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# specxy_anal

Calculates cross spectra quantities of a series.

## Prototype

```	function specxy_anal (
x    [*] : numeric,
y    [*] : numeric,
iopt  : integer,
jave  : integer,
pct   : numeric
)

return_val  :  float or double
```

## Arguments

x
y

One-dimensional arrays containing the data. x and y must be the same length, and missing values are not allowed.

iopt

A scalar representing the detrending option:

iopt = 0
Remove series mean.
iopt = 1
Remove the series mean and least squares linear trend.

jave

A scalar representing the smoothing to be performed on the periodogram estimates. This should be an odd number (>= 3). If not, the routine will force it to the next largest odd number.

jave < 3
Do no smoothing. spcx contains raw spectra estimates (periodogram).
jave >= 3
Average jave periodogram estimates together utilizing modified Daniell smoothing (good stability but may lead to large bias). All weights are 1/jave except weight(1) and weight(jave) which are 1/(2*jave). This is the recommended option. It is this number which has the most impact on the degrees of freedom.

pct

A scalar representing the percent of the series to be tapered (0.0 <= pct <= 1.0). If pct =0.0, no tapering will be done. If pct = 1.0, the whole series is affected. A value of 0.10 is common (tapering should always be done).

## Return value

The return value is a scalar representing the degrees of freedom. The scalar will be double if x, y, or pct are double, and float otherwise. See the description below for a list of attributes that are also returned.

## Description

specxy_anal returns the degrees of freedom as a scalar. It also returns the following attributes:

spcx,spcy
One-dimensional arrays of length N/2.

spcx(0) - spectral estimate at frequency = (1/N)
[N=dimsizes(x)]

spcx(N/2-1)- spectral estimate at frequency = 0.5

These spectra have been normalized so that the area under the curve:

(spcx(0)+spcx(N/2-1))*(df/2) + SUM{spcx(n)*df}
equals the variance of the detrended series, where df=(1/N)=frequency spacing and n=1 to N/2-2.

The units are variance/(unit frequency interval).

frq
A one-dimensional array of length N/2 representing frequency (cycles/time).
cospc
A one-dimensional array of length N/2 representing the cospectrum. This is the real part of the cross spectrum. It measures the extent to which there are oscillations with the same phase in the two series (or with opposite sign, i.e. with a phase shift of half a cycle). In other words, it measures the contribution of different frequencies to the total cross-covariance at zero lag.

quspc
A one-dimensional array of length N/2 representing the quadrature spectrum. This is the imaginary part of the cross spectrum. it measures the extent to which there are oscillations with a phase difference of a quarter cycle in either direction. i.e., It measures the contribution of different frequencies to the total cross-covariance of the series when all harmonics of one series are delayed a quarter cycle relative to the other relative to the other series.

coher
A one-dimensional array of length N/2 representing coherence squared. This is analogous to the square of the correlation coef except that the coherence squared is a function of frequency.

phase
A one-dimensional array of length N/2 representing the phase in degrees. A positive phase indicates that x leads y.

bw
A scalar (same type as output) representing the spectral bandwidth.

coher_probability
An array of length 4 containing the coherence corresponding to the 90, 95, 99, and 99.9% levels.

xavei
A scalar (same type as output) representing the average of the x series on input.

xvari
A scalar (same type as output) representing the variance of the x series on input.

xvaro
A scalar (same type as output) representing the variance of the x series after detrending.

xlag1
A scalar (same type as output) representing the lag-one autocorrelation of the x series after detrending.

xslope
A scalar (same type as output) representing the least-squares slope per time interval of linear trend (if iopt = 1) of the x series.

yavei
A scalar (same type as output) representing the average of the y series on input.

yvari
A scalar (same type as output) representing the variance of the y series on input.

yvaro
A scalar (same type as output) representing the variance of the y series after detrending.

ylag1
A scalar (same type as output) representing the lag-one autocorrelation of the y series after detrending.

yslope
A scalar (same type as output) representing the least-squares slope per time interval of linear trend (if iopt = 1) of the y series.

```--------------------------------------------------------------------------------
Some NCL users have asked about testing for significance. The following may help:

The coherence-squared is a statistic that can be used to examine the relation between
two signals or data sets. It allows identification of significant frequency-domain
correlation between the two time series.  Phase estimates in the cross spectrum are
only useful where significant frequency-domain correlation exists.

References:

Dennis Hartmann
http://www.atmos.washington.edu/~dennis/552_Notes_6c.pdf
See Table 6.2 , page 187 and the associated caption on page 186

Comments on the Determination of Significance Levels of the Coherence Statistic
Paul R. Julian
J. of Atm. Sci. (1975), Volume 32, pp 836-837.

Coherence Significance Levels
Rory O. R. Y. Thompson
Journal of the Atmospheric Sciences, 1979
Volume 36, pp 2020-2021
```

## Examples

Example 1

Perform cross-spectral analysis on series x and y:

```   iopt = 1    ; remove least squares linear trends from each
; series prior to tapering and computing spectra.
jave = 7    ; Average 7 periodogram estimates using modified Daniell
pct  = 0.1  ; taper 10% of the data
sdof = specxy_anal (x,y,iopt,jave,pct)
```

Example 2: Calculate the probability level of each coherence-squared returned by specxy_anal.

```    d    = 0                     ; detrending opt: 0=>remove mean 1=>remove mean and detrend
sm   = 7                     ; smoothing periodogram: should be at least 3 and odd
pct  = 0.10                  ; percent tapered: 0.10 common.

; calculate the cross-spectrum
sdof = specxy_anal(x,y,d,sm,pct)   ; sdof is a scalar quantity

c2   = spec@coher            ; c2(N)
p    = cohsq_c2p(c2, sdof)  ; p(N),  0 <= p  <= 1

; print the values
print(sdof@frq+"   "+c2+"   "+p)
```