OUT= Data Set

The SPECTRA Procedure

OUT= Data Set

The OUT= data set contains [n/2]+1 observations, if n is even, or [(n+1)/2] observations, if n is odd, where n is the number of observations in the time series.

The variables in the new data set are named according to the following conventions. Each variable to be analyzed is associated with an index. The first variable listed in the VAR statement is indexed as 01, the second variable as 02, and so on. Output variables are named by combining indexes with prefixes. The prefix always identifies the nature of the new variable, and the indices identify the original variables from which the statistics were obtained.

Variables containing spectral analysis results have names consisting of a prefix, an underscore, and the index of the variable analyzed. For example, the variable S_01 contains spectral density estimates for the first variable in the VAR statement. Variables containing cross-spectral analysis results have names consisting of a prefix, an underscore, the index of the first variable, another underscore, and the index of the second variable. For example, the variable A_01_02 contains the amplitude of the cross-spectral density estimate for the first and second variables in the VAR statement.

Table 17.1 shows the formulas and naming conventions used for the variables in the OUT= data set. Let X be variable number nn in the VAR statement list and let Y be variable number mm in the VAR statement list. Table 17.1 shows the output variables containing the results of the spectral and cross-spectral analysis of X and Y.

In Table 17.1 the following notation is used. Let W_j be the vector of 2p+1 smoothing weights given by the WEIGHTS statement, normalized to sum to ${\frac{1}{4{\pi}}}$ . The subscript of W_j runs from W_-p to W_p, so that W₀ is the middle weight in the WEIGHTS statement list. Let ${{\omega}_{k} = \frac{2{\pi}k}n}$ , where k = 0, 1, ... , floor( [n/2] ).

Table 17.1: Variables Created by PROC SPECTRA

Variable Description

FREQ frequency in radians from 0 to ${\pi}$

(Note: Cycles per observation is $\frac{{\rm FREQ}}{2{\pi}}$ .)

PERIOD period or wavelength: $\frac{2{\pi}}{{\rm FREQ}}$

(Note: PERIOD is missing for FREQ=0.)

COS_X cosine transform of X: ${ a^x_{k}=\frac{2}n \sum_{t=1}^n{X_{t} {\rm cos}({\omega}_{k}(t-1))}}$

COS_WAVE

SIN_X sine transform of X: ${ b^x_{k}=\frac{2}n \sum_{t=1}^n{X_{t} {\rm sin}({\omega}_{k}(t-1))}}$

SIN_WAVE

P_nn periodogram of X: J^x_k = [n/2] [(a^x_k)² + ( b^x_k)²]

S_nn spectral density estimate of X: ${ F^x_{k}=\sum_{j=-p}^p{W_{j} J^x_{k+j}}}$

(except across endpoints)

RP_nn_mm real part of cross-periodogram X and Y: real( J^xy_k) = [n/2] ( a^x_k a^y_k + b^x_k b^y_k)

IP_nn_mm imaginary part of cross-periodogram of X and Y:

imag( J^xy_k) = [n/2] ( a^x_k b^y_k - b^x_k a^y_k)

CS_nn_mm cospectrum estimate (real part of cross-spectrum) of X and Y:

${ C^{xy}_{k}=\sum_{j=-p}^p{W_{j}\rm{real}( J^{xy}_{k+j})}}$ (except across endpoints)

QS_nn_mm quadrature spectrum estimate (imaginary part of cross-spectrum) of X and Y:

${ Q^{xy}_{k}=\sum_{j=-p}^p{W_{j} \rm{imag}( J^{xy}_{k+j})}}$ (except across endpoints)

A_nn_mm amplitude (modulus) of cross-spectrum of X and Y: ${ A^{xy}_{k}=\sqrt{ {(C^{xy}_{k})}^2 + {(Q^{xy}_{k})}^2} }$

K_nn_mm coherency squared of X and Y: K^xy_k= (A^xy_k)² / ( F^x_k F^y_k)

PH_nn_mm phase spectrum in radians of X and Y: ${ {\Phi}^{xy}_{k}=\arctan( Q^{xy}_{k} / C^{xy}_{k} )}$

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Variable	Description

FREQ	frequency in radians from 0 to ${\pi}$
	(Note: Cycles per observation is $\frac{{\rm FREQ}}{2{\pi}}$ .)

PERIOD	period or wavelength: $\frac{2{\pi}}{{\rm FREQ}}$
	(Note: PERIOD is missing for FREQ=0.)

COS_X	cosine transform of X: ${ a^x_{k}=\frac{2}n \sum_{t=1}^n{X_{t} {\rm cos}({\omega}_{k}(t-1))}}$
COS_WAVE

SIN_X	sine transform of X: ${ b^x_{k}=\frac{2}n \sum_{t=1}^n{X_{t} {\rm sin}({\omega}_{k}(t-1))}}$
SIN_WAVE

P_nn	periodogram of X: J^x_k = [n/2] [(a^x_k)² + ( b^x_k)²]

S_nn	spectral density estimate of X: ${ F^x_{k}=\sum_{j=-p}^p{W_{j} J^x_{k+j}}}$
	(except across endpoints)

RP_nn_mm	real part of cross-periodogram X and Y: real( J^xy_k) = [n/2] ( a^x_k a^y_k + b^x_k b^y_k)

IP_nn_mm	imaginary part of cross-periodogram of X and Y:
	imag( J^xy_k) = [n/2] ( a^x_k b^y_k - b^x_k a^y_k)

CS_nn_mm	cospectrum estimate (real part of cross-spectrum) of X and Y:
	${ C^{xy}_{k}=\sum_{j=-p}^p{W_{j}\rm{real}( J^{xy}_{k+j})}}$ (except across endpoints)

QS_nn_mm	quadrature spectrum estimate (imaginary part of cross-spectrum) of X and Y:
	${ Q^{xy}_{k}=\sum_{j=-p}^p{W_{j} \rm{imag}( J^{xy}_{k+j})}}$ (except across endpoints)

A_nn_mm	amplitude (modulus) of cross-spectrum of X and Y: ${ A^{xy}_{k}=\sqrt{ {(C^{xy}_{k})}^2 + {(Q^{xy}_{k})}^2} }$

K_nn_mm	coherency squared of X and Y: K^xy_k= (A^xy_k)² / ( F^x_k F^y_k)

PH_nn_mm	phase spectrum in radians of X and Y: ${ {\Phi}^{xy}_{k}=\arctan( Q^{xy}_{k} / C^{xy}_{k} )}$