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<article id="post-数学建模之排队论" class="article article-type-post article-index" itemscope itemprop="blogPost">
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<h1 itemprop="name">
<a class="article-title" href="/2021/09/01/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1%E4%B9%8B%E6%8E%92%E9%98%9F%E8%AE%BA/">数学建模之排队论</a>
</h1>
<a href="/2021/09/01/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1%E4%B9%8B%E6%8E%92%E9%98%9F%E8%AE%BA/" class="archive-article-date">
<time datetime="2021-09-01T08:39:40.000Z" itemprop="datePublished"><i class="icon-calendar icon"></i>2021-09-01</time>
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<p>排队是在日常生活中经常遇到的现象，如顾客到商店购买物品、病人到医院看病常 常要排队。此时要求服务的数量超过服务机构（服务台、服务员等）的容量。也就是说，<strong>到达的顾客不能立即得到服务，因而出现了排队现象。</strong>这种现象不仅在个人日常生活中出现，电话局的占线问题，车站、码头等交通枢纽的车船堵塞和疏导，故障机器的停机待修，水库的存贮调节等都是有形或无形的排队现象。</p>
<p><strong>排队论</strong>（Queuing Theory）也称随机服务系统理论，就是为解决上述问题而发展 的一门学科。它研究的内容有下列三部分：</p>
<ol>
<li>性态问题，即研究各种排队系统的概率规律性，主要是研究队长分布、等待时间分布和忙期分布等，包括了瞬态和稳态两种情形。</li>
<li>优化问题，又分静态优和动态优，前者指优设计。后者指现有排队系统的优运营。</li>
<li>排队系统的统计推断，即判断一个给定的排队系统符合于哪种模型，以便根据排队理论进行分析研究。</li>
</ol>
<h1 id="1-基本概念"><a href="#1-基本概念" class="headerlink" title="1 基本概念"></a>1 基本概念</h1><h2 id="1-1-排队过程的一般表示"><a href="#1-1-排队过程的一般表示" class="headerlink" title="1.1 排队过程的一般表示"></a>1.1 排队过程的一般表示</h2><p>下图是排队论的一般模型:<br><img src="2021-09-01-16-41-41.png"><br>图中虚线所包含的部分为排队系统。各个顾客从顾客源出发，随机地来到服务机构，按一定的排队规则等待服务，直到按一定的服务规则接受完服务后离开排队系统。</p>
<p>凡要求服务的对象统称为顾客，为顾客服务的人或物称为服务员，由顾客和服务员 组成服务系统。对于一个服务系统来说，如果服务机构过小，以致不能满足要求服务的 众多顾客的需要，那么就会产生拥挤现象而使服务质量降低。 因此，顾客总希望服务机构越大越好，但是，如果服务机构过大，人力和物力方面的开支也就相应增加，从而 会造成浪费，因此研究排队模型的目的就是<strong>要在顾客需要和服务机构的规模之间进行权衡决策</strong>，使其达到合理的平衡。</p>
<h2 id="1-2-排队系统的组成和特征"><a href="#1-2-排队系统的组成和特征" class="headerlink" title="1.2 排队系统的组成和特征"></a>1.2 排队系统的组成和特征</h2><p>一般的排队过程都由输入过程、排队规则、服务过程三部分组成</p>
<h3 id="1-2-1-输入过程"><a href="#1-2-1-输入过程" class="headerlink" title="1.2.1 输入过程"></a>1.2.1 输入过程</h3><p>输入过程是指顾客到来时间的规律性，可能有下列不同情况：</p>
<ol>
<li>顾客的组成可能是有限的，也可能是无限的。</li>
<li>顾客到达的方式可能是一个—个的，也可能是成批的。</li>
<li>顾客到达可以是相互独立的，即以前的到达情况对以后的到达没有影响； 否则是相关的。</li>
<li>输入过程可以是平稳的，即相继到达的间隔时间分布及其数学期望、方差等 数字特征都与时间无关，否则是非平稳的。</li>
</ol>
<h3 id="1-2-2-排队规则"><a href="#1-2-2-排队规则" class="headerlink" title="1.2.2 排队规则"></a>1.2.2 排队规则</h3><p>排队规则指到达排队系统的顾客按怎样的规则排队等待，可分为损失制，等待制和 混合制三种.</p>
<ol>
<li>损失制（消失制）。当顾客到达时，所有的服务台均被占用，顾客随即离去。</li>
<li>等待制。当顾客到达时，所有的服务台均被占用，顾客就排队等待，直到接 受完服务才离去。例如出故障的机器排队等待维修就是这种情况。</li>
</ol>
<p>排队方式还分为单列、多列和循环队列。</p>
<h3 id="1-2-3-服务过程"><a href="#1-2-3-服务过程" class="headerlink" title="1.2.3 服务过程"></a>1.2.3 服务过程</h3><ol>
<li>服务机构。主要有以下几种类型：单服务台；多服务台并联（每个服务台同 时为不同顾客服务）；多服务台串联（多服务台依次为同一顾客服务）；混合型。 </li>
<li>服务规则。按为顾客服务的次序采用以下几种规则：<ul>
<li>先到先服务，这是通常的情形。</li>
<li>后到先服务，如情报系统中，后到的情报信息往往有价值，因而常被优先处理。 </li>
<li>随机服务，服务台从等待的顾客中随机地取其一进行服务，而不管到达的先后。</li>
<li>优先服务，如医疗系统对病情严重的病人给予优先治疗。</li>
</ul>
</li>
</ol>
<h2 id="1-3-排队模型的符号表示"><a href="#1-3-排队模型的符号表示" class="headerlink" title="1.3 排队模型的符号表示"></a>1.3 排队模型的符号表示</h2><p>排队模型用六个符号表示，在符号之间用斜线隔开，即 X/Y/Z/A/B/C 。第一 个符号 X 表示顾客到达流或顾客到达间隔时间的分布；第二个符号Y 表示服务时间的 分布；第三个符号Z 表示服务台数目；第四个符号 A是系统容量限制；第五个符号B 是 顾客源数目；第六个符号C 是服务规则，如先到先服务 FCFS，后到先服务 LCFS 等。并 约定，如略去后三项，即指X/Y/Z/∞/∞/FCFS的情形。我们只讨论先到先服务 FCFS 的情形，所以略去第六项。</p>
<p>表示顾客到达间隔时间和服务时间的分布的约定符号为：</p>
<ul>
<li>M —指数分布（M 是 Markov 的字头，因为指数分布具有无记忆性，即 Markov 性）；</li>
<li>D—确定型（Deterministic）；</li>
<li>Ek —k 阶爱尔朗(Erlang)分布；</li>
<li>G —一般（general）服务时间的分布；</li>
<li>GI —一般相互独立（General Independent）的时间间隔的分布。</li>
</ul>
<p>例如，M/M/1表示相继到达间隔时间为指数分布、服务时间为指数分布、单服务台、等待制系统。</p>
<p>D/M/c/表示确定的到达时间、服务时间为指数分布、c个平行服务台（但顾客是一队）的模型。</p>
<h2 id="1-4-排队系统的运行指标"><a href="#1-4-排队系统的运行指标" class="headerlink" title="1.4 排队系统的运行指标"></a>1.4 排队系统的运行指标</h2><p>为了研究排队系统运行的效率，估计其服务质量，确定系统的优参数，评价系统 的结构是否合理并研究其改进的措施，必须确定用以判断系统运行优劣的基本数量指标，这些数量指标通常是：</p>
<ul>
<li> 平均队长：指系统内顾客数（包括正被服务的顾客与排队等待服务的顾客）的数学期望，记作Ls 。</li>
<li>平均排队长:指系统内等待服务的顾客数的数学期望，记作 Lq 。</li>
<li>平均逗留时间：顾客在系统内逗留时间（包括排队等待的时间和接受服务的时间）的数学期望，记作Ws 。</li>
<li>平均等待时间：指一个顾客在排队系统中排队等待时间的数学期望，记作Wq 。</li>
<li>平均忙期：指服务机构连续繁忙时间（顾客到达空闲服务机构起，到服务机构再次空闲止的时间）长度的数学期望，记为 Tb</li>
</ul>
<h1 id="2-输入过程与服务时间的分布"><a href="#2-输入过程与服务时间的分布" class="headerlink" title="2 输入过程与服务时间的分布"></a>2 输入过程与服务时间的分布</h1><p>排队系统中的事件流包括顾客到达流和服务时间流。由于顾客到达的间隔时间和服 务时间不可能是负值，因此，它的分布是非负随机变量的分布。常用的分布有泊松分布、确定型分布，指数分布和爱尔朗分布。</p>
