In statistical methods, such as statistical static timing analysis, Gaussian mixture model (GMM) is a useful tool for representing a non-Gaussian distribution and handling correlation easily. In order to repeat various statistical operations such as summation and maximum for GMMs efficiently, the number of components should be restricted around two. In this paper, we propose a method for reducing the number of components of a given GMM to two (2-GMM). Moreover, since the distribution of each component is represented often by a linear combination of some explanatory variables, we propose a method to compute the covariance between each explanatory variable and the obtained 2-GMM, that is, the sensitivity of 2-GMM to each explanatory variable. In order to evaluate the performance of the proposed methods, we show some experimental results. The proposed methods minimize the normalized integral square error of probability density function of 2-GMM by the sacrifice of the accuracy of sensitivities of 2-GMM.

In statistical approaches such as statistical static timing analysis, the distribution of the maximum of plural distributions is computed by repeating a maximum operation of two distributions. Moreover, since each distribution is represented by a linear combination of several explanatory random variables so as to handle correlations efficiently, sensitivity of the maximum of two distributions to each explanatory random variable, that is, covariance between the maximum and an explanatory random variable, must be calculated in every maximum operation. Since distribution of the maximum of two Gaussian distributions is not a Gaussian, Gaussian mixture model is used for representing a distribution. However, if Gaussian mixture models are used, then it is not always possible to make both variance and covariance of the maximum correct simultaneously. We propose a new algorithm to determine covariance without deteriorating the accuracy of variance of the maximum, and show experimental results to evaluate its performance.