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February 21st, 2016, 09:58 PM   #1
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From: denver

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How to quantify differences/ similarities between groups of like objects?

First let me apologize if this is not an appropriate forum for this question. Researching all the available SE sites led me here. Also, my background is not in mathematics, so layman's terms are appreciated

The problem:

Given a set of items within a category (e.g. types of fruit), and

Given a table providing similarity values for each item as compared to each other item (how like an apple is a pear?), and

Given a control set, X (a fruit basket containing, e.g. 60 apples and 40 oranges), and

Given an arbitrary group of comparison sets (a fruit basket A containing 30 apples, 10 pears, and 60 strawberries)

Construct a method for determining each comparison set's similarity to the control. It may be assumed that the total number of pieces of fruit in each set, including the control, will be exactly 100.

Continuing the fruit example for five types of fruit (and considering visual likeness alone, i.e. how each type of fruit looks like another), here is the similarity table:

Apple Orange Pear Banana Strawberry
Apple 1.0 0.8 0.6 0.2 0.1
Orange 1.0 0.5 0.2 0.05
Pear 1.0 0.25 0.15
Banana 1.0 0.1
Strawberry 1.0

Control set X: {60A, 40B}

Comparison sets:

A: {100A}

B: {40A, 40P, 20S}

C: {80A, 20O}

D: {60A, 30B, 10P}

As a first approach I thought a simple solution would be to take each type of fruit within a set, compare it with an identical type of fruit in the control, and take the difference for each type. To make this somewhat easier, I fill each set with imaginary fruit where necessary:

A => {100A, 0B, 0P, 0O, 0S}
then simply take the absolute difference between each type of fruit and the control:

dA => {100 - 60A, 40 - 0B, 0P, 0O, 0S}
=> {40A, 40B, 0P, 0O, 0S}
and so on for B, C, D. This works ok for identical types (apples to apples) but ignores the fact that some types of fruit are visually quite similar (apples to oranges). So my next approach was to say "how alike is each type of fruit?". For set B:

X: {60A, 40B, 0O, 0P, 0S}
B: {40A, 0B, 0O, 20P, 20S}
We can compare apples precisely since they appear in both sets, but for the other fruit, we can say "how alike are these?" and "what fruit is most like this one?". But here is where it gets a little sticky. Since there are pears in set B but not in X, and bananas in X but not in B, we could compare pears against bananas:

Pears to bananas: (40B - 20P) * 0.25 = 5
but it seems just as valid to compare everything against apples (or whatever the predominant fruit is in the control).

I think I can also break the methodology down into two pieces: 1:how alike are the individual types of fruit, and 2: how alike are the distributions of each type. I ran into problems with that approach however where certain distributions of un-like fruit ranked higher than less-similar distributions of more-like fruit - which on the face of it seems wrong.

So my question is: is what I'm trying to do just complete madness? Or is this one of those "well understood, just not by me" problems? And if the latter, please help me to understand.

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