def H_1PL(theta, r, f, P, a, b, c=0, Pstar=None):
"""Get the Hessian Matrix of the log likelihood for the 1PL model
(2nd derivative wrt b)
"""
theta, r, f = expand_dims(P, theta, r, f)
rmfP = r - f * P
Q = (1 - P)
Pstar, Prat = ((P, 1) if Pstar is None else (Pstar, Pstar / P))
EL22 = -a**2 * f * Prat * Pstar * Q
return np.nansum([[EL22]], axis=2)
def H_1PL(theta, r, f, P, zeta, lam, c=0, Pstar=None):
"""Get the Hessian Matrix of the log likelihood for the 1PL model
(2nd derivative wrt zeta)
"""
theta, r, f = expand_dims(P, theta, r, f)
Pstar = P if Pstar is None else Pstar
W = Pstar * (1 - Pstar)
rP2 = r / P**2
rP2cmf = rP2 * c - f
EL11 = rP2cmf * W
return np.nansum([[EL11]], axis=2)
def J_1PL(theta, r, f, P, zeta, lam, c=0, Pstar=None):
"""Get the Jacobian of the log likelihood for the 1PL model
(1st derivative wrt zeta)
a here is an ARRAY, not a scalar
"""
theta, r, f = expand_dims(P, theta, r, f)
rmfP = r - f * P
Prat = 1 if Pstar is None else Pstar / P
L1 = Prat * rmfP
return np.nansum([L1], axis=1)
def J_3PL(theta, r, f, P, zeta, lam, c, Pstar):
"""Get the Jacobian of the log likelihood for the 3PL model
(1st derivatives wrt zeta, lam, c)
a here is an ARRAY, not a scalar
"""
theta, r, f = expand_dims(P, theta, r, f)
rmfP = r - f * P
Prat = 1 if Pstar is None else Pstar / P
iP1mc = 1 / (P * (1 - c))
L1, L2, L3 = np.array([Prat, Prat * theta, iP1mc]) * rmfP
return np.nansum([L1, L2, L3], axis=1)
def J_1PL(theta, r, f, P, a, b, c=0, Pstar=None):
"""Get the Jacobian of the log likelihood for the 1PL model
(1st derivative wrt b)
a here is an ARRAY, not a scalar
"""
a = a * np.ones(P.shape) # force a to be an array
theta, r, f = expand_dims(P, theta, r, f)
rmfP = r - f * P
Prat = 1 if Pstar is None else Pstar / P
L2 = -a * rmfP * Prat
return np.nansum([L2], axis=1)
def H_2PL(theta, r, f, P, a, b, c=0, Pstar=None):
"""Get the Hessian Matrix of the log likelihood for the 2PL model
(2nd derivative wrt a,b)
"""
theta, r, f = expand_dims(P, theta, r, f)
rmfP = r - f * P
Q = (1 - P)
Pstar, Prat = ((P, 1) if Pstar is None else (Pstar, Pstar / P))
thmb = theta - b
EL11, EL22, EL12 = np.array([thmb**2, -a**2, a * thmb
]) * f * Prat * Pstar * Q
return np.nansum([[EL11, EL12], [EL12, EL22]], axis=2)
def J_3PL(theta, r, f, P, a, b, c, Pstar):
"""Get the Jacobian of the log likelihood for the 3PL model
(1st derivatives wrt a,b,c)
a here is an ARRAY, not a scalar
"""
a = a * np.ones(P.shape) # force a to be an array
theta, r, f = expand_dims(P, theta, r, f)
rmfP = r - f * P
iPc = 1 / (P - c)
Prat = Pstar / P
thmb = theta - b
L1, L2, L3 = np.array([thmb, -a, iPc]) * rmfP * Prat
return np.nansum([L1, L2, L3], axis=1)
def H_3PL(theta, r, f, P, zeta, lam, c, Pstar):
"""Get the Hessian Matrix of the log likelihood for the 3PL model
(2nd derivative wrt zeta, lam, c)
"""
theta, r, f = expand_dims(P, theta, r, f)
Pstar = P if Pstar is None else Pstar
W = Pstar * (1 - Pstar)
rP2 = r / P**2
rP2cmf = rP2 * c - f
EL11, EL22, EL33 = [
rP2cmf * W, rP2cmf * W * theta**2, (rP2 * (2 * P - 1) - f) / (1 - c)**2
]
EL12, EL13, EL23 = [rP2cmf * W * theta, -rP2 * W, -rP2 * W * theta]
return np.nansum(
[[EL11, EL12, EL13], [EL12, EL22, EL23], [EL13, EL23, EL33]], axis=2)
def H_3PL(theta, r, f, P, a, b, c, Pstar):
"""Get the Hessian Matrix of the log likelihood for the 3PL model
(2nd derivative wrt a,b,c)
"""
theta, r, f = expand_dims(P, theta, r, f)
rmfP = r - f * P
iPc = 1 / (P - c)
Q = (1 - P)
Qic = Q / (1 - c)
Prat = Pstar / P
thmb = theta - b
EL11, EL22, EL33 = np.array(
[-P * Q * thmb**2 * Prat, -a**2 * P * Q * Prat, Qic * iPc]) * f * Prat
EL12, EL13, EL23 = np.array(
[a * thmb * P * Q * Prat, -thmb * Qic, a * Qic]) * f * Prat
return np.nansum(
[[EL11, EL12, EL13], [EL12, EL22, EL23], [EL13, EL23, EL33]], axis=2)