Sample average approximation

A remarkable property of the normal distribution is that, almost regardless of the distribution of x, sample averages x will approach the gaussian distribution as n gets large. Even for relatively small values of n, of about 10, the approximation in most cases is quite close. For example, sample averages of size 10 from the uniform distribution will have essentially a gaussian distribution. 

The t distribution is also symmetric and centered at zero. It is said to be robust in the sense that even when the individual obseiwations x are not normally distributed, sample averages o x have distributions which tend toward normahty as n gets large. Even for small n of 5 through 10, the approximation is usually relatively accurate. 

In Chapter 3 of this book we discussed the problem of multisite refinery integration under deterministic conditions. In this chapter, we extend the analysis to account for different parameter uncertainty. Robustness is quantified based on both model robustness and solution robustness, where each measure is assigned a scaling factor to analyze the sensitivity of the refinery plan and integration network due to variations. We make use of the sample average approximation (SAA) method with statistical bounding techniques to generate different scenarios. 

This section presents computational results of the models and the sampling scheme proposed in this chapter. The refinery examples considered represent industrial-scale size refineries and an actual configuration that can be found in many industrial sites around the world. In the presentation of the results, we focus on demonstrating the sample average approximation computational results as we vary the sample sizes and compare their solution accuracy and the CPU time required for solving the models. 

Kleywegt, A.J., Shapiro, A., and Homem-De-Mello, T. (2001) The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization, 12, 479. 

The sample average approximation method applied to stochastic routing problems ... 

Wei, J. and Realff, M.J. (2004) Sample average approximation methods for stochastic MINLPs. Computers el Chemical Engineering, 28, 336. 

Due to the complexity of numerical integration and the exponential increase in sample size with the increase of the random variables, we employ an approximation scheme know as the sample average approximation (SAA) method, also known as stochastic counterpart. The SAA problem can be written as ... 

The quality assurance sample used is an aliquot of a urine pool prepared by mixing equal proportions of 24-h samples from 20 men and women. Thus, the values obtained on GC-MS analysis should approximate true average steroid excretions. We use these values to adjust the calibration of steroids without authentic materials in the external standard. We have determined that the excretion of 18-OH-THA in the quality assurance sample should be 90 pg/24 h THAldo, 30 pg/24 h, and 18-oxo-THF, 5 pg/24 h. The amount of these steroids in clinical samples is thus calibrated against a normal excretion. 

The problem of sampling the piecewise constant of a random succession (5.31) is reduced to classification of distribution functions with identical averages. To approximate sample Zy by a linear, broken, randomized function, we perform the following operations. First, we find the difference... 

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