def frechnet_dev(ev, trans_mat, obs_costs, disc_fac):
"""
Calculating the Frechnet derivative of the contraction mapping.
Parameters
----------
ev : numpy.array
see :ref:`ev`
trans_mat : numpy.array
see :ref:`trans_mat`
obs_costs : numpy.array
see :ref:`costs`
disc_fac : numpy.float
see :ref:`disc_fac`
Returns
-------
t_prime : numpy.array
A num_states x num_states matrix containing the frechnet derivative of the
contraction mapping. For details see Rust (2000).
"""
choice_probs = choice_prob_gumbel(ev, obs_costs, disc_fac)
t_prime_pre = trans_mat[:, 1:] * choice_probs[1:, 0]
t_prime = disc_fac * np.column_stack(
(1 - np.sum(t_prime_pre, axis=1), t_prime_pre))
return t_prime
def loglike_cost_params_individual(
params,
maint_func,
maint_func_dev,
num_states,
trans_mat,
state_mat,
decision_mat,
disc_fac,
scale,
alg_details,
):
"""
This is the individual logliklihood function for the estimation of the cost parameters
needed for the BHHH optimizer.
Parameters
----------
params : pandas.DataFrame
see :ref:`params`
maint_func: func
see :ref:`maint_func`
num_states : int
The size of the state space.
disc_fac : numpy.float
see :ref:`disc_fac`
trans_mat : numpy.array
see :ref:`trans_mat`
state_mat : numpy.array
see :ref:`state_mat`
decision_mat : numpy.array
see :ref:`decision_mat`
Returns
-------
log_like : numpy.array
A num_buses times num_periods dimensional array containing the negative
log-likelihood contributions of the individuals.
"""
params = params["value"].to_numpy()
costs = calc_obs_costs(num_states, maint_func, params, scale)
ev, contr_step_count, newt_kant_step_count = get_ev(
params, trans_mat, costs, disc_fac, alg_details)
config.total_contr_count += contr_step_count
config.total_newt_kant_count += newt_kant_step_count
p_choice = choice_prob_gumbel(ev, costs, disc_fac)
log_like = like_hood_data_individual(np.log(p_choice), decision_mat,
state_mat)
return log_like
def mpec_loglike_cost_params_derivative(
maint_func,
maint_func_dev,
num_states,
num_params,
disc_fac,
scale,
decision_mat,
state_mat,
mpec_params,
):
"""
Computing the analytical gradient of the objective function for MPEC.
Parameters
----------
maint_func: func
see :ref:`maint_func`
maint_func_dev: func
see :ref:`maint_func`
num_states : int
The size of the state space.
num_params : int
Length of cost parameter vector.
disc_fac : numpy.float
see :ref:`disc_fac`
scale : numpy.float
see :ref:`scale`
decision_mat : numpy.array
see :ref:`decision_mat`
state_mat : numpy.array
see :ref:`state_mat`
mpec_params : numpy.array
see :ref:`mpec_params`
Returns
-------
gradient : numpy.array
Vector that holds the derivative of the negative log likelihood function
to the parameters.
"""
# Calculate choice probabilities
costs = calc_obs_costs(num_states, maint_func, mpec_params[num_states:],
scale)
p_choice = choice_prob_gumbel(mpec_params[0:num_states], costs, disc_fac)
# calculate the derivative based on the model
derivative_both = mpec_loglike_cost_params_derivative_model(
num_states, num_params, disc_fac, scale, maint_func_dev, p_choice)
# Calculate actual gradient depending on the given data
# get decision matrix into the needed shape
decision_mat_temp = np.vstack((
np.tile(decision_mat[0], (num_states + num_params, 1)),
np.tile(decision_mat[1], (num_states + num_params, 1)),
))
# calculate the gradient
gradient_temp = -np.sum(
decision_mat_temp * np.dot(derivative_both, state_mat), axis=1)
# bring the calculated gradient into the correct shape
gradient = np.reshape(gradient_temp, (num_states + num_params, 2),
order="F").sum(axis=1)
return gradient
def mpec_loglike_cost_params(
maint_func,
maint_func_dev,
num_states,
num_params,
state_mat,
decision_mat,
disc_fac,
scale,
gradient,
mpec_params,
grad,
):
"""
Calculate the negative partial log likelihood for MPEC depending on cost parameters
as well as the discretized expected values.
