Many years ago, I had a ride of a Kawasaki 500 Mach III 2-stroke motorcycle, which along with its even more horrendous 750cc version was known as the ‘widow-maker’. It was incredibly fast in a straight line, but if it went around corners at all, the rider had long since fallen (or jumped) off!
It also had a very narrow ‘power band’ http://en.wikipedia.org/wiki/Power_band, in that it would have no real power until about 7,000 revs per minute, and then all of a sudden it would whoop and holler like the proverbial bat out of hell, the front wheel would lift, the rider’s jaw drop, and well, you get the idea! In statistical terms, this was a nonlinear relationship between twisting the throttle and the available power.
A somewhat less dramatic example of a nonlinear effect is the Yerkes-Dodson ‘law’ http://en.wikipedia.org/wiki/Yerkes%E2%80%93Dodson_law, in which optimum task performance is associated with medium levels of arousal (too much arousal = the ‘heebie-jeebies’, too little = ‘half asleep’).
Various simple & esoteric methods for finding global (follows a standard pattern such as a U shape, or upside down U) or local (different parts of the data might be better explained by different models, rather than ‘one size fits all’) relationships exist. A popular ‘local’ method is known as a ‘spline’ after the flexible metal ruler that draftspeople once fitted curves with. The ‘GT’ version, Multivariate Adaptive Regression Splines http://en.wikipedia.org/wiki/Multivariate_adaptive_regression_splines. is available in R (itself a little reminiscent of a Mach III cycle at times!), the big-iron ‘1960’s 390 cubic inch Ford Galaxie V8′ of the SAS statistical package and the original, sleek ‘Ferrari V12’ Salford Systems version.
Other nonlinear methods are available http://en.wikipedia.org/wiki/Loess_curve, but the thing to remember is that life doesn’t always fit within the lines, or follow some human’s idea of a ‘natural law’.
For example, freak or rogue waves, that can literally break supertankers in half, were observed for centuries by mariners but are only recently accepted by shore-bound scientists, similarly the black swans (actually native to Australia) of the stock market http://www.fooledbyrandomness.com/
When analysing data, fitting models, (or riding motorcycles), please be careful!
Synergistic Statistical Consulting, Analysis, Arboronics 