From 9c19ff11a0612197acd280198ef75f6433ef8c7a Mon Sep 17 00:00:00 2001 From: jiangzhonglian Date: Tue, 11 Apr 2017 01:00:46 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B06.SVM=E5=BA=94=E7=94=A8?= =?UTF-8?q?=E7=9A=84=E4=B8=80=E8=88=AC=E6=A1=86=E6=9E=B6=E7=9A=84=E5=86=85?= =?UTF-8?q?=E5=AE=B9?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 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$XO2?=h&-66R9U;zRI2of{|3otmrEg`ri_&{)* z2{!1wN&f$HPSvUVzW04!-CK2E)$N(yv!?g%)%{z&cK2E>$nVHifLK*gMG-(l0{~^z z2S7pr83kW^8vsyO2Oa?cfCHe#*aMg-3iSv;gNguvUV#3O6by?3jDOPIzi%QB08uSF zS1(sjJ6AU*0Uka;R8B=5^EU@n_(QY(A&IBy6MiHEb(lM$xNr9na>2+o^hgzXc{43- zO+}Tb3V$kv?q%-g=JY$Wvx}F9wvrr^zJVbV_9B1}5CEirI6!G`>FFk`rS70N#51j&D+b`gNsSU z#gd1MiJ$K=-|y#sx7qJs065nE(-vqHWB*EXeGdR)EC7I!@UJwB3;<|q2LP(C|4L)c z0{|ik0B9U_Gxsq6iy!KWc82+X`tNBY!2joYGy*vQz*E->($Jft4 zA~GsECN>V7n)W6=BlB%mc41L*Nom=K@{e`(kcP&l=9bo;-oE~U!J#i-C#R-oX6L^D zn1`;d|J>Nz+TPiP|2jT7Jv&EST>i#|24MUJ3w8Yq*}vf;LE%Ej#KgeF{*4O_-3Jvh zNHDSPKfXgM`wZLM{T?%a7!H|S%KO@GTowUs7`cVV1Re#eAe0UM8`>Xa|L=f>|Nlbv zPhkJTwE)Ni=zjtRIywdx1_lP!9V}G1gLCJ1z`?`$6Y%~H1b@Qa-+}0#0f}OQhT?&V ziHVK+6653I6aRk>$R$+gD1w{^2r$r4%7j4zfPkAt>3guI8?=2S0Am2%KFC4>IAu|t z+eEfVAQX-S;6f7&ZwBr^%b8;TRlL-I0iNaRVD$B$Hgn7{3`%eDPEg?@`FP|RMZK^% zn~HGl;ZAH-k6W5LCkcrP_PaKRkM$kRt97*wMzoW_eyuqRPw8Gsm<2V8p65JNyeW#t zJd^sE-^)m_)9N#jOn_kTXP|)d#(1j~Im^wf>{VoA3fD>?NeFlrbH0)e zxvIIO(Mf*Pa_)Ut1#^AcmOZH5I_~W9Qo{71=?pzl5ht;Nok2~nl$vJdiz((wSFTlp zNp*BtE(bEJjn^bybaNZ*gEeS7bz+3M1hhq@GTA?1r`b+1xgQ=gy( z>o8o5XhQYdJy&l)UAI_py_iy1^m~Waeaea6R!(Onn=GF?C*Cg)+;CYomcEAd+ltL6 zd_24DT+M!7G-4{)h6FM%wu39|p9PA{2@VNWJz7w2scqrjc>1Zo_AIpw^TW2{qpJRu zo*Z?Xh2A;N5~Xre78Qa!qCTw;L`Viihib-hCFI9f&PXP&oy!-7y?vl+QmTiuf^Dsk z#Ny+PHO;!=dwds4!4i7?v!*mT1|4;XG|97Ph2PW26Pm<34-GAiXEYA|}e7ebnDxA6Q*cuu|Lb~L51`3Hge zhIm4{Yi2lBpvjd`C5_MiRS>+h`;vSF3B=V@@ji1h|J>dVwt-~7NVG`WwAV0tx$|Wd zD{u$)e66OR5W9D84&)`+z6er^19YizTjifIs#lI$cV>|EQjZT3w5S zxa0ERE>BElAQPPX+bjQj_9FG)s(0wz`#-1OoxQIb?&d$+ScKhM#rdpu(wkOFdinUd zjgeg*POs{4Vz=V5dx{wtRCf-BxFVijlTkm`EkuFSIHau##xtaC6O;uTrT z+`&neFg2~i=dZG)Y1{ar0%keE#IUA*-nbC5Kpi+UBVAeO^-IB%D6eTX$DH1>9)33O z6xJT)Ze?qLrhJ{c-4iZ<<^dA{x4+D|!)