I have a set of n points $P_i$ $(i=1..n)$ in an d-dimensional space. I can calculate the distances of every point to each other point.
What is the best way to represent the distances of these points?
What is the best way to see whether some of the points group together?
(Primarily i'm searching for the idea how this can be done in general, not specifically for a software implementation.)
Example:
Let n=5, d=4: $$P_1=\begin{pmatrix}0\\0\\0\\0\end{pmatrix}, P_2=\begin{pmatrix}0\\0\\0\\1\end{pmatrix}, P_3=\begin{pmatrix}0\\5\\0\\0\end{pmatrix}, P_4=\begin{pmatrix}0\\6\\0\\0\end{pmatrix}, P_5=\begin{pmatrix}0\\5.5\\1\\0\end{pmatrix}$$
I can simply calculate the distance $\Delta_{i,j}=||P_i - P_j||_2$, and get $$\Delta^2_{1,2}=1, \Delta^2_{1,3}=25, \Delta^2_{1,4}=36, \Delta^2_{1,5}=31.25,\\ \Delta^2_{2,3}=26, \Delta^2_{2,4}=37, \Delta^2_{2,5}=32.25\\ \Delta^2_{3,4}=1, \Delta^2_{3,5}=1.25,\\ \Delta^2_{4,5}=1.25$$
Here, $P_1$ and $P_2$ as well as $P_3$, $P_4$, $P_5$ are grouped together (their mutual distance is small).
Is there a way to represent this behaviour graphically?
Converting Roy's comment into an answer so this question can be removed from the "Unanswered" queue
This question may be NP-Complete. There is an NP-Complete problem called CLUSTERING which is similar to this.