1. 首页
  2. 编程语言

机器学习第5章支持向量机

支持向量机(简称SVM)是一个功能强大并且全面的机器学习模型,它能够执行线性或非线性分类、回归,甚至是异常值检测任务。它是机器学习领域最受欢迎的模型之一,任何对机器学习感兴趣的人都应该在工具箱中配备一个。SVM特别适用于中小型复杂数据集的分类。

  1. 保存图片
    from __future__ import division, print_function, unicode_literals
    import numpy as np
    import matplotlib as mpl
    import matplotlib.pyplot as plt
    import os
    np.random.seed(42)
    
    mpl.rc('axes', labelsize=14)
    mpl.rc('xtick', labelsize=12)
    mpl.rc('ytick', labelsize=12)
    
    # Where to save the figures
    PROJECT_ROOT_DIR = "images"
    CHAPTER_ID = "traininglinearmodels"
    
    def save_fig(fig_id, tight_layout=True):
        path = os.path.join(PROJECT_ROOT_DIR, CHAPTER_ID, fig_id + ".png")
        print("Saving figure", fig_id)
        if tight_layout:
            plt.tight_layout()
        plt.savefig(path, format='png', dpi=600)
    

线性SVM分类

  1. 加载鸢尾花数据集,缩放特征,然后训练一个线性SVM模型(使用LinearSVC类,C=0.1,用即将介绍的hinge损失函数)用来检测Virginica鸢尾花。
    from sklearn.svm import SVC
    from sklearn import datasets
    
    iris = datasets.load_iris()
    X = iris["data"][:, (2, 3)]  # petal length, petal width
    y = iris["target"]
    
    setosa_or_versicolor = (y == 0) | (y == 1)
    X = X[setosa_or_versicolor]
    y = y[setosa_or_versicolor]
    
    # SVM Classifier model
    svm_clf = SVC(kernel="linear", C=float("inf"))
    print(svm_clf.fit(X, y))
    
    # Bad models
    x0 = np.linspace(0, 5.5, 200)
    pred_1 = 5*x0 - 20
    pred_2 = x0 - 1.8
    pred_3 = 0.1 * x0 + 0.5
    
    def plot_svc_decision_boundary(svm_clf, xmin, xmax):
        w = svm_clf.coef_[0]
        b = svm_clf.intercept_[0]
    
        # At the decision boundary, w0*x0 + w1*x1 + b = 0
        # => x1 = -w0/w1 * x0 - b/w1
        x0 = np.linspace(xmin, xmax, 200)
        decision_boundary = -w[0]/w[1] * x0 - b/w[1]
    
        margin = 1/w[1]
        gutter_up = decision_boundary + margin
        gutter_down = decision_boundary - margin
    
        svs = svm_clf.support_vectors_
        plt.scatter(svs[:, 0], svs[:, 1], s=180, facecolors='#FFAAAA')
        plt.plot(x0, decision_boundary, "k-", linewidth=2)
        plt.plot(x0, gutter_up, "k--", linewidth=2)
        plt.plot(x0, gutter_down, "k--", linewidth=2)
    
    plt.figure(figsize=(12,2.7))
    
    plt.subplot(121)
    plt.plot(x0, pred_1, "g--", linewidth=2)
    plt.plot(x0, pred_2, "m-", linewidth=2)
    plt.plot(x0, pred_3, "r-", linewidth=2)
    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", label="Iris-Versicolor")
    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", label="Iris-Setosa")
    plt.xlabel("Petal length", fontsize=14)
    plt.ylabel("Petal width", fontsize=14)
    plt.legend(loc="upper left", fontsize=14)
    plt.axis([0, 5.5, 0, 2])
    
    plt.subplot(122)
    plot_svc_decision_boundary(svm_clf, 0, 5.5)
    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs")
    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo")
    plt.xlabel("Petal length", fontsize=14)
    plt.axis([0, 5.5, 0, 2])
    
    save_fig("large_margin_classification_plot较少间隔违例和大间隔对比")
    plt.show()
    

非线性SVM分类

  1. 通过添加特征使数据集线性可分离
    X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
    X2D = np.c_[X1D, X1D**2]
    y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
    '''
    plt.figure(figsize=(11, 4))
    
    plt.subplot(121)
    plt.grid(True, which='both')
    plt.axhline(y=0, color='k')
    plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
    plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
    plt.gca().get_yaxis().set_ticks([])
    plt.xlabel(r"$x_1$", fontsize=20)
    plt.axis([-4.5, 4.5, -0.2, 0.2])
    
    plt.subplot(122)
    plt.grid(True, which='both')
    plt.axhline(y=0, color='k')
    plt.axvline(x=0, color='k')
    plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
    plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
    plt.xlabel(r"$x_1$", fontsize=20)
    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
    plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
    plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
    plt.axis([-4.5, 4.5, -1, 17])
    
    plt.subplots_adjust(right=1)
    
    save_fig("higher_dimensions_plot", tight_layout=False)
    plt.show()
    
