Publication Date:
2011-01-26
Description:
In 2007, Castellaro and Bormann (2007) studied the performance of various two-dimensional (2D) regressions between different magnitude scales by mathematical simulations. The study consisted in (1) generating sets of magnitude pairs (chi (sub i) ,gamma (sub i) ) with given true slope beta (sub true) following a Gutenberg-Richter distribution with b=1 and by adding initial errors (u (sub i) ,e (sub i) ); (2) studying how far the slopes beta obtained in those data sets by standard beta (sub SR) , inverted standard beta (sub ISR) , orthogonal beta (sub OR) , and generalized orthogonal beta (sub GOR) regressions were from beta (sub true) . Studies assessing the best regression method are important because the misuse of the common standard regression easily leads to magnitude conversion errors of 0.2-0.3 units. A different approach to the magnitude conversion problem was proposed before Castellaro and Bormann's (2007) work and was based on the chi (super 2) method, which is based on the theory of independent and normally distributed errors (u (sub i) ,e (sub i) ) and x (sub i) . In this work, we derive mathematical explanations for the results of Castellaro and Bormann (2007) in terms of the chi (super 2) method and find that results agree for mean initial errors 〈0.5 magnitude units. Our results demonstrate the importance of knowing and taking into consideration the true initial errors in regression analysis.
Print ISSN:
0037-1106
Electronic ISSN:
1943-3573
Topics:
Geosciences
,
Physics
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