Record Curves Details
This is a follow up to the Time Ratings page. It provides details on the curve fit procedure as well as links to the fit parameters.
The ratings calculation shown here was inspired by the one used by the Great Bay Masters swim team, as explained here and applied to the New England LMSC. Our system is similar but we have made some modifications that result in slightly different ratings for a given performance.
Curve Fitting Procedure
The record curves are a fitted exponential function of the world best times:
record = b0 + b1*eb2*age
where the record is in seconds and the age is in years. This is a nonlinear regression model; the Solver add-in was used in Excel 2003 to obtain least-squares estimates of b0, b1 and b2. The Excel Solver uses a Generalized Reduced Gradient nonlinear optimization developed by Lasdon and Waren.
For SCY, the USMS national records were used to construct the record curves. It was not attempted to determine the exact ages of the record-holders at the times of their swims; instead, the "middle" age for each age group was used (e.g., 42 years for the 40-44 age group). For SCM and LCM records, the better of the FINA World Records or the USMS National Records was used to construct the record curves.
In the SCY data, only records from 25-29 thru 85-89 age groups were used. It was felt that the records from 18-24 and 90-94 (and older) age groups had too much uncertainty; in other words, there are generally fewer swims for most events in these age groups. Since records in the very young and very old age groups have high leverage (ie, great influence on the least-squares estimates), the decision was made to omit them.
In the SCM and LCM data, the youngest age group used was 25-29, the youngest age group recognized by FINA. The oldest age group used in constructing the fitted record curves was either the 85-89 or 90-94 age group. The decision on whether to include the latter age group was made on a case-by-case basis, based on a subjective evaluation of the effect of inclusion on the fitted curve (via the sum of square residuals). No attempt was made to develop an objective basis for inclusion (e.g., the value of the "leave-one-out" residual).
In all cases, the decision to omit extreme (young or old) age groups was made in an attempt to improve the fit significantly for the remaining age groups.
The fits of the record curves to the records may be examined for SCY, SCM, and LCM (Acrobat Reader required). If one knows to look for it, it is usually fairly obvious when one of the extreme points has been omitted from the fit.
Record Curve Parameters
These were obtained by records that were established by 8-8-07. They are collected in this Excel file.
For More Information
Please address any questions or comments to Chris Stevenson.