Questions tagged [particle-filter]
Particle filters (or sequential Monte Carlo) is a form of genetic simulation algorithm used for filtering problems in signal analysis and time series analysis.
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How to show that the variance of Sequential Importance Sampling estimates increase with the dimension?
I am trying to understand the Particle Filter and the motivation to use it over the regular Sequential Importance Sampling. As far as I understand until now:
1- We try to estimate the expectation of ...
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Carré du champ operator is a quadratic variation
Let $X_t$ be a real valued Markov process (starting at $x$) with generator $L$. Let $\Gamma(f)$ denote Carré du champ operator i.e. $L(f^2) - 2f \cdot L (f)$. As far as I know under suitable ...
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boostrap particle filter marginal likelihood
I want to calculate the marginal likelihood $p(y|\Theta)$ of the parameters of a Markov state space model with unknown parameters $\Theta$ that I am trying to estimate the marginal likelihood (...
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Soft Question: What background do I need to understand Feynmann Kac Formulae by Pierre Del Moral?
I am attempting to understand Sequential Monte Carlo(SMC) deeply, but with little theoretical background on probability theory and stochastic processes. Usually, the 'statistics' perspective of markov ...
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How to generalize Particle Filters (w.r.t. multiple states)
I'm using particle filters for inference in a hidden markov model with an infinite state-space.
My current state-variable is multidimensional and there are interdependencies between some dimensions. I ...
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Rao-Blackwellising state space for a (marginalised) particle filter
I am starting to look at particle filtering for a problem that I have. In particular, I would like to reduce the dimensionality of the particles.
The model that I have is able to be partitioned. ...
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Slice sampling in Particle Gibbs with Ancestral Sampling
Bear with me as I am not from statistical background. My question is about the implementation of PGAS algorithm as given in Lindsten et. al 2014 concerning sampling in state-space models. The two ...
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Do Particle Filters actually approximate the posterior distribution?
Im reading a tutorial paper about particle filters (Link) in which it is stated that as the number of samples tends to infinity the approximated posterior density given by
$p(x_k|z_{1:k}) \approx \...
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Particle Filtering: Derivation that mean of weights is the marginal likelihood
I see everywhere the following (for the Bootstrap Filter)
$$
p(y_t \mid y_{1:t-1}) \approx \frac{1}{N} \sum_{i=1}^N W(x_{0:t}^i)
$$
where $W(x_{0:t}^i)$ are the normalized weights defined as
$$W(x_{0:...
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Particle Filter for structural credit risk model
Kwon (2012)* proposes a structural credit risk model where the asset value process and the noise are estimated based on the observed equity prices:
$S$ - equity prices
$V$ - value of the assets
$Z$ - ...
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Predictions after SMC
I have a statistical model given by
$$
y_t\sim p(y_t|x_t, \theta)\\
x_t\sim p(x_t|x_{t-1},\theta)\\
\theta\sim p(\theta)
$$
where $y$ is the only observed component. Using a sequential Monte Carlo ...
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Sequential Monte Carlo: Joint Smoothing vs Filtering
\begin{equation}
\begin{split}
p(x_{1:t}|y_{1:t})& = \frac{p(y_t|x_{t})
p(x_{t}|x_{t-1}) p(x_{1:t-1}|y_{1:t-1})}{p(y_t|y_{1:t-1})}
\end{split}
\end{equation}
\begin{align} \label{eq:...
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Computing Monte Carlo Error: Particle Filters
I want to ask a question about the Monte Carlo error of a particle filter.
Assume we have information of our of the process of our true states, $x_t \forall t$ and hence, we generate our data $y_t$. (...
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Discrete Kernel for Sequential Monte Carlo (population monte carlo)
I'm attempting understand, and use, the population Monte Carlo algorithm found here https://arxiv.org/abs/0805.2256 for approximate Bayesian computation. However I think this is a general SMC question,...
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KL divergence in Sequential Monte Carlo
Suppose at step $t$ the particle approximation of SMC in $d$ dimensions is given by $\sum_{k=1}^N w_k\delta(\vec{x}-\vec{x}_k)$, and at the subsequent step, $t+1$ (after using Bayes' law to update ...
