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esccv

Computes sample cross-covariances.

Prototype

	function esccv (
		x         : numeric,  
		y         : numeric,  
		mxlag [1] : integer   
	)

	return_val  :  numeric

Arguments

x

An array of any numeric type or size. The rightmost dimension is usually time.

y

An array of any numeric type or size. The rightmost dimension is usually time. The size of the rightmost dimension must be the same as x.

mxlag

A scalar integer. It is recommended that 0 <= mxlag <= N/4. This is because the correlation algorithm(s) use N rather than (N-k) values in the denominator (Chatfield Chapter 4).

Description

Computes sample cross-covariances using the equations found in Chatfield [The Analysis of Time Series, 1982, Chapman and Hall]. Missing values are allowed.

Currently, to calculate positive and negative lags, one must invoke this function twice. (See example 3 below.)

Algorithm: Here, q(t) and q(t+k) refer to the rightmost dimension. k runs from 0 to mxlag.

     c(k) = SUM [(q(t)-qAve)*(s(t+k)-sAve)}]/N 
     
The dimension sizes(s) of c are a function of the dimension sizes of the x and y arrays. The following illustrates dimensioning:

        x(N), y(N)          c(mxlag+1)
        x(N), y(K,M,N)      c(K,M,mxlag+1)
      x(I,N), y(K,M,N)      c(I,K,M,mxlag+1)
    x(J,I,N), y(L,K,M,N)    c(J,I,L,K,M,mxlag+1)
    
Special case when dimensions of all x and y are identical:
    x(J,I,N), y(J,I,N)      c(J,I,mxlag+1)
    
When calculating lag cross-covariances, Chatfield (pp. 60-62, p. 173) recommends using the entire series (i.e. all non-missing values) to estimate mean and standard deviation rather than (N-k) values. The reason is better mean-square error properties.

There are trade-offs to be made. For example, it is possible that covariance coefficients calculated using qAve and qStd based on the entire series can lead to covariance coefficients that are > 1. or < -1. This is because the subset (N-k) points might be a series with slightly different statistical characteristics.

Thus, the qAve, sAve, qStd, sStd, qVar estimates are calculated using the entire series. In the case of, say, esccv, the qAve, qStd could be based upon L non-missing values while the sAve, sStd could be based upon M non-missing values. It is felt that this approach yields the best statistical estimates of these quantities.

If the user has two one-dimensional series with missing values where the lag cross-covariance at zero lag is desired and the user wishes the lag-0 covariances to be calculated based upon indices when q and s are both present, then use the following approach:

    good = ind(.not.ismissing(q) .and. .not.ismissing(s))  
    esccv(q(good),s(good),0))
    
This only works at lag zero.

See Also

esacv, esacr, esccr, escorc, escovc

Examples

Example 1:

The following will calculate the cross-covariance for a two one-dimensional arrays x(N) and y(N) at 11 lags (0->10). The result is a one-dimensional array of length 11.

        acv = esccv(x,y,10)   ; acv(0:10)
     
Example 2:

The following will calculate the cross-covariance for two three-dimensional arrays x(lat,lon,time) and y(time,lat,lon) at mxlag + 1 lags (0->mxlag).

     mxlag = 10
     acv   = esccv(x,y(lat|:,lon|:,time|:),mxlag) ; acv(nlat,nlon,mxlat+1)
     
Example 3:

To calculate both positive and negative lags, it is necessary to do the calculation twice.

     mxlag    = 9
     x_Lead_y = esccv(x,y,mxlag)
     y_Lead_x = esccv(y,x,mxlag)    ; switch the order of the series

     ccv = new ( 2*mxlag+1, float)    
     ccv(0:mxlag-1) = x_Lead_y(1:mxlag:-1)  ; "negative lag", -1 reverses order
     ccv(mxlag:)    = y_Lead_x(0:mxlag)     ; "positive lag"
     

Remember that NCL is C based and does not allow negative subscripts like Fortran does (in Fortran, ccr(-mxlag:mxlag)). Thus in the above example ccr(0:mxlag-1) corresponds to lags -9,-8,...,-2,-1 and ccr(mxlag:2*mxlag) corresponds to lags 0,1,2,...,8,9.