1、278Chapter 1111.1 (a) The linear discriminant function given in (11-19) isA (_ - )8-1 AIY = Xl - X2 pooled X = a XwhereSmoo = ( _: -: iso the the linear discriminant function is(: i - (: iH -: -: 1 z=-2 =-2Xi(b)A l(A A) l(AI AI) 8m = - Yl + Y2 = - a Xl + a X2 =-2 2Assign x to 11 ifYo = (2 7)xo r = -
2、8and assign Xo to 12 otherwise.Since (-2 O)xo = -4 is greater than r = -8, assign x to population 11-27911.2 (a) 11 = Riding-mower owners; 1T2 = NonownersHere are some summary statistics for the data in Example 11.1:Z - (I:,: 1 5, - ( : -I: 1 _(.276.675 -7.204 i8 pooled - ,-7.204 4.273Z2 - 1: 182 =
3、( 200.705 -2.589 1-2;589 4.464( .00378 AJ063718-1pooled =.00637 .24475The linear classification function for the data in Example 11.1 using (11-19)is( (109.475 i -( 87.400 i) i ( ,0037820.267 17.633 .00637where.006371 r Jx = L .100 .785 :.244751 1ri = 2“(Yl + Y2) = 2“(xi + X2) = 24.719280(b) Assign
4、an observation x to 11 if0.100x +0.785xi 24.72Otherwise, assign x to 12Here are the observations and their classifications:Owners NonownersObservation axo Classification Observation a/xo Classification1 23.44 nonowner 1 25.886 owner2 24.738 owner 2 24.608 nonowner3 26.436 owner 3 22.982 nonowner4 25
5、.478 owner 4 23.334 nonowner5 30.2261 owner 5 25.216 owner6 29.082 owner 6 21. 736 nonowner7 27.616 owner 7 21.500 nonowner8 28.864 owner 8 24.044 nonowner9 25.600 owner 9 20.614 nonowner10 28.628 owner 10 21.058 nonowner11 25.370 owner 11 19.090 nonowner12 26.800 owner 12 20.918 nonownerFrom this,
6、we can construct the confusion matrix:Actualmembership: jPredictedMembership11 1211 12 10Total1212(c) The apparent error rate is 1i2 = 0.125(d) The assumptions are that the observations from 7i and 72 are from multi-variate normal distributions;with equal covariance matrices, Li = L2 = .L.11.3 l,Ne
7、ned.t-o Shuw that the regiuns Ri and R2 that minimize the ECM are defid281by the values x for which the following inequalities hold:Ri : fi(x) ;: (C(lj2) (P2)h(x) - c(211) PiR2 : fiex) (cC112) (P2)h(x) c(211) PiSubstituting the expressions for P(211) and p(ij2) into (11-5) givesECM = c(211)Pi r fi()
8、dx + c(li2)p2 r h(x)dxJ R2 J RiAnd since n = Ri U R2,1 = r h(x)dx + r h(x)dx J Ri J R2and thus,ECM = c(211)Pi (1 - k.i fi(x)dx) + c(112)p2 i h(x)rixSince both of the integrals above are over the same. region, we haveECM = r (c(112)p2h(x)dx - c(21 l)pifi (x)ldx + c(21)PiJRiThe minimum is obtained whe
9、n Ri is chosen to be the regon where the term inbrackets is less than or equal to O. So choose Ri so thatc(211)pifi( x) ;: c(112)pd2(:i )Ur282h() )0 (C(112) ) (P2)h(x) - c(2j1) Pi11.4 (8) The minimum ECM rule is given by assigning an observation :i to 11 iffi() )0 (C(112) .(pi) = (100) () = .5h(x) -
10、 c(211) Pi 50.8and assigning x to 12 iffi(x) (C(112)(!) = .(100) (.2) = .5f2(x) c(211) Pi 50.8(b) Since fi(x) = .3 and f2(x) = .5,fi(x) = 6;: 5hex) . -and assign x to 1111.5 - (-1)t-1(-1) + (-2)lt1(:-2) =1 1 1 1 - 1 1+- 1 l +-1 1- 2(lr :-2:r +r, :i-t :+2:2+ :-2+ 21 i - 1 l l- 1 1,- 1 J= - 2(-2(:1-:2
