基于Fisher準(zhǔn)則線性分類器設(shè)計(jì)

1、基于 Fisher 準(zhǔn)則線性分類器設(shè)計(jì)專 業(yè)：電子信息工程學(xué)生姓名：李子龍學(xué) 號(hào)： 2一、實(shí)驗(yàn)類型設(shè)計(jì)型：線性分類器設(shè)計(jì)（ Fisher 準(zhǔn)則）二、實(shí)驗(yàn)?zāi)康谋緦?shí)驗(yàn)旨在讓同學(xué)進(jìn)一步了解分類器的設(shè)計(jì)概念， 能夠根據(jù)自己的設(shè)計(jì)對(duì)線性分類器有 更深刻地認(rèn)識(shí)， 理解 Fisher 準(zhǔn)則方法確定最佳線性分界面方法的原理， 以及 Lagrande 乘子 求解的原理。三、實(shí)驗(yàn)條件matlab 軟件四、實(shí)驗(yàn)原理線性判別函數(shù)的一般形式可表示成g (X) WTX Wo其中XiXXd根據(jù)FiSher選擇投影方向W的原則，即使原樣本向量在該方向上的投影能兼顧類間分JF(W)(m m2)22S12布盡可能分開，類內(nèi)樣本投

2、影盡可能密集的要求，用以評(píng)價(jià)投影方向W的函數(shù)為：* 1WSw (mi m2)上面的公式是使用 FiSher準(zhǔn)則求最佳法線向量的解，該式比較重要。另外，該式這種 形式的運(yùn)算，我們稱為線性變換，其中mi m2式一個(gè)向量，SWI是SW的逆矩陣，如m1 m2 是d維，SW和SWI都是d× d維，得到的 W*也是一個(gè)d維的向量。向量W*就是使FiSher準(zhǔn)則函數(shù)JF(W)達(dá)極大值的解，也就是按FiSher準(zhǔn)則將d維X 空間投影到一維 Y空間的最佳投影方向， 該向量W*的各分量值是對(duì)原 d維特征向量求加權(quán) 和的權(quán)值。以上討論了線性判別函數(shù)加權(quán)向量W的確定方法，并討論了使 FiSher準(zhǔn)則函數(shù)極大

3、的d維向量W*的計(jì)算方法，但是判別函數(shù)中的另一項(xiàng)WO尚未確定，一般可采用以下幾種方法確定W0如W)¾m2或者W0N而IN2m2Ni N2或當(dāng)p()1與p( )2已知時(shí)可用Womi 2In p( 1)/ p( 2)2N1 N2 2當(dāng)W確定之后，則可按以下規(guī)則分類，WTXWoX1WTXWoX2使用FiSher準(zhǔn)則方法確定最佳線性分界面的方法是一個(gè)著名的方法，盡管提出該方法 的時(shí)間比較早，仍見有人使用。五、實(shí)驗(yàn)內(nèi)容已知有兩類數(shù)據(jù)1和2二者的概率已知 p( )1=O6， p( )2 =0.4 。1中數(shù)據(jù)點(diǎn)的坐標(biāo)對(duì)應(yīng)一一如下：數(shù)據(jù)：x1 =0.23311.52070.64990.77571.0

4、5241.19740.29080.25180.66820.56220.90230.1333-0.54310.9407-0.21260.0507-0.08100.73150.33451.0650 -0.02470.10430.31220.66550.5838 1.1653 1.2653 0.8137 -0.3399 0.51520.7226x2 =-0.20150.4070-0.1717-1.0573-0.20992.33852.19461.67301.63651.78442.01552.06812.12132.47971.51181.96921.83401.87042.29481.77142.

