The paper for BIRCH (a clustering algorithm) contains definitions of a Clustering Feature (CF) where the notation is unclear (cf. PDF page 3 / section 4).
A cluster contains N d-dimensional entries $ \{ \vec{X}_1, \vec{X}_2, \dots, \vec{X}_N \} $ and its CF is composed of three values.
- N the number of elements (straight forward)
- $ \vec{LS} $ "linear sum of the elements" $ \sum_{i=1}^N \vec{X}_i $
- $ SS $ "square sum of the N data points" $ \sum_{i=1}^N \vec{X}_i^2 $
The third value seems to cause confusion and is throwing me off. Let's start with the paper's definition.
I am parsing the notation as $ \sum_{i=1}^N \vec{X}_i^2 = \vec{X}_1^2 + \vec{X}_2^2 + \dots + \vec{X}_N^2 $. Now I am left with $ \vec{X}^2 $ and to my current understanding a dot/scalar product has to be applied, e.g. $ \vec{X}^2 = \vec{X} \cdot \vec{X} = \sum_{j=1}^d x_j^2 $ where $x_j$ is an Element in $ \vec{X} = ( x_1, x_2, \dots , x_d ) $.
On the other hand, there are plenty of lectures/slides providing examples in which the value is not a scalar, but the element-wise squared sum, while the notation in the respective definition does not reflect this. The German Wikipedia article even states $ SS $ is a "d-dimensional vector".
Basically they are doing it like this, using the earlier notation: $ \vec{X}_i = ( x_{i,1}, x_{i,2}, \dots , x_{i,d} ) $ $ \vec{SS} = (\sum_{i=1}^N x_{i,1}, \sum_{i=1}^N x_{i,2}, \dots, \sum_{i=1}^N x_{i,d}) = (x_{1,1}^2 + x_{2,1}^2 + \dots + x_{N,1}^2, ... ) $.
In the end the sum of the components would be the same.
Why is are those examples doing it differently? Am I missing something?
Examples describing SS as a scalar
Examples describing SS as a d-dimensional vector
- Illinois lecture recording (youtube, around 2:44)
- Data Mining: Concepts and Techniques (Book, page 25)
- Italian lecture slides (page 69)
Edit: Added links to paper and examples, fixed typos.
