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Statistical Tools for Multivariate Six Sigma

Dr. Neil W. PolhemusCTO & Director of DevelopmentStatPoint, Inc.

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

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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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Important Tools Statistical Process Control

Multivariate capability analysis Multivariate control charts

Statistical Model Building* Data Mining - dimensionality reduction DOE - multivariate optimization

* Regression and classification.

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Example #1

Textile fiber

Characteristic #1: tensile strength (115.0 ± 1.0)

Characteristic #2: diameter (1.05 ± 0.01)

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Individuals Charts

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Capability Analysis (each separately)

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Scatterplot

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Multivariate Normal Distribution

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Control Ellipse

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Multivariate CapabilityDetermines 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 Variables

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Hotelling’s T-Squared

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

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T-Squared Chart

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T-Squared Decomposition

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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.

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Example #2

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Matrix Plot

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Multiple Regression

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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%)

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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.

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Principal Components Analysis

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Scree Plot

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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

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Interpretation

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PC Regression

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Contour Plot

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PLS Model Selection

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PLS Coefficients

Selecting to extract 3 components:

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Interpretation

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Neural Networks

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Bayesian Classifier

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Classification

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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 (20-run central composite design):

Response variable Goal

Conversion percentage maximize

Thermal activity Maintain between 55 and 60

Input factor Low High

time 8 minutes 17 minutes

temperature 160˚ C 210˚ C

catalyst 1.5% 3.5%

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Optimize Conversion

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Optimize Activity

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Desirability Functions

Maximization

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Desirability Functions

Hit a target

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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).

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Desirability ContoursMax D=0.959 at time=11.14, temperature=210.0, and catalyst = 2.20.

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Desirability Surface

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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