Statistical Process Control Using Modified Robust Hotelling's T² Control Charts

Hotelling’s T² chart is a popular tool for monitoring statistical process control. However, this chart is sensitive to outliers. To alleviate the problem, three approaches to the robust Hotelling’s T² chart namely trimming, Winsorizing and median based were proposed. These approaches used robust location and scale estimators to substitute for the usual mean and covariance matrix, respectively. For each approach, three robust scale estimators: MADn, Sn and Tn were introduced, and these estimators functioned accordingly to the approach. The first approach, denoted as T²t, applied the concept of trimming via Mahalanobis distance. The robust scale estimator was used to replace the covariance matrix in Mahalanobis distance. The trimmed mean and trimmed covariance matrix were the location and scale estimators for the T²t chart. The second approach,, T²w, employed each scale estimator as the Winsorized criterion. This approach applied Winsorized modified one step M-estimator and its corresponding Winsorized covariance as the location and the scale matrix for T²w chart, respectively. Meanwhile, in the third approach, T²н, the robust scale estimator took the role of the scale matrix with Hodges-Lehman as the location estimator. This approach worked with the original data without any trimming or Winsorizing. Altogether, nine robust control charts were proposed. The performance of each robust control chart was assessed based on false alarm rates and probability of detection. To investigate on the strengths and weaknesses of the proposed charts, various conditions were created by manipulating four variables, namely number of quality characteristics, proportion of outliers, degree of mean shifts, and nature of quality characteristics (independent and dependent). In general, the proposed charts performed well in terms of false alarm rates. With respect to probability of detection, all the proposed charts outperformed the traditional Hotelling's T² charts. The overall findings showed that, the proposed robust Hotelling's T² control charts are viable alternatives to the disputed traditional charts

Bivariate modified hotelling’s T2 charts using bootstrap data

The conventional Hotelling’s  charts are evidently inefficient as it has resulted in disorganized data with outliers, and therefore, this study proposed the application of a novel alternative robust Hotelling’s  charts approach. For the robust scale estimator , this approach encompasses the use of the Hodges-Lehmann vector and the covariance matrix in place of the arithmetic mean vector and the covariance matrix, respectively.  The proposed chart was examined performance wise. For the purpose, simulated bivariate bootstrap datasets were used in two conditions, namely independent variables and dependent variables. Then, assessment was made to the modified chart in terms of its robustness. For the purpose, the likelihood of outliers’ detection and false alarms were computed. From the outcomes from the computations made, the proposed charts demonstrated superiority over the conventional ones for all the cases tested

Modified hotelling’s T2 control charts using modified mahalanobis distance

This paper proposed new adjusted Hotelling’s T^2 control chart for individual observations. For this objective, bootstrap method for producing the individual observations were employed. To do so, both arithmetic mean vector and the covariance matrix in the traditional Hotelling’s T^2 chart were substituted by the trimmed mean vector and the covariance matrix of the robust scale estimators〖 Q〗_n, respectively which, in turn, its performance is carried out by simulated. In fact, the calculation of false alarms and the probability of detection outlier is used for determining the validity of this modified chart. The findings revealed a considerable significance in its performance

Robust Linear Discriminant Analysis with Highest Breakdown Point Estimator

Linear Discriminant Analysis (LDA) is a supervised classification technique concerned with the relationship between a categorical variable and a set of interrelated variables.The main objective of LDA is to create a rule to distinguish between populations and allocating future observations to previously defined populations.The LDA yields optimal discriminant rule between two or more groups under the assumptions of normality and homoscedasticity.Nevertheless, the classical estimates, sample mean and sample covariance matrix, are highly affected when the ideal conditions are violated.To abate these problems, a new robust LDA rule using high breakdown point estimators has been proposed in this article.A winsorized approach used to estimate the location measure while the multiplication of Spearman’s rho and the rescaled median absolute deviation were used to estimate the scatter measure to replace the sample mean and sample covariance matrix, respectively.Simulation and real data study were conducted to evaluate the performance of the proposed model measured in terms of misclassification error rates.The computational results showed that the proposed LDA is always better than the classical LDA and were comparable with the existing robust LDAs

