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I am not sure if this question makes sense or is even ideal, but here is my problem: Many proposed distance formulas only give a distance in terms of 1-dimensions. A common distance formula is the Euclidean distance (the most obvious one). Euclidean distance $d_{ecdn}$ of two points located at $(x_1, x_2, x_3, ... , x_n)$ and $(y_1, y_2, y_3, ... , y_n)$, respectively, is given by

$d_{ecdn} = \sqrt{\displaystyle\sum_{i=1}^{n} (x_i-y_i)^2}$.

Geometrically, in any number of dimensions (but imagine 3 dimensions as an example), if you were to draw a line between the two points, whose length expresses the Euclidean distance, the line itself exists in 1 dimension. Why is this a problem? I am doing a project involve data mining and information retrieval. Basically, each document in a corpus is mapped to a location in an $n$-dimensional space where $n$ is the total number of terms within the whole set of documents. They say that if document A is geometrically closer to document B than document C, then document A is more similar to B than C. Euclidean distance gives the wrong impression of similarity since each dimension matters.

I am wondering if anyone has any good algorithms that will calculate the "distance" between two points, but with a weight on each dimensions. An example would be the Pearson Product-Moment Correlation Coefficient. This formula is given by

pearson

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migrated from math.stackexchange.com Aug 22 '12 at 5:53

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Note, lines are in 1 dimension. –  Karolis Juodelė Aug 22 '12 at 5:52
    
Fixed! Stupid error. Thank you. –  Sidd Aug 22 '12 at 5:57
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I think you are looking for Mahalanobis Distance. –  TenaliRaman Aug 22 '12 at 6:27
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1) Weighted version of euclidean distance is straightforward, see the formula. 2) Why do you think Pearson r incorporates weights? I don't see any weighting. –  ttnphns Aug 22 '12 at 7:34
    
@TenaliRaman +1. That probably would have been appropriate enough for an answer, but regardless, thank you for that! –  Sidd Aug 22 '12 at 13:30

1 Answer 1

Just to create an official answer for this topic (based on @TenaliRaman's comment), you want the Mahalanobis distance.

This is computed similarly to the Euclidian distance, (in fact, the Euclidian distance formula is a special case of the Mahalanobis distance formula) except each squared difference term is 'corrected' based on an inverted covariance matrix.

From the Wikipedia page:

 Mahalanobis distance is thus unitless and scale-invariant, 
 and takes into account the correlations of the data set.
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