## Spectral proper orthogonal decomposition (SPOD)

# Spectral Proper Orthogonal Decomposition in Matlab

SPOD() is a Matlab implementation of the frequency domain form of proper orthogonal decomposition (POD, also known as principle component analysis or Karhunen-Loève decomposition) called spectral proper orthogonal decomposition (SPOD). SPOD is derived from a space-time POD problem for stationary flows [1,2] and leads to modes that each oscillate at a single frequency. SPOD modes represent dynamic structures that optimally account for the statistical variability of stationary random processes.

The large-eddy simulation data provided along with this example is a subset of the database of a Mach 0.9 turbulent jet described in [3] and was calculated using the unstructured flow solver Charles developed at Cascade Technologies. If you are using the database in your research or teaching, please include explicit mention of Brès et al. [3]. The test database consists of 5000 snapshots of the symmetric component (m=0) of a round turbulent jet.

`spod.m`

is a stand-alone Matlab function with no toolbox dependencies. All other Matlab files contained in this repository are related to the six examples that demonstrate the functionality of the code (see file descriptions below). A physical interpretation of the results obtained from the examples can be found in [4]. The reference for the frequency-time analsyis is [5].

## Download

### Using your browser

Repository zip file with examples (81.5 MB): https://github.com/SpectralPOD/spod_matlab/archive/master.zip

Matlab function only (15 KB): https://raw.githubusercontent.com/SpectralPOD/spod_matlab/master/spod.m

### Using Git in the terminal

git clone https://github.com/SpectralPOD/spod_matlab.git

## Files

File | Description |
---|---|

spod.m | Spectral proper orthogonal decomposition in Matlab |

example_1.m | Inspect data and plot SPOD spectrum |

example_2.m | Plot SPOD spectrum and inspect SPOD modes |

example_3.m | Specify spectral estimation parameters and use weighted inner product |

example_4.m | Calculate the SPOD of large data and save results on hard drive |

example_5.m | Calculate full SPOD spectrum of large data |

example_6.m | Calculate and plot confidence intervals for SPOD eigenvalues |

example_7_FTanalysis.m | Frequency-time analysis |

tcoeffs.m | Time-continuous expansion coefficients via convolution |

jet_data/getjet.m | Interfaces external data source with SPOD() (examples 4-5) |

utils/trapzWeightsPolar.m | Integration weight matrix for cylindrical coordinates (examples 3-6) |

utils/jetLES.mat | Mach 0.9 turbulent jet test database |

LICENSE.txt | License |

## Usage

[L,P,F] = SPOD(X) returns the spectral proper orthogonal decomposition
of the data matrix X whose first dimension is time. X can have any
number of additional spatial dimensions or variable indices.
The columns of L contain the modal energy spectra. P contains the SPOD
modes whose spatial dimensions are identical to those

of X. The first index of P is the frequency and
the last one the mode number ranked in descending order
by modal energy. F is the frequency vector. If DT is not specified, a
unit frequency sampling is assumed. For real-valued data, one-sided spectra
are returned. Although SPOD(X) automatically chooses default spectral
estimation parameters, it is strongly encouraged to manually specify
problem-dependent parameters on a case-to-case basis.

[L,P,F] = SPOD(X,WINDOW) uses a temporal window. If WINDOW is a vector, X is divided into segments of the same length as WINDOW. Each segment is then weighted (pointwise multiplied) by WINDOW. If WINDOW is a scalar, a Hamming window of length WINDOW is used. If WINDOW is omitted or set as empty, a Hamming window is used.

[L,P,F] = SPOD(X,WINDOW,WEIGHT) uses a spatial inner product weight in which the SPOD modes are optimally ranked and orthogonal at each frequency. WEIGHT must have the same spatial dimensions as X.

[L,P,F] = SPOD(X,WINDOW,WEIGHT,NOVERLAP) increases the number of segments by overlapping consecutive blocks by NOVERLAP snapshots. NOVERLAP defaults to 50% of the length of WINDOW if not specified.

[L,P,F] = SPOD(X,WINDOW,WEIGHT,NOVERLAP,DT) uses the time step DT between consecutive snapshots to determine a physical frequency F.

[L,P,F] = SPOD(XFUN,...,OPTS) accepts a function handle XFUN that provides the i-th snapshot as x(i) = XFUN(i). Like the data matrix X, x(i) can have any dimension. It is recommended to specify the total number of snaphots in OPTS.nt (see below). If not specified, OPTS.nt defaults to 10000. OPTS.isreal should be specified if a two-sided spectrum is desired even though the data is real-valued, or if the data is initially real-valued, but complex-valued-valued for later snaphots.

[L,P,F] = SPOD(X,WINDOW,WEIGHT,NOVERLAP,DT,OPTS) specifies options: OPTS.savefft: save FFT blocks to avoid storing all data in memory [{false} | true] OPTS.deletefft: delete FFT blocks after calculation is completed [{true} | false] OPTS.savedir: directory where FFT blocks and results are saved [ string | {'results'}] OPTS.savefreqs: save results for specified frequencies only [ vector | {all} ] OPTS.mean: provide a mean that is subtracted from each snapshot [ array of size X | {temporal mean of X; 0 if XFUN} ] OPTS.nsave: number of most energtic modes to be saved [ integer | {all} ] OPTS.isreal: complex-valuedity of X or represented by XFUN [{determined from X or first snapshot if XFUN is used} | logical ] OPTS.nt: number of snapshots [ integer | {determined from X; defaults to 10000 if XFUN is used}] OPTS.conflvl: confidence interval level [ scalar between 0 and 1 | {0.95} ]

[L,PFUN,F] = SPOD(...,OPTS) returns a function PFUN instead of the SPOD data matrix P if OPTS.savefft is true. The function returns the j-th most energetic SPOD mode at the i-th frequency as p = PFUN(i,j) by reading the modes from the saved files. Saving the data on the hard drive avoids memory problems when P is large.

[L,P,F,Lc] = SPOD(...) returns the confidence interval Lc of L. By default, the lower and upper 95% confidence levels of the j-th most energetic SPOD mode at the i-th frequency are returned in Lc(i,j,1) and Lc(i,j,2), respectively. The OPTS.conflvl*100% confidence interval is returned if OPTS.conflvl is set. For example, by setting OPTS.conflvl = 0.99 we obtain the 99% confidence interval. A chi-squared distribution is used, i.e. we assume a standard normal distribution of the SPOD eigenvalues.

## References

[1] Towne, A., Schmidt, O. T., Colonius, T., *Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis*, J. of Fluid Mech. 847, 821–867, 2018

[2] Lumley, J. L., *Stochastic tools in turbulence*, Academic Press, 1970

[3] G. A. Brès, P. Jordan, M. Le Rallic, V. Jaunet, A. V. G. Cavalieri, A. Towne, S. K. Lele, T. Colonius, O. T. Schmidt, *Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets*, J. of Fluid Mech. 851, 83-124, 2018

[4] Schmidt, O. T., Towne, A., Rigas, G., Colonius, T., Bres, G. A., *Spectral analysis of jet turbulence*, J. of Fluid Mech. 855, 953–982, 2018

[5] Nekkanti, A. and Schmidt, O. T., *Frequency–time analysis, low-rank reconstruction and denoising of turbulent flows using SPOD* , J. of Fluid Mech. 926, A26, DOI 10.1017/jfm.2021.681, 2021

### Cite As

Towne, A., Schmidt, O. T., Colonius, T., Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis, J. of Fluid Mech. 847, 821–867, 2018

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