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Statistical Tools for Multivariate Six Sigma. Dr. Neil W. Polhemus CTO & Director of Development StatPoint, Inc. Revised talk: www.statgraphics.comdocuments.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
Statistical Tools for Multivariate Six Sigma
  • Dr. Neil W. Polhemus
  • CTO & Director of Development
  • StatPoint, Inc.
  • Revised talk:
the challenge
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
The Solution
  • Proper analysis of data from such processes requires the use of multivariate statistical techniques.
important tools
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
Example #1
  • Textile fiber
  • Characteristic #1: tensile strength (115.0 ± 1.0)
    • Characteristic #2: diameter (1.05 ± 0.01)
multivariate capability
Multivariate Capability

Determines joint probability of being within

the specification limits on all characteristics.

mult capability indices
Mult. Capability Indices
  • Defined to give the
  • same DPM as in the
  • univariate case.
hotelling s t squared
Hotelling’s T-Squared
  • Measures the distance of each point from the centroid of the data (or the assumed distribution).
statistical model building
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
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


dimensionality reduction
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
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
PLS Coefficients
  • Selecting to extract 3 components:
design of experiments
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
Example #3
  • Myers and Montgomery (2002) describe an experiment on a chemical process (20-run central composite design):
combined desirability
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
Desirability Contours
  • Max D=0.959 at time=11.14, temperature=210.0, and catalyst = 2.20.
  • 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:\documents.htm