<h2 id="2-1-泊松流与指数分布"><a href="#2-1-泊松流与指数分布" class="headerlink" title="2.1 泊松流与指数分布"></a>2.1 泊松流与指数分布</h2><p><img src="2021-09-01-16-45-35.png"></p>
<h1 id="4-M-M-s等待制排队模型"><a href="#4-M-M-s等待制排队模型" class="headerlink" title="4 M/M/s等待制排队模型"></a>4 M/M/s等待制排队模型</h1><h2 id="4-1-单服务台模型"><a href="#4-1-单服务台模型" class="headerlink" title="4.1 单服务台模型"></a>4.1 单服务台模型</h2><p>单服务台等待制模型M/M/1/∞是指：顾客的相继到达时间服从参数为λ的负指数分布，服务台个数为1，服务时间V服从参数为μ的负指数分布，系统空间无限，允许无限排队，这是一类简单的排队系统。</p>
<h3 id="4-1-1-队长的分布"><a href="#4-1-1-队长的分布" class="headerlink" title="4.1.1 队长的分布"></a>4.1.1 队长的分布</h3><p><img src="2021-09-01-16-45-54.png"><br>公式（7）和（8）给出了在平衡条件下系统中顾客数为n的概率。由式（7）不难看出，ρ 是系统中至少有一个顾客的概率，也就是服务台处于忙的状态的概率，因而也称 ρ 为服务强度，它反映了系统繁忙的程度。此外，（8）式只有在ρ=λ/μ<=1的条件下才能得到，即要求顾客的平均到达率小于系统的平均服务率，才能使系统达到统计平衡.</p>
<h2 id="4-2-与排队论模型有关的-LINGO-函数"><a href="#4-2-与排队论模型有关的-LINGO-函数" class="headerlink" title="4.2 与排队论模型有关的 LINGO 函数"></a>4.2 与排队论模型有关的 LINGO 函数</h2><ul>
<li>@peb(load,S) 该函数的返回值是当到达负荷为 load，服务系统中有 S 个服务台且允许排队时系 统繁忙的概率，也就是顾客等待的概率。</li>
<li>@pel（load,S）该函数的返回值是当到达负荷为 load，服务系统中有 S 个服务台且不允许排队时系统损失概率，也就是顾客得不到服务离开的概率。 （3）@pfs(load,S,K) 该函数的返回值是当到达负荷为 load，顾客数为 K，平行服务台数量为 S 时，有限 源的 Poisson 服务系统等待或返修顾客数的期望值</li>
</ul>
<h2 id="4-3-多服务台模型（-M-M-s-∞）"><a href="#4-3-多服务台模型（-M-M-s-∞）" class="headerlink" title="4.3 多服务台模型（ M/M/s/∞）"></a>4.3 多服务台模型（ M/M/s/∞）</h2><p>设顾客单个到达，相继到达时间间隔服从参数为λ 的负指数分布，系统中共有s个 服务台，每个服务台的服务时间相互独立，且服从参数为 μ 的负指数分布。当顾客到达时，若有空闲的服务台则马上接受服务，否则便排成一个队列等待，等待时间为无限。<br><img src="2021-09-01-16-46-39.png"></p>
<h1 id="5-排队模型的计算机模拟"><a href="#5-排队模型的计算机模拟" class="headerlink" title="5 排队模型的计算机模拟"></a>5 排队模型的计算机模拟</h1><h2 id="5-1-确定随机变量概率分布的常用方法"><a href="#5-1-确定随机变量概率分布的常用方法" class="headerlink" title="5.1 确定随机变量概率分布的常用方法"></a>5.1 确定随机变量概率分布的常用方法</h2><p>在模拟一个带有随机因素的实际系统时，究竟用什么样的概率分布描述问题中的随 机变量，是我们总是要碰到的一个问题，下面简单介绍确定分布的常用方法：</p>
<ol>
<li>根据一般知识和经验，可以假定其概率分布的形式，如顾客到达间隔服从指数分布Exp(λ)；产品需求量服从正态分布N(μ,σ22) ；订票后但未能按时前往机场登机的人数服从二项分布B(n,p) 。然后由实际数据估计分布的参数 σ,μ,λ 等，参数估计 可用极大似然估计、矩估计等方法。</li>
<li>直接由大量的实际数据作直方图，得到经验分布，再通过假设检验，拟合分布 函数，可用χ22检验等方法。</li>
<li>既缺少先验知识，又缺少数据时，对区间(a,b)内变化的随机变量，可选用Beta 分布（包括均匀分布）。先根据经验确定随机变量的均值μ和频率高时的数值（即密度函数的大值点）m ，则 Beta 分布中的参数α1,α2可由以下关系求出：<br><img src="2021-09-01-16-46-56.png"></li>
</ol>
<h2 id="5-2-计算机模拟"><a href="#5-2-计算机模拟" class="headerlink" title="5.2 计算机模拟"></a>5.2 计算机模拟</h2><p>当排队系统的到达间隔时间和服务时间的概率分布很复杂时，或不能用公式给出 时，那么就不能用解析法求解。这就需用随机模拟法求解，现举例说明。</p>
<p><strong>例1:</strong><br>设某仓库前有一卸货场，货车一般是夜间到达，白天卸货，每天只能卸货 2 车，若一天内到达数超过 2 车，那么就推迟到次日卸货。根据表 3 所示的数据，货车到 达数的概率分布（相对频率）平均为 1.5 车/天，求每天推迟卸货的平均车数。<br><img src="2021-09-01-16-47-25.png"><br>解: 这是单服务台的排队系统，可验证到达车数不服从泊松分布，服务时间也不服从指数分布（这是定长服务时间）。随机模拟法首先要求事件能按历史的概率分布规律出现。模拟时产生的随机数与事件的对应关系如表4<br><img src="2021-09-01-16-47-31.png"><br>用 a1 表示产生的随机数，a2 表示到达的车数，a3 表示需要卸货车数，a4 表 示实际卸货车数，a5 表示推迟卸货车数。编写程序如下：</p>
<figure class="highlight matlab"><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">clear </span><br><span class="line"><span class="built_in">rand</span>(<span class="string">'state'</span>,sum(<span class="number">100</span>*clock)); </span><br><span class="line">n=<span class="number">50000</span>; </span><br><span class="line">m=<span class="number">2</span> </span><br><span class="line">a1=<span class="built_in">rand</span>(n,<span class="number">1</span>); </span><br><span class="line">a2=a1; <span class="comment">%a2初始化 </span></span><br><span class="line">a2(<span class="built_in">find</span>(a1<<span class="number">0.23</span>))=<span class="number">0</span>; </span><br><span class="line">a2(<span class="built_in">find</span>(<span class="number">0.23</span><=a1&a1<<span class="number">0.53</span>))=<span class="number">1</span>; </span><br><span class="line">a2(<span class="built_in">find</span>(<span class="number">0.53</span><=a1&a1<<span class="number">0.83</span>))=<span class="number">2</span>; </span><br><span class="line">a2(<span class="built_in">find</span>(<span class="number">0.83</span><=a1&a1<<span class="number">0.93</span>),<span class="number">1</span>)=<span class="number">3</span>; </span><br><span class="line">a2(<span class="built_in">find</span>(<span class="number">0.93</span><=a1&a1<<span class="number">0.98</span>),<span class="number">1</span>)=<span class="number">4</span>; </span><br><span class="line">a2(<span class="built_in">find</span>(a1>=<span class="number">0.98</span>))=<span