Low-EFFourth: A MATLAB-based computational framework for generating and studying multilevel model ensembles in low-dimensional systems (source code)

This repository contains a preliminary but fully-operational version (v0.0) of Low-EFFourth (LEF4). See also de Melo Viríssimo (2025).

1. ABOUT LEF4: LEF4 is a MATLAB-based computational framework designed to run different types of ensembles using systems of equations written as ODEs.

LEF4 stands for:

1. Low-dimensional: focused on running low-dimensional systems, being computationally cheap and effortless. 2. Ensemble Framework: a framework that allows you to run ensembles of different shapes and sizes, including very large ensembles. 3. Fourth: allows you to run four different levels of ensemble: initial condition, parametric, multi-model and multi-numerical.

LEF4 allows you to:

1. Systematically study conceptual models in different fields, such as climate, epidemiology, economics, etc. 2. Address theoretical and practical questions on the design, implementation and interpretation of ensembles in different fields using low-dimensional models. 3. Test/trial different uncertainty quantification (UQ) strategies for both deterministic and stochastic models. 4. Use as an educational tool in the classroom for undergraduate and/or postgraduate students.

2. GETTING STARTED WITH LEF4: Please refer to the README file, which contains a step-by-step description on how to set up and use LEF4.

Please note (see also README.txt):

- This current version (v.0.0) of LEF4 was developed in MATLAB_2023b. It should work in other MATLAB versions as well, but minor code tweaks might be required depending on the version. - LEF4 has been written in MacOS. If you are a Windows user, you will need to swap the slash (/) by backslash (\) in the paths written in some of the files. - In order to use the data analysis codes codes provided, you will need to download and install the cmocean colormap (Thyng et al., 2016). See README.txt for instructions.

3. CITING LEF4: If you use LEF4 in your work, I kindly ask you to cite the this code and its accompanying preprint, including DOI:

- F. de Melo Viríssimo (2025), Low-EFFourth: A computational framework for generating and studying multilevel model ensembles in low-dimensional systems. arXiv:2506.03313. https://doi.org/10.48550/arXiv.2506.03313 - F. de Melo Viríssimo (2025), Low-EFFourth: A computational framework for generating and studying multilevel model ensembles in low-dimensional systems. (v0.0). Zenodo. https://doi.org/10.5281/zenodo.15566109

You might want to cite the publications below as well, which shows how LEF4 have been used in practice:

- F. de Melo Viríssimo, D. Stainforth (2023), A low-dimensional dynamical systems approach to climate ensemble design and interpretation. EGU General Assembly 2023, EGU23-14755. https://doi.org/10.5194/egusphere-egu23-14755 - F. de Melo Viríssimo, D. A. Stainforth, J. Bröcker (2024), The evolution of a non-autonomous chaotic system under non-periodic forcing: A climate change example. Chaos, 34 (1): 013136. https://doi.org/10.1063/5.0180870 - F. de Melo Viríssimo, D. A. Stainforth (2025), Micro and macro parametric uncertainty in climate change prediction: a large ensemble perspective. Bull. Amer. Meteor. Soc. (in press), BAMS-D-24-0064.1. https://doi.org/10.1175/BAMS-D-24-0064.1

Should you have any questions, comments and/or feedback, please feel free to get in touch (check manuscripts above for my latest email address).

Have fun! :o)