Monte Carlo-Based Yield Estimation: New Methodology



Today’s analog IC sizing and optimization tools are mostly simulation based due to the results accuracy brought by commercial electrical simulators. Additionally, most of the optimization kernels, adopted by those tools, are based on evolutionary optimization algorithms or other metaheuristic techniques because of the large search space that must be explored to find the optimal solutions. The development of an accurate MC-based yield estimation technique that enables evolutionary-based analog IC sizing and optimization tools to search for more robust solutions is a challenging task that must be achieved. The early prediction of variability effect, particularly at the new nanometer technology nodes that are very sensitive to variability effects, is one of the keys to improve production costs. The addition of a large number of MC simulations, to accurately estimate the solutions yield, may increase, beyond an acceptable value, the search time for optimal IC solutions when evolutionary-based optimization algorithms are adopted. This chapter addresses this problem and presents a new MC-based yield estimation methodology with a reduced time impact in the overall optimization processes, which allows its adoption in today’s state-of-the-art evolutionary-based analog IC sizing tools.


Parametric yield estimation Clustering techniques Monte Carlo analysis 


  1. 1.
    N. García-Pedrajas, J. Pérez-Rodríguez, Multi-selection of instances: a straightforward way to improve evolutionary instance selection. Appl. Soft Comput. 12(11), 3590–3602 (2012)CrossRefGoogle Scholar
  2. 2.
    C. Ding, X. He, K-means clustering via principal component analysis, in Proc. 21st Int. Conf. Mach. Learn., Banff, Alberta, Canada, 2004Google Scholar
  3. 3.
    A. Fred, Similarity measures and clustering of string patterns, in Pattern Recognition and String Matching, (Springer, Boston, MA, 2003), pp. 155–193CrossRefGoogle Scholar
  4. 4.
    S. Theodoridis, K. Koutroumbas, Chapter 11—Clustering: basic concepts, in Pattern Recognition, 4th edn., (Academic Press, Boston, MA, 2009), pp. 595–625CrossRefzbMATHGoogle Scholar
  5. 5.
    K.-L. Wu, J. Yu, M.-S. Yang, A novel fuzzy clustering algorithm based on a fuzzy scatter matrix with optimality tests. Pattern Recogn. Lett. 26(3), 639–652 (2005)CrossRefGoogle Scholar
  6. 6.
    T. Roughgarden, J.R. Wang, The complexity of the k-means method, in 24th Annual European Symposium on Algorithms (ESA 2016), Aarhus, Denmark, 2016Google Scholar
  7. 7.
    M. Masjed-Jamei, M.A. Jafari, H.M. Srivastava, Some applications of the stirling numbers of the first and second kind. J. Appl. Math. Comput. 47(1), 153–174 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    D. Steinley, K-means clustering: a half-century synthesis. Br. J. Math. Stat. Psychol. 59(1), 1–34 (2006)CrossRefMathSciNetGoogle Scholar
  9. 9.
    J. MacQueen, Some methods for classification and analysis of multivariate observations, in Proc. 5th Berkeley Symp. Math. Stat. Probability, 1967Google Scholar
  10. 10.
    S. Theodoridis, K. Koutroumbas, Chapter 5—Feature selection, in Pattern Recognition, 2nd edn., (Elsevier—Academic Press, San Diego, CA, 2003), pp. 163–205zbMATHGoogle Scholar
  11. 11.
    C.D. Manning, P. Raghavan, H. Schütze, Introduction to Information Retrieval (Cambridge University Press, Cambridge, 2008)CrossRefzbMATHGoogle Scholar
  12. 12.
    L. Kaufman, P.J. Rousseeuw, Finding Groups in Data: An Introduction to Cluster Analysis (Wiley, New York, 1990)CrossRefzbMATHGoogle Scholar
  13. 13.
    R.T. Ng, J. Han, Efficient and effective clustering methods for spatial data mining, in Proc. 20th Int. Conf. Very Large Data Bases (VLDB’94), Santiago de Chile, Chile, 1994Google Scholar
  14. 14.
