text:
#!/usr/local/bin/env python

# ==============================================================================
# MODULE DOCSTRING
# ==============================================================================

"""
MultiStateAnalyzers
===================

Analysis tools and module for MultiStateSampler simulations. Provides programmatic and automatic
"best practices" integration to determine free energy and other observables.

Fully extensible to support new samplers and observables.

"""

# =============================================================================================
# MODULE IMPORTS
# =============================================================================================

import abc
import copy
import inspect
import logging
import re
from typing import Optional, NamedTuple, Union

import mdtraj
import numpy as np
try:
    import openmm
    import openmm.unit as units
except ImportError:  # OpenMM < 7.6
    from simtk import openmm
    import simtk.unit as units
from scipy.special import logsumexp
from pymbar import MBAR, timeseries

from openmmtools import multistate, utils, forces

ABC = abc.ABC
logger = logging.getLogger(__name__)

__all__ = [
    'PhaseAnalyzer',
    'MultiStateSamplerAnalyzer',
    'MultiPhaseAnalyzer',
    'ObservablesRegistry',
    'default_observables_registry'
]

# =============================================================================================
# GLOBAL VARIABLES
# =============================================================================================

kB = units.BOLTZMANN_CONSTANT_kB * units.AVOGADRO_CONSTANT_NA

_OPENMM_ENERGY_UNIT = units.kilojoules_per_mole
_MDTRAJ_DISTANCE_UNIT = units.nanometers

# =============================================================================================
# UTILITY FUNCTIONS
# =============================================================================================

def compute_centroid_distance(positions_group1, positions_group2, weights_group1, weights_group2):
    """Compute the distance between the centers of mass of the two groups.

    The two positions given must have the same units.

    Parameters
    ----------
    positions_group1 : numpy.array
        The positions of the particles in the first CustomCentroidBondForce group.
    positions_group2 : numpy.array
        The positions of the particles in the second CustomCentroidBondForce group.
    weights_group1 : list of float
        The mass of the particle in the first CustomCentroidBondForce group.
    weights_group2 : list of float
        The mass of the particles in the second CustomCentroidBondForce group.

    """
    assert len(positions_group1) == len(weights_group1)
    assert len(positions_group2) == len(weights_group2)
    # Compute center of mass for each group.
    com_group1 = np.average(positions_group1, axis=0, weights=weights_group1)
    com_group2 = np.average(positions_group2, axis=0, weights=weights_group2)
    # Compute distance between centers of mass.
    distance = np.linalg.norm(com_group1 - com_group2)
    return distance

# =============================================================================================
# MODULE CLASSES
# =============================================================================================

class ObservablesRegistry(object):
    """
    Registry of computable observables.

    This is a class accessed by the :class:`PhaseAnalyzer` objects to check
    which observables can be computed, and then provide a regular categorization of them.

    This registry is a required linked component of any PhaseAnalyzer and especially of the MultiPhaseAnalyzer.
    This is not an internal class to the PhaseAnalyzer however because it can be instanced, extended, and customized
    as part of the API for this module.

    To define your own methods:
    1) Choose a unique observable name.
    2) Categorize the observable in one of the following ways by adding to the list in the "observables_X" method:

        2a) "defined_by_phase":
            Depends on the Phase as a whole (state independent)

        2b) "defined_by_single_state":
            Computed entirely from one state, e.g. Radius of Gyration

        2c) "defined_by_two_states":
            Property is relative to some reference state, such as Free Energy Difference

    3) Optionally categorize the error category calculation in the "observables_with_error_adding_Y" methods
       If not placed in an error category, the observable will be assumed not to carry error
       Examples: A, B, C are the observable in 3 phases, eA, eB, eC are the error of the observable in each phase

        3a) "linear": Error between phases adds linearly.
            If C = A + B, eC = eA + eB

        3b) "quadrature": Error between phases adds in the square.
            If C = A + B, eC = sqrt(eA^2 + eB^2)

    4) Finally, to add this observable to the phase, implement a "get_{method name}" method to the subclass of
       :class:`YankPhaseAnalyzer`. Any :class:`MultiPhaseAnalyzer` composed of this phase will automatically have the
       "get_{method name}" if all other phases in the :class:`MultiPhaseAnalyzer` have the same method.
    """

    def __init__(self):
        """Register Defaults"""
        # Create empty registry
        self._observables = {'two_state': set(),
                             'one_state': set(),
                             'phase': set()}
        self._errors = {'quad': set(),
                        'linear': set(),
                        None: set()}

    def register_two_state_observable(self, name: str,
                                      error_class: Optional[str]=None,
                                      re_register: bool=False):
        """
        Register a new two state observable, or re-register an existing one.

        Parameters
        ----------
        name: str
            Name of the observable, will be cast to all lower case and spaces replaced with underscores
        error_class: "quad", "linear", or None
            How the error of the observable is computed when added with other errors from the same observable.

            * "quad": Adds in the quadrature, Observable C = A + B, Error eC = sqrt(eA**2 + eB**2)

            * "linear": Adds linearly,  Observable C = A + B, Error eC = eA + eB

            * None: Does not carry error

        re_register: bool, optional, Default: False
            Re-register an existing observable
        """

        self._register_observable(name, "two_state", error_class, re_register=re_register)

    def register_one_state_observable(self, name: str,
                                      error_class: Optional[str]=None,
                                      re_register: bool=False):
        """
        Register a new one state observable, or re-register an existing one.

        Parameters
        ----------
        name: str
            Name of the observable, will be cast to all lower case and spaces replaced with underscores
        error_class: "quad", "linear", or None
            How the error of the observable is computed when added with other errors from the same observable.

            * "quad": Adds in the quadrature, Observable C = A + B, Error eC = sqrt(eA**2 + eB**2)

            * "linear": Adds linearly,  Observable C = A + B, Error eC = eA + eB

            * None: Does not carry error

        re_register: bool, optional, Default: False
            Re-register an existing observable
        """

        self._register_observable(name, "one_state", error_class, re_register=re_register)

    def register_phase_observable(self, name: str,
                                  error_class: Optional[str]=None,
                                  re_register: bool=False):
        """
        Register a new observable defined by phaee, or re-register an existing one.

        Parameters
        ----------
        name: str
            Name of the observable, will be cast to all lower case and spaces replaced with underscores
        error_class: 'quad', 'linear', or None
            How the error of the observable is computed when added with other errors from the same observable.

            * 'quad': Adds in the quadrature, Observable C = A + B, Error eC = sqrt(eA**2 + eB**2)

            * 'linear': Adds linearly,  Observable C = A + B, Error eC = eA + eB

            * None: Does not carry error

        re_register: bool, optional, Default: False
            Re-register an existing observable

        """

        self._register_observable(name, "phase", error_class, re_register=re_register)

    ########################
    # Define the observables
    ########################
    @property
    def observables(self):
        """
        Set of observables which are derived from the subsets below
        """
        observables = set()
        for subset_key in self._observables:
            observables |= self._observables[subset_key]
        return tuple(observables)

    # ------------------------------------------------
    # Exclusive Observable categories
    # The intersection of these should be the null set
    # ------------------------------------------------

    @property
    def observables_defined_by_two_states(self):
        """
        Observables that require an i and a j state to define the observable accurately between phases
        """
        return self._get_observables('two_state')

    @property
    def observables_defined_by_single_state(self):
        """
        Defined observables which are fully defined by a single state, and not by multiple states such as differences
        """
        return self._get_observables('one_state')

    @property
    def observables_defined_by_phase(self):
        """
        Observables which are defined by the phase as a whole, and not defined by any 1 or more states
        e.g. Standard State Correction
        """
        return self._get_observables('phase')

    ##########################################
    # Define the observables which carry error
    # This should be a subset of observables
    ##########################################

    @property
    def observables_with_error(self):
        """Determine which observables have error by inspecting the the error subsets"""
        observables = set()
        for subset_key in self._errors:
            if subset_key is not None:
                observables |= self._errors[subset_key]
        return tuple(observables)

    # ------------------------------------------------
    # Exclusive Error categories
    # The intersection of these should be the null set
    # ------------------------------------------------

    @property
    def observables_with_error_adding_quadrature(self):
        """Observable C = A + B, Error eC = sqrt(eA**2 + eB**2)"""
        return self._get_errors('quad')

    @property
    def observables_with_error_adding_linear(self):
        """Observable C = A + B, Error eC = eA + eB"""
        return self._get_errors('linear')

    @property
    def observables_without_error(self):
        return self._get_errors(None)

    # ------------------
    # Internal functions
    # ------------------

    def _get_observables(self, key):
        return tuple(self._observables[key])

    def _get_errors(self, key):
        return tuple(self._errors[key])

    @staticmethod
    def _cast_observable_name(name) -> str:
        return re.sub(" +", "_", name.lower())

    def _register_observable(self, obs_name: str,
                             obs_calc_class: str,
                             obs_error_class: Union[None, str],
                             re_register: bool=False):
        obs_name = self._cast_observable_name(obs_name)
        if not re_register and obs_name in self.observables:
            raise ValueError("{} is already a registered observable! "
                             "Consider setting re_register key!".format(obs_name))
        self._check_obs_class(obs_calc_class)
        self._check_obs_error_class(obs_error_class)
        obs_name_set = {obs_name}  # set(single_object) throws an error, set(string) splits each char
        # Throw out existing observable if present (set difference)
        for obs_key in self._observables:
            self._observables[obs_key] -= obs_name_set
        for obs_err_key in self._errors:
            self._errors[obs_err_key] -= obs_name_set
        # Add new observable to correct classifiers (set union)
        self._observables[obs_calc_class] |= obs_name_set
        self._errors[obs_error_class] |= obs_name_set

    def _check_obs_class(self, obs_class):
        assert obs_class in self._observables, "{} not a known observable class!".format(obs_class)

    def _check_obs_error_class(self, obs_error):
        assert obs_error is None or obs_error in self._errors, \
            "{} not a known observable error class!".format(obs_error)

# Create a default registry and register some stock values
default_observables_registry = ObservablesRegistry()
default_observables_registry.register_two_state_observable('free_energy', error_class='quad')
default_observables_registry.register_two_state_observable('entropy', error_class='quad')
default_observables_registry.register_two_state_observable('enthalpy', error_class='quad')

# -----------------------------------------------------------------------------
# EXCEPTIONS.
# -----------------------------------------------------------------------------

class InsufficientData(Exception):
    """Raised when the data is not sufficient perform the requested analysis."""
    pass

# -----------------------------------------------------------------------------
# CACHED PROPERTIES DESCRIPTOR.
# -----------------------------------------------------------------------------

class CachedProperty(object):
    """Analyzer helper descriptor of a cached value with a dependency graph.

    Automatically takes care of invalidating the values of the cache
    that depend on this property.

    Parameters
    ----------
    name : str
        The name of the parameter in the cache.
    dependencies : iterable of str
        List of cached properties on which this property depends.
    check_changes : bool, optional
        If True, the cache dependencies will be invalidated only if
        the new value differs from the old one (default is False).
    default : object, optional
        The default value in case the cache doesn't contain a value
        for this. If a callable, this function must have the signature
        ``default(self, instance)``. It is also possible to define a
        callable default through the ``default`` decorator. After the
        first cache miss, the default value is cached. By default,
        AttributeError is raised on a cache miss.
    validator : callable, optional
        A function to call before setting a new value with signature
        ``validator(self, instance, new_value)``. It is also possible
        to define this through the ``validator`` decorator.

