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Statistical Tools for Multivariate Six Sigma

Statistical Tools for Multivariate Six Sigma. Dr. Neil W. Polhemus CTO & Director of Development StatPoint, Inc. The Challenge. The quality of an item or service usually depends on more than one characteristic.

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Statistical Tools for Multivariate Six Sigma

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  1. Statistical Tools for Multivariate Six Sigma Dr. Neil W. Polhemus CTO & Director of Development StatPoint, Inc.

  2. The Challenge The quality of an item or service usually depends on more than one characteristic. When the characteristics are not independent, considering each characteristic separately can give a misleading estimate of overall performance.

  3. The Solution Proper analysis of data from such processes requires the use of multivariate statistical techniques.

  4. Outline • Multivariate SPC • Multivariate control charts • Multivariate capability analysis • Data exploration and modeling • Principal components analysis (PCA) • Partial least squares (PLS) • Neural network classifiers • Design of experiments (DOE) • Multivariate optimization

  5. Example #1 Textile fiber Characteristic #1: tensile strength - 115 ± 1 Characteristic #2: diameter - 1.05 ± 0.05

  6. Sample Data n = 100

  7. Individuals Chart - strength

  8. Individuals Chart - diameter

  9. Capability Analysis - strength

  10. Capability Analysis - diameter

  11. Scatterplot

  12. Multivariate Normal Distribution

  13. Control Ellipse

  14. Multivariate Capability Determines joint probability of being within the specification limits on all characteristics

  15. Multivariate Capability

  16. Capability Ellipse

  17. Mult. Capability Indices Defined to give the same DPM as in the univariate case.

  18. Test for Normality

  19. More than 2 Characteristics Calculate T-squared: where S = sample covariance matrix = vector of sample means

  20. T-Squared Chart

  21. T-Squared Decomposition Subtracts the value of T-squared if each variable is removed. Large values indicate that a variable has an important contribution.

  22. Control Ellipsoid

  23. Multivariate EWMA Chart

  24. Generalized Variance Chart Plots the determinant of the variance-covariance matrix for data that is sampled in subgroups.

  25. Data Exploration and Modeling When the number of variables is large, the dimensionality of the problem often makes it difficult to determine the underlying relationships. Reduction of dimensionality can be very helpful.

  26. Example #2

  27. Matrix Plot

  28. Analysis Methods • Predicting certain characteristics based on others (regression and ANOVA) • Separating items into groups (classification) • Detecting unusual items

  29. Multiple Regression

  30. Principal Components The goal of a principal components analysis (PCA) is to construct k linear combinations of the p variables X that contain the greatest variance.

  31. Scree Plot Shows the number of significant components.

  32. Percentage Explained

  33. Components

  34. Interpretation

  35. Principal Component Regression

  36. Partial Least Squares (PLS) Similar to PCA, except that it finds components that minimize the variance in both the X’s and the Y’s. May be used with many X variables, even exceeding n.

  37. Component Extraction Starts with number of components equal to the minimum of p and (n-1).

  38. Coefficient Plot

  39. Model in Original Units

  40. Classification Principal components can also be used to classify new observations. A useful method for classification is a Bayesian classifier, which can be expressed as a neural network.

  41. 6 Types of Automobiles

  42. Neural Networks

  43. Bayesian Classifier • Begins with prior probabilities for membership in each group • Uses a Parzen-like density estimator of the density function for each group

  44. Options • The prior probabilities may be determined in several ways. • A training set is usually used to find a good value for s.

  45. Output

  46. Classification Regions

  47. Changing Sigma

  48. Overlay Plot

  49. Outlier Detection

  50. Cluster Analysis

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