Karhunen-Lo`eve Decomposition of Extensive Chaos

We show that the number of KLD (Karhunen-Lo`eve decomposition) modes D_KLD(f) needed to capture a fraction f of the total variance of an extensively chaotic state scales extensively with subsystem volume V. This allows a correlation length xi_KLD(f) to be defined that is easily calculated from spatially localized data. We show that xi_KLD(f) has a parametric dependence similar to that of the dimension correlation length and demonstrate that this length can be used to characterize high-dimensional inhomogeneous spatiotemporal chaos.Comment: 12 pages including 4 figures, uses REVTeX macros. To appear in Phys. Rev. Let

On dimension reduction in Gaussian filters

A priori dimension reduction is a widely adopted technique for reducing the computational complexity of stationary inverse problems. In this setting, the solution of an inverse problem is parameterized by a low-dimensional basis that is often obtained from the truncated Karhunen-Loeve expansion of the prior distribution. For high-dimensional inverse problems equipped with smoothing priors, this technique can lead to drastic reductions in parameter dimension and significant computational savings. In this paper, we extend the concept of a priori dimension reduction to non-stationary inverse problems, in which the goal is to sequentially infer the state of a dynamical system. Our approach proceeds in an offline-online fashion. We first identify a low-dimensional subspace in the state space before solving the inverse problem (the offline phase), using either the method of "snapshots" or regularized covariance estimation. Then this subspace is used to reduce the computational complexity of various filtering algorithms - including the Kalman filter, extended Kalman filter, and ensemble Kalman filter - within a novel subspace-constrained Bayesian prediction-and-update procedure (the online phase). We demonstrate the performance of our new dimension reduction approach on various numerical examples. In some test cases, our approach reduces the dimensionality of the original problem by orders of magnitude and yields up to two orders of magnitude in computational savings

Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices

We present a simple Coulomb gas method to calculate analytically the probability of rare events where the maximum eigenvalue of a random matrix is much larger than its typical value. The large deviation function that characterizes this probability is computed explicitly for Wishart and Gaussian ensembles. The method is quite general and applies to other related problems, e.g. the joint large deviation function for large fluctuations of top eigenvalues. Our results are relevant to widely employed data compression techniques, namely the principal components analysis. Analytical predictions are verified by extensive numerical simulations.Comment: 4 pages, 3 .eps figures include

Statistics of Atmospheric Correlations

For a large class of quantum systems the statistical properties of their spectrum show remarkable agreement with random matrix predictions. Recent advances show that the scope of random matrix theory is much wider. In this work, we show that the random matrix approach can be beneficially applied to a completely different classical domain, namely, to the empirical correlation matrices obtained from the analysis of the basic atmospheric parameters that characterise the state of atmosphere. We show that the spectrum of atmospheric correlation matrices satisfy the random matrix prescription. In particular, the eigenmodes of the atmospheric empirical correlation matrices that have physical significance are marked by deviations from the eigenvector distribution.Comment: 8 pages, 9 figs, revtex; To appear in Phys. Rev.

Wishart and Anti-Wishart random matrices

We provide a compact exact representation for the distribution of the matrix elements of the Wishart-type random matrices $A^\dagger A$, for any finite number of rows and columns of $A$, without any large N approximations. In particular we treat the case when the Wishart-type random matrix contains redundant, non-random information, which is a new result. This representation is of interest for a procedure of reconstructing the redundant information hidden in Wishart matrices, with potential applications to numerous models based on biological, social and artificial intelligence networks.Comment: 11 pages; v2: references updated + some clarifications added; v3: version to appear in J. Phys. A, Special Issue on Random Matrix Theor

