CAGE User's Guide

Inverting One-Dimensional Tables

To invert a one-dimensional table:

1. Ensure that your session contains two tables:
1. The first table from your forward strategy, filled
2. The second table from your backward strategy, which you want to fill
2. Highlight the second table.
3. Click  or select Table -> Inversion.

1. The lower pane now acts as a wizard.

4. In the lower pane, highlight the table that you want to invert.
5. Click Next. The next page asks what CAGE should do if it encounters multiple values. The options are
• Maximum selects the uppermost range if a given number has two possible inverses (like selecting the positive square root of a number).
• Minimum selects the lower of the two if a given number has two possible inverses (like selecting the negative square root of a number).
• Intermediate selects the middle range if a given number has more than two possible inverses.
• Automatic selects the range that produces the least error (see below; the last page of the wizard plots the error metric).

1. For example, the function y = x² is impossible to invert over the range -1 to 1. You can specify to invert the range from 0 to 1, sacrificing the inversion in the lower range, or the reverse. To select the range from 0 to 1, highlight Maximum.

   The display shows a comparison between the table (green) and the function x = f^-1(f(x)).

6. Highlight the part of the table to invert, then click Next.

1. The last page of the wizard has a comparison plot that shows how successful the inversion has been. If your forward function is y = f(x), and your inverse function is x = g(y), then, combining these, in an ideal world, you should have x = g(f(x)). The plot then displays a red line showing x against x and a green line showing x against g(f(x)). The closeness of these two lines indicates how good the inversion has been: a perfect inverse would show the lines exactly on top of each other. In the following example, the lines are together and then diverge; this plot can show you which part of your table has not successfully inverted and where you should try a different routine.

   

Note    The automatic inversion routine tries to minimize the total distance between these lines. This can sometimes lead to unexpected results. For example, given the function f(x) = x^2 between -1 and 1, if you select either positive or negative square root as the inverse, this induces a large error in the combined inverse. If you choose g(y) = sqrt(y), then g(f(-1)) = 1, an error of 2. To minimize this, the automatic routine might choose to send everything to zero and accept a medium error over the whole range rather than a large error over half the range. The more knowledge you have of the form of the "forward" table will help you make an informed choice about which routine to select.

7. Click Finish to accept the inversion or Cancel to ignore the result and return to the original table.

Inverting a Table

Inverting Two-Dimensional Tables