def log_likelihood(fluo, A_log, v, noise, pi0_log, alpha, beta):
"""
:param fluo: List of fluo time series
:param A_log: Log of transition matrix
:param v: Emission States
:param noise: Noise (stddev of signal)
:param pi0_log: Log of initial state PDF
:param alpha: Forward matrix
:param beta: Backward matrix
:return: Log Probability
"""
l_score = 0
K = len(v)
for f, fluo_vec in enumerate(fluo):
# Get log likelihood of sequence
p_x = logsumexp(alpha[f][:, -1])
for t in xrange(len(fluo_vec)):
for k in xrange(K):
# Likelihood of observing F(t)
l_score += math.exp(alpha[f][k, t] + beta[f][k, t] -
p_x) * log_L_fluo(fluo_vec[t], v[k], noise)
if t == 0:
# Likelihood of sequencce starting with k
for k in xrange(K):
l_score += math.exp(alpha[f][k, t] + beta[f][k, t] -
p_x) * (pi0_log[k] + alpha[f][k, t] +
beta[f][k, t])
else:
# Likelihood of transition TO l FROM k
for k in xrange(K):
for l in xrange(K):
l_score += math.exp(alpha[f][l, t] + beta[f][l, t] +
alpha[f][k, t - 1] +
beta[f][k, t - 1] - p_x) * A_log[l,
k]
return l_score
def beta_alg_cp(fluo_vec, A_log, v, w, noise, pi0_log, pointers, alpha_states,
cp_fluo, alpha_stack):
"""
:param fluo_vec: a single time series of fluorescence values
:param A: current estimate of A
:param v: current estimate of v
:param w: system memory
:param noise: system noise
:param pi0: initial state PDF
:param pointers: transition pointers from alpha calculation
:alpha_states: list of states visited by alpha alg
:cp_fluo: fluorescence corresponding to each state
:alpha_stack: list of final state vectors from alpha calc...starting point for beta
:return: K x T vector of log probabilities
"""
T = len(fluo_vec)
K = len(v)
max_stack = len(alpha_stack)
# Truncated beta array
beta_array = np.zeros((max_stack, T), dtype=float) - np.Inf
# initialize--We basically ignore this step
beta_array[:, -1] = np.log(1.0)
#Initialize Stack
Stack = alpha_stack
# Iteration
steps = np.arange(T - 1)
steps = steps[::-1]
for t in steps:
for l, state in enumerate(alpha_states[t + 1]):
for p in pointers[t][l]:
beta_array[p, t] = logsumexp([
beta_array[p, t], beta_array[l, t + 1] +
log_L_fluo(fluo_vec[t + 1], cp_fluo[t + 1][l], noise) +
A_log[state, alpha_states[t][p]]
])
return beta_array
def cpEM_BW(fluo,
A_init,
v_init,
noise,
pi0,
w,
max_stack=100,
max_iter=1000,
eps=10e-4):
"""
:param fluo: time series of fluorescent intensities (list of lists)
:param A_init: Initial guess at the system's transition probability matrix (KxK)
:param v_init: Initial guess at emissions vector (K)
:param noise: Standard Deviation of fluo emissions (Taken as given at this stage)
:param pi0: Initial state PDF (Taken as given)
:param w: memory of system
:param max_stack: Max num states tracked per time step (<= K^w)
:param max_iter: Maximum number of iterations permitted
:param eps: Termination criteria--minimum permissible percent change in estimated params
:return: Infers A and v for set of sequences
"""
pi0_log = np.log(pi0)
K = len(v_init)
A_list = [np.log(A_init)]
v_list = [v_init]
logL_list = [-10e7]
stack_depth = min(K**w, max_stack)
#Initialize variable to track percent change in likelihood
delta = 1
iter = 1
total_time = 0
while iter < max_iter and abs(delta) > eps:
loop_start_time = time.time()
setup_start = time.time()
v_curr = v_list[iter - 1]
A_log = A_list[iter - 1]
#--------------------------Fwd Bkwd Algorithm for Each Sequence----------------------------------------------------#
#store likelihood of each sequence given current parameter estimates
seq_log_probs = []
#store alpha and beta matrices
alpha_arrays = []
beta_arrays = []
#track recent state and transition pointer info
pointer_list = []
state_list = []
cp_fluo_list = []
