Saturday, March 22, 2008

Interval Math

While doing research for the Numerics Chapter of my forthcoming Mathematica Cookbook I came across a site devoted to research on Interval Math. Interval Math is an approach from the domain of Numerical Analysis that deals with the fact that all measurements are imprecise by abandoning the representation of measured values by numbers. Instead of numbers, it defines all mathematical operations on intervals.

Mathematica (as of version 5) support real (but not complex) interval math where intervals take the form Interval[{min1,max1}...]. All of the typical mathematical operations and functions are defined for intervals.

Interval math is important for computer systems that must act intelligently in the real world. All sensors are approximate. This is true for man-made devices as well as for our own eyes and ears. If a sensor on a robot returns a particular value there is always an inherent error. Rather than deal with errors by sampling and averaging, interval math allows the error to directly be represented in the values that enter downstream computations. This means all intermediate results track the propagation of errors from multiple sources to yield better information. There also seems to be a relationship between interval computation and fuzzy sets but it I have not located any resources except on paid content sites.

It seems that although the study of Interval math began in the US it is largely forgotten while in Germany it is there are conferences and it is part of the qualifying exams for studies in numerical methods.

Some of the less technical resources on the earlier mentioned site are this introduction, an article from American Scientist and even a movie.

3 comments:

serhat said...

I think the relation between fuzzy logic and interval math is interval-valued fuzzy sets which is a simplified form of type-2 fuzzy sets. In type-2 fuzzy sets the membership grade is also fuzzy. Interval-valued fuzzy sets relax this requirement and map membership grade to an interval in [0,1]. If you interested, www.irit.fr/~Didier.Dubois/Papers/cloudeus.pdf , provides a good summary on interval-based fuzzy sets and its connection with possibility theory.

serhat said...

I think the relation between fuzzy logic and interval math is interval-valued fuzzy sets which is a simplified form of type-2 fuzzy sets. In type-2 fuzzy sets the membership grade is also fuzzy. Interval-valued fuzzy sets relax this requirement and map membership grade to an interval in [0,1]. If you interested, www.irit.fr/~Didier.Dubois/Papers/cloudeus.pdf , provides a good summary on interval-based fuzzy sets and its connection with possibility theory.

Sal Mangano said...

Thanks serhat. I'll add that paper to me reading list.