class="number">5</span>; </span><br><span class="line">a3=<span class="built_in">zeros</span>(n,<span class="number">1</span>);</span><br><span class="line">a4=<span class="built_in">zeros</span>(n,<span class="number">1</span>);</span><br><span class="line">a5=<span class="built_in">zeros</span>(n,<span class="number">1</span>); <span class="comment">%a2初始化 </span></span><br><span class="line">a3(<span class="number">1</span>)=a2(<span class="number">1</span>); </span><br><span class="line"><span class="keyword">if</span> a3(<span class="number">1</span>)<=m </span><br><span class="line"> a4(<span class="number">1</span>)=a3(<span class="number">1</span>);</span><br><span class="line"> a5(<span class="number">1</span>)=<span class="number">0</span>; </span><br><span class="line"><span class="keyword">else</span> </span><br><span class="line"> a4(<span class="number">1</span>)=m;</span><br><span class="line"><span class="keyword">end</span> </span><br><span class="line"><span class="keyword">for</span> <span class="built_in">i</span>=<span class="number">2</span>:n </span><br><span class="line"> a3(<span class="built_in">i</span>)=a2(<span class="built_in">i</span>)+a5(<span class="built_in">i</span><span class="number">-1</span>); </span><br><span class="line"> <span class="keyword">if</span> a3(<span class="built_in">i</span>)<=m </span><br><span class="line"> a4(<span class="built_in">i</span>)=a3(<span class="built_in">i</span>);</span><br><span class="line"> <span class="keyword">else</span> </span><br><span class="line"> a4(<span class="built_in">i</span>)=m;</span><br><span class="line"> <span class="keyword">end</span> </span><br><span class="line"><span class="keyword">end</span> </span><br><span class="line">a=[a1,a2,a3,a4,a5]; </span><br><span class="line">sum(a)/n</span><br></pre></td></tr></table></figure>
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<article id="post-熵权法确定权重" class="article article-type-post article-index" itemscope itemprop="blogPost">
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<a class="article-title" href="/2021/08/30/%E7%86%B5%E6%9D%83%E6%B3%95%E7%A1%AE%E5%AE%9A%E6%9D%83%E9%87%8D/">熵权法确定权重</a>
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<h1 id="一、基本原理"><a href="#一、基本原理" class="headerlink" title="一、基本原理"></a>一、基本原理</h1><p>在信息论中，熵是对不确定性的一种度量。不确定性越大，熵就越大，包含的信息量越大；不确定性越小，熵就越小，包含的信息量就越小。</p>
<p>根据熵的特性，可以通过计算熵值来判断一个事件的随机性及无序程度，也可以用熵值来判断某个指标的离散程度，指标的离散程度越大，该指标对综合评价的影响（权重）越大。比如样本数据在某指标下取值都相等，则该指标对总体评价的影响为0，权值为0.</p>
<p>熵权法是一种客观赋权法，因为它仅依赖于数据本身的离散性。</p>
<h1 id="二、熵值法步骤"><a href="#二、熵值法步骤" class="headerlink" title="二、熵值法步骤"></a>二、熵值法步骤</h1><ol>
<li>对 $n$ 个样本， $m$ 个指标，则 $x_{ij}$ 为第 $i$ 个样本的第 $j$ 个指标的数值（ $i=1,···,n; j=1,···,m$ ）；</li>
</ol>
<ol start="2">
<li>指标的归一化处理：异质指标同质化</li>
</ol>
<p>由于各项指标的计量单位并不统一，因此在用它们计算综合指标前，先要进行标准化处理，即把指标的绝对值转化为相对值，从而解决各项不同质指标值的同质化问题。</p>
<p>另外，正向指标和负向指标数值代表的含义不同（正向指标数值越高越好，负向指标数值越低越好），因此，对于正向负向指标需要采用不同的算法进行数据标准化处理：</p>
<p><img src="2021-08-30-15-45-56.png"></p>
<p>为了方便起见，归一化后的数据 $x_{ij}^{\prime}$ 仍记为 $x_{ij}$ ;</p>
<ol>
<li>计算第 $j$ 项指标下第 $i$ 个样本值占该指标的比重：</li>
</ol>
<p>$p_{ij}=\frac{x_{ij}}{\Sigma_{i=1}^nx_{ij}},i=1,···,n,j=1,···,m$</p>
<ol start="4">
<li>计算第 $j$ 项指标的熵值：</li>
</ol>
<p>$e_j=-k\Sigma_{i=1}^np_{ij}ln(p_{ij}),j=1,···,m$</p>
<p>其中， $k=1/ln(n)>0$. 满足 $e_j\ge0$ ;</p>
<ol start="5">
<li>计算信息熵冗余度（差异）：</li>
</ol>
<p>$d_j=1-e_j, j=1, ···,m$</p>
<ol start="6">
<li>计算各项指标的权重：</li>
</ol>
<p>$w_j=\frac{d_j}{\Sigma_{j=1}^mw_jx_{ij}}, i=1, ···, n$</p>
<ol start="7">
<li>计算各样本的综合得分：</li>
</ol>
<p>$s_i=\Sigma_{j=1}^mw_jx_{ij}$</p>
<p>其中， $x_{ij}$ 为标准化后的数据。</p>
<p>三、Matlab实现</p>
<p>按上述算法步骤，编写Matlab函数：shang.m</p>
