# Scaling Cumulative Probability Distribution function values

We have a cumulative probability distribution function (cdf), we want to scale it down for using it in anomaly detection. The mapping should look like this.

CDF value: 0.1 ... 0.5 ... 0.9 ... 0.99 ... 0.999 ... 0.9999 ........

Mapped to: 0.001 ... 0.002 ... 0.01 ... 0.2 ... 0.3 ... 0.5 ........

So CDF greater than 0.9 is relevant and then the values should start increasing rapidly. The mapped scale is between 0 to 1, with 0.2 being strong anomaly and 1 being extreme anomaly.

Is there any standard function to scale CDF values, in above described manner?

Also is this a standard Statistics approach, or this approach is used in Anomaly Detection? (Any references will be helpful)

• How about polynomal mappings? $x\mapsto x^2$ comes to mind as the most usual, but increasing the exponent will make it increasingly sharp. Your values are quite extreme so maybe $x^{20}$ provides a good scaling. – AlexR Mar 4 '15 at 23:00
• It gives a good approximation but not exactly the same thing, as you can see 0.999 and 0.9999 have high difference in their mapping values. Using simple polynomial mappings, can't help it. – rg41 Mar 4 '15 at 23:04
• Roots, maybe? Think of $1-\sqrt[n]{1-x}$ will not even be differentiable at $1$. – AlexR Mar 4 '15 at 23:06
• This seems to be plausible solution (not exact kind of results though), but are there any standard approaches for this? Also, are there any references where people use this for anomaly detection? – rg41 Mar 4 '15 at 23:11
• I guess most of the time you'll just highlight significance levels of a certain magnitude. Not an expert on the subject, though. – AlexR Mar 4 '15 at 23:12