# Statistical Tools for Multivariate Six Sigma

• View
24

2

Embed Size (px)

DESCRIPTION

Statistical Tools for Multivariate Six Sigma. Dr. Neil W. Polhemus CTO & Director of Development StatPoint, Inc. The Challenge. The quality of an item or service usually depends on more than one characteristic. - PowerPoint PPT Presentation

### Text of Statistical Tools for Multivariate Six Sigma

• Statistical Tools for Multivariate Six SigmaDr. Neil W. PolhemusCTO & Director of DevelopmentStatPoint, Inc.

• The ChallengeThe 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 SolutionProper analysis of data from such processes requires the use of multivariate statistical techniques.

• OutlineMultivariate SPC Multivariate control charts Multivariate capability analysisData exploration and modelingPrincipal components analysis (PCA)Partial least squares (PLS) Neural network classifiersDesign of experiments (DOE)Multivariate optimization

• Example #1Textile fiber

Characteristic #1: tensile strength - 115 1

Characteristic #2: diameter - 1.05 0.05

• Sample Datan = 100

• Individuals Chart - strength

• Individuals Chart - diameter

• Capability Analysis - strength

• Capability Analysis - diameter

• Scatterplot

• Multivariate Normal Distribution

• Control Ellipse

• Multivariate CapabilityDetermines joint probability of being within the specification limits on all characteristics

Observed

Estimated

Estimated

Variable

Beyond Spec.

Beyond Spec.

DPM

strength

0.0%

0.00307572%

30.7572

diameter

0.0%

0.00445939%

44.5939

Joint

0.0%

0.00703461%

70.3461

• Multivariate Capability

• Capability Ellipse

• Mult. Capability IndicesDefined to give the same DPM as in the univariate case.

Capability Indices

Index

Estimate

MCP

1.27

MCR

78.80

DPM

70.3461

Z

3.80696

SQL

5.30696

• Test for Normality

P-Values

Shapiro-Wilk

strength

0.408004

diameter

0.615164

• More than 2 CharacteristicsCalculate T-squared:

where

S = sample covariance matrix

= vector of sample means

• T-Squared Chart

• T-Squared DecompositionSubtracts the value of T-squared if each variable is removed.

Large values indicate that a variable has an important contribution.

T-Squared Decomposition

Relative Contribution to T-Squared Signal

Observation

T-Squared

diameter

strength

17

26.3659

22.9655

25.951

• Control Ellipsoid

• Multivariate EWMA Chart

• Generalized Variance ChartPlots the determinant of the variance-covariance matrix for data that is sampled in subgroups.

• Data Exploration and ModelingWhen the number of variables is large, the dimensionality of the problem often makes it difficult to determine the underlying relationships.

Reduction of dimensionality can be very helpful.

• Example #2

• Matrix Plot

• Analysis MethodsPredicting certain characteristics based on others (regression and ANOVA)

Separating items into groups (classification)

Detecting unusual items

• Multiple Regression

MPG City = 29.6315 + 0.28816*Engine Size - 0.00688362*Horsepower - 0.297446*Passengers - 0.0365723*Length + 0.280224*Wheelbase + 0.111526*Width - 0.139763*U Turn Space - 0.00984486*Weight

Standard

T

Parameter

Estimate

Error

Statistic

P-Value

CONSTANT

29.6315

12.9763

2.28351

0.0249

Engine Size

0.28816

0.722918

0.398607

0.6912

Horsepower

-0.00688362

0.0134153

-0.513119

0.6092

Passengers

-0.297446

0.54754

-0.543241

0.5884

Length

-0.0365723

0.0447211

-0.817786

0.4158

Wheelbase

0.280224

0.124837

2.24472

0.0274

Width

0.111526

0.218893

0.5095

0.6117

U Turn Space

-0.139763

0.17926

-0.779668

0.4378

Weight

-0.00984486

0.00192619

-5.11104

0.0000

R-squared = 73.544 percent

R-squared (adjusted for d.f.) = 71.0244 percent

Standard Error of Est. = 3.02509

Mean absolute error = 1.99256

• Principal ComponentsThe goal of a principal components analysis (PCA) is to construct k linear combinations of the p variables X that contain the greatest variance.

_1210832918.unknown

_1210832919.unknown

_1210832917.unknown

• Scree PlotShows the number of significant components.

