基于Python 3.9版本演示

一、写在前面

最近看了一篇在Lancet子刊《eClinicalMedicine》上发表的机器学习分类的文章：《Development of a novel dementia risk prediction model in the general population: A large, longitudinal, population-based machine-learning study》。

学到一种叫做“概率校准”的骚操作，顺手利用GPT系统学习学习。

文章中用的技术是：保序回归（Isotonic regression）。

为了体现举一反三，顺便问了GPT还有哪些方法也可以实现概率校准。它给我列举了很多，那么就一个一个学习吧。

这一期，介绍一个叫做 Bayesian Binning into Quantiles 的方法。

二、Bayesian Binning into Quantiles

Bayesian Binning into Quantiles (BBQ)是一种用于概率校准的技术，旨在将模型输出的概率转化为更准确的预测概率。BBQ方法通过将预测概率分成多个区间（即bins），并在每个区间内进行贝叶斯估计来校准这些概率。其基本思想是通过分箱和贝叶斯推断来估计每个箱中的实际概率分布，从而提高分类模型的预测概率的准确性。

（1）主要步骤

1）分箱（Binning）：将预测概率值按照一定的规则分成多个区间，通常是通过分位数（quantiles）进行分箱。这样可以保证每个箱中的样本数量大致相同。

2）贝叶斯估计：在每个区间内，通过贝叶斯估计来计算实际的概率分布。具体做法是使用先验分布和观测数据来更新后验分布，从而得到更加可靠的概率估计。

3）校准：将每个区间的后验概率分布作为校准后的概率值，用于调整模型的原始预测概率。

三、Bayesian Binning into Quantiles代码实现

下面，我编一个1比3的不太平衡的数据进行测试，对照组使用不进行校准的SVM模型，实验组就是加入校准的SVM模型，看看性能能够提高多少？

（1）不进行校准的SVM模型（默认参数）

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.svm import SVC
from sklearn.metrics import confusion_matrix, roc_auc_score, roc_curve

# 加载数据
dataset = pd.read_csv('8PSMjianmo.csv')
X = dataset.iloc[:, 1:20].values
Y = dataset.iloc[:, 0].values

# 分割数据集
X_train, X_test, y_train, y_test = train_test_split(X, Y, test_size=0.30, random_state=666)

# 标准化数据
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)

# 使用SVM分类器
classifier = SVC(kernel='linear', probability=True)
classifier.fit(X_train, y_train)

# 预测结果
y_pred = classifier.predict(X_test)
y_testprba = classifier.decision_function(X_test)

y_trainpred = classifier.predict(X_train)
y_trainprba = classifier.decision_function(X_train)

# 混淆矩阵
cm_test = confusion_matrix(y_test, y_pred)
cm_train = confusion_matrix(y_train, y_trainpred)
print(cm_train)
print(cm_test)

# 绘制测试集混淆矩阵
classes = list(set(y_test))
classes.sort()
plt.imshow(cm_test, cmap=plt.cm.Blues)
indices = range(len(cm_test))
plt.xticks(indices, classes)
plt.yticks(indices, classes)
plt.colorbar()
plt.xlabel('Predicted')
plt.ylabel('Actual')
for first_index in range(len(cm_test)):
    for second_index in range(len(cm_test[first_index])):
        plt.text(first_index, second_index, cm_test[first_index][second_index])

plt.show()

# 绘制训练集混淆矩阵
classes = list(set(y_train))
classes.sort()
plt.imshow(cm_train, cmap=plt.cm.Blues)
indices = range(len(cm_train))
plt.xticks(indices, classes)
plt.yticks(indices, classes)
plt.colorbar()
plt.xlabel('Predicted')
plt.ylabel('Actual')
for first_index in range(len(cm_train)):
    for second_index in range(len(cm_train[first_index])):
        plt.text(first_index, second_index, cm_train[first_index][second_index])

plt.show()

