def fixed_lag_smoothing(e_t, HMM, d, ev, t):
"""
[Figure 15.6]
Smoothing algorithm with a fixed time lag of 'd' steps.
Online algorithm that outputs the new smoothed estimate if observation
for new time step is given."""
ev.insert(0, None)
T_model = HMM.transition_model
f = HMM.prior
B = [[1, 0], [0, 1]]
evidence = []
evidence.append(e_t)
O_t = vector_to_diagonal(HMM.sensor_dist(e_t))
if t > d:
f = forward(HMM, f, e_t)
O_tmd = vector_to_diagonal(HMM.sensor_dist(ev[t - d]))
B = matrix_multiplication(inverse_matrix(O_tmd), inverse_matrix(T_model), B, T_model, O_t)
else:
B = matrix_multiplication(B, T_model, O_t)
t += 1
if t > d:
# always returns a 1x2 matrix
return [normalize(i) for i in matrix_multiplication([f], B)][0]
else:
return None
def fixed_lag_smoothing(e_t, HMM, d, ev, t):
"""[Figure 15.6]
Smoothing algorithm with a fixed time lag of 'd' steps.
Online algorithm that outputs the new smoothed estimate if observation
for new time step is given."""
ev.insert(0, None)
T_model = HMM.transition_model
f = HMM.prior
B = [[1, 0], [0, 1]]
evidence = []
evidence.append(e_t)
O_t = vector_to_diagonal(HMM.sensor_dist(e_t))
if t > d:
f = forward(HMM, f, e_t)
O_tmd = vector_to_diagonal(HMM.sensor_dist(ev[t- d]))
B = matrix_multiplication(inverse_matrix(O_tmd), inverse_matrix(T_model), B, T_model, O_t)
else:
B = matrix_multiplication(B, T_model, O_t)
t = t + 1
if t > d:
# always returns a 1x2 matrix
return([normalize(i) for i in matrix_multiplication([f], B)][0])
else:
return None
def fixed_lag_smoothing(e_t, HMM, d, ev, t):
"""Algoritmo de suavização com um intervalo de tempo fixo de passos 'd'.
Algoritmo online que produz a nova estimativa suavizada se a observação
Para novo passo de tempo é dado."""
ev.insert(0, None)
T_model = HMM.transition_model
f = HMM.prior
B = [[1, 0], [0, 1]]
evidence = []
evidence.append(e_t)
O_t = vector_to_diagonal(HMM.sensor_dist(e_t))
if t > d:
f = forward(HMM, f, e_t)
O_tmd = vector_to_diagonal(HMM.sensor_dist(ev[t - d]))
B = matrix_multiplication(inverse_matrix(O_tmd),
inverse_matrix(T_model), B, T_model, O_t)
else:
B = matrix_multiplication(B, T_model, O_t)
t = t + 1
if t > d:
return [normalize(i) for i in matrix_multiplication([f], B)][0]
else:
return None
def truncated_svd(X, num_val=2, max_iter=1000):
"""Computes the first component of SVD"""
def normalize_vec(X, n = 2):
"""Normalizes two parts (:m and m:) of the vector"""
X_m = X[:m]
X_n = X[m:]
norm_X_m = norm(X_m, n)
Y_m = [x/norm_X_m for x in X_m]
norm_X_n = norm(X_n, n)
Y_n = [x/norm_X_n for x in X_n]
return Y_m + Y_n
def remove_component(X):
"""Removes components of already obtained eigen vectors from X"""
X_m = X[:m]
X_n = X[m:]
for eivec in eivec_m:
coeff = dotproduct(X_m, eivec)
X_m = [x1 - coeff*x2 for x1, x2 in zip(X_m, eivec)]
for eivec in eivec_n:
coeff = dotproduct(X_n, eivec)
X_n = [x1 - coeff*x2 for x1, x2 in zip(X_n, eivec)]
return X_m + X_n
m, n = len(X), len(X[0])
A = [[0 for _ in range(n + m)] for _ in range(n + m)]
for i in range(m):
for j in range(n):
A[i][m + j] = A[m + j][i] = X[i][j]
eivec_m = []
eivec_n = []
eivals = []
for _ in range(num_val):
X = [random.random() for _ in range(m + n)]
X = remove_component(X)
X = normalize_vec(X)
for _ in range(max_iter):
old_X = X
X = matrix_multiplication(A, [[x] for x in X])
X = [x[0] for x in X]
X = remove_component(X)
X = normalize_vec(X)
# check for convergence
if norm([x1 - x2 for x1, x2 in zip(old_X, X)]) <= 1e-10:
break
projected_X = matrix_multiplication(A, [[x] for x in X])
projected_X = [x[0] for x in projected_X]
eivals.append(norm(projected_X, 1)/norm(X, 1))
eivec_m.append(X[:m])
eivec_n.append(X[m:])
return (eivec_m, eivec_n, eivals)
def truncated_svd(X, num_val=2, max_iter=1000):
"""Computes the first component of SVD"""
def normalize_vec(X, n = 2):
"""Normalizes two parts (:m and m:) of the vector"""
X_m = X[:m]
X_n = X[m:]
norm_X_m = norm(X_m, n)
Y_m = [x/norm_X_m for x in X_m]
norm_X_n = norm(X_n, n)
Y_n = [x/norm_X_n for x in X_n]
return Y_m + Y_n
def remove_component(X):
"""Removes components of already obtained eigen vectors from X"""
X_m = X[:m]
X_n = X[m:]
for eivec in eivec_m:
coeff = dotproduct(X_m, eivec)
X_m = [x1 - coeff*x2 for x1, x2 in zip(X_m, eivec)]
for eivec in eivec_n:
coeff = dotproduct(X_n, eivec)
X_n = [x1 - coeff*x2 for x1, x2 in zip(X_n, eivec)]
return X_m + X_n
m, n = len(X), len(X[0])
A = [[0 for _ in range(n + m)] for _ in range(n + m)]
for i in range(m):
for j in range(n):
A[i][m + j] = A[m + j][i] = X[i][j]
eivec_m = []
eivec_n = []
eivals = []
for _ in range(num_val):
X = [random.random() for _ in range(m + n)]
X = remove_component(X)
X = normalize_vec(X)
for _ in range(max_iter):
old_X = X
X = matrix_multiplication(A, [[x] for x in X])
X = [x[0] for x in X]
X = remove_component(X)
X = normalize_vec(X)
# check for convergence
if norm([x1 - x2 for x1, x2 in zip(old_X, X)]) <= 1e-10:
break
projected_X = matrix_multiplication(A, [[x] for x in X])
projected_X = [x[0] for x in projected_X]
eivals.append(norm(projected_X, 1)/norm(X, 1))
eivec_m.append(X[:m])
eivec_n.append(X[m:])
return (eivec_m, eivec_n, eivals)