cp_state_list = []
for f, fluo_vec in enumerate(fluo):
alpha_array, s_list, p_list, cf_list, Stack, F_list = alpha_alg_cp(
fluo_vec=fluo_vec,
A_log=A_log,
v=v,
w=w,
noise=noise,
pi0_log=pi0_log,
max_stack=stack_depth)
beta_array = beta_alg_cp(fluo_vec=fluo_vec,
A_log=A_log,
v=v,
w=w,
noise=noise,
pi0_log=pi0_log,
pointers=p_list,
alpha_states=s_list,
cp_fluo=cf_list,
alpha_stack=Stack)
#use last values of alpha array to calculate series probability
p_seq = logsumexp(alpha_array[:, -1])
#Store Results
alpha_arrays.append(alpha_array)
beta_arrays.append(beta_array)
seq_log_probs.append(p_seq)
pointer_list.append(p_list)
state_list.append(s_list)
cp_fluo_list.append(cf_list)
cp_state_list.append(F_list)
A_start = time.time()
#---------------------------------------Calculate Updated A and v--------------------------------------------------#
#Update A
#List of Lists to store transition events
#Index scheme: K*row + col
event_list = []
event_id = []
for f, fluo_vec in enumerate(fluo):
T = len(fluo_vec)
a = alpha_arrays[f]
b = beta_arrays[f]
p = pointer_list[f]
s = state_list[f]
i = cp_fluo_list[f]
sp = seq_log_probs[f]
for t in xrange(0, T - 1):
for row in xrange(len(p[t])):
for r in p[t][row]:
from_state = s[t][r]
to_state = s[t + 1][row]
event = a[r, t] + b[row, t + 1] + A_log[
to_state, from_state] + log_L_fluo(
fluo=fluo_vec[t + 1],
fluo_est=i[t + 1][row],
noise=noise)
event_list.append(event - sp)
event_id.append(K * to_state + from_state)
event_list = np.array(event_list)
event_id = np.array(event_id)
A_log_new = np.zeros((K, K))
for k in xrange(K**2):
A_log_new[k / K, k % K] = logsumexp(
event_list[np.where(event_id == k)[0]])
A_log_new = A_log_new - np.tile(logsumexp(A_log_new, axis=0), (K, 1))
A_new = np.exp(A_log_new)
#Update v
wt_full = []
for f, fluo_vec in enumerate(fluo):
#Transpose alpha beta arrays to keep format compatible with F counts
wt_full += np.reshape(
np.transpose(alpha_arrays[f] + beta_arrays[f] -
seq_log_probs[f]).tolist(),
(len(fluo_vec) * stack_depth)).tolist()
#Convert to arrays
F_full = np.array(list(chain(*chain(*cp_state_list))))
wt_full = np.exp(np.array(wt_full))
b_full = np.repeat(list(chain(*fluo)), stack_depth)
F_square = np.zeros((K, K))
b_vec = np.zeros(K)
for k in xrange(K):
F_square[k, :] = np.sum(F_full * F_full[:, k][:, np.newaxis] *
wt_full[:, np.newaxis],
axis=0)
b_vec[k] = np.sum(F_full[:, k] * wt_full * b_full)
v_new = np.linalg.solve(F_square, b_vec)
v_list.append(v_new)
A_list.append(A_log_new)
logL = np.sum(seq_log_probs)
logL_list.append(logL)
#Check % improvement in logL
delta = (logL_list[iter - 1] - logL) / logL_list[iter - 1]
if delta < 0:
print("Warning: Non-monotonic behavior in likelihood")
#sys.exit(1)
if iter % 1 == 0:
print(logL)
print(abs(delta))
print(v_new)
print(A_new)
loop_time = time.time() - loop_start_time
print(loop_time)
total_time += loop_time
iter += 1
print(total_time)
return (A_list, v_list, logL_list)
def viterbi_compound(fluo, A_log, v, noise, pi0_log, w, cp_array, to_from,
cp_init):
"""
:param fluo: list of fluorescence time series
:param A_log: Log of transition matrix
:param v: State emission vector
:param noise: system noise
:param pi0_log: log of initation PDF
:param w: memory in time steps
:return: most likely series of promoter states and compound emissions for each fluo vector
"""
#Get state count
K = len(v)
# Calculate fluo values for each state
cp_e_vec = np.zeros(K**w)
for s in xrange(K**w):
cp_e_vec[s] = np.sum(np.array([v[cp_array[s, i]] for i in xrange(w)]))
#Initialize list to store fits
cp_state_fits = []
cp_fluo_fits = []
state_fits = []
fluo_fits = []
logL_list = []
for f, fluo_vec in enumerate(fluo):
T = len(fluo_vec)
