The Detection of Weekly Preferential Occurrences with an Application to Rainfall

Abstract

The detection of weekly preferential occurrences in atmospheric and hydrologic processes has recently attracted much attention as a way to identify the signature of anthropogenic climatic changes. The interpretation of previous analyses, however, is not unequivocal, in part as a result of a lack of widely accepted statistical criteria. Here, a general and exact method to detect the presence of weekly preferential occurrences is developed and applied to long rainfall observations in Marghera, Italy; Philadelphia, Pennsylvania; and Portland, Maine. The method makes use of the fact that, under the null hypothesis of stationarity, the process of event occurrence in the different days of the week is equivalent to the random distribution of a number of balls (the wet days) in a set of boxes (the days of the week). The departure from a homogeneous distribution is then characterized through the probability of the maximum number of balls in a box, which can be computed exactly with no ad hoc assumptions. The new method shows that (i) preferential rainfall weekly occurrences emerge in all cases in the most recent period analyzed (1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006), while they are absent-or are too weak to be detected-in previous years (before 1989); and (ii) the balls-in-boxes approach appears to be more sensitive than Pearson's test when deviations from homogeneity are associated with just one day of the week, a common occurrence in connection with day-of-the-week effects. The results presented help to reconcile previous contrasting studies and to contribute compelling evidence that anthropogenic changes in the local climate have occurred over the past century in urban and industrial areas.

Assessing large-scale weekly cycles in meteorological variables: a review

“…Other authors used some alternative testing procedures with less restrictive assumptions for the analysis of WCs (Hendricks Franssen, 2008;Laux and Kunstmann, 2008;Barmet et al, 2009;Hendricks Franssen et al, 2009;Marani, 2010). Monte Carlo bootstrapping methods can deal with arbitrary probability distributions and dependencies between time series observed at different meteorological stations.…”

Detection of weekly cycles in atmospheric data

“…So, to reduce ambiguity, Marani's test is referred to here as the max{ n i } test. Marani () recognizes that the χ 2 test spreads its power thinly, and that the max{ n i } test improves power for a targeted alternative, namely that 1 day of the week has significantly more rainfall days over a threshold than the other six. However, which day is different is not specified, whereas in practice there may be one or two specific days (most likely at the weekend) for which differences from the other days are of greatest interest.…”

“…For the relatively unlikely case that there is no particular idea of the form of a possible weekly cycle, then the usual χ 2 test is recommended. It spreads its power thinly but widely, so has some potential for detecting all types of deviation from equal occurrences on all days of the week. If the main alternative hypothesis of interest is that 1 day will have a higher rate of occurrence than the other 6 days, but it is unknown which day is the different one, then the max{ n i } test proposed by Marani () is recommended. It will be more powerful than usual χ 2 test for this alternative. If the alternative hypotheses can be expressed in terms of one or more ‘contrasts’ as defined in Section 2, then the test proposed by Jolliffe and Primo () will be more powerful than the usual χ 2 test against those alternatives and is recommended.…”

“…Marani () describes and implements two statistical tests for detecting weekly cycles. The setting for these tests is very similar to that found for the rank histogram, which is used for evaluating forecasts produced by ensemble forecasting systems.…”

“…Daniel et al (2012) have cautionary discussions regarding the use of statistical techniques when tests are carried out for multiple spatial and temporal locations, and this is relevant to all types of data. Coakley (2000) shows how autocorrelation can cause the appearance of spurious 7-day cycles, but the format of his data is very different from that considered by Marani (2010). Marani (2010) describes and implements two statistical tests for detecting weekly cycles.…”