Skip to main content
SpringerLink
Log in

On the misleading signals in simultaneous schemes for the mean vector and covariance matrix of multivariate i.i.d. output

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

The performance of a product often depends on several quality characteristics. Simultaneous schemes for the process mean vector \((\varvec{\mu })\) and the covariance matrix (\(\varvec{\Sigma }\)) are essential to determine if unusual variation in the location and dispersion of a multivariate normal vector of quality characteristics has occurred. Misleading signals (MS) are likely to happen while using such simultaneous schemes and correspond to valid signals that lead to a misinterpretation of a shift in \(\varvec{\mu }\) (resp. \(\varvec{\Sigma }\)) as a shift in \(\varvec{\Sigma }\) (resp. \(\varvec{\mu }\)). This paper focuses on numerical illustrations that show that MS are fairly frequent, and on the use of stochastic ordering to qualitatively assess the impact of changes in \(\varvec{\mu }\) and \(\varvec{\Sigma }\) in the probabilities of misleading signals in simultaneous schemes for these parameters while dealing with multivariate normal i.i.d. output.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Alt FB (1984) Multivariate quality control. In: Kotz S, Johnson NL, Read CR (eds) Encyclopedia of statistical sciences. Wiley, New York, pp 5312–5323

    Google Scholar 

  • Anderson TW (1984) An introduction to multivariate statistical analysis. Wiley, New York

    MATH  Google Scholar 

  • Antunes C (2009) Assessment of the impact of the correlation on misleading signals in joint schemes for the mean and variance (in Portuguese). MSc Dissertation, Instituto Superior Técnico, Technical University of Lisbon, Lisbon

  • Aparisi F, Jabaloyes J, Carrión A (1999) Statistical properties of the \(|\text{ S }|\) multivariate control chart. Commun Stat 28:2671–2686

    Article  MathSciNet  MATH  Google Scholar 

  • Bersimis S, Psarakis S, Panaretos J (2007) Multivariate statistical process control charts: an overview. Qual Reliab Eng Int 23:517–543

    Article  Google Scholar 

  • Brook D, Evans DA (1972) An approach to the probability distribution of CUSUM run length. Biometrika 59:539–549

    Article  MathSciNet  MATH  Google Scholar 

  • Chen G, Cheng SW, Xie H (2005) A new multivariate control chart for monitoring both location and dispersion. Commun Stat 34:203–217

    Article  MathSciNet  MATH  Google Scholar 

  • Conte SD, de Boor C (1980) Elementary numerical analysis: an algorithmic approach, 3rd edn. MacGraw-Hill, London

    MATH  Google Scholar 

  • Das N, Prakash V (2008) Interpreting the out-of-control signal in multivariate control chart—a comparative study. Int J Adv Manuf Technol 37:966–979

    Article  Google Scholar 

  • Fuchs C, Kenett RS (1998) Multivariate quality control: theory and applications. Marcel Dekker, New York

    Google Scholar 

  • Hawkins DM (1991) Multivariate quality control based on regression-adjusted variables. Technometrics 33:61–75

    Google Scholar 

  • Hawkins DM, Maboudou-Tchao EM (2008) Multivariate exponentially weighted moving covariance matrix. Technometrics 50:155–166

    Article  MathSciNet  Google Scholar 

  • Hotelling H (1947) Multivariate quality control illustrated by the air testing of sample bombsights. In: Eisenhart C, Hastay MW, Wallis WA (eds) Techniques of statistical analysis. McGraw Hill, New York, pp 111–184

    Google Scholar 

  • Huwang L, Yeh AB, Wu C-W (2007) Monitoring multivariate process variability for individual observations. J Qual Technol 39:258–278

    Google Scholar 

  • Karr AF (1993) Probability. Springer, New York

    Book  MATH  Google Scholar 

  • Knoth S, Morais MC, Pacheco A, Schmid W (2009) Misleading signals in simultaneous residual schemes for the mean and variance of a stationary process. Commun Stat 38:2923–2943

