Various exact methods, as well as approximate methods, have been proposed.
Using statistical distributions rather than distance metrics to define clusters helps in evaluating whether a model with a particular number of clusters is able to fit the data, since tests can be performed to observed (ni) versus model expected values (mi), using exact methods as recommended [ 18, 19].
Comparison of several means was performed using analysis of variance and the Newman-Keuls post-hoc test, using the Kruskall Wallis test with Bonferroni correction, or using Fisher's exact method with Bonferroni correction, as appropriate.
For different correlation ratios we evaluated the control of the Type-I error, the power, the Mean Square Error (MSE) of the estimated p v a l u e (p v a l u e from the exact method was used as a reference), and the rate of good decision (same decision as for the exact method).
Twelve numerical examples are solved using the ∈-constraint method as an exact method, the designed AMOSA and NSGA-II as a prominent multi-objective genetic algorithm in order to investigate the efficiency of the designed AMOSA algorithm.
For calculation of the 95% CIs, either the normal approximation or the exact method was used as appropriate.
Statistical evaluations were carried out with the Fisher's Exact Test, the Clopper-Pearson method, as well as a Proportional Odds Model.
Then the exact method of omitting genes is as below.
We first discuss the application of the adaptive epsilon-constraint method as an exact method for solving this model.
At first, to investigate the performance of the developed AMOSA algorithm to reach optimal Pareto front, the efficient solutions obtained by ∈-constraint method as an exact method for the three numerical examples are compared with the best Pareto solutions obtained by AMOSA.
