def make_multi_Gaussian_HMM(t_1, t_2, m_1, m_2, s_1, s_2): K = 2 A = np.array([[t_1, 1. - t_1], [1. - t_2, t_2]]) pi = np.array([0.5, 0.5]) D = [mnorm([m_1, s_1]), mnorm([m_2, s_2])] return HMM(K, A, pi, D)
def easy_case(eps):
"""
When K = 2, the transition matrix is 0.5 everywhere,
and state 0 emits only 0 and state 1 emits only 1s,
forward backward should guess 0's (p = 1) whenever it sees
a 0 and a 1 (p = 1) whenever it sees a 1.
"""
#set hyperparams
K = 2
T = 10
#initialize estimates
x = [[0 for j in range(0,T)] + [1 for j in range(0,T)]]
B = [mn(np.array([.999, 0.001])), mn(np.array([0.001, .999]))]
A = np.array([[0.5,0.5],[0.5,0.5]])
pi = [0.5,0.5]
M = HMM(2, A, pi, B)
S = States(M, x)
S.e_step()
gamma = np.exp(np.array(S.gamma[0]))
xi = np.exp(np.array(S.xi[0]))
gamma_correct = np.array([[1.0,0.0] for j in range(0,T)]
+ [[0.0,1.0] for j in range(0,T)])
xi_correct = ([np.array([[1,0], [0,0]]) for j
in range(0, T - 1)] +
[np.array([[0,1], [0,0]])] +
[np.array([[0,0], [0,1]]) for j in range(0, T - 1)])
gamma_score = check_gamma(gamma, gamma_correct, eps)
xi_score = True
for j in range(0, len(xi)):
xi_score = xi_score and check_gamma(xi[j], xi_correct[j], eps)
return gamma_score and xi_score
def make_HMM(): t_1 = 0.9 t_2 = 0.1 m_1 = np.array([0., 0.]) m_2 = np.array([10., 10.]) s_1 = np.eye(2) s_2 = np.eye(2) K = 2 A = np.array([[t_1, 1. - t_1], [1. - t_2, t_2]]) pi = np.array([0.5, 0.5]) D = [mnorm([m_1, s_1]), mnorm([m_2, s_2])] return HMM(K, A, pi, D)
def biased_coins_HMM(p, q, t_1, t_2):
"""
This method returns a 2 state HMM, where state 1 is a p-biased coin,
state 2 is a q-biased coin, the probability of staying in state
1 is t_1, and the probability of staying in state 2 is t_2.
The initial distribution pi is uniform.
"""
K = 2
A = np.array([[t_1, 1. - t_1], [1. - t_2, t_2]])
eigval, eigvec = np.linalg.eig(A)
maxind = max((l, i) for i, l in enumerate(eigval))[1]
pi = eigvec[:, maxind]
if np.min(pi) < 0:
pi = -pi
pi = (1. / np.sum(pi)) * pi
D = [mn([p, 1. - p]), mn([q, 1. - q])]
return HMM(K, A, pi, D)
def umbrella_example(eps):
"""
Umbrella example from wikipedia
"""
x = [[0, 0, 1, 0, 0]]
A = np.array([[0.7,0.3], [0.3,0.7]])
D = [mn([0.9, 0.1]), mn([0.2, 0.8])]
pi = np.array([0.5,0.5])
M = HMM(2, A, pi, D)
S = States(M, x)
S.e_step()
gamma = np.exp(np.array(S.gamma[0]))
xi = np.exp(np.array(S.xi[0]))
gamma_correct = np.array([[0.8673,0.1327],[0.82,.18],
[0.31,0.70],[0.82,0.18],[0.87,0.13]])
xi_correct = np.array([[.75,.12],[.07,.06]])
gamma_score = check_gamma(gamma, gamma_correct, eps)
xi_score = check_gamma(xi[0], xi_correct, eps)
#if not gamma_score:
# print(" gamma computation failed")
#if not xi_score:
# print(" xi computation failed")
return gamma_score and xi_score