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README.md

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## Description
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**PySPOD** is a Python package that implements the so-called **Spectral Proper Orthgonal Decomposition** whose name was first conied by (picard-&-delville-2000), and goes back to the original work by [(Lumley 1970)](#lumley-1970). The implementation proposed here follows the original contributions by [(Towne et al. 2018)](#towne-et-al.-2018), [(Schmidt & Towne 2019)](#schmidt-&-towne-2019).
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**PySPOD** is a Python package that implements the so-called **Spectral Proper Orthgonal Decomposition** whose name was first conied by (picard-&-delville-2000), and goes back to the original work by [(Lumley 1970)](#lumley-1970). The implementation proposed here follows the original contributions by [(Towne et al. 2018)](#towne-et-al-2018), [(Schmidt and Towne 2019)](#schmidt-and-towne-2019).
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**Spectral Proper Orthgonal Decomposition (SPOD)** has been extensively used in the past few years to identify spatio-temporal coherent pattern in a variety of datasets, mainly in the fluidmechanics and climate communities. In fluidmechanics it was applied to jets (Schmidt et al. 2017), wakes [(Araya et al. 2017)](#araya-et-al.-2017), and boundary layers [(Tutkun & George 2017)](#tutkun-&-george-2017), among others, while in weather and climate it was applied to ECMWF reanalysis datasets under the name Spectral Empirical Orthogonal Function, or SEOF, [(Schmidt et al. 2019)](#schmidt-et-al.-2019).
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**Spectral Proper Orthgonal Decomposition (SPOD)** has been extensively used in the past few years to identify spatio-temporal coherent pattern in a variety of datasets, mainly in the fluidmechanics and climate communities. In fluidmechanics it was applied to jets [(Schmidt et al. 2017)](#schmidt-et-al-2017), wakes [(Araya et al. 2017)](#araya-et-al-2017), and boundary layers [(Tutkun and George 2017)](#tutkun-and-george-2017), among others, while in weather and climate it was applied to ECMWF reanalysis datasets under the name Spectral Empirical Orthogonal Function, or SEOF, [(Schmidt et al. 2019)](#schmidt-et-al-2019).
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The SPOD approach targets statistically stationary problems and involves the decomposition of the cross-spectral density tensor. This means that the SPOD leads to a set of spatial modes that oscillate in time at a single frequency and that optimally capture the variance of an ensemble of stochastic data [(Towne et al. 2018)](#towne-et-al.-2018). Therefore, given a dataset that is statistically stationary, one is able to capture the optimal spatio-temporal coherent structures that explain the variance in the dataset.
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The SPOD approach targets statistically stationary problems and involves the decomposition of the cross-spectral density tensor. This means that the SPOD leads to a set of spatial modes that oscillate in time at a single frequency and that optimally capture the variance of an ensemble of stochastic data [(Towne et al. 2018)](#towne-et-al-2018). Therefore, given a dataset that is statistically stationary, one is able to capture the optimal spatio-temporal coherent structures that explain the variance in the dataset.
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This can help identifying relations to multiple variables or understanding the reduced order behavior of a given phenomenon of interest and represent a powerful tool for the data-driven analysis of nonlinear dynamical systems. The SPOD approach shares some relationships with the dynamic mode decomposition (DMD), and the resolvent analysis, [(Towne et al. 2018)](#Towne-et-al.-2018), that are also widely used approaches for the data-driven analysis of nonlinear systems. SPOD can be used for both experimental and simulation data, and a general description of its key parameters can be found in [(Schmidt & Colonius 2020)](#schmidt-&-colonius-2020).
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This can help identifying relations to multiple variables or understanding the reduced order behavior of a given phenomenon of interest and represent a powerful tool for the data-driven analysis of nonlinear dynamical systems. The SPOD approach shares some relationships with the dynamic mode decomposition (DMD), and the resolvent analysis, [(Towne et al. 2018)](#Towne-et-al-2018), that are also widely used approaches for the data-driven analysis of nonlinear systems. SPOD can be used for both experimental and simulation data, and a general description of its key parameters can be found in [(Schmidt and Colonius 2020)](#schmidt-and-colonius-2020).
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In this package we implement three version of SPOD
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- SPOD_low_storage: that is intended for large RAM machines or small datasets
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- SPOD_low_ram: that is intended for small RAM machines or large datasets, and
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- SPOD_streaming: that is the algorithm presented in [(Schmidt & Towne 2019)](schmidt-&-towne-2019), and it is intended for large datasets.
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- SPOD_streaming: that is the algorithm presented in [(Schmidt and Towne 2019)](schmidt-and-towne-2019), and it is intended for large datasets.
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To see how to use the **PySPOD** package and its user-friendly interface, you can look at the [**Tutorials**](tutorials/README.md).
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*Stochastic Tools in Turbulence.*
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[[DOI](https://www.elsevier.com/books/stochastic-tools-in-turbulence/lumey/978-0-12-395772-6?aaref=https%3A%2F%2Fwww.google.com)]
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#### (Picard & Delville 2000)
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#### (Picard and Delville 2000)
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*Pressure velocity coupling in a subsonic round jet.*
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[[DOI](https://www.sciencedirect.com/science/article/abs/pii/S0142727X00000217)]
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#### (Tutkun & George 2017)
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#### (Tutkun and George 2017)
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*Lumley decomposition of turbulent boundary layer at high Reynolds numbers.*
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[[DOI](https://aip.scitation.org/doi/10.1063/1.4974746)]
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#### (Schmidt et al. 2017)
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#### (Schmidt et al 2017)
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*Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability.*
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[[DOI](https://doi.org/10.1017/jfm.2017.407)]
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#### (Araya et al. 2017)
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#### (Araya et al 2017)
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*Transition to bluff-body dynamics in the wake of vertical-axis wind turbines.*
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[[DOI]( https://doi.org/10.1017/jfm.2016.862)]
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#### (Taira et al. 2017)
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#### (Taira et al 2017)
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*Modal analysis of fluid flows: An overview.*
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[[DOI](https://doi.org/10.2514/1.J056060)]
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#### (Towne et al. 2018)
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#### (Towne et al 2018)
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*Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis.*
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[[DOI]( https://doi.org/10.1017/jfm.2018.283)]
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#### (Schmidt and Towne 2019)
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*An efficient streaming algorithm for spectral proper orthogonal decomposition.*
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[[DOI](https://doi.org/10.1016/j.cpc.2018.11.009)]
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#### (Schmidt et al. 2019)
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#### (Schmidt et al 2019)
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*Spectral empirical orthogonal function analysis of weather and climate data.*
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[[DOI](https://doi.org/10.1175/MWR-D-18-0337.1)]
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#### (Schmidt & Colonius 2020)
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#### (Schmidt and Colonius 2020)
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*Guide to spectral proper orthogonal decomposition.*
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[[DOI](https://doi.org/10.2514/1.J058809)]
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