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題名 電路設計中電流值之罕見事件的統計估計探討
A study of statistical method on estimating rare event in IC Current作者 彭亞凌
Peng, Ya Ling貢獻者 余清祥<br>蔡紋琦
Yue, Jack C.<br>Tsai, Wen Chi
彭亞凌
Peng, Ya Ling關鍵詞 罕見事件
電流估計
加權迴歸
變數轉換
極值理論
Rare Event
Estimating Current
Weighted Least Squares
Data Transformation
Extreme Value Theory日期 2011 上傳時間 30-Oct-2012 10:58:18 (UTC+8) 摘要 距離期望值4至6倍標準差以外的罕見機率電流值，是當前積體電路設計品質的關鍵之一，但隨著精確度的標準提升，實務上以蒙地卡羅方法模擬電路資料，因曠日廢時愈發不可行，而過去透過參數模型外插估計或迴歸分析方法，也因變數蒐集不易、操作電壓減小使得電流值尾端估計產生偏差，上述原因使得尾端電流值估計困難。因此本文引進統計方法改善罕見機率電流值的估計：先以Box-Cox轉換觀察值為近似常態，改善尾端分配值的估計，再以加權迴歸方法估計罕見電流值，其中迴歸解釋變數為Log或Z分數轉換的經驗累積機率，而加權方法採用Down-weight加重極值樣本資訊的重要性，此外，本研究也考慮能蒐集完整變數的情況，改以電路資料作為解釋變數進行加權迴歸。另一方面，本研究也採用極值理論作為估計方法。本文先以電腦模擬評估各方法的優劣，假設母體分配為常態、T分配、Gamma分配，以均方誤差作為衡量指標，模擬結果驗證了加權迴歸方法的可行性。而後參考模擬結果決定篩選樣本方式進行實證研究，資料來源為新竹某科技公司，實證結果顯示加權迴歸配合Box-Cox轉換能以十萬筆樣本數，準確估計左、右尾機率10^(-4) 、10^(-5)、10^(-6)、10^(-7)極端電流值。其中右尾部分的加權迴歸解釋變數採用對數轉換，而左尾部分的加權迴歸解釋變數採用Z分數轉換，估計結果較為準確，又若能蒐集電路資訊作為解釋變數，在左尾部份可以有最準確的估計結果；而篩選樣本尾端1%和整筆資料的方式對於不同方法的估計準確度各有利弊，皆可考慮。另外，1%門檻值比例的極值理論能穩定且中等程度的估計不同電壓下的電流值，且有短程估計最準的趨勢。
To obtain the tail distribution of current beyond 4 to 6 sigma is nowadays a key issue in integrated circuit (IC) design and computer simulation is a popular tool to estimate the tail values. Since creating rare events via simulation is time-consuming, often the linear extrapolation methods (such as regression analysis) are applied to enhance efficiency. However, it is shown from past work that the tail values is likely to behave differently if the operating voltage is getting lower. In this study, a statistical method is introduced to deal with the lower voltage case. The data are evaluated via the Box-Cox (or power) transformation and see if they need to be transformed into normally distributed data, following by weighted regression to extrapolate the tail values. In specific, the independent variable is the empirical CDF with logarithm or z-score transformation, and the weight is down-weight in order to emphasize the information of extreme values observations. In addition to regression analysis, Extreme Value Theory (EVT) is also adopted in the research.The computer simulation and data sets from a famous IC manufacturer in Hsinchu are used to evaluate the proposed method, with respect to mean squared error. In computer simulation, the data are assumed to be generated from normal, student t, or Gamma distribution. For empirical data, there are 10^8 observations and tail values with probabilities 10^(-4),10^(-5),10^(-6),10^(-7) are set to be the study goal given that only 10^5 observations are available. Comparing to the traditional methods and EVT, the proposed method has the best performance in estimating the tail probabilities. If the IC current is produced from regression equation and the information of independent variables can be provided, using the weighted regression can reach the best estimation for the left-tailed rare events. Also, using EVT can also produce accurate estimates provided that the tail probabilities to be estimated and the observations available are on the similar scale, e.g., probabilities 10^(-5)~10^(-7) vs.10^5 observations.