Multispecies occupancy models estimate dependence among multiple species of interest from patterns of co‐occurrence, but problems associated with separation and boundary estimates can lead to unreasonably large estimates of parameters and associated standard errors when species are rarely observed at the same site or when data are sparse. In this paper, we overcome these issues by implementing a penalized likelihood, which introduces a small bias in parameter estimates in exchange for a potentially large reduction in variance. We compare parameter estimates obtained from both penalized and unpenalized multispecies occupancy models fit to simulated data that exhibit various degrees of separation and to a real‐word data set of bird surveys with little apparent overlap between potentially interacting species. Our simulation results demonstrate that penalized multispecies occupancy models did not exhibit boundary estimates and produced lower bias, lower mean squared error, and improved inference relative to unpenalized models. When applied to real‐world data, our penalized multispecies occupancy model constrained boundary estimates and allowed for meaningful inference related to the interactions of two species of conservation concern. To facilitate the use of our penalized multispecies occupancy model, the techniques demonstrated in this paper have been integrated into the
- Award ID(s):
- 1633587
- NSF-PAR ID:
- 10295722
- Date Published:
- Journal Name:
- Technometrics
- ISSN:
- 0040-1706
- Page Range / eLocation ID:
- 1 to 11
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Purpose To improve the performance of neural networks for parameter estimation in quantitative MRI, in particular when the noise propagation varies throughout the space of biophysical parameters.
Theory and Methods A theoretically well‐founded loss function is proposed that normalizes the squared error of each estimate with respective Cramér–Rao bound (CRB)—a theoretical lower bound for the variance of an unbiased estimator. This avoids a dominance of hard‐to‐estimate parameters and areas in parameter space, which are often of little interest. The normalization with corresponding CRB balances the large errors of fundamentally more noisy estimates and the small errors of fundamentally less noisy estimates, allowing the network to better learn to estimate the latter. Further, proposed loss function provides an absolute evaluation metric for performance: A network has an average loss of 1 if it is a maximally efficient unbiased estimator, which can be considered the ideal performance. The performance gain with proposed loss function is demonstrated at the example of an eight‐parameter magnetization transfer model that is fitted to phantom and in vivo data.
Results Networks trained with proposed loss function perform close to optimal, that is, their loss converges to approximately 1, and their performance is superior to networks trained with the standard mean‐squared error (MSE). The proposed loss function reduces the bias of the estimates compared to the MSE loss, and improves the match of the noise variance to the CRB. This performance gain translates to in vivo maps that align better with the literature.
Conclusion Normalizing the squared error with the CRB during the training of neural networks improves their performance in estimating biophysical parameters.
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1080/00401706.2021.1905070
