(Partial) coherence for a given frequency

Let $\underline{X}=(X_1,\ldots,X_m)$ be a $m$-variate point process. For a frequency $\boldsymbol\omega$, compute its squared magnitude coherence $\hat{R}^{(a,b)}(\boldsymbol\omega)$ and partial coherence $\hat{D}^ {(a,b)}(\boldsymbol\omega)$ between two point processes $X_a$ and $X_b$ as follows: $$\hat{R}^{(a,b)}(\boldsymbol\omega) = \frac{|\hat{F}^{(a,b)}(\boldsymbol \omega)|^2}{\hat{F}^{(a,a)}(\boldsymbol\omega)\hat{F}^{(b,b)}(\boldsymbol\omega)} \quad \text{and} \quad \hat{D}^{(a,b)}(\boldsymbol\omega) = \frac{|\hat{F}^{-(a,b)}(\boldsymbol \omega)|^2}{\hat{F}^{-(a,a)}(\boldsymbol\omega)\hat{F}^{-(b,b)}(\boldsymbol\omega)},$$ where $\hat{F}^{-(a,b)}$ is the $(a,b)$th element of the inverse spectrum estimator $\hat{F}(\boldsymbol\omega)^{-1}$.

Usage

CohByFreq(
  w1,
  w2,
  sp.est,
  type = "partial",
  i = NULL,
  j = NULL,
  sp.IR = NULL,
  H.list = NULL,
  ppp = NULL
)

Arguments

w1, w2: Frequency vector (only allow frequency values evaluated in sp.est)
sp.est: A list of kernel spectral estimate matrices of the point pattern.
type: If type = 'partial', compute partial coherence. If type = 'normal', compute coherence.
i, j: Optional. Index of the multivariate point process.
sp.IR: Optional. Spectrum estimate of the intensity reweighted process, which is calculated by IRspec(). If this argument is specified, then w1, w2, H.list, ppp are not required.
H.list: Optional. A list from Hmatrix(). This argument is only required if sp.IR is unspecified.
ppp: Optional. Point pattern. This argument is only required if H.list is unspecified.

Value

If both i and j are not specified (default), return a $m \times m$ matrix. If they are specified, return a value.