def get_ll(M1,C1,M2,C2,beta,s,y):
'''
Compute negative log-likelihood up to a constant.
Parameters
----------
M1: prior mean
C1: prior covariance
M2: posterior mean
C2: posterior covariance
beta: projection onto rate
s: offset
y: point-process observation count
'''
M1 = assertfinitereal(ascolumn(M1))
M2 = assertfinitereal(ascolumn(M2))
C1 = assertfinitereal(assquare(C1))
C2 = assertfinitereal(assquare(C2))
beta = assertfinitereal(ascolumn(beta))
logr = beta.T.dot(M1)+s
logPyx = y*logr - sexp(logr)
Ch = trychol(C1,1e-6)
RR = linv(Ch,M2-M1)
ll = logPyx + np.sum(slog(np.diag(chol(C2)))) - np.sum(slog(np.diag(Ch))) - 0.5*RR.T.dot(RR)
return scalar(ll)
def univariate_lgp_update_laplace(m,v,y,s,dt,
tol = 1e-6,
maxiter = 20,
eps = 1e-12):
'''
Optimize using Laplace approximation
'''
v = max(v,eps)
scale = dt*np.exp(s)
def objective(mu):
rate = scale*sexp(mu)
if not np.isfinite(mu) or not np.isfinite(rate):
return np.inf
return -y*mu+rate+0.5*(mu-m)**2/v
def gradient(mu):
rate = scale*sexp(mu)
if not np.isfinite(mu) or not np.isfinite(rate):
return np.inf
return -y+rate+(mu-m)/v
def hessian(mu):
rate = scale*sexp(mu)
if not np.isfinite(mu) or not np.isfinite(rate):
return np.inf
return rate+1/v
mu = minimize_retry(objective,m,gradient,hessian,tol=tol,show_progress=False,printerrors=False)
vv = 1/hessian(mu)
# Get likelihood at posterior mode
logr = mu+s+slog(dt)
logPyx = y*logr-sexp(logr)
ll = logPyx + 0.5*slog(vv/v) - 0.5*(mu-m)**2/v
return mu, vv, ll
def univariate_lgp_update_moment(m,v,y,s,dt,
tol = 1e-3,
maxiter = 100,
eps = 1e-7,
minlrate = -200,
maxlrate = 20,
ngrid = 50,
minprec = 1e-6,
maxrange = 150):
'''
Update a log-Gaussian distribution with a Poisson measurement by
integrating to extract the posterior mean and variance.
'''
# Get moments by integrating
v = max(v,eps)
t = 1/v
m = np.clip(m,minlrate,maxlrate)
# Set integration limits
m0,s0 = (m,np.sqrt(v)) if t>minprec else (slog(y+0.25),np.sqrt(1/(y+1)))
m0 = np.clip(m0,minlrate,maxlrate)
delta = min(4*s0,maxrange)
x = np.linspace(m0-delta,m0+delta,ngrid)
# Calculate likelihood contribution
r = x + s + slog(dt)
ll = y*r-sexp(r)
# Calculate prior contribution
lq = -.5*((x-m)**2/v+slog(2*np.pi*v))
# "clean up" prior (numerical stability)
q = np.maximum(eps,sexp(lq))
q = q/np.sum(q)
lq = slog(q)
# Normalize to prevent overflow, calculate log-posterior
nn = np.max(ll)
lp = (ll - nn) + lq
# Estimate posterior
p = np.maximum(eps,sexp(lp))
s = np.sum(p)
p /= s
# Integrate to get posterior moments and likelihood
pm = np.sum(x*p)
pv = np.sum((x-pm)**2*p)
ll = scipy.misc.logsumexp(ll+lq)
assertfinitereal(pm)
assertfinitereal(pv)
assertfinitereal(ll)
return pm,pv,ll
def univariate_lgp_update_variational_so(m,v,y,s,dt,
tol = 1e-6,
maxiter = 20,
eps = 1e-12):
'''
Optimize variational approximation
Mean and variance must be optimized jointly
2nd order Gaussian/Poisson model approximation
'''
v = max(v,eps)
scale = dt*sexp(s)
# Use minimization to solve for variational solution
def objective(parameters):
mq,vq = parameters
rate = scale*np.exp(mq)*(1 + 0.5*vq)
if vq<eps or not np.isfinite(mq) or not np.isfinite(vq) or not np.isfinite(rate):
return np.inf
return -y*mq + rate + 0.5*( -slog(vq) + vq/v + (mq-m)**2/v )
def gradient(parameters):
mq,vq = parameters
rate = scale*np.exp(mq)*(1 + 0.5*vq)
if vq<eps or not np.isfinite(mq) or not np.isfinite(vq) or not np.isfinite(rate):
return (np.NaN,np.NaN)
dm = -y + rate + (mq-m)/v
dv = rate*0.5 + 0.5*(1/v-1/vq)
return np.array([dm, dv]).squeeze()
def hessian(parameters):
mq,vq = parameters
rate = scale*np.exp(mq)*(1 + 0.5*vq)
if vq<eps or not np.isfinite(mq) or not np.isfinite(vq) or not np.isfinite(rate):
return [[np.NaN,np.NaN],[np.NaN,np.NaN]]
dmdm = rate + 1/v
dvdm = rate*0.5
dvdv = rate*0.25 + 0.5/vq**2
return np.array([[dmdm,dvdm],[dvdm,dvdv]]).squeeze()
mp,vp = minimize_retry(objective,[m,v],gradient,hessian,tol=tol,show_progress=False,printerrors=False)
# Get likelihood using second order assumption
logr = mp+s+slog(dt)
logPyx = y*logr-sexp(logr)*(1+0.5*vp)
ll = logPyx + 0.5*slog(vp/v) - 0.5*(mp-m)**2/v
return mp, vp, ll
def get_ll_univariate(m1,v1,m2,v2,beta,s,y):
'''
Compute negative log-likelihood up to a constant
Parameters
----------
m1: prior mean
v1: prior covariance
m2: posterior mean
v2: posterior covariance
beta: projection onto rate
s: offset
y: point-process observation count
'''
m1 = scalar(m1)
v1 = scalar(v1)
m2 = scalar(m2)
v2 = scalar(v2)
logr = m2+s
logPyx = y*logr - sexp(logr)
ll = logPyx + 0.5*slog(v2/v1) - 0.5*(m2-m1)**2/v1
return scalar(ll)
def filter_moments(stim,Y,A,beta,C,m,
dt = 1.0,
oversample = 10,
maxrate = 500,
maxvcorr = 2000,
method = "moment_closure",
int_method = "euler",
measurement = "moment",
reg_cov = 0,
reg_rate = 0,
return_surrogates = False,
use_surrogates = None,
initial_conditions = None,
progress = False,
safe = True):
'''
Parameters
----------
stim : zero-lage effective input (filtered stimulus plus mean offset)
Y : point-process count observations, same length as stim
A : forward operator for delay-line evolution
C : projection of current state onto delay-like
beta : basis history weights
m : log-rate bias parameter, log-rates are regularized toward this value
Other Parameters
----------------
dt : time step
oversample : int
Integration steps per time step. Should be larger if using
Gaussian moment closure, which is stiff. Can be small if using
second-order approximations, which are less stiff.
maxrate :
maximum rate tolerated
maxvcorr:
Maximum variance correction ('convexity correction' in some literature)
tolerated during the moment closure.
method :
Moment-closure method. Can be "LNA" for mean-field with linear
noise approximation, "moment_closure" for Gaussian moment-closure
on the history process, or "second_order", which discards higher
moments of the rate which emerge when exponentiating.
int_method:
Integration method. Can be either "euler" for forward-Euler, or
"exponential", which integrates the locally-linearized system
forward using matrix exponentiation (slower).
measurement:
"moment", "laplace", or "variational"
reg_cov:
Diagonal covariance regularization
reg_rate:
Small regularization toward log mean-rate; This parameter reflects
the precision of a Gaussian prior about the log mean-rate, applied
at every measurement update.
return_surrogates: bool
If true, Gaussian approximations of measurement likelihoods are
returned.
use_surrogates: None or tuple
Can be set as tuple of (means, variances) for Gaussian
approximations of meausrement updates.
initial_conditions: None or tuple
Can be set to a tuple (M1,M2) of initial conditions for moment
filtering.
progress: boolean
Whether to report progress
Returns
-------
allLR : single-time marginal mean of log-rate
allLV : single-time marginal variance of log-rate
allM1 : low-dimensional approximation of history process, mean
allM2 : low-dimensional approximation of history process, covariance
nll : negative log-likelihood
'''
# check arguments
stim = asvector(stim)
Y = asvector(Y)
A = assquare(A)
if oversample<1:
raise ValueError('oversample must be non-negative integer')
if method=="moment_closure" and measurement=="variational":
warnings.warn("There are unresolved numerical stability issues "\
"when using the log-Gaussian variational update with Gaussian "\
"moment closure. Suggest using the second-order moment closure "\
"instead")
# Precompute constants
maxlogr = np.log(maxrate)
maxratemc = maxvcorr*maxrate
dtfine = dt/oversample
T = len(stim)
K = beta.size
I = np.eye(K)
Cb = C.dot(beta.T)
CC = C.dot(C.T)
Adt = A*dtfine
if not use_surrogates is None:
MR,VR = use_surrogates
# Get measurement update function
measurement = get_measurement(measurement)
# Buid moment integrator functions
mean_update, cov_update = get_moment_integrator(int_method,Adt)
# Get update function (computes expected rate from moments)
update = get_update_function(method,Cb,Adt,maxvcorr)
# Accumulate negative log-likelihood up to a constant
nll = 0
llrescale = 1.0/len(stim)
if initial_conditions is None:
# Initial condition for moments
M1 = np.zeros((K,1))
M2 = np.eye(K)*1e-6
else:
M1,M2 = initial_conditions
# Store moments
allM1 = np.zeros((T,K))
allM2 = np.zeros((T,K,K))
allLR = np.zeros((T))
allLV = np.zeros((T))
allmr = np.zeros((T))
allvr = np.zeros((T))
if progress:
last_shown = current_milli_time()
for i,s in enumerate(stim):
# Regularize
if reg_cov>0:
strength = reg_cov+max(0,-np.min(np.diag(M2)))
M2 = 0.5*(M2+M2.T) + strength*np.eye(K)
# Integrate moments forward
for j in range(oversample):
logv = beta.T.dot(M2).dot(beta)
logx = min(beta.T.dot(M1)+s,maxlogr)
R0 = sexp(logx)
R0 = min(maxrate,R0)
R0 *= dtfine
Rm,J = update(logx,logv,R0,M1,M2)
M2 = cov_update(M2,J) + CC*Rm
M1 = mean_update(M1) + C *Rm
if safe:
M1 = np.clip(M1,-100,100)
M2 = np.clip(M2,-100,100)
# Measurement update
pM1,pM2 = M1,M2
if use_surrogates is None:
# Use specified approximation method to handle non-conjugate
# log-Gaussian Poisson measurement update
M1,M2,ll,mr,vr = measurement_update_projected_gaussian(\
M1,M2,Y[i],beta,s,dt,m,reg_rate,measurement)
allmr[i] = mr
allvr[i] = vr
else:
# Use specified Gaussian approximations (MR,VR) to the
# measurement likelihoods.
M1,M2,ll = measurement_update_projected_gaussian_surrogate(\
M1,M2,Y[i],beta,s,dt,m,reg_rate,measurement,
return_surrogate=False,
surrogate=(MR[i],VR[i]))
if safe:
M1 = np.clip(M1,-100,100)
M2 = np.clip(M2,-100,100)
# Store moments
allM1[i] = M1[:,0].copy()
allM2[i] = M2.copy()
allLR[i] = min(beta.T.dot(M1)+s,maxlogr)
allLV[i] = beta.T.dot(M2).dot(beta)
nll -= ll*llrescale
if safe:
# Heuristic: detect numerical failure and exit early
failed = np.any(M1)<-1e5
failed|= logx>100*maxlogr
failed|= nll<-1e10
if failed:
nll = np.inf
break
if progress and current_milli_time()-last_shown>500:
sys.stdout.write('\r%02.02f%%'%(i*100/T))
sys.stdout.flush()
last_shown = current_milli_time()
if progress:
sys.stdout.write('\r100.00%')
sys.stdout.flush()
sys.stdout.write('\n')
if return_surrogates:
return allLR,allLV,allM1,allM2,nll,allmr,allvr
else:
return allLR,allLV,allM1,allM2,nll
def integrate_moments(stim,A,beta,C,
dt = 1.0,
oversample = 10,
maxrate = 500,
maxvcorr = 2000,
method = "moment_closure",
int_method = "euler",
reg_cov = 1e-6,
safemode = "assert"):
'''
Integrate moment equations for autoregressive PPGLM
Parameters
----------
stim : zero-lage effective input (filtered stimulus plus mean offset)
A : forward operator for delay-line evolution
beta : basis history weights
C : projection of current state onto delay-like
Other Parameters
----------------
dt : time step
oversample : int
Integration steps per time step. Should be larger if using
Gaussian moment closure, which is stiff. Can be small if using
second-order approximations, which are less stiff.
maxrate :
maximum rate tolerated
maxvcorr:
Maximum variance correction ('convexity correction' in some literature)
tolerated during the moment closure.
method :
Moment-closure method. Can be "LNA" for mean-field with linear
noise approximation, "moment_closure" for Gaussian moment-closure
on the history process, or "second_order", which discards higher
moments of the rate which emerge when exponentiating.
int_method:
Integration method. Can be either "euler" for forward-Euler, or
"exponential", which integrates the locally-linearized system
forward using matrix exponentiation (slower).
reg_cov:
Small diagonal regularization for covariance matrix
safemode: string
Whether to check, repair NaNs and singular covariance.
if "assert", NaNs will trigger assertion error,
if "repair", NaNs will be silently corrected
Returns
-------
allLR : single-time marginal mean of log-rate
allLV : single-time marginal variance of log-rate
allM1 : low-dimensional approximation of history process, mean
allM2 : low-dimensional approximation of history process, covariance
'''
maxlogr = np.log(maxrate)
maxratemc = maxvcorr*maxrate
dtfine = dt/oversample
T = len(stim)
K = beta.size
# Precompute constants
Cb = C.dot(beta.T)
CC = C.dot(C.T)
Adt = A*dtfine
F1 = scipy.linalg.expm(Adt)
# Buid moment integrator functions
mean_update, cov_update = get_moment_integrator(int_method,Adt)
# Get update function (computes expected rate from moments)
update = get_update_function(method,Cb,Adt,maxvcorr)
# Initial condition for moments
M1 = np.zeros((K,1))
M2 = np.eye(K)*1e-6
# Store moments
allM1 = np.zeros((T,K))
allM2 = np.zeros((T,K,K))
allLR = np.zeros((T))
allLV = np.zeros((T))
# Integrate
for i,s in enumerate(stim):
for j in range(oversample):
if safemode=="assert":
assert(np.all(np.isfinite(M1)) and np.all(np.isfinite(M2)))
elif safemode=="repair":
M1[~np.isfinite(M1)] = 0
M2[~np.isfinite(M2)] = 0
# Marginal variance and mean of the predicted log-rate
# (This is a linear projection of the process history)
logv = beta.T.dot(M2).dot(beta)
logx = min(beta.T.dot(M1)+s,maxlogr)
# Uncorrected rate estimate
R0 = sexp(logx)*dtfine
# Covariance corrected rate estimate Rm
# And system Jacobian J
Rm,J = update(logx,logv,R0,M1,M2)
# Moment updates with new mean and variance
# proportional to corrected rate Rm
M2 = cov_update(M2,J) + CC*Rm
M1 = mean_update(M1) + C *Rm
if safemode=="repair":
M2 = repair_covariance(M2,reg_cov)
allM1[i] = M1[:,0]
allM2[i] = M2
allLR[i] = logx
allLV[i] = beta.T.dot(M2).dot(beta)
return allLR,allLV,allM1,allM2
def update(logx,logv,R0,M1,M2):
Rm = R0 * min(sexp(0.5*logv),maxvcorr)
J = Cb*Rm+Adt
return Rm,J
def measurement_update_projected_gaussian_surrogate(m1,m2,y,b,s,dt,pm,pt,
univariate_method = univariate_lgp_update_moment,
return_surrogate=False,
surrogate=None,
eps=1e-12,
safe=False):
'''
Please see `measurement_update_projected_gaussian`.
