Optimum Design of Experiments

Posts Tagged ‘optimal

What is Optimal Design of Experiments?

In a nutshell: Optimal Design of Experiments (ODOE) is a variance reduction technique to reduce the variance of estimators.
The variance of the standard error { \rm SE}_{ \bar x} of the mean \bar x of a random variable x with standard error {\rm SE}_x scales as
{\rm SE}_{\bar x} = \frac{ {\rm SE}_x}{\sqrt{N_m}}
where N_m is the number of repeated measurements.

History

It is often believed that Galileo Galilei’s merit was his insistance on the heliocentric world view. However, this is a persistent misbelief. Rather, Galilei was the first scientist who verified his mathematical description of reality by experiment. Not without reason Galilei is called the father of modern science.

According to a well-known legend he dropped objects of varying weight from the leaning tower of pisa to verify that the gravitational acceleration is independent of the weight. But why did Galilei supposedly climb the leaning tower of pisa and not his kitchen table? It is quite likely that he has tried that. However, it is not so easy to find out which object hits the floor first: the measurement has an error. If Galilei had known about error analysis he could have repeated this experiment many times. The theory predicts that to reduce the error of the estimated value by a factor 10 one has to do 100 times more measurements. One hundred, that seems reasonable, but to reduce the error by a factor 100 means 10000 measurements!!

Thus, Galilei had find a possibility to reduce the influence of his measurement error. His idea was to increase the falling time. It is verified historically that Galilei used a sloping plane. He measured the distance that cannon balls of varying weight pass in a certain time. This is basically the same idea as to climb the leaning tower. The important point is: Galilei optimized his experiment to be able to estimate the gravitational acceleration with as few experiments as possible even when the measurement devices at hand were very inaccurate (Galilei used his pulse to measure time).

Related Fields:

Active Learning (part of Machine Learning):
http://hunch.net/?p=49

In psychology:
http://www.phil.uni-sb.de/~jakobs/seminar/vpl/prolog.htm

In Chemistry:
http://staff.chemeng.lth.se/~BerntN

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  • Isaac Gouy: Note the benchmarks game also shows full source code and build & run logs for all the programs, for example http://shootout.alioth.debian.org/g