pfjax.models.lotvol_model

Module Contents

Classes

LotVolModel

Lotka-Volterra predator-prey model.

class pfjax.models.lotvol_model.LotVolModel(dt, n_res)

Bases: pfjax.sde.SDEModel

Lotka-Volterra predator-prey model.

The model is:

exp(x_m0) = exp( (logH_m0, logL_m0) ) ~ pi(x_m0) \propto 1
logH_mt ~ N(logH_{m,t-1} + (alpha - beta exp(logL_{m,t-1})) dt/m, sigma_H^2 dt/m)
logL_mt ~ N(logL_{m,t-1} + (-gamma + delta exp(logH_{m,t-1})) dt/m, sigma_L^2 dt/m)
y_t ~ N( exp(x_{m,mt}), diag(tau_H^2, tau_L^2) )

• Model parameters: theta = (alpha, beta, gamma, delta, sigma_H, sigma_L, tau_H, tau_L).

• Global constants: dt and n_res, i.e., m.

• State dimensions: n_state = (n_res, 2).

• Measurement dimensions: n_meas = 2.

Notes:

• The measurement y_t corresponds to x_t = (x_{m,(t-1)m+1}, …, x_{m,tm}), i.e., aligns with the last element of x_t.

• The prior is such that p(x_0 | y_0, theta) is given by:

      x_{m,n} = 0 for n = -m+1, ..., -1,
  exp(x_{m0}) ~ TruncatedNormal( y_0, diag(tau_H^2, tau_L^2) ),
  

  where

       z ~ TruncatedNormal(mu, diag(sigma^2))
  <=>  z = mu + diag(sigma) Z_0,   Z_0 ~iid N(0,1) truncated at -mu.
  

Parameters

• dt – SDE interobservation time.

• n_res – SDE resolution number. There are n_res latent variables per observation, equally spaced with interobservation time dt/n_res.

drift(x, theta)

Calculates the SDE drift function.

diff(x, theta)

Calculates the SDE diffusion function.

meas_lpdf(y_curr, x_curr, theta)

Log-density of p(y_curr | x_curr, theta).

Parameters

• y_curr – Measurement variable at current time t.

• x_curr – State variable at current time t.

• theta – Parameter value.

Returns

The log-density of p(y_curr | x_curr, theta).

meas_sample(key, x_curr, theta)

Sample from p(y_curr | x_curr, theta).

Parameters

• key – PRNG key.

• x_curr – State variable at current time t.

• theta – Parameter value.

Returns

y_curr ~ p(y_curr | x_curr, theta).

Return type

Sample of the measurement variable at current time t

pf_init(key, y_init, theta)

Importance sampler for x_init.

See file comments for exact sampling distribution of p(x_init | y_init, theta), i.e., we have a “perfect” importance sampler with logw = CONST(theta).

Parameters

• key – PRNG key.

• y_init – Measurement variable at initial time t = 0.

• theta – Parameter value.

Returns

• x_init: A sample from the proposal distribution for x_init.

• logw: The log-weight of x_init.

Return type

Tuple