...from the Microcomputer Power website.

CANOCO 3.15  |  <Previous

For further study?

1. 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:

1020 + 1 - 1020

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?

2. 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.

3. 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?

4. 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).