Tham khảo tài liệu 'advanced biomedical engineering part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Multivariate Models and Algorithms for Learning Correlation Structures from Replicated Molecular Profiling Data 51 Fig. 2. Comparison of the multivariate blind-case model and bivariate Pearson s correlation estimator. In the figure the x-axis corresponds to data quality and y-axis represents MSE ratio which is the ratio MSE from Pearson s estimator MSE from blind-case model. Pair of genes each with 4 replicated measurements across 20 samples were considered in the comparison. The between molecular correlation parameter rho was set at low and medium respectively. the unconstrained EM algorithm presented above may not necessarily converge to the MLE Ỹ. To reduce various problems associated with the convergence of EM algorithm remedies have been proposed by constraining the eigenvalues of the component correlation matrices Ingrassia 2004 Ingrassia Rocci 2007 . For example the constrained EM algorithm presented in Ingrassia 2004 considers two strictly positive constants a and b such that a b c where c Ễ 0 1 . In each iteration of the EM algorithm if the eigenvalues of the component correlation matrices are smaller than a they are replaced with a and if they greater than b they are replaced with b. Indeed if the eigenvalues of the component correlation matrices satisfy a Xj Lj b for i 1 2 j 1 2 . i 1 mi then the condition Amin S1 L-1 c Hathaway 1985 is also satisfied and results in constrained global maximization of the likelihood. 5. Results Simulations In this section we evaluate the performance of multivariate and bivariate correlation estimators using synthetic replicated data. In Figure 2 we compare multivariate blind-case model and bivariate Pearson s correlation estimator by simulating 1000 synthetic data sets corresponding to a pair of genes each with 4 replicated measurements and 20 observations. 52 Advanced Biomedical Engineering Fig. 3. Comparison of the multivariate blind-case model and informed-case model with increasing data quality and sample