<figure class="highlight matlab"><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></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">function</span> <span class="params">[s,w]</span>=<span class="title">shang</span><span class="params">(x,ind)</span></span></span><br><span class="line"><span class="comment">%实现用熵值法求各指标(列）的权重及各数据行的得分</span></span><br><span class="line"><span class="comment">%x为原始数据矩阵, 一行代表一个样本, 每列对应一个指标</span></span><br><span class="line"><span class="comment">%ind指示向量，指示各列正向指标还是负向指标，1表示正向指标，2表示负向指标</span></span><br><span class="line"><span class="comment">%s返回各行（样本）得分，w返回各列权重</span></span><br><span class="line">[n,m]=<span class="built_in">size</span>(x); <span class="comment">% n个样本, m个指标</span></span><br><span class="line"><span class="comment">%%数据的归一化处理</span></span><br><span class="line"><span class="keyword">for</span> <span class="built_in">i</span>=<span class="number">1</span>:m</span><br><span class="line"> <span class="keyword">if</span> ind(<span class="built_in">i</span>)==<span class="number">1</span> <span class="comment">%正向指标归一化</span></span><br><span class="line"> X(:,<span class="built_in">i</span>)=guiyi(x(:,<span class="built_in">i</span>),<span class="number">1</span>,<span class="number">0.002</span>,<span class="number">0.996</span>); <span class="comment">%若归一化到[0,1], 0会出问题</span></span><br><span class="line"> <span class="keyword">else</span> <span class="comment">%负向指标归一化</span></span><br><span class="line"> X(:,<span class="built_in">i</span>)=guiyi(x(:,<span class="built_in">i</span>),<span class="number">2</span>,<span class="number">0.002</span>,<span class="number">0.996</span>);</span><br><span class="line"> <span class="keyword">end</span></span><br><span class="line"><span class="keyword">end</span></span><br><span class="line"><span class="comment">%%计算第j个指标下，第i个样本占该指标的比重p(i,j)</span></span><br><span class="line"><span class="keyword">for</span> <span class="built_in">i</span>=<span class="number">1</span>:n</span><br><span class="line"> <span class="keyword">for</span> <span class="built_in">j</span>=<span class="number">1</span>:m</span><br><span class="line"> p(<span class="built_in">i</span>,<span class="built_in">j</span>)=X(<span class="built_in">i</span>,<span class="built_in">j</span>)/sum(X(:,<span class="built_in">j</span>));</span><br><span class="line"> <span class="keyword">end</span></span><br><span class="line"><span class="keyword">end</span></span><br><span class="line"><span class="comment">%%计算第j个指标的熵值e(j)</span></span><br><span class="line">k=<span class="number">1</span>/<span class="built_in">log</span>(n);</span><br><span class="line"><span class="keyword">for</span> <span class="built_in">j</span>=<span class="number">1</span>:m</span><br><span class="line"> e(<span class="built_in">j</span>)=-k*sum(p(:,<span class="built_in">j</span>).*<span class="built_in">log</span>(p(:,<span class="built_in">j</span>)));</span><br><span class="line"><span class="keyword">end</span></span><br><span class="line">d=<span class="built_in">ones</span>(<span class="number">1</span>,m)-e; <span class="comment">%计算信息熵冗余度</span></span><br><span class="line">w=d./sum(d); <span class="comment">%求权值w</span></span><br><span class="line">s=<span class="number">100</span>*w*X'; <span class="comment">%求综合得分</span></span><br></pre></td></tr></table></figure>
<p>用到的归一化函数：</p>
<figure class="highlight matlab"><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></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">function</span> <span class="title">y</span>=<span class="title">guiyi</span><span class="params">(x,type,ymin,ymax)</span></span></span><br><span class="line"><span class="comment">%实现正向或负向指标归一化，返回归一化后的数据矩阵</span></span><br><span class="line"><span class="comment">%x为原始数据矩阵, 一行代表一个样本, 每列对应一个指标</span></span><br><span class="line"><span class="comment">%type设定正向指标1,负向指标2</span></span><br><span class="line"><span class="comment">%ymin,ymax为归一化的区间端点</span></span><br><span class="line">[n,m]=<span class="built_in">size</span>(x);</span><br><span class="line">y=<span class="built_in">zeros</span>(n,m);</span><br><span class="line">xmin=<span class="built_in">min</span>(x);</span><br><span class="line">xmax=<span class="built_in">max</span>(x);</span><br><span class="line"><span class="keyword">switch</span> <span class="built_in">type</span></span><br><span class="line"> <span class="keyword">case</span> <span class="number">1</span></span><br><span class="line"> <span class="keyword">for</span> <span class="built_in">j</span>=<span class="number">1</span>:m</span><br><span class="line"> y(:,<span class="built_in">j</span>)=(ymax-ymin)*(x(:,<span class="built_in">j</span>)-xmin(<span class="built_in">j</span>))/(xmax(<span class="built_in">j</span>)-xmin(<span class="built_in">j</span>))+ymin;</span><br><span class="line"> <span class="keyword">end</span></span><br><span class="line"> <span class="keyword">case</span> <span class="number">2</span></span><br><span class="line"> <span class="keyword">for</span> <span class="built_in">j</span>=<span class="number">1</span>:m</span><br><span class="line"> y(:,<span class="built_in">j</span>)=(ymax-ymin)*(xmax(<span class="built_in">j</span>)-x(:,<span class="built_in">j</span>))/(xmax(<span class="built_in">j</span>)-xmin(<span class="built_in">j</span>))+ymin;</span><br><span class="line"> <span class="keyword">end</span></span><br><span class="line"><span class="keyword">end</span></span><br></pre></td></tr></table></figure>