11、) +l :1-:2 :2i -1 i ( ) i l- 1 ( )= (:1-:2) t : -2 :-:2 If :1+2.28311.6 a) E(II7ii) -aa = .:!:l - m = l!:i - l(i + !2J= 1 I (i - !2) = (!:i - !:2) i r i (i -!) 0 s ; neer1 is positive definite.b) E ( ,1 lir 2) - II = 1!:2 - m = l l (2 - 1)_ 1 ( ),-1 (- - 2 l - 2“ l - 2) 0 .11.7 (a.) Here are the den
12、sities:1.0 1.0R_1 -1/3 R_20.6 0.6x-0.2 0.2-0.2 R_1 1/4 R_2 -0.2-1.0 -0.5 0.0 0.5 1.0 1.5 -1.0 -0.5 0.0 0.5 1.0 1.5x x284(b) When Pi = P2 and c(112) = c(211), the classification regions areR . hex) 1i . hex) - !i(x)R2 : h (x) 1These regions are given by Ri : -1 x .25 and R2 : .25 x 1.5.(c.) When Pi =
13、 .2, P2 = .8, and c(112) = c(211), the clasification regions areRi : fi(x) ;: P2 = .4hex) - Pifiex)R2 : h (x) .4These regions are given by Ri : -1 x -1/3 and R2 : -1/3 x 1.5.11.8 (al Here are the densities:i.ciCci-,.ci,.ciiR_2 -1/2 R_1 1/6 R_2-1 o 1 2x(b) When Pi = P2 and c(112) = c(2Il), the classi
14、fication regions areR . h(x) ;: 11 . h(x) - !i(x)R2 : hex) .( 111.9285These regions are given byRi : -1/2 =: x 1/6 and R2 = -1.5 x -1/2, 1/6 x 2.5aB ,uaa/La!(1-)(1-) + (2-)(2-),J=a1ta- -hI, + ) Thus “_1 - u-_ = l(2. ll_l - U_2) and 11_2 - =w ere = 2 1 2. tt 2 - l ) soaB ,ua =a/La! I (1-2)(1-2) I Ial
15、a- -,2811.10 (a) Hotellngs two-sample T2-statistic isT2 - (:Vi - X2) f (i + n) Spooled J -i (Xi - X2)- (-3 - 2j (I + 112) l-: -: If L : I = 14.52Under Ho : l.i = 1J2, T2“, (ni + n2- 2)p F. . .+ 1 p,nl+n2-p-lni n2 - P -Since T2 = 14.52 ii-; F2,2o(.1) = 5.44, we reject the null hypothesisHo : J.i = J.
16、2 at the Q = 0.1 level of significance.(b) Fishers linear discriminant function isYo = xo = -.49Xi - .53x2(c) Here, m, = -.25. Assign x to 1i if -A9xi - .53x2 + .25 O. Otherwise. iassign Xo to 12.For x = (0 1), Yo = -.53(1) = -.53 and Yo - m = -.28 o. Thus, assignXo to 12.28711.11 Assuming equal pri
17、or probabilitis Pi = P2 = l, and equal misclasification costsc(2Il) = c(112) $10:Expectedc P(BlIA2) P(B2IAl) P(A2 and Bl) peAl and B2) P( error) cost9 .006 .691 .346 .D03 .349 3.4910 .023 .500 .250 .011 .261 2.6111 .067 .309 .154 .033 .188 1.8812 .159 .159 .079 .079 .159 1.59!13 .309 .067 .033 .154
18、.188 1.8814 .500 .023 .011 .250 .261 2.61Using (11- 5) ) the expected cost is minimized for c = 12 and the minimum expectedcost is $1.59.1i.2 Assuming equal prior probabiltiesPi = P2 = l, and misclassificationcosts c(2Il) =$5 and c(112) = $10,expected cost = $5P(A1 and B2) + $15P(A2 and B1).Expected
19、c P(BlIA2) P(B2/A1) P(A2 and Bl) P(AI and B2) P(error) cost9 0.006 0.691 0.346 0.003 0.349 1.7810 0.023 0.500 0.250 0.011 0.261 1.4211 0.067 0.309 0.154 0.033 0.188 1.2712 0.159 0.159 0.079 0.079 0.159 1.5913 0.309 0.067 0.033 0.154 0.188 2.4814 0.500 0.023 0.011 0.250 0.261 3.81 .Using (11- 5) , th
20、e expected cost is minimized for c = 10.90 and the minimumexpected cost is $1.27.11.13 Assuming prior probabilties peAl) = 0.25 and P(A2) = 0.715, and misoassIca-tion costs c(2Il) = $5 and c(lj2) = $10,expecte cost = $5PB2jAl)(.25) + $15P(BIIA2)(.75).288Expectedc P(Bl/A2) P(B2/A1) P(A2 and Bl) P(A1
21、and B2) P(error) cost9 0.006 0.691 0.173 0.005 0.178 0.9310 0.023 0.500 0.125 0.017 0.142 0.8811 0.067 0.309 0.077 0.050 0.127 1.1412 0.159 0.159 0.040 0.119 0.159 1.9813 0.309 0.067 0.017 0.231 0.248 3.5614 0.500 0.023 0.006 0.375 0.381 5.65Using (11- 5) , the expected cost is minimized for c = 9.8
