I've a set of data points $S = \{ x | x\in [0,1]\}$ (i.e. real values from the unit interval). In some cases I've big clusters in the data and I want to spread the values in between the unit interval somewhat more evenly, through some normalizing method.
I want to construct a normalization function $\psi: [0,1] \rightarrow [0,1]$ s.t.
- $x_i < x_j \Rightarrow \psi(x_i) < \psi(x_j)$
- $|x_i - x_j|< |x_m - x_n| \Rightarrow |\psi(x_i) - \psi(x_j)|< |\psi(x_m) - \psi(x_n)|$
So in crude words, I want to maintain the relative order of the points and also the relative ranking of the distances between the points.
Is such a transformation possible? As of now I think it's, however I haven't been able to prove it yet.
I'd highly appreciate a proof or a counter-proof.