    Article  MathSciNet  MATH  Google Scholar 

  • Khoo MBC (2005) A new bivariate control chart to monitor the multivariate process mean and variance simultaneously. Qual Eng 17:109–118

    Article  Google Scholar 

  • Liu RY (1995) Control charts for multivariate processes. J Am Stat Assoc 90:1380–1387

    Article  MathSciNet  MATH  Google Scholar 

  • Lou J, Reynolds MR (2012) Maximum-likelihood-based diagnostics after a signal from control charts. J Qual Technol 44:321–347

    Google Scholar 

  • Lowry CA, Montgomery DC (2009) A review of multivariate control charts. IIE Trans 27:800–810

    Article  Google Scholar 

  • Lowry CA, Woodall WH, Champ CW, Rigdon SE (1992) A multivariate exponentially weighted moving average control chart. Technometrics 34:46–53

    Article  MATH  Google Scholar 

  • Mason RL, Young JC (1987) Multivariate statistical process control with industrial applications. In: ASA-SIAM series on statistics and applied probability, Society for industrial and applied mathematics, Philadelphia

  • Mathai AM, Provost SB (1992) Quadratic forms in random variables. Marcel Dekker, New York

    MATH  Google Scholar 

  • Mittman B (1994) Die Anwendung multivariater Regelkarten und Systeme univariater Regelkarten unter besonderer Berücksichtigung einer modifizierten multivariaten EWMA-Regelkarte. Akademischer Verlag, München

    Google Scholar 

  • Montgomery DC (2008) Introduction to statistical quality control, 6th edn. Wiley, New York

    MATH  Google Scholar 

  • Montgomery DC, Wadsworth HM (1972) Some techniques for multivariate quality control applications. In: Transactions of the ASQC American society for quality control, Washington DC

  • Morais MJC (2002) Stochastic ordering in the performance analysis of quality control schemes. PhD Dissertation, Instituto Superior Técnico, Technical University of Lisbon, Lisbon

  • Morais MC, Pacheco A (2000) On the performance of combined EWMA schemes for \(\mu \) and \(\sigma \): a Markovian approach. Commun Stat 29:153–174

    Article  MathSciNet  MATH  Google Scholar 

  • Morais MC, Okhrin Y, Pacheco A, Schmid W (2008) EWMA charts for multivariate output: some stochastic ordering results. Commun Stat 37:2653–2663

    Article  MathSciNet  MATH  Google Scholar 

  • Ramos PAACFP (2013) Performance analysis of simultaneous control schemes for the process mean (vector) and (co)variance (matrix). PhD Dissertation, Instituto Superior Técnico, Technical University of Lisbon, Lisbon

  • Ramos PF, Morais MC, Pacheco A, Schmid W (2012) Assessing the impact of autocorrelation in misleading signals in simultaneous residual schemes for the process mean and variance: a stochastic ordering approach. In: Lenz HJ, Wilrich P-T, Schmid W (eds) Front Stat Qual Control, vol 10. Physica-Verlag, Heidelberg, pp 35–52

    Chapter  Google Scholar 

  • Ramos PF, Morais MC, Pacheco A, Schmid W (2013a) Misleading signals in simultaneous schemes for the mean vector and the covariance matrix of a bivariate process. In: Oliveira PE, Temido MG, Vichi M, Henriques C (eds) Recent developments in modeling and applications in statistics, studies in theoretical and applied statistics. Springer-Verlag, Berlin/Heidelberg, pp 225–235

    Google Scholar 

  • Ramos PF, Morais MC, Pacheco A (2013b) Misleading signals in simultaneous residual schemes for the process mean and variance of AR(1) processes: a stochastic ordering approach. In: Silva JL, Caeiro F, Natário I, Braumann CA (eds) Advances in regression, survival analysis, extreme values, Markov processes and other statistical applications. Springer-Verlag, Berlin and Heidelberg, pp 161–170

