# One formula for success

## New mathematical model has implications for medicine and engineering

By Karen Herland

Yogendra Chaubey has taught mathematics for more than three decades.

It’s not every day that you can revamp a 70-year-old mathematical formula and get published in the Proceedings of the Royal Society, produced by the oldest scientific society in the world.

“It was really exciting, Sir Isaac Newton was one of the former presidents of the society,” said Yogendra Chaubey of his publication in Series A of the Proceedings, addressing mathematics and the physical sciences.

Chaubey has been teaching in the Department of Mathematics and Statistics since 1979. His primary interest is in survival analysis: essentially, mathematical modeling to determine the life expectancy of, well, anything. The field develops models capable of accounting for a number of, sometimes interrelated, variables.

In engineering, similar modeling is used in reliability analysis to determine how long a particular infrastructure can function in relation to all of its parts. In medicine, the principles are used in survival distribution to determine life expectancy. Actuarial mathematics uses the concept in insurance calculations.

Chaubey was studying the theory behind the modeling distribution while on sabbatical in 2008. “I’ve been interested in this for a long time.”

He was building on Khintchine’s theorem, developed by a Russian mathematician in 1938, that deals with representation of distributional patterns of symmetric data where the numbers will plot on the positive side of the scale symmetrically (mirror-image) to the negative side with 0 being the central point. A symmetrical model means that having information related to only positive side of the scale, the full pattern can still be determined. The Gaussian or normal distribution is a prime example of such distribution that forms the basis of a lot of statistical theory.

Last year this theory got some further attention when Govind Mudholkar, at the University of Rochester, developed a concept of different kind of symmetry for values on the positive side of the line (the half-line), without the negative values. This is much more in keeping with the kind of statistical applications used for survival analysis, where the survival data do not have negative values.

Chaubey took the analysis one step further. He was able to develop a model between the values of 0 and 1, relating to the model developed by Mudholkar, that parallels Khintchine’s representation for the symmetric unimodal distributions. In other words, his contribution creates a model within a finite scale, so that the whole pattern can be easily visualized. “We can’t see the pattern all the way to infinity, but we can see it to 1.” Establishing a complete pattern allows those working on probabilistic forecasting to be that much more precise.

Consulting with others in the field he initially thought his contribution might be a footnote to Mudholkar’s work. Instead, there was enough excitement for a new paper to be co-authored, combining their work with another expert on the subject, M.C. Jones, of the Open University in Milton Keynes, U.K.

The implications for this will be seen in the coming years as the model is adapted across disciplines, “We can generate many new distribution patterns and estimate survival information,” said Chaubey, adding that the model could facilitate certain calculations in engineering or medicine.