    """
    def __init__(self, name, dependencies=(), check_changes=False,
                 default=AttributeError, validator=None):
        # Reserved names.
        # TODO make observables CachedProperties?
        assert name != 'observables'
        assert name != 'reporter'
        # TODO use __setname__() when dropping Python 3.5 support.
        self.name = name
        self.dependencies = dependencies
        self._default = default
        self._validator = validator
        self._check_changes = check_changes

    def __get__(self, instance, owner_class=None):
        # If called as a class descriptor, return the descriptor.
        if instance is None:
            return self
        # Check if the value is cached and fall back to default value.
        try:
            value = instance._cache[self.name]
        except KeyError:
            value = self._get_default(instance)
            # Cache default value for next use.
            instance._update_cache(self.name, value, self._check_changes)
        return value

    def __set__(self, instance, new_value):
        if self._validator is not None:
            new_value = self._validator(self, instance, new_value)
        instance._update_cache(self.name, new_value, self._check_changes)

    def validator(self, validator):
        return type(self)(self.name, self.dependencies, self._check_changes, self._default, validator)

    def default(self, default):
        return type(self)(self.name, self.dependencies, self._check_changes, default, self._validator)

    def _get_default(self, instance):
        if self._default is AttributeError:
            err_msg = 'Reference before assignment {}.{}'.format(instance, self.name)
            raise AttributeError(err_msg)
        elif callable(self._default):
            value = self._default(self, instance)
        else:
            value = self._default
        return value

# ---------------------------------------------------------------------------------------------
# Phase Analyzers
# ---------------------------------------------------------------------------------------------

class PhaseAnalyzer(ABC):
    """
    Analyzer for a single phase of a MultiState simulation.

    Uses the reporter from the simulation to determine the location
    of all variables.

    To compute a specific observable in an implementation of this class, add it to the ObservableRegistry and then
    implement a ``get_X`` where ``X`` is the name of the observable you want to compute. See the ObservablesRegistry for
    information about formatting the observables.

    Analyzer works in units of kT unless specifically stated otherwise. To convert back to a unit set, just multiply by
    the .kT property.

    A PhaseAnalyzer also needs an ObservablesRegistry to track how to handle each observable given implemented within
    for things like error and cross-phase analysis.

    Parameters
    ----------
    reporter : multistate.MultiStateReporter instance
        Reporter from MultiState which ties to the simulation data on disk.
    name : str, Optional
        Unique name you want to assign this phase, this is the name that will appear in :class:`MultiPhaseAnalyzer`'s.
        If not set, it will be given the arbitrary name "phase#" where # is an integer, chosen in order that it is
        assigned to the :class:`MultiPhaseAnalyzer`.
    max_n_iterations : int, optional
        The maximum number of iterations to analyze. If not provided, all
        the iterations will be analyzed.
    reference_states : tuple of ints, length 2, Optional, Default: (0,-1)
        Integers ``i`` and ``j`` of the state that is used for reference in observables, "O". These values are only used
        when reporting single numbers or combining observables through :class:`MultiPhaseAnalyzer` (since the number of
        states between phases can be different). Calls to functions such as ``get_free_energy`` in a single Phase
        results in the O being returned for all states.

            For O completely defined by the state itself (i.e. no differences between states, e.g. Temperature),
            only O[i] is used

            For O where differences between states are required (e.g. Free Energy): O[i,j] = O[j] - O[i]

            For O defined by the phase as a whole, the reference states are not needed.

    analysis_kwargs : None or dict, optional
        Dictionary of extra keyword arguments to pass into the analysis tool, typically MBAR.
        For instance, the initial guess of relative free energies to give to MBAR would be something like:
        ``{'initial_f_k':[0,1,2,3]}``

    use_online_data : bool, optional, Default: True
        Attempt to read online analysis data as a way to hot-start the analysis computation. This will attempt to
        read the data stored by the MultiStateAnalyzer.

        If this is set to ``False``, the online analysis data is not read.

        If this is set to ``False`` after being initialized, the :class:`CachedProperty` dependencies are all
        invalidated and properties will be computed from scratch on next observables

        If no online data is found, this setting has no effect

    use_full_trajectory : bool, optional, Default: False
        Force the analysis to use the full trajectory when automatically computing equilibration and decorrelation.

        Normal behavior (when this is False) is to discard the initial trajectory due to automatic equilibration
        detection, and then subsample that data to generate decorelated samples. Setting this to ``True`` ignores
        this effect, even if the equilibration data is computed.

        This can be changed after the ``PhaseAnalyzer`` has been created to re-compute properties with or without
        the full trajectory.

    registry : ObservablesRegistry instance
        Instanced ObservablesRegistry with all observables implemented through a ``get_X`` function classified and
        registered. Any cross-phase analysis must use the same instance of an ObservablesRegistry

    Attributes
    ----------
    name
    observables
    max_n_iterations
    reference_states
    n_iterations
    n_replicas
    n_states
    kT
    reporter
    registry
    use_online_data
    use_full_trajectory

    See Also
    --------
    ObservablesRegistry

    """
    def __init__(self, reporter, name=None, reference_states=(0, -1),
                 max_n_iterations=None, analysis_kwargs=None,
                 registry=default_observables_registry,
                 use_online_data=True,
                 use_full_trajectory=False):
        """
        The reporter provides the hook into how to read the data, all other options control where differences are
        measured from and how each phase interfaces with other phases.
        """
        # Arguments validation.
        if not type(reporter) is multistate.MultiStateReporter:
            raise ValueError('reporter must be a MultiStateReporter instance')
        if not isinstance(registry, ObservablesRegistry):
            raise ValueError("Registry must be an instanced ObservablesRegistry")
        if analysis_kwargs is None:
            analysis_kwargs = {}
        elif not isinstance(analysis_kwargs, dict):
            raise ValueError('analysis_kwargs must be either None or a dictionary')

        self.registry = registry
        if not reporter.is_open():
            reporter.open(mode='r')
        self._reporter = reporter

        # Initialize cached observables so the phase can be retrieved once computed.
        self._computed_observables = {observable: None for observable in self.observables}

        # Internal properties
        self._name = name
        # Start as default sign +, handle all sign conversion at preparation time
        self._sign = '+'
        self._reference_states = None  # Initialize the cache object.
        self.reference_states = reference_states
        self._user_extra_analysis_kwargs = analysis_kwargs  # Store the user-specified (higher priority) keywords

        # Initialize cached values that are read or derived from the Reporter.
        self._cache = {}  # This cache should be always set with _update_cache().
        self.clear()
        self.max_n_iterations = max_n_iterations

        # Use full trajectory store or not
        self.use_full_trajectory = use_full_trajectory

        # Check the online data
        self._online_data = None  # Init the object
        self._use_online_data = use_online_data
        self._read_online_data_if_present()

    def __del__(self):
        # Explicitly close storage
        self.clear()
        if self._reporter is not None:
            del self._reporter

    def clear(self):
        """Reset all cached objects.

        This must to be called if the information in the reporter changes
        after analysis.
        """
        # Reset cached values that are read directly from the Reporter.
        self._n_iterations = None
        self._n_replicas = None
        self._end_thermodynamic_states = None
        self._kT = None
        # Reset cached values that are derived from the reporter.
        self._invalidate_cache_values('reporter')

    @property
    def name(self):
        """User-readable string name of the phase"""
        return self._name

    @name.setter
    def name(self, value):
        self._name = value

    @property
    def observables(self):
        """List of observables that the instanced analyzer can compute/fetch."""
        # Auto-determine the computable observables by inspection of non-flagged methods
        # We determine valid observables by negation instead of just having each child
        # implement the method to enforce uniform function naming conventions.
        observables = []
        for observable in self.registry.observables:
            if hasattr(self, "get_" + observable):
                observables.append(observable)
        # Cast observables to an immutable.
        return tuple(observables)

    @property
    def reference_states(self):
        """Tuple of reference states ``i`` and ``j`` for :class:`MultiPhaseAnalyzer` instances"""
        return self._reference_states

    @reference_states.setter
    def reference_states(self, value):
        """Provide a way to re-assign the ``i, j`` states in a protected way"""
        i, j = value[0], value[1]
        if type(i) is not int or type(j) is not int:
            raise ValueError("reference_states must be a length 2 iterable of ints")
        self._reference_states = (i, j)

    @property
    def n_iterations(self):
        """int: The total number of iterations of the phase."""
        if self._n_iterations is None:
            # The + 1 accounts for iteration 0.
            self._n_iterations = self._reporter.read_last_iteration(last_checkpoint=False)
        return self._n_iterations

    @property
    def n_replicas(self):
        """int: Number of replicas."""
        if self._n_replicas is None:
            replica_state_indices = self._reporter.read_replica_thermodynamic_states(iteration=0)
            self._n_replicas = len(replica_state_indices)
        return self._n_replicas

    @property
    def n_states(self):
        """int: Number of sampled thermodynamic states."""
        return self._reporter.n_states

    def _get_end_thermodynamic_states(self):
        """Read thermodynamic states at the ends of the protocol.

        Returns
        -------
        end_thermodynamic_states : list of ThermodynamicState
            The end thermodynamic states

        """
        if self._end_thermodynamic_states is None:
            self._end_thermodynamic_states = self._reporter.read_end_thermodynamic_states()
            # Cache other useful informations since we have already read this.
            # TODO should we read temperatures of all the states and let kT property depend on reference_states?
            self._kT = self._end_thermodynamic_states[0].kT

        return self._end_thermodynamic_states

    @property
    def kT(self):
        """
        Quantity of boltzmann constant times temperature of the phase in units of energy per mol

        Allows conversion between dimensionless energy and unit bearing energy
        """
        if self._kT is None:
            self._get_end_thermodynamic_states()
        return self._kT

    @property
    def reporter(self):
        """Sampler Reporter tied to this object."""
        return self._reporter

    @reporter.setter
    def reporter(self, value):
        """Make sure users cannot overwrite the reporter."""
        raise ValueError("You cannot re-assign the reporter for this analyzer!")

    @property
    def use_online_data(self):
        """Get the online data flag"""
        return self._use_online_data

    @use_online_data.setter
    def use_online_data(self, value):
        """Set the online data boolean"""
        if type(value) != bool:
            raise ValueError("use_online_data must be a boolean!")
        if self._use_online_data is False and value is True:
            # Re-read the online data
            self._read_online_data_if_present()
        elif self._use_online_data is True and value is False:
            # Invalidate online data
            self._online_data = None
            # Re-form dict to prevent variables from referencing the same object
            self._extra_analysis_kwargs = {**self._user_extra_analysis_kwargs}
        self._use_online_data = value

    # -------------------------------------------------------------------------
    # Cached properties functions/classes.
    # -------------------------------------------------------------------------

    @classmethod
    def _get_cache_dependency_graph(cls):
        """dict: cached_value -> list of cache values to invalidate."""
        # Retrieve all cached properties.
        cached_properties = {value for name, value in inspect.getmembers(cls)
                             if isinstance(value, CachedProperty)}
        # Build the dependency graph.
        dependency_graph = {}
        for cached_property in cached_properties:
            for dependency in cached_property.dependencies:
                try:
                    dependency_graph[dependency].add(cached_property.name)
                except KeyError:
                    dependency_graph[dependency] = {cached_property.name}
        # Hard-code observable dependency since those are not CachedProperties.
        # TODO make observables CachedProperties?
        dependency_graph['mbar'] = {'observables'}
        return dependency_graph

    def _update_cache(self, key, new_value, check_changes=False):
        """Update the cache entry and invalidate the values that depend on it.

        Parameters
        ----------
        key : str
            The name of the value to update.
        new_value : object
            The new value of the key.
        check_changes : bool, optional
            If True and the new value is equal to the current one,
            the dependent cache values are not invalidated.

        """
        invalidate_cache = True
        try:
            old_value = self._cache[key]
        except KeyError:
            invalidate_cache = False
        else:
            if check_changes and self._check_equal(old_value, new_value):
                invalidate_cache = False
        # Update value and invalidate the cache.
        self._cache[key] = new_value
        if invalidate_cache:
            self._invalidate_cache_values(key)

    def _invalidate_cache_values(self, key):
        """Invalidate all the cache dependencies of key.

