Vectorize numpy code with operation depending on previous value

Question

The following code models a system that can sample 3 different states at any time, and the constant transition probability between those states is given by the matrix prob_nor. Threrefore, each point in trace depends on the previous state.

n_states, n_frames = 3, 1000
state_val = np.linspace(0, 1, n_states)

prob = np.random.randint(1, 10, size=(n_states,)*2)
prob[np.diag_indices(n_states)] += 50

prob_nor = prob/prob.sum(1)[:,None] # transition probability matrix, 
                                    # row sum normalized to 1.0

state_idx = range(n_states) # states is a list of integers 0, 1, 2...
current_state = np.random.choice(state_idx)

trace = []      
sigma = 0.1     
for _ in range(n_frames):
    trace.append(np.random.normal(loc=state_val[current_state], scale=sigma))
    current_state = np.random.choice(state_idx, p=prob_nor[current_state, :])

The loop in the above code makes it run pretty slow, specially when I have to model millions of data points. Is there any way to vectorize/accelerate it?

Answer 1

Offload the computation of probabilities as soon as possible:

possible_paths = np.vstack(
    np.random.choice(state_idx, p=prob_nor[curr_state, :], size=n_frames)
    for curr_state in range(n_states)
)

Then you can simply do a lookup to follow your path:

path_trace = [None]*n_frames
for step in range(n_frames):
    path_trace[step] = possible_paths[current_state, step]
    current_state = possible_paths[current_state, step]

Once you have your path, you can compute your trace:

sigma = 0.1
trace = np.random.normal(loc=state_val[path_trace], scale=sigma, size=n_frames)

Comparing timings:

Pure python for loop

%%timeit
trace_list = []
current_state = np.random.choice(state_idx)
for _ in range(n_frames):
    trace_list.append(np.random.normal(loc=state_val[current_state], scale=sigma))
    current_state = np.random.choice(state_idx, p=prob_nor[current_state, :])

Results:

30.1 ms ± 436 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)

Vectorized lookup:

%%timeit
current_state = np.random.choice(state_idx)
path_trace = [None]*n_frames
possible_paths = np.vstack(
    np.random.choice(state_idx, p=prob_nor[curr_state, :], size=n_frames)
    for curr_state in range(n_states)
)
for step in range(n_frames):
    path_trace[step] = possible_paths[current_state, step]
    current_state = possible_paths[current_state, step]
trace = np.random.normal(loc=state_val[path_trace], scale=sigma, size=n_frames)

Results:

641 µs ± 6.03 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

A speedup of approximately 50x.

Answer 2

Maybe I'm missing something, but I think you can create the current_states as a list and then vectorise the remaining steps:

# Make list of states (slow part)
states = []
current_state = np.random.choice(state_idx)
for _ in range(n_frames):
    states.append(current_state)
    current_state = np.random.choice(state_idx, p=prob_nor[current_state, :])

# Vectorised part
state_vals = state_val[states]   # alternatively np.array(states) / (n_states - 1)
trace = np.random.normal(loc=states, scale=sigma)

I believe this method works and will lead to a modest speed improvement while using some extra memory (3 lists/arrays are created instead of one). @PMende's solution leads to much larger speed improvement.