<p>程序测试，现有数据shang_datas.mat, 为2014年31个省份的就业与劳动保障数据, 包含5个指标：社会养老保险参保率、医疗保险参保率、失业保险参保率、工伤保险参保率、工伤事故发生率, 其中第5个指标为负向指标。</p>
<p>执行代码：</p>
<figure class="highlight matlab"><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><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><span class="line">load shang_datas</span><br><span class="line"></span><br><span class="line">Ind=[<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="comment">%指定各指标的正向or负向</span></span><br><span class="line"></span><br><span class="line">[S,W]=shang(X,Ind)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">运行结果：</span><br><span class="line"></span><br><span class="line">S =</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">Columns <span class="number">1</span> through <span class="number">9</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="number">3.2624</span> <span class="number">11.7822</span> <span class="number">5.2952</span> <span class="number">1.3521</span> <span class="number">2.5262</span> <span class="number">2.0425</span> <span class="number">4.4215</span> <span class="number">2.6953</span> <span class="number">2.5208</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">Columns <span class="number">10</span> through <span class="number">18</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="number">7.8834</span> <span class="number">4.7410</span> <span class="number">7.4679</span> <span class="number">1.4645</span> <span class="number">3.2541</span> <span class="number">1.6292</span> <span class="number">3.3035</span> <span class="number">1.6282</span> <span class="number">1.9043</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">Columns <span class="number">19</span> through <span class="number">27</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="number">2.0526</span> <span class="number">7.8582</span> <span class="number">0.7256</span> <span class="number">3.8943</span> <span class="number">3.7992</span> <span class="number">1.7187</span> <span class="number">0.7766</span> <span class="number">0.6445</span> <span class="number">2.0167</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">Columns <span class="number">28</span> through <span class="number">31</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="number">0.7801</span> <span class="number">1.3353</span> <span class="number">2.2775</span> <span class="number">2.9462</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">W = <span class="number">0.0896</span> <span class="number">0.2195</span> <span class="number">0.3330</span> <span class="number">0.3073</span> <span class="number">0.0506</span></span><br></pre></td></tr></table></figure>
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<h1 id="方式1：通过连续的"><a href="#方式1：通过连续的" class="headerlink" title="方式1：通过连续的[ ]"></a>方式1：通过连续的[ ]</h1><p>这种方式在各种编程语言中都很常见，即数组的索引，但是这种方式只能取到某一具体维度的数值，不能随心所欲的固定间隔或者非固定间隔的切片</p>
<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></pre></td><td class="code"><pre><span class="line">a = tf.ones([<span class="number">1</span>,<span class="number">5</span>,<span class="number">5</span>,<span class="number">3</span>])</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">0</span>][<span class="number">0</span>])</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">0</span>][<span class="number">0</span>][<span class="number">0</span>])</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">0</span>][<span class="number">0</span>][<span class="number">0</span>][<span class="number">2</span>])</span><br></pre></td></tr></table></figure>
<p><img src="https://img2018.cnblogs.com/i-beta/1717633/202001/1717633-20200117204201634-920546460.png"></p>