  2. 要使用Scikit-Learn实现这个想法,可以搭建一条流水线:一个PolynomialFeatures转换器,接着一个StandardScaler,然后是LinearSVC。我们用卫星数据集来测试一下
    from sklearn.preprocessing import StandardScaler
    from sklearn.svm import LinearSVC
    from sklearn.datasets import make_moons
    X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
    
    def plot_dataset(X, y, axes):
        plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
        plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
        plt.axis(axes)
        plt.grid(True, which='both')
        plt.xlabel(r"$x_1$", fontsize=20)
        plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
    
    #plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
    #plt.show()
    
    from sklearn.datasets import make_moons
    from sklearn.pipeline import Pipeline
    from sklearn.preprocessing import PolynomialFeatures
    
    polynomial_svm_clf = Pipeline([
            ("poly_features", PolynomialFeatures(degree=3)),
            ("scaler", StandardScaler()),
            ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
        ])
    
    polynomial_svm_clf.fit(X, y)
    
    def plot_predictions(clf, axes):
        x0s = np.linspace(axes[0], axes[1], 100)
        x1s = np.linspace(axes[2], axes[3], 100)
        x0, x1 = np.meshgrid(x0s, x1s)
        X = np.c_[x0.ravel(), x1.ravel()]
        y_pred = clf.predict(X).reshape(x0.shape)
        y_decision = clf.decision_function(X).reshape(x0.shape)
        plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
        plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
    
    plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
    
    save_fig("moons_polynomial_svc_plot")
    plt.show()
    
  3. 多项式核
    from sklearn.svm import SVC
    
    poly_kernel_svm_clf = Pipeline([
            ("scaler", StandardScaler()),
            ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
        ])
    poly_kernel_svm_clf.fit(X, y)
    #print(poly_kernel_svm_clf.fit(X, y))
    
    poly100_kernel_svm_clf = Pipeline([
            ("scaler", StandardScaler()),
            ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
        ])
    poly100_kernel_svm_clf.fit(X, y)
    #print(poly100_kernel_svm_clf.fit(X, y))
    
    plt.figure(figsize=(11, 4))
    
    plt.subplot(121)
    plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
    plt.title(r"$d=3, r=1, C=5$", fontsize=18)
    
    plt.subplot(122)
    plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
    plt.title(r"$d=10, r=100, C=5$", fontsize=18)
    
    save_fig("moons_kernelized_polynomial_svc_plot")
    plt.show()
    
  4. 添加相似特征
    def gaussian_rbf(x, landmark, gamma):
        return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
    
    gamma = 0.3
    
    x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
    x2s = gaussian_rbf(x1s, -2, gamma)
    x3s = gaussian_rbf(x1s, 1, gamma)
    
    XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
    yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
    
    plt.figure(figsize=(11, 4))
    
    plt.subplot(121)
    plt.grid(True, which='both')
    plt.axhline(y=0, color='k')
    plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
    plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
    plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
    plt.plot(x1s, x2s, "g--")
    plt.plot(x1s, x3s, "b:")
    plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
    plt.xlabel(r"$x_1$", fontsize=20)
    plt.ylabel(r"Similarity", fontsize=14)
    plt.annotate(r'$\mathbf{x}$',
                xy=(X1D[3, 0], 0),
                xytext=(-0.5, 0.20),
                ha="center",
                arrowprops=dict(facecolor='black', shrink=0.1),
                fontsize=18,
                )
    plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
    plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
    plt.axis([-4.5, 4.5, -0.1, 1.1])
    
    plt.subplot(122)
    plt.grid(True, which='both')
    plt.axhline(y=0, color='k')
    plt.axvline(x=0, color='k')
    plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
    plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
    plt.xlabel(r"$x_2$", fontsize=20)
    plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
    plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
                xy=(XK[3, 0], XK[3, 1]),
                xytext=(0.65, 0.50),
                ha="center",
                arrowprops=dict(facecolor='black', shrink=0.1),
                fontsize=18,
                )
    plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
    plt.axis([-0.1, 1.1, -0.1, 1.1])
        
    plt.subplots_adjust(right=1)
    
