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:\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

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)

Individuals charts

Individuals Charts

Capability analysis each separately

Capability Analysis (each separately)



Multivariate normal distribution

Multivariate Normal Distribution

Control ellipse

Control Ellipse

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.

More than 2 variables

More than 2 Variables

Hotelling s t squared

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 Chart

T squared decomposition

T-Squared Decomposition

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.

Example 2

Example #2

Matrix plot

Matrix Plot

Multiple regression

Multiple Regression

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.

Principal components analysis

Principal Components Analysis

Scree plot

Scree Plot

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



Pc regression

PC Regression

Contour plot

Contour Plot

Pls model selection

PLS Model Selection

Pls coefficients

PLS Coefficients

  • Selecting to extract 3 components:



Neural networks

Neural Networks

Bayesian classifier

Bayesian Classifier



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):

Optimize conversion

Optimize Conversion

Optimize activity

Optimize Activity

Desirability functions

Desirability Functions

  • Maximization

Desirability functions1

Desirability Functions

  • Hit a target

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.

Desirability surface

Desirability Surface



  • 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

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