The Circular Normal Distribution: Theory and Tables

'The recognition of periodicities, their amplitude and length, is one of the main tasks of the application of statistics to long-range data... The question of how to recognize a cycle remains open, since the complicated mathematical methods used in the periodogram and correlogram analysis may g...

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Veröffentlicht in:Journal of the American Statistical Association 1953, Vol.48 (261), p.131-152
Hauptverfasser: Gumbel, E. J., Greenwood, J. Arthur, Durand, David
Format: Artikel
Sprache:eng
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Zusammenfassung:'The recognition of periodicities, their amplitude and length, is one of the main tasks of the application of statistics to long-range data... The question of how to recognize a cycle remains open, since the complicated mathematical methods used in the periodogram and correlogram analysis may generate ... fictitious cycles and conceal real ones.' An examination of a specialized distribution function which can be used with small chance of generating false cycles if used in connection with data of known periodicity. The type of data for which circular distributions are applicable is exemplified by certain types of events occurring during a year: the date of each event is considered as the random variable, alpha, and the distribution of events over the year may be considered a circular distribution of an angle (the angle is a mapping of alpha into the interval 0 - 2*(pi)). A circular distribution has been derived by R. V. Mises applying Gauss' method of deriving the normal distribution to a circular variate. This is called the circular normal distribution because of its similarity in derivation to the normal distribution, and because of certain properties which it alone shares with the linear normal distribution. For this distribution, the following tables are presented in extensive form: (1) the concentration parameter (the value of which gives evidence of a cycle) as a function of the vector strength at the mode; (2) the probability density function; and (3) the areas of the distribution function. J. Coleman.
ISSN:0162-1459
1537-274X
DOI:10.1080/01621459.1953.10483462