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IB.py
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IB.py
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import numpy as np
import pandas as pd
import time
import math
vlog = np.vectorize(math.log)
vexp = np.vectorize(math.exp)
# A word on notation: for probability variables, an underscore here means a
# conditioning, so read _ as |.
def verify_inputs(pxy,beta,alpha,Tmax,p0,ctol_abs,ctol_rel,ptol,zeroLtol,clamp,compact,verbose):
"""Helper function for IB which checks for validity of input parameters."""
if not(isinstance(pxy,np.ndarray)):
raise ValueError('pxy must be a numpy array')
if np.any(pxy<0) or np.any(pxy>1):
raise ValueError('entries of pxy must be between 0 and 1')
if abs(np.sum(pxy)-1)>ptol:
raise ValueError('pxy must be normalized')
if not(beta>0) or not(isinstance(beta,(int,float))):
raise ValueError('beta must be a positive scalar')
if alpha<0 or not(isinstance(alpha,(int,float))):
raise ValueError('alpha must be a non-negative scalar')
if Tmax<1:
raise ValueError('Tmax must be a positive integer (or infinity)')
if p0<-1 or p0>1 or not(isinstance(p0,(int,float))):
raise ValueError('p0 must be a float/int between -1 and 1')
if not(ctol_abs>=0) or not(isinstance(ctol_abs,float)):
raise ValueError('ctol_abs must be a non-negative float')
if not(ctol_rel>=0) or not(isinstance(ctol_rel,float)):
raise ValueError('ctol_rel must be a non-negative float')
if (ctol_rel==0) and (ctol_abs==0):
raise ValueError('One of ctol_rel and ctol_abs must be postive')
if not(ptol>0) or not(isinstance(ptol,float)):
raise ValueError('ptol must be a positive float')
if zeroLtol<0:
raise ValueError('zeroLtol must be positive')
if not(isinstance(clamp,bool)):
raise ValueError('clamp must be a boolean')
if not(verbose in (0,1,2)):
raise ValueError('verbose should be 0, 1, or 2')
if not(compact in (0,1,2)):
raise ValueError('compact should be 0, 1, or 2')
return 0
def entropy_term(x):
"""Helper function for entropy_single: calculates one term in the sum."""
if x==0:
return 0.0
else:
return -x*math.log2(x)
def entropy_single(p):
"""Returns entropy of p: H(p)=-sum(p*log(p)). (in bits)"""
ventropy_term = np.vectorize(entropy_term)
vec = ventropy_term(p)
h = np.sum(vec)
return h
def entropy(P):
"""Returns entropy of a distribution, or series of distributions.
For the input array P [=] M x N, treats each col as a prob distribution
(over M elements), and thus returns N entropies. If P is a vector, treats
P as a single distribution and returns its entropy."""
if P.ndim==1:
return entropy_single(P)
else:
M,N = P.shape
H = np.zeros(N)
for n in range(N):
H[n] = entropy_single(P[:,n])
return H
def process_pxy(pxy,verbose=1):
"""Helper function for IB that preprocesses p(x,y) and computes metrics."""
if pxy.dtype!='float':
pxy = pxy.astype(float)
Xorig = pxy.shape[0]
Yorig = pxy.shape[1]
px = pxy.sum(axis=1)
py = pxy.sum(axis=0)
nzx = px>0 # find nonzero-prob entries
nzy = py>0
zx = np.where(px<=0)
zx = zx[0]
zy = np.where(py<=0)
zy = zy[0]
px = px[nzx] # drop zero-prob entries
py = py[nzy]
X = len(px)
Y = len(py)
if verbose>0 and (Xorig-X)>0:
print('%i of %i Xs dropped due to zero prob; size now %i. Dropped IDs:' % (Xorig-X,Xorig,X))
print(zx)
if verbose>0 and (Yorig-Y)>0:
print('%i of %i Ys dropped due to zero prob; size now %i. Dropped IDs:' % (Yorig-Y,Yorig,Y))
print(zy)
pxy_orig = pxy
tmp = pxy_orig[nzx,:]
pxy = tmp[:,nzy] # pxy_orig with zero-prob x,y removed
py_x = np.multiply(pxy.T,np.tile(1./px,(Y,1)))
hx = entropy(px)
hy = entropy(py)
hy_x = np.dot(px,entropy(py_x))
ixy = hy-hy_x
return pxy, px, py_x, hx, hy, hy_x, ixy, X, Y, zx, zy
def kl_term(x,y):
"""Helper function for kl: calculates one term in the sum."""
if x>0 and y>0:
return x*math.log2(x/y)
elif x==0:
return 0.0
else:
return math.inf
def kl_single(p,q):
"""Returns KL divergence of p and q: KL(p,q)=sum(p*log(p/q)). (in bits)"""
vkl_term = np.vectorize(kl_term)
vec = vkl_term(p,q)
dkl = np.sum(vec)
return dkl
def kl(P,Q):
"""Returns KL divergence of one or more pairs of distributions.
