The Wisdom of Crowds or Can many mediocre measurements produce a single good one?

In the introduction of his excellent book "The Wisdom of Crowds" James Surowiecki tells the story of the British Scientist Francis Galton who went to a country fair in 1906. There, a guessing game took place where one had to guess the weight of a bull.
800 people purchased a ticket and delivered their estimates on paper. Galton, who was curious about all kinds of things (I already told you he was a scientist, right) borrowed the tickets afterward and analyzed the results. He was expecting the average of the guesses to be far off because for each expert in the crowd (like for example a butcher) there were for sure a couple of inexperienced people. However, to his great surprise, the average guess of 1197 pounds was very close to the real result of 1198 pounds! This story, by the way, also documents the increase in weight of livestock - today a bull would be around twice as heavy!

If you are interested in the further aspects of collective human intelligence, I recommend reading the book:

James Surowiecki, The Wisdom of Crowds, Anchor, 2005.

But I was less interested in people, instead, I was wondering if this can be used for combining sensor measurements.

A couple of years ago I worked on a method for combining measurements from sensors with different accuracy. Translated to the story above this would mean if we know who are the experts and who are not, should we even bother to include the results of the latter? Actually, the answer is yes - given that the estimates have low correlation! But other than in the story above, the best way is to do a weighted average of values. The weights are derived from the error variance of the estimates, so a sensor with high error variance should get a low weight and a sensor with low error variance should get a high weight.

The resulting formula is surprisingly easy:

The paper explaining the approach in detail and showing how this can be integrated into a sensor network can be found here (link leads to freely accessible PDF):

W. Elmenreich. Fusion of continuous-valued sensor measurements using confidence-weighted averaging. Journal of Vibration and Control, 13(9-10):1303–1312, 2007. (doi:10.1177/1077546307077457)

So, just in case you have to guess the weight of a bull at a country fair, remember this approach :-)