In this research, we investigate a tropical principal component analysis (PCA) as a bestfit Stiefel tropical linear space to a given sample over the tropical projective torus for its dimensionality reduction and visualization. Especially, we characterize the bestfit Stiefel tropical linear space to a sample generated from a mixture of Gaussian distributions as the variances of the Gaussians go to zero. For a single Gaussian distribution, we show that the sum of residuals in terms of the tropical metric with the maxplus algebra over a given sample to a fitted Stiefel tropical linear space converges to zero by giving an upper bound for its convergence rate. Meanwhile, for a mixtures of Gaussian distribution, we show that the bestfit tropical linear space can be determined uniquely when we send variances to zero. We briefly consider the bestfit topical polynomial as an extension for the mixture of more than two Gaussians over the tropical projective space of dimension three. We show some geometric properties of these tropical linear spaces and polynomials.
more » « less NSFPAR ID:
 10394550
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Information Geometry
 Volume:
 6
 Issue:
 1
 ISSN:
 25112481
 Format(s):
 Medium: X Size: p. 171201
 Size(s):
 ["p. 171201"]
 Sponsoring Org:
 National Science Foundation
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Unmodelled sources of random effect variance had predictable effects on variance component estimates. The pattern is best viewed as a cascade of hierarchical grouping factors. Variances trickle down the hierarchy such that missing higher‐level random effect variances pool at lower levels and missing lower‐level and crossed random effect variances manifest as residual variance.
Overall, our results show remarkable robustness of mixed‐effects models that should allow researchers to use mixed‐effects models even if the distributional assumptions are objectively violated. However, this does not free researchers from careful evaluation of the model. Estimates that are based on data that show clear violations of key assumptions should be treated with caution because individual datasets might give highly imprecise estimates, even if they will be unbiased on average across datasets.