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  <title>露从今夜白，月是故乡明</title>
  
  <subtitle>关注科技与人文，在这个世界留下点痕迹</subtitle>
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  <link href="http://tobinzuo.github.io/"/>
  <updated>2018-11-19T11:27:03.974Z</updated>
  <id>http://tobinzuo.github.io/</id>
  
  <author>
    <name>Tobin</name>
    
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  <entry>
    <title>关于我</title>
    <link href="http://tobinzuo.github.io/2018/11/17/aboutme/"/>
    <id>http://tobinzuo.github.io/2018/11/17/aboutme/</id>
    <published>2018-11-17T15:48:12.000Z</published>
    <updated>2018-11-19T11:27:03.974Z</updated>
    
    <content type="html"><![CDATA[<p>&emsp;&emsp;你好，我现在硕士就读于<strong><em>浙江大学计算机科学与技术学院</em></strong>。正在从事<strong><em>机器学习，数据挖掘，深度学习</em></strong>相关的学习和工作。本博客使用的是<a href="https://hexo.io/zh-cn/" target="_blank" rel="noopener">hexo</a>开源工具，主题为<a href="https://github.com/yscoder/hexo-theme-indigo" target="_blank" rel="noopener">indigo</a>，感谢这个开源的世界。我建这个博客的初衷想有个记录自己想法的平台，Just for fun。<br>&emsp;&emsp;<strong> 欢迎联系我哈，世界会因为我们的存在而变得更美好~ </strong></p><div align="center"><img src="/2018/11/17/aboutme/weixin.png" width="40%" height="40%"></div>]]></content>
    
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        &lt;p&gt;&amp;emsp;&amp;emsp;你好，我现在硕士就读于&lt;strong&gt;&lt;em&gt;浙江大学计算机科学与技术学院&lt;/em&gt;&lt;/strong&gt;。正在从事&lt;strong&gt;&lt;em&gt;机器学习，数据挖掘，深度学习&lt;/em&gt;&lt;/strong&gt;相关的学习和工作。本博客使用的是&lt;a href=&quot;http
      
    
    </summary>
    
    
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  <entry>
    <title>[机器学习] 感知机</title>
    <link href="http://tobinzuo.github.io/2018/11/16/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0%E4%B9%8B%E6%84%9F%E7%9F%A5%E6%9C%BA/"/>
    <id>http://tobinzuo.github.io/2018/11/16/机器学习之感知机/</id>
    <published>2018-11-16T15:48:12.000Z</published>
    <updated>2018-11-17T05:39:10.918Z</updated>
    
