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Regression problems for magnitudes

  • Author(s): Castellaro, Silvia;
  • Mulargia, Francesco;
  • Kagan, Yan Y
  • et al.

Least-squares linear regression is so popular that it is sometimes applied without checking whether its basic requirements are satisfied. In particular, in studying earthquake phenomena, the conditions (a) that the uncertainty on the independent variable is at least one order of magnitude smaller than the one on the dependent variable, (b) that both data and uncertainties are normally distributed and (c) that residuals are constant are at times disregarded. This may easily lead to wrong results. As an alternative to least squares, when the ratio between errors on the independent and the dependent variable can be estimated, orthogonal regression can be applied. We test the performance of orthogonal regression in its general form against Gaussian and non-Gaussian data and error distributions and compare it with standard least-square regression. General orthogonal regression is found to be superior or equal to the standard least squares in all the cases investigated and its use is recommended. We also compare the performance of orthogonal regression versus standard regression when, as often happens in the literature, the ratio between errors on the independent and the dependent variables cannot be estimated and is arbitrarily set to 1. We apply these results to magnitude scale conversion, which is a common problem in seismology, with important implications in seismic hazard evaluation, and analyse it through specific tests. Our analysis concludes that the commonly used standard regression may induce systematic errors in magnitude conversion as high as 0.3-0.4, and, even more importantly, this can introduce apparent catalogue incompleteness, as well as a heavy bias in estimates of the slope of the frequency-magnitude distributions. All this can be avoided by using the general orthogonal regression in magnitude conversions.

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