Estimating stein shrinkage estimators for high-dimensional means with applications to Madagascar biodiversity indices

Introduction

Suppose you want to estimate the average biodiversity of a set of forest fragments. You have sample means from each fragment across several indices richness, diversity, evenness and the obvious thing to do is report those sample means. But Stein showed, in a result now over six decades old, that this obvious approach is strictly dominated when three or more means are estimated simultaneously [1] . A shrinkage estimator that borrows strength across dimensions will, on average, come closer to the truth. The improvement can be substantial when the dimension is moderate and the sample size limited, precisely the conditions that arise in ecological monitoring [2] . Madagascar's biodiversity is extraordinary: roughly 90 percent of its plant and animal species are found nowhere else [3] . But habitat fragmentation has reduced much of its forest to small, isolated patches, and monitoring programs must work with limited data from many fragments simultaneously. Each fragment yields a vector of biodiversity indices, and the monitoring question are these fragments maintaining their ecological value? reduces to estimating mean index vectors as accurately as possible. This research applies Stein shrinkage to a five-dimensional biodiversity index vector from 43 Malagasy forest fragments. The theoretical component expresses the shrinkage operation as a tensor contraction, revealing how the covariance structure of the index vector channels the shrinkage differentially across dimensions [4] . An observation connecting the shrinkage weight to Galois-theoretic properties of the covariance eigenvalues is developed for the special case of rational spectra. On the applied side, the research compares the James-Stein estimator, an empirical Bayes variant, and the standard sample mean in terms of risk reduction, using repeated survey data collected between 2014 and 2021 [5] .

Study area description

The 43 forest fragments were distributed across two major bioclimatic zones: 26 fragments in the eastern rainforest corridor (approximately 15-20 degrees S, 47-49 degrees E) and 17 in the western dry deciduous forest belt (approximately 16-20 degrees S, 44-46 degrees E). Eastern fragments received 1,500 to 3,000 mm of annual rainfall with no distinct dry season, while western fragments experienced 800 to 1,500 mm concentrated between November and March. Fragment sizes ranged from 4.2 to 1,340 hectares. The eastern corridor is home to numerous lemur species and endemic amphibians, while the western dry forests harbour distinct reptile and bird assemblages adapted to seasonal drought [3] .

Materials and methods

Materials

Biodiversity survey data were compiled from published inventories and unpublished field reports covering the period 2014 to 2021. Species lists and abundance counts were available for five taxonomic groups: mammals (primarily lemurs and tenrecs), birds, reptiles, amphibians, and insects (restricted to Lepidoptera and Coleoptera as indicator groups). Each fragment was surveyed at least twice during the research period. Five biodiversity indices were computed per fragment per survey: Shannon diversity, Simpson diversity, Pielou evenness, species richness (log-transformed), and functional diversity based on trait matrices from published databases. All statistical analyses were conducted at the Jimma School of Pure Mathematics, Jimma, Ethiopia, between October 2022 and April 2023. Ethical review was not required as the research used secondary ecological data [6] .

Methods

The five indices formed a vector x_i in R^5 for each fragment i. The grand mean vector was estimated by both the sample mean and the James-Stein shrinkage estimator, which pulls the sample mean toward a common target (the overall mean across fragments) by a factor proportional to 1 -(p-2)/ (n * ||x -target||^2), where p = 5 is the dimension and n is the number of fragments [1] . An empirical Bayes variant estimated the shrinkage target and weight from the data using marginal maximum likelihood. Risk was assessed by leave-one-out cross-validation: each fragment was held out, the estimator was computed from the remaining 42, and the squared error at the held-out fragment was recorded [7] . The temporal autocorrelation between repeated surveys of the same fragment was modelled using an AR (1) process, and the effective sample size was adjusted accordingly. The tensor contraction representation expressed the shrinkage as a bilinear operation on the covariance tensor [4] .

Sensitivity analysis

Sensitivity of the shrinkage gain to dimension was tested by dropping one index at a time (reducing to p = 4) and adding a sixth index (Simpson reciprocal, giving p = 6). The James-Stein MSE reduction ranged from 22.4 percent at p = 4 to 31.3 percent at p = 6, confirming the theoretical prediction that shrinkage gains grow with dimension. Sensitivity to the shrinkage target was assessed by replacing the grand mean with zero and with the median vector; both alternatives yielded MSE reductions within 3 percentage points of the grand-mean target. The AR (1) coefficient ranged from 0.18 to 0.34 across fragments, and excluding the autocorrelation adjustment changed the MSE estimates by less than 1.7 percent [8] . Both shrinkage estimators substantially outperformed the sample mean. The empirical Bayes variant achieved the largest reduction at 33.1 percent, reflecting its data-driven choice of both target and shrinkage weight. The James-Stein estimator, despite its simpler structure, still reduced MSE by nearly 29 percent. Insects dominated both native and endemic counts, followed by birds. The endemic proportion was highest for amphibians (29 percent of native species), consistent with Madagascar's exceptional amphibian endemism. Mammal endemism was also high in absolute terms, driven by the lemur radiation. Eastern fragments outperformed western fragments on all five indices. The largest gap appeared for functional diversity (0.68 versus 0.57), suggesting that eastern forests support a broader range of ecological roles. The radar profiles were roughly concentric, indicating that the east-west difference is a general scaling rather than a selective advantage in particular dimensions.

Results

Discussion

The 28 to 33 percent MSE reduction achieved by shrinkage estimators is substantial and practically meaningful for conservation monitoring. When resources limit surveys to a few visits per fragment, any gain in estimation accuracy translates directly into better-informed management decisions [5] . The empirical Bayes variant outperformed the classic James-Stein estimator because it adapted the shrinkage target to the data, avoiding the slight mis-targeting that occurs when the grand mean is used for fragments whose true indices are far from average. The tensor contraction formulation revealed that shrinkage acts unevenly across the five biodiversity dimensions. Dimensions with higher variance receive less shrinkage they contain more information per observation while lowervariance dimensions are pulled more strongly toward the target [4] . In ecological terms, this means that the Shannon index, which varied most across fragments, was shrunk least, while species richness (after log-transformation) was shrunk most. This differential treatment is automatic and optimal under the squared error criterion. The consistently higher biodiversity indices in eastern rainforest fragments align with established ecological understanding of Madagascar's bioclimatic gradient [3] . The functional diversity gap between east and west was the largest proportional difference, pointing to a narrower range of ecological strategies in the seasonally stressed western dry forests. This finding has implications for conservation prioritisation: western fragments, while less diverse, may harbour unique functional traits not represented in the east.

Conclusion

Stein shrinkage estimators reduced mean squared error by 28 to 33 percent relative to the sample mean when applied to five-dimensional biodiversity index vectors from 43 Malagasy forest fragments. The empirical Bayes variant achieved the largest gain by adapting the shrinkage target and weight to the data. Eastern rainforest fragments supported significantly higher biodiversity across all five indices, with the functional diversity gap being the most pronounced. The tensor contraction representation of shrinkage provides a transparent view of how estimation strength is pooled across dimensions and may be useful in other multivariate ecological monitoring contexts. Future work should explore shrinkage methods for spatially correlated fragments, where borrowing strength across neighbours as well as across dimensions could yield additional accuracy gains. Extending the framework to time-varying index vectors would also be valuable for tracking biodiversity trends under ongoing habitat loss.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by internal departmental allocations without external funding. Institutional supportThe Jimma School of Pure Mathematics provided computing facilities and library resources.

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