# 计算并打印性能参数
def calculate_metrics(cm, y_true, y_pred_prob):
    a = cm[0, 0]
    b = cm[0, 1]
    c = cm[1, 0]
    d = cm[1, 1]
    acc = (a + d) / (a + b + c + d)
    error_rate = 1 - acc
    sen = d / (d + c)
    sep = a / (a + b)
    precision = d / (b + d)
    F1 = (2 * precision * sen) / (precision + sen)
    MCC = (d * a - b * c) / (np.sqrt((d + b) * (d + c) * (a + b) * (a + c)))
    auc_score = roc_auc_score(y_true, y_pred_prob)
    
    metrics = {
        "Accuracy": acc,
        "Error Rate": error_rate,
        "Sensitivity": sen,
        "Specificity": sep,
        "Precision": precision,
        "F1 Score": F1,
        "MCC": MCC,
        "AUC": auc_score
    }
    return metrics

metrics_test = calculate_metrics(cm_test, y_test, y_testprba)
metrics_train = calculate_metrics(cm_train, y_train, y_trainprba)

print("Performance Metrics (Test):")
for key, value in metrics_test.items():
    print(f"{key}: {value:.4f}")

print("\nPerformance Metrics (Train):")
for key, value in metrics_train.items():
print(f"{key}: {value:.4f}")

结果输出：

记住这些个数字。

这个参数的SVM还没有LR好。

（2）进行校准的SVM模型（默认参数）

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.svm import SVC
from sklearn.metrics import confusion_matrix, roc_auc_score, brier_score_loss
from sklearn.calibration import calibration_curve

# 加载数据
dataset = pd.read_csv('8PSMjianmo.csv')
X = dataset.iloc[:, 1:20].values
Y = dataset.iloc[:, 0].values

# 分割数据集
X_train, X_test, y_train, y_test = train_test_split(X, Y, test_size=0.30, random_state=666)

# 标准化数据
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)

# 使用SVM分类器
classifier = SVC(kernel='rbf', C=0.1, probability=True)
classifier.fit(X_train, y_train)

# 获取未校准的概率预测
y_train_probs = classifier.predict_proba(X_train)[:, 1]
y_test_probs = classifier.predict_proba(X_test)[:, 1]

# Bayesian Binning into Quantiles (BBQ) 校准函数
def bayesian_binning_into_quantiles(probabilities, y_true, n_bins=10):
    bin_edges = np.linspace(0, 1, n_bins + 1)
    bin_indices = np.digitize(probabilities, bin_edges) - 1
    bin_sums = np.zeros(n_bins)
    bin_true_sums = np.zeros(n_bins)
    bin_counts = np.zeros(n_bins)

    for i in range(n_bins):
        bin_counts[i] = np.sum(bin_indices == i)
        if bin_counts[i] > 0:
            bin_sums[i] = np.sum(probabilities[bin_indices == i])
            bin_true_sums[i] = np.sum(y_true[bin_indices == i])
    
    bin_probs = np.zeros_like(probabilities)
    for i in range(n_bins):
        if bin_counts[i] > 0:
            alpha = bin_true_sums[i] + 1  # 使用Beta分布的参数
            beta = bin_counts[i] - bin_true_sums[i] + 1
            bin_probs[bin_indices == i] = alpha / (alpha + beta)
    
    return bin_probs

# 进行BBQ校准
calibrated_train_probs = bayesian_binning_into_quantiles(y_train_probs, y_train)
calibrated_test_probs = bayesian_binning_into_quantiles(y_test_probs, y_test)

# 预测结果
y_train_pred = (calibrated_train_probs >= 0.5).astype(int)
y_test_pred = (calibrated_test_probs >= 0.5).astype(int)

# 混淆矩阵
cm_test = confusion_matrix(y_test, y_test_pred)
cm_train = confusion_matrix(y_train, y_train_pred)
print(cm_train)
print(cm_test)

# 绘制混淆矩阵函数
def plot_confusion_matrix(cm, classes, title='Confusion Matrix'):
    plt.imshow(cm, cmap=plt.cm.Blues)
    indices = range(len(cm))
    plt.xticks(indices, classes)
    plt.yticks(indices, classes)
    plt.colorbar()
    plt.xlabel('Predicted')
    plt.ylabel('Actual')
    for first_index in range(len(cm)):
        for second_index in range(len(cm[first_index])):
            plt.text(second_index, first_index, cm[first_index][second_index])
    plt.title(title)
    plt.show()

