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What does CHRONO do ?
The chronological clustering proposed by
Legendre, Dallot
& Legendre (1985) is computed by program CHRONO. This clustering
method, which had first been described for multivariate time series,
can also be used to segment spatial series
(Galzin & Legendre,
1987). The non-hierarchical method uses a hierarchical
proportional-link linkage algorithm whose connectedness level
(Co) is determined by the user as an answer to a question of
the program; it is the test of significance, described in the next
paragraph, that makes the method non-hierarchical. The constraint of
spatial or temporal contiguity imposed to the clustering results means
that only objects or object groups that are adjacent along the
series may eventually groupements. Notice that it is unlikely that changing
the connectedness would produce a major change in the clustering
results, as can be seen in the examples of the
Legendre, Dallot &
Legendre (1985) paper.
At each step of the agglomeration, a permutation test is performed
to decide whether a fusion should be made between the two groups whose
fusion is proposed by the agglomerative algorithm. The null hypothesis
of that test is explicitly described in the output of versions CMS and
VMS:
H is the probability that the null hypothesis is true. The null hypothesis says that the two groups being tested are an artifact and should be fused in a single group. Fusion occurs if H is larger than the probability level ALPHA set by the user (above).Answering a question of the program, the user must determine the alpha rejection level of the null hypothesis (often chosen values are 0.01, 0.05 or 0.10; one may choose to use a higher level in order to identify singletons -- see below, as well as the example). One must realize, though, that this is not a genuine test of statistical hypothesis, since the data used during the test are the same as those from which the hypothesis of division into groups has been generated. Numerical simulations described in the main reference have shown, however, that for random data sets, the probability for this test of producing a significant result is equal to the preselected alpha value. The program allows to identify singletons, which are aberrant samples found along the data series. Because of the constraint of contiguity imposed on the algorithm, the presence of a singleton may prevent the formation of a group that should have included objects from both sides of the aberrant sample. At least three reasons may produce such aberrant samples: (1) random events, such as modified strata in sediment cores, or else movements of water masses during repeated samplings at the same station in aquatic environment; (2) improper sampling or inadequate preservation of the samples before they are analyzed; (3) extreme stochastic variations, which lead to rejecting the null hypothesis while no break has occurred in the succession (type II error). If the user requests to identify the singletons, the clustering will be interrupted, and started again from the beginning after removing the singleton (see example); the only exceptions to this rule are the singletons located at the beginning or the end of the data series, since no group is interrupted by their presence. It is unlikely that singletons will be identified if the alpha level is low (less than 10%), because it then becomes difficult, when testing a single object against p, to obtain a value which is smaller than that in the first column of Table 1. Final rule: if an object has a similarity of zero with all its immediate neighbors, the agglomerative algorithm does not go down to level S = 0 to force such an object to pertain to a group; these unclustered objects are represented by dashes (-) in the final solution, or by a white square in the Macintosh output graph. It is recommended to check the data for any object coming out with that symbol; if its presence in the series seems to have interrupted a group, this object may either be removed from the analysis if it is considered aberrant or exceptional (which may have given it a null similarity with its neighbors).
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