5、39391.56481.93292.20272.45681.75231.69912.48831.72592.04662.02262.37571.79872.08282.07981.94492.38012.23732.16141.92352.2604x3 =0.53380.85141.08310.41641.11760.55360.60710.44390.49280.59011.09271.07561.00720.42720.43530.98690.48411.09921.02990.71271.01240.45760.85441.12750.77050.41291.00850.76760.84

6、180.87840.97510.78400.41581.03150.75330.95482 數(shù)據(jù)點(diǎn)的對(duì)應(yīng)的三維坐標(biāo)為x1 =1.4010 1.2301 2.0814 1.1655 1.3740 1.18291.76321.97392.41522.58902.84721.95391.25001.28641.26142.00712.18311.79091.33221.14661.70871.59202.93531.46642.93131.83491.83402.50962.71982.31482.03532.60301.23272.14651.56732.9414x2 =1.02980.96110

7、.91541.49010.82000.93991.14051.06780.80501.28891.46011.43340.70911.29421.37440.93871.22661.18330.87980.55920.51500.99830.91200.71261.28331.10291.26800.71401.24461.33921.18080.55031.47081.14350.76791.1288x3 =0.62101.36560.54980.67080.89321.43420.95080.73240.57841.49431.09150.76441.21591.30491.14080.9

8、3980.61970.66031.3928 1.4084 0.6909 0.8400 0.5381 1.37290.77310.73191.3439 0.81420.9586 0.73791.36991.14580.7548 0.7393 0.6739 0.8651數(shù)據(jù)的樣本點(diǎn)分布如下圖:*22511110.50.5-23圖1 :樣本點(diǎn)分布圖六、實(shí)驗(yàn)要求1）請(qǐng)把數(shù)據(jù)作為樣本，根據(jù) Fisher 選擇投影方向 W 的原則，使原樣本向量在 該方向上的投影能兼顧類間分布盡可能分開， 類內(nèi)樣本投影盡可能密集的要 求，求出評(píng)價(jià)投影方向 W 的函數(shù)，并在圖形表示出來(lái)。并在實(shí)驗(yàn)報(bào)告中表 示出來(lái)，并求使 J

9、F （w） 取極大值的 w* 。用 matlab 完成 Fisher 線性分類器 的設(shè)計(jì)，程序的語(yǔ)句要求有注釋。2）根據(jù)上述的結(jié)果并判斷 （1，1.5 ，0.6 ）（1.2 ，1.0 ，0.55） ，（2.0 ，0.9 ，0.68） ， （1.2 ， 1.5 ，0.89） ，（0.23 ，2.33 ，1.43 ），屬于哪個(gè)類別，并畫出數(shù)據(jù)分類 相應(yīng)的結(jié)果圖，要求畫出其在 W 上的投影。3）回答如下問題， 分析一下 W 的比例因子對(duì)于 Fisher 判別函數(shù)沒有影響的原 因。七、實(shí)驗(yàn)結(jié)果1、源代碼x1=0.23311.52070.64990.77571.05241.19740.29080.251

10、80.66820.56220.90230.1333-0.54310.9407-0.21260.0507-0.08100.73150.33451.0650-0.02470.10430.31220.66550.58381.16531.26530.8137-0.33990.51520.7226-0.20150.4070-0.1717-1.0573-0.2099'y1=2.33852.19461.67301.63651.78442.01552.06812.12132.47971.51181.96921.83401.8704 2.2948 1.7714 2.3939 1.5648 1.93292

11、.2027 2.4568 1.7523 1.6991 2.4883 1.72592.04662.02262.37571.79872.08282.07981.94492.38012.23732.16141.92352.2604'z1=0.53380.85141.08310.41641.11760.55360.60710.44390.49280.59011.09271.07561.00720.42720.43530.98690.48411.09921.02990.71271.01240.45760.85441.12750.77050.41291.00850.76760.84180.8784

12、0.97510.78400.41581.03150.75330.9548'存儲(chǔ)第一類點(diǎn)x2=1.40101.23012.08141.16551.37401.18291.76321.97392.41522.58902.84721.95391.25001.28641.26142.00712.18311.79091.33221.14661.70871.59202.93531.46642.93131.83491.83402.50962.71982.31482.03532.60301.23272.14651.56732.9414'y2=1.02980.96110.91541.49010.

13、82000.93991.14051.06780.80501.28891.46011.43340.70911.29421.37440.93871.22661.18330.87980.55920.51500.99830.91200.71261.28331.10291.26800.71401.24461.33921.1808 0.5503 1.4708 1.1435 0.7679 1.1288'z2=0.6210 1.3656 0.5498 0.6708 0.8932 1.43420.95080.73240.57841.49431.09150.76441.21591.30491.14080.

