The SPECTRA Procedure

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.

\begin{eqnarray*} \hat{J}_{i}(\textrm{l}(q)) = \sum _{{\tau } = -\textrm{l}(q)}^{\textrm{l}(q)} {\textrm{w}\left( \frac{{\tau }}{\textrm{l}(q)} \right) \tilde{J}_{i+{\tau }}} \nonumber \end{eqnarray*}

where ${\mr{w}(x)}$ is the kernel or weight function. At the endpoints, the moving average is computed cyclically; that is,

\begin{eqnarray*} \tilde{J}_{i+{\tau }} = \begin{cases} J_{i+{\tau }} & 0 <= i+\tau <= q \\ J_{-(i+{\tau })} & i+\tau < 0 \\ J_{q-(i+{\tau })} & i+\tau > q \end{cases}\nonumber \end{eqnarray*}

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

Bartlett: KERNEL BART

\begin{eqnarray*} \textrm{w}(x) & =& \begin{cases} 1-{|x|} & {|x|}{\le }1 \\ 0 & \mr{otherwise} \end{cases}\\ \mr{l}(q) & =& \frac{1}{2} q^{1 / 3} \nonumber \end{eqnarray*}

Parzen: KERNEL PARZEN

\begin{eqnarray*} \textrm{w}(x) & =& \begin{cases} 1-6{|x|}^{2} + 6{|x|}^{3} & {0{\le }{|x|}{\le }\frac{1}{2}} \\ 2(1-{|x|})^{3} & {\frac{1}{2}{\le }{|x|}{\le }1} \\ 0 & \mr{otherwise} \end{cases}\\ \mr{l}(q) & =& q^{1 / 5} \nonumber \end{eqnarray*}

Quadratic spectral: KERNEL QS

\begin{eqnarray*} \mr{w}(x) & =& \frac{25}{12{\pi }^{2} x^{2}} \left( \frac{{sin}(6{\pi }x/5)}{6{\pi }x/5} - {cos}(6{\pi }x/5) \right) \\ \textrm{l}(q) & =& \frac{1}{2} q^{1 / 5} \nonumber \end{eqnarray*}

Tukey-Hanning: KERNEL TUKEY

\begin{eqnarray*} \textrm{w}(x) & =& \begin{cases} (1+{cos}( {\pi } x))/2 & {|x|}{\le }1 \\ 0 & \mr{otherwise} \end{cases}\\ \mr{l}(q) & =& \frac{2}{3} q^{1 / 5} \nonumber \end{eqnarray*}

Truncated: KERNEL TRUNCAT

\begin{eqnarray*} \textrm{w}(x) & =& \begin{cases} 1 & {|x|}{\le }1 \\ 0 & \mr{otherwise} \end{cases}\\ \mr{l}(q) & =& \frac{1}{4} q^{1 / 5} \nonumber \end{eqnarray*}

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

$1/2$

$1/3$

Parzen

1

$1/5$

quadratic

$1/2$

$1/5$

Tukey-Hanning

$2/3$

$1/5$

truncated

$1/4$

$1/5$


Figure 26.1: Kernels for Smoothing

Kernels for Smoothing


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