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CANOCO 3.15
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For further study?

Sources of Order Dependence
Order dependence of the input is disconcerting. It cannot always be avoided
in practice. Order is important when performing addition on your calculator!
Try this little problem:
10^{20} + 1  10^{20}
Does this matter? In most applications, your calculator will give you an
answer which is just fine for practical purposes. Differences between large
numbers is a famous calculator example and it is helpful to have a sense
of when things can go wrong.
In ordinations based on Eigen methods, close eigenvalues is a place to be
careful. Are there others readily characterized?

Ability to converge
CANOCO 3.15 has increased the maximum number of iterations to 999. If CANOCO
(whatever the convergence criteria it uses) does not converge for an axis,
it may converge on an equivalent data set in which the samples are shuffled.
The reshuffling should then result in a slightly higher eigenvalue, at
least in theory [but see the Twins
analogy]. With convergence problems it is always wise to plot the TWIN
axes together.

Unit of Least Precision (ULP)
Ecological data are often very difficult to gather with precision. Ideally
the analyses we perform should be resiliant to such noise. Is the unambiguous
choice of an eigenvalue/eigenvector thanks to strict convergence rendered
ambiguous again if we allow for some noise in the data set?

Analysis of the convergence problems in DCA
A problem in DCA is that twins go undetected, because the eigenvalue problem
solved for the first axis, for example, differs from the eigenvalue problem
solved for the second axis. The second eigenvalue in the eigenvalue problem
for axis 1 may be equal to the first eigenvalue, whereas the second DCA
axis is much lower. Unstable axes in DCA may therefore go unnoticed. In
other words, the reason why there is slow convergence in DCA is not obvious
from the CANOCO output, but is likely to be close eigenvalues at a particular
stage of the algorithm. The Shuffle
software available off the Oksanen web page
may help detecting such unstable eigenvalues. This feature is not unique
to DCA; it is also known in multivariate PLS regression (ter Braak, C.J.F.
& de Jong, S. The objective function of partial least squares regression,
Journal of Chemometrics, in press).
Comments or suggestions about this list? Mail me at <FurnasR@microcomputerpower.com>.
Thanks!
CANOCO 3.15
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