def inference(params):
# Read data from file
input_data = loadmat(params['input_file'])
x_sparse = np.asarray(input_data[params['data_field']], dtype=np.uint32)
if 'theta' in input_data:
theta_true = input_data['theta']
else:
theta_true = None
params['N'], params['T'] = np.max(x_sparse[:,1]), np.max(x_sparse[:,0])
params['T'] = min(params['T'], params['max_T'])
params['N'] = min(params['N'], params['max_N'])
x_sparse -= 1
theta_dim = (params['N'],params['N'],params['L'])
# Push sparse data into a dictionary
print 'Preprocessing sparse data'
x_dict = {}
for t in range(params['T']):
x_dict[t] = []
for t, i in x_sparse:
if not (t < params['T'] and i < params['N']): continue
x_dict[t].append(i)
for t in x_dict:
x_dict[t] = tuple(sorted(x_dict[t]))
# Define function for building window as needed
def make_window(cols):
w = np.zeros((params['N'], len(cols)), dtype=np.bool)
for o, col in enumerate(cols):
for i in col:
w[i, o] = 1
return w
# Generate S (calling it "windows" in code)
print 'Counting window permutations'
def n_perms(cols):
n = 0
denom = 1
for col in cols:
n += cols[col]
denom *= factorial(cols[col])
return factorial(n) / denom
windows, n_w, l_w = [], [], []
t_start = 0
while t_start < params['T']:
cols_seen = {}
t_end = t_start
while t_end < params['T']:
new_col = x_dict[t_end]
t_end += 1
if not new_col in cols_seen:
cols_seen[new_col] = 0
cols_seen[new_col] += 1
if t_end - t_start <= params['L']:
n_perm = n_perms(cols_seen)
continue
n_perm_new = n_perms(cols_seen)
if n_perm_new > params['perm_max']:
t_end -= 1
break
n_perm = n_perm_new
windows.append((t_start, t_end))
n_w.append(n_perm)
l_w.append(t_end - t_start)
t_start = t_end
params['M'] = len(n_w)
# Initialize theta
theta_init = np.zeros(theta_dim)
# Precompute statistics
print 'Precomputing statistics'
hits = [np.empty((n_w[0],)+theta_dim)]
hits_observed = np.zeros(theta_dim)
s_padded = np.zeros((params['N'],params['L']+l_w[0]), dtype=np.bool)
w_start, w_end = windows[0]
window = [x_dict[t] for t in range(w_start, w_end)]
for w, z in enumerate(permute(window)):
s = make_window(z)
s_padded[:,params['L']:(params['L']+l_w[0])] = s
for l in range(params['L']):
tmin, tmax = params['L']-(l+1), (params['L']+l_w[0])-(l+1)
s_lagged = s_padded[:,tmin:tmax]
hit = np.tensordot(s_lagged, s, axes = (1,1))
hits[0][w,:,:,l] = hit
if z == window:
hits_observed += hits[0][w]
for k in range(1, params['M']):
hits.append(np.empty((n_w[k-1],n_w[k])+theta_dim))
s_padded = np.empty((params['N'],l_w[k-1]+l_w[k]), dtype=np.bool)
w_prev_start, w_prev_end = windows[k-1]
w_start, w_end = windows[k]
window_prev = [x_dict[t] for t in range(w_prev_start, w_prev_end)]
window = [x_dict[t] for t in range(w_start, w_end)]
for w_prev, z_prev in enumerate(permute(window_prev)):
s_prev = make_window(z_prev)
s_padded[:,0:l_w[k-1]] = s_prev
for w, z in enumerate(permute(window)):
s = make_window(z)
s_padded[:,l_w[k-1]:(l_w[k-1]+l_w[k])] = s
for l in range(params['L']):
tmin, tmax = l_w[k-1]-(l+1), (l_w[k-1]+l_w[k])-(l+1)
s_lagged = s_padded[:,tmin:tmax]
hit = np.tensordot(s_lagged, s, axes = (1,1))
hits[k][w_prev,w,:,:,l] = hit
if z_prev == window_prev and z == window:
hits_observed += hits[k][w_prev,w]
del x_dict
# Common DP code used for likelihood and gradient calculations
def dp(theta):
h = [None] * params['M']
h[0] = np.empty(n_w[0])
for w in range(n_w[0]):
h[0][w] = np.sum(theta * hits[0][w])
for k in range(1, params['M']):
h[k] = np.empty((n_w[k-1], n_w[k]))
for w_prev in range(n_w[k-1]):
for w in range(n_w[k]):
h[k][w_prev,w] = np.sum(theta * hits[k][w_prev,w])
b = [None] * (params['M']+1)
