Correction: Kernel methods and their derivatives: Concept and perspectives for the earth system sciences

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Abstract

The affiliation for the fourth author is incorrect. The correct affiliations are not indicated. Miguel D. Mahecha is not affiliated with #1 but with: Remote Sensing Centre for Earth System Research, Leipzig University, Germany and Helmholtz Centre for Environmental Research, UFZ, Leipzig Germany.

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Kernel methods and their derivatives: Concept and perspectives for the earth system sciences

J. Johnson, Valero Laparra, Adrián Pérez-Suay … (2020)

Kernel methods are powerful machine learning techniques which use generic non-linear functions to solve complex tasks. They have a solid mathematical foundation and exhibit excellent performance in practice. However, kernel machines are still considered black-box models as the kernel feature mapping cannot be accessed directly thus making the kernels difficult to interpret. The aim of this work is to show that it is indeed possible to interpret the functions learned by various kernel methods as they can be intuitive despite their complexity. Specifically, we show that derivatives of these functions have a simple mathematical formulation, are easy to compute, and can be applied to various problems. The model function derivatives in kernel machines is proportional to the kernel function derivative and we provide the explicit analytic form of the first and second derivatives of the most common kernel functions with regard to the inputs as well as generic formulas to compute higher order derivatives. We use them to analyze the most used supervised and unsupervised kernel learning methods: Gaussian Processes for regression, Support Vector Machines for classification, Kernel Entropy Component Analysis for density estimation, and the Hilbert-Schmidt Independence Criterion for estimating the dependency between random variables. For all cases we expressed the derivative of the learned function as a linear combination of the kernel function derivative. Moreover we provide intuitive explanations through illustrative toy examples and show how these same kernel methods can be applied to applications in the context of spatio-temporal Earth system data cubes. This work reflects on the observation that function derivatives may play a crucial role in kernel methods analysis and understanding.

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Journal

Journal ID (nlm-ta): PLoS One

Journal ID (iso-abbrev): PLoS One

Journal ID (publisher-id): plos

Journal ID (pmc): plosone

Title: PLoS ONE

Publisher: Public Library of Science (San Francisco, CA USA )

ISSN (Electronic): 1932-6203

Publication date (Electronic): 3 February 2021

Publication date Collection: 2021

Publication date PMC-release: 3 February 2021

Volume: 16

Issue: 2

Electronic Location Identifier: e0246775

Article

Publisher ID: PONE-D-21-02883

DOI: 10.1371/journal.pone.0246775

PMC ID: 7857590

PubMed ID: 33534865

SO-VID: fa6dd5c9-0770-4357-bb30-d5ed42c9fc9b

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This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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