When can I say there is no clusters?

I am trying to do cluster analysis for a very small data set (<100) in higher dimensional feature space. I tried K-means and Hierarchical clustering, but I found no 'elbow' and also the silhouette score was never above 15%. To be honest, I do not know if I could use other algorithms as I am not sure which is best for my use case.

I tried to use PCA, but I can come up with examples where the 2D embedding has clusters but not the original feature space, and vice versa.

My question is:

When can I safely conclude that there is no clusters in my data?

Update In this notebook, I try to generate examples where PCA is misleading.

• There is no absolute safety here, or almost anywhere else in statistics, but as a rule of thumb I would say that if clear clusters exist they will usually be evident on a plot of PC2 and PC1 scores. Experts will love the challenge to produce counterexamples. Jan 23, 2022 at 15:23
• A plane (meaning a winged vehicle) hiding just beneath another plane might be an image to think about. Jan 23, 2022 at 16:51
• "cluster analysis" or "clustering", not "clustering analysis", is the right term. Jan 24, 2022 at 10:42
• Theoretically/philosophically, no one can answer your question in a general way, because there is no general definition of what is a cluster: how a "cluster structure" begins to be different from a "no-cluster structure". Jan 24, 2022 at 10:47
• Some internal clustering criteria, such as Gap statistic, attempt to check the "no-cluster" vs "there are clusters" hypothesis. But this is not assumption-free check. Jan 24, 2022 at 10:52

2. A statistic that measures the degree of clustering. For $$k$$-means this can well be the $$k$$-means objective function, but it might also be something else such as the Average Silhouette Width.
By the way, this is the idea of the gap statistic (using a uniform null and the log objective function of $$k$$-means), which gives you a formal rule for choosing "no clustering" (i.e., number of clusters 1) as opposed to the Silhouette Width or the elbow rule.