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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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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Determine the parameters of a particle filter that best fit observations

I am wondering is there any established framework to optimize the parameter $\lambda$ of a particle filter such that $p(O|\lambda)$ is maximized, where $O$ is the observation sequence. For HMM and ...
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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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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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Why do we want to minimise the variance of our importance weights in SIS with respect to the proposal distribution

Is there a clear and precise explanation of why minimising the variance of the weights in SIS with respect to a proposal ensures that the samples generated from the empirical distribution induced by ...
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What is the point of the re-sampling step in a particle filter?

My general understanding of particle filters is that you represent your state as a collection of discrete particles which you then transform using your state propagation equation. What I don't ...
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Bayesian evidence with Sequential Monte Carlo and an unnormalized likelihood function: a contradiction?

There is a contradiction in my understanding of Sequential Monte Carlo for estimating Bayesian evidence for model comparison: Marginal likelihood (aka normalizing constant, aka Bayesian evidence) ...
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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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ABC, make tolerance threshold $\epsilon$ adaptive

Briefly the Approximate Bayesian Computation instead of using the exact likelihood function $L(\theta;x)$ tries to approximate this function with the use of the observed summary statistics $s(x_{obs})$...
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Sampling from transition model in particle filter

I am reading the foundational paper about Bayesian bootstrap particle filter (Gordon, Salmond, Smith, 1993) and they are solving the following discrete time estimation problem: $x_k\in R^n$ , $$f_k:\...
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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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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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Particle Filter Derivation based on Forward Algorithm

I have been studying the particle filter, sequential monte carlo methods, and sequential importance sampling. I am interested in apply the particle filter equations to the standard forward algorithm: $...
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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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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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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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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 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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Particle filter for likelihood evaluation

I struggle to implement a particle filter to evaluate the likelihood of a textbook example. I got the following process: $x_t = \alpha + \beta x_{t-1}/(1+x_{t-1}^2) + w_t$ where $w_t \sim \mathcal{N}(...
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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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Sample from a distribution and plot in python

I am trying to understand Particle Filter and Importance Sampling Principle from a UniFreiburg Course and this USNA document on particle filters. Simultaneously, I am also trying to write a document ...
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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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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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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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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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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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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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Calculating the observation density

In the context of link. For a state space model $$ x_{k+1} = f(x_{k}, u_k, w_k) $$ $$ y_k = H x_k + v_k $$ where the measurement function is assumed linear and Gaussian and the state transition is ...
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Particle filter: Evaluating Optimal importance density

NOTE I posted this in the math stack exchange but I realized this may be the more appropriate place, old post here. I'm not sure if I should delete one of them so I just linked them in both? I am ...
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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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Bootstrap Particle Filter (Gordon, Salmond, Smith, 2003) - Importance Weights

So, my endeavor to apply the is just for my own edificationI am currently struggling with an attempt to apply a bootstrap particle filter (Gordon, Salmond, Smith, 2003) to a linear, Gaussian state-...
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extension of sequential probability ratio tests to particle filters?

I've been wondering if there are extensions of the sequential probability ratio test to account for particle filters. I ask because, in my research, I'm working with distributions that cannot be ...
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Finite grid approximation to the Bayesian filtering problem

I need some hints for solving Ecercise 4.4 from Bayesian Filtering & Smoothing by Simo Särkkä: Select a finite interval in the state space, say, $x \in [-10, 10]$ and discretize it evenly to N ...
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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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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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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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Posterior as prior for correlated parameters [closed]

I want to use the posterior distribution of the model parameters $\theta$ given data in the time frame $[0,t]$ days, $P(\theta|y_{0:t})$; as a prior for the parameters in the time frame $[t+1, t+n]$ ...
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sequential Monte Carlo sampler, why the extended space and backward kernel?

Hello cross validated, I am currently studying sequential Monte Carlo samplers. My current understanding is as follows: We are interested in the marginal distribution of some sequence of joint ...
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Kernel for MCMC moves in sequential monte carlo

I'm trying to understand how to employ MCMC moves in a sequential Monte Carlo procedure for estimating static parameters as in the setting described by Chopin. He proposes, for example, the usage of a ...
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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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Why is it necessary to perform resampling step in particle filtering (or sequential monte carlo)?

I read the Wikipedia page on particle filter, it says that during 'prediction-updating', the samples from the distribution are weighted by a likelihood that represents the probability of that particle ...
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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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Particle filter - expectations

I've recently been implementing some particle filter algorithms and I've realized there is a small detail I might have been doing incorrectly. Unfortunately the descriptions of the algorithms in ...
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Sequential monte carlo, resampling

In particle filters when one is doing sequential importance sampling, the quantity of interest that is being approximated is usually a weighted sum: $$\hat x_t = \sum_{i=1}^M \Bigl [f(v^{(i)}_{t}) \...
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Are the Sequential Monte Carlo algorithm invariant to the step at which we resample?

In a usual textual description (according to SMC in Practice book ) of a SMC algorithm for State-Space models, we usually expand the particles according to the distribution from the transition ...
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A doubt on the formula for updating the weights in Sequential Importance Sampling in a State-Space model

Let $x_{0:t}^{(i)}$ be the states from time $0$ to $t$ from sample $i$. Similarly for the observations $y_{1:t}$. The normalized weights are updated according to Where does the term $p(y_t|x_t^{(i)})...
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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 ...
machine424's user avatar
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Particle filter maximum likelihood with a discrete (Bernoulli) state variable, non-smooth loglikelihood

My model looks like this \begin{align} \begin{split} dY_{t} & = \sigma_{t} dW_{t} + Z_t dN_t \\ d\lambda_t & = \alpha(\lambda_\infty - \lambda_t)dt + \beta dN_t \end{split} \end{...
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