22、0 and the minimumexpected cost is $0.88.11.14 Using (11-21), (. 79 1ai - - and m*i = -0.10A* - v - -.61Since xo = -0.14 ri = -0.1, classify Xo as 7i2Using (11-22),aA 2* - a -( 1.00 i and m; = -0.121 -.77Since ;xo = -0.18 m; = -0.12, classify Xo as 12.These results are consistent with the classificat
23、ion obtained for the case of equalprior probabilties in Example 11.3. These two clasification r.eults should beidentical to those of Example 11.3.11.15289f1 (xl (C(lIZl P2JfZ() l eT Pi defines the same region asrc(1IZ) PzJ1n fi() -In f2() l 1n Le-pi . For a multivariatenormal distribution1n f.(x) =
24、_12 ln It.1 _.22 ln 2rr - 21(x-ii,.)r(x-ii.), i=1,2, - 1 - - , -1so1 n f1 () - , n f 2 (:) = - (:-1) i 1 (:-, )1 ( ) ,+- , 1 ( I t i I)+ 2 -!:2 +2 (-Z) - 2 1n M_ 1 ( ,.,-1 + -1 , +-1- - i : “1 : - 2rl1 : + 1 “1 1, t - , 1 +- 1 ,- 1 1 ( U i/ )- 2 + 2!:212 - !:2+2 !:2) - 21n iW1 1(+-1 +-1) (,+-1 ,+-1)
25、= - 2 1 - 12 + 1+1 - 2“2 - kwhere 1k=21n(iii/) 1 i -1 , -1iW + I!i+1 1 - 2i2 2) .29011.16(f (X)JQ = In fi(x) = - i lnl+il - i(:-l) ti1 (-1)1 l -1+ 2 In!t21 + 2(-2) t, (-2)1 , (-1 t- 1 ) i +-, 1+- i= - -2 x +i -+2 X + X t II - _X 12 ll_Z - k - 11where k 1 (1 (I t ii ) 1- t-1 J= 2 n ii + , 11 i - 2T2
26、2 .When ti = h = t,Q i -i- 1 1+-1 1 ( i t- 1 1+-1)= l 1 -: +2 - 2 i T 1 - !:21 ZIt-()( 1+-1 = l 1 - LZ - 2 l:i - e2) l (1 +!:Z)11.17 Assuming equal prior probabilties and misclassification costs c2Ii) = $W andc(1/2) = $73.89. In the table below,1Q _ i (“(-i “(-i) (i “(-i i -i)- 2 Xo LJi - 2 Xo + J.i
27、 i - 112:E2 :to-l (IEil) _ ( i-l i -1 )2 n 121 2 1-1 i 1-1 - 1-22 1-2291x P(1ilx) P (12 I x) Q Clasification(10, 15) 1. 00000 0 18.54 1i(12, 17) 0.99991 0.00009 9.36 1i(14, 19) 0.95254 0.04745 3.00 11(16, 21) 0.36731 0.63269 -0.54 ii2(18, 23) 0.21947 0.78053 -1.27 12(20, 25) 0.69517 0.30483 0.87 1l2
28、(22, 27) 0.99678 0.00322 5.74 1i(24, 291 1. 00000 0.00000 13.46 1i(26, 31) 1. 00000 0.00000 24.01 1i(28, 331 1. 00000 0.00000 37.38 11(30, 35) 1.00000 0.00000 53.56 1iThe quadratic discriminator was used to classify the observations in the abovetable. An observation x is classified as 11 fQ In r(C(1
29、12) (P2)J = In (73.89) = 2.0L c(211) Pi 10Otherwise, classify x as 12.For (a), (b), (c) and (d), see the following plot.50400030 000Cx 20100o 10 20 30)L129211.18 The vector: is an (unsealed) eigenl.ector of ;-1B sincet-l t-l 1B: = t c(1-2)(1-2)IC+- (1-2)= c2t-l (1-2) (1-2) i t-1 (1-2)where= A t-1 (1
30、-2) = A :A = e2 (1-2) t-l (!:1-2) .11.19 (a) The calculated values agree with those in Example 11.7.(b) Fishers linear discriminant function isA AI 1 2Yo = a Xo = -Xl + -X23 3where17 10 27Yl = -; Y2 = -; r = - = 4.53 3 6Assign x to 1i if -lxi + X2 - 4.5 0Otherwise assign x to 12.1i 12Observation “i
31、Classification Observation -I Classificationa Xo - m a Xo - m1 2.83 11 1 -1.50 1122 0.83 1i 2 0.50 7(13 -0.17 12 3 -2.50 72293The results from this table verify the confusion matrix given in Example 11.7.(c) This is the table of squared distances Jt( x) for the observations, whereD;(x) = (x - xd8;oi