For the input arrays P [=] M x N and Q [=] M x L, calculates KL of each col
of P with each col of Q, yielding the KL matrix DKL [=] N x L. If P=Q=1,
returns a single KL divergence."""
if P.ndim==1 and Q.ndim==1:
return kl_single(P,Q)
elif P.ndim==1 and Q.ndim!=1: # handle vector P case
M = len(P)
N = 1
M2,L = Q.shape
if M!=M2:
raise ValueError("P and Q must have same number of columns")
DKL = np.zeros((1,L))
for l in range(L):
DKL[0,l] = kl_single(P,Q[:,l])
elif P.ndim!=1 and Q.ndim==1: # handle vector Q case
M,N = P.shape
M2 = len(Q)
L = 1
if M!=M2:
raise ValueError("P and Q must have same number of columns")
DKL = np.zeros((N,1))
for n in range(N):
DKL[n,0] = kl_single(P[:,n],Q)
else:
M,N = P.shape
M2,L = Q.shape
if M!=M2:
raise ValueError("P and Q must have same number of columns")
DKL = np.zeros((N,L))
for n in range(N):
for l in range(L):
DKL[n,l] = kl_single(P[:,n],Q[:,l])
return DKL
def qt_step(qt_x,px,ptol,verbose):
"""Peforms q(t) update step for generalized Information Bottleneck."""
T, X = qt_x.shape
qt = np.dot(qt_x,px)
dropped = qt<=ptol # clusters to drop due to near-zero prob
if any(dropped):
qt = qt[~dropped] # drop ununsed clusters
qt_x = qt_x[~dropped,:]
T = len(qt) # update number of clusters
qt_x = np.multiply(qt_x,np.tile(1./np.sum(qt_x,axis=0),(T,1))) # renormalize
qt = np.dot(qt_x,px)
if verbose==2:
print('%i cluster(s) dropped. Down to %i cluster(s).' % (np.sum(dropped),T))
return qt_x,qt,T
def qy_t_step(qt_x,qt,px,py_x):
"""Peforms q(y|t) update step for generalized Information Bottleneck."""
qy_t = np.dot(py_x,np.multiply(qt_x,np.outer(1./qt,px)).T)
return qy_t
def qt_x_step(qt,py_x,qy_t,T,X,alpha,beta,verbose):
"""Peforms q(t|x) update step for generalized Information Bottleneck."""
if T==1: # no need for computation
qt_x = np.ones((1,X))
else:
qt_x = np.zeros((T,X))
for x in range(X):
l = vlog(qt)-beta*kl(py_x[:,x],qy_t) # [=] T x 1 # scales like X*Y*T
if alpha==0: # DIB
qt_x[np.argmax(l),x] = 1
else: # IB and interpolations
qt_x[:,x] = vexp(l/alpha)/np.sum(vexp(l/alpha)) # note: l/alpha<-745 is where underflow creeps in
return qt_x
def init_qt_x(alpha,X,T,p0):
"""Initializes q(t|x) for generalized Information Bottleneck."""
if alpha==0: # DIB: spread points evenly across clusters
n = math.ceil(float(X)/float(T)) # approx number points per cluster
I = np.repeat(np.arange(0,T),n).astype("int") # data-to-cluster assignment vector
np.random.shuffle(I)
qt_x = np.zeros((T,X))
for i in range(X):
qt_x[I[i],i] = 1
else: # not DIB
if p0==0: # normalized uniform random vector
qt_x = np.random.rand(T,X)
qt_x = np.multiply(qt_x,np.tile(1./np.sum(qt_x,axis=0),(T,1))) # renormalize
# others are spike plus wavy: if pos, spread peaks around like DIB init;
# if neg, put all peaks on same cluster
elif p0>0: # noisy approx to DIB init
wavy = False
if wavy:
# insert wavy noise part
f = .25; # max percent variation of flat part of dist around mean
qt_x = np.ones((T,X))+2*(np.random.rand(T,X)-.5)*f # 1+-f%
# choose clusters for each x to get spikes
n = math.ceil(float(X)/float(T)) # approx number points per cluster
I = np.repeat(np.arange(0,T),n).astype("int") # data-to-cluster assignment vector
np.random.shuffle(I)
for i in range(X):
qt_x[I[i],i] = 0 # zero out that cluster
qt_x[:,i] = (1-p0)*qt_x[:,i]/np.sum(qt_x[:,i]) # normalize others to 1-p0
qt_x[I[i],i] = p0 # insert p0 spike
else: # uniform random vector instead of wavy
qt_x = np.zeros((T,X))
# choose clusters for each x to get spikes
n = math.ceil(float(X)/float(T)) # approx number points per cluster
I = np.repeat(np.arange(0,T),n).astype("int") # data-to-cluster assignment vector
np.random.shuffle(I)
for i in range(X):
u = np.random.rand(T)
u[I[i]] = 0
u = (1-p0)*u/np.sum(u)
u[I[i]] = p0
qt_x[:,i] = u
else: # same as above, but all x assigned same cluster for spike
wavy = False
if wavy:
p0 = -p0
# p0 = prob on spike, mean prob elsewhere = (1-p0)/(T-1)
f = .25; # max percent variation of flat part of dist around mean
qt_x = np.ones((T,X))+2*(np.random.rand(T,X)-.5)*f # 1+-f%
t = np.random.randint(T) # pick cluster to get delta spike
qt_x[t,:] = np.zeros((1,X)) # zero out that cluster
qt_x = np.multiply(qt_x,np.tile(1./np.sum(qt_x,axis=0),(T,1))) # normalize the rest...
qt_x = (1-p0)*qt_x # ...to 1-p0
qt_x[t,:] = p0*np.ones((1,X)) # put in delta spike
else: # uniform random vector instead of wavy
p0 = -p0
qt_x = np.zeros((T,X))
# choose clusters for each x to get spikes
t = np.random.randint(T) # pick cluster to get delta spike
for i in range(X):
u = np.random.rand(T)
u[t] = 0
u = (1-p0)*u/np.sum(u)
u[t] = p0
qt_x[:,i] = u
return qt_x
def calc_IB_metrics(qt_x,qt,qy_t,px,hy,alpha,beta):
"""Calculates IB performance metrics."""
ht = entropy(qt)
hy_t = np.dot(qt,entropy(qy_t))
iyt = hy-hy_t
ht_x = np.dot(px,entropy(qt_x))
ixt = ht-ht_x
L = ht-alpha*ht_x-beta*iyt
return ht, hy_t, iyt, ht_x, ixt, L
def IB_single(pxy,beta,alpha,Tmax,p0,ctol_abs,ctol_rel,ptol,zeroLtol,clamp,compact,verbose):
"""Performs the generalized Information Bottleneck on the joint p(x,y).
