Computes sample auto-covariances
function esacv ( x : numeric, mxlag  : integer ) return_val : numeric
An array of any numeric type or size. The rightmost dimension is usually time.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).
An array of the same size as x except that the rightmost dimension has been replaced by mxlag+1. Double if x is double, float otherwise.
Computes sample auto-covariances using the equations found in Chatfield
[The Analysis of Time Series, 1982, Chapman and Hall].
Missing values are allowed
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)*(q(t+k)-qAve)]/(N-1) ; autoThe 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) x(N), y(K,M,N) c(K,M,mxlag) x(I,N), y(K,M,N) c(I,K,M,mxlag) x(J,I,N), y(L,K,M,N) c(J,I,L,K,M,mxlag)special case when dimensions of all x and y are identical:
x(J,I,N), y(J,I,N) c(J,I,mxlag)When calculating lag auto-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.
The following will calculate the auto-covariance for a one dimensional array at 11 lags (0->10). The result is a one-dimensional array of length 11.
acv = esacv(x,10) ; acv(0:10)Example 2: The following will calculate the auto-covariance for a three-dimensional array x(nlat,nlon,time) at mxlag + 1 lags (0->mxlag). The result is a three-dimensional array of size(nlat,nlon,mxlag+1)
mxlag = 10 acv = esacv(x,mxlag) ; acv(nlat,nlon,mxlag+1)