Embed time series using neural networks that penalize spurious neighbors.

Given a one-dimensional or few-dimensional time series, this package uses the time history of the system to find additional hidden variables that describe the process in a higher-dimensional space. In this sense, the embedding attempts to invert a projection of the higher-dimensional dynamics induced by a measurement.

This package implements many standard time series embedding techniques originally developed for embedding strange attractors of chaotic systems. These methods include average mutual information, Eigen-time-delay coordinates (ETD), time-lagged independent component analysis (tICA). The neural network models use a regularizer inspired by the false-nearest-neighbors method of Kennel et al..

The neural network and regularizer are further described in the following paper. If using this code for published work, please consider citing the paper.

William Gilpin. "Deep reconstruction of strange attractors from time series" 2020. Advances in Neural Information Processing Systems (NeurIPS) 2020 https://arxiv.org/abs/2002.05909

This library requires the following packages

• Numpy
• Scipy
• Tensorflow 2.0 or greater
• Scikit-learn
• Matplotlib (demos only)
• Jupyter Notebook (demos only)

To use this repository, directly download the source:

git clone https://github.com/williamgilpin/fnn

Test that everything is working:

python tests/test_models.py 

Models follow the scikit-learn API

from fnn.models import MLPEmbedding

model = MLPEmbedding(3, time_window=10)
time_series = np.loadtxt("datasets/lorenz.csv.gz", delimiter=",") # load 1D time series

embedding = model.fit_transform(time_series) # make 3D embedding

More detailed examples may be found in the various tutorial notebooks in the demos directory

• demos.ipynb shows the step-by-step process of constructing embeddings of the Lorenz attractor, experimental measurements of a double pendulum, a quasiperiodic torus, the Rössler attractor, and a high-dimensional chaotic ecosystem.
• compare.ipynb trains an LSTM and MLP with the FNN regularizer, as well as comparison models with tICA, ETD, and Average Mutual Information embeddings.
• exploratory.ipynb applies the embedding technique to several real-world time series datasets, such as electrocardiogram measurements, neural spiking, and measurements of eruptions of the Old Faithful geyser.

A great summary of the work in this repository, and the broader topic, has been written by Sigrid Keydana on the RStudio blog. The post includes an R implementation of the fnn regularizer.

In order to provide a baseline embedding technique for comparison, the file tica.py has been extracted and modifed from the molecular dynamics suite MSMBuilder. We include the modified file directly, in order to reduce dependencies. If using tICA in any work, please credit the original MSMBuilder authors and repository.

The datasets folder contains abridged versions of several time series datasets used for testing and evaluating the code. We summarize these files, and provide their original sources, here:

• geyser_train_test.csv corresponds to detrended temperature readings from the main runoff pool of the Old Faithful geyser in Yellowstone National Park, downloaded from the GeyserTimes database. Temperature measurements start on April 13, 2015 and occur in one-minute increments.
• electricity_train_test.csv corresponds to average power consumption by 321 Portuguese households between 2012 and 2014, in units of kilowatts consumed in fifteen minute increments. This dataset is from the UCI machine learning database.
• pendulum_train.csv and pendulum_test.csv correspond to two different double pendulum experiments, taken from a series of experiments by Asseman et al.. In Asseman et al.'s original study, pendula were filmed and segmented to produce (x, y) positions of centroids over time. Here, we have converted the dataset into the four canonical Hamiltonian coordinates (theta1, theta2, p1, p2).
• ecg_train.csv and ecg_test.csv correspond to ECG measurements for two different patients, taken from the PhysioNet QT database
• mouse.csv A time series of spiking rates for a neuron in a mouse thalamus. Raw spike data was obtained from CRCNS and processed with the authors' code in order to generate a spike rate time series.
• roaming_worm1.csv and dwelling_worm1.csv are time series of the first five principal components of C. elegans body curvature during crawling, taken from Ahamed et al 2019
• gait_marker_trackers_patient1_speed1.csv and gait_force_patient1_speed1.csv are marker positions and force recordings for a patient running on a treadmill, from the GaitPhase database
• accelerometer_subject1.csv contains smartphone accelerometer recordings of a walking individual, taken from Vajdi et al 2019