    J.C. Bezdek, R. Ehrlich, W. Full, FCM: the fuzzy c-means clustering algorithm. Comput. Geosci. 10(2), 191–203 (1984)CrossRefGoogle Scholar
  15. 15.
    A. Stetco, X.-J. Zeng, J. Keane, Fuzzy C-means++. Expert Syst. Appl. 42(21), 7541–7548 (2015)CrossRefGoogle Scholar
  16. 16.
    C.H. Li, B.C. Kuo, C.T. Lin, LDA-based clustering algorithm and its application to an unsupervised feature extraction. IEEE Trans. Fuzzy Syst. 19(1), 152–163 (2011)CrossRefGoogle Scholar
  17. 17.
    M.-S. Yang, A survey of fuzzy clustering. Math. Comput. Model. 18(11), 1–16 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    L. Bai, J. Liang, C. Dang, F. Cao, A novel fuzzy clustering algorithm with between-cluster information for categorical data. Fuzzy Sets Syst. 215, 55–73 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    V. Schwämmle, O.N. Jensen, A simple and fast method to determine the parameters for fuzzy c–means cluster analysis. Bioinformatics 26(22), 2841–2848 (2010)CrossRefGoogle Scholar
  20. 20.
    V. Torra, On the selection of m for Fuzzy c-Means, in 2015 Conf. Int. Fuzzy Syst. Assoc. European Soc. Fuzzy Logic Technol. (IFSA-EUSFLAT-15), 2015Google Scholar
  21. 21.
    K.-L. Wu, Analysis of parameter selections for fuzzy c-means. Pattern Recogn. 45(1), 407–415 (2012)CrossRefzbMATHGoogle Scholar
  22. 22.
    S. Ghosh, S.K. Dubey, Comparative analysis of K-means and fuzzy C-means algorithms. Int. J. Adv. Comput. Sci. Appl. 4(4) (2013)Google Scholar
  23. 23.
    D.J. Ketchen, C.L. Shook, The application of cluster analysis in strategic management research: an analysis and critique. Strat. Manag. J. 17, 441–458 (1996)CrossRefGoogle Scholar
  24. 24.
    P.J. Rousseeuw, Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math. 20, 53–65 (1987)CrossRefzbMATHGoogle Scholar
  25. 25.
    D.T. Pham, S.S. Dimov, C.D. Nguyen, Selection of K in K-means clustering. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 219(1), 103–119 (2005)CrossRefGoogle Scholar
  26. 26.
    M. Halkidi, Y. Batistakis, M. Vazirgiannis, On clustering validation techniques. J. Intell. Inform. Syst. 17(2), 107–145 (2001)CrossRefzbMATHGoogle Scholar
  27. 27.
    J.C. Bezdek, Numerical taxonomy with fuzzy sets. J. Math. Biol. 1, 57–71 (1974)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    J.C. Bezdek, Cluster validity with fuzzy sets. J. Cybernet. 3, 58–74 (1974)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Y. Zhang, W. Wang, X. Zhang, Y. Li, A cluster validity index for fuzzy clustering. J. Inform. Sci. 178(4), 1205–1218 (2008)CrossRefzbMATHGoogle Scholar
  30. 30.
    D. Campo, G. Stegmayer, D. Milone, A new index for clustering validation with overlapped clusters. Expert Syst. Appl. 64, 549–556 (2016)CrossRefGoogle Scholar
  31. 31.
    E. Lord, M. Willems, F.-J. Lapointe, V. Makarenkov, Using the stability of objects to determine the number of clusters in datasets. J. Inform. Sci. 393, 29–46 (2017)CrossRefGoogle Scholar
  32. 32.
    J. Wang, A linear assignment clustering algorithm based on the least similar cluster representatives, in Int. Conf. Syst. Man, Cybern., Orlando, FL, 1997Google Scholar
  33. 33.