<h1 id="方式2：通过-……"><a href="#方式2：通过-……" class="headerlink" title="方式2：通过[ , , ,……]"></a>方式2：通过[ , , ,……]</h1><p>这种方式其实是numpy对方式1的一种在可读性方面的优化，和方式1相比，可读性明显提高</p>
<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></pre></td><td class="code"><pre><span class="line">a = tf.random.normal([<span class="number">4</span>,<span class="number">28</span>,<span class="number">28</span>,<span class="number">3</span>])</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">1</span>].shape)</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">1</span>,<span class="number">2</span>].shape)</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">1</span>,<span class="number">2</span>,<span class="number">3</span>].shape)</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">1</span>,<span class="number">2</span>,<span class="number">3</span>,<span class="number">2</span>].shape)</span><br></pre></td></tr></table></figure>
<p><img src="https://img2018.cnblogs.com/i-beta/1717633/202001/1717633-20200117204654543-764698301.png"></p>
<h1 id="方式3：一维tensor可通过-："><a href="#方式3：一维tensor可通过-：" class="headerlink" title="方式3：一维tensor可通过[ ：]"></a>方式3：一维tensor可通过[ ：]</h1><p>这种方式也是python中比较常用的数组切片方式，切片范围[ A:B)</p>
<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></pre></td><td class="code"><pre><span class="line">a=tf.<span class="built_in">range</span>(<span class="number">10</span>)</span><br><span class="line"><span class="built_in">print</span>(a)</span><br><span class="line"><span class="built_in">print</span>(a[-<span class="number">1</span>:])</span><br><span class="line"><span class="built_in">print</span>(a[-<span class="number">2</span>:])</span><br><span class="line"><span class="built_in">print</span>(a[:<span class="number">2</span>])</span><br><span class="line"><span class="built_in">print</span>(a[:-<span class="number">1</span>])</span><br></pre></td></tr></table></figure>
<p><img src="https://img2018.cnblogs.com/i-beta/1717633/202001/1717633-20200117205052028-1609835816.png"></p>
<h1 id="方式4：对于多维tensor可通过-，：，：，：……"><a href="#方式4：对于多维tensor可通过-，：，：，：……" class="headerlink" title="方式4：对于多维tensor可通过[ ，：，：，：……]"></a>方式4：对于多维tensor可通过[ ，：，：，：……]</h1><p>相对前面几种切片方式都更加丰富，也可以完成多样的切片（跳过某一维度）start：end</p>
<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></pre></td><td class="code"><pre><span class="line">a = tf.random.normal([<span class="number">4</span>,<span class="number">28</span>,<span class="number">28</span>,<span class="number">3</span>])</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">0</span>,:,:,:].shape)</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">0</span>,<span class="number">1</span>,:,:].shape)</span><br><span class="line"><span class="built_in">print</span>(a[:,:,:,<span class="number">0</span>].shape)</span><br><span class="line"><span class="built_in">print</span>(a[:,:,:,<span class="number">2</span>].shape)</span><br><span class="line"><span class="built_in">print</span>(a[:,<span class="number">0</span>,:,:].shape)</span><br></pre></td></tr></table></figure>
<p><img src="https://img2018.cnblogs.com/i-beta/1717633/202001/1717633-20200117205939712-43108729.png"></p>
<h1 id="方式5：隔行采样-……"><a href="#方式5：隔行采样-……" class="headerlink" title="方式5：隔行采样[ : : ,: : ,: : ,……]"></a>方式5：隔行采样[ : : ,: : ,: : ,……]</h1><p>通过增加了一个：,使用方式start：end：step进行间隔采样（：：step代表从最开始到最末尾以步长step间隔采样）</p>
<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></pre></td><td class="code"><pre><span class="line">a = tf.random.normal([<span class="number">4</span>,<span class="number">28</span>,<span class="number">28</span>,<span class="number">3</span>])</span><br><span class="line"><span class="built_in">print</span>(a[:,<span class="number">0</span>:<span class="number">28</span>:<span class="number">2</span>,<span class="number">0</span>:<span class="number">28</span>:<span class="number">2</span>,:].shape)</span><br><span class="line"><span class="built_in">print</span>(a[:,:<span class="number">14</span>,:<span class="number">14</span>,:].shape)</span><br><span class="line"><span class="built_in">print</span>(a[:,<span class="number">14</span>:,<span class="number">14</span>:,:].shape)</span><br><span class="line"><span class="built_in">print</span>(a[:,::<span class="number">2</span>,::<span class="number">2</span>,:].shape)</span><br></pre></td></tr></table></figure>
