nonlinear_fitting.py

# This file is part of BurnMan - a thermoelastic and thermodynamic toolkit for the Earth and Planetary Sciences
# Copyright (C) 2012 - 2015 by the BurnMan team, released under the GNU
# GPL v2 or later.

from __future__ import absolute_import
from __future__ import print_function

import numpy as np
from scipy.stats import t, norm, genextreme
import copy

from .tools import unit_normalize
import matplotlib.pyplot as plt
import matplotlib.colors as colors
from matplotlib.patches import Ellipse


def nonlinear_least_squares_fit(model,
                                lm_damping=0.,
                                param_tolerance=1.e-7,
                                max_lm_iterations=100,
                                verbose=False):
    """
    Function to compute the "best-fit" parameters for a model
    by nonlinear least squares fitting.

    The nonlinear least squares algorithm closely follows the logic in
    Section 23.1 of Bayesian Probability Theory
    (von der Linden et al., 2014; Cambridge University Press).

    Parameters
    ----------
    model : class instance
        Must have the following attributes:
            data : 2D numpy array.
                Elements of x[i][j] contain the observed position of
                data point i

            data_covariances : 3D numpy array
                Elements of cov[i][j][k] contain the covariance matrix
                of data point i

            mle_tolerances : numpy array
                The iterations to find the maximum likelihood estimator
                for each observed data point will stop when mle_tolerances[i] <
                np.linalg.norm(data_mle[i] - model.function(data_mle[i], flag))

            delta_params : numpy array
                parameter perturbations used to compute the jacobian

        Must also contain the following functions:
            set_params(self, param_values):
                Function to set parameters

            get_params(self):
                Function to get current model parameters

            function(self, x):
                Returns value of model function evaluated at x

            normal(self, x):
                Returns value of normal to the model function
                evaluated at x

    lm_damping : float (optional, default: 0)
        Levenberg-Marquardt parameter for least squares minimization

    param_tolerance : float (optional, default: 1.e-5)
        Levenberg-Marquardt iterations are terminated when
        the maximum fractional change in any of the parameters
        during an iteration drops below this value

    max_lm_iterations : integer (optional, default: 100)
        Maximum number of Levenberg-Marquardt iterations

    verbose : bool
        Print some information to standard output


    Attributes added to model
    ----------
    n_dof : integer
        Degrees of freedom of the system
    data_mle : 2D numpy array
        Maximum likelihood estimates of the observed data points
        on the best-fit curve
    jacobian : 2D numpy array
        d(weighted_residuals)/d(parameter)
    weighted_residuals : numpy array
        Weighted residuals
    weights : numpy array
        1/(data variances normal to the best fit curve)
    WSS : float
        Weighted sum of squares residuals
    popt : numpy array
        Optimized parameters
    pcov : 2D numpy array
        Covariance matrix of optimized parameters
    noise_variance : float
        Estimate of the variance of the data normal to the curve

    This function is available as ``burnman.nonlinear_least_squares_fit``.
    """

    def _mle_estimate(x, x_m, cov, flag):
        n = model.normal(x_m, flag)
        var_n = abs_line_project(cov, n)
        d = (x_m - x).dot(n)
        x_mle = x + d*((n.dot(cov)).T)/var_n
        return x_mle, d, var_n

    def _find_mle():
        x_mle_arr = np.empty_like(model.data)
        residual_arr = np.empty(n_data)
        var_arr = np.empty(n_data)
        for i, (x, cov, flag) in enumerate(zip(*[model.data,
                                                 model.data_covariances,
                                                 model.flags])):
            x_mle_arr[i] = model.function(x, flag)
            x_mle_est, residual_arr[i], var_arr[i] = _mle_estimate(
                x, x_mle_arr[i], cov, flag)
            delta_x = x_mle_arr[i] - x

            while np.linalg.norm(delta_x) > model.mle_tolerances[i]:
                x_mle_est, residual_arr[i], var_arr[i] = _mle_estimate(
                    x, x_mle_arr[i], cov, flag)
                x_mle_arr[i] = model.function(x_mle_est, flag)
                delta_x = x_mle_arr[i] - x_mle_est

        return x_mle_arr, residual_arr/np.sqrt(var_arr), 1./var_arr

    def calculate_jacobian():
        model.jacobian = np.empty((n_data, n_params))
        diag_delta = np.diag(model.delta_params)
        param_values = model.get_params()
        for prm_i, value in enumerate(param_values):

            model.set_params(param_values - diag_delta[prm_i])
            x_mle_arr, residual_arr_0, weights_0 = _find_mle()

            model.set_params(param_values + diag_delta[prm_i])
            x_mle_arr, residual_arr_1, weights_1 = _find_mle()

            model.jacobian[:, prm_i] = ((residual_arr_1 - residual_arr_0)
                                        / (2.*diag_delta[prm_i][prm_i]))
        model.set_params(param_values)  # reset params

    def _update_beta(lmbda):
        # Performs a Levenberg-Marquardt iteration
        # Note that if lambda = 0, this is a simple Gauss-Newton iteration
        calculate_jacobian()
        model.data_mle, model.weighted_residuals, model.weights = _find_mle()

        J = model.jacobian  # this the weighted Jacobian
        JTJ = J.T.dot(J)
        delta_beta = np.linalg.inv(
            JTJ + lmbda*np.diag(JTJ)).dot(J.T).dot(model.weighted_residuals)
        new_params = model.get_params() - delta_beta
        f_delta_beta = delta_beta/new_params

        model.set_params(new_params)

        return f_delta_beta

    n_data = len(model.data)
    n_params = len(model.get_params())
    model.dof = n_data - n_params

    if not hasattr(model, 'flags'):
        model.flags = [None] * n_data

    for n_it in range(max_lm_iterations):
        f_delta_beta = _update_beta(lm_damping)
        max_f = np.max(np.abs(f_delta_beta))
        if verbose is True:
            print('Iteration {0:d}: {1}. Max change in param: {2}'.format(
                n_it, model.get_params(), max_f))
        if max_f < param_tolerance:
            break

    J = model.jacobian
    r = model.weighted_residuals
    model.WSS = r.dot(r.T)

    model.popt = model.get_params()
    model.pcov = np.linalg.inv(J.T.dot(J))*r.dot(r.T)/model.dof

    # Estimate the noise variance normal to the curve
    model.goodness_of_fit = model.WSS/model.dof
    model.noise_variance = r.dot(np.diag(1./model.weights)).dot(r.T)/model.dof

    if verbose is True:
        if n_it == max_lm_iterations - 1:
            print('Max iterations ({0:d}) reached (param tolerance = {1:1e})'.format(
                max_lm_iterations, param_tolerance))
        else:
            print('Converged in {0:d} iterations'.format(n_it))
        print('\nOptimised parameter values:')
        print(model.popt)
        print('\nParameter covariance matrix:')
        print(model.pcov)
        print('')


def confidence_prediction_bands(model, x_array, confidence_interval, f,
                                flag=None):
    """
    This function calculates the confidence and prediction bands of
    the function f(x) from a best-fit model with uncertainties in its
    parameters as calculated (for example) by
    the function nonlinear_least_squares_fit().

    The values are calculated via the delta method, which estimates
    the variance of f evaluated at x as var(f(x)) = df(x)/dB var(B) df(x)/dB
    where df(x)/dB is the vector of partial derivatives of f(x)
    with respect to B


    Parameters
    ----------
'    model : class instance
        As modified (for example) by the function
        nonlinear_least_squares_fit(). Should contain the following functions:
            get_params, set_params, function, normal
        And attributes:
            delta_params, pcov, dof, noise_variance

    x_array : 2D numpy array
        coordinates at which to evaluate the bounds

    confidence_interval : float
        Probability level of finding the true model (confidence bound) or
        any new data point (probability bound). For example, the 95% confidence
        bounds should be calculated using a confidence interval of 0.95.

    f : function
        This is the function defining the variable y=f(x) for which the
        confidence and prediction bounds are desired

    flag : variable type
        This (optional) flag is passed to model.function to control how the
        modified position of x is calculated. This value is then used by f(x)

    Output
    ------
    bounds : 2D numpy array
        An element of bounds[i][j] gives the lower and upper confidence
        (i=0, i=1) and prediction (i=2, i=3) bounds for the jth data point.
    """

    # Check array dimensions
    n_dimensions = len(model.data[0])
    if len(x_array[0]) != n_dimensions:
        raise Exception('Dimensions of each point must be the same as the '
                        'total number of dimensions')

    param_values = model.get_params()
    x_m_0s = np.empty_like(x_array)
    f_m_0s = np.empty_like(x_array[:, 0])
    for i, x in enumerate(x_array):
        x_m_0s[i] = model.function(x, flag)
        f_m_0s[i] = f(x)

    diag_delta = np.diag(model.delta_params)
    dxdbeta = np.empty([len(param_values), len(x_array)])

    for i, value in enumerate(param_values):
        model.set_params(param_values + diag_delta[i])

        for j, x_m_0 in enumerate(x_m_0s):
            x_m_1 = model.function(x_m_0, flag)
            dxdbeta[i][j] = (f(x_m_1) - f_m_0s[j])/diag_delta[i][i]

    model.set_params(param_values)  # reset params

    variance = np.empty(len(x_array))
    for i, Gprime in enumerate(dxdbeta.T):
        variance[i] = Gprime.T.dot(model.pcov).dot(Gprime)

    critical_value = t.isf(0.5*(confidence_interval + 1.), model.dof)

    confidence_half_widths = critical_value*np.sqrt(variance)
    prediction_half_widths = critical_value*np.sqrt(variance
                                                    + model.noise_variance)

    confidence_bound_0 = f_m_0s - confidence_half_widths
    confidence_bound_1 = f_m_0s + confidence_half_widths
    prediction_bound_0 = f_m_0s - prediction_half_widths
    prediction_bound_1 = f_m_0s + prediction_half_widths

    return np.array([confidence_bound_0, confidence_bound_1,
                     prediction_bound_0, prediction_bound_1])


def abs_line_project(M, n):
    n = unit_normalize(n)
    return n.dot(M).dot(n.T)


def plot_cov_ellipse(cov, pos, nstd=2, ax=None, **kwargs):
    """
    Plots an `nstd` sigma error ellipse based on the specified covariance
    matrix (`cov`). Additional keyword arguments are passed on to the
    ellipse patch artist.

    Parameters
    ----------
        cov : The 2x2 covariance matrix to base the ellipse on
        pos : The location of the center of the ellipse. Expects a 2-element
            sequence of [x0, y0].
        nstd : The radius of the ellipse in numbers of standard deviations.
            Defaults to 2 standard deviations.
        ax : The axis that the ellipse will be plotted on. Defaults to the
            current axis.
        Additional keyword arguments are pass on to the ellipse patch.

    Returns
    -------
        A matplotlib ellipse artist
    """
    def eigsorted(cov):
        vals, vecs = np.linalg.eigh(cov)
        order = vals.argsort()[::-1]
        return vals[order], vecs[:, order]

    if ax is None:
        ax = plt.gca()

    vals, vecs = eigsorted(cov)
    theta = np.degrees(np.arctan2(*vecs[:, 0][::-1]))

    # Width and height are "full" widths, not radius
    width, height = 2 * nstd * np.sqrt(vals)
    ellip = Ellipse(xy=pos, width=width, height=height, angle=theta, **kwargs)

    ax.add_artist(ellip)
    return ellip


def corner_plot(popt, pcov, param_names=[], n_std=1.):
    """
    Creates a corner plot of covariances

    Parameters
    ----------

    popt : numpy array
        Optimized parameters

    pcov : 2D numpy array
        Covariance matrix of the parameters

    param_names : optional list
        Parameter names

    n_std : float
        Number of standard deviations for ellipse

    Returns
    -------
    fig : matplotlib.pyplot.figure object

    ax_array : list of matplotlib Axes objects

    """

    if len(pcov[0]) != len(pcov[:, 0]):
        raise Exception('Covariance matrices must be square')

    n_params = len(pcov[0])
    if n_params < 2:
        raise Exception('Covariance matrix must be at least 2x2 for '
                        'a corner plot to be plotted')

    # ellipse plotting is prone to rounding errors, so we scale the plots here
    scaling = 1./np.power(10., np.around(np.log10(np.abs(popt)) - 0.5))
    scaling = np.outer(scaling, scaling)

    fig, ax_array = plt.subplots(n_params-1, n_params-1)
    fig.set_size_inches(3.*(n_params-1), 3.*(n_params-1))

    for i in range(n_params-1):
        for j in range(1, i+1):
            fig.delaxes(ax_array[j-1][i])

        for j in range(i+1, n_params):
            indices = np.array([i, j])
            projected_cov = (pcov*scaling)[indices[:, None], indices]

            scaled_pos = np.array([popt[i]*np.sqrt(scaling[i][i]),
                                   popt[j]*np.sqrt(scaling[j][j])])

            plot_cov_ellipse(cov=projected_cov, pos=scaled_pos,
                             nstd=n_std, ax=ax_array[j-1][i], color='grey')
            maxx = 1.5*n_std*np.sqrt(projected_cov[0][0])
            maxy = 1.5*n_std*np.sqrt(projected_cov[1][1])
            ax_array[j-1][i].set_xlim(scaled_pos[0]-maxx, scaled_pos[0]+maxx)
            ax_array[j-1][i].set_ylim(scaled_pos[1]-maxy, scaled_pos[1]+maxy)

    if param_names != []:
        for i in range(n_params-1):
            ax_array[n_params-2][i].set_xlabel('{0:s} (x 10^{1:d})'.format(
                param_names[i], -int(np.log10(np.sqrt(scaling[i][i])))))

        for j in range(1, n_params):
            ax_array[j-1][0].set_ylabel('{0:s} (x 10^{1:d})'.format(
                param_names[j], -int(np.log10(np.sqrt(scaling[j][j])))))

    fig.tight_layout()
    return fig, ax_array


def weighted_residual_plot(ax, model, flag=None, sd_limit=3,
                           cmap=plt.cm.RdYlBu,
                           plot_axes=[0, 1], scale_axes=[1., 1.]):
    """
    Creates a plot of the weighted residuals
    The user can choose the projection axes, and scaling to apply to those axes
    The chosen color palette (cmap) is discretised by standard deviation up
    to a cut off value of sd_limit.

    Parameters
    ----------

    ax : matplotlib Axes object

    model : user-defined object
        A model as used by nonlinear_least_squares_fit
        Must contain the attributes model.data,
        model.weighted_residuals and
        model.flags (if flag is not None).

    flag : string
        String to determine which data to plot.
        Finds matches with model.flags.

    sd_limit : float
        Data with weighted residuals exceeding this
        limit are plotted in black

    cmap : matplotlib color palette

    plot_axes : list of integers
        Data axes to use as plot axes

    scale_axes : list of floats
        Plot axes are scaled by multiplication of the data by these values

    Returns
    -------
    ax : matplotlib Axes object

    """
    if flag is None:
        mask = range(len(model.data[:, 0]))
    else:
        mask = [i for i, flg in enumerate(model.flags) if flg == flag]

    cmap_cp = copy.copy(cmap)
    cmap_cp.set_under('k')
    cmap_cp.set_over('k')
    bounds = np.linspace(-sd_limit, sd_limit, sd_limit*2+1)
    norm = colors.BoundaryNorm(bounds, cmap_cp.N)

    im = ax.scatter(model.data[:, plot_axes[0]][mask] * scale_axes[0],
                    model.data[:, plot_axes[1]][mask] * scale_axes[1],
                    c=model.weighted_residuals[mask],
                    cmap=cmap_cp, norm=norm, s=50)
    plt.colorbar(im, ax=ax, label='Misfit (standard deviations)')


def extreme_values(weighted_residuals, confidence_interval):
    '''
    This function uses extreme value theory to calculate the number of
    standard deviations away from the mean at which we should expect to bracket
    *all* of our n data points at a certain confidence level.

    It then uses that value to identify which (if any) of the data points
    lie outside that region, and calculates the corresponding probabilities
    of finding a data point at least that many standard deviations away.


    Parameters
    ----------

    weighted_residuals : array of floats
        Array of residuals weighted by the square root of their
        variances wr_i = r_i/sqrt(var_i)

    confidence_interval : float
        Probability at which all the weighted residuals lie
        within the confidence bounds

    Returns
    -------
    confidence_bound : float
        Number of standard deviations at which we should expect to encompass
        all data at the user-defined confidence interval.

    indices : array of floats
        Indices of weighted residuals exceeding the confidence_interval
        defined by the user

    probabilities : array of floats
        The probabilities that the extreme data point of the distribution lies
        further from the mean than the observed position wr_i for each i in
        the "indices" output array.
    '''

    n = len(weighted_residuals)
    mean = norm.isf(1./n)
    # good approximation for > 10 data points
    scale = 0.8/np.power(np.log(n), 1./2.)
    # good approximation for > 10 data points
    c = 0.33/np.power(np.log(n), 3./4.)

    # We now need a 1-tailed probability from the given confidence_interval
    # p_total = 1. - confidence_interval = p_upper + p_lower - p_upper*p_lower
    # p_total = 1. - confidence_interval = 2p - p^2, therefore:
    p = 1. - np.sqrt(confidence_interval)
    confidence_bound = genextreme.isf(p, c, loc=mean, scale=scale)

    indices = [i for i, r in enumerate(weighted_residuals)
               if np.abs(r) > confidence_bound]
    # Convert back to 2-tailed probabilities
    probabilities = 1. - \
        np.power(genextreme.sf(
            np.abs(weighted_residuals[indices]), c, loc=mean, scale=scale) - 1., 2.)

    return confidence_bound, indices, probabilities


def plot_residuals(ax, weighted_residuals, n_bins=None, flags=[]):

    if flags == []:
        flags = [''] * len(weighted_residuals)
        list_flags = ['']
    else:
        list_flags = list(set(flags))

    if n_bins is None:
        try:  # Only works for recent versions of numpy
            bin_heights, bin_bounds = np.histogram(weighted_residuals,
                                                   bins='auto',
                                                   density=True)
            n_bins = len(bin_heights)
        except:
            n_bins = 11.

    mask = [i for i, f in enumerate(flags)]
    for flag in list_flags:
        binwidth = np.ptp(weighted_residuals)/n_bins
        dmin = min(weighted_residuals) - binwidth
        dmax = max(weighted_residuals) + binwidth
        bins = np.linspace(dmin, dmax, n_bins)
        bin_heights, bin_bounds = np.histogram(weighted_residuals[mask],
                                               bins=bins,
                                               density=True)

        normalisation = (float(len(weighted_residuals[mask]))
                         / float(len(weighted_residuals)))
        bin_centers = (bin_bounds[:-1] + bin_bounds[1:])/2.
        bin_heights = bin_heights*normalisation
        bin_widths = bin_bounds[1] - bin_bounds[0]
        plt.bar(bin_centers, bin_heights,
                width=bin_widths, label=flag, alpha=0.2)
        mask = [i for i, f in enumerate(flags) if f != flag and i in mask]

        x = np.linspace(bin_bounds[0], bin_bounds[-1], 1001)
        ax.plot(x, norm.pdf(x)*normalisation)

    ax.set_title('Residual plot versus expected normal distribution')
    ax.set_xlabel('Number of standard deviations from the mean')
    ax.set_ylabel('Probability')
    ax.legend(loc='upper right')