Is Bill Gates as Rich as He is Tall?
Instinctively, we know that Bill Gates’ wealth is much further from ‘normal’ than is his height. But how?
- How can we compare income and height if they share no common units?
- How can we compare the biodiversity of sites in the tropics with those of sub-Arctic areas given that there are different numbers of species to begin with?
We need:
- Ways to make different dimensions comparable, and
- Ways to remove unit effects from distance measures.
Distance in 1D
\[
d(i,j) = |(i_{1}-j_{1})|
\]
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Distance in 2D
\[
d(i,j) = \sqrt{(i_{1}-j_{1})^{2}+(i_{2}-j_{2})^{2}}
\]
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Distance in 3D… or More
We can keep adding dimensions…
\[
d(i,j) = \sqrt{(i_{1}-j_{1})^{2}+(i_{2}-j_{2})^{2}+(i_{3}-j_{3})^{2}}
\]
You continue adding dimensions indefinitely, but from here on out you are dealing with hyperspaces!
Thinking in Data Space
We can write the coordinates of an observation with 3 attributes (e.g. height, weight, income) as:
\[
x_{i} = { {x_{i1}, x_{i2}, x_{i3} } }
\]
Something with 8 attributes (e.g. height, weight, income, age, year of birth, …) ‘occupies’ an 8-dimensional space…
Two Propositions
- That geographical space is no different from any other dimension in a data set.
- That geographical space is still special when it comes to thinking about relationships.
Implication
If you can shift from thinking in columns of data, to thinking of a data space then you’ll have a much easier time dealing with dimensionality reduction and clustering.