    <content type="html"><![CDATA[<h3 id="什么是感知机"><a href="#什么是感知机" class="headerlink" title="什么是感知机"></a>什么是感知机</h3><p>定义：假设输入空间（特征空间）$X \subseteq R^n$,输出空间是$Y={-1,+1}$,输入$x \in X$,表示实例的特征向量，对于输入空间（特征空间）中的点，输出$y \in Y$是表示实例的类别。由输入空间到输出空间的如下函数，$f(x)=sign(w \cdot x +b)$称为感知机，其中$w$和$b$称为感知机的参数。$sign$函数如下。<br>$$sign(x)=\begin{cases} +1, &amp; x&gt;=0\\-1, &amp; x&lt;0 \end{cases}$$</p><h3 id="感知机数学原理"><a href="#感知机数学原理" class="headerlink" title="感知机数学原理"></a>感知机数学原理</h3><h4 id="线性可分"><a href="#线性可分" class="headerlink" title="线性可分"></a>线性可分</h4><p>对于给定的一个数据集$T={(x_1,y_1),(x_2,y_2),…(x_n,y_n)}$,其中,$x_i \in X \subseteq R^n , y \in Y={-1, +1}, i=1,2,…,N$,如果存在某个超平面$S: w \cdot x+b = 0$,能将正负样例分到$S$两侧，则说明数据集$T$是线性可分的，那么接下来就是要求这个超平面$S$的表达式。</p><h4 id="损失函数"><a href="#损失函数" class="headerlink" title="损失函数"></a>损失函数</h4><p>数据集$T$中任意一个误分类点到这个超平面$S$的距离之和为:<br>$$\frac{1}{\Vert w \Vert}\sum_{x_i \in M}{\vert w \cdot x_i+b \vert}$$<br>其中$M$为误分类点的集合。如何去掉绝对值呢？我们注意到$y_i$的取值是有好处的。对于误分类点来说$y_i$的符号和$w \cdot x_i+b$相反。且$y_i$取值+1，-1。则我们可以将上式转换成：<br>$$- \frac{1}{\Vert w \Vert}\sum_{x_i \in M}{y_i (w \cdot x_i+b)}$$<br>确定损失函数时，不考虑$\frac{1}{\Vert w \Vert}$。</p><ul><li><strong><em>Q:感知机损失函数确定过程中的系数$\frac{1}{\Vert w \Vert}$，为什么可以忽略？</em></strong></li><li>1）感知机的损失函数时基于误分类的，$L2$范数项不会影响$y_i (w \cdot x_i+b)$的正负判断，所以$\frac{1}{\Vert w \Vert}$对感知机学习算法的中间过程可有可无</li><li>2）$\frac{1}{\Vert w \Vert}$不影响感知机学习算法的最终结果，当所有输入被正确分类时，不存在误分类的点，此时损失函数为零，缺少此项并不会影响感知机损失函数的收敛</li><li>3）我思考的是即使缺少这个项，损失函数也能对线性可分数据在有限次学习后正确分开。李航书上有证明。</li><li>4）权重$w$是一个向量,$\frac{1}{\Vert w \Vert}$的大小不会影响向量的方向，确定超平面是通过确定法向量$w$和截距$b$来确定的，而的大对权重$w$的方向没有任何影响，所以可以固定$\frac{1}{\Vert w \Vert}$|为1或者不考虑。</li></ul><h3 id="算法步骤"><a href="#算法步骤" class="headerlink" title="算法步骤"></a>算法步骤</h3><p>输入：训练数据$T={(x_1,y_1),(x_2,y_2),…(x_n,y_n)}$,其中,$x_i \in X \subseteq R^n , y \in Y={-1, +1}, i=1,2,…,N$,；学习率$\eta(0&lt;\eta&lt;=1)$；<br>输出：$w，b$；感知机模型$f(x)=sign(w \cdot x+b)$。</p><ul><li>1）选取初值$w_0，b_0$</li><li>2）在训练集中选取数据$（x_i，y_i）$</li><li>3）如果$y_i (w \cdot x_i+b)&lt;0$，$w \leftarrow w+ \eta y_ix_i；b \leftarrow b+ \eta y_i $</li><li>4）转至 2），直至训练集中没有误分类点</li></ul><h3 id="代码实现"><a href="#代码实现" class="headerlink" title="代码实现"></a>代码实现</h3><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">#初始化</span></span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">import</span> random</span><br><span class="line"><span class="keyword">import</span> matplotlib <span class="keyword">as</span> mpl</span><br><span class="line"><span class="keyword">import</span> matplotlib.pyplot <span class="keyword">as</span> plt</span><br><span class="line">datas = [[(<span class="number">1</span>,<span class="number">2</span>),<span class="number">-1</span>],[(<span class="number">2</span>,<span class="number">1</span>),<span class="number">-1</span>],[(<span class="number">2</span>,<span class="number">2</span>),<span class="number">-1</span>],[(<span class="number">1</span>,<span class="number">4</span>),<span class="number">1</span>],[(<span class="number">3</span>,<span class="number">3</span>),<span class="number">1</span>],[(<span class="number">5</span>,<span class="number">4</span>),<span class="number">1</span>],[(<span class="number">3</span>, <span class="number">3</span>), <span class="number">1</span>], [(<span class="number">4</span>, <span class="number">3</span>), <span class="number">1</span>], [(<span class="number">1</span>, <span class="number">1</span>), <span class="number">-1</span>],[(<span class="number">2</span>, <span class="number">3</span>), <span class="number">-1</span>], [(<span class="number">4</span>, <span class="number">2</span>), <span class="number">1</span>]]</span><br><span class="line"></span><br><span class="line"><span class="comment">#数据展示</span></span><br><span class="line">fig = plt.figure(<span class="string">'Input Figure'</span>)</span><br><span class="line">mpl.rcParams[<span class="string">'xtick.labelsize'</span>] = <span class="number">16</span> <span class="comment"># x轴的字体大小</span></span><br><span class="line">mpl.rcParams[<span class="string">'ytick.labelsize'</span>] = <span class="number">16</span></span><br><span class="line">plt.rcParams[<span class="string">'font.sans-serif'</span>]=[<span class="string">'SimHei'</span>] <span class="comment"># 用来正常显示中文标签</span></span><br><span class="line"></span><br><span class="line">xArr = np.array([x[<span class="number">0</span>] <span class="keyword">for</span> x <span class="keyword">in</span> datas])</span><br><span class="line">yArr = np.array([x[<span class="number">1</span>] <span class="keyword">for</span> x <span class="keyword">in</span> datas])</span><br><span class="line">xPlotx,xPlotx_,xPloty,xPloty_ = [],[],[],[]</span><br><span class="line"><span class="keyword">for</span> i <span class="keyword">in</span> range(len(datas)):</span><br><span class="line">    y = yArr[i]</span><br><span class="line">    <span class="keyword">if</span> y&gt;<span class="number">0</span>:</span><br><span class="line">        xPlotx.append(xArr[i][<span class="number">0</span>])</span><br><span class="line">        