interface_dnm.py

import numpy as np
import birdepy.utility as ut
from birdepy.interface_probability import probability
import warnings


def bld_ll_fun(data, likelihood, model, z_trunc, options):
    """
    Builds a log-likelihood function using the alternative methods for approximating
    transition probabilities.

    This function is used in :func:`bd.interface_dnm.discrete_est_dnm()` (where
    it is maximized).
    """
    # A different data structure is used depending on the likelihood method.
    # This is because for some likelihood methods each call of the function 'probability' naturally
    # generate sets of transition probabilities, while for others only a single transition
    # probability is generated.
    if likelihood in ['da', 'gwa', 'gwasa', 'ilt', 'oua']:
        def ll_fun(param):
            ll = 0
            for i in data:
                pr = probability(i[0], i[1], i[2], param, method=likelihood,
                                 model=model, z_trunc=z_trunc,
                                 options=options)[0][0]
                if np.isnan(pr):
                    warnings.warn("Computation of a transition probability has failed and been "
                                  "replaced by 1e-100. The results may be unreliable. ",
                                  category=RuntimeWarning)
                    ll += data[i] * np.log(1e-100)
                else:
                    ll += data[i] * np.log(np.maximum(1e-100, pr))
            return ll
    elif likelihood in ['Erlang', 'expm', 'uniform']:
        sorted_data = ut.data_sort_2(data)
        z_min, z_max = z_trunc

        def ll_fun(param):
            ll = 0
            for t in sorted_data:
                p_mat = probability(np.arange(z_min, z_max + 1, 1),
                                    np.arange(z_min, z_max+1, 1), t, param,
                                    method=likelihood, model=model,
                                    z_trunc=z_trunc, options=options)
                for i in sorted_data[t]:
                    ll += sorted_data[t][i] * np.log(
                        np.maximum(1e-100, p_mat[i[0] - z_min, i[1] - z_min]))
            return ll

    else:
        raise TypeError("Argument likelihood has an unknown value.")

    return ll_fun


def discrete_est_dnm(data, likelihood, model, z_trunc, idx_known_p, known_p,
                     p0, p_bounds, con, opt_method, options):
    """Parameter estimation for discretely observed continuous-time
    birth-and-death processes using direct numerical maximization of
    approximate likelihood functions.
    See :ref:`here <Direct Numerical Maximization>` for more information.

    To use this function call :func:`birdepy.estimate` with `framework` set to
    'dnm'::

        bd.estimate(t_data, p_data, p0, p_bounds, framework='dnm', likelihood='expm', z_trunc=())

    The parameters associated with this framework (listed below) can be
    accessed using kwargs in :func:`birdepy.estimate()`. See documentation of
    :func:`birdepy.estimate` for the main arguments.

    Parameters
    ----------
    likelihood : string, optional
        Likelihood approximation method. Should be one of
        (alphabetical ordering):

            - 'da' (see :ref:`here <birdepy.probability(method='da')>`)
            - 'Erlang' (see :ref:`here <birdepy.probability(method='Erlang')>`)
            - 'expm'  (default) (see :ref:`here <birdepy.probability(method='expm')>`)
            - 'gwa' (see :ref:`here <birdepy.probability(method='gwa')>`)
            - 'gwasa' (see :ref:`here <birdepy.probability(method='gwasa')>`)
            - 'ilt' (see :ref:`here <birdepy.probability(method='ilt')>`)
            - 'oua' (see :ref:`here <birdepy.probability(method='oua')>`)
            - 'uniform' (see :ref:`here <birdepy.probability(method='uniform')>`)
        The links point to the documentation of the relevant `method` in
        :func:`birdepy.probability`. The arguments associated with each of
        these methods may be used as a kwarg in :func:`birdepy.estimate()`
        when `likelihood` is set to use the method.

    z_trunc : array_like, optional
        Truncation thresholds, i.e., minimum and maximum states of process
        considered. Array of real elements of size (2,) by default\
        ``z_trunc=[z_min, z_max]`` where
        ``z_min=max(0, min(p_data) - 2*z_range)``
        and ``z_max=max(p_data) + 2*z_range`` where
        ``z_range=max(p_data)-min(p_data)``. Only applicable to `likelihood`
        methods 'Erlang', 'expm' and 'uniform'.

    Examples
    --------
    Simulate a sample path and estimate the parameters using the various
    likelihood approximation methods: ::

        import birdepy as bd
        t_data = list(range(100))
        p_data = bd.simulate.discrete([0.75, 0.25, 0.02, 1], 'Ricker', 10, t_data,
                                      survival=True, seed=2021)
        for likelihood in ['da', 'Erlang', 'expm', 'gwa', 'gwasa', 'ilt', 'oua', 'uniform']:
            likelihood = 'gwasa'
            est = bd.estimate(t_data, p_data, [0.5, 0.5, 0.05], [[1e-6,1], [1e-6,1], [1e-6, 0.1]],
                             framework='dnm', model='Ricker', idx_known_p=[3], known_p=[1],
                              likelihood=likelihood)
            print('dnm estimate using', likelihood, 'is', est.p, ', with standard errors',
                  est.se, 'computed in ', est.compute_time, 'seconds.')