參考文獻 Abu-Rahma, M., Chen,Y., Deshmukh, P., Gao, Y., Garg, M., Ge, L., Han, M., Liu, P., Sani, M., Sy, W., Terzioglu, E., Yang, S., Yeap, G., Yoon, S.S., and Wang, J., (2011). Non-Gaussian distribution of SRAM read current and design impact to low power memory using Voltage Acceleration Method, Symposium on VLSI Technology (VLSIT), pp.220-221, 14-16 June 2011Amirante, E., Einfeld, J., Fischer, T., Huber, P., Nirschl, T., Olbrich, A., Otte, C., Ostermayr, M., and Schmitt-Landsiedel, D., (2007). A 1 Mbit SRAM test structure to analyze local mismatch beyond 5 sigma variation, IEEE International Conference on Microelectronic Test Structures, (ICMTS `07), pp.63-66, 19-22 March 2007.Balkema, A.A., and de Haan, L., (1974). Residual life time at great age, Annals of Probability, 2, pp.792-804.Box, G.E.P., and Cox, D.R. (1964). An analysis of transformations, (with discussion), Journal of the Royal Statistical Society, series B, 26, pp.211-246.Casella, G., and Berger, R., (2002). Statistical Inference. Duxbury, 2nd Ed.Chiao C.H., Chang, H.W., Chou Y.F., Kwai, D.M., and Liao H.J., (2000). Detection of SRAM cell stability by lowering array supply voltage, Ninth Asian Test Symposium (ATS`00), pp.268.Croon, J.A., Di Bucchianico, A., Doorn, T.S., ter Maten, E.J.W., and Wittich, O., (2008). Importance sampling Monte Carlo simulations for accurate estimation of SRAM yield, Solid-State Circuits Conference (ESSCIRC) 34th European, pp.230-233, 15-19 Sept. 2008.Donald W.K. A. (2004). The Block-Block Bootstrap：Improved Asymptotic Refinements, Econometrica, vol. 72, No. 3, pp.673-700.Fisher, R., and Tippett, L.H.C., (1928). Limiting forms of the frequency distribution of largest or smallest member of a sample, Proceedings of the Cambridge philosophical society, vol. 24, pp180-190.Frey, R., and McNeil, A. J., (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance, 7(3-4), pp.271-300.Gnedenko, B.V., (1943). Sur la distribution limite du terme maximum d`une serie aleatoire, Annals of Mathematics, 44, pp.423-453.Gilli, M., and Këllezi, E., (2006). An Application of Extreme Value Theory for Measuring Financial Risk, Computational Economics 27.2-3 (2006), pp.207-228.Hardy, M.R., Li, J.S.H., and Tan, K.S., (2008). Threshold life tables and their applications, North American Actuarial Journal,12, pp. 99-115.Hsiao, C.H., and Kwai, D.M. (2005). Measurement and characterization of 6T SRAM cell current, Proc. of the 2005 IEEE International Workshop on Memory Technology, Design, and Testing (MTDT`05), pp.140-145, Aug. 2005Jenkinson, A.F., (1955). The frequency distribution of the annual maximum (minimum) values of meteorological events, Quarterly Journal of the Royal Meteorological Society, 81, pp.158-172.Marimoutou, V., Raggad, B., and Trabelsi, A., (2006). Extreme Value Theory and Value at Risk: Application to Oil Market, GREQAM Working Paper,38Pickands, J.I., (1975). Statistical inference using extreme value order statistics, Annals of Statististics, 3, pp.119-131.Weisberg, S., (2005). Applied Linear Regression, WILEY, 3rd Ed. 描述 碩士
國立政治大學
統計研究所
99354004
100資料來源 http://thesis.lib.nccu.edu.tw/record/#G0099354004 資料類型 thesis dc.contributor.advisor 余清祥<br>蔡紋琦 zh_TW dc.contributor.advisor Yue, Jack C.<br>Tsai, Wen Chi en_US dc.contributor.author (Authors) 彭亞凌 zh_TW dc.contributor.author (Authors) Peng, Ya Ling en_US dc.creator (作者) 彭亞凌 zh_TW dc.creator (作者) Peng, Ya Ling en_US dc.date (日期) 2011 en_US dc.date.accessioned 30-Oct-2012 10:58:18 (UTC+8) - dc.date.available 30-Oct-2012 10:58:18 (UTC+8) - dc.date.issued (上傳時間) 30-Oct-2012 10:58:18 (UTC+8) - dc.identifier (Other Identifiers) G0099354004 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/54407 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 統計研究所 zh_TW dc.description (描述) 99354004 zh_TW dc.description (描述) 100 zh_TW dc.description.abstract (摘要) 距離期望值4至6倍標準差以外的罕見機率電流值，是當前積體電路設計品質的關鍵之一，但隨著精確度的標準提升，實務上以蒙地卡羅方法模擬電路資料，因曠日廢時愈發不可行，而過去透過參數模型外插估計或迴歸分析方法，也因變數蒐集不易、操作電壓減小使得電流值尾端估計產生偏差，上述原因使得尾端電流值估計困難。