# 绘制测试集混淆矩阵
plot_confusion_matrix(cm_test, list(set(y_test)), 'Confusion Matrix (Test)')

# 绘制训练集混淆矩阵
plot_confusion_matrix(cm_train, list(set(y_train)), 'Confusion Matrix (Train)')

# 计算并打印性能参数
def calculate_metrics(cm, y_true, y_pred_prob):
    a = cm[0, 0]
    b = cm[0, 1]
    c = cm[1, 0]
    d = cm[1, 1]
    acc = (a + d) / (a + b + c + d)
    error_rate = 1 - acc
    sen = d / (d + c)
    sep = a / (a + b)
    precision = d / (b + d)
    F1 = (2 * precision * sen) / (precision + sen)
    MCC = (d * a - b * c) / (np.sqrt((d + b) * (d + c) * (a + b) * (a + c)))
    auc_score = roc_auc_score(y_true, y_pred_prob)
    brier_score = brier_score_loss(y_true, y_pred_prob)
    
    metrics = {
        "Accuracy": acc,
        "Error Rate": error_rate,
        "Sensitivity": sen,
        "Specificity": sep,
        "Precision": precision,
        "F1 Score": F1,
        "MCC": MCC,
        "AUC": auc_score,
        "Brier Score": brier_score
    }
    return metrics

metrics_test = calculate_metrics(cm_test, y_test, calibrated_test_probs)
metrics_train = calculate_metrics(cm_train, y_train, calibrated_train_probs)

print("Performance Metrics (Test):")
for key, value in metrics_test.items():
    print(f"{key}: {value:.4f}")

print("\nPerformance Metrics (Train):")
for key, value in metrics_train.items():
    print(f"{key}: {value:.4f}")

看看结果：

大同小异吧。

四、换个策略

参考那篇文章的策略：采用五折交叉验证来建立和评估模型，其中四折用于训练，一折用于评估，在训练集中，其中三折用于建立SVM模型，另一折采用Bayesian Binning into Quantiles概率校正，在训练集内部采用交叉验证对超参数进行调参。

代码：

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.model_selection import train_test_split, GridSearchCV, KFold
from sklearn.preprocessing import StandardScaler
from sklearn.svm import SVC
from sklearn.metrics import confusion_matrix, roc_auc_score, brier_score_loss
from sklearn.calibration import calibration_curve

# 加载数据
dataset = pd.read_csv('8PSMjianmo.csv')
X = dataset.iloc[:, 1:20].values
Y = dataset.iloc[:, 0].values

# 分割数据集
X_train, X_test, y_train, y_test = train_test_split(X, Y, test_size=0.30, random_state=666)

# 标准化数据
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)

# 定义五折交叉验证
kf = KFold(n_splits=5, shuffle=True, random_state=666)
calibrated_probs = []
true_labels = []

def bayesian_binning_into_quantiles(probabilities, y_true, n_bins=10):
    bin_edges = np.linspace(0, 1, n_bins + 1)
    bin_indices = np.digitize(probabilities, bin_edges) - 1
    bin_sums = np.zeros(n_bins)
    bin_true_sums = np.zeros(n_bins)
    bin_counts = np.zeros(n_bins)

    for i in range(n_bins):
        bin_counts[i] = np.sum(bin_indices == i)
        if bin_counts[i] > 0:
            bin_sums[i] = np.sum(probabilities[bin_indices == i])
            bin_true_sums[i] = np.sum(y_true[bin_indices == i])

    bin_probs = np.zeros_like(probabilities)
    for i in range(n_bins):
        if bin_counts[i] > 0:
            alpha = bin_true_sums[i] + 1  # 使用Beta分布的参数
            beta = bin_counts[i] - bin_true_sums[i] + 1
            bin_probs[bin_indices == i] = alpha / (alpha + beta)

    return bin_probs

best_params = None  # 用于存储最优参数

for train_index, test_index in kf.split(X_train):
    X_train_fold, X_val_fold = X_train[train_index], X_train[test_index]
    y_train_fold, y_val_fold = y_train[train_index], y_train[test_index]
    
    # 内部三折交叉验证用于超参数调优
    inner_kf = KFold(n_splits=3, shuffle=True, random_state=666)
    param_grid = {'C': [0.01, 0.1, 1, 10, 100], 'kernel': ['rbf']}
    svm = SVC(probability=True)
    clf = GridSearchCV(svm, param_grid, cv=inner_kf, scoring='roc_auc')
    clf.fit(X_train_fold, y_train_fold)
    best_params = clf.best_params_
    
    # 使用最佳参数训练SVM
    classifier = SVC(kernel=best_params['kernel'], C=best_params['C'], probability=True)
    classifier.fit(X_train_fold, y_train_fold)
    
