Conditional-Gaussian nonlinear system

Constructs a conditional-Gaussian nonlinear system, in which the observed drift is affine in the hidden state and the hidden drift is linear in it, so that the conditional statistics of the hidden component are Gaussian in closed form. The noise is described by up to two shared Wiener channels, so that correlated observation and signal noise can be represented.

Usage

aci_model(
  Lx,
  fx,
  Ly,
  fy,
  Sx1,
  Sx2 = NULL,
  Sy1 = NULL,
  Sy2,
  k,
  l,
  name = NULL,
  meta = list()
)

Arguments

Lx

Function of (t, x) giving the coupling of the hidden state into the observed drift, a k by l matrix.

fx

Function of (t, x) giving the remaining observed drift.

Ly

Function of (t, x) giving the hidden self-drift, an l by l matrix.

fy

Function of (t, x) giving the remaining hidden drift.

Sx1

Function of (t, x) giving the observed diffusion on the first Wiener channel.

Sx2

Optional function of (t, x) giving the observed diffusion on the second Wiener channel; NULL is a zero block matched to Sy2.

Sy1

Optional function of (t, x) giving the hidden diffusion on the first Wiener channel; NULL is a zero block matched to Sx1.

Sy2

Function of (t, x) giving the hidden diffusion on the second Wiener channel.

k

Observed dimension; a positive whole number.

l

Hidden dimension; a positive whole number.

name

Optional 1-length character label for the model.

meta

Optional named list of metadata carried on the object.

Value

An object of class cgns_model, which also inherits from stochastic_model.

Details

CGNS coefficient functions are mathematical coefficients: for a fixed (t, x) they must return deterministic values with stable shapes and diffusion-channel counts. Random-number generation or result-changing mutable state inside a coefficient function is outside the model contract. Closed-form execution may realise each coefficient once on the observation grid and reuse that realised path.

A model is fixed once constructed. Its derived drifts capture the coefficient functions supplied to the constructor, so assigning a new coefficient into an existing model changes only the field assigned and leaves simulation, filtering and every realised path on the original coefficients. Change a parameter by rebuilding the model with aci_model(), not by modifying one.

See also

aci_model_from_affine(), aci_filter(), aci()

Examples

aci_model(
  Lx = function(t, x) matrix(1, 1, 1),
  fx = function(t, x) -0.5 * x,
  Ly = function(t, x) matrix(-0.5, 1, 1),
  fy = function(t, x) 0,
  Sx1 = function(t, x) matrix(0.5, 1, 1),
  Sy2 = function(t, x) matrix(1, 1, 1),
  k = 1, l = 1)
#> <cgns_model> 'cgns_model': k = 1 observed, l = 1 hidden

# Coefficients are fixed for the model's lifetime: change a parameter by
# rebuilding, not by assigning into an existing model.
make_dyadish <- function(lambda)
  aci_model(Lx = function(t, x) matrix(1, 1, 1),
            fx = function(t, x) lambda * x,
            Ly = function(t, x) matrix(-0.5, 1, 1),
            fy = function(t, x) 0,
            Sx1 = function(t, x) matrix(0.5, 1, 1),
            Sy2 = function(t, x) matrix(1, 1, 1),
            k = 1, l = 1)
m_a <- make_dyadish(-0.5)
m_b <- make_dyadish(-1)