Positions along transect difference should be large at points where the left-hand and right-hand halves of the window contain values that are appreciably different, i.e. where discontinuities occur in the series. Various statistics may be used in the computations:
• With univariate data, calculate the absolute value of the difference between the means of the values in the left-hand and right-hand halves of the window: Statistic = \x1 - x2|.
• With univariate data again, one may choose to compute the absolute value of a t statistic comparing the two halves of the window: Statistic = \x1- x2|/sx. If one uses the standard deviation of the whole series as the best estimate of the standard deviations in the two halves (assuming homoscedasticity), this statistic is linearly related to the previous one. Alternatively, one could use the regular t-statistic formula for t-tests, estimating the variance in each window from the few values that it contains; this may lead to very unstable estimates when windows are narrow, which is often the case with this method.
• For multivariate series, compare the two halves of the window using either the Mahalanobis generalized distance (D5 or D5, eq. 7.40), which is the multivariate equivalent of a t statistic, or the coefficient of racial likeness (D12, eq. 7.52).
The width of the window is an empirical decision made by the investigator. It is recommended to try different window widths and compare the results. The window width is limited, of course, by the spacing of observations, considering the approximate interval between the expected discontinuities. Webster's method works best with equispaced observations, but some departure from equal spacing, or missing data points, are allowed, because of the empirical nature of the method.
Numerical example. A series of 40 observations was generated using a normal pseudorandom number generator N (5,1). The values of observations 11 to 30 were increased by adding 3 to the generated values in order to artificially create discontinuities between observations 10 and 11, on the one hand, and observations 30 and 31, on the other. It so happened that the first of these discontinuities was sharp whereas the second was rather smooth (Fig. 12.20b).
Webster's method for univariate data series was used with two window widths. The first window had a width of 4 observations, i.e. 2 observations in each half; the second window had a width of 8 observations, i.e. 4 in each half. Both the absolute value of the difference between means and the absolute value of the t statistic were computed. The overall standard deviation of the series was used as the denominator of t, so that this statistic was a linear transformation of the difference-between-means statistic. Results (Fig. 12.20c, d) are reported at the positions occupied by the centre of the window.
The sharp discontinuity between observations 10 and 11 was clearly identified by the two statistics and window widths. This was not the case for the second discontinuity, between observations 30 and 31. The narrow window (Fig. 12.20c) estimated its position correctly, but did not allow one to distinguish it from other fluctuations in the series, found between observations 20 and 21 for instance (remember, observations are randomly-generated numbers; so there is no structure in this part of the series). The wider window (Fig. 12.20d) brought out
D5 to the centroid the second discontinuity more clearly (higher values of the statistics), but its exact position was not estimated precisely.
Ibanez (1981) proposed a related method to detect discontinuities in multivariate records (e.g. simultaneous records of temperature, salinity, in vivo fluorescence, etc. in aquatic environments). He called the method D5 to the centroid. For every sampling site, the method computes a generalized distance D5 (eq. 7.40) between the new multivariate observation and the centroid (i.e. multidimensional mean) of the m Window previously recorded observations, m defining the width of a window. Using simulated and real multivariate data series, Ibanez showed that changes in D25 to the centroid, drawn on a graph like Figs. 12.20c or d, allowed one to detect discontinuities. For multi-species data, however, the method of Ibanez suffers from the same drawback as the segmentation method of Hawkins & Merriam: since the simultaneous absence of species is taken as an indication of similarity, it could prevent changes occurring in the frequencies of other species from producing high, detectable distances.
McCoy et al. (1986) proposed a segmentation method somewhat similar to that of Webster, for species occurrence data along a transect. A matrix of Raup & Crick similarities is first computed among sites (S27, eq. 7.33) from the species presence-absence data. A "+" sign is attached to a similarity found to be significant in the upper tail (i.e. when ahi is significantly larger than expected under the random sprinkling hypothesis) and a "-" sign to a similarity which is significant in the lower tail (i.e. when ahi is significantly smaller than expected under that null hypothesis). The number of significant pluses and minuses is analysed graphically, using a rather complex empirical method, to identify the most informative boundaries in the series.
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