Source code for convoys.regression

from convoys import autograd_scipy_monkeypatch  # NOQA
import autograd
from autograd_gamma import gammainc
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 progressbar
import scipy.optimize
import warnings


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


def generalized_gamma_loss(x, X, B, T, W, fix_k, fix_p,
                           hierarchical, flavor, callback=None):
    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)

    # 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)

    if flavor == 'logistic':  # Log-likelihood with sigmoid
        c = expit(dot(X, beta)+b)
        LL_observed = log(c) + log_pdf
        LL_censored = log((1 - c) + c * (1 - cdf))
    elif flavor == 'linear':  # L2 loss, linear
        c = dot(X, beta)+b
        LL_observed = -(1 - c)**2 + log_pdf
        LL_censored = -(c*cdf)**2

    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
    if callback is not None:
        callback(LL)
    return LL


class RegressionModel(object):
    pass


[docs]class GeneralizedGamma(RegressionModel): ''' Generalization of Gamma, Weibull, and Exponential :param ci: boolean, defaults to False. Whether to use MCMC to sample from the posterior so that a confidence interval can be estimated later (see :meth:`cdf`). :param hierarchical: boolean denoting whether we have a (Normal) prior on the alpha and beta parameters to regularize. The variance of the normal distribution is in itself assumed to be an inverse gamma distribution (1, 1). :param flavor: defaults to logistic. If set to 'linear', then an linear model is fit, where the beta params will be completely additive. This creates a much more interpretable model, with some minor loss of accuracy. 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. :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 SLSQP 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, fix_k=None, fix_p=None, hierarchical=True, flavor='logistic'): self._ci = ci self._fix_k = fix_k self._fix_p = fix_p self._hierarchical = hierarchical self._flavor = flavor
[docs] def fit(self, X, B, T, W=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 = numpy.ones(len(X)) X, B, T, W = (Z if type(Z) == numpy.ndarray else numpy.array(Z) for Z in (X, B, T, W)) keep_indexes = (T > 0) & (B >= 0) & (B <= 1) & (W >= 0) if sum(keep_indexes) < X.shape[0]: n_removed = X.shape[0] - sum(keep_indexes) 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 = (Z[keep_indexes] for Z in (X, B, T, W)) 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 self._fix_k is None else log(self._fix_k) x0[1] = -1 if self._fix_p is None else log(self._fix_p) args = (X, B, T, W, self._fix_k, self._fix_p, self._hierarchical, self._flavor) # Set up progressbar and callback bar = progressbar.ProgressBar(widgets=[ progressbar.Variable('loss', width=15, precision=9), ' ', progressbar.BouncingBar(), ' ', progressbar.Counter(width=6), ' [', progressbar.Timer(), ']']) def callback(LL, value_history=[]): value_history.append(LL) bar.update(len(value_history), loss=LL) # Define objective and use automatic differentiation f = lambda x: -generalized_gamma_loss(x, *args, callback=callback) jac = autograd.grad(lambda x: -generalized_gamma_loss(x, *args)) # Find the maximum a posteriori of the distribution res = scipy.optimize.minimize(f, x0, jac=jac, method='SLSQP', options={'maxiter': 9999}) if not res.success: raise Exception('Optimization failed with message: %s' % res.message) result = {'map': res.x} # TODO: should not use fixed k/p as search parameters if self._fix_k: result['map'][0] = log(self._fix_k) if self._fix_p: result['map'][1] = log(self._fix_p) # Make sure we're in a local minimum gradient = jac(result['map']) gradient_norm = numpy.dot(gradient, gradient) if gradient_norm >= 1e-2 * len(X): warnings.warn('Might not have found a local minimum! ' 'Norm of gradient is %f' % gradient_norm) # 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, ndim=dim, log_prob_fn=generalized_gamma_loss, 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 = 100 n_steps = numpy.ceil(2000. / n_walkers) n_iterations = n_burnin + n_steps bar = progressbar.ProgressBar(max_value=n_iterations, widgets=[ progressbar.Percentage(), ' ', progressbar.Bar(), ' %d walkers [' % n_walkers, progressbar.AdaptiveETA(), ']']) for i, _ in enumerate(sampler.sample(p0, iterations=n_iterations)): bar.update(i+1) result['samples'] = sampler.chain[:, n_burnin:, :] \ .reshape((-1, dim)).T if self._fix_k: result['samples'][0, :] = log(self._fix_k) if self._fix_p: result['samples'][1, :] = log(self._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()}
[docs] def cdf(self, x, t, ci=None): '''Returns the value of the cumulative distribution function for a fitted model. TODO: this should probably be renamed "predict" in the future to follow the scikit-learn convention. :param x: feature vector (or matrix) :param t: time :param ci: if this is provided, and the model was fit with `ci = True`, then the return value will contain one more dimension, and the last dimension will have size 3, containing the mean, the lower bound of the confidence interval, and the upper bound of the confidence interval. If this is not provided, then the max a posteriori prediction will be used. ''' x = numpy.array(x) t = numpy.array(t) if ci is None: params = self.params['map'] else: assert self._ci params = self.params['samples'] t = numpy.expand_dims(t, -1) lambd = exp(dot(x, params['alpha'].T) + params['a']) if self._flavor == 'logistic': c = expit(dot(x, params['beta'].T) + params['b']) elif self._flavor == 'linear': c = dot(x, params['beta'].T) + params['b'] M = c * gammainc( params['k'], (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
[docs]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 __init__(self, *args, **kwargs): kwargs.update(dict(fix_k=1, fix_p=1)) super(Exponential, self).__init__(*args, **kwargs)
[docs]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 __init__(self, *args, **kwargs): kwargs.update(dict(fix_k=1)) super(Weibull, self).__init__(*args, **kwargs)
[docs]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. The probability density function is: :math:`f(t) = \\lambda^k t^{k-1} \\exp(-x\\lambda) / \\Gamma(k)` See documentation for :class:`GeneralizedGamma`.''' def __init__(self, *args, **kwargs): kwargs.update(dict(fix_p=1)) super(Gamma, self).__init__(*args, **kwargs)