3.2 Two Specific Control Variates 5 1.The first idea is to utilize the error rate computed by LDA as erroree based on the assumption of equal covariance matrices for all the groups.The error rate error is calculated by QDA from N random realizations drawn from the group densities.To get Emc(error)we generate W such error rates and average.Therefore we used N x W random vectors.Now we take the same random vectors and apply LDA with the same,so-called pooled,covariance matrix for all groups to calculate error rates erroree.If Ei is the assumed covariance matrix for group i,then ((N-1))/(N-g)is the pooled covariance matrix,where M is the number of realizations in group i.The differences of the w corresponding estimates error and errorc are used to calculate E(error-errore).At last we calculate E(errorce)in an exact manner (so that we have no variance)by numerical integration based on the densities with pooled covariance matrices.We now have all the ingredients we need for an efficiency comparison with the naive Monte Carlo estimator.The variance of the naive estimator is calculated by the sample of size W of the estimated error rates error and the variance of the control variate estimator by the sample of size W of(error-error).This approach has the drawback that we have to calculate an exact integral in a projection space which might be two dimensional or of even higher dimension with rather ugly borderlines. 2.A second possibility to determine the error rate error is to use another con- trol variate:the error rate of an "optimal"one dimensional projection.This can be obtained by the largest eigenvalue and the corresponding eigenvector of QDA in the original space or by direct minimization of the error rate.We do the same as in 1 to obtain Evc(error).But then we project the random vectors onto the optimally discriminating direction taking into account the different covariance structures and build the differences of corresponding er- ror estimates to compute E(error-error).Now,the exact calculation of E(error)is simply a one dimensional integration with clearcut intersection points.This speeds up the procedure compared to 1.To construct the opti- mally discriminating one dimensional projection we follow an idea in [5]where it was proposed to project on the first eigenvector v of MMT,where M=(h-4,,2-41,2g-1,,2-) (12) where the i are the group means and the Ei are (again)the group covariance matrices,i=1,...,g.The projected means,variances and feature vectors then have the form:=i=ofiv and x*=of'r. In order to represent adequate control variates the additional computation time of procedures 1 and 2 have to be small relative to the computation time of naive Monte Carlo.That this is the case not considering the computation of the exact expected error rates should be clear by the following arguments. Naive Monte Carlo estimates the means and the covariance matrices of the groups,and evaluates the corresponding estimated group densities for each observation.3.2 Two Specic Control Variates 5 1. The rst idea is to utilize the error rate computed by LDA as errorcv based on the assumption of equal covariance matrices for all the groups. The error rate error is calculated by QDA from N random realizations drawn from the group densities. To get E^ MC (error) we generate W such error rates and average. Therefore we used N W random vectors. Now we take the same random vectors and apply LDA with the same, so-called pooled, covariance matrix for all groups to calculate error rates errorcv. If i is the assumed covariance matrix for group i, then (Pg i=1(Ni ￾ 1)i)=(N ￾ g) is the pooled covariance matrix, where Ni is the number of realizations in group i. The dierences of the W corresponding estimates error and errorcv are used to calculate E( ^ error ￾ errorcv). At last we calculate E(errorcv) in an exact manner (so that we have no variance) by numerical integration based on the densities with pooled covariance matrices. We now have all the ingredients we need for an eciency comparison with the naive Monte Carlo estimator. The variance of the naive estimator is calculated by the sample of size W of the estimated error rates error and the variance of the control variate estimator by the sample of size W of (error ￾ errorcv). This approach has the drawback that we have to calculate an exact integral in a pro jection space which might be two dimensional or of even higher dimension with rather ugly borderlines. 2. A second possibility to determine the error rate error is to use another con￾trol variate: the error rate of an "optimal" one dimensional pro jection. This can be obtained by the largest eigenvalue and the corresponding eigenvector of QDA in the original space or by direct minimization of the error rate. We do the same as in 1 to obtain E^ MC (error). But then we pro ject the random vectors onto the optimally discriminating direction taking into account the dierent covariance structures and build the dierences of corresponding er￾ror estimates to compute E( ^ error ￾ errorcv). Now, the exact calculation of E(errorcv) is simply a one dimensional integration with clearcut intersection points. This speeds up the procedure compared to 1. To construct the opti￾mally discriminating one dimensional pro jection we follow an idea in [5] where it was proposed to pro ject on the rst eigenvector v1 of MMT , where M = ( g ￾ 1; :::; 2 ￾ 1; g ￾ 1; :::; 2 ￾ 1) (12) where the i are the group means and the i are (again) the group covariance matrices, i = 1, ..., g. The pro jected means, variances and feature vectors then have the form:  i = v T 1 i, i = v T 1 iv1 and x = v T 1 x . In order to represent adequate control variates the additional computation time of procedures 1 and 2 have to be small relative to the computation time of naive Monte Carlo. That this is the case not considering the computation of the exact expected error rates should be clear by the following arguments.  Naive Monte Carlo estimates the means and the covariance matrices of the groups, and evaluates the corresponding estimated group densities for each observation.