Clustering and smoothing choices
Topic: Hierarchical clustering and kernel density estimation · more questions on this topic
‹ A tempting wrong turn
A two-dimensional dataset contains two dense groups joined by a thin chain of observations. You consider agglomerative clustering and a kernel density estimate.
Explain why single linkage can merge the two groups early. Contrast it with complete linkage.
Explain the role of bandwidth in kernel density estimation and describe the consequences of a very small and a very large bandwidth.
Would a peak in an estimated density necessarily identify a distinct generating population? Explain.
Single linkage uses the smallest inter-cluster distance, so a chain of short links can join distant dense regions. Complete linkage uses the largest inter-cluster distance and favours clusters of smaller diameter. Neither method is guaranteed to recover the groups that generated the sample.
\widehat f_h(\mathbf{x})=n^{-1}\sum_{i=1}^{n}h^{-p}K\bigl((\mathbf{x}-\mathbf{x}_i)/h\bigr).
A very small h produces noisy peaks around observations. A very large h can merge features and hide modes. The two regimes are the two sides of the bias–variance trade-off in mean integrated squared error.
- Peaks depend on the smoothing choice and on sample variability. A multimodal population, or two overlapping populations, breaks a reading in which each peak is its own generating population.