Note: fixed distributions denoted by p; optimized ones by q.
INPUTS
*** see IB function documentation below ***
OUTPUTS
metrics_stepwise = dataframe of scalar metrics for each fit step:
L = objective function value [=] scalar
ixt = I(X,T) [=] scalar
iyt = I(Y,T) [=] scalar
ht = H(T) [=] scalar
T = number of clusters used [=] pos integer
ht_x = H(T|X) [=] scalar
hy_t = H(Y|T) [=] scalar
hx = H(X) [=] scalar
ixy = I(X,Y) [=] scalar
step = index of fit step [=] pos integer
step_time = time to complete this step (in s) [=] pos scalar
hx = H(X) [=] scalar
ixy = I(X,Y) [=] scalar
Tmax
beta
alpha
p0
ctol
ptol
distributions_stepwise = dataframe of optimized distributions for each step:
qt_x = q(t|x) = [=] T x X (note: size T changes during iterations)
qt = q(t) [=] T x 1 (note: size T changes during iterations)
qy_t = q(y|t) [=] Y x T (note: size T changes during iterations)
step = index of fit step [=] pos integer
Tmax
beta
alpha
p0
ctol
ptol
metrics_converged = dataframe of last (converged) step for each Tmax/fit above:
step_time -> conv_time = time to run all steps (in s)
step -> conv_steps = number of steps to converge
conv_condition = string indicating reason for convergence [=] {cost_func_inc,small_changes,single_cluster,cost_func_NaN}
distributions_converged = dataframe of last (converged) step for each Tmax/fit above:
step_time -> conv_time = time to run all steps (in s)
step -> conv_steps = number of steps to converge
conv_condition = string indicating reason for convergence"""
verify_inputs(pxy,beta,alpha,Tmax,p0,ctol_abs,ctol_rel,ptol,zeroLtol,clamp,compact,verbose)
conv_thresh = 1 # steps in a row of small change to consider converged
# process inputs
if isinstance(alpha,int):
alpha = float(alpha)
if isinstance(beta,int):
beta = float(beta)
pxy, px, py_x, hx, hy, hy_x, ixy, X, Y, zx, zy = process_pxy(pxy,verbose)
if Tmax==math.inf:
Tmax = X
if verbose==2:
print('Tmax set to %i based on X' % Tmax)
elif Tmax>X:
if verbose==2:
print('Reduced Tmax from %i to %i based on X' % (Tmax,X))
Tmax = X
else:
Tmax = int(Tmax)
# initialize dataframes
metrics_stepwise = pd.DataFrame(columns=['L','ixt','iyt','ht','T','ht_x',
'hy_t','step','step_time'])
if compact>1:
distributions_stepwise = pd.DataFrame(columns=['qt_x','qt','qy_t','step'])
# initialize other stuff
T = Tmax
step_start_time = time.time()
# STEP 0: INITIALIZE
# initialize q(t|x)
qt_x = init_qt_x(alpha,X,T,p0)
# initialize q(t) given q(t|x)
qt_x,qt,T = qt_step(qt_x,px,ptol,verbose)
# initialize q(y|t) given q(t|x) and q(t)
qy_t = qy_t_step(qt_x,qt,px,py_x)
# calculate and print metrics
if verbose==2:
print('IB initialized')
ht, hy_t, iyt, ht_x, ixt, L = calc_IB_metrics(qt_x,qt,qy_t,px,hy,alpha,beta)
if verbose==2:
print('I(X,T) = %.6f, H(T) = %.6f, H(X) = %.6f, I(Y,T) = %.6f, I(X,Y) = %.6f, L = %.6f' % (ixt,ht,hx,iyt,ixy,L))
step_time = time.time() - step_start_time
if len(metrics_stepwise.index)==0:
this_index = 0
else:
this_index = max(metrics_stepwise.index)+1
metrics_stepwise = metrics_stepwise.append(pd.DataFrame(data={
'L': L, 'ixt': ixt, 'iyt': iyt, 'ht': ht,
'T': T, 'ht_x': ht_x, 'hy_t': hy_t,
'step_time': step_time, 'step': 0},
index=[this_index]))
if compact>1:
distributions_stepwise = distributions_stepwise.append(pd.DataFrame(data={
'qt_x': [qt_x], 'qt': [qt], 'qy_t': [qy_t],
'step': 0},
index=[this_index]))
del this_index
# ITERATE STEPS 1-3 TO CONVERGENCE
converged = 0
if T==1:
converged = conv_thresh
if verbose>0:
print('Converged due to reduction to single cluster')
conv_condition = 'single_cluster'
else:
