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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Using Bayesian statistics in time series forecasting
I would like to forecast demand count time series of taxi fleets at different locations on the map at different points in time. I.e. multivariate demand Time series forecasting.
Given hierarchinal ...
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How can mahalanobis and chi2-test be used to determine of an observation is acceptable?
Assume that you have a model
$$\dot x = Ax + Bu$$
$$y = Cx$$
And this model is SISO. Single input and single output. You got the mission to determine of an observation is acceptable for the kalman ...
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Assessing probability that one set of measurements extends the other with Kalman smoother
I have two sets of N-dimensional measurements following each other with a certain time gap in between. Let's name those sets $A$ and $B$, respectively. All observations have constant Gaussian white ...
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Exact diffuse initialization of the Kalman Filter: what does the design matrix look like?
I am using Python (statsmodels) to create a dynamic factor model on which I apply the Kalman filter. Thanks to earlier questions on this forum, I landed upon using exact diffuse initialization. My ...
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Kalman Filter Terminology: Prediction vs Estimation
The linear dynamical system model underlying the Kalman filter technique involves a random process $(x_{0}, v_{1}, w_{1}, x_{1}, z_{1}, v_{2}, w_{2}, x_{2}, z_{2}, \ldots)$ where $x_{k}$ represents ...
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Updating Dynamic Factor Model within a time-period
I have the following question.
Assume that we have the standard Dynamic Factor Model:
$$
X_{i,q} = \beta_i F_q + \epsilon_{i,q}, \qquad \epsilon_{i,q} \sim \mathcal{N}(0, \sigma_i^2),
$$
and
$$
F_q = \...
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Some Problems in Auxiliary Particle Filter
recently I am studying PF. And I am stuck in APF for a few days, though I derived many times. Here is my question:
I followed the framework of this paper. The APF is defined in Algorithm 1:
The ...
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Kalman filter) Observation matrix of measurement equation and what is a good signal?
I am trying to use a Kalman filter, but my data are somewhat deviating from the assumptions.
The noises in my measurement equation are not normally distributed. First of all, they are not zero-mean. ...
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How to handle exact diffuse initialization of a Kalman filter?
This is partially a coding question so I hope I'm on the right platform for this.
I am fitting a dynamic factor model using the state space framework.
I don't know the initial distribution of the ...
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Inferences with Filtered and Smoothed state estimates from a tracking problem
I am working on a synthetic tracking problem in a two-dimensional space, where the start and end positions are known and noisy measurements of the state variables are given at discrete time steps. As ...
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How to treat non-normal measurement noise in a Kalman filter? Plus, how to treat non-zero mean of noise in Kalman filter?
Typically in textbooks, it is assumed that measurement noise $\nu_{t}$ is normally distributed. Suppose that $S_{t}$ is a signal. Then a measurement equation is
$$
S_{t} = x_{t} +\nu_{t} $$
where $$ \...
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Why do popular ML and statistical packages simply ignore classical estimation and detection algorithms for statistical signal processing? [closed]
For those who had a hard time to study and understand classical estimation and detection algorithms, and unfortunately realized that these algorithms are simply ignored by many packages that have the ...
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Derivation of posterior distribution of Kalman filter
I am following this article on Understanding the Kalman filter by Meinhold, and I can understand all the derivation up until equation (4.4). However, I cannot understand how they arrive at the ...
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Proving consistent/inconsistency of a fusion of KF estimates
I have a distributed fusion scenario with a single target where two sensor nodes $i,j$ estimate the true state $\mathbf{x}$ using a local Kalman filter. The (linear, Gaussian) measurement errors of ...
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Smoothing of GPS tracks - remove noise and stop-go clusters
I know there are several posts about this, but I could not find exactly what I need. I have GPS track data (from an underwater vehicle) for short intervals of 1 second (time-stamps on data). The data ...
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Instantaneous propagation of process covariance matrix modification's effect on state
I am trying to build a zero-delay kalman filter which udates its process noise covariance matrix $Q_k$ depending on the value of the residues $z_k - H\cdot x_k$. My problem is, I have adopted a 1D ...
