Data Reduction Techniques 



OVERLAY METHOD 



Since it was expected that the probability density- 

 curves obtained with the PDA would have a gaussian or 

 nearly gaussian distribution, an overlay with a gaussian 

 curve was used. The curve had parameters of a mean 

 equal to zero and a standard deviation equal to one. Figure 

 5 illustrates the use of this method with two curves, one 

 judged to be gaussian and the other non-gaussian. Some 

 probability density curves obtained with the PDA were 

 judged to be very nearly gaussian. 



One disadvantage of the overlay method is that de- 

 cisions about how well a particular curve compares with 

 the overlay are purely subjective. Skewness and kurtosis 

 can be detected, but the magnitudes of these moments cannot 

 be estimated with accuracy. An extension of the overlay 

 method which will allow estimates of skewness and kurtosis 

 is described here. 



The extension is an overlay with several curves in- 

 stead of just one. Each curve has a different set of values 

 for skewness and kurtosis. The curves are positioned 

 over the actual probability density curve and the parameters 

 are estimated by interpolation between the two closest 

 curves. The curves of the overlay can be computed with 

 the use of Edge worth's series approximation for nearly 

 gaussian distributions. 3 The first four terms of this series 

 are 



fix) = h(x)-—h 3 (x)+-^hUx)+— h e (x) (3) 



where h(x) is the normalized gaussian distribution, h (x) 

 is the nth derivative of h(x), g is the standardized skew- 

 ness, and g is the standardized kurtosis. 



14 



