Error Amplification in Failure Probability Estimates of Small Errors in Response Surface Approximations

2007 ◽  
Author(s):  
Palaniappan Ramu ◽  
Nam H. Kim ◽  
Raphael T. Haftka
Keyword(s):  
Response Surface ◽  
Failure Probability ◽  
Response Surface Approximations ◽  
Probability Estimates ◽  
Error Amplification

FORWARD APPORTIONMENT OF CENSORED COUNTS FOR DISCRETE NONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION OF FAILURE PROBABILITIES

2009 ◽  
Vol 16 (03) ◽  
pp. 213-234
Author(s):  
MINNIE H. PATEL ◽  
H.-S. JACOB TSAO
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Closed Form ◽  
Failure Probability ◽  
Likelihood Estimation ◽  
Lifetime Distribution ◽  
Nonparametric Maximum Likelihood ◽  
Kaplan Meier ◽  
Probability Estimates ◽  
Failure Probabilities

Empirical cumulative lifetime distribution function is often required for selecting lifetime distribution. When some test items are censored from testing before failure, this function needs to be estimated, often via the approach of discrete nonparametric maximum likelihood estimation (DN-MLE). In this approach, this empirical function is expressed as a discrete set of failure-probability estimates. Kaplan and Meier used this approach and obtained a product-limit estimate for the survivor function, in terms exclusively of the hazard probabilities, and the equivalent failure-probability estimates. They cleverly expressed the likelihood function as the product of terms each of which involves only one hazard probability ease of derivation, but the estimates for failure probabilities are complex functions of hazard probabilities. Because there are no closed-form expressions for the failure probabilities, the estimates have been calculated numerically. More importantly, it has been difficult to study the behavior of the failure probability estimates, e.g., the standard errors, particularly when the sample size is not very large. This paper first derives closed-form expressions for the failure probabilities. For the special case of no censoring, the DN-MLE estimates for the failure probabilities are in closed forms and have an obvious, intuitive interpretation. However, the Kaplan–Meier failure-probability estimates for cases involving censored data defy interpretation and intuition. This paper then develops a simple algorithm that not only produces these estimates but also provides a clear, intuitive justification for the estimates. We prove that the algorithm indeed produces the DN-MLE estimates and demonstrate numerically their equivalence to the Kaplan–Meier-based estimates. We also provide an alternative algorithm.

scholarly journals Gaussian Process-Based Response Surface Method for Slope Reliability Analysis

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Bin Hu ◽  
Guo-shao Su ◽  
Jianqing Jiang ◽  
Yilong Xiao
Keyword(s):  
Gaussian Process ◽  
Reliability Analysis ◽  
Response Surface ◽  
Response Surface Method ◽  
Failure Probability ◽  
Limit State ◽  
State Function ◽  
Limit State Function ◽  
Surface Method ◽  
Gp Model

A new response surface method (RSM) for slope reliability analysis was proposed based on Gaussian process (GP) machine learning technology. The method involves the approximation of limit state function by the trained GP model and estimation of failure probability using the first-order reliability method (FORM). A small amount of training samples were firstly built by the limited equilibrium method for training the GP model. Then, the implicit limit state function of slope was approximated by the trained GP model. Thus, the implicit limit state function and its derivatives for slope stability analysis were approximated by the GP model with the explicit formulation. Furthermore, an iterative algorithm was presented to improve the precision of approximation of the limit state function at the region near the design point which contributes significantly to the failure probability. Results of four case studies including one nonslope and three slope problems indicate that the proposed method is more efficient to achieve reasonable accuracy for slope reliability analysis than the traditional RSM.

Determination of the Optimal Locations for Injection Molding Gates with Higher Order Response Surface Approximations

2013 ◽  
Vol 479-480 ◽  
pp. 126-130 ◽  
Author(s):  
Kun Nan Chen ◽  
Wen Der Ueng
Keyword(s):  
Injection Molding ◽  
Pressure Distribution ◽  
Response Surface ◽  
Injection Pressure ◽  
Optimization Method ◽  
Higher Order ◽  
Programming Technique ◽  
Location Optimization ◽  
Gate Location ◽  
Response Surface Approximations

This paper proposed a gate location optimization scheme to minimize the maximum injection pressure in plastic injection molding. The method utilized a series of higher order response surface approximations (RSA) to model the maximum injection pressure distribution with respect to gate locations, and the global minimum of these response surface models were subsequently sought by a global optimization method based on a multi-start sequential quadratic programming technique. The design points for RSA were evaluated by the finite element method. After a sequence of repetitions of RSA and optimization, the converged minimizer would represent the optimal gate location. A rectangular plate with two segments of different thicknesses was selected to demonstrate the effectiveness of the procedure. The variation of the thicknesses causes the optimal gate location to deviate from the center and induce multiple valleys in the maximum injection pressure distribution, which is ideal for the application of the higher order RSA and a global searching technique.

Sign in / Sign up

Export Citation Format

Share Document