When is logistic error growth a spectral reduction?
Abstract
Forecast-error growth has been studied through complementary approaches: scalar models describe the evolution of bulk error, whereas spectral theories follow decorrelation and error transfer across scales. Their connection matters when a fitted growth curve is used to infer a physical mechanism or a limit of predictability. We establish a conditional link between these approaches. Within a bilinear spectral exchange model, assume a stationary reference spectrum and a remainder that vanishes on the proposed common-shape evolution. For a common initial decorrelation fraction strictly between zero and one, the spectral shape is preserved and its amplitude grows logistically exactly when every participating scale shares one positive growth coefficient. The converse inference fails: a logistic aggregate need not imply proportional decorrelation or a common coefficient. An exact two-shell construction demonstrates this failure with identical aggregate curves and different spectral dynamics. For heterogeneous amplitudes, an exact averaging identity separates the contributions of growth-rate mismatch, shape variance, rate--shape covariance, evolving spectral weights, and residual tendencies. A finite-time bound relates their combined effect to departure from a reference logistic curve. Forced surface-quasigeostrophic simulations illustrate how similar retrospective agreement with a logistic reference can coexist with heterogeneous effective shell rates and substantial instantaneous tendency discrepancies. Thus a scalar growth curve alone cannot identify how uncertainty is distributed across scales or justify interpreting a fitted rate as a common spectral growth coefficient. The framework specifies the additional spectral and budget evidence needed to give scalar predictability models that interpretation.
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BibTeX
@misc{pena2026when,
title = {{When is logistic error growth a spectral reduction?}},
author = {Malaquias Peña},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.26492},
eprint = {2608.26492},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26492},
}
RIS
TY - UNPB
TI - When is logistic error growth a spectral reduction?
AU - Malaquias Peña
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2608.26492
UR - https://arxiv.org/abs/2608.26492
AB - Forecast-error growth has been studied through complementary approaches: scalar models describe the evolution of bulk error, whereas spectral theories follow decorrelation and error transfer across scales. Their connection matters when a fitted growth curve is used to infer a physical mechanism or a limit of predictability. We establish a conditional link between these approaches. Within a bilinear spectral exchange model, assume a stationary reference spectrum and a remainder that vanishes on the proposed common-shape evolution. For a common initial decorrelation fraction strictly between zero and one, the spectral shape is preserved and its amplitude grows logistically exactly when every participating scale shares one positive growth coefficient. The converse inference fails: a logistic aggregate need not imply proportional decorrelation or a common coefficient. An exact two-shell construction demonstrates this failure with identical aggregate curves and different spectral dynamics. For heterogeneous amplitudes, an exact averaging identity separates the contributions of growth-rate mismatch, shape variance, rate--shape covariance, evolving spectral weights, and residual tendencies. A finite-time bound relates their combined effect to departure from a reference logistic curve. Forced surface-quasigeostrophic simulations illustrate how similar retrospective agreement with a logistic reference can coexist with heterogeneous effective shell rates and substantial instantaneous tendency discrepancies. Thus a scalar growth curve alone cannot identify how uncertainty is distributed across scales or justify interpreting a fitted rate as a common spectral growth coefficient. The framework specifies the additional spectral and budget evidence needed to give scalar predictability models that interpretation.
ER -