Curve45: An Inflection-Point-Bound Function

An S-curve, Curve45, is defined that exhibits several distinguishing properties. It comprises the arcs of two circles making it particularly “regular.” Both the original X values and the resulting Y values are restricted to a common range. The lower value of the range is zero, the upper value is a positive value that is specifiable. The inflection point of Curve45 is bound to the 45° diagonal. Its three parameters are:

The left-hand concave up arc intersects the origin (X = 0, Y = 0) and the 45° diagonal at the specified inflection point. The right-hand concave down arc intersects that inflection point and (X = specified maximum value, Y = specified maximum value).

Curve45 may be particularly applicable in situations where, say, a mean (or median or first quartile, etc.) X value is to serve as a benchmark with X values above the mean yielding disproportionately (compared to linear) large Y values and X values below the mean yielding disproportionately small Y values. That mean X value, of course, defines the inflection point. Curve45 is especially useful where that mean value is determined dynamically for the specific data set, this being done automatically (i.e., computerized) rather than explicit specification by the user.

One example is where annually (or by location or by application type, etc.) disbursements up to some maximum, e.g., $10,000, are to be made. Applications are scored (the scores eventually being transformed to 0–1, i.e., the X values) with the mean score defining the Curve45 inflection point. The overall merit of the applications from one year to the next may change. Using Curve45 ensures that the disbursement to an application with the mean score for the year’s applications will receive a proportion of the maximum amount exactly commensurate with the prevailing overall merit. Applications scored above the mean receive disproportionately greater proportions and those below the mean receive disproportionately lesser proportions. (The Y values are the proportions for the respective applications.) Curve45 may also be suitable for computer-based business simulation games where its regularity might be more assimilable to students than less regularly shaped functions. S-curves, of course, are also used to model actual market response data. Whether a specific S-curve “fits” is an empirical issue.