A Python implimentation of Sparse Random Mode Decomposition
The goal of this algorithm is to provide a sparse time-frequency representation of a signal to decompose the signal into intrinsic mode-like functions.
Easiest method:
pip install git+https://github.com/GiangTTran/SparseRandomModeDecomposition.git#egg=srmdpy
The code can also be accessed by cloning this repository
git clone <https://github.com/GiangTTran/SparseRandomModeDecomposition>
or downloading the source by clicking Code -> Download ZIP, and then installed via
python setup.py install
The algorithm can be broken into three main steps:
- Randomly generate many time-frequency features
- Find a sparse representation of the input with respect to the random features
- Cluster near-by features into modes
The features used are (Gaussian) windowed sinusoids with random phases, although any wavelet-like feature could be used. The sparse representation is found by solving the associated Basic Pursuit Denoising Problem (BPDP) using the SPGL1 algorithm. And the clustering is performed by scikit-learn's implimentation of the DBSCAN algorithm.
Given a time series y=[y1, ..., ym] sampled at time points t=[t1, ..., tm],
SRMD recovers the modes s1, ..., sK and possibly denoise the input signal where
y(t) = s1(t) + s2(t) + ... + sK(t) + noise.
This method is most effective when the modes sk(t) occupy connected curves or regions that are mutualy disjoint in time-frequency space such as non-intersecting intrinsic mode functions.
Specificly, SRMD(y, ...) first solves the problem of representing a time-series y as a sum of random features
y(t) ~ sum_j(c_j * exp(-0.5 * ((t-tau_j)/w)^2) * sin(2*pi*frq_j*t + phs_j)),
where the learned weights vector c is sparse, by solving BPDN
argmin ||c||_1 s.t. ||Ac - y||_2 < r*||y||_2.
Here, (A)_ij = exp(-0.5 * ((t_i-tau_j)/w)^2) * sin(2*pi*frq_j*t_i + phs_j).
The features are clustered with DBSCAN which groups features with near-by
(tau_j, frq_j*frq_scale) in terms of their l2 distance to each other.
srmdpy.py:
SRMD(y, t):
Decomposes a time series y(t) = sum_k s_k(t) into modes s_k(t) that are well
connected in time-frequency space, and mutualy disjoint from each other.
utils.py:
- Hosts the helper functions for generating the features used by SRMD
generate_features(N, t):
Generates N random features on the time points t.
features_from_parameters(t, tau, frq, phs):
Generates features from the given parameters
window(t, w):
Creates a Gaussian window function of width w on the time points t.
visualization.py:
- Contains useful plotting functions for visualizating decomposition results.
signal(t, y):
Plots a time series y sampled at time points t.
all_modes(t, modes):
Plots each mode_i = modes[:, i] sampeled at t.
modes_with_cluster(t, modes, tau, frq, labels):
Plots each mode_i and the time-frequency pairs (tau_n, frq_n) coloured
according to labels.
weights(tau, frq, weights):
Creates a spectrogram-like plot where time-frequency pairs (tau_n, frq_n)
are coloured accoring to weights_n.
constants.py:
- Store helpful constants and default values used by SRMD and helper functions.
twopi = 2 * pi
default_w = 0.1
default_r = 0.05
default_min_samples = 4
minimal.ipynb:
- Decomposes a composite signal with two pure sinusoids at 5 Hz and 20 Hz into its two modes.
frequency_estimation.ipynb:
- Decomposes a composite signal similar to minimal.ipynb. Results are also plotted and the center frequencies are estimated.
graviational.ipynb:
- Visualizes the time-frequency chirp from a black hole merger
music.ipynb:
- Decomposes a flute and guitar clip into each instrument
discontinuous.ipynb:
- Showcases the visualization tools in srmdpy on a challanging signal. This is similar to the example shown in Section 3.1 of the SRMD paper.
all_parameters.ipynb:
- Showcases all the hyperparameters avalible on an intersecting signal. This is similar to the example shown in Section 3.2 of the SRMD paper.
synthetic.ipynb:
- Creates the synthetic data from the SRMD paper
music:
- flute.wav, guitar.wav, both.wav
gravitational:
- numerical_gravitational_data.txt, observed_gravitational_data.txt
Created by Nicholas Richardson March 2022 based on the decomposition algorithm developed by Nicholas Richardson, Hayden Schaeffer, and Giang Tran.
Feel free to reach out to njericha@uwaterloo.ca if you have any questions or just to say hi. Any suggestions and contributions are also welcome!
If you found this package useful in any way, please cite the associated paper:
@article{richardson2022srmd,
doi = {10.48550/ARXIV.2204.06108},
url = {\url{https://arxiv.org/abs/2204.06108}},
author = {Richardson, Nicholas and Schaeffer, Hayden and Tran, Giang},
title = {{SRMD}: Sparse Random Mode Decomposition},
publisher = {arXiv},
year = {2022},
copyright = {arXiv.org perpetual, non-exclusive license},
journal = {arXiv preprint arXiv:2204.06108}
}