32、ed(X - Xi)11 12Obs. ,JI(x) J (x) Classification Obs. J (x) JH x ) Clasification1 i 21 1i 1 13 i 7f23 3 3 32 i J! 1i 2 l i 7fi3 3 333 4 3 12 3 19 4 7f23 3 3 3The classification results are identical to those obtained in (b)11.20 The result obtained from this matrix identity is identical to the result
33、 of Example11.7.11.23 (a) Here ar the normal probabiHty plotS for each of the vaables Xi,X2,Xa, X4,XS294-2 .1 o 2295-2 -1 0 2a_300 00 ocPx 250 200 00/0 0.2 -1 0 2 .2 -1 0 280 060IIx4020,i.III.ID.ooO 00.2 -1 0 1 2Standard Normal QuantilesVariables Xi, xa, and Xs appear to be nonnormaL. The transforma
34、tions Inxi) , In,(x3 + 1),and In(xs + 1) appear to slightly improve normality.(b) Using the original data, the linear discriminant function is:y = x = 0.023xi - O.034x2 + O.2lx3 - 0.08X4 - 0.25xswhereri = -23.23Thus, we allocate Xo to 1i (NMS group) if296xo - r = 0.023xi - 0.034x2 + 0.2lx3 - 0.08X4
35、- 0.25xs + 23.23 ;: 0Otherwise, allocate Xo to 12 (MS group).( c) Confusion matrix:ActualmembershipAPER= 69 = .102; jPredictedMembership1i 1266 37 22This is the holdout confusion matrix:Actualmembership(AER) = 69 = .133; jPredictedMembership,1i 1264 58 21Totalt Totalt :11.24 (a) Here are the scatter
36、plots for the pairs of observations (xi, X2),tXi, X3), andXl X4):297+0.1 0 bankrupt 0 Q. +* +:+ nonbankrupt +it+ +0.0 + +lt +o 0.0.1 + ceC)( 0-0.2 0-0.3 00 0-0.4-0.6 -0.4 -0.2 0.0 0.2 0.4 0.65 + +4 +3 + +C“)( + +; +2 +0 + + 00+ +0oOi 8(31 0 000 +0 +-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60.80 0 J + +0.6 0 +-a
37、 + + +)( 0 + +0.4 0 o C + +0 + + q.+ 0 0 0 0 Ll0.2 0 0 0+ +-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6x1The data in the above plot appear to form fairly ellptical shapes, so bivaatenorma1ty -does not seem like an unreasonable asumption.298(b) 11 = bankrupt firms, 12 = nonbankrupt firms. For (Xi,X2):( -0,0688 i (
38、 0,0442 0.02847 JXi - 8i -0.0819 0.02847 0.02092X2 -( 0.2354 i 82 -lOM735 0.O37 J0.0551 0.00837 0.00231(c), (d), (e) See the tables of part (g)(f)( 0.045948 pooled =0.017510.01751 J0.01077Fishers linear discriminant function isy = x = -4.67xi - 5.l2x2wherer = -.32Thus, we allocate Xo to 1i (Bankrupt
39、 group) ifxo - r = -4.67xi - 5.12x2 + .32 0Otherwise, allocate Xo to 12 (Nonbankrupt group).APER= :6 = .196.299Since 8i and 82 look quite different, Fishers linear discriminant functionFor the various classification rules and error rates for these variable pairs, seethe following tables.This is the
40、table of quadratic functions for the variable pairs .(Xb X2),Xb X3),and (Xb xs), both with Pi = 0.5 and Pi = 0.05. The classification rule for anyof thee functions is to classify a new observation into 1ii (bankrupt firms)if the quadratic function is 0, and to classify the new observation into30012
41、(nonbankrupt firms) otherwise. Notice in the table below that only theconstant term changes when the prior probabilties hange.Variables Prior Quadratic functionPi = 0.5 -61.77xi + 35.84xiX2 + 407.20x + .s.64xi - 30.60X2 - 0.17(Xi,X2) Pi = 0.05 - 3.11Pi = 0.5 -i.55x + 3.S9xiXa - 3.08x3 - 10.69xi + 7.