因此本文引進統計方法改善罕見機率電流值的估計：先以Box-Cox轉換觀察值為近似常態，改善尾端分配值的估計，再以加權迴歸方法估計罕見電流值，其中迴歸解釋變數為Log或Z分數轉換的經驗累積機率，而加權方法採用Down-weight加重極值樣本資訊的重要性，此外，本研究也考慮能蒐集完整變數的情況，改以電路資料作為解釋變數進行加權迴歸。另一方面，本研究也採用極值理論作為估計方法。本文先以電腦模擬評估各方法的優劣，假設母體分配為常態、T分配、Gamma分配，以均方誤差作為衡量指標，模擬結果驗證了加權迴歸方法的可行性。而後參考模擬結果決定篩選樣本方式進行實證研究，資料來源為新竹某科技公司，實證結果顯示加權迴歸配合Box-Cox轉換能以十萬筆樣本數，準確估計左、右尾機率10^(-4) 、10^(-5)、10^(-6)、10^(-7)極端電流值。其中右尾部分的加權迴歸解釋變數採用對數轉換，而左尾部分的加權迴歸解釋變數採用Z分數轉換，估計結果較為準確，又若能蒐集電路資訊作為解釋變數，在左尾部份可以有最準確的估計結果；而篩選樣本尾端1%和整筆資料的方式對於不同方法的估計準確度各有利弊，皆可考慮。另外，1%門檻值比例的極值理論能穩定且中等程度的估計不同電壓下的電流值，且有短程估計最準的趨勢。 zh_TW dc.description.abstract (摘要) To obtain the tail distribution of current beyond 4 to 6 sigma is nowadays a key issue in integrated circuit (IC) design and computer simulation is a popular tool to estimate the tail values. Since creating rare events via simulation is time-consuming, often the linear extrapolation methods (such as regression analysis) are applied to enhance efficiency. However, it is shown from past work that the tail values is likely to behave differently if the operating voltage is getting lower. In this study, a statistical method is introduced to deal with the lower voltage case. The data are evaluated via the Box-Cox (or power) transformation and see if they need to be transformed into normally distributed data, following by weighted regression to extrapolate the tail values. In specific, the independent variable is the empirical CDF with logarithm or z-score transformation, and the weight is down-weight in order to emphasize the information of extreme values observations. In addition to regression analysis, Extreme Value Theory (EVT) is also adopted in the research.The computer simulation and data sets from a famous IC manufacturer in Hsinchu are used to evaluate the proposed method, with respect to mean squared error. In computer simulation, the data are assumed to be generated from normal, student t, or Gamma distribution. For empirical data, there are 10^8 observations and tail values with probabilities 10^(-4),10^(-5),10^(-6),10^(-7) are set to be the study goal given that only 10^5 observations are available. Comparing to the traditional methods and EVT, the proposed method has the best performance in estimating the tail probabilities. If the IC current is produced from regression equation and the information of independent variables can be provided, using the weighted regression can reach the best estimation for the left-tailed rare events. Also, using EVT can also produce accurate estimates provided that the tail probabilities to be estimated and the observations available are on the similar scale, e.g., probabilities 10^(-5)~10^(-7) vs.10^5 observations. en_US dc.description.tableofcontents 第一章 緒論 .....1第一節 研究動機 .....1第二節 研究目的 .....2第二章 文獻探討 .....4第一節 背景文獻探討 .....4第二節 統計方法探討 .....6第三節 極值理論 .....9第三章 研究方法與模擬 .....12第一節 研究方法 .....12第二節 模擬設定 .....17第三節 模擬結果 .....20第四章 實證研究 .....25第一節 資料來源與目標 .....25第二節 實證結果－無解釋變數.....26第三節 實證結果－有解釋變數.....34第四節 實證結論 .....36第五章 結論與建議 .....37第一節 結論 .....37第二節 討論與建議 .....39參考文獻 .....41附錄 .....44 zh_TW dc.language.iso en_US - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0099354004 en_US dc.subject (關鍵詞) 罕見事件 zh_TW dc.subject (關鍵詞) 電流估計 zh_TW dc.subject (關鍵詞) 加權迴歸 zh_TW dc.subject (關鍵詞) 變數轉換 zh_TW