机器学习第5章支持向量机

参考：作者的jupyter notebook
chapter 5 – support vector machines

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

2. 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

3. 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()
   

4. 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()
   

5. 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()
   

6. 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()
   

7. 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))
   

8. 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

9. 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()
   

10. 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()
    

11. 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()
    

12. 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()