regression.py

import autograd
import emcee
import numpy
from scipy.special import gammaincinv
from autograd.scipy.special import expit, gammaln
from autograd.numpy import isnan, exp, dot, log, sum
import scipy.optimize
import sys
import warnings
from convoys.gamma import gammainc


__all__ = ['Exponential',
           'Weibull',
           'Gamma',
           'GeneralizedGamma']


def generalized_gamma_LL(x, X, B, T, W, fix_k, fix_p, hierarchical):
    k = exp(x[0]) if fix_k is None else fix_k
    p = exp(x[1]) if fix_p is None else fix_p
    log_sigma_alpha = x[2]
    log_sigma_beta = x[3]
    a = x[4]
    b = x[5]
    n_features = int((len(x)-6)/2)
    alpha = x[6:6+n_features]
    beta = x[6+n_features:6+2*n_features]
    lambd = exp(dot(X, alpha)+a)
    c = expit(dot(X, beta)+b)

    # PDF: p*lambda^(k*p) / gamma(k) * t^(k*p-1) * exp(-(x*lambda)^p)
    log_pdf = log(p) + (k*p) * log(lambd) - gammaln(k) \
              + (k*p-1) * log(T) - (T*lambd)**p
    cdf = gammainc(k, (T*lambd)**p)

    LL_observed = log(c) + log_pdf
    LL_censored = log((1-c) + c * (1 - cdf))

    LL_data = sum(
        W * B * LL_observed +
        W * (1 - B) * LL_censored, 0)

    if hierarchical:
        # Hierarchical model with sigmas ~ invgamma(1, 1)
        LL_prior_a = -4*log_sigma_alpha - 1/exp(log_sigma_alpha)**2 \
                     - dot(alpha, alpha) / (2*exp(log_sigma_alpha)**2) \
                     - n_features*log_sigma_alpha
        LL_prior_b = -4*log_sigma_beta - 1/exp(log_sigma_beta)**2 \
                     - dot(beta, beta) / (2**exp(log_sigma_beta**2)) \
                     - n_features*log_sigma_beta
        LL = LL_prior_a + LL_prior_b + LL_data
    else:
        LL = LL_data

    if isnan(LL):
        return -numpy.inf
    return LL


class RegressionModel(object):
    pass


class GeneralizedGamma(RegressionModel):
    ''' Generalization of Gamma, Weibull, and Exponential

    This mostly follows the `Wikipedia article
    <https://en.wikipedia.org/wiki/Generalized_gamma_distribution>`_, although
    our notation is slightly different. Also see `this paper
    <http://data.princeton.edu/pop509/ParametricSurvival.pdf>`_ for an overview.

    **Shape of the probability function**

    The cumulative density function is:

    :math:`F(t) = P(k, (t\\lambda)^p)`

    where :math:`P(a, x) = \\gamma(a, x) / \\Gamma(a)` is the lower regularized
    incomplete gamma function. See :meth:`convoys.gamma.gammainc`.
    :math:`\\gamma(a, x)` is the incomplete gamma function and :math:`\\Gamma(a)`
    is the standard gamma function.

    The probability density function is:

    :math:`f(t) = p\\lambda^{kp} t^{kp-1} \exp(-(t\\lambda)^p) / \\Gamma(k)`

    **Modeling conversion rate**

    Since our goal is to model the conversion rate, we assume the conversion
    rate converges to a final value

    :math:`c = \\sigma(\mathbf{\\beta^Tx} + b)`

    where :math:`\\sigma(z) = 1/(1+e^{-z})` is the sigmoid function,
    :math:`\\mathbf{\\beta}` is an unknown vector we are solving for (with
    corresponding  intercept :math:`b`), and :math:`\\mathbf{x}` are the
    feature vector (inputs).

    We also assume that the rate parameter :math:`\\lambda` is determined by

    :math:`\\lambda = exp(\mathbf{\\alpha^Tx} + a)`

    where :math:`\\mathrm{\\alpha}` is another unknown vector we are
    trying to solve for (with corresponding intercept :math:`a`).

    We also assume that the :math:`\\mathbf{\\alpha}, \\mathbf{\\beta}`
    vectors have a normal distribution

    :math:`\\alpha_i \sim \\mathcal{N}(0, \\sigma_{\\alpha})`,
    :math:`\\beta_i \sim \\mathcal{N}(0, \\sigma_{\\beta})`

    where hyperparameters :math:`\\sigma_{\\alpha}^2, \\sigma_{\\beta}^2`
    are drawn from an inverse gamma distribution

    :math:`\\sigma_{\\alpha}^2 \sim \\text{inv-gamma}(1, 1)`,
    :math:`\\sigma_{\\beta}^2 \sim \\text{inv-gamma}(1, 1)`

    **List of parameters**

    The full model fits vectors :math:`\\mathbf{\\alpha, \\beta}` and scalars
    :math:`a, b, k, p, \\sigma_{\\alpha}, \\sigma_{\\beta}`.

    **Likelihood and censorship**

    For entries that convert, the contribution to the likelihood is simply
    the probability density given by the probability distribution function
    :math:`f(t)` times the final conversion rate :math:`c`.

    For entries that *did not* convert, there is two options. Either the
    entry will never convert, which has probability :math:`1-c`. Or,
    it will convert at some later point that we have not observed yet,
    with probability given by the cumulative density function
    :math:`F(t)`.

    **Solving the optimization problem**

    To find the MAP (max a posteriori), `scipy.optimize.minimize
    <https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html#scipy.optimize.minimize>`_
    with the CG method.

    If `ci == True`, then `emcee <http://dfm.io/emcee/current/>`_ is used
    to sample from the full posterior in order to generate uncertainty
    estimates for all parameters.
    '''
    def __init__(self, ci=False):
        self._ci = ci

    def fit(self, X, B, T, W=None, fix_k=None, fix_p=None):
        '''Fits the model.

        :param X: numpy matrix of shape :math:`k \cdot n`
        :param B: numpy vector of shape :math:`n`
        :param T: numpy vector of shape :math:`n`
        :param W: (optional) numpy vector of shape :math:`n`
        '''

        if W is None:
            W = [1] * len(X)
        XBTW = [(x, b, t, w) for x, b, t, w in zip(X, B, T, W)
                if t > 0 and 0 <= float(b) <= 1 and w >= 0]
        if len(XBTW) < len(X):
            n_removed = len(X) - len(XBTW)
            warnings.warn('Warning! Removed %d/%d entries from inputs where '
                          'T <= 0 or B not 0/1 or W < 0' % (n_removed, len(X)))
        X, B, T, W = (numpy.array([z[i] for z in XBTW], dtype=numpy.float32)
                      for i in range(4))
        n_features = X.shape[1]

        # scipy.optimize and emcee forces the the parameters to be a vector:
        # (log k, log p, log sigma_alpha, log sigma_beta,
        #  a, b, alpha_1...alpha_k, beta_1...beta_k)
        # Generalized Gamma is a bit sensitive to the starting point!
        x0 = numpy.zeros(6+2*n_features)
        x0[0] = +1 if fix_k is None else log(fix_k)
        x0[1] = -1 if fix_p is None else log(fix_p)
        args = (X, B, T, W, fix_k, fix_p, True)

        # Callback for progress to stdout
        sys.stdout.write('\n')

        def callback(x, x_history=[]):
            x_history.append(x)
            sys.stdout.write('Finding MAP: %13d\r' % len(x_history))
            sys.stdout.flush()

        # Find the maximum a posteriori of the distribution
        res = scipy.optimize.minimize(
            lambda x: -generalized_gamma_LL(x, *args),
            x0,
            jac=autograd.grad(lambda x: -generalized_gamma_LL(x, *args)),
            method='CG',
            callback=callback,
        )
        sys.stdout.write('\n')
        result = {'map': res.x}

        # TODO: should not use fixed k/p as search parameters
        if fix_k:
            result['map'][0] = log(fix_k)
        if fix_p:
            result['map'][1] = log(fix_p)

        # Let's sample from the posterior to compute uncertainties
        if self._ci:
            dim, = res.x.shape
            n_walkers = 5*dim
            sampler = emcee.EnsembleSampler(
                nwalkers=n_walkers,
                dim=dim,
                lnpostfn=generalized_gamma_LL,
                args=args,
            )
            mcmc_initial_noise = 1e-3
            p0 = [result['map'] + mcmc_initial_noise * numpy.random.randn(dim)
                  for i in range(n_walkers)]
            n_burnin = 40
            n_steps = numpy.ceil(1000. / n_walkers)
            n_iterations = n_burnin + n_steps
            for i, _ in enumerate(sampler.sample(p0, iterations=n_iterations)):
                sys.stdout.write('MCMC (%3d walkers): %6d/%-6d (%6.2f%%)\r' % (
                        n_walkers, i+1, n_iterations, 100.*(i+1)/n_iterations))
                sys.stdout.flush()
            sys.stdout.write('\n')
            result['samples'] = sampler.chain[:, n_burnin:, :] \
                                       .reshape((-1, dim)).T
            if fix_k:
                result['samples'][0, :] = log(fix_k)
            if fix_p:
                result['samples'][1, :] = log(fix_p)

        self.params = {k: {
            'k': exp(data[0]),
            'p': exp(data[1]),
            'a': data[4],
            'b': data[5],
            'alpha': data[6:6+n_features].T,
            'beta': data[6+n_features:6+2*n_features].T,
        } for k, data in result.items()}

    def cdf(self, x, t, ci=None):
        x = numpy.array(x)
        t = numpy.array(t)
        if ci is None:
            params = self.params['map']
        else:
            assert self._ci
            params = self.params['samples']
        lambd = exp(dot(x, params['alpha'].T) + params['a'])
        c = expit(dot(x, params['beta'].T) + params['b'])
        M = c * gammainc(
            params['k'],
            numpy.multiply.outer(t, lambd)**params['p'])

        if not ci:
            return M
        else:
            # Replace the last axis with a 3-element vector
            y = numpy.mean(M, axis=-1)
            y_lo = numpy.percentile(M, (1-ci)*50, axis=-1)
            y_hi = numpy.percentile(M, (1+ci)*50, axis=-1)
            return numpy.stack((y, y_lo, y_hi), axis=-1)

    def rvs(self, x, n_curves=1, n_samples=1, T=None):
        # Samples values from this distribution
        # T is optional and means we already observed non-conversion until T
        assert self._ci  # Need to be fit with MCMC
        if T is None:
            T = numpy.zeros((n_curves, n_samples))
        else:
            assert T.shape == (n_curves, n_samples)
        B = numpy.zeros((n_curves, n_samples), dtype=numpy.bool)
        C = numpy.zeros((n_curves, n_samples))
        params = self.params['samples']
        for i, j in enumerate(numpy.random.randint(len(params['k']),
                                                   size=n_curves)):
            k = params['k'][j]
            p = params['p'][j]
            lambd = exp(dot(x, params['alpha'][j]) + params['a'][j])
            c = expit(dot(x, params['beta'][j]) + params['b'][j])
            z = numpy.random.uniform(size=(n_samples,))
            cdf_now = c * gammainc(
                k,
                numpy.multiply.outer(T[i], lambd)**p)  # why is this outer?
            adjusted_z = cdf_now + (1 - cdf_now) * z
            B[i] = (adjusted_z < c)
            y = adjusted_z / c
            w = gammaincinv(k, y)
            # x = (t * lambd)**p
            C[i] = w**(1./p) / lambd
            C[i][~B[i]] = 0

        return B, C


class Exponential(GeneralizedGamma):
    ''' Specialization of :class:`.GeneralizedGamma` where :math:`k=1, p=1`.

    The cumulative density function is:

    :math:`F(t) = 1 - \\exp(-t\\lambda)`

    The probability density function is:

    :math:`f(t) = \\lambda\\exp(-t\\lambda)`

    The exponential distribution is the most simple distribution.
    From a conversion perspective, you can interpret it as having
    two competing final states where the probability of transitioning
    from the initial state to converted or dead is constant.

    See documentation for :class:`GeneralizedGamma`.'''
    def fit(self, X, B, T, W=None):
        super(Exponential, self).fit(X, B, T, W, fix_k=1, fix_p=1)


class Weibull(GeneralizedGamma):
    ''' Specialization of :class:`.GeneralizedGamma` where :math:`k=1`.

    The cumulative density function is:

    :math:`F(t) = 1 - \\exp(-(t\\lambda)^p)`

    The probability density function is:

    :math:`f(t) = p\\lambda(t\\lambda)^{p-1}\\exp(-(t\\lambda)^p)`

    See documentation for :class:`GeneralizedGamma`.'''
    def fit(self, X, B, T, W=None):
        super(Weibull, self).fit(X, B, T, W, fix_k=1)


class Gamma(GeneralizedGamma):
    ''' Specialization of :class:`.GeneralizedGamma` where :math:`p=1`.

    The cumulative density function is:

    :math:`F(t) = P(k, t\\lambda)`

    where :math:`P(a, x) = \\gamma(a, x) / \\Gamma(a)` is the lower regularized
    incomplete gamma function. See :meth:`convoys.gamma.gammainc`.

    The probability density function is:

    :math:`f(t) = \\lambda^k t^{k-1} \exp(-x\\lambda) / \\Gamma(k)`

    See documentation for :class:`GeneralizedGamma`.'''
    def fit(self, X, B, T, W=None):
        super(Gamma, self).fit(X, B, T, W, fix_p=1)