# compering distances between points in different dimension

I have pairs of points, each pair is in the same dimension, and I need to measure the distance between each pair:

For example:

Points:    dim:     distance:
x1,x2      2        0.8
x3,x4      8        0.2
x5,x6      12       0.5


I need to compare the distances (sort, find the smallest etc). I think they are not comparable due to the different dimensions. Is there a way to normelize the distances so I compare them?

Thanks !

## 1 Answer

Distances are measured in linear units no matter what the dimension. I can talk about two points in the plane being 1 unit apart and two points in space being 1 unit apart using the same unit. You can compare them without a problem. Whether it is useful to do so is another issue, but that depends on what your data looks like and what you are trying to do with it. Even in the same dimension, you might have very different scales in different directions. Think about if your two dimensions were age and annual income-the scales are very different.

• Hi, The scales are quite the same, but if two 12-dim points are very close isn't that more "powerful" then two 2-dim points with the same distance? does it matter what metric I'm using? in this case spearman and cosine – matlabit Aug 27 '13 at 20:20
• If $x_1,x_2$ live in 2-space and $x_5,x_6$ live in 12-space, what does it matter which is closer linearly? Are you thinking of things that live in 12-space but you don't have data on 10 of the values? You could certainly do a chi-square to compensate if you want. – Ross Millikan Aug 27 '13 at 20:44
• I don't think of the data as n-dim (where n is the largest dimension) and the rest of the data is missing. My data has many features in different dimensions. Does it makes any difference ? – matlabit Aug 28 '13 at 5:35
• I think I can assume that things live in the higher dim and treat other entries as zero, depending on the distance metric I'm using. – matlabit Aug 29 '13 at 17:21