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PyLDS: Bayesian inference for linear dynamical systems

Authors: Matt Johnson and Scott Linderman

This package supports Bayesian learning and inference via Gibbs sampling, structured mean field, and expectation maximization (EM) for dynamical systems with linear Gaussian state dynamics and either linear Gaussian or count observations. For count data, we support either Pólya-gamma augmentation or Laplace approximation. All inference algorithms benefit from fast message passing code written in Cython with direct calls to the BLAS and LAPACK routines linked to the scipy build.

Installation

To install from pypi, just run

pip install pylds

To install from a clone of the git repository, you need to install Cython. Here's one way to do it:

pip install cython
git clone https://github.com/mattjj/pylds.git
pip install -e pylds

To handle count data, you'll also need pypolyagamma, which can be installed with

pip install pypolyagamma>=1.1

Example

PyLDS exposes a variety of classes and functions for working with linear dynamical systems. For example, the following snippet will generate synthetic data from a random model:

import numpy.random as npr
from pylds.models import DefaultLDS

D_obs = 1       # Observed data dimension
D_latent = 2	# Latent state dimension
D_input = 0	    # Exogenous input dimension
T = 2000  	    # Number of time steps to simulate

true_model = DefaultLDS(D_obs, D_latent, D_input)
inputs = npr.randn(T, D_input)
data, stateseq = true_model.generate(T, inputs=inputs)

# Compute the log likelihood of the data with the true params
true_ll = true_model.log_likelihood() 

The DefaultLDS constructor initializes an LDS with a random rotational dynamics matrix. The outputs are data, a T x D_obs matrix of observations, and stateseq, a T x D_latent matrix of latent states.

Now create another LDS and try to infer the latent states and learn the parameters given the observed data.

# Create a separate model and add the observed data
test_model = DefaultLDS(D_obs, D_latent, D_input)
test_model.add_data(data)

# Run the Gibbs sampler
N_samples = 100
def update(model):
    model.resample_model()
    return model.log_likelihood()

lls = [update(test_model) for _ in range(N_samples)]

We can plot the log likelihood over iterations to assess the convergence of the sampling algorithm:

# Plot the log likelihoods
plt.figure()
plt.plot([0, N_samples], true_ll * np.ones(2), '--k', label="true")
plt.plot(np.arange(N_samples), lls, color=colors[0], label="test")
plt.xlabel("iteration")
plt.ylabel("training likelihood")
plt.legend(loc="lower right")

We can also smooth the observations with the test model.

# Smooth the data
smoothed_data = test_model.smooth(data, inputs)

plt.figure()
plt.plot(data, color=colors[0], lw=2, label="observed")
plt.plot(smoothed_data, color=colors[1], lw=1, label="smoothed")
plt.xlabel("Time")
plt.xlim(0, 500)
plt.ylabel("Smoothed Data")
plt.legend(loc="upper center", ncol=2)

Check out the examples directory for demos of other types of inference, as well as examples of how to work with count data and missing observations.

For a lower-level interface to the fast message passing functions, see lds_messages.pyx, lds_info_messages.pyx, and lds_messages_interface.py.