提升模擬實驗之準確度

Increasing the Effectiveness of Simulation Models

邱文琪(Wenchi Chiu)

指導教授 ： 桑慧敏

國立清華大學/工學院/工業工程與工程管理學系/博士(2007年)

https://doi.org/10.6843/NTHU.2007.00064

摘要

進行模擬分析時, 以下三個問題通常是值得我們去探討的: (I) 如何得到一個好的估計式? (II) 輸入值與輸出值間的關係為何? (III) 當實驗發生率太低時, 該如何解決? 本論文由三部分所構成, 第一部份探討關於平均數的變異數的估計: 如何更準確地提供估計值的品質指標; 第二部份是有關於再次模式的估計: 如何提供一個更準確的再次模式, 以了解輸入與輸出間的關係; 第三部分是關於估計數位通訊系統的失誤率, 即如何利用重點抽樣法克服實驗中發生率太低的情況。 第一部份討論對穩定的模擬實驗必須面對的一個基本問題 -- 模擬樣本平均數的變異數的估計。過去許多學者提出不同類型的估計式, 其中有許多都是以批量大小為參數, 而其評估準則大都以最小化均方差 (mean-squared-error, mse) 來衡量。我們則提出另一個評估準則 --讓估計式的偏誤 (bias) 在滿足使用者設定的條件下, 最小化估計式的變異數 -- 以滿足對於偏誤較為重要的情況。為了使目標式具體化, 我們並且提出一套偏誤察覺機制 (bias-aware mechanism), 首先透過調整線性組合估計式的批量大小來滿足使用者所設定的偏誤限制, 接著透過選擇適當的線性組合參數來降低估計式的變異數。我們更進一步透過對不同估計式(包括NBM, OBM, STS.A) 的組合來描述這個機制的使用。此外, 我們也藉由蒙地卡羅法來實 際測試這個機制。 第二部份是有關於模擬估計的再次模式, 也就是模擬試驗中, 模式輸出平均與輸入參數間的數學關係式。再次模式的估計可說是模擬的進階問題, 針對 2k 多因子實驗, 我們提出五類變異數轉移法則 (five-class variance swapping rule), 透過控制實驗點間的相關程度, 將所有估計效用的變異數分為五類, 讓估計式更能滿足一些實驗設計的設計準則: (1) 最重要的效用有最小的變異數, (2) 所有低階估計效用的變異數比最高階效用的變異數小。五類變異數轉移 法則也是目前所有被提出變異數轉移法的一般化法則。 第三部分是關估計數位通訊系統中的失誤率 (bit error rate)。由於科技不斷的進步, 失誤率非常小, 因此資料收集十分耗時, 本研究中, 我們評估一些應用重點抽樣變異數降低法估計失誤率, 我們發現選擇混合右尾與一致分配當作重點抽樣法中的偏誤分配, 能使變異數在很多情況下比原本的右尾分配小。

關鍵字

變異數估計 ； 再次模式 ； 變異數降低 ； 失誤率 ； 重點抽樣法

並列摘要

When undertaking the estimation of a performance measure, we may interested in three types of problems: (I) how should one obtain a good estimator of the performance measure and the estimator’s quality? (II) what are the relationships between inputs and outputs? (III) what can we do if the occurrence rate is too low during the experiments?
In this dissertation, we focus on increasing the quality of estimation. The first part is related to the estimation of variance of sample mean of simulation models: providing the
quality measure of the estimated values more precisely. The second part is related to the estimation of metamodel: providing a better metamodel to know the inherent relationship between input and output variables. The third part is related to estimation of the bit error
rate: using importance sampling to overcome the low occurrence rate during the experiment. Specifically, we propose a combined estimator which can more precisely estimate the variance of sample mean, a random number assigning method which can estimate a more precise
metamodel, and the mixed basing distribution for estimating bit error rate using importance
sampling technique.

For the first part, estimating the variance of the sample mean is a prototype problem in steady-state simulation. We propose a bias-aware mechanism which attempts to minimize
the variance of an estimator subject to a bias constraint – a goal that differs from that of minimizing mse (sum of variance and bias squared), in which case there would be no explicit bias constraint. Specifically, we use linear combinations of estimators based on different batch
sizes to satisfy the bias constraint; and then we minimize variance by choosing appropriate linear combination weights. We illustrate the use of this mechanism by presenting bias-aware linear combinations of several variance estimators, including non-overlapping batch means,
overlapping batch means, and standardized time series weighted area estimators. We also evaluate our mechanism with Monte Carlo examples.

For the second part, estimating the simulation metamodel, which is a functional relationship between the mean response of the simulation model and a set of simulation inputs, is an advanced simulation problem. We propose a five-class variance swapping rule, which classifies all variances of the effect estimators into five classes, for linear metamodels of 2k factorial designs. The proposed rule is a generalization of all existing variance swapping
rules (VSRs) and is a better VSR than the existing ones in that it makes a finer distinction among all effects, provided that the most important effects have possible minimal variance and all the lower-interaction effect estimators have smaller variances than that for the
highest-interaction effect.

For the third part, we are interested in estimating the BER for signal transmission in digital communication systems. Since BERs tend to be extremely small, it is difficult to obtain precise estimators based on the use of crude Monte Carlo simulation techniques. In this research, we review, expand upon, and evaluate a number of importance sampling variance reduction techniques for estimating the BER. We find that a mixture of certain “tailed”distributions with a uniform distribution produce estimators that are at least competitive with those in the literature. Our comparisons are based on analytical calculations and lay the groundwork for the evaluation of more-general mixture distributions.

並列關鍵字

variance estimation ； metamodel ； variance reduction ； bit error rate ； importance sampling

參考文獻

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