conv_condition = ''
Nsteps = 0
L_old = L
iter_start_time = time.time()
while converged<conv_thresh:
step_start_time = time.time()
Nsteps += 1
if verbose==2:
print('Beginning IB step %i' % Nsteps)
# STEP 1: UPDATE Q(T|X)
qt_x = qt_x_step(qt,py_x,qy_t,T,X,alpha,beta,verbose)
# STEP 2: UPDATE Q(T)
qt_x,qt,T = qt_step(qt_x,px,ptol,verbose)
# STEP 3: UPDATE Q(Y|T)
qy_t = qy_t_step(qt_x,qt,px,py_x)
# calculate and print metrics
ht, hy_t, iyt, ht_x, ixt, L = calc_IB_metrics(qt_x,qt,qy_t,px,hy,alpha,beta)
if verbose==2:
print('I(X,T) = %.6f, H(T) = %.6f, H(X) = %.6f, I(Y,T) = %.6f, I(X,Y) = %.6f, L = %.6f' % (ixt,ht,hx,iyt,ixy,L))
# check for convergence
L_abs_inc_flag = L>(L_old+ctol_abs)
L_rel_inc_flag = L>(L_old+(abs(L_old)*ctol_rel))
if abs(L_old-L)<ctol_abs:
converged += 1
if (converged>=conv_thresh):
conv_condition = 'small_abs_changes'
if verbose>0:
print('Converged due to small absolute changes in objective')
else:
converged = 0 # reset counter if change wasn't small
if (abs(L_old-L)/abs(L_old))<ctol_rel:
converged = conv_thresh
if verbose>0:
print('Converged due to small relative changes in objective')
if len(conv_condition)==0:
conv_condition = 'small_rel_changes'
else:
conv_condition += '_AND_small_rel_changes'
if (T==1) and not(L_abs_inc_flag) and not(L_rel_inc_flag):
converged = conv_thresh
if verbose>0:
print('Converged due to reduction to single cluster')
if len(conv_condition)==0:
conv_condition = 'single_cluster'
else:
conv_condition += '_AND_single_cluster'
if np.isnan(L):
converged = conv_thresh
if verbose>0:
print('Stopped because objective = NaN')
if len(conv_condition)==0:
conv_condition = 'cost_func_NaN'
else:
conv_condition += '_AND_cost_func_NaN'
# check if obj went up by amount above threshold (after 1st step)
if (L_abs_inc_flag or L_rel_inc_flag) and (Nsteps>1): # if so, don't store or count this step!
converged = conv_thresh
if L_abs_inc_flag:
if verbose>0:
print('Converged due to absolute increase in objective value')
if len(conv_condition)==0:
conv_condition = 'cost_func_abs_inc'
else:
conv_condition += '_AND_cost_func_abs_inc'
if L_rel_inc_flag:
if verbose>0:
print('Converged due to relative increase in objective value')
if len(conv_condition)==0:
conv_condition = 'cost_func_rel_inc'
else:
conv_condition += '_AND_cost_func_rel_inc'
# revert to metrics/distributions from last step
L = L_old
ixt = ixt_old
iyt = iyt_old
ht = ht_old
T = T_old
ht_x = ht_x_old
hy_t = hy_t_old
qt_x = qt_x_old
qt = qt_old
qy_t = qy_t_old
else:
# store stepwise data
step_time = time.time() - step_start_time
this_index = max(metrics_stepwise.index)+1
metrics_stepwise = metrics_stepwise.append(pd.DataFrame(data={
'L': L, 'ixt': ixt, 'iyt': iyt, 'ht': ht,
'T': T, 'ht_x': ht_x, 'hy_t': hy_t,
'step_time': step_time, 'step': Nsteps},
index=[this_index]))
if compact>1:
distributions_stepwise = distributions_stepwise.append(pd.DataFrame(data={
'qt_x': [qt_x], 'qt': [qt], 'qy_t': [qy_t],
'step': Nsteps},
index=[this_index]))
del this_index
L_old = L
ixt_old = ixt
iyt_old = iyt
ht_old = ht
T_old = T
ht_x_old = ht_x
hy_t_old = hy_t
qt_x_old = qt_x
qt_old = qt
qy_t_old = qy_t
# end iterative IB steps
# replace converged step with single-cluster map if better
if T>1:
if verbose>0:
print("Trying single-cluster mapping.")
step_start_time = time.time()
sqt_x = np.zeros((T,X))
sqt_x[0,:] = 1.
sqt_x,sqt,sT = qt_step(sqt_x,px,ptol,verbose)
sqy_t = qy_t_step(sqt_x,sqt,px,py_x)
sht, shy_t, siyt, sht_x, sixt, sL = calc_IB_metrics(sqt_x,sqt,sqy_t,px,hy,alpha,beta)
if sL<(L-zeroLtol): # if better fit...
