MagiPosterior {magi}

MAGI posterior density

Description

Computes the MAGI log-posterior value and gradient for an ODE model with the given inputs: the observations Y, the latent system trajectories X, the parameters \theta, the noise standard deviations \sigma, and covariance kernels.

Usage

MagiPosterior(
  y,
  xlatent,
  theta,
  sigma,
  covAllDimInput,
  odeModel,
  priorTemperatureInput = 1,
  useBand = FALSE
)

Arguments

y

data matrix of observations

xlatent

matrix of system trajectory values

theta

vector of parameter values \theta

sigma

vector of observation noise for each system component

covAllDimInput

list of covariance kernel objects for each system component. Covariance calculations may be carried out with calCov.

odeModel

list of ODE functions and inputs. See details.

priorTemperatureInput

vector of tempering factors for the GP prior, derivatives, and observations, in that order. Controls the influence of the GP prior relative to the likelihood. Recommended values: the total number of observations divided by the total number of discretization points for the GP prior and derivatives, and 1 for the observations.

useBand

logical: should the band matrix approximation be used? If TRUE, covAllDimInput must include CinvBand, mphiBand, and KinvBand as computed by calCov.

Value

A list with elements value for the value of the log-posterior density and grad for its gradient.

Examples

# Trajectories from the Fitzhugh-Nagumo equations
tvec <- seq(0, 20, 2)
Vtrue <- c(-1, 1.91, 1.38, -1.32, -1.5, 1.73, 1.66, 0.89, -1.82, -0.93, 1.89)
Rtrue <- c(1, 0.33, -0.62, -0.82, 0.5, 0.94, -0.22, -0.9, -0.08, 0.95, 0.3)

# Noisy observations
Vobs <- Vtrue + rnorm(length(tvec), sd = 0.05)
Robs <- Rtrue + rnorm(length(tvec), sd = 0.1)

# Prepare distance matrix for covariance kernel calculation
foo <- outer(tvec, t(tvec), '-')[, 1, ]
r <- abs(foo)
r2 <- r^2
signr <- -sign(foo)
  
# Choose some hyperparameter values to illustrate
rphi <- c(0.95, 3.27)
vphi <- c(1.98, 1.12)
phiTest <- cbind(vphi, rphi)

# Covariance computations
curCovV <- calCov(phiTest[,1], r, signr, kerneltype = "generalMatern")
curCovR <- calCov(phiTest[,2], r, signr, kerneltype = "generalMatern")

# Y and X inputs to MagiPosterior
yInput <- data.matrix(cbind(Vobs, Robs))
xlatentTest <- data.matrix(cbind(Vtrue, Rtrue))

# Create odeModel list for FN equations
fnmodel <- list(
  fOde = fnmodelODE,
  fOdeDx = fnmodelDx,
  fOdeDtheta = fnmodelDtheta,
  thetaLowerBound = c(0, 0, 0),
  thetaUpperBound = c(Inf, Inf, Inf)
)

MagiPosterior(yInput, xlatentTest, theta = c(0.2, 0.2, 3), sigma = c(0.05, 0.1),
    list(curCovV, curCovR), fnmodel)