14、93980.61970.66031.39281.40840.69090.84000.53811.37290.77310.73191.34390.81420.95860.73790.75480.73930.67390.86511.36991.1458'存儲(chǔ)第二類點(diǎn)Pw1=0.6Pw2=0.4%求第一類點(diǎn)的均值向量m1m1x=mean(x1(:)%全部平均m1y=mean(y1(:)%全部平均m1z=mean(z1(:)%全部平均m1=m1xm1ym1z%求第二類點(diǎn)的均值向量m2m2x=mean(x2(:)%全部平均m2y=mean(y2(:)%全部平均m2z=mean(z2(:)%全部平

15、均m2=m2xm2ym2z%求第一類類內(nèi)離散矩陣 S1S1=zeros(3,3)for i=1:36S1=S1+(x1(i),y1(i),z1(i)'-m1)*(x1(i),y1(i),z1(i)'-m1)'end%求第二類類內(nèi)離散矩陣 S2S2=zeros(3,3)for i=1:36S2=S2+(x2(i),y2(i),z2(i)'-m2)*(x2(i),y2(i),z2(i)'-m2)'end%求總類內(nèi)離散度矩陣 SwSw=S1+S2%求向量 W*W=(inv(Sw)*(m1-m2)%畫出決策面x=0:.1:2.5y=0:.1:3X,Y=m

16、eshgrid(x,y)Z=(W(1)*X+W(2)*Y)/(-W(3)mesh(X,Y,Z)%保持hold on%透視決策面hidden off%求第一類樣品的投影值均值Y1=0for i=1:36Y1=Y1+W'*x1(i),y1(i),z1(i)' endM1=Y1/36 %求第二類樣品的投影值均值Y2=0 for i=1:36Y2=Y2+W'*x2(i),y2(i),z2(i)' endM2=Y2/36 %選取閾值 Y0Y0=(M1+M2)/2+(log(Pw1)/log(Pw2)/70 %判定未知樣品類別X1=1,1.5,0.6'if W

17、9;*X1>Y0disp(' 點(diǎn) X1(1,1.5,0.6)屬于第一類 ' )plot3(1,0.5,0.6,'or'elsedisp(' 點(diǎn) X1(1,1.5,0.6)屬于第二類 ' )plot3(1,0.5,0.6,'ob'endX2=1.2,1.0,0.55' if W'*X2>Y0disp(' 點(diǎn) X2(1.2,1.0,0.55)屬于第一類 ' )plot3(1.2,1.0,0.55,'or' )elsedisp(' 點(diǎn) X2(1.2,1.0,0.55)屬

18、于第二類')plot3(1.2,1.0,0.55,'ob'endX3=2.0,0.9,0.68'if W'*X3>Y0disp(' 點(diǎn) X3(2.0,0.9,0.68)屬于第一類')plot3(2.0,0.9,0.68,'or'elsedisp(' 點(diǎn) X3(2.0,0.9,0.68)屬于第二類')plot3(2.0,0.9,0.68,'ob'endX4=1.2,1.5,0.89'if W'*X4>Y0disp(' 點(diǎn) X4(1.2,1.5,0.89)屬于

19、第一類')plot3(1.2,1.5,0.89,'or'elsedisp(' 點(diǎn) X4(1.2,1.5,0.89)屬于第二類')plot3(1.2,1.5,0.89,'ob'endX5=0.23,2.33,1.43'if W'*X5>Y0disp(' 點(diǎn) X5(0.23,2.33,1.43)屬于第一類 ' )plot3(0.23,2.33,1.43,'or' )elsedisp(點(diǎn) X5(023,2.33,143)plot3(023,2.33,143,屬于第二類')'o

20、b')end2、決策面1510圖2 :決策面(紅色代表第一類，藍(lán)色代表第二類)3、參數(shù)決策面向量W =-0.07980.2005-0.0478閾值Y0 =0.1828樣本點(diǎn)分類X1 =1.00001.50000.6000點(diǎn) X1(1,1.5,0.6) 屬于第一類X2 =1.20001.00000.5500點(diǎn) X2(1.2,1.0,0.55) 屬于第二類X3 =2.00000.90000.6800點(diǎn) X3(2.0,0.9,0.68) 屬于第二類X4 =1.20001.50000.8900點(diǎn) X4(1.2,1.5,0.89) 屬于第二類X5 =0.23002.33001.4300點(diǎn) X5(0.23,2.33,1.43) 屬于第一