# 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 (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: .

• 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:

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):

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):

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:

pf = FeynmanKacParticleFilters.generic_particle_filtering_adaptive_resampling_logweights(Mt, logGt, Nparts, RS)


To sample nsamples values from the i-th filtering distributions, do:

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.

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:

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:

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.