Scale-Invariant Clustering and Regression

The impact of a change of scale, for instance using years instead of days as the unit of measurement for one variable in a clustering problem, can be dramatic. It can result in a totally different cluster structure. Frequently, this is not a desirable property, yet it is rarely mentioned in textbooks. I think all clustering software should state in their user guide, that the algorithm is sensitive to scale.

We illustrate the problem here, and propose a scale-invariant methodology for clustering. It applies to all clustering algorithms, as it consists of normalizing the observations before classifying the data points. It is not a magic solution, and it has its own drawbacks as we will see. In the case of linear regression, there is indeed no problem, and this is one of the few strengths of this technique.

Scale-invariant clustering

The problem may not be noticeable at first glance, especially in Excel, as charts are by default always re-scaled in spreadsheets (or when using charts in R or Python, for that matter). For simplicity, we consider here two clusters, see figure below.

Original data (left), X-axis re-scaled (middle), scale-invariant clustering (right)

The middle chart is obtained after re-scaling the X-axis, and as a result, the two-clusters structure is lost. Or maybe it is the one on the left-hand side that is wrong. Or both. Astute journalists and even researchers actually exploit this issue to present misleading, usually politically motivated, analyses. Students working on a clustering problem might not even be aware of the issue.

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