Calculate the residuals of a CSI process.

csi_resid(dX, drift, acf, Sigma)

Arguments

dX: Matrix of trajectory increments.
drift: Matrix of the same size as dX of increment drift in each coordinate per timepoint (see 'Details').
acf: Autocorrelation vector of length nrow(dX) (see 'Details').
Sigma: Covariance matrix of size ncol(dX) x ncol(dX).

Value

A residual matrix of the same size as dX.

Details

The residuals are calculated as

Z = toeplitz(acf)^{-1/2} %*% (dX - drift) %*% Sigma^{-1/2},

where the "square roots" correspond to the Cholesky decomposition for the row-dependence matrix toeplitz(acf), and the eigen decomposition for the column-dependence matrix Sigma.

Examples

# simulate data from the fbm model
alpha <- .8

dt <- 1/60
N <- 1800
ndim <- 2

Xt <- csi_sim(drift = matrix(0, N-1, ndim),
              acf = fbm_acf(alpha, dt, N-1),
              Sigma = diag(ndim),
              X0 = rep(0, ndim))


# compute the residuals
Z <- csi_resid(dX = apply(Xt, 2, diff),
               drift = matrix(0, N-1, ndim),
               acf = fbm_acf(alpha, dt, N-1),
               Sigma = diag(ndim))