From 09b11615829930adc55c7a0829e08fdb8e6f1531 Mon Sep 17 00:00:00 2001 From: jiangzhonglian Date: Mon, 10 Apr 2017 13:14:00 +0800 Subject: [PATCH 1/3] =?UTF-8?q?=E6=9B=B4=E6=96=B0ApacheCN=E7=9A=84?= =?UTF-8?q?=E8=A7=86=E9=A2=91=E9=93=BE=E6=8E=A5=E5=9C=B0=E5=9D=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- README.md | 4 ++++ images/MainPage/apachecn_baiduyun.png | Bin 0 -> 62347 bytes images/MainPage/apachecn_youku.png | Bin 0 -> 138251 bytes 3 files changed, 4 insertions(+) create mode 100644 images/MainPage/apachecn_baiduyun.png create mode 100644 images/MainPage/apachecn_youku.png diff --git a/README.md b/README.md index 2b2f7a42..1f646477 100644 --- a/README.md +++ b/README.md @@ -65,3 +65,7 @@ * [ApacheCN(apache中文网) 维护更新](http://www.apache.wiki) * [关于我们](http://www.apache.wiki/pages/viewpage.action?pageId=2887240) * [加入我们](http://www.apache.wiki/pages/viewpage.action?pageId=2887239) +* [ApacheCN-ML视频-优酷地址](http://i.youku.com/apachecn) +* ![](/images/MainPage/apachecn_youku.png) +* [ApacheCN-ML视频-百度云地址](https://pan.baidu.com/s/1eS44hCu) +* ![](/images/MainPage/apachecn_baiduyun.png) diff --git a/images/MainPage/apachecn_baiduyun.png b/images/MainPage/apachecn_baiduyun.png new file mode 100644 index 0000000000000000000000000000000000000000..ea8697257a26cf10d99b407e80181b211e215eb2 GIT binary patch literal 62347 zcmeFZ2Ut`~);78k1Oya8auAfH1d*KSqevDIBuH#Yl9S}v3IdV^1eB-*$+YB5OU@u! zG7U&hO>Cg)zS}c1-^@8P&-cy$KXd>4+~?lX-PEq$yK1dk71ny!s>c1mEdbY56qFSJ zJUjqU0{;OxB=DQOm(2?RP*VfA0RSKd@Iq|>0x$2DdJ>L^uqw$ z;-xpra&pioS{e$XMtOv$OrLmmM74UTP^mWPPS@z)Ca&kOE`?B_IjVnwq;h zKX~%w@$Ys1%a7yn-)RR%_7n-O-VQ2fBDMa3rQ_3UClw}iNPXs z3s-XoFmDg$dCi?)x`Fu*V4ly@&G}cp3(Ql!1PKK5Gr#gyf64Ft%A5V3zx0b0s+ZbY za$q;of%)rZf63qZmACp!Ui??y+}hLvdjSW7 zEm`H99Gun{KtX#Fk=JetA3mpOd~00}k#z>oZU znOPzLR5t(sUC-ak*wX=kTnYfH2AoY_n*OCe@DuNt;D5SzeIX?H`}25Y4*}rv91eHD z2>?WK0B{kD!<}d1a2Hu%`=HFe+2g(e6vTLu1Zntqw}DF(c=!}}xHfVfQ= 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b/docs/6.支持向量机.md index 951c3e46..e4c9107c 100644 --- a/docs/6.支持向量机.md +++ b/docs/6.支持向量机.md @@ -1,12 +1,68 @@ - # 6) 支持向量机 -* 基本概念 +![支持向量机_首页](/images/6.SVM/SVM_1.jpg) + +## 支持向量机的概念 + +> 支持向量机(Support Vector Machines, SVM) + +* 支持向量(Support Vector)就是离分隔超平面最近的那些点。 +* 机(Machine)就是表示一种算法,而不是表示机器。 +* SVM有很多种实现,最流行的一种实现是: `序列最小优化(Sequential Minimal Optimization, SMO)算法`。 +* 下面还会介绍一种称为`核函数(kernel)`的方式将SVM扩展到更多数据集上。 +* 实战项目:回顾第1章中手写识别的案例,并考察其能否通过SVM来提高识别的效果。 +* 注意:`SVM几何含义比较直观,但其算法实现较复杂,牵扯大量数学公式的推导。` + +``` +优点:泛化错误率低,计算开销不大,结果易理解。 +缺点:对参数调节和核函数的选择敏感,原始分类器不加修改仅适合于处理二分类问题。 +使用数据类型:数值型和标称型数据。 +``` + +## 基于最大间隔分隔数据 + +* 数据可以通过画一条直线就可以将它们完全分开,这组数据叫`线性可分(linearly separable)`数据。 +* 而这条分隔直线称为`分隔超平面(separating hyperplane)`。 +* 如果数据集上升到1024维呢?那么需要1023维来分隔数据集,也就说需要N-1维的对象来分隔,这个对象叫做`超平面(hyperlane)`,也就是分类的决策边界。 +* ![分隔超平面](/images/6.SVM/SVM_2_separating-hyperplane.jpg) + +## 寻找最大间隔 + +> 为什么寻找最大间隔 + +``` +摘录地址:http://slideplayer.com/slide/8610144 (第12条信息) +Support Vector Machines: Slide 12 Copyright © 2001, 2003, Andrew W. Moore Why Maximum Margin? +denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) The maximum margin linear classifier is the linear classifier with the, um, maximum margin. +This is the simplest kind of SVM (Called an LSVM) Support Vectors are those datapoints that the margin pushes up against + +1.Intuitively this feels safest. +2.If we’ve made a small error in the location of the boundary (it’s been jolted in its perpendicular direction) this gives us least chance of causing a misclassification. +3.CV is easy since the model is immune to removal of any non-support-vector datapoints. +4.There’s some theory that this is a good thing. +5.Empirically it works very very well. + +* * * + +1. 直觉上是安全的 +2. 如果我们在边界的位置发生了一个小错误(它在垂直方向上被颠倒),这给我们最小的错误分类机会。 +3. CV很容易,因为该模型对任何非支持向量数据点的去除是免疫的。 +4. 有一些理论,这是一件好事。 +5. 通常它的工作非常好。 +``` + +* 选择D会比B、C分隔的效果要好很多,原因是上述的5个结论。 +![线性可分](/images/6.SVM/SVM_3_linearly-separable.jpg) + +> 怎么寻找最大间隔 + + + + + + + - * 假设有两组数据,我们可以用一条线将这两个数据集分隔开,这条直线称为分隔超平面 - (当维数很高时称为分隔超平面)。 - * 我们希望找到离分隔超平面最近的点,确保它们离分隔面的距离尽可能远。 - * 支持向量就是离分隔超平面最近的那些点。 * 支持向量机的目标是找出能够最大化训练集数据间隔(margin)的最优分类超平面。 diff --git a/images/6.SVM/SVM_1.jpg b/images/6.SVM/SVM_1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..37c313fa4e81b18479b55b7867cb0a10f30eb573 GIT binary patch literal 157674 zcmb@t2Q*w!`!{+R3?jOO=uC7HC3+`%@15vG8$|DdVDuVMqZ7SHkKP5*`-q8%-us>8 zz5nlB>#lFDTh@JM&N;ucXU{(S+56c~JLf#iKdb}z3NrFC0165K$RR(#!v-M!