conv_condition += '_AND_force_single'
if verbose>0:
print("Single-cluster mapping reduces L from %.6f to %.6f; replacing." % (L,sL))
# replace everything
qt_x = sqt_x
qt = sqt
T = sT
qy_t = sqy_t
ht = sht
hy_t = shy_t
iyt = siyt
ht_x = sht_x
ixt = sixt
L = sL
# store stepwise data
step_time = time.time() - step_start_time
this_index = max(metrics_stepwise.index)+1
metrics_stepwise = metrics_stepwise.append(pd.DataFrame(data={
'L': L, 'ixt': ixt, 'iyt': iyt, 'ht': ht,
'T': T, 'ht_x': ht_x, 'hy_t': hy_t,
'step_time': step_time, 'step': Nsteps+1},
index=[this_index]))
if compact>1:
distributions_stepwise = distributions_stepwise.append(pd.DataFrame(data={
'qt_x': [qt_x], 'qt': [qt], 'qy_t': [qy_t],
'step': Nsteps+1},
index=[this_index]))
del this_index
L_old = L
ixt_old = ixt
iyt_old = iyt
ht_old = ht
T_old = T
ht_x_old = ht_x
hy_t_old = hy_t
qt_x_old = qt_x
qt_old = qt
qy_t_old = qy_t
elif verbose>0:
print("Single-cluster mapping not better; increases L from %.6f to %.6f." % (L,sL))
# end single-cluster check
conv_time = time.time() - iter_start_time
metrics_converged = pd.DataFrame(data={
'L': L, 'ixt': ixt, 'iyt': iyt, 'ht': ht,
'T': T, 'ht_x': ht_x, 'hy_t': hy_t,
'conv_time': conv_time, 'conv_steps': Nsteps,
'hx': hx, 'ixy': ixy, 'Tmax': Tmax,
'beta': beta, 'alpha': alpha, 'p0': p0,
'ctol_abs': ctol_abs, 'ctol_rel': ctol_rel,
'ptol': ptol, 'zeroLtol': zeroLtol,
'conv_condition': conv_condition, 'clamp': False},
index=[0])
if compact>0:
distributions_converged = pd.DataFrame(data={
'qt_x': [qt_x], 'qt': [qt], 'qy_t': [qy_t],
'Tmax': Tmax, 'beta': beta, 'alpha': alpha,
'p0': p0, 'ctol_abs': ctol_abs, 'ctol_rel': ctol_rel,
'ptol': ptol, 'zeroLtol': zeroLtol,
'conv_condition': conv_condition, 'clamp': False},
index=[0])
# add in stuff that doesn't vary by step
metrics_stepwise['hx'] = hx
metrics_stepwise['ixy'] = ixy
metrics_stepwise['Tmax'] = Tmax
metrics_stepwise['beta'] = beta
metrics_stepwise['alpha'] = alpha
metrics_stepwise['p0'] = p0
metrics_stepwise['ctol_abs'] = ctol_abs
metrics_stepwise['ctol_rel'] = ctol_rel
metrics_stepwise['ptol'] = ptol
metrics_stepwise['zeroLtol'] = zeroLtol
if compact>1:
distributions_stepwise['Tmax'] = Tmax
distributions_stepwise['beta'] = beta
distributions_stepwise['alpha'] = alpha
distributions_stepwise['p0'] = p0
distributions_stepwise['ctol_abs'] = ctol_abs
distributions_stepwise['ctol_rel'] = ctol_rel
distributions_stepwise['ptol'] = ptol
distributions_stepwise['zeroLtol'] = zeroLtol
# optional clamping step (doesn't apply to DIB)
if (alpha>0) and clamp:
start_time = time.time()
if verbose>0:
print('****************************** Clamping IB fit with following parameters ******************************')
if Tmax == math.inf:
print('alpha = %.6f, beta = %.6f, Tmax = inf, p0 = %.6f, ctol_abs = %.6f, ctol_rel = %.6f, ptol = %.6f'\
% (alpha,beta,p0,ctol_abs,ctol_rel,ptol))
else:
print('alpha = %.6f, beta = %.6f, Tmax = %i, p0 = %.6f, ctol_abs = %.6f, ctol_rel = %.6f, ptol = %.6f'\
% (alpha,beta,Tmax,p0,ctol_abs,ctol_rel,ptol))
print('**************************************************************************************************')
# STEP 1: CLAMP Q(T|X)
for x in range(X):
tstar = np.argmax(qt_x[:,x])
qt_x[:,x] = 0
qt_x[tstar,x] = 1
del tstar
del x
# STEP 2: UPDATE Q(T)
qt_x,qt,T = qt_step(qt_x,px,ptol,verbose)
# STEP 3: UPDATE Q(Y|T)
qy_t = qy_t_step(qt_x,qt,px,py_x)
# calculate and print metrics
ht, hy_t, iyt, ht_x, ixt, L = calc_IB_metrics(qt_x,qt,qy_t,px,hy,alpha,beta)
if verbose>0:
print('***** unclamped fit *****')
print('I(X,T) = %.6f, H(T) = %.6f, H(X) = %.6f, I(Y,T) = %.6f, I(X,Y) = %.6f, L = %.6f' % (ixt_old,ht_old,hx,iyt_old,ixy,L_old))
print('***** clamped fit *****')
print('I(X,T) = %.6f, H(T) = %.6f, H(X) = %.6f, I(Y,T) = %.6f, I(X,Y) = %.6f, L = %.6f' % (ixt,ht,hx,iyt,ixy,L))
# store everything
this_step_time = time.time()-start_time
metrics_converged = metrics_converged.append(pd.DataFrame(data={
'L': L, 'ixt': ixt, 'iyt': iyt, 'ht': ht,
'T': T, 'ht_x': ht_x, 'hy_t': hy_t,
'conv_time': conv_time+this_step_time, 'conv_steps': Nsteps+1,
'hx': hx, 'ixy': ixy, 'Tmax': Tmax,
'beta': beta, 'alpha': alpha, 'p0': p0, 'zeroLtol': zeroLtol,
'ctol_abs': ctol_abs, 'ctol_rel': ctol_rel,
'ptol': ptol, 'conv_condition': conv_condition,
'clamp': True},
index=[1]))
if compact>0:
distributions_converged = distributions_converged.append(pd.DataFrame(data={
'qt_x': [qt_x], 'qt': [qt], 'qy_t': [qy_t],
'Tmax': Tmax, 'beta': beta, 'alpha': alpha,
'p0': p0, 'ctol_abs': ctol_abs, 'ctol_rel': ctol_rel,
'ptol': ptol, 'zeroLtol': zeroLtol,
'conv_condition': conv_condition, 'clamp': True},
index=[1]))
# return results
if compact>1:
return metrics_stepwise, distributions_stepwise,\
metrics_converged, distributions_converged
elif compact>0:
return metrics_stepwise,\
metrics_converged, distributions_converged
else:
return metrics_stepwise,\
metrics_converged
def refine_beta(metrics_converged,verbose=2):
"""Helper function for IB to refine beta parameter."""
