Quantifying Aggregate Vector Strength/Vector Arithmatic

2009-10-28T23:27:26Z

Say I have 5 vectors and I measure the similarity of each one to a fixed reference vector using cosine similarity. But now what I want to do is understand the aggregate or collective strength of these 5 vectors as it relates to how similar they are collectively to the reference vector. Is this simply an average of the 5 cossim values I calculate; or do I add up each scalar position of the 5 vectors together and then do a cossim; or do I average the scalar positions of the 5 vectors and then do a cossim. Or is there a better way to measure the collective similarity (or strength) of the 5 vectors to the reference vector? Thanks for any insight that you may have !

Additional Context So lets say that each of the 5 vectors represent 5 sentences in a document that are considered the most salient bits of information in the document on a topic like say Pneumonia. The question I am asking of the 5 vectors (sentences) is how likely or strong do these 5 sentences represent a positive Pneumonia case? I don't want to consider all sentences in the document because typically these documents are super noisy. What I want is a relative strength or conviction of the 5 vectors (sentences) to represent a Pneumonia case. But how do I determine their aggregate strength? I can measure each vector (sentence) relative to a reference vector for Pneumonia but how do I aggregate the separate results?

Or maybe can I somehow quantify the cluster density around the reference vector?

Answer by Martin M. W. for Quantifying Aggregate Vector Strength/Vector Arithmatic

2009-10-29T00:06:25Z

I think the answer depends on why you're computing this number. One way to think about it is to look at some toy 2D examples.

For example, suppose your reference vector is (1,0) and you have two comparison vectors, (1,1) and (1,-1). If you average their coordinates, you get (1,0) which is identical to the reference vector--the most "similar" possible; if you average their cosine similarities, you get a less "similar" result. Which would you prefer?

Some notes on what you've said already: if you're only interested in angles (or their cosines) then averaging the reference vectors will give the same result as just adding them. But the fact that you refer to the "collective strength" makes me wonder if you're interested in magnitude as well, in which case you might consider a dot product.

In any case, feel free to add more detail about your specific context, this will help people answer the question.

Quantifying Aggregate Vector Strength/Vector Arithmatic - MathOverflow

Quantifying Aggregate Vector Strength/Vector Arithmatic

Answer by Martin M. W. for Quantifying Aggregate Vector Strength/Vector Arithmatic