    Outputs: ::

        dnm estimate using da is [0.742967758293063, 0.21582082476775125, 0.022598554340938885] , with standard errors [0.16633436 0.03430501 0.00436483] computed in  12.29400086402893 seconds.
        dnm estimate using Erlang is [0.7478158070214841, 0.2150341741826537, 0.022748822721356705] , with standard errors [0.16991822 0.0345762  0.00435054] computed in  0.9690003395080566 seconds.
        dnm estimate using expm is [0.7477255598292176, 0.2150476994316206, 0.022745305129350565] , with standard errors [0.16904095 0.03443197 0.00433563] computed in  1.6919987201690674 seconds.
        dnm estimate using gwa is [0.6600230500097711, 0.16728663936008945, 0.02512248420514078] , with standard errors [0.14248815 0.02447161 0.00488879] computed in  37.52255415916443 seconds.
        dnm estimate using gwasa is [0.6604981297820195, 0.16924607541398484, 0.02492054535741541] , with standard errors [0.14244908 0.02485465 0.00488222] computed in  0.8699958324432373 seconds.
        dnm estimate using ilt is [0.7466254648849691, 0.21415145383850764, 0.022794996238547492] , with standard errors [0.10187377 0.03137803        nan] computed in  1185.0924031734467 seconds.
        dnm estimate using oua is [0.5000083585920406, 0.5, 0.05] , with standard errors [       nan        nan 0.01961143] computed in  3.466001272201538 seconds.
        dnm estimate using uniform is [0.7477293759434092, 0.215047068344254, 0.022745437226772615] , with standard errors [0.16900378 0.03443071 0.00433504] computed in  3.275972366333008 seconds.

    The constraint ``con={'type': 'ineq', 'fun': lambda p: p[0]-p[1]}`` ensures
    that p[0] > p[1] (i.e., rate of spread greater than recovery rate).

    How well methods perform varies from case to case. In this instance most methods perform well,
    while some throw errors but return useful output regardless, and some fail altogether.

    Notes
    -----
    Estimation algorithms and models are also described in [1]. If you use this
    function for published work, then please cite this paper.

    For a text book treatment on the theory of birth-and-death processes
    see [2].

    See also
    --------
    :func:`birdepy.estimate()` :func:`birdepy.probability()` :func:`birdepy.forecast()`

    :func:`birdepy.simulate.discrete()` :func:`birdepy.simulate.continuous()`

    References
    ----------
    .. [1] Hautphenne, S., & Patch, B. (2021). Birth-and-death Processes in Python:
     The BirDePy Package. arXiv preprint arXiv:2110.05067.

    .. [2] Feller, W. (1968) An introduction to probability theory and its
     applications (Volume 1) 3rd ed. John Wiley & Sons.
    """
    # Build the log-likelihood function assuming all parameters are known
    pre_ll_fun = bld_ll_fun(data, likelihood, model, z_trunc, options)

    # Adjust the log-likelihood function to account for unknown parameters and
    # negate it so that the minimize function performs maximisation
    def error_fun(p_prop):
        param = ut.p_bld(p_prop, idx_known_p, known_p)
        return -pre_ll_fun(param)

    # The actual log-likelihood is needed to determine the covariance
    def ll(p_prop):
        param = ut.p_bld(p_prop, idx_known_p, known_p)
        return pre_ll_fun(param)

    # Obtain parameter estimates
    opt = ut.minimize_(error_fun, p0, p_bounds, con, opt_method, options)

    # Obtain covariance estimates
    if p0.size > 1:
        try:
            cov = np.linalg.inv(-ut.Hessian(ll, opt.x, p_bounds))
        except:
            cov = 'Covariance matrix could not be determined.'
    else:
        try:
            cov = -1/ut.Hessian(ll, opt.x, p_bounds)
        except:
            cov = 'Covariance matrix could not be determined.'

    return opt, cov


def continuous_est_dnm(t_data, p_data, p0, b_rate, d_rate, p_bounds, con,
                       known_p, idx_known_p, opt_method, options):
    """Parameter estimation for continuously  observed continuous-time
    birth-and-death processes using direct numerical maximization of
    the likelihood.
    See :ref:`here <Continuously Observed Data>` for more information.

    To use this function call :func:`birdepy.estimate` with `scheme` set to
    'continuous'::

        bd.estimate(t_data, p_data, p0, p_bounds, scheme='continuous')


    Examples
    --------
    Simulate a continuous sample path and estimate the parameters.

    Import BirDePy: ::

        import birdepy as bd

    Simulate some synthetic data: ::

        t_data, p_data = bd.simulate.continuous([0.75, 0.25, 0.02, 1], 'Ricker', 10,
                                                100, survival=True, seed=2021)

    Estimate: ::

        est = bd.estimate(t_data, p_data, [0.5, 0.5, 0.05], [[0,1], [0,1], [0, 0.1]],
                          model='Ricker', idx_known_p=[3], known_p=[1], scheme='continuous')
        print(est.p)

    Outputs: ::

        [0.7603171062895576, 0.2514810854871476, 0.020294342655751033]

    Notes
    -----
    Estimation algorithms and models are also described in [1]. If you use this
    function for published work, then please cite this paper.

    For a text book treatment on the theory of birth-and-death processes
    see [2].

    See also
    :func:`birdepy.estimate()` :func:`birdepy.probability()` :func:`birdepy.forecast()`

    :func:`birdepy.simulate.discrete()` :func:`birdepy.simulate.continuous()`

    --------


    References
    ----------
    .. [1] Hautphenne, S., & Patch, B. (2021). Birth-and-death Processes in Python:
     The BirDePy Package. arXiv preprint arXiv:2110.05067.

    .. [2] Feller, W. (1968) An introduction to probability theory and its
     applications (Volume 1) 3rd ed. John Wiley & Sons.

    .. [3] Wolff, R. W. (1965). Problems of statistical inference for birth and
     death queuing models. Operations Research, 13(3), 343-357.
    """
    # Convert data into the format of a list of lists
    if type(t_data[0]) != list:
        t_data = [t_data]
    if type(p_data[0]) != list:
        p_data = [p_data]

    # Create a list containing inter-event (or "holding") times
    holding_times = []
    for idx in range(len(t_data)):
        holding_times.append(np.array(t_data[idx][1:]) -
                             np.array(t_data[idx][0:-1]))

    # Create a list containing population sizes (or states) corresponding to holding_times
    flattened_p_data = [item for sublist in p_data for item in sublist]

    # Create an array of all observed population sizes
    states_visited = np.arange(np.amin(flattened_p_data),
                               np.amax(flattened_p_data) + 1, 1)

    # Create arrays to store holding times and counts of up and down jumps for each population size
    holding_t_data = np.zeros(len(states_visited))
    downward_jump_counts = np.zeros(len(states_visited))
    upward_jump_counts = np.zeros(len(states_visited))

    # Determine holding times and counts of up and down jumps for each population size
    for idx1 in range(len(t_data)):
        state_record_temp = np.array(p_data[idx1])
        holding_times_temp = np.array(holding_times[idx1])
        for idx2, state in enumerate(states_visited):
            holding_t_data[idx2] += \
                np.sum(holding_times_temp[state_record_temp[:-1] == state])
            downward_jump_counts[idx2] += \
                np.sum(np.less(state_record_temp[1:], state_record_temp[0:-1])
                       & (state_record_temp[0:-1] == state))
            upward_jump_counts[idx2] += \
                np.sum(np.greater(state_record_temp[1:], state_record_temp[0:-1])
                       & (state_record_temp[0:-1] == state))

    # Determine the log-likelihood function (e.g., Equation (2) in reference [3])
    def pre_ll(param):
        ll_ = 0
        for idx_, state_ in enumerate(states_visited):
            if b_rate(state_, param) > 0:
                ll_ += np.log(b_rate(state_, param)) * upward_jump_counts[idx_]
            if d_rate(state_, param) > 0:
                ll_ += np.log(d_rate(state_, param)) * downward_jump_counts[idx_]
            ll_ -= (b_rate(state_, param) + d_rate(state_, param)) * holding_t_data[idx_]
        return ll_

    # Adjust the log-likelihood function to account for unknown parameters and
    # negate it so that the minimize function performs maximisation
    def error_fun(p_prop):
        param = ut.p_bld(p_prop, idx_known_p, known_p)
        return -pre_ll(param)

    # The actual log-likelihood is needed to determine the covariance
    def ll(p_prop):
        param = ut.p_bld(p_prop, idx_known_p, known_p)
        return pre_ll(param)

    # Obtain parameter estimates
    opt = ut.minimize_(error_fun, p0, p_bounds, con, opt_method, options)

    # Obtain covariance estimates
    if p0.size > 1:
        try:
            cov = np.linalg.inv(-ut.Hessian(ll, opt.x, p_bounds))
        except:
            cov = 'Covariance matrix could not be determined.'
    else:
        try:
            cov = -1/ut.Hessian(ll, opt.x, p_bounds)
        except:
            cov = 'Covariance matrix could not be determined.'

    return opt, cov