def L_dL_singleuser(arg):
""" calculate log likelihood and gradient wrt couplings of mIRT model
for single user """
couplings, state, options = arg
abilities = state['abilities'].copy()
correct = state['correct']
exercises_ind = state['exercises_ind']
# pad the abilities vector with a 1 to act as a bias
abilities = np.append(abilities.copy(),
np.ones((1, abilities.shape[1])),
axis=0)
# the abilities to exercise coupling parameters for this exercise
exercise_couplings = couplings[exercises_ind, :]
# calculate the probability of getting a question in this exercise correct
Y = np.dot(exercise_couplings, abilities)
Z = mirt_util.sigmoid(Y) # predicted correctness value
Zt = correct.reshape(Z.shape) # true correctness value
pdata = Zt * Z + (1 - Zt) * (1 - Z) # = 2*Zt*Z - Z + const
dLdY = ((2 * Zt - 1) * Z * (1 - Z)) / pdata
dL = np.dot(dLdY, abilities.T)
dL /= np.log(2.)
L = np.sum(np.log(pdata))
L /= np.log(2.)
return -L, -dL, exercises_ind
def L_dL_singleuser(arg):
""" calculate log likelihood and gradient wrt couplings of mIRT model
for single user """
theta, state, options = arg
abilities = state['abilities'].copy()
correct = state['correct']
exercises_ind = state['exercises_ind']
dL = mirt_util.Parameters(theta.num_abilities, len(exercises_ind))
# pad the abilities vector with a 1 to act as a bias
abilities = np.append(abilities.copy(),
np.ones((1, abilities.shape[1])),
axis=0)
# the abilities to exercise coupling parameters for this exercise
W_correct = theta.W_correct[exercises_ind, :]
# calculate the probability of getting a question in this exercise correct
Y = np.dot(W_correct, abilities)
Z = mirt_util.sigmoid(Y) # predicted correctness value
Zt = correct.reshape(Z.shape) # true correctness value
pdata = Zt * Z + (1. - Zt) * (1. - Z) # = 2*Zt*Z - Z + const
dLdY = ((2. * Zt - 1.) * Z * (1. - Z)) / pdata
L = -np.sum(np.log(pdata))
dL.W_correct = -np.dot(dLdY, abilities.T)
if not options.correct_only:
# calculate the probability of taking time response_time to answer
log_time_taken = state['log_time_taken']
# the abilities to time coupling parameters for this exercise
W_time = theta.W_time[exercises_ind, :]
sigma = theta.sigma_time[exercises_ind].reshape((-1, 1))
Y = np.dot(W_time, abilities)
err = (Y - log_time_taken.reshape((-1, 1)))
L += np.sum(err ** 2 / sigma ** 2) / 2.
dLdY = err / sigma ** 2
dL.W_time = np.dot(dLdY, abilities.T)
dL.sigma_time = (-err ** 2 / sigma ** 3).ravel()
# normalization for the Gaussian
L += np.sum(0.5 * np.log(sigma ** 2))
dL.sigma_time += 1. / sigma.ravel()
return L, dL, exercises_ind
def L_dL_singleuser(arg):
""" calculate log likelihood and gradient wrt couplings of mIRT model
for single user """
theta, state, options = arg
abilities = state['abilities'].copy()
correct = state['correct']
exercises_ind = state['exercises_ind']
dL = mirt_util.Parameters(theta.num_abilities, len(exercises_ind))
# pad the abilities vector with a 1 to act as a bias
abilities = np.append(abilities.copy(),
np.ones((1, abilities.shape[1])),
axis=0)
# the abilities to exercise coupling parameters for this exercise
W_correct = theta.W_correct[exercises_ind, :]
# calculate the probability of getting a question in this exercise correct
Y = np.dot(W_correct, abilities)
Z = mirt_util.sigmoid(Y) # predicted correctness value
Zt = correct.reshape(Z.shape) # true correctness value
pdata = Zt * Z + (1. - Zt) * (1. - Z) # = 2*Zt*Z - Z + const
dLdY = ((2. * Zt - 1.) * Z * (1. - Z)) / pdata
L = -np.sum(np.log(pdata))
dL.W_correct = -np.dot(dLdY, abilities.T)
if not options.correct_only:
# calculate the probability of taking time response_time to answer
log_time_taken = state['log_time_taken']
# the abilities to time coupling parameters for this exercise
W_time = theta.W_time[exercises_ind, :]
sigma = theta.sigma_time[exercises_ind].reshape((-1, 1))
Y = np.dot(W_time, abilities)
err = (Y - log_time_taken.reshape((-1, 1)))
L += np.sum(err ** 2 / sigma ** 2) / 2.
dLdY = err / sigma ** 2
dL.W_time = np.dot(dLdY, abilities.T)
dL.sigma_time = (-err ** 2 / sigma ** 3).ravel()
# normalization for the Gaussian
L += np.sum(0.5 * np.log(sigma ** 2))
dL.sigma_time += 1. / sigma.ravel()
return L, dL, exercises_ind
