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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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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29 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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1answer
27 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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41 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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1answer
64 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
65 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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36 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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19 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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51 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 ...
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1answer
48 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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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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14 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
42 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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19 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
66 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
36 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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36 views
Kalman-filter different sampling rates
I'm trying to implement a simple Kalman filer on GPS and Linear Acceleration in Python. I'm having trouble because I get acceleration data 100 times a second, but GPS data just once a second (so 100 ...
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1answer
35 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
81 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
106 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
30 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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43 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
31 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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30 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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59 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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41 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
48 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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37 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
68 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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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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26 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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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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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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83 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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82 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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42 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$$
...
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1answer
94 views
Negative variances in Kalman smoother (FFBS)
I have implemented the forward-filtering-backwards-sampling (ffbs) algorithm.
It consists of kalman filtering forward in time (to obtain mean and sigma). Then it uses these values and the Kalman ...
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137 views
ARIMAX and Kalman filter to impute missing data
Following this post How to use auto.arima to impute missing values, and the really comprehensive answers there:
Is it possible to implement this gap filling method with covariates, e.g. using climatic ...
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151 views
Transaction and Observation matrices in Kalman Filtering for univariate data
I am working on Kalman Filtering and I have questions about observation and transition matrices. So, I have a variable and I want to clean the noise from my variable by using Kalman Filtering. What ...
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Is a Kalman Filter applicable for irregular, infrequent measurement?
I have taken on a project previously approached by someone else, looking at sensor data. Each sensor produces about three days of data (sampling about once a second), and each day a calibration is ...
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55 views
Why are Kalman Filters nontrivial?
I'm starting to learn about Kalman Filters and its variants but am having a hard time understanding why they're interesting or nontrivial. As far as I understand, giving some observations $O_{0:t}$ ...
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1answer
37 views
MLE derivation of the Recursive Least Squares estimator
I think I'm able to derive the RLS estimate using simple properties of the likelihood/score function, assuming standard normal errors. If the model is $$Y_t = X_t\beta + W_t$$
then the likelihood ...
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1answer
220 views
Bayesian Filtering for linear but non-Gaussian estimation problems
It seems that most optimal estimation literature is divided into either linear Gaussian problems, for which you use Kalman Filter, or non linear and non Gaussian problems for which you use EKF, UKF or ...
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65 views
Using xreg function in R while using kalman filter [closed]
I am using a kalman filter on my data to both impute missing values and predict 20 samples ahead. However, I would like to use a ARIMAX model instead of Arima model using XREG as well...
this is my ...
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52 views
How does one approximate $\mu$ and $\sigma$ in an arithmetic Brownian motion using a Kalman filter?
My concern arises from the fact that in the following system:
$x_k = (\mu, \sigma)^T = x_{k-1}$
$Y_k = Y_{k-1} + \mu + \sigma Z_k \quad Z_k \sim N(0,1)$
that I cannot separate the states I want to ...
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1answer
249 views
Kalman Filter prediction using different time step
Typically Kalman Filter or any other time series forecasting methods use a single step prediction - update step.
For eg: Let us say I have sensor data collected at every 1ms.
Let z denote ...
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Including features in a Dirichlet model with Markov dynamics
I have a fantasizing about a model here, so please keep in mind that this is not even half-baked:
I have categorical time series data $y_t\sim\text{Cat}(y\ |\ \lambda_t)$ with a hidden variable $\...