    #save_fig("kernel_method_plot")
    #plt.show()
    
  5. 高斯RBF
    x1_example = X1D[3, 0]
    for landmark in (-2, 1):
        k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
        print("Phi({}, {}) = {}".format(x1_example, landmark, k))
    
  6. 使用SVC类试试高斯RBF核
    rbf_kernel_svm_clf = Pipeline([
            ("scaler", StandardScaler()),
            ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
        ])
    rbf_kernel_svm_clf.fit(X, y)
    print(rbf_kernel_svm_clf.fit(X, y))
    
    from sklearn.svm import SVC
    
    gamma1, gamma2 = 0.1, 5
    C1, C2 = 0.001, 1000
    hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
    
    svm_clfs = []
    for gamma, C in hyperparams:
        rbf_kernel_svm_clf = Pipeline([
                ("scaler", StandardScaler()),
                ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
            ])
        rbf_kernel_svm_clf.fit(X, y)
        svm_clfs.append(rbf_kernel_svm_clf)
    
    plt.figure(figsize=(11, 7))
    
    for i, svm_clf in enumerate(svm_clfs):
        plt.subplot(221 + i)
        plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
        plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
        gamma, C = hyperparams[i]
        plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
    #使用RBF核的SVM分类器
    save_fig("moons_rbf_svc_plot")
    plt.show()
    

SVM回归

  1. SVM回归
    np.random.seed(42)
    m = 50
    X = 2 * np.random.rand(m, 1)
    y = (4 + 3 * X + np.random.randn(m, 1)).ravel()
    
    from sklearn.svm import LinearSVR
    
    svm_reg = LinearSVR(epsilon=1.5, random_state=42)
    svm_reg.fit(X, y)
    
    svm_reg1 = LinearSVR(epsilon=1.5, random_state=42)
    svm_reg2 = LinearSVR(epsilon=0.5, random_state=42)
    svm_reg1.fit(X, y)
    svm_reg2.fit(X, y)
    
    def find_support_vectors(svm_reg, X, y):
        y_pred = svm_reg.predict(X)
        off_margin = (np.abs(y - y_pred) >= svm_reg.epsilon)
        return np.argwhere(off_margin)
    
    svm_reg1.support_ = find_support_vectors(svm_reg1, X, y)
    svm_reg2.support_ = find_support_vectors(svm_reg2, X, y)
    
    eps_x1 = 1
    eps_y_pred = svm_reg1.predict([[eps_x1]])
    
    def plot_svm_regression(svm_reg, X, y, axes):
        x1s = np.linspace(axes[0], axes[1], 100).reshape(100, 1)
        y_pred = svm_reg.predict(x1s)
        plt.plot(x1s, y_pred, "k-", linewidth=2, label=r"$\hat{y}$")
        plt.plot(x1s, y_pred + svm_reg.epsilon, "k--")
        plt.plot(x1s, y_pred - svm_reg.epsilon, "k--")
        plt.scatter(X[svm_reg.support_], y[svm_reg.support_], s=180, facecolors='#FFAAAA')
        plt.plot(X, y, "bo")
        plt.xlabel(r"$x_1$", fontsize=18)
        plt.legend(loc="upper left", fontsize=18)
        plt.axis(axes)
    
    plt.figure(figsize=(9, 4))
    plt.subplot(121)
    plot_svm_regression(svm_reg1, X, y, [0, 2, 3, 11])
    plt.title(r"$\epsilon = {}$".format(svm_reg1.epsilon), fontsize=18)
    plt.ylabel(r"$y$", fontsize=18, rotation=0)
    #plt.plot([eps_x1, eps_x1], [eps_y_pred, eps_y_pred - svm_reg1.epsilon], "k-", linewidth=2)
    plt.annotate(
            '', xy=(eps_x1, eps_y_pred), xycoords='data',
            xytext=(eps_x1, eps_y_pred - svm_reg1.epsilon),
            textcoords='data', arrowprops={'arrowstyle': '<->', 'linewidth': 1.5}
        )
    plt.text(0.91, 5.6, r"$\epsilon$", fontsize=20)
    plt.subplot(122)
    plot_svm_regression(svm_reg2, X, y, [0, 2, 3, 11])
    plt.title(r"$\epsilon = {}$".format(svm_reg2.epsilon), fontsize=18)
    save_fig("svm_regression_plot使用二阶多项式核的SVM回归")
    plt.show()
    
