FlowHMM: Flow-based continuous hidden Markov models

Abstract

Continuous hidden Markov models (HMMs) assume that observations are generated from a mixture of Gaussian densities, limiting their ability to model more complex distributions. In this work, we address this shortcoming and propose a novel continuous HMM that allows to learn general continuous observation densities without constraining them to follow a Gaussian distribution or their mixtures. To that end, we leverage deep flow-based architectures that model complex, non-Gaussian functions. Moreover, to simplify optimization and avoid costly expectation-maximization algorithm, we use the co-occurrence matrix of discretized observations and consider the joint distribution of pairs of co-observed values. Even though our model is trained on discretized observations, it represents a continuous variant of HMM during inference, thanks to applying a separate flow model for each hidden state. The experiments on synthetic and real datasets show that our method outperforms Gaussian baselines.

Installation

Conda environment

• Create the new conda environment

conda env create -f conda.yml

• or update the existing one

conda env update -f conda.yml --prune

Packages installation

• Install all the required packages with single command

poetry install

Examples

• synthetic dataset with the following distributions (see Example 1 in paper):
  • 2 gaussians
  • 1 uniform

python main.py -e examples/SYNTHETIC_2G_1U.yaml \
 --nr_epochs=500 \
 --training_type=Q_training # or ML \
 --add_noise=True --noise_var=0.1 \
 --show_plots \
 --extra_n=$N 

where N variable is the length of training observations. We chose N=1000, 10000, 100000; see SYNTHETIC_2G_1U.yaml for more details.

• synthetic dataset with the following distributions (see Example 2 in paper):
  • 1 beta
  • 1 uniform
  • 1 gaussian

python main.py -e examples/SYNTHETIC_1B_1U.yaml \
 --nr_epochs=1000 \
 --add_noise=True --noise_var=0.00005 \
 --training_type=Q_training # or ML \
 --show_plots \
 --extra_n=$N --extra_L=$L 

where N variable is the length of training observations and L is the number of hidden states (flow models to learn). We chose N=1000, 10000, 100000 and L=2, 3; see SYNTHETIC_1B_1U_1G.yaml for more details.

• 2D synthetic dataset with two "Moons" and one Uniform distribution (see Example 5 in paper):

python main2d.py \
 -e examples/SYNTHETIC_2d_data_1U_2Moons.yaml # variant (a) \
 # -e examples/SYNTHETIC_2d_data_1U_2Moons_A2.yaml # variant (b) \
 --nr_epochs=500 \
 --training_type=Q_training # or ML \
 --extra_n=1000 --lrate=0.01 \
 --add_noise --noise_var=0.001 \
 --show_plots

• 2D synthetic dataset with one bivariate Gaussian, one Uniform and one related to geometric Brownian motion (see Example 6 in paper):

python main2d.py \
  -e examples/SYNTHETIC_2d_data_1G_1U_1GeomBrownianMotion.yaml # variant (a) \
  # -e examples/SYNTHETIC_2d_data_1G_1U_1GeomBrownianMotion_A2.yaml # variant (b) \
  --nr_epochs=500
  --training_type=Q_training # or ML \
  --extra_n=1000 \ 
  --lrate=0.01  \
  --add_noise --noise_var=0.001 \
  --show_plots