b[params['M']] = np.zeros(n_w[params['M']-1])
for k in range(params['M']-1, 0, -1):
b[k] = np.empty(n_w[k-1])
for w_prev in range(n_w[k-1]):
b[k][w_prev] = logaddexp(h[k][w_prev] + b[k+1])
return h, b
# Define objective function, in this case, the negative log-likelihood
def neg_log_likelihood(theta_vec):
theta = np.reshape(theta_vec, theta_dim)
h, b = dp(theta)
log_kappa = logaddexp(h[0] + b[1])
nll = log_kappa
nll -= h[0][0]
for k in range(1, params['M']):
nll -= h[k][0,0]
nll += params['lambda'] * np.sum(np.abs(theta))
return nll
# Define gradient of the objective function
def grad_neg_log_likelihood(theta_vec):
theta = np.reshape(theta_vec, theta_dim)
h, b = dp(theta)
# Compute expected statistics
w_prob = unlog(h[0] + b[1])
hits_expected = fast_average(hits[0], w_prob)
for k in range(1, params['M']):
w_prob_new = np.zeros(n_w[k])
for w_prev in range(n_w[k-1]):
w_weight = unlog(h[k][w_prev,:] + b[k+1])
w_prob_new += w_weight * w_prob[w_prev]
hits_expected += (w_prob[w_prev] *
fast_average(hits[k][w_prev], w_weight))
w_prob = w_prob_new
# Adjust gradient for L1 regularization
reg = params['lambda'] * np.sign(theta)
return np.reshape(hits_expected - hits_observed + reg, theta_vec.shape)
# Callback for displaying state during optimization
def show_theta(theta_vec):
theta = np.reshape(theta_vec, (params['N'], params['N'], params['L']))
if theta_true is None:
print np.round(theta, decimals = 2)
else:
diff = np.reshape(theta - theta_true, theta_vec.shape)
print np.sqrt(np.dot(diff, diff))
if params['intermediate_viz']:
theta_viz(theta)
# Do optimization
print 'Starting optimization'
theta_opt = opt.fmin_bfgs(f = neg_log_likelihood,
fprime = grad_neg_log_likelihood,
x0 = theta_init,
callback = show_theta,
**(params['opt_params']))
# Output
print 'x'
print x_sparse
print
print 'Parameters'
for param in params:
print '%s: %s' % (param, str(params[param]))
print
print 'Inferred theta'
print np.reshape(theta_opt, (params['N'], params['N'], params['L']))
def inference(params):
# Read data from file
input_data = loadmat(params['input_file'])
x_sparse = np.asarray(input_data[params['data_field']], dtype=np.uint32)
labels, theta_true = None, None
if params['label_field'] in input_data:
labels = input_data[params['label_field']][:,0]
if params['theta_field'] in input_data:
theta_true = input_data[params['theta_field']]
params['N'], params['T'] = np.max(x_sparse[:,1]), np.max(x_sparse[:,0])
params['T'] = min(params['T'], params['max_T'])
params['N'] = min(params['N'], params['max_N'])
params['M'] = int(np.ceil(1.0 * params['T'] / params['Delta']))
params['T'] = params['Delta'] * params['M']
x_sparse -= 1
theta_dim = (params['N'],params['N'],params['L'])
# Push sparse data into a dictionary
print 'Preprocessing sparse data'
x_dict = {}
for t in range(params['T']):
x_dict[t] = []
for t, i in x_sparse:
if not (t < params['T'] and i < params['N']): continue
x_dict[t].append(i)
for t in x_dict:
x_dict[t] = tuple(sorted(x_dict[t]))
# Define function for building window as needed
def make_window(cols):
w = np.zeros((params['N'], len(cols)), dtype=np.bool)
for o, col in enumerate(cols):
for i in col:
w[i, o] = 1
return w
# Generate S (calling it "windows" in code)
print 'Counting window permutations'
n_perms_memo = {}
fact_memo = {}
def fact(n):
if n in fact_memo:
return fact_memo[n]
else:
val = factorial(n)
fact_memo[n] = val
return val
def n_perms(cols):
n = 0
denoms = []
for col in cols:
n += cols[col]
denoms.append(cols[col])
key = (n, tuple(sorted(denoms)))
if key in n_perms_memo:
return n_perms_memo[key]
else:
val = fact(n)
for denom in denoms:
