I was wondering if it is possible to transform data series with positive and negative values to only positive values and still preserve the relationship between the values?
For example if two values were 79.5 and 47.5, their relationship would be that 7.95 is 1.67 times greater than 47.5.
I tried scaling the data series from 0 to 1 but it did not preserve the 1.67 relationship. For example the scaled values were 1 and .75, and 1 is 1.32 times greater than .75.
The series is 21.5 16.5 8.5 6.0 3.5 2.0 0.5 -2.0 -10.0 -13.0 -14.0 -19.5.
The relationship between 21.5 and 16.5 is 21.5/16.5=1.30.
When I scale from 0-1 it is: 1.00 0.88 0.68 0.62 0.56 0.52 0.49 0.43 0.23 0.16 0.13 0.00
The relationship between 1.00 and 0.88 is 1/0.88=1.13.
Consider a particular case of the problem. Say there are two values $a>0$ and $b<0$ such that $\frac ab=k$, and we want to transform them into two new variables $a',b'>0$ with $\frac{a'}{b'}=k$.
This is not possible, as $k<0$ but the division of two positive numbers has a positive result (in other words, $\frac{a'}{b'}>0$). However, it is always possible to scale the series such that it lies within $[-1,1]$ and preserves the ratios between values: divide each member by the member with the largest absolute value.