Some additional information taken from SAS Stat Documentation.
One useful descriptive approach to the number-of-clusters problem is provided by Wong and Schaack (1982)
based on a kth-nearest-neighbor density estimate. The kth-nearest-neighbor clustering method developed by
Wong and Lane (1983) is applied with varying values of k. Each value of k yields an estimate of the number
of modal clusters. If the estimated number of modal clusters is constant for a wide range of k values, there
is strong evidence of at least that many modes in the population. A plot of the estimated number of modes
against k can be highly informative. Attempts to derive a formal hypothesis test from this diagnostic plot
have met with difficulties, but a simulation approach similar to Silverman (1986) does seem to work Girman
(1994). The simulation, of course, requires considerable computer time.
And you could also check CCC statistic and its disadvantage.
Sarle (1983) used extensive simulations to develop the cubic clustering criterion (CCC), which can be used
for crude hypothesis testing and estimating the number of population clusters. The CCC is based on the
assumption that a uniform distribution on a hyperrectangle will be divided into clusters shaped roughly like
hypercubes. In large samples that can be divided into the appropriate number of hypercubes, this assumption
gives very accurate results. In other cases the approximation is generally conservative. For details about the
interpretation of the CCC, consult Sarle (1983).
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