# parameters governing insertion of betas, or when there is a transition to NaNs (due to under/overflow)
l = 1 # number of betas to insert into gaps
del_R = .05 # if fractional change in I(Y;T) exceeds this between adjacent betas, insert more betas
del_C = .05 # if fractional change in H(T) or I(X;T) exceeds this between adjacent betas, insert more betas
min_abs_res = 1e-1 # if beta diff smaller than this absolute threshold, don't insert; consider as phase transition
min_rel_res = 2e-2 # if beta diff smaller than this fractional threshold, don't insert
# parameters governing insertion of betas when I(X;T) doesn't reach zero
eps0 = 1e-2 # tolerance for considering I(X;T) to be zero
l0 = 1 # number of betas to insert at low beta end
f0 = .5 # new betas will be minbeta*f0.^1:l0
# parameters governing insertion of betas when I(T;Y) doesn't reach I(X;Y)
eps1 = .99 # tolerance for considering I(T;Y) to be I(X;Y)
l1 = 1 # number of betas to insert at high beta end
f1 = 2 # new betas will be maxbeta*f0.^1:l0
max_beta_allowed = 100 # any proposed betas above this will be filtered out and replaced it max_beta_allowed
# sort by beta
metrics_converged = metrics_converged.sort_values(by='beta')
# init
new_betas = np.array([])
NaNtran = False
ixy = metrics_converged['ixy'].iloc[0]
logT = math.log2(metrics_converged['Tmax'].iloc[0])
if verbose>0:
print('-----------------------------------')
# check that smallest beta was small enough
if metrics_converged['ixt'].min()>eps0:
minbeta = metrics_converged['beta'].min()
betas_to_add = np.array([minbeta*(f0**n) for n in range(1,l0+1)])
new_betas = np.append(new_betas,betas_to_add)
if verbose==2:
print('Added %i small betas. %.6f was too large.' % (l0,minbeta))
del betas_to_add
# check for gaps to fill
for i in range(metrics_converged.shape[0]-1):
beta1 = metrics_converged['beta'].iloc[i]
beta2 = metrics_converged['beta'].iloc[i+1]
if (beta2-beta1)<min_abs_res or ((beta2-beta1)/beta1)<min_rel_res: # if beta diff small than absolute/relative resolution, just check for NaNtran
cc1 = metrics_converged['conv_condition'].iloc[i]
cc2 = metrics_converged['conv_condition'].iloc[i+1]
NaNtran = (("cost_func_NaN" not in cc1) and ("cost_func_NaN" in cc2))
else: # otherwise, do all gap checks
iyt1 = metrics_converged['iyt'].iloc[i]
iyt2 = metrics_converged['iyt'].iloc[i+1]
ixt1 = metrics_converged['ixt'].iloc[i]
ixt2 = metrics_converged['ixt'].iloc[i+1]
ht1 = metrics_converged['ht'].iloc[i]
ht2 = metrics_converged['ht'].iloc[i+1]
cc1 = metrics_converged['conv_condition'].iloc[i]
cc2 = metrics_converged['conv_condition'].iloc[i+1]
NaNtran = (("cost_func_NaN" not in cc1) and ("cost_func_NaN" in cc2))
if ((abs(iyt1-iyt2)/ixy)>del_R) or\
((abs(ixt1-ixt2)/logT)>del_C) or\
((abs(ht1-ht2)/logT)>del_C) or\
NaNtran:
betas_to_add = np.linspace(beta1,beta2,l+2)[1:l+1]
new_betas = np.append(new_betas,betas_to_add)
del betas_to_add
if verbose==2:
print('Added %i betas between %.3f and %.3f.' % (l,beta1,beta2))
if NaNtran: # stop search if there was a NaNtran
if verbose==2:
print('(...because there was a transition to NaNs.)')
break
# check that largest beta was large enough
if ((metrics_converged['iyt'].max()/ixy)<eps1) and ~NaNtran:
maxbeta = metrics_converged['beta'].max()
betas_to_add = np.array([maxbeta*(f1**n) for n in range(1,l1+1)])
new_betas = np.append(new_betas,betas_to_add)
if verbose==2:
print('Added %i large betas. %.3f was too small.' % (l1,maxbeta))
del betas_to_add
# filter out betas above max_beta_allowed
too_large_mask = new_betas>max_beta_allowed
to_keep_mask = new_betas<max_beta_allowed
max_beta_allowed_used = (metrics_converged['beta'].max()==max_beta_allowed)
if any(too_large_mask):
new_betas = new_betas[to_keep_mask]
if verbose==2:
print('Filtered out %i betas larger than max_beta_allowed.' % np.sum(too_large_mask))
if max_beta_allowed_used:
if verbose==2:
print('...and not replaced since max_beta_allowed = %i already used.' % max_beta_allowed)
else:
new_betas = np.append(new_betas,max_beta_allowed)
if verbose==2:
print('And replaced them with max_beta_allowed = %i.' % max_beta_allowed)
if verbose>0:
print('Added %i new betas.' % len(new_betas))
print('-----------------------------------')
return new_betas
def set_param(fit_param,param_name,def_val):
"""Helper function for IB to handle setting of fit parameters."""
if param_name in fit_param.index.values: # check if param included
param_val = fit_param[param_name]
if np.isnan(param_val) or math.isinf(param_val): # if so, check val
param = def_val
else:
param = param_val
else: # if not, use default
param = def_val
return param
def IB(pxy,fit_param,compact=1,verbose=2):
"""Performs many generalized IB fits to a single p(x,y).
