Spectral Proper Orthogonal Decomposition in Python

This is a port of the Towne et al. SPOD Matlab function to Python 3.

Features

• Calculate SPOD on N-dimensional data, where time is first dimension
• Same set of SPOD options as original Matlab version (window,weight,dt,noverlap,normvar)
• Keep data in memory or use little memory and keep on disk (low memory mode)
• Can save results to disk (HDF5 file), in addition to or instead of returning in-memory results

Requirements

Install the below requirements with pip install requirements.txt

• spod: numpy, scipy, h5py for low-memory mode
• examples: matplotlib, h5py

Files

File	Description
spod.py	Spectral proper orthogonal decomposition in Matlab
example_1.py	Inspect data and plot SPOD spectrum
example_2.py	Plot SPOD spectrum and inspect SPOD modes
example_3.py	Specify spectral estimation parameters and use weighted inner product
example_4.py	Calculate the SPOD of large data and save results on hard drive
example_5.py	Calculate full SPOD spectrum of large data
example_6.py	Calculate and plot confidence intervals for SPOD eigenvalues
utils.py/getjet	Interfaces external data source with SPOD() (examples 4-5)
utils.py/trapzWeightsPolar	Integration weight matrix for cylindrical coordinates (examples 3-6)
jet_data/jetLES.mat	Mach 0.9 turbulent jet test database

Usage

spod(x, window='hamming', weight=None, noverlap=None, dt=1, mean=None, isreal=None,
     nt=None, conflvl=None, normvar=False,  debug=0, lowmem=False, savefile=None,
     nmodes=None, savefreqs=None)

Parameters
----------
x : array or function object
    Data array whose first dimension is time, or function that retrieves
    one snapshot at a time like x(i).  x(i), like x, can have any dimension. 
    If x is a function, it is recommended to specify the total number of 
    snaphots in nt (see below). If not specified, nt defaults to 10000. 
    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.
window : vector, int, or string, optional
    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 none or 'hamming',
    a Hamming window is used.
weight : array, optional
    A spatial inner product weight.  SPOD modes are optimally ranked and 
    orthogonal at each frequency. WEIGHT must have the same spatial 
    dimensions as x.
noverlap : int, optional
    Number of snaptions to overlap consecutive blocks.  noverlap defaults
    to 50% of the length of WINDOW if not specified.
dt : float, optional
    Time step between consecutive snapshots to determine a physical 
    frequency F.  dt defaults to 1 if not specified.
mean : array or string, optional
    A mean that is subtracted from each snapshot.  If 'blockwise', the mean
    of each block is subtracted from itself.  If x is a function the mean
    provided is a temporal mean.
isreal : bool, optional
    Describes if x data is real.
nt : int, optional
    Number of snapshots.  If x is an array, this is determined from x
    dimensions.  If x is a function, this defaults to 10000 if not specified.
conflvl : bool or float, optional
    Calculate and return confidence interval levels of L (Lc).  If True,
    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.  If a float between 0 and 1, 
    the conflvl*100% confidence interval of L is returned. A 
    chi-squared distribution is used, i.e. we assume a standard normal 
    distribution of the SPOD eigenvalues.  Defaults to None/False.
normvar : bool, optional
    Normalize each block by pointwise variance.  Defaults to false.
debug : {0, 1, 2}, optional
    Verbosity of output.  0 hides output.  1 shows some output, 2 shows all output.
    Defaults to 0.
lowmem : bool, optional
    Specifies whether to use low-memory mode.  If True, this stores the FFT blocks in a
    temporary file on disk, and also stores all returned quantities on disk, returning
    a file handle. Default is False, keeping everything in memory.  If
    lowmem is True, savefile must be specified for the returned data.  This mode requires
    the h5py package.
savefile : string, optional
    Filename to which to save the results in HDF5 format.  If lowmem is True,
    a handle for this file is returned.  If False or None, the in-memory results 
    are returned in a dictionary.  If file exists, it is overwritten.  Defaults to None/False.
nmodes : int, optional
    Number of most energetic SPOD modes to be saved.  Defaults to all modes.
savefreqs: list of ints, optional
    List of frequency indices to calculate modes (P) and spectral energies (L).  Meant 
    to reduce size of data if not all frequences are needed.  Defaults to all frequences.

Returns
-------
output
    A dictionary containing SPOD modes (P), modal energy spectra (L), and frequency
    vector (f).  If conflvl is specified, confidence interval is returned (Lc).  If
    lowmem is True, a file object is returned which points to the on-disk HDF5 dataset.

References

[1] Towne, A., Schmidt, O. T., Colonius, T., Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis, arXiv:1708.04393, 2017

[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, submitted to JFM, 2017

[4] Schmidt, O. T., Towne, A., Rigas, G., Colonius, T., Bres, G. A., Spectral analysis of jet turbulence, arXiv:1711.06296, 2017