5-classification-DA

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Unformatted text preview: “large” & spee 18 These are called the linear discriminant functions. 2.2 Incorp orating prior probabilities C>D*'(*0>'(E If the prior probability for group k is pk then the discriminant functions ¯￿ ¯￿ ¯ Xk S−1 ed X0 − Xk S−1 ed Xk + log pk pool pool k=1 where log is F#)"&()"'@*A=A2+()"='*>(%"('$.*"#*(* away from the grou the natual log. This shifts the boundary Example: We might use the sample size in the olive oils, nS ard = 98, A%=6+.&0*<")?*(*+"&").3*'2&6.%*=;*3()(* bilities, pS ard = 98/249 = 0.39, pN th = 151/249 = 0.61. The result is to ch A="')#G −42.61(= −43.04 + log(151/98)). The training error is slightly better, 7/ R=+2)"='S*:?('@.*)?.*.#)"&()"='*&.)?=30* ('3*2#.*(*A.'(+"8.3*.#)"&().0* (1 − λ)S + λI, λ ∈ [0, 1] ¯ ¯ (Xk − X0 )￿ S−1 (Xk pooled 4 19 Statistics 503, Spring 2013, ISU a1 = β0 + β1 mxPH + β2 mnO2 + β3 Cl F$2-,'G$)*456 β7 Chla + τseason + 60 80 100 > library(penalizedLDA) > df5.plda <- PenalizedLDA(df5[,1:200], df5[,201], lambda=0.2, K=1, xte=df5.test[,1:200]) > hist(df5.plda$xproj, col="grey70", ylim=c(-10,+ β2 mnO2 + β3 Cl + β4 N a1 = β0 + β1 mxPH 100), xlab="PLD1", main="Penalized LDA") Penalized LDA β8 Chla + τseas > abline(h=0, lty=2) > text(df5.plda$xteproj, rep(-5, 40), label=df5.test$y) > table(df5.test$y,df5.plda$ypred) Frequency 12 1 20 0 2 0 20 0 20 40 a1 = β0 + β1 Cl + β2 oPO4 + β3 Chla + 2 ￿ a1 = 22 22 222 2 2 222 2 1 111111 1 1 11 11 5.27 − 0.41Cl − 0.69oPO4 − 0.55Ch -2 -1 0 1 2 PLD1 = 5.57 ISU Statistics 503, Spring 2013,− 0.41Cl − 0.69oPO4 − 0.55Ch = = 5.77 − 0.41Cl − 0.69oPO4 − 0.55Ch 20 5.76 − 0.41Cl − 0.69oPO4 − 0.55Ch Linear Discriminant Analysis F%1H$/0'12*(0$# LDA is based on the assumption that the data comes from a multivariate normal distribution with equal variance-covariance matrices. Comparing the density functions reduces the rule to: Y?.'*)?.%.*(%.*)<=*@%=2A#S Allocate a new observation, X0 to group 1 if 1 ¯ ¯ ¯ ¯ ¯ ¯ (X1 − X2)￿S−1 X0 − (X1 − X2)￿S−1 (X1 + X2) ≥ 0 pooled pooled 2 else allocate to g...
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