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44 | 44 |
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45 | 45 |
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46 | 46 | ## Description |
47 | | -**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). |
| 47 | +**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). |
48 | 48 |
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49 | | -**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). |
| 49 | +**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). |
50 | 50 |
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51 | | -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. |
| 51 | +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. |
52 | 52 |
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53 | | -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). |
| 53 | +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). |
54 | 54 |
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55 | 55 | In this package we implement three version of SPOD |
56 | 56 | - SPOD_low_storage: that is intended for large RAM machines or small datasets |
57 | 57 | - SPOD_low_ram: that is intended for small RAM machines or large datasets, and |
58 | | -- SPOD_streaming: that is the algorithm presented in [(Schmidt & Towne 2019)](schmidt-&-towne-2019), and it is intended for large datasets. |
| 58 | +- SPOD_streaming: that is the algorithm presented in [(Schmidt and Towne 2019)](schmidt-and-towne-2019), and it is intended for large datasets. |
59 | 59 |
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60 | 60 | To see how to use the **PySPOD** package and its user-friendly interface, you can look at the [**Tutorials**](tutorials/README.md). |
61 | 61 |
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@@ -208,39 +208,39 @@ IF you want to run tests locally, you can do so by: |
208 | 208 | *Stochastic Tools in Turbulence.* |
209 | 209 | [[DOI](https://www.elsevier.com/books/stochastic-tools-in-turbulence/lumey/978-0-12-395772-6?aaref=https%3A%2F%2Fwww.google.com)] |
210 | 210 |
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211 | | -#### (Picard & Delville 2000) |
| 211 | +#### (Picard and Delville 2000) |
212 | 212 | *Pressure velocity coupling in a subsonic round jet.* |
213 | 213 | [[DOI](https://www.sciencedirect.com/science/article/abs/pii/S0142727X00000217)] |
214 | 214 |
|
215 | | -#### (Tutkun & George 2017) |
| 215 | +#### (Tutkun and George 2017) |
216 | 216 | *Lumley decomposition of turbulent boundary layer at high Reynolds numbers.* |
217 | 217 | [[DOI](https://aip.scitation.org/doi/10.1063/1.4974746)] |
218 | 218 |
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219 | | -#### (Schmidt et al. 2017) |
| 219 | +#### (Schmidt et al 2017) |
220 | 220 | *Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability.* |
221 | 221 | [[DOI](https://doi.org/10.1017/jfm.2017.407)] |
222 | 222 |
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223 | | -#### (Araya et al. 2017) |
| 223 | +#### (Araya et al 2017) |
224 | 224 | *Transition to bluff-body dynamics in the wake of vertical-axis wind turbines.* |
225 | 225 | [[DOI]( https://doi.org/10.1017/jfm.2016.862)] |
226 | 226 |
|
227 | | -#### (Taira et al. 2017) |
| 227 | +#### (Taira et al 2017) |
228 | 228 | *Modal analysis of fluid flows: An overview.* |
229 | 229 | [[DOI](https://doi.org/10.2514/1.J056060)] |
230 | 230 |
|
231 | | -#### (Towne et al. 2018) |
| 231 | +#### (Towne et al 2018) |
232 | 232 | *Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis.* |
233 | 233 | [[DOI]( https://doi.org/10.1017/jfm.2018.283)] |
234 | 234 |
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235 | 235 | #### (Schmidt and Towne 2019) |
236 | 236 | *An efficient streaming algorithm for spectral proper orthogonal decomposition.* |
237 | 237 | [[DOI](https://doi.org/10.1016/j.cpc.2018.11.009)] |
238 | 238 |
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239 | | -#### (Schmidt et al. 2019) |
| 239 | +#### (Schmidt et al 2019) |
240 | 240 | *Spectral empirical orthogonal function analysis of weather and climate data.* |
241 | 241 | [[DOI](https://doi.org/10.1175/MWR-D-18-0337.1)] |
242 | 242 |
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243 | | -#### (Schmidt & Colonius 2020) |
| 243 | +#### (Schmidt and Colonius 2020) |
244 | 244 | *Guide to spectral proper orthogonal decomposition.* |
245 | 245 | [[DOI](https://doi.org/10.2514/1.J058809)] |
246 | 246 |
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