Kappa评分衡量两个评分之间的一致性。二次加权kappa是利用人工评分员分配的分数和预测分数计算的。该指标从-1（评分员之间完全不一致）到1（评分员之间完全一致）不等。κ的定义是：
其中k是类别数，oi j和ei j分别是观察矩阵和期望矩阵中的元素。wi j计算如下：
由于Cohens Kappa特性，研究人员必须仔细解释该比率。例如，如果我们考虑两对评级者具有相同的协议百分比，但评级比例不同，我们应该知道，这将极大地影响Kappa比率。另一个问题是代码的数量：随着代码数量的增加，Kappa变得更高。此外，Kappa可能较低，即使存在高水平的一致性，即使个人评级为精确。上述所有因素使Kappa成为一个可分析的波动率。使用Kappa比率的主要原因是我们无法获得验证和测试数据集的标签。这些数据集的Kappa值是通过将我们的模型和运行代码提交给Kaggle站点上的检查系统获得的。此外，我们无法明确访问测试数据集中的图像。
一般用模型获得相同评价的数量与基于可能性的期望是否有差别来分析，当两个模型相同评价的数量和基于可能性期望的数量基本一样时，kappa的值就接近于1。
举个栗子，模型A和基准的kappa：
kappa = (p0-pe) / (n-pe)
其中，P0 = 对角线单元中观测值的总和；pe = 对角线单元中期望值的总和。
根据kappa的计算方法分为简单kappa（simple kappa）和加权kappa（weighted kappa），加权kappa又分为linear weighted kappa和quadratic weighted kappa。
weighted kappa
关于linear还是quadratic weighted kappa的选择，取决于你的数据集中不同class之间差异的意义。比如对于眼底图像识别的数据，class=0为健康，class=4为疾病晚期非常严重，所以对于把class=0预测成4的行为所造成的惩罚应该远远大于把class=0预测成class=1的行为，使用quadratic的话0->4所造成的惩罚就等于16倍的0->1的惩罚。如下图是一个四分类的两个计算方法的比较。
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Python代码实现：
#! /usr/bin/env python2.7
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)