val /= fact(denom)
n_perms_memo[key] = val
return val
windows, n_w, l_w = [], [], []
for k in range(params['M']):
t_start = k * params['Delta']
t_end = min(params['T'], (k+1) * params['Delta'])
cols_seen = {}
for t in range(t_start, t_end):
new_col = x_dict[t]
if not new_col in cols_seen:
cols_seen[new_col] = 0
cols_seen[new_col] += 1
n_perm = n_perms(cols_seen)
windows.append((t_start, t_end))
n_w.append(n_perm)
l_w.append(t_end - t_start)
# Initialize theta (using sparse representation)
theta = {}
def theta_dense(theta_sparse):
theta = np.zeros(theta_dim)
for ind in theta_sparse:
theta[ind] = theta_sparse[ind]
return theta
def arrays_from_theta(theta_sparse):
inds = []
theta = []
for ind in theta_sparse:
inds.append(ind)
theta.append(theta_sparse[ind])
return inds, np.array(theta)
def theta_from_arrays(inds, vec):
theta = {}
for ind, v in zip(inds, vec):
theta[ind] = v
return theta
# Precompute statistics
print 'Precomputing statistics'
hits_pre = [np.empty((n_w[0],)+theta_dim)]
hits_observed = np.zeros(theta_dim)
s_padded = np.zeros((params['N'],params['L']+l_w[0]), dtype=np.uint32)
w_start, w_end = windows[0]
window = [x_dict[t] for t in range(w_start, w_end)]
for w, z in enumerate(permute(window)):
s = make_window(z)
s_padded[:,params['L']:(params['L']+l_w[0])] = s
for l in range(params['L']):
tmin, tmax = params['L']-(l+1), (params['L']+l_w[0])-(l+1)
s_lagged = s_padded[:,tmin:tmax]
hit = np.tensordot(s_lagged, s, axes = (1,1))
hits_pre[0][w,:,:,l] = hit
if z == window:
hits_observed += hits_pre[0][w]
for k in range(1, params['M']):
hits_pre.append(np.empty((n_w[k-1],n_w[k])+theta_dim))
s_padded = np.empty((params['N'],l_w[k-1]+l_w[k]), dtype=np.uint32)
w_prev_start, w_prev_end = windows[k-1]
w_start, w_end = windows[k]
window_prev = [x_dict[t] for t in range(w_prev_start, w_prev_end)]
window = [x_dict[t] for t in range(w_start, w_end)]
for w_prev, z_prev in enumerate(permute(window_prev)):
s_prev = make_window(z_prev)
s_padded[:,0:l_w[k-1]] = s_prev
for w, z in enumerate(permute(window)):
s = make_window(z)
s_padded[:,l_w[k-1]:(l_w[k-1]+l_w[k])] = s
for l in range(params['L']):
tmin, tmax = l_w[k-1]-(l+1), (l_w[k-1]+l_w[k])-(l+1)
s_lagged = s_padded[:,tmin:tmax]
hit = np.tensordot(s_lagged, s, axes = (1,1))
hits_pre[k][w_prev,w,:,:,l] = hit
if z_prev == window_prev and z == window:
hits_observed += hits_pre[k][w_prev,w]
del x_dict
# Common DP code used for likelihood and gradient calculations
def dp(theta_sparse):
theta = theta_dense(theta_sparse)
h = [None] * params['M']
h[0] = np.empty(n_w[0])
for w in range(n_w[0]):
h[0][w] = np.sum(theta * hits_pre[0][w])
for k in range(1, params['M']):
h[k] = np.empty((n_w[k-1], n_w[k]))
for w_prev in range(n_w[k-1]):
for w in range(n_w[k]):
h[k][w_prev,w] = np.sum(theta * hits_pre[k][w_prev,w])
b = [None] * (params['M']+1)
b[params['M']] = np.zeros(n_w[params['M']-1])
for k in range(params['M']-1, 0, -1):
b[k] = np.empty(n_w[k-1])
for w_prev in range(n_w[k-1]):
b[k][w_prev] = logsumexp(h[k][w_prev] + b[k+1])
return h, b
# Define objective function, in this case, the negative log-likelihood
def neg_log_likelihood(theta_sparse, hb = None):
if not hb is None:
h, b = hb
else:
h, b = dp(theta_sparse)
log_kappa = logsumexp(h[0] + b[1])
nll = log_kappa
nll -= h[0][0]
for k in range(1, params['M']):
nll -= h[k][0,0]
for ind in theta_sparse:
nll += params['lambda'] * np.abs(theta_sparse[ind])
return nll
# Compute expected statistics
def expected_statistics(h, b):
w_prob = unlog(h[0] + b[1])
hits_expected = fast_average(hits_pre[0], w_prob)
for k in range(1, params['M']):