+RM%w z0F;#h761S+0h9no^}W25l&PAAs*JqS>wld> zb${dH;_&y+j!y2b8nTi!din-5;AP+ufCCT$g22-^7H%#QYHCXV+Wg<{&S(Gi9hheS z+m^Nb)q-%CXHGqa!bd7WceelSU;bP8$kNKq0?9lkvdF^H&B76h9gvvK!o}4ciPMpo z-P7IWFCIiMfgD*Y8YyQ-mv`<~u52{&P8J+&G+Zw^ zU;eH4ch3GE0pLXA-&~+PnfPy+a~=Q)&;tMn_TMt|Q~+pd0|3(D|CT|r0RT@302;?# 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docs/6.支持向量机.md | 72 +++++++++++++++------ images/6.SVM/SVM_4_point2line-distance.png | Bin 0 -> 105844 bytes images/6.SVM/SVM_5_Lagrangemultiplier.png | Bin 0 -> 81895 bytes 3 files changed, 53 insertions(+), 19 deletions(-) create mode 100644 images/6.SVM/SVM_4_point2line-distance.png create mode 100644 images/6.SVM/SVM_5_Lagrangemultiplier.png diff --git a/docs/6.支持向量机.md b/docs/6.支持向量机.md index e4c9107c..277c79e6 100644 --- a/docs/6.支持向量机.md +++ b/docs/6.支持向量机.md @@ -1,4 +1,5 @@ # 6) 支持向量机 + ![支持向量机_首页](/images/6.SVM/SVM_1.jpg) @@ -54,30 +55,63 @@ This is the simplest kind of SVM (Called an LSVM) Support Vectors are those data * 选择D会比B、C分隔的效果要好很多,原因是上述的5个结论。 ![线性可分](/images/6.SVM/SVM_3_linearly-separable.jpg) -> 怎么寻找最大间隔 +### 怎么寻找最大间隔 +> 点到超平面的距离 +* 分隔超平面`函数间距`: $$y(x)=w^Tx+b$$ +* 分类的结果: $$f(x)=sign(w^Tx+b)$$ (sign表示>0为1,<0为-1,=0为0) +* 点到超平面的`几何间距`: $$d(x)=(w^Tx+b)/||w||$$ [ ||w||表示w矩阵的二范式=> sqrt(w*w^T), 点到超平面的距离也是类似的,需要复习一下向量的知识 ] +![点到直线的几何距离](/images/6.SVM/SVM_4_point2line-distance.png) +> 拉格朗日乘子法的使用 +* 类别标签用-1、1,是为了后期方便`lable*(w^Tx+b)`的标识和距离计算;如果`lable*(w^Tx+b)>0`表示预测正确,否则预测错误。 +* 现在目标很明确,就是要找到`w`和`b`,因此我们必须要找到最小间隔的数据点,也就是前面所说的`支持向量`。 + * 也就说,让最小的距离取最大.(最小的距离:就是最小间隔的数据点;最大:就是最大间距,为了找出最优超平面--最终就是支持向量) + * 怎么理解呢? 例如: 所有的点看作地雷吧,那么我们(超平面)得找到最近所有的地雷,并保证我们离它最远。 + * 目标函数:`arg max{min(lable*(w^Tx+b)/||w||)}` + * 1.如果`lable*(w^Tx+b)>0`表示预测正确,`||w||`可以理解为归一化,我们始终可以找到一个阈值让`lable*(w^Tx+b)>=1` + * 2.所以令`lable*(w^Tx+b)=1`,我们本质上是求 `arg max[关于w, b] (1/||w||)`;也就说,我们约束(前提)条件是`lable*(w^Tx+b)>=1` +* 新的目标函数求解: `arg max[关于w, b] (1/||w||)` + * => 就是求: `arg min[关于w, b] (||w||)` (求矩阵会比较麻烦,如果x只是1/2*X^2的偏导数,那么。。