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Statistical Tools for Multivariate Six Sigma. Dr. Neil W. Polhemus CTO & Director of Development StatPoint, Inc. Revised talk: www.statgraphics.com\documents.htm. 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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Statistical Tools for Multivariate Six Sigma

  • Dr. Neil W. Polhemus

  • CTO & Director of Development

  • StatPoint, Inc.

  • Revised talk:

  • www.statgraphics.com\documents.htm


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.


The Solution

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


Important Tools

  • Statistical Process Control

    • Multivariate capability analysis

    • Multivariate control charts

  • Statistical Model Building*

    • Data Mining - dimensionality reduction

    • DOE - multivariate optimization

  • * Regression and classification.


Example #1

  • Textile fiber

  • Characteristic #1: tensile strength (115.0 ± 1.0)

    • Characteristic #2: diameter (1.05 ± 0.01)


Individuals Charts


Capability Analysis (each separately)


Scatterplot


Multivariate Normal Distribution


Control Ellipse


Multivariate Capability

Determines joint probability of being within

the specification limits on all characteristics.


Mult. Capability Indices

  • Defined to give the

  • same DPM as in the

  • univariate case.


More than 2 Variables


Hotelling’s T-Squared

  • Measures the distance of each point from the centroid of the data (or the assumed distribution).


T-Squared Chart


T-Squared Decomposition


Statistical Model Building

  • Defining relationships (regression and ANOVA)

  • Classifying items

  • Detecting unusual events

  • Optimizing processes

  • When the response variables are correlated, it is important to consider the responses together.

  • When the number of variables is large, the dimensionality of the problem often makes it difficult to determine the underlying relationships.


Example #2


Matrix Plot


Multiple Regression


Reduced Models

MPG City = 29.9911 - 0.0103886*Weight + 0.233751*Wheelbase (R2=73.0%)

MPG City = 64.1402 - 0.054462*Horsepower - 1.56144*Passengers - 0.374767*Width

(R2=64.3%)


Dimensionality Reduction

  • Construction of linear combinations of the variables can often provide important insights.

  • Principal components analysis (PCA) and principal components regression (PCR): constructs linear combinations of the predictor variables X that contain the greatest variance and then uses those to predict the responses.

  • Partial least squares (PLS): finds components that minimize the variance in both the X’s and the Y’s simultaneously.


Principal Components Analysis


Scree Plot


Component Weights

C1 = 0.377*Engine Size + 0.292*Horsepower + 0.239*Passengers + 0.370*Length

+ 0.375*Wheelbase + 0.389*Width + 0.360*U Turn Space + 0.396*Weight

C2 = -0.205*Engine Size – 0.593*Horsepower + 0.731*Passengers + 0.043*Length

+ 0.260*Wheelbase – 0.042*Width – 0.026*U Turn Space – 0.030*Weight


Interpretation


PC Regression


Contour Plot


PLS Model Selection


PLS Coefficients

  • Selecting to extract 3 components:


Interpretation


Neural Networks


Bayesian Classifier


Classification


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.


Example #3

  • Myers and Montgomery (2002) describe an experiment on a chemical process (20-run central composite design):


Optimize Conversion


Optimize Activity


Desirability Functions

  • Maximization


Desirability Functions

  • Hit a target


Combined Desirability

  • di = desirability of i-th response given the settings of the m experimental factors X.

  • D ranges from 0 (least desirable) to 1 (most desirable).


Desirability Contours

  • Max D=0.959 at time=11.14, temperature=210.0, and catalyst = 2.20.


Desirability Surface


References

  • 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.

  • Revised talk: www.statgraphics.com\documents.htm


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