The Mahalanobis-Taguchi system based on statistical modeling

早大学位記番号:新7809早稲田大

H-statistic with winsorized modified one-step M-estimator as central tendency measure

Two-sample independent t-test and ANOVA are classical procedures which are widely used to test the equality of two groups and more than two groups respectively. However, these parametric procedures are easily affected by non-normality, becoming more obvious when heterogeneity of variances and unbalanced group sizes exist. It is well known that the violation in the assumption of the tests will lead to inflation in Type I error rate and decreasing in the power of test. Nonparametric procedures like Mann-Whitney and Kruskal-Wallis may be the alternative to the parametric procedures, however, loss of information occur due to the ranking data. In mitigating these problems, robust procedures can be used as the other alternative. One of the procedures is H-statistic. When used with modified one-step M-estimator (MOM), the test statistic (MOM-H) produces good control of Type I error rate even under small sample size but inconsistent under certain conditions investigated. Furthermore, power of test is low which might be due to the trimming process. In this study, MOM was winsorized (WMOM) to retain the original sample size. The Hstatistic when combines with WMOM as the central tendency measure (WMOM-H) shows better control of Type I error rate as compared to MOM-H especially under balanced design regardless of the shape of distributions. It also performs well under highly skewed and heavy tailed distribution for unbalanced design. On top of that, WMOM-H also generates better power value, as compared to MOM-H and ANOVA under most of the conditions investigated. WMOM-H also has better control of Type I error rates with no liberal value (>0.075) compared to the parametric (t-test and ANOVA) and nonparametric (Mann-Whitney and Kruskal-Wallis) procedures. In general, this study demonstrates that winsorization process (WMOM) is able to improve the performance of H-statistic in terms of controlling Type I error rate and increasing power of test

Robust hotelling's T2 statistic based on M-estimator

Hotelling’s T2 statistic is the multivariate generalization of the student’s t statistic. Hotelling’s T2 statistic is a method for testing hypotheses about multidimensional means. However, the classical Hotelling’s T2 statistic is very sensitive to the presence of outliers. In order to overcome this limitation, a modification is needed so that Hotelling’s T2 is robust. In this paper, classical Hotelling’s T2 statistic has been modified by substituting mean vector and covariance matrix with a robust estimator. M-estimator has been used for this modification. The performance of modified Hotelling’s T2 statistic has been compared with the classical Hotelling’s T 2 statistic and discussed in this paper to illustrate the advantage of modified Hotelling’s T2 statistic towards outliers. The performance of modified Hotelling’s T 2 statistic is better than classical Hotelling’s T2 when number of sample, n and dimension, p is small

Robust linear discriminant rules with coordinatewise and distance based approaches

Linear discriminant analysis (LDA) is one of the supervised classification techniques to deal with relationship between a categorical variable and a set of continuous variables. The main objective of LDA is to create a function to distinguish between groups and allocating future observations to previously defined groups. Under the assumptions of normality and homoscedasticity, the LDA yields optimal linear discriminant rule (LDR) between two or more groups. However, the optimality of LDA highly relies on the sample mean and sample covariance matrix which are known to be sensitive to outliers. To abate these conflicts, robust location and scale estimators via coordinatewise and distance based approaches have been applied in constructing new robust LDA. These robust estimators were used to replace the classical sample mean and sample covariance to form robust linear discriminant rules (RLDR). A total of six RLDR, namely four coordinatewise (RLDRM, RLDRMw, RLDRW, RLDRWw) and two distance based (RLDRV, RLDRT) approaches have been proposed and implemented in this study. Simulation and real data study were conducted to investigate on the performance of the proposed RLDR, measured in terms of misclassification error rates and computational time. Several data conditions such as non-normality, heteroscedasticity, balanced and unbalanced data set were manipulated in the simulation study to evaluate the performance of these proposed RLDR. In real data study, a set of diabetes data was used. This data set violated the assumptions of normality as well as homoscedasticity. The results showed that the novel RLDRV is the best proposed RLDR to solve classification problem since it provides as much as 91.03% accuracy in classification as shown in the real data study. The proposed RLDR are good alternatives to the classical LDR as well as existing RLDR since these RLDR perform well in classification problems even under contaminated data