Background¶

The Hopkins statistic is a test statistic for the null hypothesis of complete spatial randomness (CSR), i.e., that points are independently and uniformly distributed within a region of space (the sampling frame). Under this null, the expected value of the statistic is 0.5. Larger values indicate more clustering than expected under CSR, while smaller values indicate more regular spacing. Thus, the statistic is often used as a scalar measure of clustering tendency.

Definition¶

As noted by Wright (2022), the definition of the Hopkins statistic is a common source of confusion in both literature and software implementations. This library defaults to the formulation by Cross and Jain (1982), which generalizes the original definition by Hopkins and Skellam (1954) to data in any dimension.

Definition

Given a set \(X\) of \(n\) data points in a \(d\)-dimensional Euclidean space, choose \(m\) such that \(m \ll n\) and let

\(\lbrace x_i \rbrace_{i=1}^m\) be a simple random sample from \(X\) (without replacement), and
\(\lbrace y_i \rbrace_{i=1}^m\) be points placed uniformly at random in the sampling frame.

For each \(i \in \lbrace 1,\dots,m \rbrace\), let

\(u_i\) be the distance from \(y_i\) to its nearest neighbor in \(X\), and
\(w_i\) be the distance from \(x_i\) to its nearest neighbor in \(X \setminus \lbrace x_i \rbrace\).

Then the Hopkins statistic is defined as

\[ H = \frac{\sum_{i=1}^m u_i^d}{\sum_{i=1}^m u_i^d + \sum_{i=1}^m w_i^d}. \]

Under the CSR null hypothesis, \(H \sim \mathrm{Beta}(m,m)\).

Note

Other implementations may follow Lawson and Jurs (1990) by not raising distances to the power of \(d\), or may return \(1 - H\) instead of \(H\).

Interpretation¶

While critical values can be obtained from the \(\mathrm{Beta}(m,m)\) null distribution, the table below lists commonly used rules of thumb for interpreting \(H\).

\(H\)	Pattern	Interpretation
\(\ge 0.7\)	clustered	Suggests a departure from CSR toward clustering.
\(\approx 0.5\)	random	Consistent with complete spatial randomness (CSR).
\(\le 0.3\)	regular	Suggests a departure from CSR toward more even spacing.

Guidelines¶

Euclidean distances on non-spatial data often benefit from scaling features in X to comparable ranges.
The sample size m should typically be at least 10 to avoid small-sample problems and no more than about one tenth of \(n\) to keep the null-distribution approximations accurate.
The sampling frame defaults to the axis-aligned bounding box of X. A known rectangular frame can be specified using the frame parameter. If the underlying sampling frame is not aligned with the coordinate axes, frame can be set to 'hull' for using the convex hull of X.
To mitigate edge effects, e.g., when events may also occur outside the sampling frame, periodic boundary conditions can be applied with toroidal=True. Alternatively, buffer zones can be used by specifying a frame smaller than the full extent of X.
The exponent power applied to distances defaults to \(d\), the number of columns in X. This yields the statistic as defined above. Other values alter the null distribution.

References¶

Cross, G. R., & Jain, A. K. (1982). Measurement of clustering tendency. In Theory and Application of Digital Control (pp. 315–320). Pergamon. https://doi.org/10.1016/S1474-6670(17)63365-2
Hopkins, B., & Skellam, J. G. (1954). A new method for determining the type of distribution of plant individuals. Annals of Botany, 18(2), 213–227. https://doi.org/10.1093/oxfordjournals.aob.a083391
Lawson, R. G., & Jurs, P. C. (1990). New index for clustering tendency and its application to chemical problems. Journal of chemical information and computer sciences, 30(1), 36–41. https://doi.org/10.1021/ci00065a010
Wright, K. (2022). Will the Real Hopkins Statistic Please Stand Up? The R Journal, 14(3), 282–292. https://doi.org/10.32614/rj-2022-055