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Inferring a random walk from noisy "images"
I'm interested in the following inference / filtering problem in a hidden Markov model setting. Suppose we have a simple random walk $x_t\in\mathbb{Z}$ and observations are "images" ...
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What is the null hypothesis and p value in the lagged autocorrelation of innovations in Kaman filter
I am using statsmodels.tsa.stattools.acf to calculate the lagged autocorrelation of innovations in Kalman filter with alpha=0.05....
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Negative value of likelihood function in Kalman filter
I use kalman filter algorithm, where I minimize the value of likelihood function. But after some iteration I got negative value of likelihood function. Is that a problem?
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Extended Kalman Filter for estimating angle using tan measurement function and two measurements
I'm attempting to implement an extended Kalman filter (EKF) to estimate an angle $\alpha$ given measurements of two scalars $x$ and $y$ where the measurement function is $\alpha=atan(\frac{y}{x})$. ...
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How do I calculate the standard error of Kalman Filter parameter estimates?
I am trying to implement the Schwartz-Smith (2000) commodity pricing model from the paper Short-term variations and long-term dynamics in commodity prices
The model is estimated using the Kalman ...
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Deriving Kalman Filter equations
I am trying to model a Kalman Filter for an IMU (inertial measurement unit) with the method described by Zhou (2004)
and Filippeschi (2017, pp.11-12).
In this method, the state vector is:
$$
X = \...
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Comparison of two models with different number of parameters
I want to compare two models, which has different number of parameters. The first model is Arbitrage free Nelson-Siegel model, which has the following equation:
$y_{t}(\tau )=X_{1,t}+X_{2,t}(\frac{1-e^...
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Do we need to propagate state covariance matrix 'P' during missing observations in the Extended Kalman Filter?
Basically as the title says: in a scenario, we have missing observations where the entire state vector is unknown for consecutive time steps. Do we just run through the prediction section of the ...
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Method of evaluating the feature map of a polynomial kernel feature mapping
I'm attempting to implement an adaptive kernel Kalman filter following this paper https://arxiv.org/abs/2203.08300, but I'm struggling to find a method of evaluating the feature mapping for a ...
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Estimating a model with two unobserved components that has measurement equation, signal equations and three transition equations
I have a dataset containing CPI inflation, 10 year breakeven rate, output gap, relative import price inflation, 2-year breakeven rate, 2-year firm inflation expectations, 2 year household inflation ...
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State Space with Space Lags in R (dlm, MARSS or anything else)
** Edited to reflect on some first comments **
I am trying to estimate a state space model which does a kind of disaggregation. In particular, I am interested in estimating high-frequency unobserved ...
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How does maximum likelihood estimation from the Kalman filter work?
My understanding is
Step 1: You would run through the Kalman filter equations
with initial parameter values.
Step 2: After you run through the Kalman filter equations,
you will have innovations ...
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Can you create Kalman filter (or a recurssive state estimator) with Beta and Binomial distributions?
I have to infer the probability of a system failing from observations. Since probabilities are bounded between 0 and 1, they are sometimes modeled using Beta distribution. While the traditional Kalman ...
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Estimate the direction of the signal using RSSI indicator
I have an antenna that can read RSSI indicator of the signal emitted by the target. The antenna can pan 360. All hooked up to an Arduino. Thus at time t I can read ...
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The mean of Gaussian distribution subject to another Gaussian distribution, how to derive it？
I got a fomula
$$N(y;cx,R)N(x;\bar{x},\Sigma)=N(y;c\bar{x},S)N(x;g,F)$$
where:
\begin{align}S&=c\Sigma c^T+R\\g &=\bar{x}+\Sigma c^Ts^{-1}(y-c\bar{x})\\F &= \Sigma - \Sigma c^T s^{-1} c \...
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How can I use Kalman Filter formulation for inferring the probability of system failure?
I want to use the Kalman filter to sense the state of a machine. The machine could be either working or damaged. I'm trying to infer the probability of the machine working, given observations from ...