42、9xa - 3.14(xi, Xa) Pi = 0.05 - 6.08(Xl, X4)Pi = 0.5 -0.46xf. + 7.75xiX4 + 8.43x - 10.05xi - 8.11x4 + 2.23Pi = 0.05 - 0.71Here is a table of the APER and (AER) for the various variable pairs andprior probabilties.APER (APR)Variables Pi = 0.5 Pi = 0.05 Pi = 0.5 Pi = 0.05(Xi, X2) 0.20 0.26 0.22 0.26(Xi
43、, xa) 0.11 0.37 0.13 0.39(Xi, X4) 0.17 0.39 0.22 o ,4t)For equal priors, it appears that the (Xl, Xa) vaiable pair is the best clasifer,as it has the lowest APER. For unequal priors, Pi = 0.05 and P2 = 0.95, thevariable pair (xi, X2) has the lowet APER.301(h) When using all four variables (Xb X2l X3
44、, X4),-0.0688 0.04424 0.02847 0.03428 0.00431-0.0819 0.02847 0.02092 0.0258D () .00362Xi - , 8i -1.3675 0.03428 0.02580 0.16455 iJ.0330u0.4368 0.00431 0.00362 0.03300 0.044410.2354 0.04735 0.00837 0.07543 -u.006620.0551 0.00837 0.u023l 0.00873 D.0003lX2 - , 82 -2.5939 0.07543 0.00873 1 :04596 0.0317
45、70.4264 -0.00662 0.00031 0.03177 0.02618Assign a new observation Xo to 1i if its quadratic function .given below is lessthan 0:Prior Quadratic function-49.232 -20.657 -2.623 14.050 4.91-20.657 526.336 11.412 -52.493 -28.42Pi = 0.5 x xo+ Xo- 2.690-2.623 11.412 -3.748 1.4337 8.6514.050 -52.493 1.434 1
46、1.974 -11.80Pi = 0.05- 5.64For Pi = 0.5 : APER = ;6 = .07, (AER) = ;6 = .11For Pi = D.n5 : APER = :6 = .20, (AER) = = .2430211.25 (a) Fishers linear discriminant function isYo = a Xo - r = -4.80xi - 1.48xg + 3.33Classify Xo to 1i (bankrupt firms) ifa Xo - r ;: 0Otherwise classify Xo to 12 (nonbankru
47、pt firms).The APER is 2:l4 = .13., This is the scatterplot of the data in the (xi, Xg) coordinate system, alongwith the discriminant line.543Cx21-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6x1(b) With data point 16 for the bankrupt firms delete, Fishers linear discrimit303function is given byYo = aa;O - m = -5.93x
48、i - 1.46x3 + 3.31Classify Xo to1i (bankrupt firms) ifaxo - m, 2: 0Otherwise classify Xo to 12 (nonbankrupt firms).The APER is 1;4 = .11.With data point 13 for the nonbankrupt firms deleted, Fishers linear dis-criminant function is given byYo = axo - m = -4.35xi - i.97x3 + 4.36Classify Xo to 1i (bankrupt firms) ifa/:.o - m ;: 0Otherwise classif