#Intilize array to store state likelihoods for each step
v_array = np.zeros((K**w, T)) - np.Inf
#Initialize array to store pointers
p_array = np.zeros((K**w, T - 1), dtype='int')
for t in xrange(T):
if t == 0:
for l in cp_init:
v_array[l, t] = log_L_fluo(
fluo=fluo_vec[t], fluo_est=cp_e_vec[l],
noise=noise) + pi0_log[cp_array[l, 0]]
else:
for l in xrange(K**w):
#Convenience lookup vector to simplify indexing
lk = to_from[l, :]
#most recent state in current cp state
rc = cp_array[l, 0]
lookback = [
v_array[k, t - 1] + A_log[rc, cp_array[k, 0]]
for k in lk
]
e_prob = log_L_fluo(fluo=fluo_vec[t],
fluo_est=cp_e_vec[l],
noise=noise)
#Get probs for present time point
v_array[l, t] = np.max(e_prob + lookback)
#Get pointer to most likely previous state
p_array[l, t - 1] = lk[np.argmax(np.array(lookback))]
#Backtrack to find optimal path
#Arrays to store compound sequences
cp_v_fits = np.zeros(T, dtype='int')
cp_f_fits = np.zeros(T)
#Arrays to store simple sequences
v_fits = np.zeros(T, dtype='int')
f_fits = np.zeros(T)
cp_v_fits[T - 1] = np.argmax(v_array[:, T - 1])
cp_f_fits[T - 1] = cp_e_vec[cp_v_fits[T - 1]]
v_fits[T - 1] = cp_array[cp_v_fits[T - 1], 0]
f_fits[T - 1] = v[v_fits[T - 1]]
prev = cp_v_fits[T - 1]
for t in xrange(T - 1):
cp_v_fits[T - 2 - t] = p_array[prev, T - 2 - t]
cp_f_fits[T - 2 - t] = cp_e_vec[cp_v_fits[T - 2 - t]]
v_fits[T - 2 - t] = cp_array[cp_v_fits[T - 2 - t], 0]
f_fits[T - 2 - t] = v[v_fits[T - 2 - t]]
prev = cp_v_fits[T - 2 - t]
cp_state_fits.append(cp_v_fits)
cp_fluo_fits.append(cp_f_fits)
state_fits.append(v_fits)
fluo_fits.append(f_fits)
logL_list.append(np.max(v_array[:, T - 1]))
return (cp_state_fits, cp_fluo_fits, state_fits, fluo_fits, logL_list)
def alpha_alg_cp(fluo_vec, A_log, v, w, noise, pi0_log, max_stack):
"""
:param fluo_vec: a single time series of fluorescence values
:param A: current estimate of A
:param v: current estimate of v
:param w: system memory
:param noise: system noise
:param pi0: initial state PDF
:param max_stack: max num states tracked per step
:return: K x T vector of log probabilities
"""
K = len(v)
T = len(fluo_vec)
#List to Store Transition Pointers (len T-1). Points to positional id NOT state ID
p_list = []
#List to Store most recently added state (len T)
s_list = []
#List to store cp fluo values (for convenience, could be calculated from above arrays) (len T)
cf_list = []
#Truncated alpha array
alpha_array = np.zeros((max_stack, T), dtype=float) - np.Inf
#Stack list to track most recent N active states
Stack = []
#List of Lists to track state counte
F_list = []
# Iteration
dp_total = 0
update_total = 0
sort_total = 0
for t in xrange(0, T):
if t == 0:
s_list.append(range(K))
cf_list.append([v[s] for s in xrange(K)])
alpha_array[:K, 0] = pi0_log + np.array(
[log_L_fluo(fluo_vec[t], v[s], noise) for s in xrange(K)])
Stack = [[s] + [0] * (w - 1) for s in xrange(K)]
# [s_{t}, s_{t-1}, s_{t-2},...s_{t-w+1}
else:
#Iterate through previous list of states to determine updates
new_probs = []
new_pointers = []
new_states = []
new_fluo = []
new_stack = []
for l in xrange(K):
for k, state in enumerate(s_list[t - 1]):
dp_start = time.time()
sn = [l] + Stack[k][:-1]
if k > 0 and sn == new_stack[-1]:
new_probs[-1] = logsumexp([
log_L_fluo(fluo_vec[t], new_fluo[-1], noise) +
A_log[l, state] + alpha_array[k, t - 1],
new_probs[-1]
])
new_pointers[-1] = [
pointer for pointer in new_pointers[-1]
] + [k]
dp_total += time.time() - dp_start
else:
update_start = time.time()
if t < w:
new_fluo.append(cf_list[t - 1][k] + v[l])
else:
new_fluo.append(cf_list[t - 1][k] + v[l] -
v[Stack[k][-1]])
new_stack.append([l] + Stack[k][:-1])
# get logL of transitions into state l from each of the previous states
new_states.append(l)