    # 获取未校准的概率预测
    y_val_fold_probs = classifier.predict_proba(X_val_fold)[:, 1]
    
    # 进行BBQ校准
    calibrated_val_fold_probs = bayesian_binning_into_quantiles(y_val_fold_probs, y_val_fold)
    calibrated_probs.extend(calibrated_val_fold_probs)
    true_labels.extend(y_val_fold)

# 用于测试集的SVM模型训练和校准
classifier_final = SVC(kernel=best_params['kernel'], C=best_params['C'], probability=True)
classifier_final.fit(X_train, y_train)
y_test_probs = classifier_final.predict_proba(X_test)[:, 1]
calibrated_test_probs = bayesian_binning_into_quantiles(y_test_probs, y_test)

# 预测结果
y_train_pred = (np.array(calibrated_probs) >= 0.5).astype(int)
y_test_pred = (calibrated_test_probs >= 0.5).astype(int)

# 混淆矩阵
cm_test = confusion_matrix(y_test, y_test_pred)
cm_train = confusion_matrix(true_labels, y_train_pred)  # 注意，这里使用了校准后的训练集结果
print("Training Confusion Matrix:\n", cm_train)
print("Testing Confusion Matrix:\n", cm_test)

# 绘制混淆矩阵函数
def plot_confusion_matrix(cm, classes, title='Confusion Matrix'):
    plt.imshow(cm, cmap=plt.cm.Blues)
    indices = range(len(cm))
    plt.xticks(indices, classes)
    plt.yticks(indices, classes)
    plt.colorbar()
    plt.xlabel('Predicted')
    plt.ylabel('Actual')
    for first_index in range(len(cm)):
        for second_index in range(len(cm[first_index])):
            plt.text(second_index, first_index, cm[first_index][second_index])
    plt.title(title)
    plt.show()

# 绘制测试集混淆矩阵
plot_confusion_matrix(cm_test, list(set(y_test)), 'Confusion Matrix (Test)')

# 绘制训练集混淆矩阵
plot_confusion_matrix(cm_train, list(set(true_labels)), 'Confusion Matrix (Train)')

# 计算并打印性能参数
def calculate_metrics(cm, y_true, y_pred_prob):
    a = cm[0, 0]
    b = cm[0, 1]
    c = cm[1, 0]
    d = cm[1, 1]
    acc = (a + d) / (a + b + c + d)
    error_rate = 1 - acc
    sen = d / (d + c)
    sep = a / (a + b)
    precision = d / (b + d)
    F1 = (2 * precision * sen) / (precision + sen)
    MCC = (d * a - b * c) / (np.sqrt((d + b) * (d + c) * (a + b) * (a + c)))
    auc_score = roc_auc_score(y_true, y_pred_prob)
    brier_score = brier_score_loss(y_true, y_pred_prob)
    
    metrics = {
        "Accuracy": acc,
        "Error Rate": error_rate,
        "Sensitivity": sen,
        "Specificity": sep,
        "Precision": precision,
        "F1 Score": F1,
        "MCC": MCC,
        "AUC": auc_score,
        "Brier Score": brier_score
    }
    return metrics

metrics_test = calculate_metrics(cm_test, y_test, calibrated_test_probs)
metrics_train = calculate_metrics(cm_train, true_labels, np.array(calibrated_probs))

print("Performance Metrics (Test):")
for key, value in metrics_test.items():
    print(f"{key}: {value:.4f}")

print("\nPerformance Metrics (Train):")
for key, value in metrics_train.items():
    print(f"{key}: {value:.4f}")

# 绘制校准曲线
def plot_calibration_curve(y_true, probs, title='Calibration Curve'):
    fraction_of_positives, mean_predicted_value = calibration_curve(y_true, probs, n_bins=10)
    plt.plot(mean_predicted_value, fraction_of_positives, "s-", label="BBQ Calibration")
    plt.plot([0, 1], [0, 1], "k--")
    plt.xlabel('Mean predicted value')
    plt.ylabel('Fraction of positives')
    plt.title(title)
    plt.legend()
    plt.show()

# 绘制校准曲线
plot_calibration_curve(y_test, calibrated_test_probs, title='Calibration Curve (Test)')
plot_calibration_curve(true_labels, np.array(calibrated_probs), title='Calibration Curve (Train)')

输出：

提升也不是很明显。

五、最后

各位可以去试一试在其他数据或者在其他机器学习分类模型中使用的效果。

数据不分享啦。