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# Statistical Tools for Multivariate Six Sigma - PowerPoint PPT Presentation

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)

### 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):

• Maximization

• 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