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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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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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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32 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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24 views

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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21 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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31 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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92 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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38 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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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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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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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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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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102 views

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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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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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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49 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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130 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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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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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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166 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?
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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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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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33 views

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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37 views

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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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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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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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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182 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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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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381 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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74 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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572 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 ...