最近在做眼底圖像的無監督分類，使用的數據集辣子kaggle的Diabetic Retinopathy，簡稱DR，中文稱糖尿病型眼底疾病。
最后的評估方法是二次加權kappa。以前沒接觸過，網上也沒有具體的介紹，在這里簡單談談我的理解，如有錯誤歡迎指出。
簡介
Kappa指數用來衡量兩個模型對同一張圖片進行判斷時，判斷結果一致的程度，結果范圍從0~1，1表示評價完全相同，0表示評價完全相反。
一般用模型獲得相同評價的數量與基於可能性的期望是否有差別來分析，當兩個模型相同評價的數量和基於可能性期望的數量基本一樣時，kappa的值就接近於1。
舉個栗子，模型A和基准的kappa：
kappa = (p0-pe) / (n-pe)
其中，P0 = 對角線單元中觀測值的總和；pe = 對角線單元中期望值的總和。
根據kappa的計算方法分為簡單kappa（simple kappa）和加權kappa（weighted kappa），加權kappa又分為linear weighted kappa和quadratic weighted kappa。
weighted kappa
關於linear還是quadratic weighted kappa的選擇，取決於你的數據集中不同class之間差異的意義。比如對於眼底圖像識別的數據，class=0為健康，class=4為疾病晚期非常嚴重，所以對於把class=0預測成4的行為所造成的懲罰應該遠遠大於把class=0預測成class=1的行為，使用quadratic的話0->4所造成的懲罰就等於16倍的0->1的懲罰。如下圖是一個四分類的兩個計算方法的比較。
Python實現
參考：https://github.com/benhamner/Metrics/blob/master/Python/ml_metrics/quadratic_weighted_kappa.py
#! /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)