One fit is performed for each row of input dataframe fit_param. Columns
correspond to parameters of IB_single. See definition of IB_single for
more details.
REQUIRED INPUTS
pxy - p(x,y) [=] X x Y
fit_param = pandas df, with each row specifying a single IB fit; columns inc:
alpha = IB parameter [=] pos scalar (required)
beta = vector of IB parameters [=] vector of pos scalars
beta_search = flag indicating whether to perform automatic beta search
or use only initial beta(s) [=] boolean
Tmax = max cardinality of T / max # of clusters [=] pos integer
p0 = determines initialization of q(t|x) when alpha>0 (i.e. non-DIB fits)
for pos p0, p0 is prob mass on ~unique cluster for each input x
(i.e. q(t_i|x_i)=p0 where t_i is unique for each x_i)
for neg p0, p0 is prob mass on shared cluster for all inputs
(i.e. q(t*_x_i)=p0 for all x_i)
in both cases, the probability over the remaining clusters is set
to a normalized uniform random vector (with prob mass 1-p0)
for p0=0, q(t|x_i) is set to a normalized unirand vec for each x_i
ctol_abs = if abs(L_old-L)<ctol_abs, converge [=] non-neg scalar
ctol_rel = if abs(L_old-L)/abs(L_old)<ctol_rel, converge [=] non-neg scalar
ptol = x,y,t values dropped if prob<ptol [=] non-neg scalar
max_fits = max number of beta fits allowed for each input row [=] pos integer
max_time = max time (in seconds) allowed for fitting of each input row [=] pos scalar
repeats = repeated fits per beta / row, after which best value of L retained [=] pos int
zeroLtol = if L>zeroLtol, revert to solution mapping all x to same t (with L=0) [=] non-neg scalar
clamp = if true, for all non-DIB fits, insert a clamped version of the solution
into the results after convergence [=] boolean
*** default parameter values below ***
OPTIONAL INPUTS
verbose = integer indicating verbosity of updates [=] 0/1/2
compact = integer indicating how much data to save [=] 0/1/2
(0: only save metrics;
1: also save converged distributions;
2: also save stepwise distributions)"""
# set defaults
def_betas = np.array([.1,1,2,3,4,5,7,9,10])
def_beta_search = True
def_Tmax = math.inf
def_p0 = .75
def_ctol_abs = 10**-3
def_ctol_rel = 0.
def_ptol = 10**-8
def_max_fits = 100
def_max_time = 7*24*60*60 # 1 week
def_repeats = 1
def_zeroLtol = 1
def_clamp = True
# initialize dataframes
metrics_stepwise_allreps = pd.DataFrame(columns=['L','T','ht','ht_x','hy_t',
'ixt','iyt','step','step_time',
'hx','ixy','Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel','ptol',
'zeroLtol','repeat','repeats'])
if compact>1:
distributions_stepwise_allreps = pd.DataFrame(columns=['qt','qt_x','qy_t',
'step','Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel',
'ptol','zeroLtol',
'repeat','repeats'])
metrics_converged_allreps = pd.DataFrame(columns=['L','T','conv_steps','conv_time',
'ht','ht_x','hy_t','ixt','iyt',
'hx','ixy','Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel','ptol',
'zeroLtol','conv_condition','clamp',
'repeat','repeats'])
if compact>0:
distributions_converged_allreps = pd.DataFrame(columns=['qt','qt_x','qy_t',
'Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel',
'ptol','zeroLtol',
'conv_condition','clamp',
'repeat','repeats'])
metrics_stepwise = pd.DataFrame(columns=['L','T','ht','ht_x','hy_t',
'ixt','iyt','step','step_time',
'hx','ixy','Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel','ptol',
'zeroLtol','repeat','repeats'])
if compact>1:
distributions_stepwise = pd.DataFrame(columns=['qt','qt_x','qy_t',
'step','Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel',
'ptol','zeroLtol',
'repeat','repeats'])
metrics_converged = pd.DataFrame(columns=['L','T','conv_steps','conv_time',
'ht','ht_x','hy_t','ixt','iyt',
'hx','ixy','Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel','ptol',
'zeroLtol','conv_condition','clamp',
'repeat','repeats'])
if compact>0:
distributions_converged = pd.DataFrame(columns=['qt','qt_x','qy_t',
'Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel',
'ptol','zeroLtol',
'conv_condition','clamp',
'repeat','repeats'])
# iterate over fit parameters (besides beta, which is done below)
for irow in range(len(fit_param.index)):
# extract required parameters
this_fit = fit_param.iloc[irow]
this_alpha = this_fit['alpha']
this_betas = set_param(this_fit,'betas',def_betas)
this_beta_search = set_param(this_fit,'beta_search',def_beta_search)
# extract optional parameters
this_Tmax = set_param(this_fit,'Tmax',def_Tmax)
if this_alpha>0:
this_p0 = set_param(this_fit,'p0',def_p0)
else:
this_p0 = 1.
