/usr/lib/python2.7/dist-packages/csb/statistics/samplers/mc/singlechain.py is in python-csb 1.2.3+dfsg-1.
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Various Monte Carlo equilibrium sampling algorithms, which simulate only one Markov chain.
Here is how to sample from a PDF using the L{HMCSampler} class. In the following
snippet we draw 5000 samples from a 1D normal distribution and plot them:
>>> import numpy
>>> from csb.io.plots import Chart
>>> from csb.statistics.pdf import Normal
>>> from csb.statistics.samplers import State
>>> from csb.statistics.samplers.mc.singlechain import HMCSampler
>>> initial_state = State(numpy.array([1.]))
>>> grad = lambda q, t: q
>>> timestep = 1.5
>>> nsteps = 30
>>> nsamples = 5000
>>> sampler = HMCSampler(Normal(), initial_state, grad, timestep, nsteps)
>>> states = []
>>> for i in range(nsamples):
sampler.sample()
states.append(sampler.state)
>>> print('acceptance rate:', sampler.acceptance_rate)
0.8
>>> states = [state.position[0]for state in states]
>>> chart = Chart()
>>> chart.plot.hist([numpy.random.normal(size=5000), states], bins=20, normed=True)
>>> chart.plot.legend(['numpy.random.normal', 'HMC'])
>>> chart.show()
First, several things which are being needed are imported.
As every sampler in this module implements a Markov Chain, an initial state has to be
chosen. In the following lines, several parameters are set:
- the gradient of the negative log-probability of the PDF under consideration
- the integration timestep
- the number of integration steps to be performed in each iteration, that is, the HMC
trajectory length
- the number of samples to be drawn
The empty list states is initialized. It will serve to store the samples drawn.
In the loop, C{sampler.sample()} is repeatedly called. After each call of C{sampler.sample()},
the current state of the Markov Chain is stored in sampler.state and this state is appended
to the sample storage list.
Then the acceptance rate is printed, the numeric values are being extracted from the
L{State} objects in states, a histogram is created and finally plotted.
"""
import numpy
import csb.numeric
import csb.core
from abc import ABCMeta, abstractmethod
from csb.statistics.samplers import State
from csb.statistics.samplers.mc import AbstractMC, MCCollection, augment_state
from csb.statistics.samplers.mc.propagators import MDPropagator
from csb.statistics.samplers.mc.neqsteppropagator import NonequilibriumStepPropagator
from csb.numeric.integrators import FastLeapFrog
from csb.numeric import InvertibleMatrix
class AbstractSingleChainMC(AbstractMC):
"""
Abstract class for Monte Carlo sampling algorithms simulating
only one ensemble.
@param pdf: probability density function to sample from
@type pdf: subclass of L{csb.statistics.pdf.AbstractDensity}
@param state: Initial state
@type state: L{State}
@param temperature: Pseudo-temperature of the Boltzmann ensemble
M{p(x) = 1/N * exp(-1/T * E(x))} with the
pseudo-energy defined as M{E(x) = -log(p(x))}
where M{p(x)} is the PDF under consideration
@type temperature: float
"""
__metaclass__ = ABCMeta
def __init__(self, pdf, state, temperature=1.):
super(AbstractSingleChainMC, self).__init__(state)
self._pdf = pdf
self._temperature = temperature
self._nmoves = 0
self._accepted = 0
self._last_move_accepted = None
def _checkstate(self, state):
if not isinstance(state, State):
raise TypeError(state)
def sample(self):
"""
Draw a sample.
@rtype: L{State}
"""
proposal_communicator = self._propose()
pacc = self._calc_pacc(proposal_communicator)
accepted = None
if numpy.random.random() < pacc:
accepted = True
else:
accepted = False
if accepted == True:
self._accept_proposal(proposal_communicator.proposal_state)
self._update_statistics(accepted)
self._last_move_accepted = accepted
return self.state
@abstractmethod
def _propose(self):
"""
Calculate a new proposal state and gather additional information
needed to calculate the acceptance probability.
@rtype: L{SimpleProposalCommunicator}
"""
pass
@abstractmethod
def _calc_pacc(self, proposal_communicator):
"""
Calculate probability with which to accept the proposal.
@param proposal_communicator: Contains information about the proposal
and additional information needed to
calculate the acceptance probability
@type proposal_communicator: L{SimpleProposalCommunicator}
"""
pass
def _accept_proposal(self, proposal_state):
"""
Accept the proposal state by setting it as the current state of the sampler
object
@param proposal_state: The proposal state
@type proposal_state: L{State}
"""
self.state = proposal_state
def _update_statistics(self, accepted):
"""
Update the sampling statistics.
@param accepted: Whether or not the proposal state has been accepted
@type accepted: boolean
"""
self._nmoves += 1
self._accepted += int(accepted)
@property
def energy(self):
"""
Negative log-likelihood of the current state.
