to reconstruct our power law £orm, 



7 - ax^ 



Recalling that the method of least-squares minimizes the total sum 

 of squares of residual distance from the observed data to the resultant 

 regression curve, the method just described minimizes these distances In 

 log-log space. The solution derived In this manner does not minimize 

 the total residual distance In linear-linear space, although the esti- 

 mates might be quite close. Methods exist for performing the least- 

 square fit In either log-log or linear-linear spaces, and these are dis- 

 cussed here. The choice of method and the use of weighting schemes 

 depend on the distribution of the data being fit as well as the distri- 

 bution of the estimation error. In all cases, the error is assumed to 

 reside In the amplitude estimates, rather than In the Independent 

 variable. 



In 'fitting the energy envelope estimates produced In the delinea- 

 tion of stationary provinces (see Chapter 5), the regression must be 

 performed in linear-linear space without weighting. The errors asso- 

 ciated with the envelope estimates (dependent variable), do not depend 

 on frequency band (independent variable) and should not be log-trans- 

 formed. In the case of amplitude spectra however, it was shown by 

 Blackman and Tukey (1958) that estimation error is related to the chl- 

 squared distribution and that error bars remain constant in log-log 

 space. Under these conditions, the regression analysis is optimally 

 performed on log-transformed data. 



125 