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Kalman Filter Propagation using two previous time steps
Kalman Filters propagate using a single previous state estimate $\hat{x}$ with covariance $P$. Is there a formal way of propagating using two previous state estimates with different associated ...
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What model to use to predict in real time the mouvement of an animal?
For an experiment I’m running in my lab, I need to predict the movements of an animal (from the past positions), More precisely, head movements.
It will help me to detect when an animal is about to ...
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Kalman Filter: Question regarding Measurment Noise Covariance Matrix
I am new to the topic of state estimation and, therefore, I have the following question:
If I want to define the Measurment Noise Covariance Matrix ("R-Matrix") for a Kalman Filter from real ...
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Robustness analysis using Monte Carlo
Using a mathematical model, I have estimated some system parameters using Kalman filter. Now I have to verify whether the proposed system is robust against uncertainties. I have estimated data and ...
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What is the difference between these two observational update models in an information filter?
The Information Filter is defined as the mathematical inverse of the Kalman filter. As defined in this Wikipedia article, the observation update of the Information Matrix is defined as $$y_{k|k} = y_{...
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Parameter estimation of state-space models with hidden variables
I have a time-series analysis problem, that I am having trouble finding a suitable regression technique for.
I have a coupled linear three dimensional system
\begin{align*}
X_{t} & =\left(1+J\...
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Is there some standard way to diagnose a structural time series model (also called simple unobserved components model)?
I am dealing with a structural time series model (also called a simple unobserved components model), and I wonder if there is some standard way to diagnose this sort of models. In most reference books ...
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Is there a probabilistic or bayesian interpretation of the kalman filter gain?
The Kalman filter makes sense to me as the repeated application of Bayes' theorem - if you correctly propagate the gaussian prior at each step and then update on new observation, you get a gaussian ...
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How to apply Kalman filter to improve my model?
I'm trying to build a model that predicts humidity W
This is the data I'm working with :
...
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Kalman forecast of AR(1)
I'm trying to work out the details of the proof of the following statement:
Suppose $\xi_t = \rho \xi_{t-1} + \epsilon_t$ is an AR(1) process.
Using Kalman filter, one can prove that $\mathbb{E}_t\{\...
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What is the scope of application of Kalman filter?
Recently I learned some basics about Kalman Filter 1D
As I know, Kalman Filter is useful in Telecommunication and GPS positioning.
My estimation goal is to measure the reliability of the Circuit using ...
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Approximating a 1-d Kalman Filter with non-Gaussian Observation Noise
I'm looking for a Bayesian filter where observations are generated according to $s_t = \gamma s_{t-1} + w_p$ and $w_p \sim Normal(0, \sigma_p^2)$. Both $\gamma$ and the variance of the process noise $\...
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Is the state covariance matrix (or estimate uncertainty) in a Kalman filter in 1D equal to the variance of the current normal distribution?
I'm trying to use a Kalman-filter for some kind of anomaly detection. But I think that maybe I have misunderstood something fundamental about the filter. I'm following this "guide".
I'm ...
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(Co)variance interpretation in Kalman filter
Let's say I have a device which uses Kalman filter to fuse sensor data and produce an optimal estimate of the system parameters. As it should, it also estimates parameter covariance matrices at each ...
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Kalman filter: updating the state-transition model
I am currently reading lots of material on the Kalman filter (in order to do some experiments), and there is something that I don't get, and I can't get a clear understanding.
I'll stick to the ...
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Gaussian Process Regression vs Kalman Filter (for time series)?
I'm curious about the similarities and differences between Kalman Filter (KF) and Gaussian Process Regression (GPR). From various sources, I've pieced together that the KF is analogous to a Hidden ...
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Where is a good place to start with (hidden) state-space models?
I'm interested in (hidden) state-space models. My language here might be poorly articulated as I'm quite new to this area of math. The topic of Kalman filters has come "across my desk" a ...
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Unknown parameter - augmenting state equation (Kalman filter)
First, we have a state space model with mean reversion and $\mu$ is unknown
$y(t )= F* x_t +e_t$
$x_t- \mu = G* (x_{t-1}-\mu) +n_t$
There is a option to add unknown parameters to the state vector and ...
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