PROC SPECTRA: Kernels

Kernels

Kernels are used to smooth the periodogram by using a weighted moving average of nearby points. A smoothed periodogram is defined by the following equation.

$\text{[math]}$

where $\text{[math]}$ is the kernel or weight function. At the endpoints, the moving average is computed cyclically; that is,

$\text{[math]}$

The SPECTRA procedure supports the following kernels. They are listed with their default bandwidth functions.

Bartlett: KERNEL BART

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

Parzen: KERNEL PARZEN

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

Quadratic spectral: KERNEL QS

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

Tukey-Hanning: KERNEL TUKEY

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

Truncated: KERNEL TRUNCAT

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

A summary of the default values of the bandwidth parameters, c and e, associated with the kernel smoothers in PROC SPECTRA are listed below in Table 26.2:

Table 26.2 Bandwidth Parameters
Kernel	c	e
Bartlett	$\text{[math]}$	$\text{[math]}$
Parzen	$\text{[math]}$	$\text{[math]}$
quadratic	$\text{[math]}$	$\text{[math]}$
Tukey-Hanning	$\text{[math]}$	$\text{[math]}$
truncated	$\text{[math]}$	$\text{[math]}$

Figure 26.1 Kernels for Smoothing

See Andrews (1991) for details about the properties of these kernels.

The SPECTRA Procedure