同样是求最小值) + * => 就是求: `arg min[关于w, b] (1/2*||w||^2)` (二次函数求导,求极值,平方也方便计算) + * 本质上就是求线性不等式的二次优化问题(求分隔超平面,等价于求解相应的凸二次规划问题。) +* 通过拉格朗日乘子法,求二次优化问题 + * 假设需要求极值的目标函数 (objective function) 为 f(x,y),限制条件为 φ(x,y)=M + * 设g(x,y)=M-φ(x,y) # 临时φ(x,y)表示下文中`label*(w^Tx+b)` + * 定义一个新函数: F(x,y,λ)=f(x,y)+λg(x,y) + * a为λ,代表要引入的拉格朗日乘子(Lagrange multiplier) + * 那么: `L(w,b,a)=1/2||w||^2 + Σ a*[1-label*(w^Tx+b)]` => `L(w,b,a)=1/2||w||^2 - Σ a*[label*(w^Tx+b)-1]` + * 因为:`label*(w^Tx+b)>=1, a>0` , 所以`a*[label*(w^Tx+b)-1]>=0`, Σ 0~n的加和也都大于0 + * `max[关于a] (L(w,b,a)) = 1/2*||w||^2` + * 如果求: `min(1/2*||w||^2)`, 也就是要求: `arg min[关于w, b]{ max[关于a] (L(w,b,a)) }` +* 现在转化到对偶问题的求解 + * `min[关于w, b]{ max[关于w, b] (L(w,b,a)) } >= max[关于w, b] { min[关于w, b] (L(w,b,a)) }` + * 现在分2步 + * 先求:`min[关于w, b] (L(w,b,a))=1/2||w||^2 + Σ a*[1-label*(w^Tx+b)]` + * 就是求`L(w,b,a)`关于[w, b]的偏导数, 得到`w和b的值`,并化简为:`L和a的方程`。 + * 参考: 1.【计算拉格朗日函数的对偶函数-下图是参考小象学院】 2.如果公式推导还是不懂,也可以参考《统计学习方法》李航-P103<学习的对偶算法> + * ![计算拉格朗日函数的对偶函数](/images/6.SVM/SVM_5_Lagrangemultiplier.png) +* 终于得到课本上的公式:`max(Σ(i=>n) a[i] - 1/2*Σ(i,j=>n) a[i]*a[j]]*label(i)*label(j)*)` +* 约束条件: `a>=0 and Σ a[i]*label(i)=0` +> 松弛变量(slack variable) +* 我们知道几乎所有的数据都不那么干净, 通过引入松弛变量来允许数据点可以处于分隔面错误的一侧。 +* 约束条件: `C>=a>=0 and Σ a[i]*label(i)=0` +* 这里常量C用于控制“最大化间隔”和“保证大部分点的函数间隔小于1.0” 这两个目标的权重。 +* 常量C是一个常数,我们通过调节钙参数得到不同的结果。一旦求出了所有的alpha,那么分隔超平面就可以通过这些alpha来表示。 +* 这一结论十分直接,SVM中的主要工作就是要求解 alpha. +> SVM应用的一般框架 - -* 支持向量机的目标是找出能够最大化训练集数据间隔(margin)的最优分类超平面。 - -* 支持向量机特点 - - * 优点:泛化错误率低,计算开销不大,结果易解释。 - * 缺点:对参数调节和核函数的选择敏感,原始分类器不加修改仅适用于处理二类问题。 - * 适用数据类型:数值型和标称型数据。 - -* SVM应用的一般流程 - 1. 准备收集数据:可以使用任何方法。 - 2. 准备数据:需要数值型数据。 - 3. 分析数据:有助于可视化分隔超平面。 - 4. 训练算法:SVM的大部分时间都花在训练上,该过程主要实现两个参数的调优。 - 5. 测试算法:测试分类结果是否达到预期的目的。 - 6. 使用算法:几乎所有分类问题都可以使用SVM,值得一提的是,SVM本身是一个二类分类器, - 对多类问题应用SVM需要对代码做一些修改。 - +``` +SVM的一般流程 +收集数据:可以使用任意方法。 +准备数据:需要数值型数据。 +分析数据:有助于可视化分隔超平面。 +训练算法:SVM的大部分时间都源自训练,该过程主要实现两个参数的调优。 +测试算法:十分简单的计算过程就可以实现。 +使用算法:几乎所有分类问题都可以使用SVM,值得一提的是,SVM本身是一个二类分类器,对多类问题应用SVM需要对代码做一些修改。 +``` diff --git a/images/6.SVM/SVM_4_point2line-distance.png b/images/6.SVM/SVM_4_point2line-distance.png new file mode 100644 index 0000000000000000000000000000000000000000..bb680b31d9d2d25d528ed58bffc41aaa7e28feeb GIT binary patch literal 105844 zcmeFY1yEeiwlBP~;6Z|WkU)Y4_d$XO2?=h&-66R9U;zRI2of{|3otmrEg`ri_&{)* 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