"""
Tools for maximum likelihood fits to single trigger statistic values

For some set of values above a threshold, e.g. trigger SNRs, the functions
in this module perform maximum likelihood fits with 1-sigma uncertainties
to various simple functional forms of PDF, all normalized to 1.
You can also obtain the fitted function and its (inverse) CDF and perform
a Kolmogorov-Smirnov test.

Usage:
# call the fit function directly if the threshold is known
alpha, sigma_alpha = fit_exponential(snrs, 5.5)

# apply a threshold explicitly
alpha, sigma_alpha = fit_above_thresh('exponential', snrs, thresh=6.25)

# let the code work out the threshold from the smallest value via the default thresh=None
alpha, sigma_alpha = fit_above_thresh('exponential', snrs)

# or only fit the largest N values, i.e. tail fitting
thresh = tail_threshold(snrs, N=500)
alpha, sigma_alpha = fit_above_thresh('exponential', snrs, thresh)

# obtain the fitted function directly
xvals = numpy.xrange(5.5, 10.5, 20)
exponential_fit = expfit(xvals, alpha, thresh)

# or access function by name
exponential_fit_1 = fit_fn('exponential', xvals, alpha, thresh)

# Use weighting factors to e.g. take decimation into account
alpha, sigma_alpha = fit_above_thresh('exponential', snrs, weights=weights)

# get the KS test statistic and p-value - see scipy.stats.kstest
ks_stat, ks_pval = KS_test('exponential', snrs, alpha, thresh)

"""

# Copyright T. Dent 2015 (thomas.dent@aei.mpg.de)
#
# This program is free software; you can redistribute it and/or modify it
# under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 2 of the License, or (at your
# option) any later version.
#
# This program is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General
# Public License for more details.

import logging
import numpy
from scipy.stats import kstest

def exponential_fitalpha(vals, thresh, w):
    """
    Maximum likelihood estimator for the fit factor for
    an exponential decrease model
    """
    return 1. / (numpy.average(vals, weights=w) - thresh)


def rayleigh_fitalpha(vals, thresh, w):
    """
    Maximum likelihood estimator for the fit factor for
    a Rayleigh distribution of events
    """
    return 2. / (numpy.average(vals ** 2., weights=w) - thresh ** 2.)


def power_fitalpha(vals, thresh, w):
    """
    Maximum likelihood estimator for the fit factor for
    a power law model
    """
    return numpy.average(numpy.log(vals/thresh), weights=w) ** -1. + 1.


fitalpha_dict = {
    'exponential' : exponential_fitalpha,
    'rayleigh'    : rayleigh_fitalpha,
    'power'       : power_fitalpha
}

# measurement standard deviation = (-d^2 log L/d alpha^2)^(-1/2)
fitstd_dict = {
    'exponential' : lambda weights, alpha : alpha / sum(weights) ** 0.5,
    'rayleigh'    : lambda weights, alpha : alpha / sum(weights) ** 0.5,
    'power'       : lambda weights, alpha : (alpha - 1.) / sum(weights) ** 0.5
}

def fit_above_thresh(distr, vals, thresh=None, weights=None):
    """
    Maximum likelihood fit for the coefficient alpha

    Fitting a distribution of discrete values above a given threshold.
    Exponential  p(x) = alpha exp(-alpha (x-x_t))
    Rayleigh     p(x) = alpha x exp(-alpha (x**2-x_t**2)/2)
    Power        p(x) = ((alpha-1)/x_t) (x/x_t)**-alpha
    Values below threshold will be discarded.
    If no threshold is specified the minimum sample value will be used.

    Parameters
    ----------
    distr : {'exponential', 'rayleigh', 'power'}
        Name of distribution
    vals : sequence of floats
        Values to fit
    thresh : float
        Threshold to apply before fitting; if None, use min(vals)
    weights: sequence of floats
        Weighting factors to use for the values when fitting.
        Default=None - all the same

    Returns
    -------
    alpha : float
        Fitted value
    sigma_alpha : float
        Standard error in fitted value
    """
    vals = numpy.array(vals)
    if thresh is None:
        thresh = min(vals)
        above_thresh = numpy.ones_like(vals, dtype=bool)
    else:
        above_thresh = vals >= thresh
        if numpy.count_nonzero(above_thresh) == 0:
            # Nothing is above threshold - warn and return -1
            logging.warning("No values are above the threshold, %.2f, "
                            "maximum is %.2f.", thresh, vals.max())
            return -1., -1.

        vals = vals[above_thresh]

    # Set up the weights
    if weights is not None:
        weights = numpy.array(weights)
        w = weights[above_thresh]
    else:
        w = numpy.ones_like(vals)

    alpha = fitalpha_dict[distr](vals, thresh, w)
    return alpha, fitstd_dict[distr](w, alpha)


# Variables:
# x: the trigger stat value(s) at which to evaluate the function
# a: slope parameter of the fit
# t: lower threshold stat value
fitfn_dict = {
    'exponential' : lambda x, a, t : a * numpy.exp(-a * (x - t)),
    'rayleigh' : lambda x, a, t : (a * x * \
                                   numpy.exp(-a * (x ** 2 - t ** 2) / 2.)),
    'power' : lambda x, a, t : (a - 1.) * x ** (-a) * t ** (a - 1.)
}

def fit_fn(distr, xvals, alpha, thresh):
    """
    The fitted function normalized to 1 above threshold

    To normalize to a given total count multiply by the count.

    Parameters
    ----------
    xvals : sequence of floats
        Values where the function is to be evaluated
    alpha : float
        The fitted parameter
    thresh : float
        Threshold value applied to fitted values

    Returns
    -------
    fit : array of floats
        Fitted function at the requested xvals
    """
    xvals = numpy.array(xvals)
    fit = fitfn_dict[distr](xvals, alpha, thresh)
    # set fitted values below threshold to 0
    numpy.putmask(fit, xvals < thresh, 0.)
    return fit


cum_fndict = {
    'exponential' : lambda x, alpha, t : numpy.exp(-alpha * (x - t)),
    'rayleigh' : lambda x, alpha, t : numpy.exp(-alpha * (x ** 2. - t ** 2.) / 2.),
    'power' : lambda x, alpha, t : x ** (1. - alpha) * t ** (alpha - 1.)
}

def cum_fit(distr, xvals, alpha, thresh):
    """
    Integral of the fitted function above a given value (reverse CDF)

    The fitted function is normalized to 1 above threshold

    Parameters
    ----------
    xvals : sequence of floats
        Values where the function is to be evaluated
    alpha : float
        The fitted parameter
    thresh : float
        Threshold value applied to fitted values

    Returns
    -------
    cum_fit : array of floats
        Reverse CDF of fitted function at the requested xvals
    """
    xvals = numpy.array(xvals)
    cum_fit = cum_fndict[distr](xvals, alpha, thresh)
    # set fitted values below threshold to 0
    numpy.putmask(cum_fit, xvals < thresh, 0.)
    return cum_fit

def tail_threshold(vals, N=1000):
    """Determine a threshold above which there are N louder values"""
    vals = numpy.array(vals)
    if len(vals) < N:
        raise RuntimeError('Not enough input values to determine threshold')
    vals.sort()
    return min(vals[-N:])


def KS_test(distr, vals, alpha, thresh=None):
    """
    Perform Kolmogorov-Smirnov test for fitted distribution

    Compare the given set of discrete values above a given threshold to the
    fitted distribution function.
    If no threshold is specified, the minimum sample value will be used.
    Returns the KS test statistic and its p-value: lower p means less
    probable under the hypothesis of a perfect fit

    Parameters
    ----------
    distr : {'exponential', 'rayleigh', 'power'}
        Name of distribution
    vals : sequence of floats
        Values to compare to fit
    alpha : float
        Fitted distribution parameter
    thresh : float
        Threshold to apply before fitting; if None, use min(vals)

    Returns
    -------
    D : float
        KS test statistic
    p-value : float
        p-value, assumed to be two-tailed
    """
    vals = numpy.array(vals)
    if thresh is None:
        thresh = min(vals)
    else:
        vals = vals[vals >= thresh]
    def cdf_fn(x):
        return 1 - cum_fndict[distr](x, alpha, thresh)
    return kstest(vals, cdf_fn)


def which_bin(par, minpar, maxpar, nbins, log=False):
    """
    Helper function

    Returns bin index where a parameter value belongs (from 0 through nbins-1)
    when dividing the range between minpar and maxpar equally into bins.

    Parameters
    ----------
    par : float
        Parameter value being binned
    minpar : float
        Minimum parameter value
    maxpar : float
        Maximum parameter value
    nbins : int
        Number of bins to use
    log : boolean
        If True, use log spaced bins

    Returns
    -------
    binind : int
        Bin index
    """
    assert (par >= minpar and par <= maxpar)
    if log:
        par, minpar, maxpar = numpy.log(par), numpy.log(minpar), numpy.log(maxpar)
    # par lies some fraction of the way between min and max
    if minpar != maxpar:
        frac = float(par - minpar) / float(maxpar - minpar)
    else:
        # if they are equal there is only one size 0 bin
        # must be in that bin
        frac = 0
    # binind then lies between 0 and nbins - 1
    binind = int(frac * nbins)
    # corner case
    if par == maxpar:
        binind = nbins - 1
    return binind