locmulti: Local polynomial spectral density estimation

Description

This function implements a local polinomial estimation for the log spectral density at a point x0 $\in R^2$.

Usage

locmulti(x0, l_period, n, freq, h)

Arguments

Point $\in R^2$ at which the spectral density estimate is evaluated.

l_period

Vector of length n with the log-periodogram evaluations at the n Fourier frequencies.

Number of points in the analyzed lattice.

freq

n $\times$ 2 matrix with the n Fourier frequencies.

Kernel bandwidth.

Value

It returns a list containing the following component:
ahatlocal polynomial estimate of the log spectral density at x0.

Details

locmulti function is auxiliary for the nonparametric estimation of the sources spectral density step of the scICA function. locmulti function implements the initial estimates for the local maximum likelihood estimator of the log spectral density $m(\code{x0})$ at a point x0 $\in R^2$. To obtain an estimate of $m(\code{x0})$ the local likelihood function $$\sum_{k}\left(Y_k - a - \textbf{b}'(\boldsymbol{\omega}_k - x0) - e^{Y_k - a - \textbf{b}'(\boldsymbol{\omega}_k - x0)} \right)K_H(\boldsymbol{\omega}_k - x0)$$ is constructed, where $Y_k$ denotes the log-periodogram value at the Fourier frequency $\boldsymbol{\omega}_k$, $K_H$ a surface kernel and $H=(h,h)$. The local maximum estimator $\widehat{m}_{LK}(\code{x0})$ is $\widehat{a}$ in the maximizer $(\widehat{a},\widehat{\textbf{b}})$. The estimate is implemented directly in the scICA function through a Newton-Rapshon algorithm. The initialization for the Newton-Rapshon algorithm is derived through a local polynomial approximation implemented in this locmulti function. In particular the following function is minimized to find a local polynomial approximation for $m(\code{x0})$ $$\sum_{k}\left(Y_k - a - \textbf{b}'(\boldsymbol{\omega}_k - x0) \right)^{2}K_H(\boldsymbol{\omega}_k - x0)$$ and the minimizer $\widehat{a}$ is used as an initial value in order to obtain the local maximum likelihood estimator $\widehat{m}_{LK}(\code{x0})$.

References