Sequential Monte Carlo (SMC)

The direct algorithmic implementation of Sequential Monte Carlo. It represents the posterior probability density function of a system's state using a set of discrete random samples called particles. Each particle is a hypothesis of the true state, weighted by its likelihood given the latest observation. The key steps are:

• Prediction: Propagate particles through the state transition model.
• Update: Re-weight particles based on the new measurement.
• Resampling: Eliminate particles with negligible weights and replicate high-weight ones to prevent particle degeneracy.

A Monte Carlo technique for estimating properties of a target distribution when direct sampling is difficult. Samples are drawn from a simpler proposal distribution, and weights are assigned based on the ratio of the target density to the proposal density. In SMC, Sequential Importance Sampling (SIS) applies this recursively, but suffers from weight variance increasing over time, necessitating the resampling step to create the Sampling Importance Resampling (SIR) variant.

The statistical paradigm that underpins SMC. It provides a mathematical framework for updating the probability of a hypothesis as new evidence arrives. The core formula is Bayes' Theorem: P(State | Data) ∝ P(Data | State) × P(State). SMC approximates the intractable posterior distribution P(State | Data) recursively over time, making it ideal for online state estimation in non-linear, non-Gaussian systems like dynamic spectrum access.

A mathematical framework describing a dynamic system by two equations:

• State Equation: x_k = f(x_{k-1}, v_k) — models how the hidden state evolves (e.g., channel occupancy status).
• Observation Equation: z_k = h(x_k, n_k) — models how noisy measurements relate to the hidden state (e.g., received signal energy). SMC excels in this framework because it does not require the linear Gaussian assumptions of the Kalman Filter, allowing it to track complex primary user activity models.

An alternative class of Bayesian computation methods that construct a Markov chain whose stationary distribution is the target posterior. Unlike SMC, which processes data sequentially online, standard MCMC is inherently batch or offline. However, the two are closely related: Particle MCMC (PMCMC) algorithms combine them, using SMC to generate high-quality proposals for MCMC samplers, which is useful for offline learning of spectrum environment parameters.

A critical mechanism to combat weight degeneracy in SMC, where after a few iterations, all weight concentrates on a single particle. Resampling draws a new set of particles from the current weighted set, duplicating high-weight particles and discarding low-weight ones. Common schemes include:

• Multinomial Resampling: The standard draw with replacement.
• Systematic Resampling: A lower-variance method using a single random start.
• Stratified Resampling: Divides the weight interval into strata for more uniform sampling.