def expectation_maximization(X,
U,
config=None,
verbose_level=2,
print_every=10):
"""
Finding the 'best' parameters using the Expectation Maximization algorithm
Parameters
----------
X: [seq_count] array of [seq_length] array of ([dimension] numpy array/None)
'X[seq_idx][time_step]' = the measurement of the seq_idx sequence at time time_step
None for missing measurements
U: [seq_count] array of [seq_length] array of ([dimension] numpy array)
'U[seq_idx][time_step]' = the control of the seq_idx sequence at time time_step
config: {String: String}
specified EM config dict
verbose_level: int
0: print nothing
1: print summary at the end of all EM iterations
2: print summary for every print_every EM iteration
print_every: int
print summary for every print_every EM iteration if verbose_level = 2
Returns
-------
parameters: z_0_hat, P_0_hat, A, B, Q, R
z_0_hat: [dimension] numpy array
initial mean for every sequence
P_0_hat: [dimension, dimension] numpy array
initial covariance for every sequence
A: [dimension, dimension] numpy array
transition matrix
B: [dimension, control_dimension] numpy array
control effect matrix
Q: [dimension, dimension] numpy array
transition covariance matrix
R: [dimension, dimension] numpy array
emission covariance matrix
ll_history: [num_iterations] array of numpy float
'll_history[iteration_idx]' = the log likelihood at EM iteration iteration_idx
"""
# initialize the parameters
dimension, control_dimension = get_dimension(X), get_control_dimension(U)
if config is None:
config = EM_Config(dimension, control_dimension).get_default_config()
dimension, control_dimension = get_dimension(X), get_control_dimension(U)
mskf = MSKF(dimension, control_dimension, config['initial_z_0_hat'],
config['initial_P_0_hat'], config['initial_A'],
config['initial_B'], config['initial_Q'], config['initial_R'])
mskf.pass_in_measurement_and_control(X, U)
# set the configuration
mskf.set_em_config(config)
# EM
parameters, ll_history = mskf.em(verbose_level=verbose_level,
print_every=print_every)
return parameters, ll_history
def infer(X, U, parameters):
"""
Infer the hidden variables
Parameters
----------
X: [seq_count] array of [seq_length] array of ([dimension] numpy array/None)
'X[seq_idx][time_step]' = the measurement of the seq_idx sequence at time time_step
None for missing measurements
U: [seq_count] array of [seq_length] array of ([dimension] numpy array)
'U[seq_idx][time_step]' = the control of the seq_idx sequence at time time_step
parameters: z_0_hat, P_0_hat, A, B, Q, R
z_0_hat: [dimension] numpy array
initial mean for every sequence
P_0_hat: [dimension, dimension] numpy array
initial covariance for every sequence
A: [dimension, dimension] numpy array
transition matrix
B: [dimension, control_dimension] numpy array
control effect matrix
Q: [dimension, dimension] numpy array
transition covariance matrix
R: [dimension, dimension] numpy array
emission covariance matrix
Returns
-------
predicted: {String: array}
'predicted['mean'][seq_idx][time_step]' = the mean of the prediction
of the seq_idx sequence at time time_step
'predicted['covariance'][seq_idx][time_step]' = the covariance of prediction
of the seq_idx sequence at time time_step
filtered: {String: array}
'filtered['mean'][seq_idx][time_step]' = the mean of filtered value
of the seq_idx sequence at time time_step
'filtered['covariance'][seq_idx][time_step]' = the covariance of filtered value
of the seq_idx sequence at time time_step
smoothed: {String: array}
'smoothed['mean'][seq_idx][time_step]' = the mean of smoothed value
of the seq_idx sequence at time time_step
'smoothed['covariance'][seq_idx][time_step]' = the covariance of smoothed value
of the seq_idx sequence at time time_step
"""
# initialize the parameters
z_0_hat, P_0_hat, A, B, Q, R = parameters
dimension, control_dimension = get_dimension(X), get_control_dimension(U)
mskf = MSKF(dimension, control_dimension, z_0_hat, P_0_hat, A, B, Q, R)
mskf.pass_in_measurement_and_control(X, U)
# estimate
predicted, filtered, smoothed = mskf.estimate()
return predicted, filtered, smoothed
def calculate_log_likelihood(X, U, parameters):
"""
Calculate the log likelihood per data point given the parameters
Parameters
----------
X: [seq_count] array of [seq_length] array of ([dimension] numpy array/None)
'X[seq_idx][time_step]' = the measurement of the seq_idx sequence at time time_step
None for missing measurements
U: [seq_count] array of [seq_length] array of ([dimension] numpy array)
'U[seq_idx][time_step]' = the control of the seq_idx sequence at time time_step
parameters: z_0_hat, P_0_hat, A, B, Q, R
z_0_hat: [dimension] numpy array
initial mean for every sequence
P_0_hat: [dimension, dimension] numpy array
initial covariance for every sequence
A: [dimension, dimension] numpy array
transition matrix
B: [dimension, control_dimension] numpy array
control effect matrix
Q: [dimension, dimension] numpy array
transition covariance matrix
R: [dimension, dimension] numpy array
emission covariance matrix
Returns
-------
mskf.log_likelihood(): numpy float
the log likelihood per data point
"""
# initialize the parameters
z_0_hat, P_0_hat, A, B, Q, R = parameters
dimension, control_dimension = get_dimension(X), get_control_dimension(U)
mskf = MSKF(dimension, control_dimension, z_0_hat, P_0_hat, A, B, Q, R)
mskf.pass_in_measurement_and_control(X, U)
return mskf.log_likelihood()