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[R 3.0 icon] What does PERIODOGRAPH do ? Program PERIODOGRAPH computes and plots a contingency periodogram (Legendre et al., 1981) for a univariate space or time series. The data may be qualitative (nominal), semi-quantitative (ordinal), or quantitative. Quantitative and semi-quantitative data must first be divided into classes before computing this periodogram; the program takes care of this division, following an optimality criterion. In the periodogram itself, the contingency statistic is computed for all periods in the observation window, that is, periods T = 2 to T = n/2 where n is the length of the data series; in the Macintosh version, the user may choose a narrower computation window. Legendre & Legendre (in 1984a, vol. 2, and more briefly in their 1983 book ), as well as the above-mentioned paper, provide more details on the method. Besides its capacity to analyze semi-quantitative or qualitative data series, the method also allows to analyze short series, which is not the case with the Schuster periodogram or spectral analysis, for example. The method also allows to analyze multivariate time series, by first computing a multivariate partitioning of the data series (through clustering), followed by periodogram analysis of the resulting data partition, as proposed in the 1981 paper ; however, one should prefer to compute a Mantel correlogram (see program MANTEL) instead of a contingency periodogram in this case. Another advantage of the Mantel correlogram method is that it is not restricted to data with a constant sampling interval. Dividing a quantitative or semi-quantitative variable into classes is accomplished by a procedure which optimizes the following two criteria, in such a way as to take tied values (ex aequo) of the data series into account:
  1. For a given number of classes, one minimizes the sum of within-class sums of squares; this part of the computations is done either on the raw data, or on ranks.
  2. The program selects the number of classes that maximizes the amount of entropy per class.

A stepwise algorithm, transcribed into procedure APPROX of the program, is described in the Legendre et al. (1981: 969-973) paper. That procedure first looks for the two-class partition which minimizes the first criterion; then, keeping the first division fixed, one looks for a second cutting point which would create three classes minimizing again the minimum variance criterion; and so on until the second criterion is maximized. A second algorithm has recently been created by A. Vaudor. This method, translated into procedure EXACT of the program, finds at each step the optimal partition of the observations into k classes, independently of the class limits found during the previous step; the partition that maximizes the amount of information per class is retained. The program uses procedure EXACT whenever possible. One should notice that with this algorithm, the second criterion is often optimized for three classes. The user can always impose another number of classes if she so wishes.

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Last updated on Sunday, August 01, 2010 by Philippe Casgrain