Given a value \(\hat{I}^{(i,j)}(\boldsymbol\omega)\) for a frequency \(\boldsymbol\omega\),
this function applies kernel smoothing and returns the smoothed value as follows:
$$\hat{F}^{(i,j)}_{\boldsymbol{b}}(\boldsymbol\omega) = \frac{\sum_{\boldsymbol{k}
\in\mathbb{Z}^2}K_{\boldsymbol{b}}(\boldsymbol\omega - \boldsymbol{x_{k,\Omega}})
\hat{I}^{(i,j)}(\boldsymbol{x_{k,\Omega}})}{\sum_{\boldsymbol{k}\in\mathbb{Z}^2}K_{
\boldsymbol{b}}(\boldsymbol\omega - \boldsymbol{x_{k,\Omega}})}.$$
This function is used in periodogram_smooth() to evaluate the kernel spectral
density estimate for all frequencies.
Arguments
- w
Input frequency \(\boldsymbol\omega = (\omega_1,\omega_2)^\intercal\).
- period.mat
Matrix. The naive spectral estimate, i.e., periodogram \( \hat{I}(\cdot)\), could be complex-valued.
- w.k1, w.k2
Vectors containing the whole frequencies at horizontal and vertical directions.
- b1, b2
Numeric. Bandwidth values for horizontal and vertical directions.
- loo
Logical. If
TRUE, conduct leave-one-out kernel smoothing (the centerwwill be excluded when averaging). Otherwise, keep the centerwfor averaging.- kernel_uni
Univariate kernel function \(K\).