def populate_backward(self):
"""Populate backward likelihoods for all states/times"""
# initialise with exit probabilities
self.backward[:, -1] = self.state_transitions[1:len(self.states) + 1,
-1]
# below iterator skips first observation
# (will be used when finalising P(O|model))
# iterate backwards through observations (time) [::-1] <- reverses list
for t, observation in list(enumerate(self.observations[1:]))[::-1]:
# print(t, observation)
for state_index in range(len(self.states)):
state_number = state_index + 1
# ^ for easier reading (arrays 0-indexed, _number 1-indexed)
other_index = self.get_other_state_index(state_index)
other_number = other_index + 1 # for 1 indexing
# observation for transitions from the same state
this_state_gaussian = gaussian(
observation, self.states[state_index].mean,
self.states[state_index].std_dev)
# observation for transitions from the other state
other_state_gaussian = gaussian(
observation, self.states[other_index].mean,
self.states[other_index].std_dev)
# a * b * beta
this_from_this = self.state_transitions[
state_number,
state_number] * this_state_gaussian * self.backward[
state_index, t + 1]
other_from_this = self.state_transitions[
state_number,
other_number] * other_state_gaussian * self.backward[
other_index, t + 1]
self.backward[state_index,
t] = this_from_this + other_from_this
return self.backward
def populate_forward(self):
"""Populate forward likelihoods for all states/times"""
for t, observation in enumerate(self.observations):
# iterate through observations (time)
for state_index, state in enumerate(self.states):
# both states at each step
state_number = state_index + 1
# ^ for easier reading (arrays 0-indexed, _number 1-indexed)
if t == 0: # calcualte initial, 0 = first row = initial
self.forward[state_index, t] = self.state_transitions[
0, state_number] * gaussian(observation, state.mean,
state.std_dev)
else:
# each state for each time has two paths leading to it,
# the same state (this) and the other state (other)
other_index = self.get_other_state_index(state_index)
other_number = other_index + 1 # for 1 indexing
# previous value * prob of changing from previous state to current
this_to_this = self.forward[state_index, t -
1] * self.state_transitions[
state_number, state_number]
other_to_this = self.forward[
other_index, t -
1] * self.state_transitions[other_number, state_number]
self.forward[
state_index,
t] = (this_to_this + other_to_this) * gaussian(
observation, state.mean, state.std_dev)
return self.forward
def calculate_p_obs_backward(self):
"""Calculate, store and return P(O|model) going backwards"""
sum = 0
for state_index, initial_likelihood in enumerate(self.backward[:, 0]):
pi = self.state_transitions[0, state_index + 1]
b = gaussian(self.observations[0], self.states[state_index].mean,
self.states[state_index].std_dev)
beta = initial_likelihood
sum += pi * b * beta
self.p_obs_backward = sum
return sum
def transition_likelihood(self, from_index, to_index, t):
"""Get specific transition likelihood given state index either side and the timestep"""
#from_index = i, from equations in the notes
#to_index = j, from equations in the notes
if t == 0:
print("no transition likelihood for t == 0")
forward = self.forward[from_index, t - 1]
transition = self.state_transitions[from_index + 1, to_index + 1]
emission = gaussian(self.observations[t], self.states[to_index].mean,
self.states[to_index].std_dev)
backward = self.backward[to_index, t]
return (forward * transition * emission *
backward) / self.observation_likelihood
from markov import MarkovModel
from markovlog import LogMarkovModel
fig_dpi = 200
fig_export = False
x = np.linspace(-4, 8, 300) # x values for figures
x_label = "Observation Space"
y_label = "Probability Density"
# %% [markdown]
# State Probability Functions (1)
# ===================
# %%
state_1_y = [gaussian(i, state1.mean, state1.std_dev) for i in x]
state_2_y = [gaussian(i, state2.mean, state2.std_dev) for i in x]
plt.plot(x, state_1_y, c='r', label="State 1")
plt.plot(x, state_2_y, c='b', label="State 2")
plt.legend()
plt.title("State Probability Density Functions")
plt.xlabel(x_label)
plt.ylabel(y_label)
plt.grid(linestyle="--")
fig = matplotlib.pyplot.gcf()
fig.set_dpi(fig_dpi)
fig.set_tight_layout(True)