Scene VI Quantifying Nonlinearity, Part and a News Flash

Linearity in Calibration Act III Scene VI - Quantifying Nonlinearity, Part II, and a News Flash 

In Chapters 63 through 67 [1-5], we devised a test for the amount of nonlinearity present in a set of comparative data (e.g., as are created by any of the standard methods of calibration for spectroscopic analysis), and then discovered a flaw in the method. The concept of a measure of nonlinearity that is independent of the units that the X and Y data have is a good one. The flaw is that the nonlinearity measurement depends on the distribution of the data uniformly distributed data will provide one value, Normally distributed data will provide a different value, randomly distributed (i.e., what is commonly found in real data sets) will give still a different value, and so forth, even if the underlying relationship between the pairs of values is the same in all cases. 

Our task now is to come up with a way to quantifying the amount of nonlinearity the data exhibits, independent of the scale (i.e., units) of either variable, and even independent of the data itself. Our method of addressing this task is not unique, there are other ways to reach the goal. But we will base our solution on the methodology we have already developed. We do this by noting that the first eondition is met by converting the nonlinear component of the data to a dimensionless number (i.e., a statistic), akin to but different than the correlation coefficient, as we showed in our previous chapter first published as [5]. 

The second condition can be met by simply ignoring the data itself, once we have reached this point. What we need is a standard way to express the data so that when the statistic in computed, the standard data expression will give rise to a given value of the statistic, regardless of the nature of the original data. 

For this purpose, then, it would suffice to replace the original data with a set of synthetic data with the necessary properties. What are those properties The key properties comprise the number of data values, the range of the data values and their distribution. 

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