this_ctol_abs = set_param(this_fit,'ctol_abs',def_ctol_abs)
this_ctol_rel = set_param(this_fit,'ctol_rel',def_ctol_rel)
this_ptol = set_param(this_fit,'ptol',def_ptol)
this_max_fits = set_param(this_fit,'max_fits',def_max_fits)
this_max_time = set_param(this_fit,'max_time',def_max_time)
this_repeats = set_param(this_fit,'repeats',def_repeats)
this_zeroLtol = set_param(this_fit,'zeroLtol',def_zeroLtol)
this_clamp = set_param(this_fit,'clamp',def_clamp)
# make pre-fitting initializations
betas = this_betas # stack of betas
fit_count = 0
fit_time = 0
fit_start_time = time.time()
these_betas_metrics_converged = pd.DataFrame(columns=['ht','ixt','iyt',
'hx','ixy','Tmax','beta',
'conv_condition']) # used for beta refinement
while (fit_count<=this_max_fits) and (fit_time<=this_max_time) and (betas.size>0):
this_beta = betas[0] # use beta at front of list
# init data structures that will store the repeated fits for this particular setting of parameters
these_reps_metrics_stepwise = pd.DataFrame(columns=['L','T','ht','ht_x','hy_t',
'ixt','iyt','step','step_time',
'hx','ixy','Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel','ptol','zeroLtol',
'repeat','repeats'])
if compact>1:
these_reps_distributions_stepwise = pd.DataFrame(columns=['qt','qt_x','qy_t',
'step','Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel','ptol',
'zeroLtol','repeat','repeats'])
these_reps_metrics_converged = pd.DataFrame(columns=
['L','T','conv_steps','conv_time',
'ht','ht_x','hy_t','ixt','iyt',
'hx','ixy','Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel','ptol',
'zeroLtol','conv_condition','clamp',
'repeat','repeats'])
if compact>0:
these_reps_distributions_converged = pd.DataFrame(columns=
['qt','qt_x','qy_t',
'Tmax','beta','alpha',
'p0','ctol_abs','ctol_rel',
'ptol','zeroLtol',
'conv_condition','clamp',
'repeat','repeats'])
for repeat in range(this_repeats):
# do a single fit
if verbose>0:
print('****************************** Running IB with following parameters ******************************')
if this_Tmax == math.inf:
print('alpha = %.6f, beta = %.6f, Tmax = inf, p0 = %.6f, ctol_abs = %.6f, ctol_rel = %.6f, ptol = %.6f, zeroLtol = %.6f, repeat = %i of %i'\
% (this_alpha,this_beta,this_p0,this_ctol_abs,this_ctol_rel,this_ptol,this_zeroLtol,repeat,this_repeats))
else:
print('alpha = %.6f, beta = %.6f, Tmax = %i, p0 = %.6f, ctol_abs = %.6f, ctol_rel = %.6f, ptol = %.6f, zeroLtol = %.6f, repeat = %i of %i'\
% (this_alpha,this_beta,this_Tmax,this_p0,this_ctol_abs,this_ctol_rel,this_ptol,this_zeroLtol,repeat,this_repeats))
print('**************************************************************************************************')
if compact>1:
this_metrics_stepwise, this_distributions_stepwise, \
this_metrics_converged, this_distributions_converged = \
IB_single(pxy,this_beta,this_alpha,this_Tmax,
this_p0,this_ctol_abs,this_ctol_rel,this_ptol,this_zeroLtol,this_clamp,compact,verbose)
elif compact>0:
this_metrics_stepwise, \
this_metrics_converged, this_distributions_converged = \
IB_single(pxy,this_beta,this_alpha,this_Tmax,
this_p0,this_ctol_abs,this_ctol_rel,this_ptol,this_zeroLtol,this_clamp,compact,verbose)
else:
this_metrics_stepwise, \
this_metrics_converged = \
IB_single(pxy,this_beta,this_alpha,this_Tmax,
this_p0,this_ctol_abs,this_ctol_rel,this_ptol,this_zeroLtol,this_clamp,compact,verbose)
# add repeat labels
this_metrics_stepwise['repeat'] = repeat
this_metrics_stepwise['repeats'] = this_repeats
if compact>1:
this_distributions_stepwise['repeat'] = repeat
this_distributions_stepwise['repeats'] = this_repeats
this_metrics_converged['repeat'] = repeat
this_metrics_converged['repeats'] = this_repeats
if compact>0:
this_distributions_converged['repeat'] = repeat
this_distributions_converged['repeats'] = this_repeats
# add this repeat to these repeats
these_reps_metrics_stepwise = these_reps_metrics_stepwise.append(this_metrics_stepwise)
if compact>1:
these_reps_distributions_stepwise = these_reps_distributions_stepwise.append(this_distributions_stepwise)
these_reps_metrics_converged = these_reps_metrics_converged.append(this_metrics_converged)
if compact>0:
these_reps_distributions_converged = these_reps_distributions_converged.append(this_distributions_converged)
# end of repeat fit loop for single beta
# reindex fits to work with existing dataframe
if len(metrics_stepwise_allreps.index)>0:
num_there = max(metrics_stepwise_allreps.index)
else:
num_there = 0
num_added = len(these_reps_metrics_stepwise.index)
these_reps_metrics_stepwise.index = np.arange(num_there+1,num_there+num_added+1)
if compact>1:
these_reps_distributions_stepwise.index = np.arange(num_there+1,num_there+num_added+1)
del num_there, num_added
if len(metrics_converged_allreps.index)>0: