def eval_model(self, params, ret="model"):
"""
Evaluate the model for the requested set of parameters.
Parameters
----------
params: 1D float ndarray
The set of model fitting parameters.
ret: String
Flag to indicate what to return. Valid options:
- 'model' Return the evaluated model.
- 'chisq' Return chi-square.
- 'both' Return a list with the model and chisq.
"""
if self.wlike:
model = self.func(params[0:-3], *self.args)
else:
model = self.func(params, *self.args)
# Reject proposed iteration if any model value is infinite:
if np.any(model == np.inf):
chisq = np.inf
else:
# Calculate prioroff = params-prior:
prioroff = params[self.iprior] - self.prior
# Calculate chisq:
if self.wlike:
chisq = dwt.wlikelihood(params[-3:], model, self.data, prioroff,
self.priorlow, self.priorup)
else:
chisq = cs.chisq(model, self.data, self.uncert,
prioroff, self.priorlow, self.priorup)
# Return evaluated model if requested:
if ret == "both":
return [model, chisq]
elif ret == "chisq":
return chisq
else: # ret == "model"
return model
def eval_model(self, params, ret="model"):
"""
Evaluate the model for the requested set of parameters.
Parameters
----------
params: 1D float ndarray
The set of model fitting parameters.
ret: String
Flag to indicate what to return. Valid options:
- 'model' Return the evaluated model.
- 'chisq' Return chi-square.
- 'both' Return a list with the model and chisq.
"""
if self.wlike:
model = self.func(params[0:-3], *self.args)
else:
model = self.func(params, *self.args)
# Reject proposed iteration if any model value is infinite:
if np.any(model == np.inf):
chisq = np.inf
else:
# Calculate prioroff = params-prior:
prioroff = params[self.iprior] - self.prior
# Calculate chisq:
if self.wlike:
chisq = dwt.wlikelihood(params[-3:], model, self.data,
prioroff, self.priorlow, self.priorup)
else:
chisq = cs.chisq(model, self.data, self.uncert, prioroff,
self.priorlow, self.priorup)
# Return evaluated model if requested:
if ret == "both":
return [model, chisq]
elif ret == "chisq":
return chisq
else: # ret == "model"
return model
def mcmc(data, uncert=None, func=None, indparams=[],
parnames=None, params=None, pmin=None, pmax=None,
stepsize=None, prior=None, priorlow=None, priorup=None,
numit=10, nchains=10, walk='demc', wlike=False,
leastsq=True, chisqscale=False, grtest=True, grexit=False,
burnin=0, thinning=1, fgamma=1.0, fepsilon=0.0,
plots=False, savefile=None, savemodel=None, comm=None,
resume=False, log=None, rms=False, hsize=1,
modelper=0, p_est=np.array([0.68269, 0.95450, 0.99730])):
"""
This beautiful piece of code runs a Markov-chain Monte Carlo algoritm.
Parameters
----------
data: 1D ndarray
Dependent data fitted by func.
uncert: 1D ndarray
Uncertainty of data.
func: callable or string-iterable
The callable function that models data as:
model = func(params, *indparams)
Or an iterable (list, tuple, or ndarray) of 3 strings:
(funcname, modulename, path)
that specify the function name, function module, and module path.
If the module is already in the python-path scope, path can be omitted.
indparams: tuple
Additional arguments required by func.
params: 1D or 2D ndarray
Set of initial fitting parameters for func. If 2D, of shape
(nparams, nchains), it is assumed that it is one set for each chain.
pmin: 1D ndarray
Lower boundaries of the posteriors.
pmax: 1D ndarray
Upper boundaries of the posteriors.
stepsize: 1D ndarray
Proposal jump scale. If a values is 0, keep the parameter fixed.
Negative values indicate a shared parameter (See Note 1).
prior: 1D ndarray
Parameter prior distribution means (See Note 2).
priorlow: 1D ndarray
Lower prior uncertainty values (See Note 2).
priorup: 1D ndarray
Upper prior uncertainty values (See Note 2).
numit: Scalar
Total number of iterations.
nchains: Scalar
Number of simultaneous chains to run.
walk: String
Random walk algorithm:
- 'mrw': Metropolis random walk with Gaussian proposals.
- 'demc': Differential Evolution Markov chain.
- 'snooker': DEMC with modifications as per ter Braak & Vrugt 2008
wlike: Boolean
If True, calculate the likelihood in a wavelet-base. This requires
three additional parameters (See Note 3).
leastsq: Boolean
Perform a least-square minimization before the MCMC run.
chisqscale: Boolean
Scale the data uncertainties such that the reduced chi-squared = 1.
grtest: Boolean
Run Gelman & Rubin test.
grexit: Boolean
Exit the MCMC loop if the MCMC satisfies GR two consecutive times.
burnin: Scalar
Burned-in (discarded) number of iterations at the beginning
of the chains.
thinning: Integer
Thinning factor of the chains (use every thinning-th iteration) used
in the GR test and plots.
fgamma: Float
Proposals jump scale factor for DEMC's gamma.
The code computes: gamma = fgamma * 2.4 / sqrt(2*Nfree)
fepsilon: Float
Jump scale factor for DEMC's support distribution.
The code computes: e = fepsilon * Normal(0, stepsize)
plots: Boolean
If True plot parameter traces, pairwise-posteriors, and posterior
histograms.
savefile: String
If not None, filename to store allparams (with np.save).
savemodel: String
If not None, filename to store the values of the evaluated function
(with np.save).
comm: MPI Communicator
A communicator object to transfer data through MPI.
resume: Boolean
If True resume a previous run.
log: FILE pointer
File object to write log into.
hsize: Int
Initial samples for snooker walk.
modelper: Int
Sets how to split `savemodel` into subfiles.
If 0, does not split. If >0, splits even `modelper` iterations.
E.g., if nchains=10 and modelper=5, splits every 50 model evaluations.
p_est: array
Credible regions to estimate uncertainty.
Returns
-------
allparams: 2D ndarray
An array of shape (nfree, numit-nchains*burnin) with the MCMC
posterior distribution of the fitting parameters.
bestp: 1D ndarray
Array of the best fitting parameters.
Notes
-----
1.- To set one parameter equal to another, set its stepsize to the
negative index in params (Starting the count from 1); e.g.: to set
the second parameter equal to the first one, do: stepsize[1] = -1.
2.- If any of the fitting parameters has a prior estimate, e.g.,
param[i] = p0 +up/-low,
with up and low the 1sigma uncertainties. This information can be
considered in the MCMC run by setting:
prior[i] = p0
priorup[i] = up
priorlow[i] = low
All three: prior, priorup, and priorlow must be set and, furthermore,
priorup and priorlow must be > 0 to be considered as prior.
3.- FINDME WAVELET LIKELIHOOD
Examples
--------
>>> # See examples: https://github.com/pcubillos/MCcubed/tree/master/examples
Uncredited developers
---------------------
Kevin Stevenson (UCF)
"""
mu.msg(1, "{:s}\n Multi-Core Markov-Chain Monte Carlo (MC3).\n"
" Version {:d}.{:d}.{:d}.\n"
" Copyright (c) 2015-{:d} Patricio Cubillos and collaborators.\n"
" MC3 is open-source software under the MIT license "
"(see LICENSE).\n{:s}\n\n".
format(mu.sep, ver.MC3_VER, ver.MC3_MIN, ver.MC3_REV,
date.today().year, mu.sep), log)
# Import the model function:
if type(func) in [list, tuple, np.ndarray]:
if func[0] != 'hack':
if len(func) == 3:
sys.path.append(func[2])
exec('from %s import %s as func'%(func[1], func[0]))
elif not callable(func):
mu.error("'func' must be either, a callable, or an iterable (list, "
"tuple, or ndarray) of strings with the model function, file, "
"and path names.", log)
if np.ndim(params) == 1: # Force it to be 2D (one for each chain)
params = np.atleast_2d(params)
nparams = len(params[0]) # Number of model params
ndata = len(data) # Number of data values
# Set default uncertainties:
if uncert is None:
uncert = np.ones(ndata)
# Set default boundaries:
if pmin is None:
pmin = np.zeros(nparams) - np.inf
if pmax is None:
pmax = np.zeros(nparams) + np.inf
# Set default stepsize:
if stepsize is None:
stepsize = 0.1 * np.abs(params[0])
# Set prior parameter indices:
if (prior is None) or (priorup is None) or (priorlow is None):
prior = priorup = priorlow = np.zeros(nparams) # Zero arrays
iprior = np.where(priorlow != 0)[0]
ilog = np.where(priorlow < 0)[0]
# Check that initial values lie within the boundaries:
if np.any(np.asarray(params) < pmin):
mu.error("One or more of the initial-guess values:\n{:s}\n are smaller "
"than their lower boundaries:\n{:s}".format(str(params), str(pmin)), log)
if np.any(np.asarray(params) > pmax):
mu.error("One or more of the initial-guess values:\n{:s}\n are greater "
"than their higher boundaries:\n{:s}".format(str(params), str(pmax)), log)
nfree = np.sum(stepsize > 0) # Number of free parameters
chainsize = int(np.ceil(numit/nchains)) # Number of iterations per chain
ifree = np.where(stepsize > 0)[0] # Free parameter indices
ishare = np.where(stepsize < 0)[0] # Shared parameter indices
# Number of model parameters (excluding wavelet parameters):
if wlike:
mpars = nparams - 3
else:
mpars = nparams
if chainsize < burnin:
mu.error("The number of burned-in samples ({:d}) is greater than "
"the number of iterations per chain ({:d}).".
format(burnin, chainsize), log)
# Ensure that hsize is > nchains
if walk=='snooker' and hsize < nchains:
hsize = nchains + 1
# Intermediate steps to run GR test and print progress report:
intsteps = chainsize / 10
# Allocate arrays with variables:
numaccept = np.zeros(nchains) # Number of accepted proposal jumps
outbounds = np.zeros((nchains, nfree), np.int) # Out of bounds proposals
allparams = np.zeros((nchains, nfree, chainsize)) # Parameter's record
if savemodel is not None:
# Number of files to be saved out
if modelper > 0:
nsaves = int(np.ceil(chainsize / modelper)) # Number of files to save
nsaved = 0 # Number of model files saved out so far
neval = 0 # Number of models evaluated for current savemodel file
allmodel = np.zeros((nchains, ndata, modelper)) # Fit model
else:
allmodel = np.zeros((nchains, ndata, chainsize)) # Fit model
if resume:
oldparams = np.load(savefile)
nold = np.shape(oldparams)[2] # Number of old-run iterations
allparams = np.dstack((oldparams, allparams))
if savemodel is not None:
allmodel = np.dstack((np.load(savemodel), allmodel))
# Set params to the last-iteration state of the previous run:
params = np.repeat(params, nchains, 0)
params[:,ifree] = oldparams[:,:,-1]
# Snooker things - not currently implemented into the savefile
'''Zold = oldparams["Z"]
Zlenold = Zold.shape()[0]
Zchainold = oldparams["Zchain"]
Zlen = Zlen + Zlenold'''
else:
nold = 0
# Set MPI flag:
mpi = comm is not None
if mpi:
from mpi4py import MPI
# Send sizes info to other processes:
if walk=="snooker":
array1 = np.asarray([mpars, chainsize+hsize], np.int)
else:
array1 = np.asarray([mpars, chainsize], np.int)
mu.comm_bcast(comm, array1, MPI.INT)
# DEMC parameters:
gamma = fgamma * 2.4 / np.sqrt(2*nfree)
# Least-squares minimization:
if leastsq and walk!='unif':
fitargs = (params[0], func, data, uncert, indparams, stepsize, pmin, pmax,
prior, priorlow, priorup)
fitchisq, dummy = mf.modelfit(params[0,ifree], args=fitargs)
fitbestp = np.copy(params[0, ifree])
mu.msg(1, "Least-squares best-fitting parameters: \n{:s}\n\n".
format(str(fitbestp)), log)
# Replicate to make one set for each chain: (nchains, nparams):
if np.shape(params)[0] != nchains:
params = np.repeat(params, nchains, 0)
# Start chains with an initial jump:
for p in ifree:
# For each free param, use a normal distribution:
params[1:, p] = np.random.normal(params[0, p], stepsize[p], nchains-1)
# Stay within pmin and pmax boundaries:
params[np.where(params[:, p] < pmin[p]), p] = pmin[p]
params[np.where(params[:, p] > pmax[p]), p] = pmax[p]
# Update shared parameters:
for s in ishare:
params[:, s] = params[:, -int(stepsize[s])-1]
# Calculate chi-squared for model using current params:
models = np.zeros((nchains, ndata))
if mpi:
# Scatter (send) parameters to func:
mu.comm_scatter(comm, params[:,0:mpars].flatten(), MPI.DOUBLE)
# Gather (receive) evaluated models:
mpimodels = np.zeros(nchains*ndata, np.double)
mu.comm_gather(comm, mpimodels)
# Store them in models variable:
models = np.reshape(mpimodels, (nchains, ndata))
else:
fargs = [params[:, 0:mpars]] + indparams # List of function's arguments
models[:] = func(*fargs)
currchisq = np.zeros(nchains)
if walk!='unif':
# Calculate chi-squared for each chain:
c2 = np.zeros(nchains) # No-Jeffrey's chisq
for c in np.arange(nchains):
if wlike: # Wavelet-based likelihood (chi-squared, actually)
currchisq[c], c2[c] = dwt.wlikelihood(params[c, mpars:], models[c]-data,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
currchisq[c], c2[c] = cs.chisq(models[c], data, uncert,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
# Scale data-uncertainties such that reduced chisq = 1:
if chisqscale:
chifactor = np.sqrt(np.amin(currchisq)/(ndata-nfree))
uncert *= chifactor
# Re-calculate chisq with the new uncertainties:
for c in np.arange(nchains):
if wlike: # Wavelet-based likelihood (chi-squared, actually)
currchisq[c], c2[c] = dwt.wlikelihood(params[c,mpars:], models[c]-data,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
currchisq[c], c2[c] = cs.chisq(models[c], data, uncert,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
if leastsq:
fitchisq = currchisq[0]
# Snooker stuff - ter Braak & Vrugt 2008
if walk == "snooker":
# Initial number of samples
M0 = hsize * nchains
Zsize = hsize
# Number of Z samples per chain
nZchain = int(np.ceil(numit / nchains / thinning))
# Number of iterations per chain
niter = nZchain * thinning
# Total number of Z samples
Zlen = M0 + nZchain * nchains
# Burned samples
Zburn = int(burnin / thinning)
# Z array
Z = np.zeros((hsize+nZchain, nchains, nparams), dtype=np.float64)
# Chi-squared for Z
Zchisq = np.zeros((hsize+nZchain, nchains), dtype=np.float64)
Zc2 = np.zeros((hsize+nZchain, nchains), dtype=np.float64)
# Z models
Zmodels = np.zeros((hsize+nZchain, nchains, ndata), np.double)
# Populate Z array
Z[:, :, 0:mpars] = params[:, 0:mpars]
# Populate M0 samples in Z
for i in range(nfree):
ind = ifree[i]
Z[:hsize, :, ind] = np.random.uniform(pmin[ind], pmax[ind],
(hsize, nchains) )
# Evaluate models for initial samples of Z if using MPI
if mpi:
for i in range(hsize):
# Send params to func
mu.comm_scatter(comm, Z[i,:,0:mpars].flatten(), MPI.DOUBLE)
# Get evaluated models
mpiZmodels = np.zeros(nchains*ndata, np.double)
mu.comm_gather(comm, mpiZmodels)
# Store in `Zmodels`
Zmodels[i] = np.reshape(mpiZmodels, (nchains, ndata))
# Evaluate chi squared, and model if not using MPI
for i in range(hsize):
if not mpi:
fargs = [Z[i,:,:mpars]] + indparams
Zmodels[i] = func(*fargs)
for c in range(nchains):
# Chi squared
if wlike:
Zchisq[i,c], Zc2[i,c] = dwt.wlikelihood(Z[i,c,mpars:],
Zmodels[i,c] - data,
(Z[i,c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
Zchisq[i,c], Zc2[i,c] = cs.chisq(Zmodels[i,c], data, uncert,
(Z[i,c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
# Current best Z
Zibest = np.unravel_index(np.argmin(Zchisq[:hsize]),
Zchisq[:hsize].shape)
Zbestchisq = Zchisq[Zibest]
Zbestp = np.copy(Z[Zibest])
Zbestmodel = np.copy(Zmodels[:hsize][Zibest])
# Get lowest chi-square and best fitting parameters:
bestchisq = np.amin(c2)
bestp = np.copy(params[np.argmin(c2)])
bestmodel = np.copy(models[np.argmin(c2)])
if walk == "snooker":
if Zbestchisq < bestchisq:
bestchisq = Zbestchisq
bestp = Zbestp
bestmodel = Zbestmodel
if savemodel is not None:
allmodel[:,:,0] = models
# Set up the random walks:
if walk == "mrw":
# Generate proposal jumps from Normal Distribution for MRW:
mstep = np.random.normal(0, stepsize[ifree], (chainsize, nchains, nfree))
elif walk == "demc":
# Support random distribution:
support = np.random.normal(0, stepsize[ifree], (chainsize, nchains, nfree))
# Generate indices for the chains such that R1[c] != c:
r1 = np.random.randint(0, nchains-1, (nchains, chainsize))
for c in np.arange(nchains):
r1[c][np.where(r1[c]==c)] = nchains-1
# Make sure R2[c] != c and R2 != R1:
r2 = np.zeros((nchains, chainsize), int)
for c in np.arange(nchains):
r2[c] = (c + np.random.randint(1, nchains-1, chainsize))%nchains
r2[c][np.where(r2[c]==r1[c])] = (c-1)%nchains
elif walk == "snooker":
# Support random distribution:
support = np.random.normal(0, stepsize[ifree], (chainsize, nchains, nfree))
# Uniform random distribution for the Metropolis acceptance rule:
unif = np.random.uniform(0, 1, (chainsize, nchains))
# Uniform distribution to do full DEMC jump:
ugamma = np.random.uniform(0, 1, (chainsize, nchains))
gamma1 = np.tile(gamma, (nchains,1))
# Use Uniform distribution to determine snooker jumps
if walk == "snooker":
sjump = ugamma < 0.1
elif walk == 'unif':
unif = np.zeros((chainsize, nchains))
# Proposed iteration parameters and chi-square (per chain):
nextp = np.copy(params) # Proposed parameters
nextchisq = np.zeros(nchains) # Chi square of nextp
# Gelman-Rubin exit flag:
grflag = False
mrfactor = np.zeros(nchains)
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Start loop:
mu.msg(1, "Start MCMC chains ({:s})".format(time.ctime()), log)
for i in np.arange(chainsize):
# Proposal jump:
if walk == "mrw":
jump = mstep[i]
elif walk == "demc":
gamma1[ugamma[i]>=0.1] = gamma
gamma1[ugamma[i]< 0.1] = 0.98
jump = (gamma1 * (params[r1[:,i]]-params[r2[:,i]])[:,ifree] +
fepsilon * support[i])
elif walk == "snooker":
# Random without replacement
i1 = np.random.randint(0, (Zsize-1)*nchains, nchains)
i2 = np.random.randint(0, (Zsize-1)*nchains, nchains)
for j in range(nchains):
while i1[j] == i2[j]:
i2[j] = np.random.randint(0, (Zsize-1)*nchains)
iz1, ic1 = np.unravel_index(i1, (Zsize, nchains))
iz2, ic2 = np.unravel_index(i2, (Zsize, nchains))
# Select another chain in state z, for each chain
iz = np.random.randint(0, Zsize-1, nchains)
ic = np.random.randint(0, nchains, nchains)
z = Z[iz, ic]
# Jumps for chains
jump = np.zeros((nchains, nfree))
noproj = np.all(z == params, axis=1)
# Snooker jumps, do not project:
if np.sum(noproj*sjump[i]) != 0:
jump[noproj*sjump[i]] = np.random.uniform(1.2, 2.2,
(np.sum(noproj*sjump[i]), nfree)) * \
(Z[iz2,ic2]-Z[iz1,ic1])[noproj*sjump[i]][:,ifree]
# Snooker jumps, project:
if np.sum(~noproj*sjump[i]) != 0:
dz = (params - z)[:,ifree][~noproj*sjump[i]]
zp1 = np.sum(Z[iz1,ic1][:,ifree][~noproj*sjump[i]] * dz, axis=1)
zp2 = np.sum(Z[iz2,ic2][:,ifree][~noproj*sjump[i]] * dz, axis=1)
jump[~noproj*sjump[i]] = np.random.uniform(1.2, 2.2,
(np.sum(~noproj*sjump[i]), nfree)) * \
(zp1 - zp2).reshape(zp1.shape[0],1) / \
np.sum(dz**2, axis=1).reshape(zp1.shape[0],1) * \
dz
# Standard DEMC jumps
jump[~sjump[i]] = gamma * (Z[iz1,ic1]- Z[iz2,ic2])[~sjump[i]][:,ifree] \
+ fepsilon * support[i][~sjump[i]]
# Propose next point:
if walk == 'unif':
nextp[:,ifree] = np.random.uniform(pmin[ifree], pmax[ifree], (nchains, nfree))
else:
nextp[:,ifree] = params[:,ifree] + jump
# Check it's within boundaries:
outpars = np.asarray(((nextp < pmin) | (nextp > pmax))[:,ifree])
outflag = np.any(outpars, axis=1)
outbounds += ((nextp < pmin) | (nextp > pmax))[:,ifree]
for p in ifree:
nextp[np.where(nextp[:, p] < pmin[p]), p] = pmin[p]
nextp[np.where(nextp[:, p] > pmax[p]), p] = pmax[p]
# Update shared parameters:
for s in ishare:
nextp[:, s] = nextp[:, -int(stepsize[s])-1]
# Evaluate the models for the proposed parameters:
if mpi:
mu.comm_scatter(comm, nextp[:,0:mpars].flatten(), MPI.DOUBLE)
mu.comm_gather(comm, mpimodels)
models = np.reshape(mpimodels, (nchains, ndata))
else:
c = np.where(~outflag)[0]
if len(c) > 0:
fargs = [nextp[c, 0:mpars]] + indparams # List of function's arguments
models[c] = func(*fargs)
else:
continue
if walk!='unif':
# Calculate chisq:
for c in np.where(~outflag)[0]:
if wlike: # Wavelet-based likelihood (chi-squared, actually)
nextchisq[c], c2[c] = dwt.wlikelihood(nextp[c,mpars:], models[c]-data,
(nextp[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
nextchisq[c], c2[c] = cs.chisq(models[c], data, uncert,
(nextp[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
nextchisq[outflag] = np.inf # Reject out of bounds jumps
# Metropolis ratio of accepted projected snooker jumps
mrfactor[:] = 1.0
if walk == "snooker" and np.any(sjump[i] * ~noproj * ~outflag):
asj = sjump[i] * ~noproj * ~outflag
mrfactor[asj] = (np.linalg.norm((nextp -z)[:,ifree][asj]) /
np.linalg.norm((params-z)[:,ifree][asj]) )**(nfree-1)
# Determine accepted jumps
accept = np.ones(nchains)
if walk != 'unif':
accept = np.exp(0.5 * (currchisq - nextchisq)) * mrfactor
accepted = accept >= unif[i]
if i >= burnin:
numaccept += accepted
# Update params and chi square:
params [accepted] = nextp [accepted]
currchisq[accepted] = nextchisq[accepted]
if walk!='unif':
# Check lowest chi-square:
if np.amin(c2) < bestchisq:
bestp = np.copy(params[np.argmin(c2)])
bestmodel = np.copy(models[np.argmin(c2)])
bestchisq = np.amin(c2)
else:
bestp = np.copy(params[0])
bestchisq = 0.
# Store current iteration values:
allparams[:,:,i+nold] = params[:, ifree]
if savemodel is not None:
if modelper > 0:
models[~accepted] = allmodel[~accepted,:,neval-1]
if neval == modelper:
neval = 0 # Reset counter
# Save this set
np.save(savemodel.replace('.npy',
str(nsaved).zfill(len(str(nsaves)))+'.npy'),
allmodel)
nsaved += 1
# Reset array
allmodel = np.zeros((nchains, ndata, modelper))
# Store current iteration
allmodel[:,:,neval] = models
neval += 1
else:
models[~accepted] = allmodel[~accepted,:,i+nold-1]
allmodel[:,:,i+nold] = models
# Update Z
if walk == "snooker":
if i%thinning == 0:
Z[hsize + i//thinning][:, ifree] = np.copy(params[:, ifree])
Zchisq[hsize + i//thinning] = np.copy(currchisq)
if savemodel:
Zmodels[hsize + i//thinning] = np.copy(models)
Zsize += 1
# Print intermediate info:
if ((i+1) % intsteps == 0) and (i > 0):
mu.progressbar((i+1.0)/chainsize, log)
mu.msg(1, "Out-of-bound Trials:\n {}".
format(np.sum(outbounds, axis=0)), log)
mu.msg(1, "Best Parameters: (chisq={:.4f})\n{:s}".
format(bestchisq, str(bestp)), log)
# Gelman-Rubin statistic:
if grtest and (i+nold) > burnin:
psrf = gr.convergetest(allparams[:, :, burnin:i+nold+1:thinning])
mu.msg(1, "Gelman-Rubin statistic for free parameters:\n{:s}".
format(str(psrf)), log)
if np.all(psrf < 1.01):
mu.msg(1, "All parameters have converged to within 1% of unity.", log)
# End the MCMC if all parameters satisfy GR two consecutive times:
if grexit and grflag:
# Let the workers know that the MCMC is stopping:
if mpi:
endflag = np.tile(np.inf, nchains*mpars)
mu.comm_scatter(comm, endflag, MPI.DOUBLE)
break
grflag = True
else:
grflag = False
# Save current results:
if savefile is not None:
np.save(savefile, allparams[:,:,0:i+nold])
if savemodel is not None and modelper < 1:
np.save(savemodel, allmodel[:,:,0:i+nold])
# Stack together the chains:
chainlen = nold + i+1
allstack = allparams[0, :, burnin:chainlen]
for c in np.arange(1, nchains):
allstack = np.hstack((allstack, allparams[c, :, burnin:chainlen]))
# And the models:
if savemodel is not None:
modelstack = allmodel[0,:,burnin:chainlen]
for c in np.arange(1, nchains):
modelstack = np.hstack((modelstack, allmodel[c, :, burnin:chainlen]))
# Print out Summary:
mu.msg(1, "\nFin, MCMC Summary:\n------------------", log)
if walk!='unif':
nsample = (i+1-burnin)*nchains
ntotal = np.size(allstack[0])
BIC = bestchisq + nfree*np.log(ndata)
redchisq = bestchisq/(ndata-nfree)
sdr = np.std(bestmodel-data)
fmtlen = len(str(ntotal))
mu.msg(1, "Burned in iterations per chain: {:{}d}".
format(burnin, fmtlen), log, 1)
mu.msg(1, "Number of iterations per chain: {:{}d}".
format(i+1, fmtlen), log, 1)
mu.msg(1, "MCMC sample size: {:{}d}".
format(nsample, fmtlen), log, 1)
mu.msg(resume, "Total MCMC sample size: {:{}d}".
format(ntotal, fmtlen), log, 1)
mu.msg(1, "Acceptance rate: {:.2f}%\n ".
format(np.sum(numaccept)*100.0/nsample), log, 1)
meanp = np.mean(allstack, axis=1) # Parameters mean
uncertp = np.std(allstack, axis=1) # Parameter standard deviation
mu.msg(1, "Best-fit params Uncertainties S/N Sample "
"Mean Note", log, 1)
for i in np.arange(nparams):
if i in ifree: # Free-fitting value
unc = "{:13.7e}". format(uncertp[np.where(ifree==i)][0])
snr = "{:8.2f}". format(np.abs(bestp[i])/uncertp[np.where(ifree==i)][0])
mean = "{: 14.7e}".format(meanp [np.where(ifree==i)][0])
note = ""
elif i in ishare: # Shared value
j = int(-stepsize[i]-1)
unc = "{:13.7e}". format(uncertp[np.where(ifree==j)][0])
snr = "{:8.2f}". format(np.abs(bestp[j])/uncertp[np.where(ifree==j)][0])
mean = "{: 14.7e}".format(meanp [np.where(ifree==j)][0])
note = "Shared"
else: # Fixed value
unc = "0.0"
snr = "---"
mean = "---"
note = "Fixed"
mu.msg(1, "{: 15.7e} {:>13s} {:>8s} {:>14s} {:s}".
format(bestp[i], unc, snr, mean, note), log, 1)
if leastsq and np.any(np.abs((bestp[ifree]-fitbestp)/fitbestp) > 1e-08):
np.set_printoptions(precision=8)
mu.warning("MCMC found a better fit than the minimizer:\n"
" MCMC best-fitting parameters: (chisq={:.8g})\n {:s}\n"
" Minimizer best-fitting parameters: (chisq={:.8g})\n"
" {:s}".format(bestchisq, str(bestp[ifree]),
fitchisq, str(fitbestp)), log)
fmtl = len("%.4f"%BIC) # Length of string formatting
mu.msg(1, " ", log)
if chisqscale:
mu.msg(1, "sqrt(reduced chi-squared) factor: {:{}.4f}".
format(chifactor, fmtl), log, 1)
mu.msg(1, "Best-parameter's chi-squared: {:{}.4f}".
format(bestchisq, fmtl), log, 1)
mu.msg(1, "Bayesian Information Criterion: {:{}.4f}".
format(BIC, fmtl), log, 1)
mu.msg(1, "Reduced chi-squared: {:{}.4f}".
format(redchisq, fmtl), log, 1)
mu.msg(1, "Standard deviation of residuals: {:.6g}".format(sdr), log, 1)
# Compute credible regions
if walk != "unif":
try:
speis, ess = cr.ess(allparams[:, :, burnin:])
p_unc = cr.sig(ess, p_est)
mu.msg(1, " ", log)
mu.msg(1, "SPEIS: "+str(speis) , log, 1)
mu.msg(1, "ESS : "+str(ess)+"\n", log, 1)
mu.msg(1, " ", log)
except:
mu.msg(1, " ", log)
mu.msg(1, "Unable to determine SPEIS.", log, 1)
mu.msg(1, " ", log)
p_unc = np.ones(len(p_est)) * np.nan
outpar = np.asarray(parnames)[stepsize>0]
for n in range(allstack.shape[0]):
pdf, xpdf, CRlo, CRhi = cr.credregion(allstack[n], p_est)
creg = [' U '.join(['({: 10.4e}, {: 10.4e})'.format(
CRlo[j][k], CRhi[j][k])
for k in range(len(CRlo[j]))])
for j in range(len(CRlo))]
mu.msg(1, outpar[n]+" credible regions:\n", log, 1)
for i in range(len(creg)):
mu.msg(1, '{:0<.2f}'.format(100*p_est[i]) + " +- " + \
'{:0<.4f}'.format(100*p_unc[i]) + " %: " + \
creg[i].replace(' U ', '\n' + ' '*18 + 'U '), log, 2)
# FINDME Hardcoded 18 is for output alignment. If you
# replace this, make sure the output still looks good
mu.msg(1, " ", log)
if rms:
rms, rmse, stderr, bs = ta.binrms(bestmodel-data)
if plots:
print("Plotting figures ...")
# Extract filename from savefile:
if savefile is not None:
if savefile.rfind(".") == -1:
fname = savefile[savefile.rfind("/")+1:] # Cut out file extention.
else:
fname = savefile[savefile.rfind("/")+1:savefile.rfind(".")]
else:
fname = "MCMC"
# Trace plot:
mp.trace(allstack, parname=parnames, thinning=thinning,
savefile=fname+"_trace.png",
sep=np.size(allstack[0])/nchains)
# Pairwise posteriors:
mp.pairwise(allstack, parname=parnames, thinning=thinning,
savefile=fname+"_pairwise.png")
# Histograms:
mp.histogram(allstack, parname=parnames, thinning=thinning,
savefile=fname+"_posterior.png")
# RMS vs bin size:
if rms:
mp.RMS(bs, rms, stderr, rmse, binstep=len(bs)/500+1,
savefile=fname+"_RMS.png")
if indparams != [] and np.size(indparams[0]) == ndata:
mp.modelfit(data, uncert, indparams[0], bestmodel,
savefile=fname+"_model.png")
# Save definitive results:
if savefile is not None:
np.save(savefile, allparams[:,:,:chainlen])
if savemodel is not None:
if modelper > 0 and nsaved < nsaves:
np.save(savemodel.replace('.npy',
str(nsaved).zfill(len(str(nsaves)))+'.npy'),
allmodel)
else:
np.save(savemodel, allmodel [:,:,:chainlen])
return allstack, bestp
def mcmc(data, uncert=None, func=None, indparams=[],
params=None, pmin=None, pmax=None, stepsize=None,
prior=None, priorlow=None, priorup=None, numit=10,
nchains=10, walk='demc', wlike=False, leastsq=True,
chisqscale=False, grtest=True, grexit=False, burnin=0,
thinning=1, plots=False, savefile=None, savemodel=None,
comm=None, resume=False, log=None, rms=False):
"""
This beautiful piece of code runs a Markov-chain Monte Carlo algoritm.
Parameters
----------
data: 1D ndarray
Dependent data fitted by func.
uncert: 1D ndarray
Uncertainty of data.
func: callable or string-iterable
The callable function that models data as:
model = func(params, *indparams)
Or an iterable (list, tuple, or ndarray) of 3 strings:
(funcname, modulename, path)
that specify the function name, function module, and module path.
If the module is already in the python-path scope, path can be omitted.
indparams: tuple
Additional arguments required by func.
params: 1D or 2D ndarray
Set of initial fitting parameters for func. If 2D, of shape
(nparams, nchains), it is assumed that it is one set for each chain.
pmin: 1D ndarray
Lower boundaries of the posteriors.
pmax: 1D ndarray
Upper boundaries of the posteriors.
stepsize: 1D ndarray
Proposal jump scale. If a values is 0, keep the parameter fixed.
Negative values indicate a shared parameter (See Note 1).
prior: 1D ndarray
Parameter prior distribution means (See Note 2).
priorlow: 1D ndarray
Lower prior uncertainty values (See Note 2).
priorup: 1D ndarray
Upper prior uncertainty values (See Note 2).
numit: Scalar
Total number of iterations.
nchains: Scalar
Number of simultaneous chains to run.
walk: String
Random walk algorithm:
- 'mrw': Metropolis random walk.
- 'demc': Differential Evolution Markov chain.
wlike: Boolean
If True, calculate the likelihood in a wavelet-base. This requires
three additional parameters (See Note 3).
leastsq: Boolean
Perform a least-square minimization before the MCMC run.
chisqscale: Boolean
Scale the data uncertainties such that the reduced chi-squared = 1.
grtest: Boolean
Run Gelman & Rubin test.
grexit: Boolean
Exit the MCMC loop if the MCMC satisfies GR two consecutive times.
burnin: Scalar
Burned-in (discarded) number of iterations at the beginning
of the chains.
thinning: Integer
Thinning factor of the chains (use every thinning-th iteration) used
in the GR test and plots.
plots: Boolean
If True plot parameter traces, pairwise-posteriors, and posterior
histograms.
savefile: String
If not None, filename to store allparams (with np.save).
savemodel: String
If not None, filename to store the values of the evaluated function
(with np.save).
comm: MPI Communicator
A communicator object to transfer data through MPI.
resume: Boolean
If True resume a previous run.
log: FILE pointer
File object to write log into.
Returns
-------
allparams: 2D ndarray
An array of shape (nfree, numit-nchains*burnin) with the MCMC
posterior distribution of the fitting parameters.
bestp: 1D ndarray
Array of the best fitting parameters.
Notes
-----
1.- To set one parameter equal to another, set its stepsize to the
negative index in params (Starting the count from 1); e.g.: to set
the second parameter equal to the first one, do: stepsize[1] = -1.
2.- If any of the fitting parameters has a prior estimate, e.g.,
param[i] = p0 +up/-low,
with up and low the 1sigma uncertainties. This information can be
considered in the MCMC run by setting:
prior[i] = p0
priorup[i] = up
priorlow[i] = low
All three: prior, priorup, and priorlow must be set and, furthermore,
priorup and priorlow must be > 0 to be considered as prior.
3.- FINDME WAVELET LIKELIHOOD
Examples
--------
>>> # See examples: https://github.com/pcubillos/MCcubed/tree/master/examples
Previous (uncredited) developers
--------------------------------
Kevin Stevenson UCF [email protected]
"""
mu.msg(1, "{:s}\n Multi-Core Markov-Chain Monte Carlo (MC3).\n"
" Version {:d}.{:d}.{:d}.\n"
" Copyright (c) 2015-2016 Patricio Cubillos and collaborators.\n"
" MC3 is open-source software under the MIT license "
"(see LICENSE).\n{:s}\n\n".
format(mu.sep, ver.MC3_VER, ver.MC3_MIN, ver.MC3_REV, mu.sep), log)
# Import the model function:
if type(func) in [list, tuple, np.ndarray]:
if func[0] != 'hack':
if len(func) == 3:
sys.path.append(func[2])
exec('from %s import %s as func'%(func[1], func[0]))
elif not callable(func):
mu.error("'func' must be either, a callable, or an iterable (list, "
"tuple, or ndarray) of strings with the model function, file, "
"and path names.", log)
if np.ndim(params) == 1: # Force it to be 2D (one for each chain)
params = np.atleast_2d(params)
nparams = len(params[0]) # Number of model params
ndata = len(data) # Number of data values
# Set default uncertainties:
if uncert is None:
uncert = np.ones(ndata)
# Set default boundaries:
if pmin is None:
pmin = np.zeros(nparams) - np.inf
if pmax is None:
pmax = np.zeros(nparams) + np.inf
# Set default stepsize:
if stepsize is None:
stepsize = 0.1 * np.abs(params[0])
# Set prior parameter indices:
if (prior is None) or (priorup is None) or (priorlow is None):
prior = priorup = priorlow = np.zeros(nparams) # Zero arrays
iprior = np.where(priorlow != 0)[0]
ilog = np.where(priorlow < 0)[0]
# Check that initial values lie within the boundaries:
if np.any(np.asarray(params) < pmin):
mu.error("One or more of the initial-guess values:\n{:s}\n are smaller "
"than their lower boundaries:\n{:s}".format(str(params), str(pmin)), log)
if np.any(np.asarray(params) > pmax):
mu.error("One or more of the initial-guess values:\n{:s}\n are greater "
"than their higher boundaries:\n{:s}".format(str(params), str(pmax)), log)
nfree = np.sum(stepsize > 0) # Number of free parameters
chainsize = int(np.ceil(numit/nchains)) # Number of iterations per chain
ifree = np.where(stepsize > 0)[0] # Free parameter indices
ishare = np.where(stepsize < 0)[0] # Shared parameter indices
# Number of model parameters (excluding wavelet parameters):
if wlike:
mpars = nparams - 3
else:
mpars = nparams
if chainsize < burnin:
mu.error("The number of burned-in samples ({:d}) is greater than "
"the number of iterations per chain ({:d}).".
format(burnin, chainsize), log)
# Intermediate steps to run GR test and print progress report:
intsteps = chainsize / 10
# Allocate arrays with variables:
numaccept = np.zeros(nchains) # Number of accepted proposal jumps
outbounds = np.zeros((nchains, nfree), np.int) # Out of bounds proposals
allparams = np.zeros((nchains, nfree, chainsize)) # Parameter's record
if savemodel is not None:
allmodel = np.zeros((nchains, ndata, chainsize)) # Fit model
if resume:
oldparams = np.load(savefile)
nold = np.shape(oldparams)[2] # Number of old-run iterations
allparams = np.dstack((oldparams, allparams))
if savemodel is not None:
allmodel = np.dstack((np.load(savemodel), allmodel))
# Set params to the last-iteration state of the previous run:
params = np.repeat(params, nchains, 0)
params[:,ifree] = oldparams[:,:,-1]
else:
nold = 0
# Set MPI flag:
mpi = comm is not None
if mpi:
from mpi4py import MPI
# Send sizes info to other processes:
array1 = np.asarray([mpars, chainsize], np.int)
mu.comm_bcast(comm, array1, MPI.INT)
# DEMC parameters:
gamma = 2.4 / np.sqrt(2*nfree)
gamma2 = 0.001 # Jump scale factor of support distribution
# Least-squares minimization:
if leastsq:
fitargs = (params[0], func, data, uncert, indparams, stepsize, pmin, pmax,
prior, priorlow, priorup)
fitchisq, dummy = mf.modelfit(params[0,ifree], args=fitargs)
fitbestp = np.copy(params[0, ifree])
mu.msg(1, "Least-squares best-fitting parameters: \n{:s}\n\n".
format(str(fitbestp)), log)
# Replicate to make one set for each chain: (nchains, nparams):
if np.shape(params)[0] != nchains:
params = np.repeat(params, nchains, 0)
# Start chains with an initial jump:
for p in ifree:
# For each free param, use a normal distribution:
params[1:, p] = np.random.normal(params[0, p], stepsize[p], nchains-1)
# Stay within pmin and pmax boundaries:
params[np.where(params[:, p] < pmin[p]), p] = pmin[p]
params[np.where(params[:, p] > pmax[p]), p] = pmax[p]
# Update shared parameters:
for s in ishare:
params[:, s] = params[:, -int(stepsize[s])-1]
# Calculate chi-squared for model using current params:
models = np.zeros((nchains, ndata))
if mpi:
# Scatter (send) parameters to func:
mu.comm_scatter(comm, params[:,0:mpars].flatten(), MPI.DOUBLE)
# Gather (receive) evaluated models:
mpimodels = np.zeros(nchains*ndata, np.double)
mu.comm_gather(comm, mpimodels)
# Store them in models variable:
models = np.reshape(mpimodels, (nchains, ndata))
else:
for c in np.arange(nchains):
fargs = [params[c, 0:mpars]] + indparams # List of function's arguments
models[c] = func(*fargs)
# Calculate chi-squared for each chain:
currchisq = np.zeros(nchains)
c2 = np.zeros(nchains) # No-Jeffrey's chisq
for c in np.arange(nchains):
if wlike: # Wavelet-based likelihood (chi-squared, actually)
currchisq[c], c2[c] = dwt.wlikelihood(params[c, mpars:], models[c]-data,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
currchisq[c], c2[c] = cs.chisq(models[c], data, uncert,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
# Scale data-uncertainties such that reduced chisq = 1:
if chisqscale:
chifactor = np.sqrt(np.amin(currchisq)/(ndata-nfree))
uncert *= chifactor
# Re-calculate chisq with the new uncertainties:
for c in np.arange(nchains):
if wlike: # Wavelet-based likelihood (chi-squared, actually)
currchisq[c], c2[c] = dwt.wlikelihood(params[c,mpars:], models[c]-data,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
currchisq[c], c2[c] = cs.chisq(models[c], data, uncert,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
if leastsq:
fitchisq = currchisq[0]
# Get lowest chi-square and best fitting parameters:
bestchisq = np.amin(c2)
bestp = np.copy(params[np.argmin(c2)])
bestmodel = np.copy(models[np.argmin(c2)])
if savemodel is not None:
allmodel[:,:,0] = models
# Set up the random walks:
if walk == "mrw":
# Generate proposal jumps from Normal Distribution for MRW:
mstep = np.random.normal(0, stepsize[ifree], (chainsize, nchains, nfree))
elif walk == "demc":
# Support random distribution:
support = np.random.normal(0, stepsize[ifree], (chainsize, nchains, nfree))
# Generate indices for the chains such r[c] != c:
r1 = np.random.randint(0, nchains-1, (nchains, chainsize))
r2 = np.random.randint(0, nchains-1, (nchains, chainsize))
for c in np.arange(nchains):
r1[c][np.where(r1[c]==c)] = nchains-1
r2[c][np.where(r2[c]==c)] = nchains-1
# Uniform random distribution for the Metropolis acceptance rule:
unif = np.random.uniform(0, 1, (chainsize, nchains))
# Proposed iteration parameters and chi-square (per chain):
nextp = np.copy(params) # Proposed parameters
nextchisq = np.zeros(nchains) # Chi square of nextp
# Gelman-Rubin exit flag:
grflag = False
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Start loop:
mu.msg(1, "Start MCMC chains ({:s})".format(time.ctime()), log)
for i in np.arange(chainsize):
# Proposal jump:
if walk == "mrw":
jump = mstep[i]
elif walk == "demc":
jump = (gamma * (params[r1[:,i]]-params[r2[:,i]])[:,ifree] +
gamma2 * support[i] )
# Propose next point:
nextp[:,ifree] = params[:,ifree] + jump
# Check it's within boundaries:
outpars = np.asarray(((nextp < pmin) | (nextp > pmax))[:,ifree])
outflag = np.any(outpars, axis=1)
outbounds += ((nextp < pmin) | (nextp > pmax))[:,ifree]
for p in ifree:
nextp[np.where(nextp[:, p] < pmin[p]), p] = pmin[p]
nextp[np.where(nextp[:, p] > pmax[p]), p] = pmax[p]
# Update shared parameters:
for s in ishare:
nextp[:, s] = nextp[:, -int(stepsize[s])-1]
# Evaluate the models for the proposed parameters:
if mpi:
mu.comm_scatter(comm, nextp[:,0:mpars].flatten(), MPI.DOUBLE)
mu.comm_gather(comm, mpimodels)
models = np.reshape(mpimodels, (nchains, ndata))
else:
for c in np.where(~outflag)[0]:
fargs = [nextp[c, 0:mpars]] + indparams # List of function's arguments
models[c] = func(*fargs)
# Calculate chisq:
for c in np.where(~outflag)[0]:
if wlike: # Wavelet-based likelihood (chi-squared, actually)
nextchisq[c], c2[c] = dwt.wlikelihood(nextp[c,mpars:], models[c]-data,
(nextp[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
nextchisq[c], c2[c] = cs.chisq(models[c], data, uncert,
(nextp[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
# Reject out-of-bound jumps:
nextchisq[np.where(outflag)] = np.inf
# Evaluate which steps are accepted and update values:
accept = np.exp(0.5 * (currchisq - nextchisq))
accepted = accept >= unif[i]
if i >= burnin:
numaccept += accepted
# Update params and chi square:
params [accepted] = nextp [accepted]
currchisq[accepted] = nextchisq[accepted]
# Check lowest chi-square:
if np.amin(c2) < bestchisq:
bestp = np.copy(params[np.argmin(c2)])
bestmodel = np.copy(models[np.argmin(c2)])
bestchisq = np.amin(c2)
# Store current iteration values:
allparams[:,:,i+nold] = params[:, ifree]
if savemodel is not None:
models[~accepted] = allmodel[~accepted,:,i+nold-1]
allmodel[:,:,i+nold] = models
# Print intermediate info:
if ((i+1) % intsteps == 0) and (i > 0):
mu.progressbar((i+1.0)/chainsize, log)
mu.msg(1, "Out-of-bound Trials:\n {:s}".
format(np.sum(outbounds, axis=0)), log)
mu.msg(1, "Best Parameters: (chisq={:.4f})\n{:s}".
format(bestchisq, str(bestp)), log)
# Gelman-Rubin statistic:
if grtest and (i+nold) > burnin:
psrf = gr.convergetest(allparams[:, :, burnin:i+nold+1:thinning])
mu.msg(1, "Gelman-Rubin statistic for free parameters:\n{:s}".
format(psrf), log)
if np.all(psrf < 1.01):
mu.msg(1, "All parameters have converged to within 1% of unity.", log)
# End the MCMC if all parameters satisfy GR two consecutive times:
if grexit and grflag:
# Let the workers know that the MCMC is stopping:
if mpi:
endflag = np.tile(np.inf, nchains*mpars)
mu.comm_scatter(comm, endflag, MPI.DOUBLE)
break
grflag = True
else:
grflag = False
# Save current results:
if savefile is not None:
np.save(savefile, allparams[:,:,0:i+nold])
if savemodel is not None:
np.save(savemodel, allmodel[:,:,0:i+nold])
# Stack together the chains:
chainlen = nold + i+1
allstack = allparams[0, :, burnin:chainlen]
for c in np.arange(1, nchains):
allstack = np.hstack((allstack, allparams[c, :, burnin:chainlen]))
# And the models:
if savemodel is not None:
modelstack = allmodel[0,:,burnin:chainlen]
for c in np.arange(1, nchains):
modelstack = np.hstack((modelstack, allmodel[c, :, burnin:chainlen]))
# Print out Summary:
mu.msg(1, "\nFin, MCMC Summary:\n------------------", log)
nsample = (i+1-burnin)*nchains
ntotal = np.size(allstack[0])
BIC = bestchisq + nfree*np.log(ndata)
redchisq = bestchisq/(ndata-nfree)
sdr = np.std(bestmodel-data)
fmtlen = len(str(ntotal))
mu.msg(1, "Burned in iterations per chain: {:{}d}".
format(burnin, fmtlen), log, 1)
mu.msg(1, "Number of iterations per chain: {:{}d}".
format(i+1, fmtlen), log, 1)
mu.msg(1, "MCMC sample size: {:{}d}".
format(nsample, fmtlen), log, 1)
mu.msg(resume, "Total MCMC sample size: {:{}d}".
format(ntotal, fmtlen), log, 1)
mu.msg(1, "Acceptance rate: {:.2f}%\n ".
format(np.sum(numaccept)*100.0/nsample), log, 1)
meanp = np.mean(allstack, axis=1) # Parameters mean
uncertp = np.std(allstack, axis=1) # Parameter standard deviation
mu.msg(1, "Best-fit params Uncertainties Signal/Noise Sample "
"Mean", log, 1)
for i in np.arange(nfree):
mu.msg(1, "{: 15.7e} {: 15.7e} {:12.2f} {: 15.7e}".
format(bestp[ifree][i], uncertp[i],
np.abs(bestp[ifree][i])/uncertp[i], meanp[i]), log, 1)
if leastsq and np.any(np.abs((bestp[ifree]-fitbestp)/fitbestp) > 1e-08):
np.set_printoptions(precision=8)
mu.warning("MCMC found a better fit than the minimizer:\n"
" MCMC best-fitting parameters: (chisq={:.8g})\n {:s}\n"
" Minimizer best-fitting parameters: (chisq={:.8g})\n"
" {:s}".format(bestchisq, str(bestp[ifree]),
fitchisq, str(fitbestp)), log)
fmtl = len("%.4f"%BIC) # Length of string formatting
mu.msg(1, " ", log)
if chisqscale:
mu.msg(1, "sqrt(reduced chi-squared) factor: {:{}.4f}".
format(chifactor, fmtl), log, 1)
mu.msg(1, "Best-parameter's chi-squared: {:{}.4f}".
format(bestchisq, fmtl), log, 1)
mu.msg(1, "Bayesian Information Criterion: {:{}.4f}".
format(BIC, fmtl), log, 1)
mu.msg(1, "Reduced chi-squared: {:{}.4f}".
format(redchisq, fmtl), log, 1)
mu.msg(1, "Standard deviation of residuals: {:.6g}\n".format(sdr), log, 1)
if rms:
rms, rmse, stderr, bs = ta.binrms(bestmodel-data)
if plots:
print("Plotting figures ...")
# Extract filename from savefile:
if savefile is not None:
if savefile.rfind(".") == -1:
fname = savefile[savefile.rfind("/")+1:] # Cut out file extention.
else:
fname = savefile[savefile.rfind("/")+1:savefile.rfind(".")]
else:
fname = "MCMC"
# Trace plot:
mp.trace(allstack, thinning=thinning, savefile=fname+"_trace.png",
sep=np.size(allstack[0])/nchains)
# Pairwise posteriors:
mp.pairwise(allstack, thinning=thinning, savefile=fname+"_pairwise.png")
# Histograms:
mp.histogram(allstack, thinning=thinning, savefile=fname+"_posterior.png")
# RMS vs bin size:
if rms:
mp.RMS(bs, rms, stderr, rmse, binstep=len(bs)/500+1,
savefile=fname+"_RMS.png")
if indparams != [] and np.size(indparams[0]) == ndata:
mp.modelfit(data, uncert, indparams[0], bestmodel,
savefile=fname+"_model.png")
# Save definitive results:
if savefile is not None:
np.save(savefile, allparams[:,:,:chainlen])
if savemodel is not None:
np.save(savemodel, allmodel [:,:,:chainlen])
return allstack, bestp
def mcmc(data, uncert=None, func=None, indparams=[],
params=None, pmin=None, pmax=None, stepsize=None,
prior=None, priorlow=None, priorup=None,
numit=10, nchains=10, walk='demc', wlike=False,
leastsq=True, chisqscale=False, grtest=True, burnin=0,
thinning=1, plots=False, savefile=None, savemodel=None,
comm=None, resume=False, log=None, rms=False):
"""
This beautiful piece of code runs a Markov-chain Monte Carlo algoritm.
Parameters:
-----------
data: 1D ndarray
Dependent data fitted by func.
uncert: 1D ndarray
Uncertainty of data.
func: callable or string-iterable
The callable function that models data as:
model = func(params, *indparams)
Or an iterable (list, tuple, or ndarray) of 3 strings:
(funcname, modulename, path)
that specify the function name, function module, and module path.
If the module is already in the python-path scope, path can be omitted.
indparams: tuple
Additional arguments required by func.
params: 1D or 2D ndarray
Set of initial fitting parameters for func. If 2D, of shape
(nparams, nchains), it is assumed that it is one set for each chain.
pmin: 1D ndarray
Lower boundaries of the posteriors.
pmax: 1D ndarray
Upper boundaries of the posteriors.
stepsize: 1D ndarray
Proposal jump scale. If a values is 0, keep the parameter fixed.
Negative values indicate a shared parameter (See Note 1).
prior: 1D ndarray
Parameter prior distribution means (See Note 2).
priorlow: 1D ndarray
Lower prior uncertainty values (See Note 2).
priorup: 1D ndarray
Upper prior uncertainty values (See Note 2).
numit: Scalar
Total number of iterations.
nchains: Scalar
Number of simultaneous chains to run.
walk: String
Random walk algorithm:
- 'mrw': Metropolis random walk.
- 'demc': Differential Evolution Markov chain.
wlike: Boolean
If True, calculate the likelihood in a wavelet-base. This requires
three additional parameters (See Note 3).
leastsq: Boolean
Perform a least-square minimization before the MCMC run.
chisqscale: Boolean
Scale the data uncertainties such that the reduced chi-squared = 1.
grtest: Boolean
Run Gelman & Rubin test.
burnin: Scalar
Burned-in (discarded) number of iterations at the beginning
of the chains.
thinning: Integer
Thinning factor of the chains (use every thinning-th iteration) used
in the GR test and plots.
plots: Boolean
If True plot parameter traces, pairwise-posteriors, and posterior
histograms.
savefile: String
If not None, filename to store allparams (with np.save).
savemodel: String
If not None, filename to store the values of the evaluated function
(with np.save).
comm: MPI Communicator
A communicator object to transfer data through MPI.
resume: Boolean
If True resume a previous run.
log: FILE pointer
File object to write log into.
Returns:
--------
allparams: 2D ndarray
An array of shape (nfree, numit-nchains*burnin) with the MCMC
posterior distribution of the fitting parameters.
bestp: 1D ndarray
Array of the best fitting parameters.
Notes:
------
1.- To set one parameter equal to another, set its stepsize to the
negative index in params (Starting the count from 1); e.g.: to set
the second parameter equal to the first one, do: stepsize[1] = -1.
2.- If any of the fitting parameters has a prior estimate, e.g.,
param[i] = p0 +up/-low,
with up and low the 1sigma uncertainties. This information can be
considered in the MCMC run by setting:
prior[i] = p0
priorup[i] = up
priorlow[i] = low
All three: prior, priorup, and priorlow must be set and, furthermore,
priorup and priorlow must be > 0 to be considered as prior.
3.- FINDME WAVELET LIKELIHOOD
Examples:
---------
>>> # See examples: https://github.com/pcubillos/MCcubed/tree/master/examples
Developers:
-----------
Kevin Stevenson UCF [email protected]
Patricio Cubillos UCF [email protected]
Modification History:
---------------------
2008-05-02 kevin Initial implementation
2008-06-21 kevin Finished updating
2009-11-01 kevin Updated for multi events:
2010-06-09 kevin Updated for ipspline, nnint & bilinint
2011-07-06 kevin Updated for Gelman-Rubin statistic
2011-07-22 kevin Added principal component analysis
2011-10-11 kevin Added priors
2012-09-03 patricio Added Differential Evolution MC. Documented.
2013-01-31 patricio Modified for general purposes.
2013-02-21 patricio Added support distribution for DEMC.
2014-03-31 patricio Modified to be completely agnostic of the
fitting function, updated documentation.
2014-04-17 patricio Revamped use of 'func': no longer requires a
wrapper. Alternatively, can take a string list with
the function, module, and path names.
2014-04-19 patricio Added savefile, thinning, plots, and mpi arguments.
2014-05-04 patricio Added Summary print out.
2014-05-09 patricio Added Wavelet-likelihood calculation.
2014-05-09 patricio Changed figure types from pdf to png, because it's
much faster.
2014-05-26 patricio Changed mpi bool argument by comm. Re-engineered
MPI communications to make direct calls to func.
2014-06-09 patricio Fixed glitch with leastsq+informative priors.
2014-10-17 patricio Added savemodel argument.
2014-10-23 patricio Added support for func hack.
2015-02-04 patricio Added resume argument.
2015-05-15 patricio Added log argument.
"""
# Import the model function:
if type(func) in [list, tuple, np.ndarray]:
if func[0] != 'hack':
if len(func) == 3:
sys.path.append(func[2])
exec('from %s import %s as func'%(func[1], func[0]))
elif not callable(func):
mu.error("'func' must be either, a callable, or an iterable (list, "
"tuple, or ndarray) of strings with the model function, file, "
"and path names.", log)
if np.ndim(params) == 1: # Force it to be 2D (one for each chain)
params = np.atleast_2d(params)
nparams = len(params[0]) # Number of model params
ndata = len(data) # Number of data values
# Set default uncertainties:
if uncert is None:
uncert = np.ones(ndata)
# Set default boundaries:
if pmin is None:
pmin = np.zeros(nparams) - np.inf
if pmax is None:
pmax = np.zeros(nparams) + np.inf
# Set default stepsize:
if stepsize is None:
stepsize = 0.1 * np.abs(params[0])
# Set prior parameter indices:
if (prior is None) or (priorup is None) or (priorlow is None):
prior = priorup = priorlow = np.zeros(nparams) # Zero arrays
iprior = np.where(priorlow != 0)[0]
ilog = np.where(priorlow < 0)[0]
nfree = np.sum(stepsize > 0) # Number of free parameters
chainlen = int(np.ceil(numit/nchains)) # Number of iterations per chain
ifree = np.where(stepsize > 0)[0] # Free parameter indices
ishare = np.where(stepsize < 0)[0] # Shared parameter indices
# Number of model parameters (excluding wavelet parameters):
if wlike:
mpars = nparams - 3
else:
mpars = nparams
# Intermediate steps to run GR test and print progress report:
intsteps = chainlen / 10
# Allocate arrays with variables:
numaccept = np.zeros(nchains) # Number of accepted proposal jumps
outbounds = np.zeros((nchains, nfree), np.int) # Out of bounds proposals
allparams = np.zeros((nchains, nfree, chainlen)) # Parameter's record
if savemodel is not None:
allmodel = np.zeros((nchains, ndata, chainlen)) # Fit model
if resume:
oldparams = np.load(savefile)
nold = np.shape(oldparams)[2] # Number of old-run iterations
allparams = np.dstack((oldparams, allparams))
if savemodel is not None:
allmodel = np.dstack((np.load(savemodel), allmodel))
# Set params to the last-iteration state of the previous run:
params = np.repeat(params, nchains, 0)
params[:,ifree] = oldparams[:,:,-1]
else:
nold = 0
# Set MPI flag:
mpi = comm is not None
if mpi:
from mpi4py import MPI
# Send sizes info to other processes:
array1 = np.asarray([mpars, chainlen], np.int)
mu.comm_bcast(comm, array1, MPI.INT)
# DEMC parameters:
gamma = 2.4 / np.sqrt(2*nfree)
gamma2 = 0.001 # Jump scale factor of support distribution
# Least-squares minimization:
if leastsq:
fitargs = (params[0], func, data, uncert, indparams, stepsize, pmin, pmax,
prior, priorlow, priorup)
fitchisq, dummy = mf.modelfit(params[0,ifree], args=fitargs)
fitbestp = np.copy(params[0, ifree])
mu.msg(1, "Least-squares best-fitting parameters: \n{:s}\n\n".
format(str(fitbestp)), log)
# Replicate to make one set for each chain: (nchains, nparams):
if np.shape(params)[0] != nchains:
params = np.repeat(params, nchains, 0)
# Start chains with an initial jump:
for p in ifree:
# For each free param, use a normal distribution:
params[1:, p] = np.random.normal(params[0, p], stepsize[p], nchains-1)
# Stay within pmin and pmax boundaries:
params[np.where(params[:, p] < pmin[p]), p] = pmin[p]
params[np.where(params[:, p] > pmax[p]), p] = pmax[p]
# Update shared parameters:
for s in ishare:
params[:, s] = params[:, -int(stepsize[s])-1]
# Calculate chi-squared for model using current params:
models = np.zeros((nchains, ndata))
if mpi:
# Scatter (send) parameters to func:
mu.comm_scatter(comm, params[:,0:mpars].flatten(), MPI.DOUBLE)
# Gather (receive) evaluated models:
mpimodels = np.zeros(nchains*ndata, np.double)
mu.comm_gather(comm, mpimodels)
# Store them in models variable:
models = np.reshape(mpimodels, (nchains, ndata))
else:
for c in np.arange(nchains):
fargs = [params[c, 0:mpars]] + indparams # List of function's arguments
models[c] = func(*fargs)
# Calculate chi-squared for each chain:
currchisq = np.zeros(nchains)
c2 = np.zeros(nchains) # No-Jeffrey's chisq
for c in np.arange(nchains):
if wlike: # Wavelet-based likelihood (chi-squared, actually)
currchisq[c], c2[c] = dwt.wlikelihood(params[c, mpars:], models[c]-data,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
currchisq[c], c2[c] = cs.chisq(models[c], data, uncert,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
# Scale data-uncertainties such that reduced chisq = 1:
if chisqscale:
chifactor = np.sqrt(np.amin(currchisq)/(ndata-nfree))
uncert *= chifactor
# Re-calculate chisq with the new uncertainties:
for c in np.arange(nchains):
if wlike: # Wavelet-based likelihood (chi-squared, actually)
currchisq[c], c2[c] = dwt.wlikelihood(params[c,mpars:], models[c]-data,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
currchisq[c], c2[c] = cs.chisq(models[c], data, uncert,
(params[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
if leastsq:
fitchisq = currchisq[0]
# Get lowest chi-square and best fitting parameters:
bestchisq = np.amin(c2)
bestp = np.copy(params[np.argmin(c2)])
bestmodel = np.copy(models[np.argmin(c2)])
if savemodel is not None:
allmodel[:,:,0] = models
# Set up the random walks:
if walk == "mrw":
# Generate proposal jumps from Normal Distribution for MRW:
mstep = np.random.normal(0, stepsize[ifree], (chainlen, nchains, nfree))
elif walk == "demc":
# Support random distribution:
support = np.random.normal(0, stepsize[ifree], (chainlen, nchains, nfree))
# Generate indices for the chains such r[c] != c:
r1 = np.random.randint(0, nchains-1, (nchains, chainlen))
r2 = np.random.randint(0, nchains-1, (nchains, chainlen))
for c in np.arange(nchains):
r1[c][np.where(r1[c]==c)] = nchains-1
r2[c][np.where(r2[c]==c)] = nchains-1
# Uniform random distribution for the Metropolis acceptance rule:
unif = np.random.uniform(0, 1, (chainlen, nchains))
# Proposed iteration parameters and chi-square (per chain):
nextp = np.copy(params) # Proposed parameters
nextchisq = np.zeros(nchains) # Chi square of nextp
# Start loop:
mu.msg(1, "Start MCMC chains ({:s})".format(time.ctime()), log)
for i in np.arange(chainlen):
# Proposal jump:
if walk == "mrw":
jump = mstep[i]
elif walk == "demc":
jump = (gamma * (params[r1[:,i]]-params[r2[:,i]])[:,ifree] +
gamma2 * support[i] )
# Propose next point:
nextp[:,ifree] = params[:,ifree] + jump
# Check it's within boundaries:
outpars = np.asarray(((nextp < pmin) | (nextp > pmax))[:,ifree])
outflag = np.any(outpars, axis=1)
outbounds += ((nextp < pmin) | (nextp > pmax))[:,ifree]
for p in ifree:
nextp[np.where(nextp[:, p] < pmin[p]), p] = pmin[p]
nextp[np.where(nextp[:, p] > pmax[p]), p] = pmax[p]
# Update shared parameters:
for s in ishare:
nextp[:, s] = nextp[:, -int(stepsize[s])-1]
# Evaluate the models for the proposed parameters:
if mpi:
mu.comm_scatter(comm, nextp[:,0:mpars].flatten(), MPI.DOUBLE)
mu.comm_gather(comm, mpimodels)
models = np.reshape(mpimodels, (nchains, ndata))
else:
for c in np.where(~outflag)[0]:
fargs = [nextp[c, 0:mpars]] + indparams # List of function's arguments
models[c] = func(*fargs)
# Calculate chisq:
for c in np.where(~outflag)[0]:
if wlike: # Wavelet-based likelihood (chi-squared, actually)
nextchisq[c], c2[c] = dwt.wlikelihood(nextp[c,mpars:], models[c]-data,
(nextp[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
else:
nextchisq[c], c2[c] = cs.chisq(models[c], data, uncert,
(nextp[c]-prior)[iprior], priorlow[iprior], priorlow[iprior])
# Reject out-of-bound jumps:
nextchisq[np.where(outflag)] = np.inf
# Evaluate which steps are accepted and update values:
accept = np.exp(0.5 * (currchisq - nextchisq))
accepted = accept >= unif[i]
if i >= burnin:
numaccept += accepted
# Update params and chi square:
params [accepted] = nextp [accepted]
currchisq[accepted] = nextchisq[accepted]
# Check lowest chi-square:
if np.amin(c2) < bestchisq:
bestp = np.copy(params[np.argmin(c2)])
bestmodel = np.copy(models[np.argmin(c2)])
bestchisq = np.amin(c2)
# Store current iteration values:
allparams[:,:,i+nold] = params[:, ifree]
if savemodel is not None:
models[~accepted] = allmodel[~accepted,:,i+nold-1]
allmodel[:,:,i+nold] = models
# Print intermediate info:
if ((i+1) % intsteps == 0) and (i > 0):
mu.progressbar((i+1.0)/chainlen, log)
mu.msg(1, "Out-of-bound Trials:\n {:s}".
format(np.sum(outbounds, axis=0)), log)
mu.msg(1, "Best Parameters: (chisq={:.4f})\n{:s}".
format(bestchisq, str(bestp)), log)
# Gelman-Rubin statistic:
if grtest and (i+nold) > burnin:
psrf = gr.convergetest(allparams[:, :, burnin:i+nold+1:thinning])
mu.msg(1, "Gelman-Rubin statistic for free parameters:\n{:s}".
format(psrf), log)
if np.all(psrf < 1.01):
mu.msg(1, "All parameters have converged to within 1% of unity.", log)
# Save current results:
if savefile is not None:
np.save(savefile, allparams[:,:,0:i+nold])
if savemodel is not None:
np.save(savemodel, allmodel[:,:,0:i+nold])
# Stack together the chains:
allstack = allparams[0, :, burnin:]
for c in np.arange(1, nchains):
allstack = np.hstack((allstack, allparams[c, :, burnin:]))
# And the models:
if savemodel is not None:
modelstack = allmodel[0,:,burnin:]
for c in np.arange(1, nchains):
modelstack = np.hstack((modelstack, allmodel[c, :, burnin:]))
# Print out Summary:
mu.msg(1, "\nFin, MCMC Summary:\n------------------", log)
nsample = (chainlen-burnin)*nchains # This sample
ntotal = (nold+chainlen-burnin)*nchains
BIC = bestchisq + nfree*np.log(ndata)
redchisq = bestchisq/(ndata-nfree)
sdr = np.std(bestmodel-data)
fmtlen = len(str(ntotal))
mu.msg(1, "Burned in iterations per chain: {:{}d}".
format(burnin, fmtlen), log, 1)
mu.msg(1, "Number of iterations per chain: {:{}d}".
format(chainlen, fmtlen), log, 1)
mu.msg(1, "MCMC sample size: {:{}d}".
format(nsample, fmtlen), log, 1)
mu.msg(resume, "Total MCMC sample size: {:{}d}".
format(ntotal, fmtlen), log, 1)
mu.msg(1, "Acceptance rate: {:.2f}%\n ".
format(np.sum(numaccept)*100.0/nsample), log, 1)
meanp = np.mean(allstack, axis=1) # Parameters mean
uncertp = np.std(allstack, axis=1) # Parameter standard deviation
mu.msg(1, "Best-fit params Uncertainties Signal/Noise Sample "
"Mean", log, 1)
for i in np.arange(nfree):
mu.msg(1, "{: 15.7e} {: 15.7e} {:12.2f} {: 15.7e}".
format(bestp[ifree][i], uncertp[i],
np.abs(bestp[ifree][i])/uncertp[i], meanp[i]), log, 1)
if leastsq and np.any(np.abs((bestp[ifree]-fitbestp)/fitbestp) > 1e-08):
np.set_printoptions(precision=8)
mu.warning("MCMC found a better fit than the minimizer:\n"
" MCMC best-fitting parameters: (chisq={:.8g})\n {:s}\n"
" Minimizer best-fitting parameters: (chisq={:.8g})\n"
" {:s}".format(bestchisq, str(bestp[ifree]),
fitchisq, str(fitbestp)), log)
fmtl = len("%.4f"%BIC) # Length of string formatting
mu.msg(1, " ", log)
if chisqscale:
mu.msg(1, "sqrt(reduced chi-squared) factor: {:{}.4f}".
format(chifactor, fmtl), log, 1)
mu.msg(1, "Best-parameter's chi-squared: {:{}.4f}".
format(bestchisq, fmtl), log, 1)
mu.msg(1, "Bayesian Information Criterion: {:{}.4f}".
format(BIC, fmtl), log, 1)
mu.msg(1, "Reduced chi-squared: {:{}.4f}".
format(redchisq, fmtl), log, 1)
mu.msg(1, "Standard deviation of residuals: {:.6g}\n".format(sdr), log, 1)
if rms:
rms, rmse, stderr, bs = ta.binrms(bestmodel-data)
if plots:
print("Plotting figures ...")
# Extract filename from savefile:
if savefile is not None:
if savefile.rfind(".") == -1:
fname = savefile[savefile.rfind("/")+1:] # Cut out file extention.
else:
fname = savefile[savefile.rfind("/")+1:savefile.rfind(".")]
else:
fname = "MCMC"
# Trace plot:
mp.trace(allstack, thinning=thinning, savefile=fname+"_trace.png",
sep=np.size(allstack[0])/nchains)
# Pairwise posteriors:
mp.pairwise(allstack, thinning=thinning, savefile=fname+"_pairwise.png")
# Histograms:
mp.histogram(allstack, thinning=thinning, savefile=fname+"_posterior.png")
# RMS vs bin size:
if rms:
mp.RMS(bs, rms, stderr, rmse, binstep=len(bs)/500+1,
savefile=fname+"_RMS.png")
if indparams != [] and np.size(indparams[0]) == ndata:
mp.modelfit(data, uncert, indparams[0], bestmodel,
savefile=fname+"_model.png")
# Save definitive results:
if savefile is not None:
np.save(savefile, allparams)
if savemodel is not None:
np.save(savemodel, allmodel)
return allstack, bestp
