Monday, September 8, 2014

The Art and Science of Mathematical Modeling

                           Drawing of Water Lifting Device by Leonardo da Vinci

The renown social psychologist Daniel Kahneman, co-winner of the Nobel Prize for Economics in 2002, argues that humans are hard wired to find cause and affect relationships, even when one does not exist (e.g., “clustering illusion” like random bombing of London by Germany in World War II). Humans take comfort in believing that we can discover patterns or relationships that can be used to accurately predict, exploit, and even change future outcomes.  We have a proven history of constructing mathematically-based computer models that by executing a series of instructions accurately simulate and predict systems of interest, even complex ones.  However, despite our best attempts, models remain gross simplifications of complex real-world systems, and are thus inherently prone to prediction errors. 

The huge disparity in complexity between the real world and our most advanced mathematical models is illustrated in the book “Probably Approximately Correct”, written by the pre-eminent computational theorists, Dr. Leslie Valiant.  In it, he recounts the following story:

“In 1947 John von Neumann, the famously gifted mathematician, was keynote speaker at the first annual meeting of the Association for Computing Machinery.  In his address, he said that future computers would get along with just a dozen instruction types, a number known to be adequate for expressing all of mathematics.  He went on to say that one need not be surprised at this small number, since 1,000 words were known to be adequate for most situations in real life, and mathematics was only part of life, and a very simple part at that.  The audience reacted with hilarity.  This provoked von Neumann to respond:  “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” 

Still, mathematical models are developed, sometimes at significant cost and effort for a very good reason; because even with relatively simple assumptions and mathematics, they often predict much better than “dead reckoning” or “gut feelings”.  The benefits include higher returns and security with reduced costs and risks, which transfers enormous benefits to the user and often society.  Models when developed and used properly limit inherent human biases and emotions and provide a more objective framework for simulating and understanding a system of interest.  To this end, we use models in a variety of real-world applications, including forecasting of things like weather, forest fires, and water demand.  Sometimes the models are extremely accurate; other times, we are striving only to increase our forecast from a “best guess” to a nominal improvement, where even a small increase in forecasting accuracy produces significant benefits over the long run.  
Modeling is often as much of an art as it is a science.  On the science side, a fundamental understanding of the governing physical laws or processes is necessary for not only constructing an acceptable model, but also understanding its limitations, and the conditions under which it may fail.  There are often multiple approaches and varying levels of complexity that may achieve similar modeling performance and capability.  Being able to construct an appropriate model that produces desired performance often requires a combination of expertise, experience, and flexibility.  A dose of humility also helps. 

 Like mathematics and computational theory, modeling is evolving, and as prediction needs and expectations change, new creative modeling methodologies will continue to emerge that improve and expand modeling capabilities.  Many models today are capable of providing high prediction accuracy for complex systems, which can help humans better manage resources, even helping avert possible adverse outcomes, including disasters.   In water resources, modeling has helped minimize potentially devastating consequences like saltwater intrusion in Orange County, California and land subsidence in Venice, Italy.   At its best, modeling can help safeguard our environment, our water, even our cultural and artistic treasures.