how to draw w; close issue #87

CH07/README.md

@@ -124,12 +124,10 @@ $$
 \gamma=\frac{\hat \gamma}{\|w\|}\\
 $$
 
-和感知机相比，支持向量机引入间隔，将距离
+和感知机相比，支持向量机引入间隔，将距离的大小考虑进来，而不只是符号。
 
 #### 间隔最大化
 
-
-
 #### 支持向量和间隔边界
 
 由于支持向量在确定分离超平面中起着决定作用，所以将这种分类模型称为支持向量机。
@@ -199,16 +197,13 @@ $$
 >
 > In [mathematical optimization](https://en.wikipedia.org/wiki/Mathematical_optimization) theory, **duality** or the **duality principle** is the principle that [optimization problems](https://en.wikipedia.org/wiki/Optimization_problem) may be viewed from either of two perspectives, the **primal problem** or the **dual problem**. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem.[[1\]](https://en.wikipedia.org/wiki/Duality_(optimization)#cite_note-Boyd-1) However in general the optimal values of the primal and dual problems need not be equal. Their difference is called the [duality gap](https://en.wikipedia.org/wiki/Duality_gap). For [convex optimization](https://en.wikipedia.org/wiki/Convex_optimization) problems, the duality gap is zero under a [constraint qualification](https://en.wikipedia.org/wiki/Constraint_qualification) condition.
 
-
-
 转换后的对偶问题
 $$
 \min\limits_\alpha \frac{1}{2}\sum_{i=1}^N\sum_{j=1}^N\alpha_i\alpha_jy_iy_j(x_i\cdot x_j)-\sum_{i=1}^N\alpha_i\\
 s.t. \ \ \ \sum_{i=1}^N\alpha_iy_i=0\\
 \alpha_i\geqslant0, i=1,2,\dots,N
 $$
 
-
 对于任意线性可分的两组点，他们在分类超平面上的投影都是线性不可分的。
 
 $\alpha$不为零的点对应的实例为支持向量，通过支持向量可以求得$b$值
@@ -278,6 +273,11 @@ $$
 
 线性支持向量机是线性可分支持向量机的超集。
 
+这章单元测试中包含了几个图，可以直接看代码。
+关于$w$是如何绘制的，这个添加了个测试案例，体会一下。
+
+![fig_w](assets/fig_w.png)
+
 ### 算法
 
 #### 软间隔最大化

CH07/svm.py

@@ -1,5 +1,6 @@
 #! /usr/bin/env python
-#! -*- coding=utf-8 -*-
+
+# -*- coding:utf-8 -*-
 # Project:  Lihang
 # Filename: svm
 # Date: 9/27/18

CH07/unit_test.py

@@ -1,5 +1,6 @@
 #! /usr/bin/env python
-#! -*- coding=utf-8 -*-
+
+# -*- coding:utf-8 -*-
 # Project:  Lihang
 # Filename: unit_test
 # Date: 9/27/18
@@ -28,7 +29,7 @@ def test_e71(self):
                 {'type': 'ineq', 'fun': lambda x: 4 * x[0] + 3 * x[1] + x[2] - 1},
                 {'type': 'ineq', 'fun': lambda x: -x[0] - x[1] - x[2] - 1})
         res = optimize.minimize(fun, np.ones(3), method='SLSQP', constraints=cons)
-        logger.info("\n res is \n %s \n x is \n %s\n" % (str(res), res["x"]))
+        print("\n res is \n %s \n x is \n %s\n" % (str(res), res["x"]))
 
         self.assertListEqual(res["x"].round(2).tolist(), [0.5, 0.5, -2])
 
@@ -152,6 +153,32 @@ def test_f76(self):
         # plt.savefig("fig76.png")
         plt.show()
 
+    def test_w(self):
+        # support vector
+        alpha = np.array([[0, 0], [1, 3], [2.8, 2]])
+        # hyperplane
+        kb = np.polyfit(alpha[1:].T[0], alpha[1:].T[1], 1)
+        x = np.arange(0, 4, 0.1)
+        y = kb[0]*x + kb[1]
+        y_ = -x/kb[0]
+        plt.figure(figsize=(6, 6))
+        plt.xlim(0, 4)
+        plt.ylim(0, 4)
+        # point
+        plt.scatter(alpha[:, 0], alpha[:, 1],
+                    marker="o", edgecolors="b", c="w", label="p1,p2")
+        # vector
+        plt.plot(alpha[:2].T[0], alpha[:2].T[1], label="o-p1")
+        plt.plot(alpha[[0, 2]].T[0], alpha[[0, 2]].T[1], label="o-p2")
+        # point1 - point2
+        plt.plot(alpha[1:].T[0], alpha[1:].T[1], label="o-p1-p2")
+        # hyperplane
+        plt.plot(x, y, alpha=0.3, linestyle="--", label="hyperplane")
+        # w
+        plt.plot(x, y_, alpha=0.3, label="w")
+        plt.legend()
+        # plt.plot(x, y)
+        plt.show()
 
 if __name__ == '__main__':
     logging.basicConfig(level=logging.INFO, format='%(asctime)s - %(name)s - %(levelname)s - %(message)s')