<p><img src="https://img2018.cnblogs.com/i-beta/1717633/202001/1717633-20200117211102432-1630988448.png"></p>
<p>注：若step<0则倒序采样</p>
<h1 id="方式6：用…进行采样"><a href="#方式6：用…进行采样" class="headerlink" title="方式6：用…进行采样"></a>方式6：用…进行采样</h1><p>…可以代替连续的：,增强代码的易书写性和可读性</p>
<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></pre></td><td class="code"><pre><span class="line">a = tf.random.normal([<span class="number">2</span>,<span class="number">4</span>,<span class="number">28</span>,<span class="number">28</span>,<span class="number">3</span>])</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">0</span>,:,:,:,:].shape)</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">0</span>,...].shape)</span><br><span class="line"><span class="built_in">print</span>(a[:,:,:,:,<span class="number">0</span>].shape)</span><br><span class="line"><span class="built_in">print</span>(a[...,<span class="number">0</span>].shape)</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">0</span>,...,<span class="number">2</span>].shape)</span><br><span class="line"><span class="built_in">print</span>(a[<span class="number">1</span>,<span class="number">0</span>,...,<span class="number">0</span>].shape)</span><br></pre></td></tr></table></figure>
<p><img src="https://img2018.cnblogs.com/i-beta/1717633/202001/1717633-20200117211942026-2079829726.png"></p>
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<h1 id="权重衰减（weight-decay）与学习率衰减（learning-rate-decay）"><a href="#权重衰减（weight-decay）与学习率衰减（learning-rate-decay）" class="headerlink" title="权重衰减（weight decay）与学习率衰减（learning rate decay）"></a>权重衰减（weight decay）与学习率衰减（learning rate decay）</h1><h2 id="1-权重衰减（weight-decay）"><a href="#1-权重衰减（weight-decay）" class="headerlink" title="1. 权重衰减（weight decay）"></a>1. 权重衰减（weight decay）</h2><p>L2正则化的目的就是为了让权重衰减到更小的值，在一定程度上减少模型过拟合的问题，所以权重衰减也叫L2正则化。</p>
<h3 id="1-1-L2正则化与权重衰减系数"><a href="#1-1-L2正则化与权重衰减系数" class="headerlink" title="1.1 L2正则化与权重衰减系数"></a>1.1 L2正则化与权重衰减系数</h3><p>L2正则化就是在代价函数后面再加上一个正则化项：<br><img src="https://img-blog.csdn.net/20180630161015526?watermark/2/text/aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3Byb2dyYW1fZGV2ZWxvcGVy/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70"></p>
<p>其中C0代表原始的代价函数，后面那一项就是L2正则化项，它是这样来的：所有参数w的平方的和，除以训练集的样本大小n。λ就是正则项系数，权衡正则项与C0项的比重。另外还有一个系数1/2，1/2 1/211经常会看到，主要是为了后面求导的结果方便，后面那一项求导会产生一个2，与1/2相乘刚好凑整为1。系数λ就是权重衰减系数。</p>
<h3 id="1-2-为什么可以对权重进行衰减"><a href="#1-2-为什么可以对权重进行衰减" class="headerlink" title="1.2 为什么可以对权重进行衰减"></a>1.2 为什么可以对权重进行衰减</h3><p>我们对加入L2正则化后的代价函数进行推导，先求导：</p>
<p><img src="https://img-blog.csdn.net/2018063016132630?watermark/2/text/aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3Byb2dyYW1fZGV2ZWxvcGVy/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70"></p>
<p>可以发现L2正则化项对b的更新没有影响，但是对于w的更新有影响：</p>
<p><img src="https://img-blog.csdn.net/20180630161350343?watermark/2/text/aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3Byb2dyYW1fZGV2ZWxvcGVy/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70"></p>
<p>在不使用L2正则化时，求导结果中w前系数为1，现在w前面系数为1-ηλ/n，因为η、λ、n都是正的，所以1-ηλ/n小于1，它的效果是减小w，这也就是权重衰减（weight decay）的由来。当然考虑到后面的导数项，w最终的值可能增大也可能减小。</p>
<p><img src="https://img-blog.csdn.net/20180630161538529?watermark/2/text/aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3Byb2dyYW1fZGV2ZWxvcGVy/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70"><br><img src="https://img-blog.csdn.net/20180630161545925?watermark/2/text/aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3Byb2dyYW1fZGV2ZWxvcGVy/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70"></p>
<p>另外，需要提一下，对于基于mini-batch的随机梯度下降，w和b更新的公式跟上面给出的有点不同：</p>
<p>对比上面w的更新公式，可以发现后面那一项变了，变成所有导数加和，乘以η再除以m，m是一个mini-batch中样本的个数。</p>
<h3 id="1-3-权重衰减（L2正则化）的作用"><a href="#1-3-权重衰减（L2正则化）的作用" class="headerlink" title="1.3 权重衰减（L2正则化）的作用"></a>1.3 权重衰减（L2正则化）的作用</h3><p><strong>作用</strong>： 权重衰减（L2正则化）可以避免模型过拟合问题。</p>
<p><strong>思考</strong>： L2正则化项有让w变小的效果，但是为什么w变小可以防止过拟合呢？</p>
<p><strong>原理</strong>： </p>
<ol>
<li>从模型的复杂度上解释：更小的权值w，从某种意义上说，表示网络的复杂度更低，对数据的拟合更好（这个法则也叫做奥卡姆剃刀），而在实际应用中，也验证了这一点，L2正则化的效果往往好于未经正则化的效果。</li>
<li>从数学方面的解释：过拟合的时候，拟合函数的系数往往非常大，为什么？如下图所示，过拟合，就是拟合函数需要顾忌每一个点，最终形成的拟合函数波动很大。在某些很小的区间里，函数值的变化很剧烈。这就意味着函数在某些小区间里的导数值（绝对值）非常大，由于自变量值可大可小，所以只有系数足够大，才能保证导数值很大。而正则化是通过约束参数的范数使其不要太大，所以可以在一定程度上减少过拟合情况。</li>