xPloty.append(xArr[i][<span class="number">1</span>])</span><br><span class="line">    <span class="keyword">else</span>:</span><br><span class="line">        xPlotx_.append(xArr[i][<span class="number">0</span>])</span><br><span class="line">        xPloty_.append(xArr[i][<span class="number">1</span>])</span><br><span class="line">plt.title(<span class="string">'Perception 输入数据'</span>)</span><br><span class="line">plt.grid(<span class="keyword">True</span>)</span><br><span class="line">pPlot1,pPlot2 = plt.plot(xPlotx,xPloty,<span class="string">'b+'</span>,xPlotx_,xPloty_,<span class="string">'rx'</span>)</span><br><span class="line">plt.legend(handles = [pPlot1,pPlot2],labels=[<span class="string">'Positive Sample'</span>,<span class="string">'Negtive Sample'</span>],loc=<span class="string">'upper center'</span>)</span><br><span class="line">plt.show()</span><br></pre></td></tr></table></figure><figure class="image-bubble">                <div class="img-lightbox">                    <div class="overlay"></div>                    <img src="/2018/11/16/机器学习之感知机/output_1.png" alt="output_1.png" title="">                </div>                <div class="image-caption">output_1.png</div>            </figure><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">#感知机算法</span></span><br><span class="line">w = np.array([<span class="number">1</span>, <span class="number">1</span>])</span><br><span class="line">b = <span class="number">3</span></span><br><span class="line">eta = <span class="number">0.5</span></span><br><span class="line"><span class="keyword">while</span> <span class="keyword">True</span>:</span><br><span class="line">    num = <span class="number">0</span></span><br><span class="line">    <span class="keyword">for</span> i <span class="keyword">in</span> range(len(datas)):</span><br><span class="line">        num = num + <span class="number">1</span></span><br><span class="line">        x = xArr[i]</span><br><span class="line">        y = yArr[i]</span><br><span class="line">        z = y*(np.dot(w,x)+b)</span><br><span class="line">        <span class="keyword">if</span> z &lt;= <span class="number">0</span>:</span><br><span class="line">            w = w+eta*y*x</span><br><span class="line">            b = b+eta*y</span><br><span class="line">            num = <span class="number">0</span> <span class="comment"># 只要更新一次，强制令num=0，否则到最后一个数据不满足条件更新后，会直接跳出，而此时并不满足条件。</span></span><br><span class="line">            <span class="keyword">break</span></span><br><span class="line">    <span class="keyword">if</span> num &gt;= len(datas):</span><br><span class="line">        <span class="keyword">break</span></span><br><span class="line">fig = plt.figure(<span class="string">'Output Figure'</span>)</span><br><span class="line">plt.title(<span class="string">'Perception 输出平面'</span>)</span><br><span class="line">x0 =np.linspace(<span class="number">0</span>,<span class="number">5</span>,<span class="number">100</span>)</span><br><span class="line">w0 = w[<span class="number">0</span>]</span><br><span class="line">w1 = w[<span class="number">1</span>]</span><br><span class="line">x1 = -(w0/w1)*x0-b/w1</span><br><span class="line">plt.xlabel(<span class="string">'x0'</span>,fontsize=<span class="number">15</span>)</span><br><span class="line">plt.ylabel(<span class="string">'x1'</span>,fontsize=<span class="number">15</span>)</span><br><span class="line">plt.plot(x0,x1,<span class="string">'k'</span>, lw=<span class="number">1</span>)</span><br><span class="line">pPlot1,pPlot2 = plt.plot(xPlotx,xPloty,<span class="string">'b+'</span>,xPlotx_,xPloty_,<span class="string">'rx'</span>)</span><br><span class="line">plt.legend(handles = [pPlot1,pPlot2],labels=[<span class="string">'Positive Sample'</span>,<span class="string">'Negtive Sample'</span>],loc=<span class="string">'upper right'</span>)</span><br><span class="line">plt.annotate(<span class="string">'Output Hyperplane'</span>,xy=(<span class="number">2.0</span>,<span class="number">3.0</span>),xytext=(<span class="number">1.7</span>,<span class="number">3.5</span>)) <span class="comment"># 图形注释 # xy设置箭头尖的坐标,xytext设置注释内容显示的起始位置</span></span><br><span class="line"></span><br><span class="line">plt.show()</span><br></pre></td></tr></table></figure><figure class="image-bubble">                <div class="img-lightbox">                    <div class="overlay"></div>                    <img src="/2018/11/16/机器学习之感知机/output_2.png" alt="output_2.png" title="">                </div>                <div class="image-caption">output_2.png</div>            </figure><ul><li>李航,《统计学习方法》</li><li><a href="https://blog.csdn.net/u014630987/article/details/78670258" target="_blank" rel="noopener">Hexo中如何配置mathjax</a></li><li><a href="http://kubicode.me/2016/03/18/Hexo/The-Trick-about-Hexo-Support-MutliLine-Equation-using-Mathjax/" target="_blank" rel="noopener">Hexo中如何输入跨行公式</a></li><li><a href="https://blog.csdn.net/xjm850552586/article/details/84101345" target="_blank" rel="noopener">Hexo无法加载本地图片</a></li></ul>]]></content>
    