    J. Fan, J. Wang, A two-phase fuzzy clustering algorithm based on neurodynamic optimization with its application for PolSAR image segmentation. IEEE Trans. Fuzzy. Syst. 26(1), 72–83 (2016). Scholar
  34. 34.
    K.L. Cheng, J. Fan, J. Wang, A two-pass clustering algorithm based on linear assignment initialization and k-means method, in 5th Int. Symp. Commun., Control Signal Process., Rome, 2012Google Scholar
  35. 35.
    D. Arthur, S. Vassilvitskii, k-means++: the advantages of careful seeding, in Proc. 18th Annu. ACM-SIAM Symp. Discrete Algorithms (SODA'07), New Orleans, Louisiana, 2007Google Scholar
  36. 36.
    M.E. Celebi, H.A. Kingravi, P.A. Vela, A comparative study of efficient initialization methods for the k-means clustering algorithm. Expert Syst. Appl. 40(1), 200–210 (2013)CrossRefGoogle Scholar
  37. 37.
    A.K. Jain, M.N. Murty, P.J. Flynn, Data clustering: a review. ACM Comput. Surv. 31(3), 264–323 (1999)CrossRefGoogle Scholar
  38. 38.
    K. Abirami, P. Mayilvahanan, Performance analysis of K-means and bisecting K-means algorithms in weblog data. Int. J. Emerg. Technol. Eng. Res. 4(8), 119–124 (2016)Google Scholar
  39. 39.
    R.R. Patil, A. Khan, Bisecting K-means for clustering web log data. Int. J. Comput. Appl. 116(19), 36–41 (2015)Google Scholar
  40. 40.
    P. Cimiano, A. Hotho, S. Staab, Comparing conceptual, partitional and agglomerative clustering for learning taxonomies from text, in Proc. 16th European Conf. Artificial Intell., Amsterdam, 2004Google Scholar
  41. 41.
    L. Sousa, J. Gama, The application of hierarchical clustering algorithms for recognition using biometrics of the hand. Int. J. Adv. Eng. Res. Sci. 1(7), 14–24 (2014)Google Scholar
  42. 42.
    F. Murtagh, P. Contreras, Algorithms for hierarchical clustering: an overview. WIREs Data Mining Knowl. Discov. 2(1), 86–97 (2012)CrossRefGoogle Scholar
  43. 43.
    G.W. Milligan, M.C. Cooper, An examination of procedures for determining the number of clusters in a data set. Psychometrika 50(2), 159–179 (1985)CrossRefGoogle Scholar
  44. 44.
    Y. Jung, H. Park, D.-Z. Du, B.L. Drake, A decision criterion for the optimal number of clusters in hierarchical clustering. J. Glob. Optim. 25(1), 91–111 (2003)CrossRefMathSciNetGoogle Scholar
  45. 45.
    R. Jenssen, D. Erdogmus, K.E. Hild, J.C. Principe, T. Eltoft, Information force clustering using directed trees, in Energy Minimization Methods in Computer Vision and Pattern Recognition, Lisbon, 2003.Google Scholar
  46. 46.
    G. Karypis, E.-H. Han, V. Kumar, Chameleon: hierarchical clustering using dynamic modeling. Computer 32(8), 68–75 (1999)CrossRefGoogle Scholar
  47. 47.
    M. Ren, P. Liu, Z. Wang, J. Yi, A self-adaptive fuzzy c-means algorithm for determining the optimal number of clusters. Comput. Intell. Neurosci. 2016, 12 (2016)CrossRefGoogle Scholar
  48. 48.
    X.L. Xie, G. Beni, A validity measure for fuzzy clustering. IEEE Trans. Pattern Anal. Mach. Intell. 13(8), 841–847 (1991)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Instituto Superior TécnicoInstituto de TelecomunicaçõesLisbonPortugal
  2. 2.Instituto Politécnico de TomarInstituto de TelecomunicaçõesLisbonPortugal

Personalised recommendations