Spectral analysis of the Forel-Ule ocean colour comparator scale

François Alphonse Forel (1890) and Willi Ule (1892) composed a colour comparator scale, with tints varying from indigo-blue to coca-cola brown, to quantify the colour of natural waters, like seas, lakes and rivers. For each measurement, the observer compares the colour of the water above a submersed white disc (Secchi disc) with the hand-held scale of pre-defined colours. The scale can be well reproduced from a simple recipe for twenty-one coloured chemical solutions and because the ease of its use, the Forel-Ule (FU) scale has been applied globally and intensively by oceanographers and limnologists from the year 1890. Indeed, the archived FU data belong to the oldest oceanographic data sets and do contain information on the changes in geobiophysical properties of natural waters during the last century. In this article we describe the optical properties of the FU-scale and its ability to cover the colours of natural waters, as observed by the human eye. The recipe of the scale and its reproduction is described. The spectral transmission of the tubes, with belonging chromaticity coordinates, is presented. The FU scale, in all its simplicity, is found to be an adequate ocean colour comparator scale. The scale is well characterized, is stable and observations are reproducible. This supports the idea that the large historic data base of FU measurements is coherent and well calibrated. Moreover, the scale can be coupled to contemporary multi-spectral observations with hand-held and satellite-based spectrometers

Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices

We compute analytically the probability of large fluctuations to the left of the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of positive definite random matrices. We show that the probability that all the eigenvalues of a (N x N) Wishart matrix W=X^T X (where X is a rectangular M x N matrix with independent Gaussian entries) are smaller than the mean value =N/c decreases for large N as $\sim \exp[-\frac{\beta}{2}N^2 \Phi_{-}(\frac{2}{\sqrt{c}}+1;c)]$, where \beta=1,2 correspond respectively to real and complex Wishart matrices, c=N/M < 1 and \Phi_{-}(x;c) is a large deviation function that we compute explicitly. The result for the Anti-Wishart case (M < N) simply follows by exchanging M and N. We also analytically determine the average spectral density of an ensemble of constrained Wishart matrices whose eigenvalues are forced to be smaller than a fixed barrier. The numerical simulations are in excellent agreement with the analytical predictions.Comment: Published version. References and appendix adde

Southward re-distribution of tropical tuna fisheries activity can be explained by technological and management change

There is broad evidence of climate change causing shifts in fish distribution worldwide, but less is known about the response of fisheries to these changes. Responses to climate-driven shifts in a fishery may be constrained by existing management or institutional arrangements and technological settings. In order to understand how fisheries are responding to ocean warming, we investigate purse seine fleets targeting tropical tunas in the east Atlantic Ocean using effort and sea surface temperature anomaly (SSTA) data from 1991 to 2017. An analysis of the spatial change in effort using a centre of gravity approach and empirical orthogonal functions is used to assess the spatiotemporal changes in effort anomalies and investigate links to SSTA. Both analyses indicate that effort shifts southward from the equator, while no clear pattern is seen northward from the equator. Random forest models show that while technology and institutional settings better explain total effort, SSTA is playing a role when explaining the spatiotemporal changes of effort, together with management and international agreements. These results show the potential of management to minimize the impacts of climate change on fisheries activity. Our results provide guidance for improved understanding about how climate, management and governance interact in tropical tuna fisheries, with methods that are replicable and transferable. Future actions should take into account all these elements in order to plan successful adaptation. © 2020 The Authors. Fish and Fisheries published by John Wiley & Sons Ltd.This research is supported by the project CLOCK, under the European Horizon 2020 Program, ERC Starting Grant Agreement nº679812 funded by the European Research Council. It is also supported by the Basque Government through the BERC 2018-2021 programme and by the Spanish Ministry of Economy and Competitiveness MINECO through the BC3 María de Maeztu excellence accreditation MDM- 2017-0714. We thank, without implicating, C. Palma for his helpful advice on the ICCAT database and M. Gabantxo and H. Gabantxo for their knowledge transfer about tropical tuna fisheries. Also, we thank I. Arostegui for her comments during the design of the random forest; F. Saborido, A. Tidd and H. Arrizabalaga for scientific advice and H. Murua and M. Ortiz for providing ICCAT data. Elena Ojea thanks the Xunta the Galicia GAIN Oportunius programme and Consellería de Educación (Galicia, Spain) for additional financial support