第 7 章 · 误差下降与线性边界代码实验¶

# 加载 Diabetes 经典回归数据集，使用 BMI 预测一年后的疾病进展指标。
diabetes = load_diabetes(as_frame=True)
gd_df = diabetes.frame[["bmi", "target"]].copy()
gd_df.columns = ["BMI", "疾病进展指标"]

X_gd = gd_df[["BMI"]].to_numpy()
y_gd = gd_df["疾病进展指标"].to_numpy()

display(gd_df.head(10))
display(gd_df.describe().round(3))

# 用随机梯度下降做回归，记录损失下降过程。
gd_scaler = StandardScaler()
X_gd_scaled = gd_scaler.fit_transform(X_gd)
regressor = SGDRegressor(
    loss="squared_error",
    penalty=None,
    learning_rate="invscaling",
    eta0=0.04,
    power_t=0.25,
    random_state=0,
)

gd_rows = []
for epoch in range(1, 61):
    regressor.partial_fit(X_gd_scaled, y_gd)
    pred = regressor.predict(X_gd_scaled)
    gd_rows.append({
        "轮次": epoch,
        "标准化斜率": round(regressor.coef_[0], 4),
        "截距": round(regressor.intercept_[0], 4),
        "MSE": round(mean_squared_error(y_gd, pred), 3),
    })

gd_trace = pd.DataFrame(gd_rows)
display(gd_trace.iloc[[0, 1, 2, 9, 29, 59]])

# 绘制拟合直线和 MSE 曲线。
fig, axes = plt.subplots(1, 2, figsize=(10.2, 4.5))

x_line = np.linspace(X_gd.min(), X_gd.max(), 100).reshape(-1, 1)
axes[0].scatter(X_gd[:, 0], y_gd, s=42, color="#2563eb", alpha=0.72, edgecolor="white", linewidth=0.4)
axes[0].plot(x_line[:, 0], regressor.predict(gd_scaler.transform(x_line)), color="#f97316", linewidth=2.4)
axes[0].set_title("糖尿病数据 BMI 回归", loc="left", fontweight="bold")
axes[0].set_xlabel("BMI")
axes[0].set_ylabel("疾病进展指标")
axes[0].grid(True, color="#e2e8f0", linewidth=0.8)

axes[1].plot(gd_trace["轮次"], gd_trace["MSE"], color="#2563eb", linewidth=2.4)
axes[1].set_title("损失下降", loc="left", fontweight="bold")
axes[1].set_xlabel("轮次")
axes[1].set_ylabel("均方误差")
axes[1].grid(True, color="#e2e8f0", linewidth=0.8)

plt.tight_layout()
plt.show()

# 加载鸢尾花中的 setosa 与 versicolor，使用两个花瓣特征做感知机二分类。
iris = load_iris(as_frame=True)
perceptron_df = iris.frame.copy()
iris_feature_cn_map = dict(zip(iris.feature_names, ["萼片长度", "萼片宽度", "花瓣长度", "花瓣宽度"]))
perceptron_df["品种"] = perceptron_df["target"].map(dict(enumerate(iris.target_names)))
perceptron_df = perceptron_df[perceptron_df["品种"].isin(["setosa", "versicolor"])].copy()

per_features = ["petal length (cm)", "petal width (cm)"]
X_per = perceptron_df[per_features].to_numpy()
y_per = (perceptron_df["品种"] == "versicolor").astype(int).to_numpy()
perceptron_df["二分类标签"] = y_per

display(perceptron_df.rename(columns=iris_feature_cn_map))

# 逐轮训练感知机，记录边界参数。
per_scaler = StandardScaler()
X_per_scaled = per_scaler.fit_transform(X_per)
perceptron = Perceptron(eta0=0.08, random_state=1, warm_start=True)
classes = np.array([0, 1])
per_rows = []

for epoch in range(1, 13):
    perceptron.partial_fit(X_per_scaled, y_per, classes=classes)
    pred = perceptron.predict(X_per_scaled)
    per_rows.append({
        "轮次": epoch,
        "花瓣长度权重": round(perceptron.coef_[0, 0], 4),
        "花瓣宽度权重": round(perceptron.coef_[0, 1], 4),
        "偏置": round(perceptron.intercept_[0], 4),
        "错误数": int((pred != y_per).sum()),
    })

per_trace = pd.DataFrame(per_rows)
display(per_trace)

# 绘制感知机决策边界。
fig, ax = plt.subplots(figsize=(7.6, 5.2))
colors = np.where(y_per == 1, "#2563eb", "#f97316")
ax.scatter(X_per[:, 0], X_per[:, 1], c=colors, s=130, edgecolors="white", linewidth=1.5)

x_min, x_max = X_per[:, 0].min() - 0.05, X_per[:, 0].max() + 0.05
x_line = np.linspace(x_min, x_max, 100)
w0, w1 = perceptron.coef_[0]
bias = perceptron.intercept_[0]
scaled_x = (x_line - per_scaler.mean_[0]) / per_scaler.scale_[0]
scaled_y = -(w0 * scaled_x + bias) / w1
y_line = scaled_y * per_scaler.scale_[1] + per_scaler.mean_[1]
ax.plot(x_line, y_line, color="#0f172a", linewidth=2.2)

ax.set_title("鸢尾花感知机决策边界", loc="left", fontsize=14, fontweight="bold", color="#0f172a")
ax.set_xlabel("花瓣长度（cm）")
ax.set_ylabel("花瓣宽度（cm）")
ax.set_xlim(x_min, x_max)
ax.set_ylim(X_per[:, 1].min() - 0.08, X_per[:, 1].max() + 0.12)
ax.grid(True, color="#e2e8f0", linewidth=0.8)
plt.tight_layout()
plt.show()

# 用 Iris 感知机的 decision_function 分数观察阈值变化。
y_true = y_per
y_score = perceptron.decision_function(X_per_scaled)
thresholds = np.quantile(y_score, [0.35, 0.50, 0.65])

metric_rows = []
for threshold in thresholds:
    y_hat = (y_score >= threshold).astype(int)
    precision, recall, f1, _ = precision_recall_fscore_support(
        y_true, y_hat, average="binary", zero_division=0
    )
    metric_rows.append({
        "阈值": round(float(threshold), 3),
        "预测为正": int(y_hat.sum()),
        "精确率": round(precision, 3),
        "召回率": round(recall, 3),
        "F1": round(f1, 3),
    })

display(pd.DataFrame(metric_rows))