Normalization

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Category: Math

Published on Thursday, 27 September 2012 04:50

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Normalization is the process of adjusting a set of related values that use different scales so they use the same scale.  For subjective numerical opinions, such as those gathered in a consensus meeting, it is often convenient to normalize by the maximum.  If we apply this to these data, \( [15, 60, 95, 20] \), we see that the maximum is 95.  Dividing by 95 and rounding to two significant figures, we get the new data set \( [0.16, 0.63, 1.00, 0.21] \). 

The value of this method becomes apparent when evaluators in a consensus meeting are accustom to using different scales.  If one evaluator rates our search areas as \( [20, 50, 90, 10] \), and another rates the same areas as \( [10, 25, 45, 5] \), it seems that we have a disparity.  If we then normalize those values, we can see that our fist evaluator rated the areas as \( [0.22, 0.56, 1.00, 0.11] \), and our second evaluator rated the areas as \( [0.20, 0.60, 1.00, 0.10] \).  By normalizing these values, we can see that the proportionality of these ratings is in very close agreement. 

Normalizing for Probability of Area

For the case of looking for a lost person, we can say that, assuming that this person exists, there is a 100% chance that they are either in the search area or somewhere in the Rest of the World (ROW).  To accurately describe the Probability of Area (POA) for our search segments, we must ensure that the POA for all of the search segments plus the ROW equals 100%.  The original Mattson method accomplished this by only allowing an evaluator 100 "points" to assign to segments.  If this restriction is abandon, and we normalize by the sum.

As an example of this case, we will start with a simple case of 3 values.  This would be analogous to a search area with two segments, and the ROW.  If the search segment evaluations are the set S, and our evaluator scores them, we could normalize them as follows:

\[ S=[100, 90, 80] \]

\[ S_{Normal}=\frac{S}{\sum{S_1,S_2,S_3}} \]

\[ S_{Normal}=[0.37, 0.33, 0.30] \]

\[ POA=S_{Normal}\times100\%=[37\%, 33\%, 30\%] \]

This method has the feature that it maintains the proportionality of the rankings, but puts them in a standardized format.  For the simple case above, this may seem trivial, but for a very large number of search segments, the value becomes more apparent.  For 30 search areas, 

\[ S=[65, 70, 80, 70, 70, 90, 90, 95, 65, 30, 90, 70, 30, 60, 70, 80, 30, 80, 30, 90, 10, 10, 40, 25, 60, 70, 10, 10, 15, 20] \]

\[ POA=[4\%, 4\%, 5\%, 4\%, 4\%, 6\%, 6\%, 6\%, 4\%, 2\%, 6\%, 4\%, 2\%, 4\%, 4\%, 5\%, 2\%, 5\%, 2\%, 6\%, 1\%, 1\%, 2\%, 2\%, 4\%, 4\%, 1\%, 1\%, 1\%, 1\%] \]