第100+27步 ChatGPT学习：概率校准 Temperature Scaling

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 Temperature Scaling 的方法。

二、Temperature Scaling

Temperature Scaling的核心思想是通过引入一个温度参数（temperature parameter, T）来调整模型的logits（未归一化的输出值），从而校准输出的概率分布。具体来说，它是一种后处理方法，即在训练完模型后进行校准，而不改变模型的结构或训练过程。

（1）主要步骤

1）训练模型：首先，训练你的分类模型，获得logits和初步的概率预测。

2）验证集校准：使用验证集来找到最优的温度参数T。通过最小化负对数似然（negative log-likelihood, NLL）或者期望校准误差（expected calibration error, ECE）等校准指标来找到最佳的T。

3）校准：使用找到的最优温度参数T对测试集或实际应用中的预测概率进行校准。

三、Temperature Scaling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]

# Temperature Scaling
class TemperatureScaling:
    def __init__(self):
        self.temperature = 1.0

    def fit(self, logits, true_labels):
        from scipy.optimize import minimize

        def nll_loss(T):
            scaled_logits = logits / T
            probs = np.exp(scaled_logits) / np.sum(np.exp(scaled_logits), axis=1, keepdims=True)
            log_probs = np.log(probs[np.arange(len(true_labels)), true_labels])
            return -np.mean(log_probs)

        logits = np.log(np.column_stack([1 - logits, logits]))  # 转换成logits
        result = minimize(nll_loss, [1.0], bounds=[(0.1, 10.0)])
        self.temperature = result.x[0]

    def predict_proba(self, probs):
        logits = np.log(np.column_stack([1 - probs, probs]))  # 转换成logits
        scaled_logits = logits / self.temperature
        exp_scaled_logits = np.exp(scaled_logits)
        probs = exp_scaled_logits[:, 1] / np.sum(exp_scaled_logits, axis=1, keepdims=True)[:, 0]
        return probs

# 训练Temperature Scaling模型
temp_scaling = TemperatureScaling()
temp_scaling.fit(y_train_probs, y_train)

# 进行校准
calibrated_train_probs = temp_scaling.predict_proba(y_train_probs)
calibrated_test_probs = temp_scaling.predict_proba(y_test_probs)

# 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Temperature Scaling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 = []

# Temperature Scaling
class TemperatureScaling:
    def __init__(self):
        self.temperature = 1.0

    def fit(self, logits, true_labels):
        from scipy.optimize import minimize

        def nll_loss(T):
            scaled_logits = logits / T
            probs = np.exp(scaled_logits) / np.sum(np.exp(scaled_logits), axis=1, keepdims=True)
            log_probs = np.log(probs[np.arange(len(true_labels)), true_labels])
            return -np.mean(log_probs)

        logits = np.log(np.column_stack([1 - logits, logits]))  # 转换成logits
        result = minimize(nll_loss, [1.0], bounds=[(0.1, 10.0)])
        self.temperature = result.x[0]

    def predict_proba(self, probs):
        logits = np.log(np.column_stack([1 - probs, probs]))  # 转换成logits
        scaled_logits = logits / self.temperature
        exp_scaled_logits = np.exp(scaled_logits)
        probs = exp_scaled_logits[:, 1] / np.sum(exp_scaled_logits, axis=1, keepdims=True)[:, 0]
        return probs

best_params = None  # 用于存储最优参数

for train_index, val_index in kf.split(X_train):
    X_train_fold, X_val_fold = X_train[train_index], X_train[val_index]
    y_train_fold, y_val_fold = y_train[train_index], y_train[val_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]
    
    # Temperature Scaling 校准
    temp_scaling = TemperatureScaling()
    temp_scaling.fit(y_val_fold_probs, y_val_fold)
    calibrated_val_fold_probs = temp_scaling.predict_proba(y_val_fold_probs)
    
    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]

# Temperature Scaling 校准
temp_scaling_final = TemperatureScaling()
temp_scaling_final.fit(y_test_probs, y_test)
calibrated_test_probs = temp_scaling_final.predict_proba(y_test_probs)

# 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)
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="Temperature Scaling 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：

效果一般般吧。

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。