pohmm is an implementation of the partially observable hidden Markov model, a generalization of the hidden Markov model in which the underlying system state is partially observable through event metadata at each time step.
An application that motivates usage of such a model is keystroke biometrics where the user can be in either a passive or active hidden state at each time step, and the time between key presses depends on the hidden state. In addition, the hidden state depends on the key that was pressed; thus the keys are observed symbols that partially reveal the hidden state of the user.
Much of this code is based on the hmmlearn python package.
To install the latest stable version:
$ pip install pohmm
To install the development version:
$ pip install git+git://github.com/vmonaco/pohmm.git
Given an observation sequence,
TRAIN_OBS = [4, 3, 4, 2, 3, 1]With partially observed states,
TRAIN_PSTATES = ['b', 'a', 'b', 'a', 'c', 'a']Create and train a model:
model = pohmm.Pohmm(n_hidden_states=2,
init_spread=2,
emissions=['lognormal'],
init_method='obs',
smoothing='freq',
random_state=1234)
model.fit([np.c_[TRAIN_OBS]], [TRAIN_PSTATES])To view the fitted model parameters,
np.set_printoptions(precision=3)
print(model)
# POHMM
# H-states: 2
# P-states: (3) ['a', 'b', 'c']
# Emission: ['lognormal']
# --------------------------------------------------------------------------------
# Transition probas
# ________________________________________
# . -> .
# [[ 0.5 0.5 ]
# [ 0.333 0.667]]
# ________________________________________
# a -> .
# [[ 5.000e-01 5.000e-01]
# [ 1.000e-05 1.000e+00]]
# ________________________________________
# . -> a
# [[ 0.5 0.5]
# [ 0.5 0.5]]
# ________________________________________
# b -> .
# [[ 5.000e-01 5.000e-01]
# [ 5.000e-06 1.000e+00]]
# ________________________________________
# . -> b
# [[ 5.000e-01 5.000e-01]
# [ 1.000e-05 1.000e+00]]
# ________________________________________
# c -> .
# [[ 5.000e-01 5.000e-01]
# [ 1.000e+00 1.000e-05]]
# ________________________________________
# . -> c
# [[ 5.000e-01 5.000e-01]
# [ 1.000e-05 1.000e+00]]
# ________________________________________
# a -> a
# [[ 0.5 0.5 ]
# [ 0.375 0.625]]
# ________________________________________
# a -> b
# [[ 5.000e-01 5.000e-01]
# [ 1.000e-05 1.000e+00]]
# ________________________________________
# a -> c
# [[ 5.000e-01 5.000e-01]
# [ 1.000e-05 1.000e+00]]
# ________________________________________
# b -> a
# [[ 0.5 0.5]
# [ 0.1 0.9]]
# ________________________________________
# b -> b
# [[ 0.5 0.5 ]
# [ 0.083 0.917]]
# ________________________________________
# b -> c
# [[ 0.5 0.5 ]
# [ 0.083 0.917]]
# ________________________________________
# c -> a
# [[ 0.5 0.5 ]
# [ 0.833 0.167]]
# ________________________________________
# c -> b
# [[ 0.5 0.5]
# [ 0.5 0.5]]
# ________________________________________
# c -> c
# [[ 0.5 0.5]
# [ 0.5 0.5]]
# --------------------------------------------------------------------------------
# Starting probas
# .: [ 1.000e-05 1.000e+00]
# a: [ 1.000e-05 1.000e+00]
# b: [ 1.000e-05 1.000e+00]
# c: [ 1.000e-05 1.000e+00]
# --------------------------------------------------------------------------------
# Steady probas
# .: [ 0.375 0.625]
# a: [ 0.5 0.5]
# b: [ 0.25 0.75]
# c: [ 0.25 0.75]
# --------------------------------------------------------------------------------
# Emissions
# ________________________________________
# Feature 0: lognormal
# .: logmu = [ 1.116e-05 1.093e+00]
# a: logmu = [ 4.367e-06 9.452e-01]
# b: logmu = [ 2.151e-05 1.289e+00]
# c: logmu = [ 9.213e-06 1.096e+00]
# .: logsigma = [ 0.004 0.262]
# a: logsigma = [ 0.002 0.218]
# b: logsigma = [ 0.005 0.087]
# c: logsigma = [ 0.003 0.131]
# --------------------------------------------------------------------------------And obtain the loglikelihood of a test sequence:
TEST_OBS = [2, 1, 6]
TEST_PSTATES = ['c', 'a', 'd']
print(model.score(np.c_[TEST_OBS], TEST_PSTATES))
# -4.81378034867Finally, use the model to generate data:
sample_obs, sample_pstates, sample_hstates = model.sample(n_obs=10, random_state=1234)
print('Observations:\n', sample_obs)
print('partial states:\n', sample_pstates)
print('hidden states:\n', sample_hstates)
# Observations:
# [[ 4.321]
# [ 2.579]
# [ 3.674]
# [ 1.006]
# [ 2.875]
# [ 1.002]
# [ 1.003]
# [ 2.473]
# [ 2.744]
# [ 1.002]]
# p-states:
# ['b' 'a' 'b' 'a' 'c' 'a' 'b' 'a' 'c' 'a']
# h-states:
# [1 1 1 0 1 0 0 1 1 0]Other examples can be found in the examples/ directory. To obtain results for che CMU Keystroke dataset, run
$ python examples/keystroke.py
@article{Monaco2017,
title = "The Partially Observable Hidden Markov Model and its Application to Keystroke Dynamics ",
journal = "Pattern Recognition ",
year = "2017",
issn = "0031-3203",
doi = "https://doi.org/10.1016/j.patcog.2017.11.021",
url = "https://www.sciencedirect.com/science/article/pii/S0031320317304752",
author = "John V. Monaco and Charles C. Tappert",
}