</ol>
<h2 id="学习率衰减（learning-rate-decay）"><a href="#学习率衰减（learning-rate-decay）" class="headerlink" title="学习率衰减（learning rate decay）"></a>学习率衰减（learning rate decay）</h2><p>在训练模型的时候，通常会遇到这种情况：我们平衡模型的训练速度和损失（loss）后选择了相对合适的学习率（learning rate），但是训练集的损失下降到一定的程度后就不在下降了，比如training loss一直在0.7和0.9之间来回震荡，不能进一步下降。如下图所示：</p>
<p><img src="https://img-blog.csdn.net/2018063016182747?watermark/2/text/aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3Byb2dyYW1fZGV2ZWxvcGVy/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70"></p>
<p>遇到这种情况通常可以通过适当降低学习率（learning rate）来实现。但是，降低学习率又会延长训练所需的时间。<br>学习率衰减（learning rate decay） 就是一种可以平衡这两者之间矛盾的解决方案。学习率衰减的基本思想是：学习率随着训练的进行逐渐衰减。<br>学习率衰减基本有两种实现方法：</p>
<ul>
<li>线性衰减。例如：每过5个epochs学习率减半。</li>
<li>指数衰减。例如：随着迭代轮数的增加学习率自动发生衰减，每过5个epochs将学习率乘以0.9998。具体算法如下：</li>
</ul>
<p>decayed_learning_rate=learning_rate*decay_rate^(global_step/decay_steps)<br>其中decayed_learning_rate为每一轮优化时使用的学习率，learning_rate为事先设定的初始学习率，decay_rate为衰减系数，decay_steps为衰减速度。</p>
<h1 id="残差连接（skip-connect）-（residual-connections）"><a href="#残差连接（skip-connect）-（residual-connections）" class="headerlink" title="残差连接（skip connect）/（residual connections）"></a>残差连接（skip connect）/（residual connections）</h1><p>想必做深度学习的都知道skip connect，也就是残差连接，那什么是skip connect呢？如下图<br><img src="https://img-blog.csdnimg.cn/20200502093404478.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzM5ODUyNjc2,size_16,color_FFFFFF,t_70"></p>
<p>上面是来自于resnet【1】的skip block的示意图。我们可以使用一个非线性变化函数来描述一个网络的输入输出，即输入为X，输出为F(x)，F通常包括了卷积，激活等操作。</p>
<p>当我们强行将一个输入添加到函数的输出的时候，虽然我们仍然可以用G(x)来描述输入输出的关系，但是这个G(x)却可以明确的拆分为F(x)和X的线性叠加。</p>
<p>这就是skip connect的思想，将输出表述为输入和输入的一个非线性变换的线性叠加，没用新的公式，没有新的理论，只是换了一种新的表达。</p>
<p>它解决了深层网络的训练问题，作者的原论文中达到了上千层。</p>
<p>残差连接是何的首创吗？当然不是，传统的神经网络中早就有这个概念，文【2】中则明确提出了残差的结构，这是来自于LSTM的控制门的思想。<br>$$y = H(x,WH) + X，此所谓残差连接，skip-connection。$$</p>
<p>那为什么要这么做呢？首先大家已经形成了一个通识，在一定程度上，网络越深表达能力越强，性能越好。</p>
<p>不过，好是好了，随着网络深度的增加，带来了许多问题，梯度消散，梯度爆炸；在resnet出来之前大家没想办法去解决吗？当然不是。更好的优化方法，更好的初始化策略，BN层，Relu等各种激活函数，都被用过了，但是仍然不够，改善问题的能力有限，直到残差连接被广泛使用。</p>
<p>大家都知道深度学习依靠误差的链式反向传播来进行参数更新，假如我们有这样一个函数：<br><img src="https://img-blog.csdnimg.cn/20200502093551295.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzM5ODUyNjc2,size_16,color_FFFFFF,t_70"></p>
<p>其中的f，g，k大家可以自行脑补为卷积，激活，分类器。<br>cost对f的导数为：<br><img src="https://img-blog.csdnimg.cn/20200502093606306.png"></p>
<p>它有隐患，一旦其中某一个导数很小，多次连乘后梯度可能越来越小，这就是常说的梯度消散，对于深层网络，传到浅层几乎就没了。但是如果使用了残差，每一个导数就加上了一个恒等项1，$dh/dx=d(f+x)/dx=1+df/dx$。此时就算原来的导数df/dx很小，这时候误差仍然能够有效的反向传播，这就是核心思想。</p>
<p>有这样的一个输入输出关系：</p>
<h1 id="dilated-convolution"><a href="#dilated-convolution" class="headerlink" title="dilated convolution"></a>dilated convolution</h1><p>直观来讲，Dilated Convolutions就是为了获得大的感受野。</p>
<p>插播一句：何谓感受野？ 感受野范围内的任意元素变化都会影响到输出。</p>
<p>下图很直观的介绍了什么是 Dilated Convolutions 以及 感受野。<br><img src="https://img-blog.csdn.net/20170410142647837?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvc2h1emZhbg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast"></p>
<ul>
<li><strong>(a)</strong> 原始的 3×3卷积，1-dilated，感受野为 3×3；</li>
<li><strong>(b)</strong> 在(a)的基础上进行 3×3卷积，2-dilated，感受野为 7×7；</li>
<li><strong>(c)</strong> 在(b)的基础上进行 3×3卷积，4-dilated，感受野为 15×15；</li>
</ul>
<p>由于padding和卷积的stride=1，卷积前后feature map大小可以保持不变，但每个元素的感受野指数增大。</p>
<p>Dilated/Atrous Convolution 或者是 Convolution with holes 从字面上就很好理解，是在标准的 convolution map 里注入空洞，以此来增加 reception field。相比原来的正常convolution，dilated convolution 多了一个 hyper-parameter 称之为 dilation rate 指的是kernel的间隔数量(e.g. 正常的 convolution 是 dilatation rate 1)。</p>
<p>下图为原始3x3卷积<br><img src="https://github.com/vdumoulin/conv_arithmetic/raw/master/gif/padding_strides.gif"></p>
<p>这是Dilated Convolution with a 3 x 3 kernel and dilation rate 2<br><img src="https://github.com/vdumoulin/conv_arithmetic/raw/master/gif/dilation.gif"></p>
<h1 id="dropout"><a href="#dropout" class="headerlink" title="dropout"></a>dropout</h1><h2 id="1-Dropout简介"><a href="#1-Dropout简介" class="headerlink" title="1. Dropout简介"></a>1. Dropout简介</h2><h3 id="1-1-Dropout出现的原因"><a href="#1-1-Dropout出现的原因" class="headerlink" title="1.1 Dropout出现的原因"></a>1.1 Dropout出现的原因</h3><p>在机器学习的模型中，如果模型的参数太多，而训练样本又太少，训练出来的模型很容易产生过拟合的现象。在训练神经网络的时候经常会遇到过拟合的问题，过拟合具体表现在：模型在训练数据上损失函数较小，预测准确率较高；但是在测试数据上损失函数比较大，预测准确率较低。</p>
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