new_pointers.append([k])
# scipy has a built in function for handling logsums
new_probs.append(
log_L_fluo(fluo_vec[t], new_fluo[-1], noise) +
A_log[l, state] + alpha_array[k, t - 1])
update_total += time.time() - update_start
#Sort resulting probs
sort_time = time.time()
new_args = np.argsort(new_probs)[-max_stack:]
n_arg = len(new_args)
#Now that we have ranked entries by prob, sort so that most similar stack entries are adjacent
#will be used in next iteration to (somewhat) efficiently look for new duplicate entries
stack_strings = []
for arg in new_args:
stack_strings.append(''.join(map(str, new_stack[arg][:-1])))
sort_args = np.argsort(stack_strings)
sorted_new_args = [new_args[i] for i in sort_args]
#Update lists
alpha_array[0:n_arg, t] = [new_probs[a] for a in sorted_new_args]
p_list.append([new_pointers[a] for a in sorted_new_args])
s_list.append([new_states[a] for a in sorted_new_args])
cf_list.append([new_fluo[a] for a in sorted_new_args])
Stack = [new_stack[a] for a in sorted_new_args]
sort_total += time.time() - sort_time
f = [[0, 0, 0]] * max_stack
for i, stk in enumerate(Stack):
ct = [
len(np.where(np.array(stk, dtype='int') == k)[0])
for k in xrange(K)
]
f[i] = ct
F_list.append(f)
return (alpha_array, s_list, p_list, cf_list, Stack, F_list)
def decode_cp(fluo,
A_log,
pi0_log,
v,
w,
noise,
stack_depth,
alpha=0,
log_stack=0):
"""
:param fluo: list of fluorescence vectors
:param A_log: log of transition probability matrix
:param pi0_log: log of initial state PDF
:param v: emission state vector
:param w: memory of system
:param noise: estimated system noise
:param stack_depth: max num active hypotheses to keep in stack
:param alpha: fractional num time steps needed to transcribe MS2 loops
:return: Best guess at optimal path for each input fluo vec
"""
# Get state count
K = len(v)
seq_out = []
f_out = []
v_out = []
logL_out = []
stack_register = []
# Calculate convolution kernel to apply to fluorescence
if alpha > 0:
alpha_vec = [
(float(i + 1) / alpha +
(float(i) / alpha)) / 2.0 * (i < alpha) * ((i + 1) <= alpha) +
((alpha - i) * (1 + float(i) / alpha) / 2.0 + i + 1 - alpha) *
(i < alpha) * (i + 1 > alpha) + 1 * (i >= alpha) for i in xrange(w)
]
else:
alpha_vec = np.array([1.0] * w)
kernel = np.ones(w) * alpha_vec
kernel = kernel[::-1]
iter = 0
for f, fluo_vec in enumerate(fluo):
Stack = []
T = len(fluo_vec)
# assume that fluo vectors are off prior to first obs...
for l in range(K):
#Assume initial condition is OFF
ns = [0] * (w)
ns.append(l)
ns = np.array(ns)
#z_{t-w+1} ... z_{t-1}, z_{t}
f = np.sum(ns[1:] * kernel)
ns_score = log_L_fluo(fluo=fluo_vec[0], fluo_est=f,
noise=noise) + pi0_log[l]
Stack.append([ns_score, ns, f])
Stack.sort(key=itemgetter(0))
while (len(Stack[-1][1])) < T + w:
#Get current best hypothesis
hypothesis = Stack.pop()
h_seq = hypothesis[1]
prev = h_seq[-1]
t = len(h_seq) - w
#Check possible extensions
for l in range(K):
new_seq = [0] * (len(h_seq) + 1)
new_seq[:-1] = h_seq
new_seq[-1] = l
f1 = np.array([v[z] for z in new_seq])
f1 = np.sum(kernel * f1[-w:])
ns_score = (hypothesis[0] * float(t) + log_L_fluo(
fluo=fluo_vec[t], fluo_est=f1, noise=noise) +
A_log[l, prev]) / float(t + 1)
bisect.insort(Stack, [ns_score, new_seq, f1])
#Enforce max stack length
while (len(Stack) > stack_depth):
Stack.pop(0)
if iter % 10 == 0:
if log_stack:
stack_register.append(Stack[-1][1])
iter += 1
logL_out.append(T * Stack[-1][0])
seq_out.append(Stack[-1][1][w:])
v_out.append([v[i] for i in Stack[-1][1][w:]])
emissions = [v[t] for t in Stack[-1][1]]
f_cp = np.convolve(kernel[::-1], emissions, mode='full')
f_cp = f_cp[w:-w + 1]
f_out.append(f_cp)
return (seq_out, f_out, v_out, logL_out, stack_register)