w_prob_new = np.zeros(n_w[k])
for w_prev in range(n_w[k-1]):
w_weight = unlog(h[k][w_prev,:] + b[k+1])
w_prob_new += w_weight * w_prob[w_prev]
hits_expected += (w_prob[w_prev] *
fast_average(hits_pre[k][w_prev], w_weight))
w_prob = w_prob_new
return hits_expected
# Define gradient of the objective function
def grad_neg_log_likelihood(theta_sparse, hb = None):
if not hb is None:
h, b = hb
else:
h, b = dp(theta_sparse)
hits_expected = expected_statistics(h, b)
grad_full = hits_expected - hits_observed
grad_sparse = {}
for ind in theta_sparse:
grad_sparse[ind] = grad_full[ind]
# Adjust gradient for L1 regularization
for ind in theta_sparse:
grad_sparse[ind] += params['lambda'] * np.sign(theta_sparse[ind])
return grad_sparse
# Do optimization
print 'Starting stepwise optimization'
h, b = dp(theta)
nll = neg_log_likelihood(theta, (h, b))
print 'Initial negative log-likelihood: %.2f' % nll
while True:
# Assess model at current theta
if params['intermediate_viz']:
theta_viz(theta_dense(theta), labels = labels)
# Sample at current theta
hits_sample = np.zeros((params['num_samples'],)+theta_dim)
w_samps = log_weighted_sample(h[0] + b[1], params['num_samples'])
for rep in range(params['num_samples']):
hits_sample[rep] += hits_pre[0][w_samps[rep]]
for k in range(1, params['M']):
w_samps_next = []
w_samps_next_uniques = []
w_samps_uniques, inds = np.unique(w_samps, return_inverse=True)
for i, w in enumerate(w_samps_uniques):
n = len(inds[inds == i])
w_samps_next_uniques.append(log_weighted_sample(h[k][w]+b[k+1],n))
for rep in range(params['num_samples']):
w_samps_next.append(w_samps_next_uniques[inds[rep]].pop())
hits_sample[rep] += hits_pre[k][w_samps[rep]][w_samps_next[rep]]
w_samps = w_samps_next
# Check global goodness-of-fit
hits_expected = expected_statistics(h, b)
hits_norms = np.array([la.norm(hits - hits_expected)
for hits in hits_sample])
ext = np.where(la.norm(hits_observed - hits_expected) > hits_norms)[0]
score = 1.0 * len(ext) / params['num_samples']
print 'Global score: %.2f' % score
if score < params['stopping_global']:
print 'Global goodness-of-fit criterion achieved'
break
# Find component with largest z-score
hits_sd = np.sqrt(np.mean((hits_sample-hits_expected)**2, axis=0))
z_scores = (hits_observed - hits_expected) / hits_sd
z_scores[hits_sd == 0] = 0
for ind in theta:
z_scores[ind] = 0
argmax_z = np.unravel_index(np.argmax(np.abs(z_scores)), theta_dim)
if abs(z_scores[argmax_z]) < params['stopping_z']:
print 'Largest z-score below stopping threshold'
break
print 'New component: %s (z = %.2f)' % (str(argmax_z), z_scores[argmax_z])
theta[argmax_z] = 0.0
# Make big steps in direction of new theta component
grad = grad_neg_log_likelihood(theta, (h, b))
dir_new = -np.sign(grad[argmax_z])
while True:
print 'Making big step'
old_nll = nll
theta[argmax_z] += dir_new * params['step_size']
h, b = dp(theta)
nll = neg_log_likelihood(theta, (h, b))
print 'Negative log-likelihood: %.2f' % nll
if nll > old_nll or abs(nll - old_nll) < params['opt_tol']: break
# Refit theta with new non-zero component
print 'Optimization by gradient descent'
while True:
old_nll = nll
grad = grad_neg_log_likelihood(theta, (h, b))
grad_norm = max(la.norm(np.array(grad.values())), 1.0)
for ind in grad:
theta[ind] -= (params['step_size'] / grad_norm) * grad[ind]
h, b = dp(theta)
nll = neg_log_likelihood(theta, (h, b))
print 'Negative log-likelihood: %.2f' % nll
if nll > old_nll: break
# Output
print 'x'
print x_sparse
print
print 'Parameters'
for param in params:
print '%s: %s' % (param, str(params[param]))
print
print 'Inferred theta'
print theta_dense(theta)