    Google Scholar 

  • Ramos PF, Morais MC, Pacheco A, Schmid W (2013c) Stochastic ordering in the qualitative assessment of the performance of simultaneous schemes for bivariate processes. Seq Anal 32:214–229

    Article  MathSciNet  MATH  Google Scholar 

  • Reynolds MR Jr, Stoumbos ZG (2001) Monitoring the process mean and variance using individual observations and variable sampling intervals. J Qual Technol 33:181–205

    Google Scholar 

  • Reynolds MR Jr, Cho G-Y (2006) Multivariate control charts for monitoring the mean vector and covariance matrix. J Qual Technol 38:230–253

    Google Scholar 

  • Rosołowski M, Schmid W (2003) EWMA charts for monitoring the mean and the autocovariances of stationary Gaussian processes. Seq Anal 22:257–285

    Article  MathSciNet  MATH  Google Scholar 

  • Searle SR (1982) Matrix algebra useful for statistics. Wiley, Hoboken

    MATH  Google Scholar 

  • Seber GA (2008) A matrix handbook for statisticians. Wiley, Hoboken

    MATH  Google Scholar 

  • Shaked M, Shanthikumar JG (1994) Stochastic orders and their applications. Academic Press, London

    MATH  Google Scholar 

  • St. John RC, Bragg DJ (1991) Joint X-bar and R-charts under shift in mu or sigma. ASQC Quality Congress Transactions, Milwaukee, pp 547–550

    Google Scholar 

  • Tang PF, Barnett NS (1996a) Dispersion control for multivariate processes. Aust J Stat 38:235–251

    Article  MathSciNet  MATH  Google Scholar 

  • Tang PF, Barnett NS (1996b) Dispersion control for multivariate processes—some comparisons. Aust J Stat 38:253–273

    Article  MathSciNet  MATH  Google Scholar 

  • Wierda SJ (1994) Multivariate statistical process control—recent results and direction for future research. Stat Neerl 48:147–168

    Article  MathSciNet  MATH  Google Scholar 

  • Yeh AB, Lin DK-J (2002) A new variables control chart for simultaneously monitoring multivariate process mean and variability. Int J Reliab Qual Saf Eng 9:41–59

    Article  Google Scholar 

  • Yeh AB, Lin DK-J, Zhou H, Venkataramani C (2003) A multivariate exponentially weighted moving average control chart for monitoring process variability. J Appl Stat 30:507–536

    Article  MathSciNet  MATH  Google Scholar 

  • Yeh AB, Lin DK-J, McGrath RN (2006) Multivariate control charts for monitoring covariance matrix: a review. Qual Technol Quant Manag 3:415–436

    MathSciNet  Google Scholar 

  • Zhang J, Li Z, Wang Z (2010) A multivariate control chart for simultaneously monitoring process mean and variability. Comput Stat Data Anal 54:2244–2252

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The first author was supported by Grant SFRH/BD/35739/2007 of Fundação para a Ciência e a Tecnologia (FCT). This work received financial support from Portuguese National Funds through FCT within the scope of the Projects PEst-OE/MAT/UI0822/2011 and PTDC/MAT-STA/3169/2012. We are grateful to the Editor-in-Chief and the anonymous referees for all the valuable suggestions, which led to an improved version of the original draft of this paper.

Author information

Authors and Affiliations

  1. CEMAT, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Patrícia Ferreira Ramos

  2. Department of Mathematics and CEMAT, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Manuel Cabral Morais & António Pacheco

  3. Department of Statistics, European University Viadrina, Postfach 1786, 15207, Frankfurt (Oder), Germany

    Wolfgang Schmid

Authors
  1. Patrícia Ferreira Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel Cabral Morais
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. António Pacheco
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wolfgang Schmid
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrícia Ferreira Ramos.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ramos, P.F., Morais, M.C., Pacheco, A. et al. On the misleading signals in simultaneous schemes for the mean vector and covariance matrix of multivariate i.i.d. output. Stat Papers 57, 471–498 (2016). https://doi.org/10.1007/s00362-015-0663-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-015-0663-5

Keywords

Search

Navigation