@rtype: float
"""
return -self._pdf.log_prob(self.state.position)
@property
def acceptance_rate(self):
"""
Acceptance rate.
"""
if self._nmoves > 0:
return float(self._accepted) / float(self._nmoves)
else:
return 0.0
@property
def last_move_accepted(self):
"""
Information whether the last MC move was accepted or not.
"""
return self._last_move_accepted
@property
def temperature(self):
return self._temperature
class HMCSampler(AbstractSingleChainMC):
"""
Hamilton Monte Carlo (HMC, also called Hybrid Monte Carlo by the inventors,
Duane, Kennedy, Pendleton, Duncan 1987).
@param pdf: Probability density function to be sampled from
@type pdf: L{csb.statistics.pdf.AbstractDensity}
@param state: Inital state
@type state: L{State}
@param gradient: Gradient of the negative log-probability
@type gradient: L{AbstractGradient}
@param timestep: Timestep used for integration
@type timestep: float
@param nsteps: Number of integration steps to be performed in
each iteration
@type nsteps: int
@param mass_matrix: Mass matrix
@type mass_matrix: n-dimensional L{InvertibleMatrix} with n being the dimension
of the configuration space, that is, the dimension of
the position / momentum vectors
@param integrator: Subclass of L{AbstractIntegrator} to be used for
integrating Hamiltionian equations of motion
@type integrator: L{AbstractIntegrator}
@param temperature: Pseudo-temperature of the Boltzmann ensemble
M{p(x) = 1/N * exp(-1/T * E(x))} with the
pseudo-energy defined as M{E(x) = -log(p(x))}
where M{p(x)} is the PDF under consideration
@type temperature: float
"""
def __init__(self, pdf, state, gradient, timestep, nsteps,
mass_matrix=None, integrator=FastLeapFrog, temperature=1.):
super(HMCSampler, self).__init__(pdf, state, temperature)
self._timestep = None
self.timestep = timestep
self._nsteps = None
self.nsteps = nsteps
self._mass_matrix = None
self._momentum_covariance_matrix = None
self._integrator = integrator
self._gradient = gradient
self._setup_matrices(mass_matrix)
self._propagator = self._propagator_factory()
def _setup_matrices(self, mass_matrix):
self._d = len(self.state.position)
self._mass_matrix = mass_matrix
if self.mass_matrix is None:
self.mass_matrix = InvertibleMatrix(numpy.eye(self._d), numpy.eye(self._d))
self._momentum_covariance_matrix = self._temperature * self.mass_matrix
def _propagator_factory(self):
"""
Factory which produces a L{MDPropagator} object initialized with
the MD parameters given in __init__().
@return: L{MDPropagator} instance
@rtype: L{MDPropagator}
"""
return MDPropagator(self._gradient, self._timestep,
mass_matrix=self._mass_matrix,
integrator=self._integrator)
def _propose(self):
current_state = self.state.clone()
current_state = augment_state(current_state, self.temperature, self.mass_matrix)
proposal_state = self._propagator.generate(current_state, self._nsteps).final
return SimpleProposalCommunicator(current_state, proposal_state)
def _hamiltonian(self, state):
"""
Evaluates the Hamiltonian consisting of the negative log-probability
and a quadratic kinetic term.
@param state: State on which the Hamiltonian should be evaluated
@type state: L{State}
@return: Value of the Hamiltonian (total energy)
@rtype: float
"""
V = lambda q: -self._pdf.log_prob(q)
T = lambda p: 0.5 * numpy.dot(p.T, numpy.dot(self.mass_matrix.inverse, p))
return T(state.momentum) + V(state.position)
def _calc_pacc(self, proposal_communicator):
cs = proposal_communicator.current_state
ps = proposal_communicator.proposal_state
pacc = csb.numeric.exp(-(self._hamiltonian(ps) - self._hamiltonian(cs)) /
self.temperature)
if self.state.momentum is None:
proposal_communicator.proposal_state.momentum = None
else:
proposal_communicator.proposal_state.momentum = self.state.momentum
return pacc
@property
def timestep(self):
return self._timestep
@timestep.setter
def timestep(self, value):
self._timestep = float(value)
if "_propagator" in dir(self):
self._propagator.timestep = self._timestep
@property
def nsteps(self):
return self._nsteps
@nsteps.setter
def nsteps(self, value):
self._nsteps = int(value)
@property
def mass_matrix(self):
return self._mass_matrix
@mass_matrix.setter
def mass_matrix(self, value):
self._mass_matrix = value
if "_propagator" in dir(self):
self._propagator.mass_matrix = self._mass_matrix
class RWMCSampler(AbstractSingleChainMC):
"""
Random Walk Metropolis Monte Carlo implementation
(Metropolis, Rosenbluth, Teller, Teller 1953; Hastings, 1970).
@param pdf: Probability density function to be sampled from
@type pdf: L{csb.statistics.pdf.AbstractDensity}
@param state: Inital state
@type state: L{State}
@param stepsize: Serves to set the step size in
proposal_density, e.g. for automatic acceptance
rate adaption
@type stepsize: float
@param proposal_density: The proposal density as a function f(x, s)
of the current state x and the stepsize s.
By default, the proposal density is uniform,
centered around x, and has width s.
@type proposal_density: callable
@param temperature: Pseudo-temperature of the Boltzmann ensemble
M{p(x) = 1/N * exp(-1/T * E(x))} with the
pseudo-energy defined as M{E(x) = -log(p(x))}
where M{p(x)} is the PDF under consideration
@type temperature: float
"""
def __init__(self, pdf, state, stepsize=1., proposal_density=None, temperature=1.):
super(RWMCSampler, self).__init__(pdf, state, temperature)
self._stepsize = None
self.stepsize = stepsize
if proposal_density == None:
self._proposal_density = lambda x, s: x.position + \
s * numpy.random.uniform(size=x.position.shape,
low=-1., high=1.)
else:
self._proposal_density = proposal_density
def _propose(self):
current_state = self.state.clone()
proposal_state = self.state.clone()
proposal_state.position = self._proposal_density(current_state, self.stepsize)
return SimpleProposalCommunicator(current_state, proposal_state)
def _calc_pacc(self, proposal_communicator):
current_state = proposal_communicator.current_state
proposal_state = proposal_communicator.proposal_state
E = lambda x: -self._pdf.log_prob(x)
pacc = csb.numeric.exp((-(E(proposal_state.position) - E(current_state.position))) /
self.temperature)
return pacc
@property
def stepsize(self):
return self._stepsize
@stepsize.setter
def stepsize(self, value):
self._stepsize = float(value)
class AbstractNCMCSampler(AbstractSingleChainMC):
"""
Implementation of the NCMC sampling algorithm (Nilmeier et al., "Nonequilibrium candidate Monte
Carlo is an efficient tool for equilibrium simulation", PNAS 2011) for sampling from one
ensemble only.
Subclasses have to specify the acceptance probability, which depends on the kind of
perturbations and propagations in the protocol.
@param state: Inital state
@type state: L{State}
@param protocol: Nonequilibrium protocol with alternating perturbations and propagations
@type protocol: L{Protocol}
@param reverse_protocol: The reversed version of the protocol, that is, the order of
perturbations and propagations in each step is reversed
@type reverse_protocol: L{Protocol}
"""
__metaclass__ = ABCMeta
def __init__(self, state, protocol, reverse_protocol):
self._protocol = None
self.protocol = protocol
self._reverse_protocol = None
self.reverse_protocol = reverse_protocol
pdf = self.protocol.steps[0].perturbation.sys_before.hamiltonian
temperature = self.protocol.steps[0].perturbation.sys_before.hamiltonian.temperature
super(AbstractNCMCSampler, self).__init__(pdf, state, temperature)
def _pick_protocol(self):
"""
Picks either the protocol or the reversed protocol with equal probability.
@return: Either the protocol or the reversed protocol
@rtype: L{Protocol}
"""
if numpy.random.random() < 0.5:
return self.protocol
else:
return self.reverse_protocol
def _propose(self):
protocol = self._pick_protocol()
gen = NonequilibriumStepPropagator(protocol)
traj = gen.generate(self.state)
return NCMCProposalCommunicator(traj)
def _accept_proposal(self, proposal_state):
if self.state.momentum is not None:
proposal_state.momentum *= -1.0
else:
proposal_state.momentum = None
super(AbstractNCMCSampler, self)._accept_proposal(proposal_state)
@property
def protocol(self):
return self._protocol
@protocol.setter
def protocol(self, value):
self._protocol = value
@property
def reverse_protocol(self):
return self._reverse_protocol
@reverse_protocol.setter
def reverse_protocol(self, value):
self._reverse_protocol = value
class SimpleProposalCommunicator(object):
"""
This holds all the information needed to calculate the acceptance
probability in both the L{RWMCSampler} and L{HMCSampler} classes,
that is, only the proposal state.
For more advanced algorithms, one may derive classes capable of
holding the neccessary additional information from this class.
@param current_state: Current state
@type current_state: L{State}
@param proposal_state: Proposal state
@type proposal_state: L{State}
"""
__metaclass__ = ABCMeta
def __init__(self, current_state, proposal_state):
self._current_state = current_state
self._proposal_state = proposal_state
@property
def current_state(self):
return self._current_state
@property
def proposal_state(self):
return self._proposal_state
class NCMCProposalCommunicator(SimpleProposalCommunicator):
"""
Holds all information (that is, the trajectory with heat, work, Hamiltonian difference
and jacbian) needed to calculate the acceptance probability in the AbstractNCMCSampler class.
@param traj: Non-equilibrium trajectory stemming from a stepwise protocol
@type traj: NCMCTrajectory
"""
def __init__(self, traj):
self._traj = None
self.traj = traj
super(NCMCProposalCommunicator, self).__init__(traj.initial, traj.final)
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