        Parameters
        ----------
        key : str
            The name of the cached whose dependencies must be invalidated.

        """
        dependency_graph = self._get_cache_dependency_graph()
        for k in dependency_graph[key]:
            # Invalidate observables that are in a separate cache.
            if k == 'observables':
                for observable in self.observables:
                    self._computed_observables[observable] = None
            else:
                # Invalidate dependencies of k.
                self._invalidate_cache_values(k)
                # Remove k.
                self._cache.pop(k, None)

    @staticmethod
    def _check_equal(old_value, new_value):
        """Broader equality check"""
        try:
            np.testing.assert_equal(old_value, new_value)
        except AssertionError:
            return False
        return True

    # -------------------------------------------------------------------------
    # Cached properties.
    # -------------------------------------------------------------------------

    max_n_iterations = CachedProperty('max_n_iterations', check_changes=True)

    @max_n_iterations.validator
    def max_n_iterations(self, instance, new_value):
        """The maximum allowed value for max_n_iterations is n_iterations."""
        if new_value is None or new_value > instance.n_iterations:
            new_value = instance.n_iterations
        return new_value

    use_full_trajectory = CachedProperty('use_full_trajectory', check_changes=True)

    @use_full_trajectory.validator
    def use_full_trajectory(self, _, new_value):
        if type(new_value) is not bool:
            raise ValueError("use_full_trajectory must be a boolean!")
        return new_value

    _extra_analysis_kwargs = CachedProperty('_extra_analysis_kwargs', check_changes=True)

    # -------------------------------------------------------------------------
    # Abstract methods.
    # -------------------------------------------------------------------------

    def read_energies(self):
        """
        Extract energies from the ncfile and order them by replica, state, iteration.

        Returns
        -------
        sampled_energy_matrix : np.ndarray of shape [n_replicas, n_states, n_iterations]
            Potential energy matrix of the sampled states.
        unsampled_energy_matrix : np.ndarray of shape [n_replicas, n_unsamped_states, n_iterations]
            Potential energy matrix of the unsampled states.
            Energy from each drawn sample n, evaluated at unsampled state l.
            If no unsampled states were drawn, this will be shape (0,N).
        neighborhoods : np.ndarray of shape [n_replicas, n_states, n_iterations]
            Neighborhood energies were computed at, uses a boolean mask over the energy_matrix.
        replica_state_indices : np.ndarray of shape [n_replicas, n_iterations]
            States sampled by the replicas in the energy_matrix

        """
        # TODO: should we keep it unified and always truncate to max_n_iterations?
        return self._read_energies(truncate_max_n_iterations=False)

    def _read_energies(self, truncate_max_n_iterations=False):
        """
        Extract energies from the ncfile and order them by replica, state, iteration.

        Parameters
        ----------
        truncate_max_n_iterations : bool, optional, default=False
            If True, will truncate the data to self.max_n_iterations.

        Returns
        -------
        sampled_energy_matrix : numpy.ndarray with shape (n_replicas, n_states, n_iterations)
            ``sampled_energy_matrix[replica, state, iteration]`` is the reduced potential of replica ``replica`` at sampled state ``state`` for iteration ``iteration``
        unsampled_energy_matrix : numpy.ndarray with shape (n_replicas, n_states, n_iterations)
            ``unsampled_energy_matrix[replica, state, iteration]`` is the reduced potential of replica i at unsampled state j for iteration ``iteration``
        neighborhoods : numpy.ndarray with shape (n_replicas, n_states, n_iterations)
            ``neighborhoods[replica, state, iteration]`` is 1 if the energy for replica ``replica`` at iteration ``iteration`` was computed for state ``state``, 0 otherwise
        replica_state_indices : numpy.ndarray with shape (n_replicas, n_iterations)
            ``replica_state_indices[replica, iteration]`` is the thermodynamic state index sampled by replica ``replica`` at iteration ``iteration``
        """
        logger.debug("Reading energies...")
        # reporter_energies is [energy_sampled_states, neighborhoods, energy_unsampled_states].
        energy_data = list(self._reporter.read_energies())
        energy_data.append(self._reporter.read_replica_thermodynamic_states())
        logger.debug("Done.")

        # Truncate the number of iterations to self.max_n_iterations if requested.
        if truncate_max_n_iterations:
            for i, energies in enumerate(energy_data):
                # The +1 accounts for minimization iteration.
                energy_data[i] = energies[:self.max_n_iterations+1]

        # Convert from (n_iterations, n_replicas, n_states) to (n_replicas, n_states, n_iterations).
        for i, energies in enumerate(energy_data):
            energy_data[i] = np.moveaxis(energies, 0, -1)

        # Unpack.
        sampled_energy_matrix, neighborhoods, unsampled_energy_matrix, replicas_state_indices = energy_data
        # TODO: Figure out what format we need the data in to be useful for both global and local MBAR/WHAM
        # For now, we simply can't handle analysis of non-global calculations.
        if np.any(neighborhoods == 0):
            raise Exception('Non-global MBAR analysis not implemented yet.')

        return sampled_energy_matrix, unsampled_energy_matrix, neighborhoods, replicas_state_indices

    @property
    def has_log_weights(self):
        """
        Return True if the storage has log weights, False otherwise
        """
        try:
            # Check that logZ and log_weights have per-iteration data
            # If either of these return a ValueError, then no history data are available
            _ = self._reporter.read_logZ(0)
            _ = self._reporter.read_online_analysis_data(0, 'log_weights')
            return True
        except ValueError:
            return False

    def read_log_weights(self):
        """
        Extract log weights from the ncfile, if present.
        Returns ValueError if not present.

        Returns
        -------
        log_weights : np.ndarray of shape [n_states, n_iterations]
            log_weights[l,n] is the log weight applied to state ``l``
            during the collection of samples at iteration ``n``

        """
        log_weights = np.array(
            self._reporter.read_online_analysis_data(slice(None, None), 'log_weights')['log_weights'])

        log_weights = np.moveaxis(log_weights, 0, -1)
        return log_weights

    def read_logZ(self, iteration=None):
        """
        Extract logZ estimates from the ncfile, if present.
        Returns ValueError if not present.

        Parameters
        ----------
        iteration : int or slice, optional, default=None
            If specified, iteration or slice of iterations to extract

        Returns
        -------
        logZ : np.ndarray of shape [n_states, n_iterations]
            logZ[l,n] is the online logZ estimate for state ``l`` at iteration ``n``

        """
        if iteration == -1:
            log_z = self._reporter.read_logZ(iteration)
        else:
            if iteration is not None:
                log_z = self._reporter.read_online_analysis_data(iteration, "logZ")["logZ"]
            else:
                log_z = self._reporter.read_online_analysis_data(slice(0, None), "logZ")["logZ"]
            log_z = np.moveaxis(log_z, 0, -1)

        # We don't want logZ to be a masked array
        log_z = np.array(log_z)

        return log_z

    def get_effective_energy_timeseries(self, energies=None, replica_state_indices=None):
        """
        Generate the effective energy (negative log deviance) timeseries that is generated for this phase

        The effective energy for a series of samples x_n, n = 1..N, is defined as

        u_n = - \ln \pi(x_n) + c

        where \pi(x) is the probability density being sampled, and c is an arbitrary constant.

        Parameters
        ----------
        energies : ndarray of shape (K,L,N), optional, Default: None
            Energies from replicas K, sampled states L, and iterations N
            If provided, then states input_sampled_states must also be provided
        replica_state_indices : ndarray of shape (K,N), optional, Default: None
            Integer indices of each sampled state (matching L dimension in input_energy)
            that each replica K sampled every iteration N.
            If provided, then states input_energies must also be provided

        Returns
        -------
        u_n : ndarray of shape (N,)
            u_n[n] is the negative log deviance of the same from iteration ``n``
            Timeseries used to determine equilibration time and statistical inefficiency.

        """

        raise NotImplementedError("This class has not implemented this function")

    # -------------------------------------------------------------------------
    # MBAR routines.
    # -------------------------------------------------------------------------

    @staticmethod
    def reformat_energies_for_mbar(u_kln: np.ndarray, n_k: Optional[np.ndarray]=None):
        """
        Convert [replica, state, iteration] data into [state, total_iteration] data

        This method assumes that the first dimension are all samplers,
        the second dimension are all the thermodynamic states energies were evaluated at
        and an equal number of samples were drawn from each k'th sampler, UNLESS n_k is specified.

        Parameters
        ----------
        u_kln : np.ndarray of shape (K,L,N')
            K = number of replica samplers
            L = number of thermodynamic states,
            N' = number of iterations from state k
        n_k : np.ndarray of shape K or None
            Number of samples each _SAMPLER_ (k) has drawn
            This allows you to have trailing entries on a given kth row in the n'th (n prime) index
            which do not contribute to the conversion.

            If this is None, assumes ALL samplers have the same number of samples
            such that N_k = N' for all k

            **WARNING**: N_k is number of samples the SAMPLER drew in total,
            NOT how many samples were drawn from each thermodynamic state L.
            This method knows nothing of how many samples were drawn from each state.

        Returns
        -------
        u_ln : np.ndarray of shape (L, N)
            Reduced, non-sparse data format
            L = number of thermodynamic states
            N = \sum_k N_k. note this is not N'
        """
        k, l, n = u_kln.shape
        if n_k is None:
            n_k = np.ones(k, dtype=np.int32)*n
        u_ln = np.zeros([l, n_k.sum()])
        n_counter = 0
        for k_index in range(k):
            u_ln[:, n_counter:n_counter + n_k[k_index]] = u_kln[k_index, :, :n_k[k_index]]
            n_counter += n_k[k_index]
        return u_ln

    # Private Class Methods
    def _create_mbar(self, energy_matrix, samples_per_state):
        """
        Initialize MBAR for Free Energy and Enthalpy estimates, this may take a while.
        This function is helpful for those who want to create a slightly different mbar object with different
        parameters.

        This function is hidden from the user unless they really, really need to create their own mbar object

        Parameters
        ----------
        energy_matrix : array of numpy.float64, optional, default=None
           Reduced potential energies of the replicas.
        samples_per_state : array of ints, optional, default=None
           Number of samples drawn from each kth state.

        """
        # Initialize MBAR (computing free energy estimates, which may take a while)
        logger.debug("Computing free energy differences...")
        self.mbar = MBAR(energy_matrix, samples_per_state, **self._extra_analysis_kwargs)
        logger.debug('Done.')
        return self.mbar

    def _read_online_data_if_present(self):
        """
        Attempt to read the online analysis data needed to hot-start the output
        """
        try:
            online_f_k = self.reporter.read_online_analysis_data(None, 'f_k')['f_k']
            self._online_data = {'initial_f_k': online_f_k}
        except ValueError:
            self._online_data = None

    # -------------------------------------------------------------------------
    # Analysis combination.
    # -------------------------------------------------------------------------

    def _combine_phases(self, other, operator='+'):
        """
        Workhorse function when creating a :class:`MultiPhaseAnalyzer` object by combining single
        :class:`PhaseAnalyzer`s
        """
        phases = [self]
        names = []
        signs = [self._sign]
        # Reset self._sign
        self._sign = '+'
        if self.name is None:
            names.append(multistate.utils.generate_phase_name(self.name, []))
        else:
            names.append(self.name)
        if isinstance(other, MultiPhaseAnalyzer):
            new_phases = other.phases
            new_signs = other.signs
            new_names = other.names
            final_new_names = []
            for name in new_names:
                other_names = [n for n in new_names if n != name]
                final_new_names.append(multistate.utils.generate_phase_name(name, other_names + names))
            names.extend(final_new_names)
            for new_sign in new_signs:
                if operator != '+' and new_sign == '+':
                    signs.append('-')
                else:
                    signs.append('+')
            phases.extend(new_phases)
        elif isinstance(other, PhaseAnalyzer):
            names.append(multistate.utils.generate_phase_name(other.name, names))
            if operator != '+' and other._sign == '+':
                signs.append('-')
            else:
                signs.append('+')
            # Reset the other's sign if it got set to negative
            other._sign = '+'
            phases.append(other)
        else:
            base_err = "cannot {} 'PhaseAnalyzer' and '{}' objects"
            if operator == '+':
                err = base_err.format('add', type(other))
            else:
                err = base_err.format('subtract', type(other))
            raise TypeError(err)
        phase_pass = {'phases': phases, 'signs': signs, 'names': names}
        return MultiPhaseAnalyzer(phase_pass)

    def __add__(self, other):
        return self._combine_phases(other, operator='+')

    def __sub__(self, other):
        return self._combine_phases(other, operator='-')

    def __neg__(self):
        """Internally handle the internal sign"""
        if self._sign == '+':
            self._sign = '-'
        else:
            self._sign = '+'
        return self

class MultiStateSamplerAnalyzer(PhaseAnalyzer):
    """
    The MultiStateSamplerAnalyzer is the analyzer for a simulation generated from a MultiStateSampler simulation,
    implemented as an instance of the :class:`PhaseAnalyzer`.