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Difference between particle filter (PF) and recurrent neural network (RNN) for time series
Both method are used to estimate time series from data. The question is, when should I use one method or other? Is any advantage to use one instead of the other?
I know that in a PF there is a hidden ...
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Particle Filter and Gaussian Mixture
Let an observation model be given as $f(y_t|x_t)$ - this pdf is assumed to be nontrivial (not normal, not linear). The observation model is assumed to be known.
Despite there is a state evolution ...
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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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Computing mean of filtering and smoothing distributions from a particle filter
Suppose I have a model with latent states $x_1, x_2, \ldots x_T$ and observations $y_1, y_2, \ldots y_T$. I run a sequential monte carlo algorithm to give me the following approximation to $p(x_{1:T} |...
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How does Particle Filters work?
I'm trying to figure out how particle filter works.
Assume that I have selected propability function called $a \sim Gauss(\mu, \sigma)$. We call it proposial (Gaussian) Distribution.
Then we have ...
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Computing Posterior via Particle Filter
I have a question regarding the computation of the posterior using particle filter. I start reporting details from https://cse.sc.edu/~terejanu/files/tutorialMC.pdf
Consider random processes $(Q_t)_{t ...
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Particle Gibbs Sampler For Regime-Switching Nonlinear Gaussian SSM
I'm reading this paper on using a non-linear Gaussian SSM for measuring regime-switching leverage effect using stock market data. I'm using it as jump-off point for an undergraduate paper. My advisor ...
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Particle filter/SMC - dynamic rotation in ICA (independent component analysis)
I struggle with the applicability of the bootstrap particle filter within dynamic rotations in independent component analysis.
To be clear, suppose the following:
$$Y_{t} = R(\delta_{t})\epsilon_{t}$$
...
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Modelling Ball Movement using Delayed Measurements with Known Latency
I am a hobby programmer currently developing an algorithm to combine measurements of a dynamically moving ball position (and velocity) from multiple robots. Each robot measure and calculate the ...
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Inference for Maximum Likelihood Estimator Using Particle Filter
How does one compute standard errors for the MLE when using a particle filter approximation to the likelihood? I know that the estimator is asymptotically normal and that the variance-covariance ...
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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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Stochastic models: freedom in choosing error type?
I have a general question concerning probabilistic models. Imagine that I have a system with a high-dimensional state $x$ with a pdf $p(x)$. The state $x$ is time-dependent, and a propagating ...
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Estimating Gamma PDF parameters from data with negative increments
Say we have collected data, and from a physical perspective we know that the collected data should increase positively with time. However the data looks more like this:
This data shown in the figure ...
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Calculate projectile trajectory from 3d points
I am trying to calculate the trajectory of a moving object (specifically, a thrown object) through a series of video frames. My tracking algorithm can reliably detect ~90% of the object occurrences ...
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What is computational complexity of Monte-Carlo sampling?
Monte-Carlo method is basically used to integrate multivariate functions.
If we use deterministic methods (e.g. Riemann integration), estimation error has an order of $O(m^{-\frac{1}{n}})$, where $m$ -...
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Deriving the particle filter with driving-force/inputs/control-signal
Whenever the particle filter is derived (I used a different condition for $u_t$ as a solution to the nonlinear filtering problem;
$x_{t+1} \mid x_t \sim f_{\theta}(x_{t+1} \mid x_t,u_t) \\
y_{t} \...
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Why the bootstrap filter is inefficient when the prior is vague or the likelihood picky?
I am reading these notes about Sequentially Monte Carlo for state space model.
On page 14 it says that "The SMC algorithm discussed earlier is very inefficient. This is particularly true for vague ...
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Cross Validation and Sequential Monte Carlo (SMC)
Please bear with me, I am not a statistician.
I have a likelihood
$$\Pr (d|\vec{x},\vec{e})$$
where $d$ is a single datum, $\vec{x}$ are the model parameters, and $\vec{e}$ are known experiment ...
