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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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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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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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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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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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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24 views

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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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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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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16 views

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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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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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
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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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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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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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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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1answer
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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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32 views

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
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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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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1answer
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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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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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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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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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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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69 views

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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36 views

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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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
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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1answer
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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
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Two time-varying coefficients in Kalman filter with DLM package [closed]

I am trying to estimate a model that has two time varying coefficients in R using the "DLM" package. My measurement equation would be = Yt = F1tx1t + F2tx2t + v The state equations are: F1t = F1t-...
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2answers
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When forecasting, is it better to remove the outliers or just to transform them?

I am forecasting the number of logins. I have a dataset with the number of logins for each hour. First, I use LOF (local outlier factor) to find the outliers and then I remove them. Second, I use ...
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1answer
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How to Derive SISO Kalman Filter Update Equation Using only Probability Density Functions

I'm trying to prove to myself that a single state/single measurement kalman update can be derived using bayes theorem (as proof of concept for a more complicated task) only using the PDF. I am able ...
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1answer
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Do we need to stationarize a time series signal when using Kalman filter?

I am working on forecasting the number of logins. I know that before using ARIMA, it is important to remove trend and seasonality. But in the case of Kalman filter, I am not sure. After all it is a ...
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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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Books on systematic risk(beta)

I am looking for some reference material on beta (systematic risk in market model regression). I want to calculate time varying beta for stocks using kalman filter. First I would like to go through ...
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3answers
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Kalman filter parameter estimation

From what I've known about Kalman filter, it requires all the parameters of the underlying state space model. Say the state space model is: $$\xi_{t+1} = F\xi_t + v_{t+1}$$ $$y_t = H\xi_t + w_{t}$$ ...
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How do I tell if the sensors that feed a Kalman filter has diverged?

I have a time varying variable $x$ that I want to estimate. I have two sensors A and B that measure $x$. I feed their measurements to a Kalman filter. Sometimes, one of the sensors degrades for a ...
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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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32 views

Kalman filter on stock sentiment time series

I was wondering if & how I can use a Kalman filter on my dataset which contains closing prices of stocks + sentiment scores of tweets about that stock for each day in a timeframe of 1 month. e.g....
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77 views

DLM regression with parameter restriction

Good afternoon, I am attempting to fit a state space regression model of the form: $Y_{t} = \beta_{1}Y_{t-1} + (1-\beta_{1})[i^* + \beta_{2}X_{t}] + \epsilon_{1,t}$ $i^* = i^*_{t-1} + \epsilon_{2,t}...
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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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Smoothing detected vehicle positions from camera given my own vehicle's location, velocity and acceleration

I have a dashcam which detects the position and estimated depth of other vehicles on the road relative to my own. Given my own vehicle's global co-ordinates, I can convert the detected vehicle's ...
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207 views

Kalman Filter with MLE giving bad estimates

I am trying to learn and implement the Kalman filter. Yesterday I successfully implemented a non linear kalman filter of the form: $$ x_t = a(x_{t-1}) + u_t \\ y_t = Gy_{t-1} + v_t $$ $u_t$ and $v_t$...
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53 views

Kalman/HMM for (short) multivariate time series from a sample with missing values

The problem in short: I want to estimate (?) a lag-1 Markovian hidden process for offline multi-variate discrete-time time series with continuous distributions via smoothing, with no dimensionality ...
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26 views

Calculating travel distance from GPS updates

I am learning about Kalman filters but struggle to apply them for the following problem: tracking the distance traveled from GPS data. The GPS provides position updates every second and an estimate of ...
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386 views

Statsmodels Kalman Filter: simple equivalent to pykalman set up (partly answered)

Following some examples on Chad Fulton's blog and in statsmodels' tests, I have tried to come up with an equivalent of a pykalman implementation. The original question was deemed unclear and was ...
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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?