Statistical background

There are three kinds of lies: lies, damn lies and statistics.

Benjamin Disraeli (1804-81)

There is a fundamental presentational problem in discussing how to examine the evidence of cyclic behaviour in any stream of data recorded at regular intervals. This is that such examination usually requires some ferocious statistical analysis. Whether we are considering weather data or any other data recorded at set times (e.g. economic series), there is no way we can avoid this statistical approach. But to make it easier to present the underlying analytical techniques, the mathematics will be kept to a minimum in this chapter. This approach does, however, run the risk of glossing over the complexities of the analysis and giving the impression that the statistics can be put on one side. So to understand the problems of sifting through the evidence it is necessary to consider not only the description provided in this chapter but also the basic mathematics given in Appendix A. Failure to recognise the need for mathematical rigour can result in the reader being led up the garden path. It is important to belabour this unpalatable fact, as many of the published examples of 'weather cycles' have wittingly or unwittingly been the product of superficial or selective analysis of the available data.

Bearing in mind these words of warning, we must now consider the examination of the evidence ofweather cycles. As explainedin Chapter i,the scale of the effort that has been devoted to the search for cycles is massive. So in addressing the results of this work and deciding what conclusions can be drawn from this effort, we have to consider first the nature of the data that has been collected and how it can be analysed. To do this we must start with the basic properties of time series.1

2.1 Time series

Any physical variable that is sampled at set constant time intervals can produce a time series. In the case of the weather, a series could consist of temperature or pressure measurements sampled continuously or every so many minutes or hours, or the amount of precipitation in successive equal time intervals. For the purposes of this book, which deals principally with cycles of periods longer than a year, we will be looking at series consisting of data that has been averaged over periods of a month or longer. So we will mostly be considering monthly or annual figures of average temperatures, pressure values or rainfall amounts for given locations or geographical areas.

The use of average figures in inevitable when we turn to indirect (proxy) data. Tree-ring widths, ice-core measurements, sedimentary records, cereal prices and wine harvest dates all by their nature will usually contain information about the integrated effect of a number of meteorological variables over a year or more. While it is possible to extract some seasonal information from variations of the form of, say, individual tree rings or the amount of snow making up a single annual layer in an ice core, and its properties, the amount of fine detail is inevitably limited. Given the emphasis on finding evidence of cycles of periods longer than a year, this is not a major drawback. It does, however, impose limitations on what can be extracted from the data, and it is essential that the effect of the form of the series is fully understood, otherwise there is a danger of falling into elementary traps.

The starting point for considering how much information can be extracted from recorded data is the fundamental property of time series. As a result of the work of the French mathematician Jean-Baptiste

1 Much of this chapter consists of an attempt to encapsulate the essence of a variety of more detailed analyses of the statistics of time series and their spectra. So, rather than attempting to provide detailed references to these texts in this chapter the reader is advised to check the sources identified in the statistics section of the annotated bibliography. Only where recent new developments have been introduced into the study of time series will specific references be cited.

1st harmonic +

2nd harmonic +

3rd harmonic

Time series

4th harmonic +

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