This function is the same, except that it can return "surrogate"
gaussian approximations of the measurement updates, which can then
be used in subsequent filtering of the same data for much more rapid
measurement updates.
If the parameters do not change too much, these surrogate updates
will remain approximately correct. This provides a path toward an
approximate EM-style algorithm for optimizing the likelihood using
moment-closures as an additional likelihood penalty (regularizer) for
slow dynamics.
Parameters
----------
m1 : first moment (mean of multivariate gaussian)
m2 : covariance of multivariate gaussian
y : spiking observatoin (scalar)
b : projection from multivariate gaussian onto univariate log-rate distribution
s : stimulus or bias term for log-rate
dt : time-step scaling of rate (can also be used as generic gain parameter)
pm : regularizing prior mean log-rate
pt : regularizing prior precision
univariate_method : function, one of
univariate_lgp_update_moment
univariate_lgp_update_variational
univariate_lgp_update_laplace
'''
if surrogate is None:
return measurement_update_projected_gaussian(m1,m2,y,b,s,dt,pm,pt,
univariate_method = univariate_method,
eps=eps,
safe=safe)
(mr, vr) = surrogate
# Validate arguments
if safe:
m2 = assertfinitereal(assertsquare(m2))
m1 = assertfinitereal(assertcolumn(m1))
b = assertfinitereal(assertcolumn(b))
# Precompute constants
m2 = 0.5*(m2+m2.T)
# Gaussian state prior on log-rate
m2b = m2.dot(b)
if safe:
v = scalar(b.T.dot(m2b))
m = scalar(b.T.dot(m1))
v = max(eps,(b.T.dot(m2b))[0,0])
m = (b.T.dot(m1))[0,0]
t = 1/v
# Regularizing Gaussian prior on log-rate
tq = pt + t
mq = (m*t+pm*pt)/tq
vq = 1/tq
tr = 1/vr
tp = tq + tr
mp = (mr*tr + mq*tq)/tp
vp = 1/tp
if safe:
mp = scalar(assertfinitereal(mp))
vp = scalar(assertfinitereal(vp))
# Futher optimized
K = m2b/(vr+v)
m2p = m2 - K.dot(m2b.T)
m1p = m1 + K*(mr-m)
# Also compute log-likelihood from univariate
logr = mp+s
logPyx = y*logr-sexp(logr)
ll = logPyx + 0.5*slog(vp/v) - 0.5*(mp-m)**2/v
return m1p, m2p, scalar(ll)
def hessian(mu):
rate = scale*sexp(mu)
if not np.isfinite(mu) or not np.isfinite(rate):
return np.inf
return rate+1/v
def gradient(mu):
rate = scale*sexp(mu)
if not np.isfinite(mu) or not np.isfinite(rate):
return np.inf
return -y+rate+(mu-m)/v
def objective(mu):
rate = scale*sexp(mu)
if not np.isfinite(mu) or not np.isfinite(rate):
return np.inf
return -y*mu+rate+0.5*(mu-m)**2/v