    <summary type="html">
    
      
      
        &lt;h3 id=&quot;什么是感知机&quot;&gt;&lt;a href=&quot;#什么是感知机&quot; class=&quot;headerlink&quot; title=&quot;什么是感知机&quot;&gt;&lt;/a&gt;什么是感知机&lt;/h3&gt;&lt;p&gt;定义：假设输入空间（特征空间）$X \subseteq R^n$,输出空间是$Y={-1,+1}$,输入$x
      
    
    </summary>
    
    
      <category term="机器学习" scheme="http://tobinzuo.github.io/tags/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/"/>
    
      <category term="神经网络" scheme="http://tobinzuo.github.io/tags/%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C/"/>
    
  </entry>
  
  <entry>
    <title>更新日志</title>
    <link href="http://tobinzuo.github.io/2018/11/15/%E6%9B%B4%E6%96%B0%E6%97%A5%E5%BF%97/"/>
    <id>http://tobinzuo.github.io/2018/11/15/更新日志/</id>
    <published>2018-11-15T15:48:12.000Z</published>
    <updated>2018-11-18T08:26:11.629Z</updated>
    
    <content type="html"><![CDATA[<p><strong> 2018/11/17 </strong></p><ul><li>上线评论系统，增加了gitment enable ，更改了gitment.ejs，按照page.date来区分评论。</li><li>缩减了部分图片大小。</li><li>解决了部分日志无法登陆评论的问题，原因是文件名称中带有括号<br><a href="https://www.jianshu.com/p/57afa4844aaa" target="_blank" rel="noopener">Gitment评论功能接入踩坑教程</a><br><a href="https://blog.csdn.net/wardseptember/article/details/82828391" target="_blank" rel="noopener">解决gitment无法登陆评论问题（Object ProgressEvent）</a><br><a href="https://jingyan.baidu.com/article/2f9b480de2b5b341cb6cc2be.html" target="_blank" rel="noopener">NexT主题使用gitment评论系统</a></li></ul><p><strong> 2018/11/18 </strong></p><ul><li>上线valine评论插件，gitment使用不便。</li><li>Valine 目前使用的是Gravatar 作为评论列表头像。请自行登录或注册Gravatar，然后修改自己的头像。评论的时候，留下在Gravatar注册时所使用的邮箱即可。<br><a href="https://valine.js.org/quickstart.html" target="_blank" rel="noopener">Valine</a><br><a href="https://en.gravatar.com/" target="_blank" rel="noopener">Gravatar</a></li></ul>]]></content>
    
    <summary type="html">
    
      
      
        &lt;p&gt;&lt;strong&gt; 2018/11/17 &lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;上线评论系统，增加了gitment enable ，更改了gitment.ejs，按照page.date来区分评论。&lt;/li&gt;
&lt;li&gt;缩减了部分图片大小。&lt;/li&gt;
&lt;li&gt;解决了部分日
      
    
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