• Percentage Explained

Principal Components Analysis

Component

Percent of

Cumulative

Number

Eigenvalue

Variance

Percentage

1

5.8263

72.829

72.829

2

1.09626

13.703

86.532

3

0.339796

4.247

90.779

4

0.270321

3.379

94.158

5

0.179286

2.241

96.400

6

0.12342

1.543

97.942

7

0.109412

1.368

99.310

8

0.0552072

0.690

100.000

• Components

Table of Component Weights

Component

Component

1

2

Engine Size

0.376856

-0.205144

Horsepower

0.292144

-0.592729

Passengers

0.239193

0.730749

Length

0.369908

0.0429221

Wheelbase

0.374826

0.259648

Width

0.38949

-0.0422083

U Turn Space

0.359702

-0.0256716

Weight

0.396236

-0.0298902

First component

0.376856*Engine Size + 0.292144*Horsepower + 0.239193*Passengers + 0.369908*Length

+ 0.374826*Wheelbase + 0.38949*Width + 0.359702*U Turn Space + 0.396236*Weight

Second component

-0.205144*Engine Size 0.592729*Horsepower + 0.730749*Passengers + 0.0429221*Length

+ 0.259648*Wheelbase - 0.0422083*Width - 0.0256716*U Turn Space 0.0298902*Weight

• Interpretation

• Principal Component Regression

MPG City = 22.3656 - 1.84685*size + 0.567176*unsportiness

Standard

T

Parameter

Estimate

Error

Statistic

P-Value

CONSTANT

22.3656

0.353316

63.302

0.0000

size

-1.84685

0.147168

-12.5492

0.0000

unsportiness

0.567176

0.339277

1.67172

0.0981

R-squared = 64.0399 percent

R-squared (adjusted for d.f.) = 63.2408 percent

Standard Error of Est. = 3.40726

Mean absolute error = 2.26553

• Partial Least Squares (PLS)Similar to PCA, except that it finds components that minimize the variance in both the Xs and the Ys.

May be used with many X variables, even exceeding n.

• Component ExtractionStarts with number of components equal to the minimum of p and (n-1).

• Coefficient Plot

• Model in Original Units

MPG City = 50.0593 0.214083*Engine Size - 0.0347708*Horsepower

- 0.884181*Passengers + 0.0294622*Length - 0.0362471*Wheelbase

- 0.0882233*Width - 0.0282326*U Turn Space - 0.00391616*Weight

• ClassificationPrincipal components can also be used to classify new observations.

A useful method for classification is a Bayesian classifier, which can be expressed as a neural network.

• 6 Types of Automobiles

• Neural Networks

• Bayesian ClassifierBegins with prior probabilities for membership in each group

Uses a Parzen-like density estimator of the density function for each group

• OptionsThe prior probabilities may be determined in several ways.A training set is usually used to find a good value for s.

• Output

Number of cases in training set: 93

Number of cases in validation set: 0

Spacing parameter used: 0.0109375 (optimized by jackknifing during training)

Training Set

Percent Correctly

Type

Members

Classified

Compact

16

75.0

Large

11

100.0

Midsize

22

77.2727

Small

21

76.1905

Sporty

14

85.7143

Van

9

100.0

Total

93

82.7957

• Classification Regions

• Changing Sigma

• Overlay Plot

• Outlier Detection

• Cluster Analysis

• Design of ExperimentsWhen 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 #3Myers and Montgomery (2002) describe an experiment on a chemical process:

• Experiment

run

time

temperature

catalyst

conversion

activity

(minutes )

(degrees C )

(percent )

1

10.0

170.0

2.0

74.0

53.2

2

15.0

170.0

2.0

51.0

62.9

3

10.0

200.0

2.0

88.0

53.4

4

15.0

200.0

2.0

70.0

62.6

5

10.0

170.0

3.0

71.0

57.3

6

15.0

170.0

3.0

90.0

67.9

7

10.0

200.0

3.0

66.0

59.8

8

15.0

200.0

3.0

97.0

67.8

9

8.3

185.0

2.5

76.0

59.1

10

16.7

185.0

2.5

79.0

65.9

11

12.5

160.0

2.5

85.0

60.0 ##### CONSULTING, STATISTICAL 141 MULTIVARIATE ANALYSIS STATISTICAL INFERENCE) · PDF file 2016-03-23 · MULTIVARIATE ANALYSIS STATISTICAL INFERENCE) CONSULTING, STATISTICAL DEFINITION
Documents ##### 1 Statistical Tools for Multivariate Six Sigma Dr. Neil W. Polhemus CTO & Director of Development StatPoint, Inc
Documents ##### Multivariate Statistical Process Control: an .1 Multivariate Statistical Process Control: an introduction
Documents ##### Applied Multivariate Statistical 2008-03-24¢  on Applied Multivariate Statistical Analysis presents
Documents ##### Multivariate Statistical Process .APPLICATIONS â€¢ Apply multivariate statistical methods quality
Documents ##### Comparing Multivariate Statistical Techniques and ... Comparing Multivariate Statistical Techniques
Documents ##### Mass++ : Statistical and multivariate analysis ... Mass++ : Statistical and multivariate analysis functions
Documents Documents