  2. 使用二阶多项式核的SVM回归
    np.random.seed(42)
    m = 100
    X = 2 * np.random.rand(m, 1) - 1
    y = (0.2 + 0.1 * X + 0.5 * X**2 + np.random.randn(m, 1)/10).ravel()
    #SVR类是SVC类的回归等价物,LinearSVR类也是LinearSVC类的回归等价物。LinearSVR与训练集的大小线性相关
    #(跟LinearSVC一样),而SVR则在训练集变大时,变得很慢(SVC也是一样)。
    
    from sklearn.svm import SVR
    
    svm_poly_reg1 = SVR(kernel="poly", degree=2, C=100, epsilon=0.1, gamma="auto")
    svm_poly_reg2 = SVR(kernel="poly", degree=2, C=0.01, epsilon=0.1, gamma="auto")
    svm_poly_reg1.fit(X, y)
    svm_poly_reg2.fit(X, y)
    
    plt.figure(figsize=(9, 4))
    plt.subplot(121)
    plot_svm_regression(svm_poly_reg1, X, y, [-1, 1, 0, 1])
    plt.title(r"$degree={}, C={}, \epsilon = {}$".format(svm_poly_reg1.degree, svm_poly_reg1.C, svm_poly_reg1.epsilon), fontsize=18)
    plt.ylabel(r"$y$", fontsize=18, rotation=0)
    plt.subplot(122)
    plot_svm_regression(svm_poly_reg2, X, y, [-1, 1, 0, 1])
    plt.title(r"$degree={}, C={}, \epsilon = {}$".format(svm_poly_reg2.degree, svm_poly_reg2.C, svm_poly_reg2.epsilon), fontsize=18)
    save_fig("svm_with_polynomial_kernel_plot使用二阶多项式核的SVM回归")
    plt.show()
    
  3. 鸢尾花数据集的决策函数
    scaler = StandardScaler()
    svm_clf1 = LinearSVC(C=1, loss="hinge", random_state=42)
    svm_clf2 = LinearSVC(C=100, loss="hinge", random_state=42)
    
    scaled_svm_clf1 = Pipeline([
            ("scaler", scaler),
            ("linear_svc", svm_clf1),
        ])
    scaled_svm_clf2 = Pipeline([
            ("scaler", scaler),
            ("linear_svc", svm_clf2),
        ])
    
    scaled_svm_clf1.fit(X, y)
    scaled_svm_clf2.fit(X, y)
    
    # Convert to unscaled parameters
    b1 = svm_clf1.decision_function([-scaler.mean_ / scaler.scale_])
    b2 = svm_clf2.decision_function([-scaler.mean_ / scaler.scale_])
    w1 = svm_clf1.coef_[0] / scaler.scale_
    w2 = svm_clf2.coef_[0] / scaler.scale_
    svm_clf1.intercept_ = np.array([b1])
    svm_clf2.intercept_ = np.array([b2])
    svm_clf1.coef_ = np.array([w1])
    svm_clf2.coef_ = np.array([w2])
    
    # Find support vectors (LinearSVC does not do this automatically)
    t = y * 2 - 1
    support_vectors_idx1 = (t * (X.dot(w1) + b1) < 1).ravel()
    support_vectors_idx2 = (t * (X.dot(w2) + b2) < 1).ravel()
    svm_clf1.support_vectors_ = X[support_vectors_idx1]
    svm_clf2.support_vectors_ = X[support_vectors_idx2]
    
    from sklearn import datasets
    
    iris = datasets.load_iris()
    X = iris["data"][:, (2, 3)]  # petal length, petal width
    y = (iris["target"] == 2).astype(np.float64)  # Iris-Virginica
    