dc.subject (關鍵詞) 極值理論 zh_TW dc.subject (關鍵詞) Rare Event en_US dc.subject (關鍵詞) Estimating Current en_US dc.subject (關鍵詞) Weighted Least Squares en_US dc.subject (關鍵詞) Data Transformation en_US dc.subject (關鍵詞) Extreme Value Theory en_US dc.title (題名) 電路設計中電流值之罕見事件的統計估計探討 zh_TW dc.title (題名) A study of statistical method on estimating rare event in IC Current en_US dc.type (資料類型) thesis en dc.relation.reference (參考文獻) Abu-Rahma, M., Chen,Y., Deshmukh, P., Gao, Y., Garg, M., Ge, L., Han, M., Liu, P., Sani, M., Sy, W., Terzioglu, E., Yang, S., Yeap, G., Yoon, S.S., and Wang, J., (2011). Non-Gaussian distribution of SRAM read current and design impact to low power memory using Voltage Acceleration Method, Symposium on VLSI Technology (VLSIT), pp.220-221, 14-16 June 2011Amirante, E., Einfeld, J., Fischer, T., Huber, P., Nirschl, T., Olbrich, A., Otte, C., Ostermayr, M., and Schmitt-Landsiedel, D., (2007). A 1 Mbit SRAM test structure to analyze local mismatch beyond 5 sigma variation, IEEE International Conference on Microelectronic Test Structures, (ICMTS `07), pp.63-66, 19-22 March 2007.Balkema, A.A., and de Haan, L., (1974). Residual life time at great age, Annals of Probability, 2, pp.792-804.Box, G.E.P., and Cox, D.R. (1964). An analysis of transformations, (with discussion), Journal of the Royal Statistical Society, series B, 26, pp.211-246.Casella, G., and Berger, R., (2002). Statistical Inference. Duxbury, 2nd Ed.Chiao C.H., Chang, H.W., Chou Y.F., Kwai, D.M., and Liao H.J., (2000). Detection of SRAM cell stability by lowering array supply voltage, Ninth Asian Test Symposium (ATS`00), pp.268.Croon, J.A., Di Bucchianico, A., Doorn, T.S., ter Maten, E.J.W., and Wittich, O., (2008). Importance sampling Monte Carlo simulations for accurate estimation of SRAM yield, Solid-State Circuits Conference (ESSCIRC) 34th European, pp.230-233, 15-19 Sept. 2008.Donald W.K. A. (2004). The Block-Block Bootstrap：Improved Asymptotic Refinements, Econometrica, vol. 72, No. 3, pp.673-700.Fisher, R., and Tippett, L.H.C., (1928). Limiting forms of the frequency distribution of largest or smallest member of a sample, Proceedings of the Cambridge philosophical society, vol. 24, pp180-190.Frey, R., and McNeil, A. J., (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance, 7(3-4), pp.271-300.Gnedenko, B.V., (1943). Sur la distribution limite du terme maximum d`une serie aleatoire, Annals of Mathematics, 44, pp.423-453.Gilli, M., and Këllezi, E., (2006). An Application of Extreme Value Theory for Measuring Financial Risk, Computational Economics 27.2-3 (2006), pp.207-228.Hardy, M.R., Li, J.S.H., and Tan, K.S., (2008). Threshold life tables and their applications, North American Actuarial Journal,12, pp. 99-115.Hsiao, C.H., and Kwai, D.M. (2005). Measurement and characterization of 6T SRAM cell current, Proc. of the 2005 IEEE International Workshop on Memory Technology, Design, and Testing (MTDT`05), pp.140-145, Aug. 2005Jenkinson, A.F., (1955). The frequency distribution of the annual maximum (minimum) values of meteorological events, Quarterly Journal of the Royal Meteorological Society, 81, pp.158-172.Marimoutou, V., Raggad, B., and Trabelsi, A., (2006). Extreme Value Theory and Value at Risk: Application to Oil Market, GREQAM Working Paper,38Pickands, J.I., (1975). Statistical inference using extreme value order statistics, Annals of Statististics, 3, pp.119-131.Weisberg, S., (2005). Applied Linear Regression, WILEY, 3rd Ed. zh_TW
