Synopsis

Persistent Homology on Embedded Time-Series [DOI: 10.5281/zenodo.3528239]

This package offers high-level tools for exploration and visualization of delay coordinate embedding and persistent homology. It is used to investigate the utilization of these together as a signal processing technique. Also included is a dataset of time-series (mostly musical instrument recordings) and higher dimensional trajectories as .txt files.

PHETS uses Perseus to compute persistent homology.

PHETS was originally developed for the research detailed here. If you find PHETS useful in your research work, please cite N. Sanderson, E. Shugerman, S. Molnar, J. Meiss, and E. Bradley, "Computational Topology Techniques for Characterizing Time-Series Data", IDA-17 (Proceedings of the 13th International Symposium on Intelligent Data Analysis), London, October 2017:

Please direct inquires to [email protected].

Installation

Dependencies

• Python 2.7
• ffmpeg
• gnuplot (with pngcairo)

Debian-based

sudo apt-get install ffmpeg
sudo apt-get install gnuplot-x11

macOS

brew install ffmpeg
brew install gnuplot --with-cairo

If you're running into an error about compiling find_landmarks.c, see the trouble shooting section of documentation.pdf for information about C compiler / library dependencies on macOS.

Install PHETS

git clone https://github.com/eeshugerman/PHETS.git
cd PHETS
pip install -r requirements.txt

Demo

import numpy as np
from signals import TimeSeries
from utilities import idx_to_freq
from DCE.movies import slide_window
from PH import Filtration
from config import default_filtration_params as filt_params

ts = TimeSeries(
    'datasets/time_series/C135B/49-C135B.txt',
    crop=(0, 5),
    num_windows=250,
    window_length=.05,
    time_units='seconds'    # defaults to 'samples'
)

# the slide_window function will create a movie showing an embedding for each
# window of the time series and return the trajectory

tau = (1 / idx_to_freq(49)) / np.pi      # first, choose tau = period / pi 

traj = slide_window(ts, 'output/demo/embed_movie.mp4', m=2, tau=tau)

# alternatively, we could skip the movie and embed explicitly:
traj = ts.embed(m=2, tau=tau)

# now, lets build a filtration from the trajectory that is shown in the 100th 
# frame of the slide_window movie
traj_window = traj.windows[100]

# parameters used to build the filtration:
filt_params.update(
    {
        'ds_rate': 25,
        'num_divisions': 10,                # number of epsilon vals in filtration
        # 'max_filtration_param': .05,      # if > 0, explicit
        'max_filtration_param': -10,        # if < 0, stops st first 10 dim simplex
        # 'use_cliques': True,
    }
)

# build the filtration:
filt = Filtration(traj_window, filt_params)

filt.movie('output/demo/filt_movie.mp4')

A filtration can be summarized by its homology, which may be expressed as a persistence rank function:

Or as a persistence rank function:

filt.plot_prf('output/demo/prf.png')        # plot the persistence rank function

Persistence rank functions are amenable to statistical analysis. prfstats.L2Classifier, upon initialization, computes a mean PRF and variance from a set of training PRFs; subsequently, L2Classifier.predict(PRF, k) returns True if the L2 distance from PRF to the mean PRF is smaller than k times the standard deviation. prfstats.plot_l2rocs takes two pre-windowed Trajectorys, traj1 and traj2, and partitions the windows roughly as follows:

windows1, windows2 = traj1.windows, = traj2.windows
train1, test1 = windows1[1::2], windows1[::2]
train2, test2 = windows2[1::2], windows2[::2]

Two L2Classifiers are initialized:

clf1 = L2Classifier(train1)
clf2 = L2Classifier(train2)

clf1.predict and clf2.predict are each called on both test1 and test2 for a range of k, and the results are plotted as ROC curves.

traj1 = TimeSeries(
    'datasets/time_series/clarinet/sustained/high_quality/40-clarinet-HQ.txt',
    crop=(75000, 180000),
    num_windows=50,
    window_length=1500,
    vol_norm=(0, 0, 1)  # (full, crop, windows)
).embed(tau=32, m=2)


traj2 = TimeSeries(
    'datasets/time_series/viol/40-viol.txt',
    crop=(35000, 140000),
    num_windows=50,
    window_length=1500,
    vol_norm=(0, 0, 1)
).embed(tau=32, m=2)


filt_params.update({
    'max_filtration_param': -21,
    'num_divisions': 20,
    'ds_rate': 20
})

plot_l2rocs(
    traj1, traj2,
    'clarinet', 'viol',
    'output/demo/ROCs.png',
    filt_params,
    k=(0, 10.01, .01),
)

For this case, at least, the classifiers preform very well: