quadratic_weighted_kappa.py

#! /usr/bin/env python2.7
# Code credits: https://github.com/benhamner/Metrics
# Copyright (c) 2012, Ben Hamner
# Author: Ben Hamner (ben@benhamner.com)
# All rights reserved.
#
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# modification, are permitted provided that the following conditions are met:
#
# 1. Redistributions of source code must retain the above copyright notice, this
#    list of conditions and the following disclaimer.
# 2. Redistributions in binary form must reproduce the above copyright notice,
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# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
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# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
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import numpy as np


def confusion_matrix(rater_a, rater_b, min_rating=None, max_rating=None):
    """
    Returns the confusion matrix between rater's ratings
    """
    assert(len(rater_a) == len(rater_b))
    if min_rating is None:
        min_rating = min(rater_a + rater_b)
    if max_rating is None:
        max_rating = max(rater_a + rater_b)
    num_ratings = int(max_rating - min_rating + 1)
    conf_mat = [[0 for i in range(num_ratings)]
                for j in range(num_ratings)]
    for a, b in zip(rater_a, rater_b):
        conf_mat[a - min_rating][b - min_rating] += 1
    return conf_mat


def histogram(ratings, min_rating=None, max_rating=None):
    """
    Returns the counts of each type of rating that a rater made
    """
    if min_rating is None:
        min_rating = min(ratings)
    if max_rating is None:
        max_rating = max(ratings)
    num_ratings = int(max_rating - min_rating + 1)
    hist_ratings = [0 for x in range(num_ratings)]
    for r in ratings:
        hist_ratings[r - min_rating] += 1
    return hist_ratings


def quadratic_weighted_kappa(rater_a, rater_b, min_rating=None, max_rating=None):
    """
    Calculates the quadratic weighted kappa
    quadratic_weighted_kappa calculates the quadratic weighted kappa
    value, which is a measure of inter-rater agreement between two raters
    that provide discrete numeric ratings.  Potential values range from -1
    (representing complete disagreement) to 1 (representing complete
    agreement).  A kappa value of 0 is expected if all agreement is due to
    chance.

    quadratic_weighted_kappa(rater_a, rater_b), where rater_a and rater_b
    each correspond to a list of integer ratings.  These lists must have the
    same length.

    The ratings should be integers, and it is assumed that they contain
    the complete range of possible ratings.

    quadratic_weighted_kappa(X, min_rating, max_rating), where min_rating
    is the minimum possible rating, and max_rating is the maximum possible
    rating
    """
    rater_a = np.array(rater_a, dtype=int)
    rater_b = np.array(rater_b, dtype=int)
    assert(len(rater_a) == len(rater_b))
    if min_rating is None:
        min_rating = min(min(rater_a), min(rater_b))
    if max_rating is None:
        max_rating = max(max(rater_a), max(rater_b))
    conf_mat = confusion_matrix(rater_a, rater_b,
                                min_rating, max_rating)
    num_ratings = len(conf_mat)
    num_scored_items = float(len(rater_a))

    hist_rater_a = histogram(rater_a, min_rating, max_rating)
    hist_rater_b = histogram(rater_b, min_rating, max_rating)

    numerator = 0.0
    denominator = 0.0

    for i in range(num_ratings):
        for j in range(num_ratings):
            expected_count = (hist_rater_a[i] * hist_rater_b[j]
                              / num_scored_items)
            d = pow(i - j, 2.0) / pow(num_ratings - 1, 2.0)
            numerator += d * conf_mat[i][j] / num_scored_items
            denominator += d * expected_count / num_scored_items

    return 1.0 - numerator / denominator


def linear_weighted_kappa(rater_a, rater_b, min_rating=None, max_rating=None):
    """
    Calculates the linear weighted kappa
    linear_weighted_kappa calculates the linear weighted kappa
    value, which is a measure of inter-rater agreement between two raters
    that provide discrete numeric ratings.  Potential values range from -1
    (representing complete disagreement) to 1 (representing complete
    agreement).  A kappa value of 0 is expected if all agreement is due to
    chance.

    linear_weighted_kappa(rater_a, rater_b), where rater_a and rater_b
    each correspond to a list of integer ratings.  These lists must have the
    same length.

    The ratings should be integers, and it is assumed that they contain
    the complete range of possible ratings.

    linear_weighted_kappa(X, min_rating, max_rating), where min_rating
    is the minimum possible rating, and max_rating is the maximum possible
    rating
    """
    assert(len(rater_a) == len(rater_b))
    if min_rating is None:
        min_rating = min(rater_a + rater_b)
    if max_rating is None:
        max_rating = max(rater_a + rater_b)
    conf_mat = confusion_matrix(rater_a, rater_b,
                                min_rating, max_rating)
    num_ratings = len(conf_mat)
    num_scored_items = float(len(rater_a))

    hist_rater_a = histogram(rater_a, min_rating, max_rating)
    hist_rater_b = histogram(rater_b, min_rating, max_rating)

    numerator = 0.0
    denominator = 0.0

    for i in range(num_ratings):
        for j in range(num_ratings):
            expected_count = (hist_rater_a[i] * hist_rater_b[j]
                              / num_scored_items)
            d = abs(i - j) / float(num_ratings - 1)
            numerator += d * conf_mat[i][j] / num_scored_items
            denominator += d * expected_count / num_scored_items

    return 1.0 - numerator / denominator


def kappa(rater_a, rater_b, min_rating=None, max_rating=None):
    """
    Calculates the kappa
    kappa calculates the kappa
    value, which is a measure of inter-rater agreement between two raters
    that provide discrete numeric ratings.  Potential values range from -1
    (representing complete disagreement) to 1 (representing complete
    agreement).  A kappa value of 0 is expected if all agreement is due to
    chance.

    kappa(rater_a, rater_b), where rater_a and rater_b
    each correspond to a list of integer ratings.  These lists must have the
    same length.

    The ratings should be integers, and it is assumed that they contain
    the complete range of possible ratings.

    kappa(X, min_rating, max_rating), where min_rating
    is the minimum possible rating, and max_rating is the maximum possible
    rating
    """
    assert(len(rater_a) == len(rater_b))
    if min_rating is None:
        min_rating = min(rater_a + rater_b)
    if max_rating is None:
        max_rating = max(rater_a + rater_b)
    conf_mat = confusion_matrix(rater_a, rater_b,
                                min_rating, max_rating)
    num_ratings = len(conf_mat)
    num_scored_items = float(len(rater_a))

    hist_rater_a = histogram(rater_a, min_rating, max_rating)
    hist_rater_b = histogram(rater_b, min_rating, max_rating)

    numerator = 0.0
    denominator = 0.0

    for i in range(num_ratings):
        for j in range(num_ratings):
            expected_count = (hist_rater_a[i] * hist_rater_b[j]
                              / num_scored_items)
            if i == j:
                d = 0.0
            else:
                d = 1.0
            numerator += d * conf_mat[i][j] / num_scored_items
            denominator += d * expected_count / num_scored_items

    return 1.0 - numerator / denominator


def mean_quadratic_weighted_kappa(kappas, weights=None):
    """
    Calculates the mean of the quadratic
    weighted kappas after applying Fisher's r-to-z transform, which is
    approximately a variance-stabilizing transformation.  This
    transformation is undefined if one of the kappas is 1.0, so all kappa
    values are capped in the range (-0.999, 0.999).  The reverse
    transformation is then applied before returning the result.

    mean_quadratic_weighted_kappa(kappas), where kappas is a vector of
    kappa values

    mean_quadratic_weighted_kappa(kappas, weights), where weights is a vector
    of weights that is the same size as kappas.  Weights are applied in the
    z-space
    """
    kappas = np.array(kappas, dtype=float)
    if weights is None:
        weights = np.ones(np.shape(kappas))
    else:
        weights = weights / np.mean(weights)

    # ensure that kappas are in the range [-.999, .999]
    kappas = np.array([min(x, .999) for x in kappas])
    kappas = np.array([max(x, -.999) for x in kappas])

    z = 0.5 * np.log((1 + kappas) / (1 - kappas)) * weights
    z = np.mean(z)
    return (np.exp(2 * z) - 1) / (np.exp(2 * z) + 1)


def weighted_mean_quadratic_weighted_kappa(solution, submission):
    predicted_score = submission[submission.columns[-1]].copy()
    predicted_score.name = "predicted_score"
    if predicted_score.index[0] == 0:
        predicted_score = predicted_score[:len(solution)]
        predicted_score.index = solution.index
    combined = solution.join(predicted_score, how="left")
    groups = combined.groupby(by="essay_set")
    kappas = [quadratic_weighted_kappa(group[1]["essay_score"], group[1]["predicted_score"]) for group in groups]
    weights = [group[1]["essay_weight"].irow(0) for group in groups]
    return mean_quadratic_weighted_kappa(kappas, weights=weights)