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# crosscov()

Usage:
 ```ccvf <- crosscov(y [,nlags] [,degree:d] [,auto:T] [,cor:T]\ [,full:T or center:T] [,nfreq:M]), REAL matrix y, integer nlags > 0, integer scalar or vector d, integer M > 0 ccvf <- crosscov(y, i, j [,nlags] [,degree:d] [,cor:T] \ [,full:T or center:T] [,nfreq:M]), integers i > 0, j > 0 ```

Keywords: autocorrelation, autocovariance, time domain
```c_yy <- crosscov(y, nlags) computes sample auto- and crosscovariance
functions ccvf(h,y[,i],y[,j]) for lags h = 0, 1, ..., nlags.

The columns of y, a N by ny REAL matrix with no MISSING elements, are
interpreted as time series defined at equally spaced time points.

c_yy will be a nlags+1 by ny by ny REAL array, with c_yy[h+1,i,j] =
ccvf(h,y[,i],y[,j]), h = 0, ..., nlags, where
ccvf(h,y[,i],y[,j]) =
sum((y[l+h,i]-ybar[i])*(y[l,j]-ybar[j]),l=1,...,N-h)/N.
Note the divisor is N, not N-h or N-h-1.  ybar[j] = sum(y[,j])/N is the
sample mean of y[,j].

Note that y[,j] lags h behind y[,i].  Covariances where y[,j] leads h
ahead of y[,i] are in c_yy[h+1,j,i].

crosscov(y) is the same as crosscov(y,nrows(y)-1), computing all
non-degenerate sample autocovariances.

c_yy <- crosscov(y, i, j [,nlags]) computes ccvf(k,i,j) for k =
-nlags,...,0,...,nlags as a vector of length 2*nlags+1.  Without
center:T (see below), results for lags 0, ..., nlags are in rows 1
through nlags+1 and those for lags -1, ..., -nlags are in rows
2*nlags+1, 2*nlags, ..., nlags+2.  If you want ccvf(k,i,j) only for k >=
0, use full:F as an argument.

c_yy <- crosscov(y [,nlags], auto:T) computes only autocovariances.  The
result is a matrix with ny columns with c_yy[h+1,j] = ccvf(h,
y[,j],y[,j]), h = 0,...,nlags, j = 1,...,ny.

crosscov(y, i, j [,nlags], auto:T) is legal only when i = j.

c_yy <- crosscov(y [,nlags] full:T) does the same, except auto- and
cross-covariances are also computed for negative lags.

c_yy will be a 2*nlags+1 by ny by ny array, with results for lags 0
through nlags in c_yy[1,,], c_yy[2,,],..., c_yy[nlags+1,,], and results
for lags -1 through -nlags in c_yy[2*nlags+1,,], c_yy[2*nlags,,],...,
c_yy[nlags+2,,].

c_yy <- crosscov(y [,nlags] center:T) does the same as with full:T,
except the auto- and cross-covariance functions are centered at row
nlags+1.  Specifically, results for lags -nlags, -nlags+1, ..., -1, 0,
1, ..., nlags are in c_yy[1,,], c_yy[2,,], ..., c_yy[2*nlags+1,,]

With auto:T, with either full:T or center:T, c_yy will be a 2*nlags+1 by
ny matrix.

You can specify that columns of y be detrended by subtracting a
polynomial in time using keyword phrase 'degree:d', where d is an
integer scalar or vector of length ny = ncols(y).  A scalar d is
interpreted as rep(d,ny).

c_yy <- crosscov(y [,nlags], degree:d) computes auto- and cross-
covariances of the residuals from polynomial trends fit by least squares
to each column of y.  The values of the fitted polynomial replace ybar
in the definition of ccvf.

d[j] is the degree of the polynomial fit to column j.  When d[j] = 0,
only the sample mean is subtracted (default).  When d[j] < 0, nothing is
subtracted.

You can use keyword phrases 'full:T', 'center:T' and 'auto:T' with
'degree:d'

r_yy <- crosscov(y [,i,j] [,nlags], cor:T ...) computes auto- and cross-
correlations instead of auto- and cross-covariances.  You can use any of
the other keywords with 'cor:T'.

Note that, unlike autocor(), the result includes lag 0 correlations.

The necessary sums of lagged products are computed using discrete
Fourier transforms (DFTs) of length >= N + nlags, the actual length
chosen being M = goodfactors(N + nlags).  See goodfactors().

When working with long time series, it may be possible to find M1 > M
with more small factors resulting in faster DFT computation.  For
example, when N + nlags = 24388, M = goodfactors(24388) = 24389 = 29^3,
but a DFT of length M1 = 24576 = 3*2^13 takes only about 40% as long to
compute as a DFT of length M.

c_yy <- crosscov(y[,i,j] [,nlags], nfreq:M1 ...) uses DFTs of length M1.