Log-log plot

Abstract: In this paper, we show that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is a universal approximator of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named log-sum-exp ( $\mathrm {LSE}_{T}$ ). Under a suitable exponential transformation, the class of $\mathrm {LSE}_{T}$ functions maps to a family of generalized posynomials $\mathrm {GPOS}_{T}$ , which we similarly show to be universal approximators for log-log-convex functions. A key feature of an $\mathrm {LSE}_{T}$ network is that, once it is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a $\mathrm {GPOS}_{T}$ model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using geometric programming (GP). The proposed methodology is illustrated by two numerical examples, in which, first, models are constructed from simulation data of the two physical processes (namely, the level of vibration in a vehicle suspension system, and the peak power generated by the combustion of propane), and then optimization-based design is performed on these models.

Abstract: The PLP (power-law process) or the Duane model is a simple model that can be used for both reliability growth and reliability deterioration. GOF (goodness-of-fit) tests for the PLP have attracted much attention. However, the practical use of the PLP model is its graphical analysis or the Duane plot, which is a log-log plot of the cumulative number of failures versus time. This has been commonly used for model validation and parameter estimation. When a plot is made, and the coefficient of determination, R/sup 2/, of the regression line is computed, the model can be tested based on this value. This paper introduces a statistical test, based on this simple procedure. The distribution of R/sup 2/ under the PLP hypothesis is shown not to depend on the true model parameters. Hence, it is possible to build a statistical GOF test for the PLP. The critical values of the test depend only on the sample size. Simulations show that this test is reasonably powerful compared with the usual PLP GOF tests. It is sometimes more powerful, especially for deteriorating systems. Implementing this test needs only the computation of a coefficient of determination. It is much easier than, for example, computing an Anderson-Darling statistic. Further study is needed to compare more precisely this new test with the existing ones. But the R/sup 2/ test provides a very simple and useful objective approach for decision making with regard to model validation.

Abstract: In this paper, we propose a new distribution, named unit log–log distribution, defined on the bounded (0,1) interval. Basic distributional properties such as model shapes, stochastic ordering, quan...