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A Note on the Usefulness of Constrained Fourth-Order Latent Differential Equation Models in the Case of Small T.

ABSTRACT: Constrained fourth-order latent differential equation (FOLDE) models have been proposed (e.g., Boker et al. 2020) as alternative to second-order latent differential equation (SOLDE) models to estimate second-order linear differential equation systems such as the damped linear oscillator model. When, however, only a relatively small number of measurement occasions T are available (i.e., [Formula: see text]), the recommendation of which model to use is not clear (Boker et al. 2020). Based on a data set, which consists of [Formula: see text] observations of daily stress for [Formula: see text] individuals, we illustrate that FOLDE can help to choose an embedding dimension, even in the case of a small T. This is of great importance, as parameter estimates depend on the embedding dimension as well as on the latent differential equations model. Consequently, the wavelength as quantity of potential substantive interest may vary considerably. We extend the modeling approaches used in past research by including multiple subjects, by accounting for individual differences in equilibrium, and by including multiple instead of one single observed indicator.

SUBMITTER: Hardt K

PROVIDER: S-EPMC7826318 | biostudies-literature | 2020 Dec

REPOSITORIES: biostudies-literature

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A Note on the Usefulness of Constrained Fourth-Order Latent Differential Equation Models in the Case of Small T.

Hardt Katinka K Boker Steven M SM Bergeman Cindy S CS

Psychometrika 20201220 4

Constrained fourth-order latent differential equation (FOLDE) models have been proposed (e.g., Boker et al. 2020) as alternative to second-order latent differential equation (SOLDE) models to estimate second-order linear differential equation systems such as the damped linear oscillator model. When, however, only a relatively small number of measurement occasions T are available (i.e., [Formula: see text]), the recommendation of which model to use is not clear (Boker et al. 2020). Based on a dat ...[more]

PMID: 33341912

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Project description:Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

Dataset Information

A Note on the Usefulness of Constrained Fourth-Order Latent Differential Equation Models in the Case of Small T.

Publications

A Note on the Usefulness of Constrained Fourth-Order Latent Differential Equation Models in the Case of Small T.

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