Nowadays, the observations in many scientific fields, including biological sciences, are often high dimensional, meaning the number of variables exceeds the number of samples. One of the problems in model-based clustering of these data types is the estimation of too many parameters. To overcome this problem, the dimension of data must be first reduced before clustering, which can be done through dimension reduction methods. In this context, a recent approach that is recently receiving more attention is the random Projections method. This method has been studied from theoretical and practical perspectives in this paper. Its superiority over some conventional approaches such as principal component analysis and variable selection method was shown in analyzing three real data sets.

This paper considers series and parallel systems with independent and identically distributed component lifetimes. The reliability of these systems can be improved by using the reduction method. In the reduction method, system reliability is increased by reducing the failure rates of some of its components by a factor 0<ρ<1, called the equivalent reliability factor. Closed formulas are obtained for some reliability equivalence factors. In comparisons among the performance of the systems, these factors are helpful. We discuss that the reduction method can be considered as a particular case of the proportional hazard rates (PHR) model. Sufficient conditions for the relative aging comparison of the improved series and parallel systems under the PHR model and reduction method are also developed.