Euclidean distance
D(X1, X2) = sqrt( (x11 - x21)^2 + (x12 - x22)^2 + ... + (x1n - x2n)^2 )
For assignment, compare the squared value and skip the root.
Any continuous-scale similarity: k-means assignment, KNN, load-distance. Standardise the columns first or the widest one decides the answer.
- x1j, x2j
- Value of variable j for observations 1 and 2
- n
- Number of variables
Standardising a column
z = (x - mean) / standard deviation (z-score)
or x' = (x - min) / (max - min) (min-max, range 0 to 1)
Before any distance-based method. z-scoring in Excel is =(x - AVERAGE(col)) / STDEV.S(col); in R it is scale(data).
- mean, standard deviation
- Computed down the column, not across the row
- min, max
- Smallest and largest values in the column
Centroid and WCSS
centroid_j = mean of variable j over the cluster's members
WCSS = SUM over clusters SUM over members ( distance to own centroid )^2
The update step of k-means, and the elbow plot. WCSS falls monotonically in k and is zero at k = n, so read the bend rather than minimising it.
- centroid_j
- The cluster's mean on variable j
- k
- Number of clusters, fixed before k-means runs