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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1answer
23 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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24 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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13 views
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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14 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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10 views
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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36 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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15 views
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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1answer
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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33 views
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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69 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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19 views
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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1answer
75 views
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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1answer
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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1answer
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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24 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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55 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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18 views
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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1answer
49 views
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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79 views
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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15 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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1answer
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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25 views
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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1answer
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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2answers
42 views
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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1answer
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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1answer
89 views
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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1answer
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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1answer
31 views
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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53 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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1answer
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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49 views
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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75 views
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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54 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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1answer
52 views
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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27 views
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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57 views
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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1answer
73 views
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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46 views
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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22 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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199 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-...
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96 views
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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44 views
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$$
...
3
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1answer
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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137 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 ...