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How to compute initial weights of SMC, when you initialize the algorithm with MCMC draws
Let $$y_t | \alpha_t \sim N(0, \exp(\alpha_t)), \\
\alpha_t = \phi_0 + \phi_1\alpha_{t-1} + \sigma \epsilon_t, \\
\text{ where } \epsilon_t \sim N(0,1) \text{ i.i.d.},\\
t=1,2,\cdots,T$$
The unknown ...
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convergence of MSE (mean square error) using Sequential monte carlo
I am using sequential monte carlo method for a regression problem with bayesian estimation . I am trying to find a measure to confirm that my distribution has converged to the actual posterior ...
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Is it a problem to have non homogeneus sampling time in Bayes Filter?
I have a doubt related with Recursive State Estimation using Bayes Filter (actually using an aproximation to that through Particle Filters)
This algorithm is explained in several sources with ...
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Covariance update from Jacobian of transition function
In this paper on particle filtering with gradient descent, the authors sample Xk+1 through gradient descent, then update the covariance matrix P associated with Xk+1 as follows:
Pi+1(k + 1|k + 1) = ...
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A motion model to track a moving object using the condensation algorithm
I have implemented the condensation algorithm in order to track a moving object in video sequences, however the predictive step does not work properly, so the samples moves excessively compared to the ...
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Particle Filtering with Nonlinear Observation Equation of 2 set of variables
I am stuck with this problem in my research.
I am having a State Space Model like the below mentioned one:
State Equation: $\mathbf{d}_k = \mathbf{d}_{k-1} + \mathbf{u}_k + \boldsymbol{\epsilon}_k$ ...
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Setting up a particle filter for a deterministic system with stochastic, time-discrete observations
I have a deterministic process $x(t)>0$ for $0 < t < T$, governed by an ODE for which I want to do parameter inference in a Bayesian sense. The process is hidden but I have $n$ stochastic ...
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Setting the observation likelihood threshold for outlier detection if you know know the percentage of outliers
Let's assume I have a sensor that gives me measurements $z$ and I know that $50\%$ of the measurements I read are outliers (more than 3 standard deviations away from the real measurement distribution)....
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What filter should I use for non gaussian distribution?
I have a process that measureing distance between 10-100mm and I currently measuring at 11-18mm with a fixed distance. I want to improve this measurement by adding a filter.
Here is the distribution ...
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Do I need to know the distribution of the noise before I'm using Monte Carlo Sampling?
I'm going to use Particle Filter, which is a Monte Carlo Sampling.
My simple question is: Do I need to know the distribution of the noise before I'm using Monte Carlo Sampling?
Or can I just use a ...
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Particle Filter for navigation through known map
I have some issues with understanding the Particle Filter for navigation through a known map. So, consider a situation where I want to write a Particle filter to navigate through a maze or a map that ...
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Monte Carlo probability approximation vs Histogram
I am trying to learn the sequential Monte Carlo method (particle filter) in data assimilation. In this method, the aim is to approximate the CDF of the target variable having a random sample of the ...
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Simplying Bayes Theorem expression: SIS particle filter posteriori
In the book Beyond the Kalman Filter: Particle Filters for Tracking Applications on page 39 the weight update equation for the particle filter is derived.
The derivations begins by introducing the ...
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Evaluating Likelihood in Bootstrap Particle Filter
I am currently struggling with an attempt to apply a bootstrap particle filter to a linear, Gaussian state-space model
$$s_t=A\,s_{t-1}+B\,\nu_t\qquad\text{( transition equation )}$$
$$\qquad z_t=C\,...
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Particle filter for diagnosis
I have two annual measurements taken on medical images depicting a lung cancer tumor 's condition. I have likelihood function that taken in the measurement values and estimates malignancy of the tumor....
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Updating a belief using a particle filter
I am using a particle filter to update a belief (the context is the POMCP algorithm found in Silver & Veness, "Monte-Carlo Planning in Large POMDPs"). A belief is represented as a probability mass ...
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How can independence be represented efficiently?
Consider a probability distribution over a high dimensional space. We would like to find an encoding describing the distribution, so that we can approximately compute the expected value of most random ...