Parameters
----------
maint_func: func
see :ref:`maint_func`
maint_func_dev: func
see :ref:`maint_func`
num_states : int
The size of the state space.
num_params : int
Length of cost parameter vector.
state_mat : numpy.array
see :ref:`state_mat`
decision_mat : numpy.array
see :ref:`decision_mat`
disc_fac : numpy.float
see :ref:`disc_fac`
scale : numpy.float
see :ref:`scale`
gradient : str
Indicates whether analytical or numerical gradient should be used.
mpec_params : numpy.array
see :ref:`mpec_params`
grad : numpy.array, optional
The gradient of the function. The default is np.array([]).
Returns
-------
log_like: float
Contains the negative partial log likelihood for the given parameters.
"""
if grad.size > 0:
if gradient == "No":
# numerical gradient
partial_loglike_mpec = partial(
mpec_loglike_cost_params,
maint_func,
maint_func_dev,
num_states,
num_params,
state_mat,
decision_mat,
disc_fac,
scale,
gradient,
grad=np.array([]),
)
grad[:] = approx_derivative(partial_loglike_mpec,
mpec_params,
method="2-point")
else:
# analytical gradient
grad[:] = mpec_loglike_cost_params_derivative(
maint_func,
maint_func_dev,
num_states,
num_params,
disc_fac,
scale,
decision_mat,
state_mat,
mpec_params,
)
costs = calc_obs_costs(num_states, maint_func, mpec_params[num_states:],
scale)
p_choice = choice_prob_gumbel(mpec_params[0:num_states], costs, disc_fac)
log_like = like_hood_data(np.log(p_choice), decision_mat, state_mat)
return float(log_like)
def test_choice_probs(inputs, outputs):
assert_array_almost_equal(
choice_prob_gumbel(outputs["fixp"], outputs["costs"],
inputs["disc_fac"]),
outputs["choice_probs"],
)
def get_demand(init_dict, demand_dict, demand_params):
"""
Calculates the implied demand for a range of replacement costs
for a certain number of buses over a certain time period.
Parameters
----------
init_dict : dict
see :ref:`init_dict`.
demand_dict : dict
see :ref:`demand_dict`.
demand_params : np.array
see :ref:`demand_params`
Returns
-------
demand_results : pd.DataFrame
see :ref:`demand_results`
"""
params = demand_params.copy()
(
disc_fac,
num_states,
maint_func,
maint_func_dev,
num_params,
scale,
) = select_model_parameters(init_dict)
# Initialize the loop over the replacement costs
rc_range = np.linspace(
demand_dict["RC_lower_bound"],
demand_dict["RC_upper_bound"],
demand_dict["demand_evaluations"],
)
demand_results = pd.DataFrame(index=rc_range, columns=["demand", "success"])
demand_results.index.name = "RC"
for rc in rc_range:
params[-num_params] = rc
demand_results.loc[(rc), "success"] = "No"
# solve the model for the given paramaters
trans_mat = create_transition_matrix(num_states, params[:-num_params])
obs_costs = calc_obs_costs(num_states, maint_func, params[-num_params:], scale)
ev = calc_fixp(trans_mat, obs_costs, disc_fac)[0]
p_choice = choice_prob_gumbel(ev, obs_costs, disc_fac)
# calculate initial guess for pi and run contraction iterations
pi_new = np.full((num_states, 2), 1 / (2 * num_states))
tol = 1
iteration = 1
while tol >= demand_dict["tolerance"]:
pi = pi_new
pi_new = p_choice * (
np.dot(trans_mat.T, pi[:, 0])
+ np.dot(np.tile(trans_mat[0, :], (num_states, 1)).T, pi[:, 1])
).reshape((num_states, 1))
tol = np.max(np.abs(pi_new - pi))
iteration = +1
if iteration > 200:
break
if tol < demand_dict["tolerance"]:
demand_results.loc[(rc), "success"] = "Yes"
demand_results.loc[(rc), "demand"] = (
demand_dict["num_buses"] * demand_dict["num_periods"] * pi_new[:, 1].sum()
)
return demand_results
def derivative_loglike_cost_params_individual(
params,
maint_func,
maint_func_dev,
num_states,
trans_mat,
state_mat,
decision_mat,
disc_fac,
scale,
alg_details,
):
"""
This is the Jacobian of the individual log likelihood function of the cost
parameter estimation with respect to all cost parameters needed for the BHHH.