    from mpl_toolkits.mplot3d import Axes3D
    
    def plot_3D_decision_function(ax, w, b, x1_lim=[4, 6], x2_lim=[0.8, 2.8]):
        x1_in_bounds = (X[:, 0] > x1_lim[0]) & (X[:, 0] < x1_lim[1])
        X_crop = X[x1_in_bounds]
        y_crop = y[x1_in_bounds]
        x1s = np.linspace(x1_lim[0], x1_lim[1], 20)
        x2s = np.linspace(x2_lim[0], x2_lim[1], 20)
        x1, x2 = np.meshgrid(x1s, x2s)
        xs = np.c_[x1.ravel(), x2.ravel()]
        df = (xs.dot(w) + b).reshape(x1.shape)
        m = 1 / np.linalg.norm(w)
        boundary_x2s = -x1s*(w[0]/w[1])-b/w[1]
        margin_x2s_1 = -x1s*(w[0]/w[1])-(b-1)/w[1]
        margin_x2s_2 = -x1s*(w[0]/w[1])-(b+1)/w[1]
        ax.plot_surface(x1s, x2, np.zeros_like(x1),
                        color="b", alpha=0.2, cstride=100, rstride=100)
        ax.plot(x1s, boundary_x2s, 0, "k-", linewidth=2, label=r"$h=0$")
        ax.plot(x1s, margin_x2s_1, 0, "k--", linewidth=2, label=r"$h=\pm 1$")
        ax.plot(x1s, margin_x2s_2, 0, "k--", linewidth=2)
        ax.plot(X_crop[:, 0][y_crop==1], X_crop[:, 1][y_crop==1], 0, "g^")
        ax.plot_wireframe(x1, x2, df, alpha=0.3, color="k")
        ax.plot(X_crop[:, 0][y_crop==0], X_crop[:, 1][y_crop==0], 0, "bs")
        ax.axis(x1_lim + x2_lim)
        ax.text(4.5, 2.5, 3.8, "Decision function $h$", fontsize=15)
        ax.set_xlabel(r"Petal length", fontsize=15)
        ax.set_ylabel(r"Petal width", fontsize=15)
        ax.set_zlabel(r"$h = \mathbf{w}^T \mathbf{x} + b$", fontsize=18)
        ax.legend(loc="upper left", fontsize=16)
    
    fig = plt.figure(figsize=(11, 6))
    ax1 = fig.add_subplot(111, projection='3d')
    plot_3D_decision_function(ax1, w=svm_clf2.coef_[0], b=svm_clf2.intercept_[0])
    
    #save_fig("iris_3D_plot鸢尾花数据集的决策函数")
    #plt.show()
    
  4. 权重向量越小,间隔越大
    def plot_2D_decision_function(w, b, ylabel=True, x1_lim=[-3, 3]):
        x1 = np.linspace(x1_lim[0], x1_lim[1], 200)
        y = w * x1 + b
        m = 1 / w
    
        plt.plot(x1, y)
        plt.plot(x1_lim, [1, 1], "k:")
        plt.plot(x1_lim, [-1, -1], "k:")
        plt.axhline(y=0, color='k')
        plt.axvline(x=0, color='k')
        plt.plot([m, m], [0, 1], "k--")
        plt.plot([-m, -m], [0, -1], "k--")
        plt.plot([-m, m], [0, 0], "k-o", linewidth=3)
        plt.axis(x1_lim + [-2, 2])
        plt.xlabel(r"$x_1$", fontsize=16)
        if ylabel:
            plt.ylabel(r"$w_1 x_1$  ", rotation=0, fontsize=16)
        plt.title(r"$w_1 = {}$".format(w), fontsize=16)
    
    plt.figure(figsize=(12, 3.2))
    plt.subplot(121)
    plot_2D_decision_function(1, 0)
    plt.subplot(122)
    plot_2D_decision_function(0.5, 0, ylabel=False)
    save_fig("small_w_large_margin_plot")
    plt.show()
    '''
    
    from sklearn.svm import SVC
    from sklearn import datasets
    
    iris = datasets.load_iris()
    X = iris["data"][:, (2, 3)] # petal length, petal width
    y = (iris["target"] == 2).astype(np.float64) # Iris-Virginica
    
    svm_clf = SVC(kernel="linear", C=1)
    svm_clf.fit(X, y)
    svm_clf.predict([[5.3, 1.3]])
    #print(svm_clf.predict([[5.3, 1.3]]))
    
    #Hinge loss
    t = np.linspace(-2, 4, 200)
    h = np.where(1 - t < 0, 0, 1 - t)  # max(0, 1-t)
    
    plt.figure(figsize=(5,2.8))
    plt.plot(t, h, "b-", linewidth=2, label="$max(0, 1 - t)$")
    plt.grid(True, which='both')
    plt.axhline(y=0, color='k')
    plt.axvline(x=0, color='k')
    plt.yticks(np.arange(-1, 2.5, 1))
    plt.xlabel("$t$", fontsize=16)
    plt.axis([-2, 4, -1, 2.5])
    plt.legend(loc="upper right", fontsize=16)
    save_fig("hinge_plot")
    plt.show()

本文来自投稿,不代表程序员编程网立场,如若转载,请注明出处:http://www.cxybcw.com/189255.html

联系我们

13687733322

在线咨询:点击这里给我发消息

邮件:1877088071@qq.com

工作时间:周一至周五,9:30-18:30,节假日休息

QR code