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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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Model selection with this model of a large number of components

I have a discrete time Markov Chain $\{X_n: n \in \mathbb{N}_0\}$ with unknown transition matrix $P \in \mathbb{R}^{M \times M}$ on the state space $\mathcal{S}_X = \{1,2, \dots, M\}$, with $M \geq 2$....
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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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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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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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1answer
62 views

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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50 views

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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1answer
94 views

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 ...
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1answer
65 views

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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1answer
135 views

Importance weight of conditioned particle in conditional SMC

In a generic particle filter, I understand the importance weights for each particle are calculated as $w_t^s \propto w_{t-1}^s \frac{p(y_t \mid z_t^s) p(z_t^s \mid z_{t-1}^s)}{q(z_t^s \mid z_{t-1}^s, ...
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460 views

Understanding Sequential Importance Sampling and Particle Filtering

I am struggling with SIS for particle filtering in the following aspect: In particle filtering (as per this book), the objective is to estimate the full posterior $p( x_{0:k} \mid y_{1:k} )$ rather ...
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1answer
74 views

notation conditional normal distribution [duplicate]

I'm describing parameter search using a particle filter, for which I use West M. (1993) Approximating Posterior Distributions by Mixture. On page 8 of the document, he states "and $p(\theta)$ is (...
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51 views

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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222 views

How to calculate importance weights for update step of an SIR (Sequential Importance Resampling) Particle filter?

I understand that one may use a particle filter to solve the filtering problem (estimating the hidden state of a system which can be described as a Hidden Markov Model). If I have a system where I ...
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2answers
311 views

Resample-move algorithm for Sequential Monte Carlo

I'm reading about the resample-move strategy, originally by Gilks and Berzuini, but my question will use the slightly more verbose description from the review of Doucet and Johansen, section 4.4, PDF ...
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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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248 views

Stochastic volatility: particle filter vs Metropolis-Hastings

In many of the papers on particle filter I've read (e.g. Douc, Moulines and Olsson, 2007), stochastic volatility is a common example to show that a newly-proposed filter is working. At the same time, ...
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69 views

Multi-Target Tracking Filters

I am trying to solve a multi-target tracking problem, which is in some parts different to some filters I have already researched such as the PHD filter. I am asking for advise which filters to start ...
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1answer
138 views

Bootstrap filter

I am trying to implement bootstrap filter and I'm trying to understand it based on Bootstrap filter/ Particle filter algorithm(Understanding) EDIT: Following is the example that I'm trying to solve: ...
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25 views

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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1answer
582 views

What is the best way to apply the log-sum-exp trick in this situation?

I am aware of the "log-sum-exp" trick for calculating the logarithm of sums that handles overflow and underflow issues. However, I would like to know more about how it works. In particular, I am ...
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1answer
92 views

weights versus shifted log-weights

I'm having a hard time checking an equality in https://arxiv.org/pdf/1511.01707.pdf. It is unnumbered, immediately before (24), on page 13. Any help would be appreciated. Here is some simplified ...
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643 views

SMC (Particle Filtering) code [closed]

Does anyone know where I can find particle filtering code for R? In particular I'm looking for code for filtering a forward-rate curve.
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1answer
159 views

Techniques to estimate constant states with particle filter?

I have an application where some of my states are constant and therefore have no process noise. Over the course of the estimation process, the uncertainty in these states drops several orders of ...
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70 views

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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165 views

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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1answer
51 views

How to sample from discrete 3D distribution

I am trying to implement a particle filter to track multiple objects. During the propagation phase I need to take N samples from a three dimensional probability distribution, which does not fit known ...
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507 views

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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63 views

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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1answer
383 views

Particle Filter: Confidence Intervals

Context This is a basic question about confidence intervals. So the standard way to estimate a confidence interval.Assuming we have a set of $N$ random variables $\{X^i\}$ such that all of them are i....
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427 views

Particle Degeneracy: Variance of the weights

This is a qualitative question. In the literature of particle filters/ sequential monte carlo, particle degeneracy is unavoidable. However, frequented cases where it may happen is when the likelihood ...
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146 views

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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1answer
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Hamiltonian Monte Carlo vs. Sequential Monte Carlo

I am trying to get a feel for the relative merits and drawbacks, as well as different application domains of these two MCMC schemes. When would you use which and why? When might one fail but the ...
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1answer
240 views

Empirical Covariance for Set of Particles and Weights

In the context of particle filtering. We assume a standard state space model where k is time and i is the particle index.Note: $w_k^i$ are normalised weights. Assume I have a set $\{x_k^i, w_k^i\}_{i=...
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446 views

Sampling from Likelihood: Likelihood Particle Filter

I am particularly baffled with the idea of sampling from the likelihood. To give some context, I am studying particle filters and am investigating the "Likelihood Particle Filter". I am reading this ...
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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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1answer
113 views

Exact Particle Filtering Derivation

I am very confused with one of the derivations used by this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.409.2389&rep=rep1&type=pdf in pg 1021 of this paper. It is a simple ...
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1answer
156 views

Application of the law of large numbers

In the book by Christian Robert and George Casella on Monte Carlo Statistical Methods, they use an argument of LLN on pages 551 and 552. I'm attaching the argument in this screen shot $t$ is ...
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161 views

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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1answer
33 views

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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2answers
153 views

throwing away all Gibbs samples after approximation

This is more of a theory question, consider: $$P(w_1|D)=\int P(w_1|S)P(S|D)d(S)$$ which we approximate via Gibbs sampling $S$ (assume the initial state of the Gibbs sampler is denoted by $M_0$), ...
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1answer
42 views

Doucet Notation- d in expectation estimate

In the book Sequential Monte Carlo Methods in Practice, the beginning of section 1.3.1. I am confused by the notation used $$P_N(dx_{0:t}|y_{0:t})=\frac{1}{N}\sum_{i=1}^{N}{\delta_{x_{0:t}^{(i)}}(...
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34 views

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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1answer
207 views

Does the Auxiliary Particle Filter not ordinarily yield estimates for $p(y_t|y_{1:t-1})$

Background Say you're implementing basic importance sampling. To approximate a distribution $p(x)$, you utilize an importance density $q(x)$ that you can sample from. Also, assume that you can only ...
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1answer
3k views

Bootstrap filter/ Particle filter algorithm(Understanding)

I truly have a lack of understanding of how the bootstrap filter works. I roughly know the concepts but I fail to grasps certain details. This question is for me to clear up the clutter. Here I will ...
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4answers
831 views

Role of Dirac function in particle filters

Particle approximations to probability densities are often introduced as a weighted sum of Dirac functions $$p(x) \approx \sum_{i=1}^N \omega^i \delta(x-x^i)$$ with the weights $$\omega^i \propto \...
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1answer
346 views

Particle filter reducing the covariance matrix of joint normal distribution too much [closed]

I am working on developing a particle filter to improve the results of an Unscented Kalman filter (UKF) to satellite attitude determination. The UKF outputs a 12-dimensional joint normal distribution (...
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639 views

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 ...