    Parameters
    ----------
    unbias_restraint : bool, optional
        If True and a radially-symmetric restraint was used in the simulation,
        the analyzer will remove the bias introduced by the restraint by
        reweighting each of the end-points to a state using a square-well
        potential restraint.

    restraint_energy_cutoff : float or 'auto', optional
        When the restraint is unbiased, the analyzer discards all the samples
        for which the restrain potential energy (in kT) is above this cutoff.
        Effectively, this is equivalent to placing a hard wall potential at a
        restraint distance such that the restraint potential energy is equal to
        ``restraint_energy_cutoff``.

        If ``'auto'`` and ``restraint_distance_cutoff`` is ``None``, this will
        be set to the 99.9-percentile of the distribution of the restraint energies
        in the bound state.

    restraint_distance_cutoff : openmm.unit.Quantity or 'auto', optional
        When the restraint is unbiased, the analyzer discards all the samples
        for which the distance between the restrained atoms is above this cutoff.
        Effectively, this is equivalent to placing a hard wall potential at a
        restraint distance ``restraint_distance_cutoff``.

        If ``'auto'`` and ``restraint_energy_cutoff`` is not specified, this will
        be set to the 99.9-percentile of the distribution of the restraint distances
        in the bound state.

    Attributes
    ----------
    unbias_restraint
    restraint_energy_cutoff
    restraint_distance_cutoff
    mbar
    n_equilibration_iterations
    statistical_inefficiency

    See Also
    --------
    PhaseAnalyzer

    """

    def __init__(self, *args, unbias_restraint=True, restraint_energy_cutoff='auto',
                 restraint_distance_cutoff='auto', **kwargs):

        # Warn that API is experimental
        logger.warn('Warning: The openmmtools.multistate API is experimental and may change in future releases')

        # super() calls clear() that initialize the cached variables.
        super().__init__(*args, **kwargs)

        # Cached values with dependencies.
        self.unbias_restraint = unbias_restraint
        self.restraint_energy_cutoff = restraint_energy_cutoff
        self.restraint_distance_cutoff = restraint_distance_cutoff

    # TODO use class syntax and add docstring after dropping python 3.5 support.
    _MixingStatistics = NamedTuple('MixingStatistics', [
        ('transition_matrix', np.ndarray),
        ('eigenvalues', np.ndarray),
        ('statistical_inefficiency', np.ndarray)
    ])

    def clear(self):
        """Reset all cached objects.

        This must to be called if the information in the reporter changes
        after analysis.
        """
        # Reset cached values that are read directly from the Reporter.
        # super() takes care of invalidating the cached properties.
        super().clear()
        self._radially_symmetric_restraint_data = None
        self._restraint_energies = {}
        self._restraint_distances = {}

    def generate_mixing_statistics(self, number_equilibrated: Union[int, None] = None) -> NamedTuple:
        """
        Compute and return replica mixing statistics.

        Compute the transition state matrix, its eigenvalues sorted from
        greatest to least, and the state index correlation function.

        Parameters
        ----------
        number_equilibrated : int, optional, default=None
            If specified, only samples ``number_equilibrated:end`` will
            be used in analysis. If not specified, automatically retrieves
            the number from equilibration data or generates it from the
            internal energy.

        Returns
        -------
        mixing_statistics : namedtuple
            A namedtuple containing the following attributes:
            - ``transition_matrix``: (nstates by nstates ``np.array``)
            - ``eigenvalues``: (nstates-dimensional ``np.array``)
            - ``statistical_inefficiency``: float
        """
        # Read data from disk
        if number_equilibrated is None:
            number_equilibrated = self.n_equilibration_iterations
        states = self._reporter.read_replica_thermodynamic_states()
        n_states = self._reporter.n_states
        n_ij = np.zeros([n_states, n_states], np.int64)

        # Compute empirical transition count matrix.
        for iteration in range(number_equilibrated, self.max_n_iterations - 1):
            for i_replica in range(self.n_replicas):
                i_state = states[iteration, i_replica]
                j_state = states[iteration + 1, i_replica]
                n_ij[i_state, j_state] += 1

        # Compute transition matrix estimate.
        # TODO: Replace with maximum likelihood reversible count estimator from msmbuilder or pyemma.
        t_ij = np.zeros([n_states, n_states], np.float64)
        for i_state in range(n_states):
            # Cast to float to ensure we don't get integer division
            denominator = float((n_ij[i_state, :].sum() + n_ij[:, i_state].sum()))
            if denominator > 0:
                for j_state in range(n_states):
                    t_ij[i_state, j_state] = (n_ij[i_state, j_state] + n_ij[j_state, i_state]) / denominator
            else:
                t_ij[i_state, i_state] = 1.0

        # Estimate eigenvalues
        mu = np.linalg.eigvals(t_ij)
        mu = -np.sort(-mu)  # Sort in descending order

        # Compute state index statistical inefficiency of stationary data.
        # states[n][k] is the state index of replica k at iteration n, but
        # the functions wants a list of timeseries states[k][n].
        states_kn = np.transpose(states[number_equilibrated:self.max_n_iterations,])
        g = timeseries.statisticalInefficiencyMultiple(states_kn)

        return self._MixingStatistics(transition_matrix=t_ij, eigenvalues=mu,
                                      statistical_inefficiency=g)

    def show_mixing_statistics(self, cutoff=0.05, number_equilibrated=None):
        """
        Print summary of mixing statistics. Passes information off to generate_mixing_statistics then prints it out to
        the logger

        Parameters
        ----------
        cutoff : float, optional, default=0.05
           Only transition probabilities above 'cutoff' will be printed
        number_equilibrated : int, optional, default=None
           If specified, only samples number_equilibrated:end will be used in analysis
           If not specified, it uses the internally held statistics best

        """

        mixing_statistics = self.generate_mixing_statistics(number_equilibrated=number_equilibrated)

        # Print observed transition probabilities.
        nstates = mixing_statistics.transition_matrix.shape[1]
        logger.info("Cumulative symmetrized state mixing transition matrix:")
        str_row = "{:6s}".format("")
        for jstate in range(nstates):
            str_row += "{:6d}".format(jstate)
        logger.info(str_row)

        for istate in range(nstates):
            str_row = ""
            str_row += "{:-6d}".format(istate)
            for jstate in range(nstates):
                P = mixing_statistics.transition_matrix[istate, jstate]
                if P >= cutoff:
                    str_row += "{:6.3f}".format(P)
                else:
                    str_row += "{:6s}".format("")
            logger.info(str_row)

        # Estimate second eigenvalue and equilibration time.
        perron_eigenvalue = mixing_statistics.eigenvalues[1]
        if perron_eigenvalue >= 1:
            logger.info('Perron eigenvalue is unity; Markov chain is decomposable.')
        else:
            equilibration_timescale = 1.0 / (1.0 - perron_eigenvalue)
            logger.info('Perron eigenvalue is {0:.5f}; state equilibration timescale '
                        'is ~ {1:.1f} iterations'.format(perron_eigenvalue, equilibration_timescale)
            )

        # Print information about replica state index statistical efficiency.
        logger.info('Replica state index statistical inefficiency is '
                    '{:.3f}'.format(mixing_statistics.statistical_inefficiency))

    def _get_radially_symmetric_restraint_data(self):
        """Return the radially-symmetric restraint force used in the bound state.

        Returns
        -------
        restraint_force : forces.RadiallySymmetricRestraintForce
            The restraint force used in the bound state.
        weights_group1 : list of openmm.unit.Quantity
            The masses of the restrained atoms in the first centroid group.
        weights_group2 : list of openmm.unit.Quantity
            The masses of the restrained atoms in the second centroid group.

        Raises
        ------
        TypeError
            If the end states don't have lambda_restraints set to 1.
        forces.NoForceFoundError
            If there are no radially-symmetric restraints in the bound state.

        """
        logger.debug('Trying to get radially symmetric restraint data...')

        # Check cached value.
        if self._radially_symmetric_restraint_data is not None:
            return self._radially_symmetric_restraint_data

        # Isolate the end states.
        logger.debug('Retrieving end thermodynamic states...')
        end_states = self._get_end_thermodynamic_states()

        # Isolate restraint force.
        logger.debug('Isolating restraint force...')
        system = end_states[0].system
        restraint_parent_class = forces.RadiallySymmetricRestraintForce
        # This raises forces.NoForceFoundError if there's no restraint to unbias.
        force_idx, restraint_force = forces.find_forces(system, force_type=restraint_parent_class,
                                                                only_one=True, include_subclasses=True)
        # The force is owned by the System, we have to copy to avoid the memory to be deallocated.
        logger.debug('Deep copying restraint force...')
        restraint_force = copy.deepcopy(restraint_force)

        # Check that the restraint was turned on at the end states.
        if end_states[0].lambda_restraints != 1.0 or end_states[-1].lambda_restraints != 1.0:
            raise TypeError('Cannot unbias a restraint that is turned off at one of the end states.')

        # Read the centroid weights (mass) of the restrained particles.
        logger.debug('Retrieving particle masses...')
        weights_group1 = [system.getParticleMass(i) for i in restraint_force.restrained_atom_indices1]
        weights_group2 = [system.getParticleMass(i) for i in restraint_force.restrained_atom_indices2]

        # Cache value so that we won't have to deserialize the system again.
        self._radially_symmetric_restraint_data = restraint_force, weights_group1, weights_group2
        logger.debug('Done.')
        return self._radially_symmetric_restraint_data

    # -------------------------------------------------------------------------
    # MBAR creation.
    # -------------------------------------------------------------------------

    def get_effective_energy_timeseries(self, energies=None, neighborhoods=None, replica_state_indices=None):
        """
        Generate the effective energy (negative log deviance) timeseries that is generated for this phase.

        The effective energy for a series of samples x_n, n = 1..N, is defined as

        u_n = - \ln \pi(x_n) + c

        where \pi(x) is the probability density being sampled, and c is an arbitrary constant.

        Parameters
        ----------
        energies : ndarray of shape (K,L,N), optional, Default: None
            Energies from replicas K, sampled states L, and iterations N.
            If provided, then states input_sampled_states must also be provided.
        neighborhoods : list of list of int, optional, Default: None
            neighborhoods[iteration] is the list of states in the neighborhood of the currently sampled state
        replica_state_indices : ndarray of shape (K,N), optional, Default: None
            Integer indices of each sampled state (matching L dimension in input_energy).
            that each replica K sampled every iteration N.
            If provided, then states input_energies must also be provided.

        Returns
        -------
        u_n : ndarray of shape (N,)
            u_n[n] is the negative log deviance of the same from iteration ``n``
            Timeseries used to determine equilibration time and statistical inefficiency.

        """
        if energies is None and replica_state_indices is None:
            # Case where no input is provided
            energies, _, neighborhoods, replica_state_indices = self._read_energies(truncate_max_n_iterations=True)
        elif (energies is not None) != (replica_state_indices is not None):
            # XOR operator
            raise ValueError("If input_energy or input_sampled_states are provided, "
                             "then the other cannot be None due to ambiguity!")

        n_replicas, n_states, n_iterations = energies.shape

        logger.debug("Assembling effective timeseries...")
        # Check for log weights
        has_log_weights = False
        if self.has_log_weights:
            has_log_weights = True
            log_weights = self.read_log_weights()
            f_l = - self.read_logZ(iteration=-1)  # use last (best) estimate of free energies
            logger.debug("log_weights: {}".format(log_weights))
            logger.debug("f_k: {}".format(f_l))

        u_n = np.zeros([n_iterations], np.float64)
        # Slice of all replicas, have to use this as : is too greedy
        replicas_slice = range(n_replicas)
        for iteration in range(n_iterations):
            # Slice the current sampled states by those replicas.
            states_slice = replica_state_indices[:, iteration]
            u_n[iteration] = np.sum(energies[replicas_slice, states_slice, iteration])

            # Correct for potentially-changing log weights
            if has_log_weights:
                u_n[iteration] += - np.sum(log_weights[states_slice, iteration]) \
                    + logsumexp(-f_l[:] + log_weights[:, iteration])

        logger.debug("Done.")
        return u_n

    def _compute_mbar_decorrelated_energies(self):
        """Return an MBAR-ready decorrelated energy matrix.

        The data is returned after discarding equilibration and truncating
        the iterations to self.max_n_iterations.

        Returns
        -------
        energy_matrix : energy matrix of shape (K,N) indexed by k,n
            K is the total number of states observables are desired.
            N is the total number of samples drawn from ALL states.
            The nth configuration is the energy evaluated in the kth thermodynamic state.
        samples_per_state : 1-D iterable of shape K
            The number of samples drawn from each kth state.
            The \sum samples_per_state = N.
        """
        # energy_data is [energy_sampled, energy_unsampled, neighborhood, replicas_state_indices]
        energy_data = list(self._read_energies(truncate_max_n_iterations=True))

        # Use the cached information to generate the equilibration data.
        sampled_energy_matrix, unsampled_energy_matrix, neighborhoods, replicas_state_indices = energy_data
        number_equilibrated, g_t, Neff_max = self._get_equilibration_data(sampled_energy_matrix, neighborhoods, replicas_state_indices)

        logger.debug("Assembling uncorrelated energies...")

        if not self.use_full_trajectory:
            for i, energies in enumerate(energy_data):
                # Discard equilibration iterations.
                energies = multistate.utils.remove_unequilibrated_data(energies, number_equilibrated, -1)
                # Subsample along the decorrelation data.
                energy_data[i] = multistate.utils.subsample_data_along_axis(energies, g_t, -1)
        sampled_energy_matrix, unsampled_energy_matrix, neighborhood, replicas_state_indices = energy_data

        # Initialize the MBAR matrices in ln form.
        n_replicas, n_sampled_states, n_iterations = sampled_energy_matrix.shape
        _, n_unsampled_states, _ = unsampled_energy_matrix.shape
        n_total_states = n_sampled_states + n_unsampled_states
        energy_matrix = np.zeros([n_total_states, n_iterations*n_replicas])
        samples_per_state = np.zeros([n_total_states], dtype=int)

        # Compute shift index for how many unsampled states there were.
        # This assume that we set an equal number of unsampled states at the end points.
        first_sampled_state = int(n_unsampled_states/2.0)
        last_sampled_state = n_total_states - first_sampled_state

        # Cast the sampled energy matrix from kln' to ln form.
        energy_matrix[first_sampled_state:last_sampled_state, :] = self.reformat_energies_for_mbar(sampled_energy_matrix)
        # Determine how many samples and which states they were drawn from.
        unique_sampled_states, counts = np.unique(replicas_state_indices, return_counts=True)
        # Assign those counts to the correct range of states.
        samples_per_state[first_sampled_state:last_sampled_state][unique_sampled_states] = counts
        # Add energies of unsampled states to the end points.
        if n_unsampled_states > 0:
            energy_matrix[[0, -1], :] = self.reformat_energies_for_mbar(unsampled_energy_matrix)
            logger.debug("Found expanded cutoff states in the energies!")
            logger.debug("Free energies will be reported relative to them instead!")
        if self.use_online_data and self._online_data is not None:
            # Do online data only if present and already exists as stored in self._online_data
            temp_online = copy.deepcopy(self._online_data)
            f_k_i = np.zeros(n_sampled_states + n_unsampled_states)
            online_f_k = temp_online['initial_f_k']
            f_k_i[first_sampled_state:last_sampled_state] = online_f_k
            temp_online['initial_f_k'] = f_k_i
            # Re-set the extra analysis_
            self._extra_analysis_kwargs = {**temp_online, **self._user_extra_analysis_kwargs}
        else:
            # Possible reset of the final key if online was true then false to invalidate any online data.
            self._extra_analysis_kwargs = {**self._user_extra_analysis_kwargs}

        # These cached values speed up considerably the computation of the
        # free energy profile along the restraint distance/energy cutoff.
        self._decorrelated_u_ln = energy_matrix
        self._decorrelated_N_l = samples_per_state

        logger.debug('Done.')
        return self._decorrelated_u_ln, self._decorrelated_N_l

    def _compute_mbar_unbiased_energies(self):
        """Unbias the restraint, and apply restraint energy/distance cutoffs.

        When there is a restraint to unbias, the function adds two extra unbiased
        states at the end points of the energy matrix. Otherwise, the return value
        is identical to self._compute_mbar_decorrelated_energies().

        Returns
        -------
        unbiased_decorrelated_u_ln : np.array
            A n_states x (n_sampled_states * n_unbiased_decorrelated_iterations)
            array of energies (in kT), where n_unbiased_decorrelated_iterations
            is generally <= n_decorrelated_iterations whe a restraint cutoff is
            set.
        unbiased_decorrelated_N_l : np.array
            The total number of samples drawn from each state (including the
            unbiased states).
        """
        logger.debug('Checking if we need to unbias the restraint...')

        # Check if we need to unbias the restraint.
        unbias_restraint = self.unbias_restraint
        if unbias_restraint:
            try:
                restraint_data = self._get_radially_symmetric_restraint_data()
            except (TypeError, forces.NoForceFoundError) as e:
                # If we don't need to unbias the restraint there's nothing else to do.
                logger.debug(str(e) + ' The restraint will not be unbiased.')
                unbias_restraint = False
        if not unbias_restraint:
            self._unbiased_decorrelated_u_ln = self._decorrelated_u_ln
            self._unbiased_decorrelated_N_l = self._decorrelated_N_l
            return self._unbiased_decorrelated_u_ln, self._unbiased_decorrelated_N_l

        # Compute the restraint energies/distances.
        restraint_force, weights_group1, weights_group2 = restraint_data
        logger.debug('Found {} restraint. The restraint will be unbiased.'.format(restraint_force.__class__.__name__))
        logger.debug('Receptor restrained atoms: {}'.format(restraint_force.restrained_atom_indices1))
        logger.debug('ligand restrained atoms: {}'.format(restraint_force.restrained_atom_indices2))

        # Compute restraint energies/distances.
        logger.debug('Computing restraint energies...')
        energies_ln, distances_ln = self._compute_restraint_energies(restraint_force, weights_group1,
                                                                     weights_group2)

        # Convert energies to kT unit for comparison to energy cutoff.
        energies_ln = energies_ln / self.kT
        logger.debug('Restraint energy mean: {} kT; std: {} kT'
                     ''.format(np.mean(energies_ln), np.std(energies_ln, ddof=1)))

        # Don't modify the cached decorrelated energies.
        u_ln = copy.deepcopy(self._decorrelated_u_ln)
        N_l = copy.deepcopy(self._decorrelated_N_l)
        n_decorrelated_iterations_ln = u_ln.shape[1]
        assert len(energies_ln) == n_decorrelated_iterations_ln, '{}, {}'.format(energies_ln.shape, u_ln.shape)
        assert len(self._decorrelated_state_indices_ln) == n_decorrelated_iterations_ln

        # Determine the cutoffs to use for the simulations.
        restraint_energy_cutoff, restraint_distance_cutoff = self._get_restraint_cutoffs()
        apply_energy_cutoff = restraint_energy_cutoff is not None
        apply_distance_cutoff = restraint_distance_cutoff is not None

        # We need to take into account the initial unsampled states to index correctly N_l.
        n_unsampled_states = len(u_ln) - self.n_states
        first_sampled_state_idx = int(n_unsampled_states / 2)

        # Determine which samples are outside the cutoffs or have to be truncated.
        columns_to_keep = []
        for iteration_ln_idx, state_idx in enumerate(self._decorrelated_state_indices_ln):
            if ((apply_energy_cutoff and energies_ln[iteration_ln_idx] > restraint_energy_cutoff) or
                    (apply_distance_cutoff and distances_ln[iteration_ln_idx] > restraint_distance_cutoff)):
                # Update the number of samples generated from its state.
                N_l[state_idx + first_sampled_state_idx] -= 1
            else:
                columns_to_keep.append(iteration_ln_idx)

        # Drop all columns that exceed the cutoff(s).
        n_discarded = n_decorrelated_iterations_ln - len(columns_to_keep)
        logger.debug('Discarding {}/{} samples outside the cutoffs (restraint_distance_cutoff: {}, '
                     'restraint_energy_cutoff: {}).'.format(n_discarded, n_decorrelated_iterations_ln,
                                                            restraint_distance_cutoff,
                                                            restraint_energy_cutoff))
        u_ln = u_ln[:, columns_to_keep]

        # Add new end states that don't include the restraint.
        energies_ln = energies_ln[columns_to_keep]
        n_states, n_iterations = u_ln.shape
        n_states_new = n_states + 2
        N_l_new = np.zeros(n_states_new, N_l.dtype)
        u_ln_new = np.zeros((n_states_new, n_iterations), u_ln.dtype)
        u_ln_new[0, :] = u_ln[0] - energies_ln
        u_ln_new[-1, :] = u_ln[-1] - energies_ln
        # Copy old values.
        N_l_new[1:-1] = N_l
        u_ln_new[1:-1, :] = u_ln
        # Expand the f_k_i if need be
        try:
            f_k_i_new = np.zeros(n_states_new, N_l.dtype)
            f_k_i_new[1:-1] = self._extra_analysis_kwargs['initial_f_k']  # This triggers the KeyError Trap
            self._extra_analysis_kwargs['initial_f_k'] = f_k_i_new  # This triggers the ValueError trap
        except (KeyError, ValueError):
            # Not there, move on, or already set nothing to do
            pass

        # Cache new values.
        self._unbiased_decorrelated_u_ln = u_ln_new
        self._unbiased_decorrelated_N_l = N_l_new

        logger.debug('Done.')
        return self._unbiased_decorrelated_u_ln, self._unbiased_decorrelated_N_l

    def _compute_restraint_energies(self, restraint_force, weights_group1, weights_group2):
        """Compute the restrain energies and distances for the uncorrelated iterations.

        Parameters
        ----------
        restraint_force : forces.RadiallySymmetricRestraintForce
            The restraint force.
        weights_group1 : list of float
            The mass of the particle in the first CustomCentroidBondForce group.
        weights_group2 : list of float
            The mass of the particles in the second CustomCentroidBondForce group.

        Returns
        -------
        restraint_energies_ln : openmm.unit.Quantity
            A (n_sampled_states * n_decorrelated_iterations)-long array with
            the restrain energies (units of energy/mole).
        restraint_distances_ln : openmm.unit.Quantity or None
            If we are not applying a distance cutoff, this is None. Otherwise,
            a (n_sampled_states * n_decorrelated_iterations)-long array with
            the restrain distances (units of length) for each frame.

        """
        decorrelated_iterations = self._decorrelated_iterations  # Shortcut.
        decorrelated_iterations_set = set(decorrelated_iterations)

        # Determine total number of energies/distances to compute.
        # The +1 is for the minimization iteration.
        n_frames_ln = self.n_replicas * len(decorrelated_iterations)

        # Computing the restraint energies/distances is expensive and we
        # don't want to recompute everything when _decorrelated_iterations
        # changes (e.g. when max_n_iterations changes) so we keep the cached
        # values of the iterations we have computed.
        # The dictionary instead of a masked array is for memory efficiency
        # since the matrix will be very sparse (especially with SAMS).

        def extract_decorrelated(cached_dict, dtype, unit):
            if not decorrelated_iterations_set.issubset(set(cached_dict)):
                return None
            decorrelated = np.zeros(n_frames_ln, dtype=dtype)
            for replica_idx in range(self.n_replicas):
                for iteration_idx, iteration in enumerate(decorrelated_iterations):
                    frame_idx = replica_idx*len(decorrelated_iterations) + iteration_idx
                    decorrelated[frame_idx] = cached_dict[iteration][replica_idx]
            return decorrelated * unit

        # We compute the distances only if we are using a distance cutoff.
        _, compute_distances = self._get_use_restraint_cutoff()

        # Check cached values.
        if compute_distances and decorrelated_iterations_set.issubset(set(self._restraint_distances)):
            compute_distances = False
        if decorrelated_iterations_set.issubset(set(self._restraint_energies)) and not compute_distances:
            return (extract_decorrelated(self._restraint_energies, dtype=np.float64, unit=_OPENMM_ENERGY_UNIT),
                    extract_decorrelated(self._restraint_distances, dtype=np.float32, unit=_MDTRAJ_DISTANCE_UNIT))

        # Don't modify the original restraint force.
        restraint_force = copy.deepcopy(restraint_force)
        is_periodic = restraint_force.usesPeriodicBoundaryConditions()

        # Store the indices of the restrained atoms in the reduced system.
        analysis_indices = self._reporter.analysis_particle_indices

        mapped_restrained_indices1 = restraint_force.restrained_atom_indices1
        mapped_restrained_indices2 = restraint_force.restrained_atom_indices2

        mapped_restrained_indices1 = [analysis_indices.index(index)
                                      for index in mapped_restrained_indices1]
        mapped_restrained_indices2 = [analysis_indices.index(index)
                                      for index in mapped_restrained_indices2]

        mapped_restrained_indices = (mapped_restrained_indices1 +
                                     mapped_restrained_indices2)

        # Create new system with only solute and restraint forces.
        reduced_system = openmm.System()
        for weight in weights_group1 + weights_group2:
            reduced_system.addParticle(weight)
        # Adapt the restraint force atom indices to the reduced system.
        n_atoms1 = len(weights_group1)
        n_atoms = n_atoms1 + len(weights_group2)
        restraint_force.restrained_atom_indices1 = list(range(n_atoms1))
        restraint_force.restrained_atom_indices2 = list(range(n_atoms1, n_atoms))
        reduced_system.addForce(restraint_force)

        # If we need to image the molecule, we need an MDTraj trajectory.
        if compute_distances and is_periodic:
            # Create topology with only the restrained atoms.
            serialized_topography = self._reporter.read_dict('metadata/topography')
            topography = utils.deserialize(serialized_topography)
            topology = topography.topology
            topology = topology.subset(self._reporter.analysis_particle_indices)
            # Use the receptor as an anchor molecule and image the ligand.
            anchor_molecules = [{a for a in topology.atoms if a.index in set(topography.receptor_atoms)}]
            imaged_molecules = [{a for a in topology.atoms if a.index in set(topography.ligand_atoms)}]
            # Initialize trajectory object needed for imaging molecules.
            trajectory = mdtraj.Trajectory(xyz=np.zeros((topology.n_atoms, 3)), topology=topology)

        # Create context used to compute the energies.
        integrator = openmm.VerletIntegrator(1.0*units.femtosecond)
        platform = openmm.Platform.getPlatformByName('CPU')
        context = openmm.Context(reduced_system, integrator, platform)

        # TODO: we need to provide a reporter generator to iterate over single
        # TODO:     iterations but reading automatically one chunksize at a time.
        # chunk_size = self._reporter.checkpoint_interval
        # iterations_groups = itertools.groupby(enumerate(decorrelated_iterations), key=lambda x: int(x[1] / chunk_size))

        # Pre-computing energies/distances.
        logger.debug('Computing restraint energies/distances...')
        for iteration_idx, iteration in enumerate(decorrelated_iterations):
            # Check if we have already computed this energy/distance.
            if (iteration in self._restraint_energies and
                    (not compute_distances or iteration in self._restraint_distances)):
                continue
            self._restraint_energies[iteration] = {}
            if compute_distances:
                self._restraint_distances[iteration] = {}

            # Read sampler states only if we haven't computed this iteration yet.
            # Obtain solute only sampler states.
            sampler_states = self._reporter.read_sampler_states(iteration=iteration,
                                                                analysis_particles_only=True)

            for replica_idx, sampler_state in enumerate(sampler_states):
                sliced_sampler_state = sampler_state[mapped_restrained_indices]
                sliced_sampler_state.apply_to_context(context)
                potential_energy = context.getState(getEnergy=True).getPotentialEnergy()
                self._restraint_energies[iteration][replica_idx] = potential_energy / _OPENMM_ENERGY_UNIT

                if compute_distances:
                    # Check if an analytical solution is available.
                    try:
                        distance = restraint_force.distance_at_energy(potential_energy) / _MDTRAJ_DISTANCE_UNIT
                    except (NotImplementedError, ValueError):
                        if is_periodic:
                            # Update trajectory positions/box vectors.
                            trajectory.xyz = (sampler_state.positions / _MDTRAJ_DISTANCE_UNIT).astype(np.float32)
                            trajectory.unitcell_vectors = np.array([sampler_state.box_vectors / _MDTRAJ_DISTANCE_UNIT],
                                                                   dtype=np.float32)
                            trajectory.image_molecules(inplace=True, anchor_molecules=anchor_molecules,
                                                       other_molecules=imaged_molecules)
                            positions_group1 = trajectory.xyz[0][mapped_restrained_indices1]
                            positions_group2 = trajectory.xyz[0][mapped_restrained_indices2]
                        else:
                            positions_group1 = sampler_state.positions[mapped_restrained_indices1]
                            positions_group2 = sampler_state.positions[mapped_restrained_indices2]
                            positions_group1 /= _MDTRAJ_DISTANCE_UNIT
                            positions_group2 /= _MDTRAJ_DISTANCE_UNIT

                        # Set output arrays.
                        distance = compute_centroid_distance(positions_group1, positions_group2,
                                                             weights_group1, weights_group2)
                    self._restraint_distances[iteration][replica_idx] = distance

        return (extract_decorrelated(self._restraint_energies, dtype=np.float64, unit=_OPENMM_ENERGY_UNIT),
                extract_decorrelated(self._restraint_distances, dtype=np.float32, unit=_MDTRAJ_DISTANCE_UNIT))

    def _get_use_restraint_cutoff(self):
        """Determine if we need to apply a cutoff on the restraint energies and/or distances."""
        apply_distance_cutoff = isinstance(self.restraint_distance_cutoff, units.Quantity)
        apply_energy_cutoff = isinstance(self.restraint_energy_cutoff, float)
        # When both cutoffs are auto, use distance cutoff.
        if self.restraint_distance_cutoff == 'auto' and not apply_energy_cutoff:
            apply_distance_cutoff = True
        elif self.restraint_energy_cutoff == 'auto' and self.restraint_distance_cutoff is None:
            apply_energy_cutoff = True
        return apply_energy_cutoff, apply_distance_cutoff

    def _get_restraint_energies_distances_at_state(self, state_idx, get_energies=True, get_distances=True):
        """Return the restraint energies and distances for a single state."""
        # Resolve negative indices.
        if state_idx < 0:
            state_idx = self.n_states + state_idx
        replica_state_indices = self._reporter.read_replica_thermodynamic_states()

        # Gather the state restraint energies/distances.
        state_energies = [] if get_energies else None
        state_distances = [] if get_distances else None
        for state_data, cached_data in [(state_energies, self._restraint_energies),
                                        (state_distances, self._restraint_distances)]:
            if state_data is None:
                continue
            for iteration, states_data in cached_data.items():
                # Find the replicas in this state.
                replica_indices = np.where(replica_state_indices[iteration] == state_idx)[0]
                for replica_idx in replica_indices:
                    state_data.append(states_data[replica_idx])

        # Convert to the correct units.
        if state_energies is not None:
            state_energies = np.array(state_energies) * _OPENMM_ENERGY_UNIT / self.kT
        if state_distances is not None:
            state_distances = np.array(state_distances) * _MDTRAJ_DISTANCE_UNIT
        return state_energies, state_distances

    def _determine_automatic_restraint_cutoff(self, compute_energy_cutoff=True, compute_distance_cutoff=True):
        """Automatically determine the restraint cutoffs.

        This must be called after _compute_restraint_energies(). The cutoffs are
        determine as the 99.9%-percentile of the distribution of the restraint
        energies/distances in the bound state.
        """
        # Gather the bound state restraint energies/distances.
        state0_energies, state0_distances = self._get_restraint_energies_distances_at_state(
            state_idx=0, get_energies=compute_energy_cutoff, get_distances=compute_distance_cutoff)

        # Compute cutoff as the 99.9%-percentile of the energies/distances distributions.
        energy_cutoff = None
        distance_cutoff = None
        err_msg = ('Thermodynamic state 0 has not been sampled enough to '
                   'determine automatically the restraint {} cutoff.')

        if compute_energy_cutoff:
            if len(state0_energies) == 0:
                raise InsufficientData(err_msg.format('energy'))
            energy_cutoff = np.percentile(state0_energies, 99.9)
        if compute_distance_cutoff:
            if len(state0_distances) == 0:
                raise InsufficientData(err_msg.format('distance'))
            state0_distances /= _MDTRAJ_DISTANCE_UNIT
            distance_cutoff = np.percentile(state0_distances, 99.9) * _MDTRAJ_DISTANCE_UNIT

        return energy_cutoff, distance_cutoff

    def _get_restraint_cutoffs(self):
        """Return the restraint energies and distance cutoff to be used for unbiasing."""
        apply_energy_cutoff, apply_distance_cutoff = self._get_use_restraint_cutoff()
        # Determine automatically the restraint distance cutoff is necessary.
        if apply_distance_cutoff and self.restraint_distance_cutoff == 'auto':
            _, restraint_distance_cutoff = self._determine_automatic_restraint_cutoff(compute_energy_cutoff=False)
            logger.debug('Chosen automatically a restraint distance cutoff of {}'.format(restraint_distance_cutoff))
        elif self.restraint_distance_cutoff == 'auto':
            restraint_distance_cutoff = None
        else:
            restraint_distance_cutoff = self.restraint_distance_cutoff
        # Determine automatically the restraint energy cutoff is necessary.
        if apply_energy_cutoff and self.restraint_energy_cutoff == 'auto':
            restraint_energy_cutoff, _ = self._determine_automatic_restraint_cutoff(compute_distance_cutoff=False)
            logger.debug('Chosen automatically a restraint energy cutoff of {}kT'.format(restraint_energy_cutoff))
        elif self.restraint_energy_cutoff == 'auto':
            restraint_energy_cutoff = None
        else:
            restraint_energy_cutoff = self.restraint_energy_cutoff
        return restraint_energy_cutoff, restraint_distance_cutoff

    # -------------------------------------------------------------------------
    # Observables.
    # -------------------------------------------------------------------------

    def _compute_free_energy(self):
        """
        Estimate free energies of all alchemical states.
        """
        nstates = self.mbar.N_k.size

        # Get matrix of dimensionless free energy differences and uncertainty estimate.
        logger.debug("Computing covariance matrix...")

        try:
            # pymbar 2
            (Deltaf_ij, dDeltaf_ij) = self.mbar.getFreeEnergyDifferences()
        except ValueError:
            # pymbar 3
            (Deltaf_ij, dDeltaf_ij, _) = self.mbar.getFreeEnergyDifferences()

        # Matrix of free energy differences
        logger.debug("Deltaf_ij:")
        for i in range(nstates):
            str_row = ""
            for j in range(nstates):
                str_row += "{:8.3f}".format(Deltaf_ij[i, j])
            logger.debug(str_row)

        # Matrix of uncertainties in free energy difference (expectations standard
        # deviations of the estimator about the true free energy)
        logger.debug("dDeltaf_ij:")
        for i in range(nstates):
            str_row = ""
            for j in range(nstates):
                str_row += "{:8.3f}".format(dDeltaf_ij[i, j])
            logger.debug(str_row)

        # Return free energy differences and an estimate of the covariance.
        free_energy_dict = {'value': Deltaf_ij, 'error': dDeltaf_ij}
        self._computed_observables['free_energy'] = free_energy_dict

    def get_free_energy(self):
        """
        Compute the free energy and error in free energy from the MBAR object

        Output shape changes based on if there are unsampled states detected in the sampler

        Returns
        -------
        DeltaF_ij : ndarray of floats, shape (K,K) or (K+2, K+2)
            Difference in free energy from each state relative to each other state
        dDeltaF_ij : ndarray of floats, shape (K,K) or (K+2, K+2)
            Error in the difference in free energy from each state relative to each other state
        """
        if self._computed_observables['free_energy'] is None:
            self._compute_free_energy()
        free_energy_dict = self._computed_observables['free_energy']
        return free_energy_dict['value'], free_energy_dict['error']

    def _compute_enthalpy_and_entropy(self):
        """Function to compute the cached values of enthalpy and entropy"""
        (f_k, df_k, H_k, dH_k, S_k, dS_k) = self.mbar.computeEntropyAndEnthalpy()
        enthalpy = {'value': H_k, 'error': dH_k}
        entropy = {'value': S_k, 'error': dS_k}
        self._computed_observables['enthalpy'] = enthalpy
        self._computed_observables['entropy'] = entropy

    def get_enthalpy(self):
        """
        Compute the difference in enthalpy and error in that estimate from the MBAR object

        Output shape changes based on if there are unsampled states detected in the sampler

        Returns
        -------
        DeltaH_ij : ndarray of floats, shape (K,K) or (K+2, K+2)
            Difference in enthalpy from each state relative to each other state
        dDeltaH_ij : ndarray of floats, shape (K,K) or (K+2, K+2)
            Error in the difference in enthalpy from each state relative to each other state

        """
        if self._computed_observables['enthalpy'] is None:
            self._compute_enthalpy_and_entropy()
        enthalpy_dict = self._computed_observables['enthalpy']
        return enthalpy_dict['value'], enthalpy_dict['error']

    def get_entropy(self):
        """
        Compute the difference in entropy and error in that estimate from the MBAR object

        Output shape changes based on if there are unsampled states detected in the sampler

        Returns
        -------
        DeltaH_ij : ndarray of floats, shape (K,K) or (K+2, K+2)
            Difference in enthalpy from each state relative to each other state
        dDeltaH_ij : ndarray of floats, shape (K,K) or (K+2, K+2)
            Error in the difference in enthalpy from each state relative to each other state

        """
        if self._computed_observables['entropy'] is None:
            self._compute_enthalpy_and_entropy()
        entropy_dict = self._computed_observables['entropy']
        return entropy_dict['value'], entropy_dict['error']

    def _get_equilibration_data(self, energies=None, neighborhoods=None, replica_state_indices=None):
        """Generate the equilibration data from best practices.

        Parameters
        ----------
        energies : ndarray of shape (K,L,N), optional, Default: None
            Energies from replicas K, sampled states L, and iterations N.
            If provided, then replica_state_indices must also be provided.
        neighborhoods : numpy.ndarray with shape (n_replicas, n_states, n_iterations)
            ``neighborhoods[replica, state, iteration]`` is 1 if the energy for
            replica ``replica`` at iteration ``iteration`` was computed for state ``state``,
            0 otherwise
        replica_state_indices : ndarray of shape (K,N), optional, Default: None
            Integer indices of each sampled state (matching L dimension in input_energy).
            that each replica K sampled every iteration N.
            If provided, then states input_energies must also be provided.

        Returns
        -------
        n_equilibration_iterations : int
            Number of equilibration iterations discarded
        statistical_inefficiency : float
            Statistical inefficiency of production iterations
        n_uncorrelated_iterations : float
            Effective number of uncorrelated iterations
        """
        u_n = self.get_effective_energy_timeseries(energies=energies, neighborhoods=neighborhoods, replica_state_indices=replica_state_indices)

        # For SAMS, if there is a second-stage start time, use only the asymptotically optimal data
        t0 = 1 # discard minimization frame
        try:
            iteration = len(u_n)
            data = self._reporter.read_online_analysis_data(None, 't0')
            t0 = max(t0, int(data['t0'][0]))
            logger.debug('t0 found; using initial t0 = {} instead of 1'.format(t0))
        except Exception as e:
            # No t0 found
            logger.debug('Could not find t0: {}'.format(e))
            pass

        # Discard equilibration samples.
        # TODO: if we include u_n[0] (the energy right after minimization) in the equilibration detection,
        # TODO:         then number_equilibrated is 0. Find a better way than just discarding first frame.
        i_t, g_i, n_effective_i = multistate.utils.get_equilibration_data_per_sample(u_n[t0:])
        n_effective_max = n_effective_i.max()
        i_max = n_effective_i.argmax()
        n_equilibration = i_t[i_max] + t0 # account for initially discarded frames
        g_t = g_i[i_max]

        # Store equilibration data
        self._equilibration_data = tuple([n_equilibration, g_t, n_effective_max])
        logger.debug('Equilibration data:')
        logger.debug(' number of iterations discarded to equilibration : {}'.format(n_equilibration))
        logger.debug(' statistical inefficiency of production region   : {}'.format(g_t))
        logger.debug(' effective number of uncorrelated samples        : {}'.format(n_effective_max))

        return n_equilibration, g_t, n_effective_max

    # -------------------------------------------------------------------------
    # Cached properties.
    # -------------------------------------------------------------------------

    unbias_restraint = CachedProperty('unbias_restraint', check_changes=True)
    restraint_energy_cutoff = CachedProperty('restraint_energy_cutoff', check_changes=True)
    restraint_distance_cutoff = CachedProperty('restraint_distance_cutoff', check_changes=True)

    _equilibration_data = CachedProperty(
        name='equilibration_data',
        dependencies=['reporter', 'max_n_iterations'],
        check_changes=True,
    )

    @_equilibration_data.default
    def _equilibration_data(self, instance):
        return instance._get_equilibration_data()

    _decorrelated_state_indices_ln = CachedProperty(
        name='decorrelated_state_indices_ln',
        dependencies=['equilibration_data', 'use_full_trajectory'],
    )

    @_decorrelated_state_indices_ln.default
    def _decorrelated_state_indices_ln(self, instance):
        """Compute the replica thermodynamic state indices in ln formats."""
        decorrelated_iterations = instance._decorrelated_iterations  # Shortcut.
        replica_state_indices = instance._reporter.read_replica_thermodynamic_states()
        n_correlated_iterations, instance._n_replicas = replica_state_indices.shape

        # Initialize output array.
        n_frames = instance.n_replicas * len(decorrelated_iterations)
        decorrelated_state_indices_ln = np.zeros(n_frames, dtype=np.int32)

        # Map ln columns to the state.
        for iteration_idx, iteration in enumerate(decorrelated_iterations):
            for replica_idx in range(instance.n_replicas):
                frame_idx = replica_idx*len(decorrelated_iterations) + iteration_idx
                # Set output array.
                state_idx = replica_state_indices[iteration, replica_idx]
                decorrelated_state_indices_ln[frame_idx] = state_idx
        instance._decorrelated_state_indices_ln = decorrelated_state_indices_ln
        return decorrelated_state_indices_ln

    _decorrelated_u_ln = CachedProperty(
        name='decorrelated_u_ln',
        dependencies=['equilibration_data', 'use_full_trajectory'],
    )

    @_decorrelated_u_ln.default
    def _decorrelated_u_ln(self, instance):
        return instance._compute_mbar_decorrelated_energies()[0]

    _decorrelated_N_l = CachedProperty(
        name='decorrelated_N_l',
        dependencies=['equilibration_data', 'use_full_trajectory'],
    )

    @_decorrelated_N_l.default
    def _decorrelated_N_l(self, instance):
        return instance._compute_mbar_decorrelated_energies()[1]

    _unbiased_decorrelated_u_ln = CachedProperty(
        name='unbiased_decorrelated_u_ln',
        dependencies=['unbias_restraint', 'restraint_energy_cutoff', 'restraint_distance_cutoff',
                      'decorrelated_state_indices_ln', 'decorrelated_u_ln', 'decorrelated_N_l'],
    )

    @_unbiased_decorrelated_u_ln.default
    def _unbiased_decorrelated_u_ln(self, instance):
        return instance._compute_mbar_unbiased_energies()[0]

    _unbiased_decorrelated_N_l = CachedProperty(
        name='unbiased_decorrelated_N_l',
        dependencies=['unbias_restraint', 'restraint_energy_cutoff', 'restraint_distance_cutoff',
                      'decorrelated_state_indices_ln', 'decorrelated_u_ln', 'decorrelated_N_l'],
    )

    @_unbiased_decorrelated_N_l.default
    def _unbiased_decorrelated_N_l(self, instance):
        return instance._compute_mbar_unbiased_energies()[1]

    mbar = CachedProperty(
        name='mbar',
        dependencies=['unbiased_decorrelated_u_ln', 'unbiased_decorrelated_N_l',
                      '_extra_analysis_kwargs'],
    )

    @mbar.default
    def mbar(self, instance):
        # u_ln[l,n] is the reduced potential for concatenated decorrelated snapshot n evaluated at thermodynamic state l
        # Shape is (n_states + n_unsampled_states, n_samples)
        # N_l[l] is the number of decorrelated samples sampled from thermodynamic state l, some can be 0.
        # Shape is (n_states + n_unsampled_states, )
        return instance._create_mbar(instance._unbiased_decorrelated_u_ln,
                                     instance._unbiased_decorrelated_N_l)

    # -------------------------------------------------------------------------
    # Dynamic properties.
    # -------------------------------------------------------------------------

    @property
    def n_equilibration_iterations(self):
        """int: The number of equilibration interations."""
        return self._equilibration_data[0]

    @property
    def statistical_inefficiency(self):
        """float: The statistical inefficiency of the sampler."""
        return self._equilibration_data[1]

    @property
    def _decorrelated_iterations(self):
        """list of int: the indices of the decorrelated iterations truncated to max_n_iterations."""
        if self.use_full_trajectory:
            return np.arange(self.max_n_iterations + 1, dtype=int)
        equilibrium_iterations = np.array(range(self.n_equilibration_iterations, self.max_n_iterations + 1))
        decorrelated_iterations_indices = timeseries.subsampleCorrelatedData(equilibrium_iterations,
                                                                             self.statistical_inefficiency)
        return equilibrium_iterations[decorrelated_iterations_indices]

# https://choderalab.slack.com/files/levi.naden/F4G6L9X8S/quick_diagram.png

class MultiPhaseAnalyzer(object):
    """
    Multiple Phase Analyzer creator, not to be directly called itself, but instead called by adding or subtracting
    different implemented :class:`PhaseAnalyzer` or other :class:`MultiPhaseAnalyzers`'s. The individual Phases of
    the :class:`MultiPhaseAnalyzer` are only references to existing Phase objects, not copies. All
    :class:`PhaseAnalyzer` and :class:`MultiPhaseAnalyzer` classes support ``+`` and ``-`` operations.

    The observables of this phase are determined through inspection of all the passed in phases and only observables
    which are shared can be computed. For example:

        ``PhaseA`` has ``.get_free_energy`` and ``.get_entropy``

        ``PhaseB`` has ``.get_free_energy`` and ``.get_enthalpy``,

        ``PhaseAB = PhaseA + PhaseB`` will only have a ``.get_free_energy`` method

    Because each Phase may have a different number of states, the ``reference_states`` property of each phase
    determines which states from each phase to read the data from.

    For observables defined by two states, the i'th and j'th reference states are used:

        If we define ``PhaseAB = PhaseA - PhaseB``

        Then ``PhaseAB.get_free_energy()`` is roughly equivalent to doing the following:

            ``A_i, A_j = PhaseA.reference_states``

            ``B_i, B_j = PhaseB.reference_states``

            ``PhaseA.get_free_energy()[A_i, A_j] - PhaseB.get_free_energy()[B_i, B_j]``

        The above is not exact since get_free_energy returns an error estimate as well

    For observables defined by a single state, only the i'th reference state is used

        Given ``PhaseAB = PhaseA + PhaseB``, ``PhaseAB.get_temperature()`` is equivalent to:

            ``A_i = PhaseA.reference_states[0]``

            ``B_i = PhaseB.reference_states[0]``

            ``PhaseA.get_temperature()[A_i] + PhaseB.get_temperature()[B_i]``

    For observables defined entirely by the phase, no reference states are needed.

        Given ``PhaseAB = PhaseA + PhaseB``, ``PhaseAB.get_standard_state_correction()`` gives:

            ``PhaseA.get_standard_state_correction() + PhaseB.get_standard_state_correction()``

    Each phase MUST use the same ObservablesRegistry, otherwise an error is raised

    This class is public to see its API.

    Parameters
    ----------
    phases : dict
        has keys "phases", "names", and "signs"

    Attributes
    ----------
    observables
    phases
    names
    signs
    registry

    See Also
    --------
    PhaseAnalyzer
    ObservablesRegistry

    """
    def __init__(self, phases):
        """
        Create the compound phase which is any combination of phases to generate a new MultiPhaseAnalyzer.

        """
        # Compare ObservableRegistries
        ref_registry = phases['phases'][0].registry
        for phase in phases['phases'][1:]:
            # Use is comparison since we are checking same insetance
            if phase.registry is not ref_registry:
                raise ValueError("Not all phases have the same ObservablesRegistry! Observable calculation "
                                 "will be inconsistent!")
        self.registry = ref_registry
        # Determine available observables
        observables = []
        for observable in self.registry.observables:
            shared_observable = True
            for phase in phases['phases']:
                if observable not in phase.observables:
                    shared_observable = False
                    break
            if shared_observable:
                observables.append(observable)
        if len(observables) == 0:
            raise RuntimeError("There are no shared computable observable between the phases, combining them will do "
                               "nothing.")
        self._observables = tuple(observables)
        self._phases = phases['phases']
        self._names = phases['names']
        self._signs = phases['signs']
        # Set the methods shared between both objects
        for observable in self.observables:
            setattr(self, "get_" + observable, self._spool_function(observable))

    def _spool_function(self, observable):
        """
        Dynamic observable calculator layer

        Must be in its own function to isolate the variable name space

        If you have this in the __init__, the "observable" variable colides with any others in the list, causing a
        the wrong property to be fetched.
        """
        return lambda: self._compute_observable(observable)

    @property
    def observables(self):
        """List of observables this :class:`MultiPhaseAnalyzer` can generate"""
        return self._observables

    @property
    def phases(self):
        """List of implemented :class:`PhaseAnalyzer`'s objects this :class:`MultiPhaseAnalyzer` is tied to"""
        return self._phases

    @property
    def names(self):
        """
        Unique list of string names identifying this phase. If this :class:`MultiPhaseAnalyzer` is combined with
        another, its possible that new names will be generated unique to that :class:`MultiPhaseAnalyzer`, but will
        still reference the same phase.

        When in doubt, use :func:`MultiPhaseAnalyzer.phases` to get the actual phase objects.
        """
        return self._names

    @property
    def signs(self):
        """
        List of signs that are used by the :class:`MultiPhaseAnalyzer` to
        """
        return self._signs

    def clear(self):
        """
        Clear the individual phases of their observables and estimators for re-computing quantities
        """
        for phase in self.phases:
            phase.clear()

    def _combine_phases(self, other, operator='+'):
        """
        Function to combine the phases regardless of operator to reduce code duplication. Creates a new
        :class:`MultiPhaseAnalyzer` object based on the combined phases of the other. Accepts either a
        :class:`PhaseAnalyzer` or a :class:`MultiPhaseAnalyzer`.

        If the names have collision, they are re-named with an extra digit at the end.

        Parameters
        ----------
        other : :class:`MultiPhaseAnalyzer` or :class:`PhaseAnalyzer`
        operator : sign of the operator connecting the two objects

        Returns
        -------
        output : :class:`MultiPhaseAnalyzer`
            New :class:`MultiPhaseAnalyzer` where the phases are the combined list of the individual phases from each
            component. Because the memory pointers to the individual phases are the same, changing any
            single :class:`PhaseAnalyzer`'s
            reference_state objects updates all :class:`MultiPhaseAnalyzer` objects they are tied to

        """
        phases = []
        names = []
        signs = []
        # create copies
        phases.extend(self.phases)
        names.extend(self.names)
        signs.extend(self.signs)
        if isinstance(other, MultiPhaseAnalyzer):
            new_phases = other.phases
            new_signs = other.signs
            new_names = other.names
            final_new_names = []
            for name in new_names:
                other_names = [n for n in new_names if n != name]
                final_new_names.append(multistate.utils.generate_phase_name(name, other_names + names))
            names.extend(final_new_names)
            for new_sign in new_signs:
                if (operator == '-' and new_sign == '+') or (operator == '+' and new_sign == '-'):
                    signs.append('-')
                else:
                    signs.append('+')
            signs.extend(new_signs)
            phases.extend(new_phases)
        elif isinstance(other, PhaseAnalyzer):
            names.append(multistate.utils.generate_phase_name(other.name, names))
            if (operator == '-' and other._sign == '+') or (operator == '+' and other._sign == '-'):
                signs.append('-')
            else:
                signs.append('+')
            other._sign = '+'  # Recast to positive if negated
            phases.append(other)
        else:
            baseerr = "cannot {} 'MultiPhaseAnalyzer' and '{}' objects"
            if operator == '+':
                err = baseerr.format('add', type(other))
            else:
                err = baseerr.format('subtract', type(other))
            raise TypeError(err)
        phase_pass = {'phases': phases, 'signs': signs, 'names': names}
        return MultiPhaseAnalyzer(phase_pass)

    def __add__(self, other):
        return self._combine_phases(other, operator='+')

    def __sub__(self, other):
        return self._combine_phases(other, operator='-')

    def __neg__(self):
        """
        Return a SHALLOW copy of self with negated signs so that the phase objects all still point to the same
        objects
        """
        new_signs = []
        for sign in self._signs:
            if sign == '+':
                new_signs.append('-')
            else:
                new_signs.append('+')
        # return a *shallow* copy of self with the signs reversed
        output = copy.copy(self)
        output._signs = new_signs
        return output

    def __str__(self):
        """Simplified string output"""
        header = "MultiPhaseAnalyzer<{}>"
        output_string = ""
        for phase_name, sign in zip(self.names, self.signs):
            if output_string == "" and sign == '-':
                output_string += '{}{} '.format(sign, phase_name)
            elif output_string == "":
                output_string += '{} '.format(phase_name)
            else:
                output_string += '{} {} '.format(sign, phase_name)
        return header.format(output_string)

    def __repr__(self):
        """Generate a detailed representation of the MultiPhase"""
        header = "MultiPhaseAnalyzer <\n{}>"
        output_string = ""
        for phase, phase_name, sign in zip(self.phases, self.names, self.signs):
            if output_string == "" and sign == '-':
                output_string += '{}{} ({})\n'.format(sign, phase_name, phase)
            elif output_string == "":
                output_string += '{} ({})\n'.format(phase_name, phase)
            else:
                output_string += '    {} {} ({})\n'.format(sign, phase_name, phase)
        return header.format(output_string)

    def _compute_observable(self, observable_name):
        """
        Helper function to compute arbitrary observable in both phases

        Parameters
        ----------
        observable_name : str
            Name of the observable as its defined in the ObservablesRegistry

        Returns
        -------
        observable_value
            The observable as its combined between all the phases

        """
        def prepare_phase_observable(single_phase):
            """Helper function to cast the observable in terms of observable's registry"""
            observable = getattr(single_phase, "get_" + observable_name)()
            if isinstance(single_phase, MultiPhaseAnalyzer):
                if observable_name in self.registry.observables_with_error:
                    observable_payload = dict()
                    observable_payload['value'], observable_payload['error'] = observable
                else:
                    observable_payload = observable
            else:
                raise_registry_error = False
                if observable_name in self.registry.observables_with_error:
                    observable_payload = {}
                    if observable_name in self.registry.observables_defined_by_phase:
                        observable_payload['value'], observable_payload['error'] = observable
                    elif observable_name in self.registry.observables_defined_by_single_state:
                        observable_payload['value'] = observable[0][single_phase.reference_states[0]]
                        observable_payload['error'] = observable[1][single_phase.reference_states[0]]
                    elif observable_name in self.registry.observables_defined_by_two_states:
                        observable_payload['value'] = observable[0][single_phase.reference_states[0],
                                                                    single_phase.reference_states[1]]
                        observable_payload['error'] = observable[1][single_phase.reference_states[0],
                                                                    single_phase.reference_states[1]]
                    else:
                        raise_registry_error = True
                else:  # No error
                    if observable_name in self.registry.observables_defined_by_phase:
                        observable_payload = observable
                    elif observable_name in self.registry.observables_defined_by_single_state:
                        observable_payload = observable[single_phase.reference_states[0]]
                    elif observable_name in self.registry.observables_defined_by_two_states:
                        observable_payload = observable[single_phase.reference_states[0],
                                                        single_phase.reference_states[1]]
                    else:
                        raise_registry_error = True
                if raise_registry_error:
                    raise RuntimeError("You have requested an observable that is improperly registered in the "
                                       "ObservablesRegistry!")
            return observable_payload

        def modify_final_output(passed_output, payload, sign):
            if observable_name in self.registry.observables_with_error:
                if sign == '+':
                    passed_output['value'] += payload['value']
                else:
                    passed_output['value'] -= payload['value']
                if observable_name in self.registry.observables_with_error_adding_linear:
                    passed_output['error'] += payload['error']
                elif observable_name in self.registry.observables_with_error_adding_quadrature:
                    passed_output['error'] = (passed_output['error']**2 + payload['error']**2)**0.5
            else:
                if sign == '+':
                    passed_output += payload
                else:
                    passed_output -= payload
            return passed_output

        if observable_name in self.registry.observables_with_error:
            final_output = {'value': 0, 'error': 0}
        else:
            final_output = 0
        for phase, phase_sign in zip(self.phases, self.signs):
            phase_observable = prepare_phase_observable(phase)
            final_output = modify_final_output(final_output, phase_observable, phase_sign)
        if observable_name in self.registry.observables_with_error:
            # Cast output to tuple
            final_output = (final_output['value'], final_output['error'])
        return final_output

reason: import numpy,from scipy