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https://github.com/konkam/FeynmanKacParticleFilters.jl
Particle filtering using the Feynman-Kac formalism
https://github.com/konkam/FeynmanKacParticleFilters.jl
Last synced: 3 months ago
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Particle filtering using the Feynman-Kac formalism
- Host: GitHub
- URL: https://github.com/konkam/FeynmanKacParticleFilters.jl
- Owner: konkam
- License: gpl-3.0
- Created: 2018-11-12T13:59:45.000Z (almost 6 years ago)
- Default Branch: master
- Last Pushed: 2024-01-29T15:06:06.000Z (9 months ago)
- Last Synced: 2024-07-10T18:50:12.941Z (4 months ago)
- Language: Julia
- Homepage:
- Size: 1.56 MB
- Stars: 4
- Watchers: 1
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
- awesome-sciml - konkam/FeynmanKacParticleFilters.jl: Particle filtering using the Feynman-Kac formalism
README
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# FeynmanKacParticleFilters
A package to perform particle filtering (as well as likelihood estimation and smoothing) using the Feynman-Kac formalism.
Filtering and likelihood estimation are illustrated on two stochastic diffusion equation models:
- The [Cox-Ingersoll-Ross](https://en.wikipedia.org/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model) (CIR) model
- The K dimensional continuous Wright Fisher model (continuous time, infinite population, see Jenkins & Spanò (2017) for instance)
*Particle smoothing for the Wright-Fisher model is not implemented for lack of a tractable form of the transition density.*
Outputs:
- Marginal likelihood
- Samples from the filtering distribution
- Samples from the marginal smoothing distribution
Implemented:
- Bootstrap particle filter with adaptive resampling.
- Forward Filtering Backward Sampling (FFBS) algorithm
Potentially useful functions:
- Evaluation of the transition density for the Cox-Ingersoll-Ross process (*based on the representation with the Bessel function*)
- Random trajectory generation from the Cox-Ingersoll-Ross process (*based on the Gamma Poisson expansion of the transition density*)
# Preliminary notions on the Feynman-Kac formalism
The Feynman-Kac formalism allows to formulate different types of particle filters using the same abstract elements.
The input of a generic particle filter are:
- A Feynman-Kac model M_t, G_t, where:
- G_t is a potential function which can be evaluated for all values of t
- It is possible to simulate from M_0(dx0) and M_t(x_t-1, dxt)
- The number of particles N
- The choice of an unbiased resampling scheme (e.g. multinomial), i.e. an algorithm to draw variables 
in 1:N where RS is a distribution such that: 
.
For adaptive resampling, one needs in addition:
- a scalar 
Using this formalism, the boostrap filter is expressed as:
- G_0(x_0) = f_0(y_0|x_0), where f is the emission density
- G_t(x_t-1, x_t) = f_0(y_t|x_t)
- M_0(dx0) = P_0(dx0) the prior on the hidden state
- M_t(x_t-1, dxt) = P_t(x_t-1, dxt) given by the transition function
# How to install the package
Press `]` in the Julia interpreter to enter the Pkg mode and input:
```julia
pkg> add https://github.com/konkam/FeynmanKacParticleFilters.jl
```
# How to use the package (Example with the CIR model)
The transition density of the 1-D CIR process is available as:
from which it easy to simulate.
Moreover, we consider a Poisson distribution as the emission density:
We start by simulating some data (a function to simulate from the transition density is available in the package):
```julia
using FeynmanKacParticleFilters, Distributions, Random
Random.seed!(0)
Δt = 0.1
δ = 3.
γ = 2.5
σ = 4.
Nobs = 2
Nsteps = 4
λ = 1.
Nparts = 10
α = δ/2
β = γ/σ^2
time_grid = [k*Δt for k in 0:(Nsteps-1)]
times = [k*Δt for k in 0:(Nsteps-1)]
X = FeynmanKacParticleFilters.generate_CIR_trajectory(time_grid, 3, δ*1.2, γ/1.2, σ*0.7)
Y = map(λ -> rand(Poisson(λ), Nobs), X);
data = zip(times, Y) |> Dict
4-element Array{Float64,1}:
0.0
0.1
0.2
0.30000000000000004
```
## Filtering
Now we define the (log)potential function Gt, a simulator from the transition kernel for the Cox-Ingersoll-Ross model and a resampling scheme (here multinomial):
```julia
Mt = FeynmanKacParticleFilters.create_transition_kernels_CIR(data, δ, γ, σ)
logGt = FeynmanKacParticleFilters.create_log_potential_functions_CIR(data)
RS(W) = rand(Categorical(W), length(W))
```
Running the boostrap filter algorithm is performed as follows:
```julia
pf = FeynmanKacParticleFilters.generic_particle_filtering_adaptive_resampling_logweights(Mt, logGt, Nparts, RS)
```
To sample `nsamples` values from the i-th filtering distributions, do:
```julia
n_samples = 100
i = 4
FeynmanKacParticleFilters.sample_from_filtering_distributions_logweights(pf, n_samples, i)
100-element Array{Float64,1}:
5.371960182098351
5.371960182098351
3.3924167451813956
3.3924167451813956
3.3924167451813956
⋮
```
## Smoothing
### Forward Filtering Backward Sampling (FFBS)
To perform a simple particle smoothing on the CIR process using the FFBS algorithm, we additionally need a function which evaluates the transition density of the CIR process.
```julia
transition_logdensity_CIR(Xtp1, Xt, Δtp1) = FeynmanKacParticleFilters.CIR_transition_logdensity(Xtp1, Xt, Δtp1, δ, γ, σ)
```
With the transition density, we can obtain the FFBS filter:
```julia
ps = FeynmanKacParticleFilters.generic_FFBS_algorithm_logweights(Mt, logGt, Nparts, Nparts, RS, transition_logdensity_CIR)
```
To sample `nsamples` values from the i-th smoothing distribution, do:
```julia
n_samples = 100
i = 4
FeynmanKacParticleFilters.sample_from_smoothing_distributions_logweights(ps, n_samples, i)
100-element Array{Float64,1}:
7.134633585387236
2.513540876531395
5.0555536713845814
7.983322471825221
4.651221100411266
⋮
```
**References:**
- Briers, M., Doucet, A. and Maskell, S. *Smoothing algorithms for state–space models.* Annals of the Institute of Statistical Mathematics 62.1 (2010): 61.
- Chopin, N. & Papaspiliopoulos, O. *A concise introduction to Sequential Monte Carlo*, to appear.
- Del Moral, P. (2004). *Feynman-Kac formulae. Genealogical and interacting particle
systems with applications.* Probability and its Applications. Springer Verlag, New York.
- Jenkins, P. A., & Spanò, D. (2017). Exact simulation of the Wright--Fisher diffusion. The Annals of Applied Probability, 27(3), 1478–1509.