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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)

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.

Hotelling’s T-Squared

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

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.

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.

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

PLS Coefficients

- Selecting to extract 3 components:

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):

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.

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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