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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 l.jpg

Statistical Tools for Multivariate Six Sigma

Dr. Neil W. Polhemus

CTO & Director of Development

StatPoint, Inc.

The challenge l.jpg
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.

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The Solution

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

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  • 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

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Example #1

Textile fiber

Characteristic #1: tensile strength - 115 ± 1

Characteristic #2: diameter - 1.05 ± 0.05

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Sample Data

n = 100

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Multivariate Capability

Determines joint probability of being within the specification limits on all characteristics

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Mult. Capability Indices

Defined to give the

same DPM as in the

univariate case.

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More than 2 Characteristics

Calculate T-squared:


S = sample covariance matrix

= vector of sample means

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T-Squared Decomposition

Subtracts the value of T-squared if each variable is removed.

Large values indicate that a variable has an important contribution.

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Generalized Variance Chart

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

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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.

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Analysis Methods

  • Predicting certain characteristics based on others (regression and ANOVA)

  • Separating items into groups (classification)

  • Detecting unusual items

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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.

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Scree Plot

Shows the number of significant components.

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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.

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Component Extraction

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

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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.

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Bayesian Classifier

  • Begins with prior probabilities for membership in each group

  • Uses a Parzen-like density estimator of the density function for each group

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  • The prior probabilities may be determined in several ways.

  • A training set is usually used to find a good value for s.

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Design of Experiments

When more than one characteristic is important, finding the optimal operating conditions usually requires a tradeoff of one characteristic for another.

One approach to finding a single solution is to use desirability functions.

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Example #3

Myers and Montgomery (2002) describe an experiment on a chemical process:

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Combined Desirability

where m = # of factors and 0 ≤ Ij ≤ 5. D ranges from 0 to 1.

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  • Johnson, R.A. and Wichern, D.W. (2002). Applied Multivariate Statistical Analysis. Upper Saddle River: Prentice Hall.Mason, R.L. and Young, J.C. (2002).

  • Mason and Young (2002). Multivariate Statistical Process Control with Industrial Applications. Philadelphia: SIAM.

  • Montgomery, D. C. (2005). Introduction to Statistical Quality Control, 5th edition. New York: John Wiley and Sons.

  • Myers, R. H. and Montgomery, D. C. (2002). Response Surface Methodology: Process and Product optimization Using Designed Experiments, 2nd edition. New York: John Wiley and Sons.

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PowerPoint Slides

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