t \, \text{d}t + \mathbf{G} \,\text{d} \boldsymbol{\...
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26 views

State space estimation with state dependent state variance

I am estimating a state space of the following form $$Y_t= A X_t + \epsilon_t$$ $$X_{t+1} = B X_{t} + \sigma \sqrt{( a-X_t)(X_t-b)} \eta_t$$ Considering the variance of the state error is state ...
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78 views

What is the difference between using a Kalman filter and just recursively applying the Bayes rule as new data comes in?

I'm having a hard time wrapping my head around the Kalman filter: Is it just Bayes rule applied over and over with each new measurement? Or is there more to it?
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26 views

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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78 views

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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220 views

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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33 views

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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251 views

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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54 views

Why assume controls are independent of state estimation in Kalman Filter?

Here's the classic graphical model depiction of a Kalman Filter (or any form of Bayes filter for that matter). Why do we assume that the controls are independent of the previous state estimation? For ...
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1answer
468 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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245 views

Parameters estimation by MLE and Kalman filter

I am trying to estimate the parameters of a discrete nonlinear state space model using MLE and kalman filter: \begin{equation} \begin{aligned} x_k & = f(x_{k-1},\theta)+q_{k-1}\\ y_k & = h(...
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352 views

How to ensure covariance matrix is positive semi definite in linear dynamical model learning?

I am trying to learn a linear dynamical model for a data using expectation-maximization algorithm. The model is defined as follows: $$x_0 \sim \mathcal{N}(\mu_0 ,\Sigma_0)$$ $$ x_{t+1} = Fx_t + w_t, \...
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965 views

Square root algorithm (Kalman Filter)

I am interested in implementing a Kalman Filtering and smoothing procedure in R without relaying on existing (and excellent) packages such as dlm. Hereby I run (not too surprisingly) into numerical ...
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60 views

Does it make sense to think about steady state forecast error for Kalman filter with time-varying parameters?

The environment: We have a state equation: $$ \xi_t =F\xi_{t-1} + v_t $$ and a measurement equation $$ y_t = H\xi_t + w_t $$ with $$ E\Bigg[\begin{pmatrix}v_t\\w_t\end{pmatrix}\begin{pmatrix}v_t'&...
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1answer
155 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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92 views

adaptive Kalman filtering

I am learning about Kalman filters/dynamic linear models/state-space models and I am interested in whether the following scheme is possible, in which I try to estimate distribution parameters ...
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696 views

Multi-step ahead Kalman Filter

Suppose my state-space model is: $$x_{t+k} = Ax_{t}+\eta$$ $$z_{t} = Bx_{t}+\epsilon$$ ,where $\eta\sim N(0,\Sigma)$ and $\epsilon\sim N(0,\Phi)$. Since I want a multi-step model instead of the usual ...
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142 views

How to calculate advertising copy wearout using kalman filter?

I have to calculate half life of an advertisement using Kalman filter in R. The paper 'Estimating the Half-life of Advertisements' (Naik, 1999[1]) provides the base but am unable to understand how ...
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714 views

Kalman filter observation model

I am using the following equations for Kalman filter. The state vector is $x$ and observations are $z$. Both $x$ and $z$ are multivariate vectors of same length. Forecast model: $x_{t+1} = 0.963 x_t +...
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92 views

Kalman-Bucy filter: how does a prior mean change, alter the posterior?

I have a question on Kalman-Bucy filter: The prior distribution is $g \sim N(0,σ_g^2 )$, signal is $ds=(μ+g_t )dt+σdZ_t$, posterior distribution becomes $g_t \sim N((\hat{g_t},\hatσ_t^2)$. Here,$σ_g,...
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74 views

Time variation in coefficients

Given $x_t, y_t$ ($t=1,\ldots,240$), I want to estimate $y_t = \alpha_t + \beta_t x_t$ and test $H_0: \alpha_1=\ldots=\alpha_T=0$. It is crucial to allow for time variation in the regression ...
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2k views

Linear regression with time-varying parameters

I am attempting to estimate the following model in Stata (the only statistics package I'm familiar with): $$ Y_{t}=\alpha_{t} + \beta_{t}X_{t}+\varepsilon_{t} $$ where $X$ and $Y$ are series and $\...
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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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28 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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1answer
69 views

Parameter estimation in Dynamic Linear Models

I am currently developing a DLM of the following form $$\underset{k \times 1} {y_t} = \underset{k \times n}A \underset{n \times 1}{\theta_t} + \epsilon_t$$ $$\theta_t = \mu + \underset{n \times n}B\...
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98 views

Multilevel dynamic linear models in R

I am interested in fitting a multilevel bayesian structural time series with a hierarchical structure of the dynamic regression coefficients. The reason I want to do this is is that I have a number of ...
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1answer
37 views

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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0answers
49 views

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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0answers
35 views

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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1answer
73 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 ...