Parameters
----------
params : pandas.DataFrame
see :ref:`params`
maint_func: func
see :ref:`maint_func`
num_states : int
The size of the state space.
disc_fac : numpy.float
see :ref:`disc_fac`
trans_mat : numpy.array
see :ref:`trans_mat`
state_mat : numpy.array
see :ref:`state_mat`
decision_mat : numpy.array
see :ref:`decision_mat`
Returns
-------
dev : numpy.array
A num_buses + num_periods x dim(params) matrix in form of numpy array
containing the derivative of the individual log-likelihood function for
every cost parameter.
"""
params = params["value"].to_numpy()
dev = np.zeros((decision_mat.shape[1], len(params)))
obs_costs = calc_obs_costs(num_states, maint_func, params, scale)
ev = get_ev(params, trans_mat, obs_costs, disc_fac, alg_details)[0]
p_choice = choice_prob_gumbel(ev, obs_costs, disc_fac)
maint_cost_dev = maint_func_dev(num_states, scale)
lh_values_rc = like_hood_vaules_rc(ev, obs_costs, p_choice, trans_mat,
disc_fac)
like_dev_rc = like_hood_data_individual(lh_values_rc, decision_mat,
state_mat)
dev[:, 0] = like_dev_rc
for i in range(len(params) - 1):
if len(params) == 2:
cost_dev_param = maint_cost_dev
else:
cost_dev_param = maint_cost_dev[:, i]
log_like_values_params = log_like_values_param(ev, obs_costs, p_choice,
trans_mat,
cost_dev_param,
disc_fac)
dev[:, i + 1] = like_hood_data_individual(log_like_values_params,
decision_mat, state_mat)
return dev
def loglike_cost_params_individual(
params,
maint_func,
maint_func_dev,
num_states,
trans_mat,
state_mat,
decision_mat,
disc_fac,
scale,
alg_details,
):
"""
This is the individual logliklihood function for the estimation of the cost parameters
needed for the BHHH optimizer.
Parameters
----------
params : pandas.DataFrame
see :ref:`params`
maint_func: func
see :ref:`maint_func`
num_states : int
The size of the state space.
disc_fac : numpy.float
see :ref:`disc_fac`
trans_mat : numpy.array
see :ref:`trans_mat`
state_mat : numpy.array
see :ref:`state_mat`
decision_mat : numpy.array
see :ref:`decision_mat`
Returns
-------
log_like : numpy.array
A num_buses times num_periods dimensional array containing the negative
log-likelihood contributions of the individuals.
"""
if "omega" in params.index:
omega = params.loc["omega", "value"]
cost_params = params.drop("omega")["value"].to_numpy()
p_ml = trans_mat[0, 0:3]
sample_size = 4292 / 78
rho = chi2.ppf(omega, len(p_ml) - 1) / (2 * (sample_size))
costs = calc_obs_costs(num_states, maint_func, cost_params, scale)
v_start, _, _, _ = value_function_contraction(trans_mat,
costs,
disc_fac,
threshold=1e-12,
max_it=1000000)
v_worst, worst_trans_mat, success, converge_crit_ev, num_eval = worst_value_fixp(
v_start,
trans_mat,
costs,
disc_fac,
rho,
threshold=1e-3,
max_it=1000000)
ev = np.dot(worst_trans_mat, v_worst)
if not success:
raise ValueError("Not converging.")
else:
cost_params = params["value"].to_numpy()
costs = calc_obs_costs(num_states, maint_func, cost_params, scale)
ev, contr_step_count, newt_kant_step_count = get_ev(
cost_params, trans_mat, costs, disc_fac, alg_details)
config.total_contr_count += contr_step_count
config.total_newt_kant_count += newt_kant_step_count
p_choice = choice_prob_gumbel(ev, costs, disc_fac)
log_like = like_hood_data_individual(np.log(p_choice), decision_mat,
state_mat)
return log_like