>ZWQY6)-bJJi2y0k8Y8&rTTQ+M6mXG}R@@ITNf=HD^$b^3bT~_MhZUJ-D0?YR zxGt0PGB4a@YkweGzAq4VWR@^*7UrEOZv?wrX1|Khxxh=Y>)%mSX6I@AvG+{Z$cj3s ziw7?c_sGou8gpEbWF17|d(Q~rY`L1KAth$nd__2xJ!NDZ_4Oj9WsvY)IVQttvAOrJ zmb7zir^^ui4ul%?K-$E}P+Xy;m@!yM*$u?xsJgrY{XD6}2BCRT!GVQWjO`}%c+mfv z1o7E<6$ua_29Q9HO^?T{;q~kQx8m*jm00CdzPHzB_7PTM9ysQ6*(YN^;(M3r;Cg*W z_=x8^kd8ck>yB(#)y;MjI<~y?&x5x_T%L(P7R5K~3NGeNzFUBN0!}~<4JHCz;`C__ zh9gP2mJX43GVITSyQ)FBDh)1ip+i>`N12ErE4Wp$@&31qg z@G7HgXGDHc@MtSRp$WTUHsSNWV~n?JjiGUtVGFfEMdjNLj9^OQKud&QVW0zo-j(Ze ztT$0k-X^sVo)c@XV-)+E&Q%x9O;KrBgTmqH(&YBnBK^UQSwPU>LT^t81?U*N9tm{$ zA^}8p?_2@G6B;ixEh%eXrf&#|C(h7k(OKTYYgw4izEJ8%0t_c(v7K1gAUI`DKJgqh z+hoa4gmh7!Ac0-MI`wa zE%^}o90~9_@?K*n|Lz6Sk$`PyXom>GaqYlp&~PlttdC-uJ<7VJi2^Q{?EYHUNr@&@ zhU@O*cgK00``)8sWo(ulU$3Y@usDR#ZRo5Sf`}0b+*7({4yHf?1wxs(7=uWlZ}x_q zjf(OA^b8qV7;3c2M~zm5)_;#yf+0QZP}fTggOF9pkZX0<37P+4^FOlrH{JRlz4_n2 zcmE@s|B=mqStrVf9KX4e1g{4Mtc^%?od&lj12^R2vlsCZV1Cl`lyXlJz-1v)Oyl@PfCOW}wV^HHT32f&`+gMwxM0vgm^I)bcBhXB24v z2xNF0zC19ouAfI{rP*2{i?J>798@p3+-V(ui!g&#ZAgr2-#P~0!40Tr1qpWAn8lhh zMx5!)ND_E+-f)^uay8f=GR!Wk@Nmy;=FTe1$Mz+l5fpK@&9@Ii13`Ch>Gb6*eUp2Z zdFI-gS+Zq)D!QqnYDzhqzSb$d&SC|wN0yA$S4l>cCD<;S9HC8jOI zrdfNS!5KY7Z~ri${8j5!k9Oe5dDCj(Gq`+jCs81+EmSih_)$m8g!c%$X2w^cCdH7( zte2wU)>@%>Lwd(E zRY%8wV*NvA6)igGG~dvf5L41)2o@tR{-9vR!sJRALdgF80z!8R>J%&r zn~RgYUoziO#67Z9LCsT|{hVloVSL$e^)6x1#+9f;J{A&yK7ykmf`Zo6)r_XO;+mUh zQ))MQ&_C!hdGPT`aUMAlXvr{zqhVh>0accXuJ+m(|M_jcZl6zuGJV>CYT#l(+y=Q3Xau{D@S4MFIY>>2V#)Oatr zlJ!_MLh2xid15^8?J>Tl@?-9WD{46YWQ^>*8~hlN2BoNn+4ijvtiuL!L$F}Cv&}8f zyhbDh`eU{EMVFFK(=RJO#G)7A6%}BBeGJ)XkIfEhf}6p0aLnvn-5xfcpoOgt z(GJCW09AYKSAe;o21dg!;RYUXun5sTR}kgKk0YZaEoKN3Ew@AC6UbI?*vh{r&E<=~ zdK)VF3{LdSnOAL@=$*^WF}-9u55XDcG(w(K@QPC{*XoPhx*!CBtB-%${d&a6*5ntC^pK0G5tksy@W%o+O_wV8@3Y9($Wl@ zxi)+wocfkE^vk0AZ7wC-R~@hW_pW>cE#x5GW~%70_@J&of0s|t5ZhEaIWhZp%F zZ^yAqpDB<5|BWLWa`NW75X=D-v)+Y321eb#k7FD!gOBZRANfl2#-4TJX|JgSDtJ2z z9i`UtJqafTe;buCRZ9TW{h4o8r9=yG?Fzxb0G#nUREYQTJQE4X3T=kepAmJ};D^lB zhinhlsKA*k>}XOy4GI7H>3?0?ZUiOBV{SlT6o^D}J0CXa)Q8mf`*6ymg*=9LJW-({ zD$EA9Z&?uMJ*`q3HIWUuuNIUXPmg(M{B#A*Ub=lPY0bPUzAKli_&EE*@{D-K8G;0& zLWl)NFhlI-5kCFmZzzT_{PHX$-Hji;-vaFg?d>9g=vK|7Rant-pnCmO2$ZYE&+4tg zw{wcYv<;>V<)WaE&1Nlwp96*WuK>Hc+S1f^>t-T8&H-kP%G%s1+buAjJ@awl$uDULLl0@yv&dsUJCrbsq z(<|fcn}(g`np>||*u?imXsF9;L8d0EGnT~k0;`o1#6J;eiq=TO&U=YsTSfXc$*xt} z^Llne$c+pZFgC@dB3~6mE5~F!2|Kq+)x%O^cvo2`5_v1>dx&!N&uN`pB)MJLAoU

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