Tuesday, August 13, 2013

How Vector Average Would Sum to Zero

In my post Consensus Averaging vs. Vector Averaging, I mentioned that adding two opposing vectors could produce undefined results. In a wind vane data averaging scenario, this would be extremely unlikely.

Assume the following conditions: A sample interval of 1 second, an average interval of 2 minutes, yielding 120 samples per average result. Incoming data is aligned on 16 compass headings:

It is quite possible for any two vectors to sum to zero. But we have 120 vectors to sum. In order to cancel out completely, we would require one of the following:
  • 60 vectors symmetrically distributed on either yellow or violet line.
  • 30 vectors symmetrically distributed on both the yellow and violet lines.
  • 30 vectors symmetrically distributed on any red, green or blue rectangle.
  • 15 vectors symmetrically distributed on all points.
  • Other permutations of symmetrical distributions on the various lines and rectangles and their rotations.
  • Equal numbers of samples from the northeast and northwest will result in a north vector that could be canceled by some number of south vectors. 
This list goes on, but when the wind is blowing, by far the most common scenario will have chaotic or turbulent input data in which more than half of the input data will be in the general direction of the prevailing wind, providing a dithered average with good resolution.

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