Statistical Tools for Multivariate Six Sigma. Dr. Neil W. Polhemus CTO & Director of Development StatPoint, Inc.

Transcription

1 Statistical Tools for Multivariate Six Sigma Dr. Neil W. Polhemus CTO & Director of Development StatPoint, Inc. 1

2 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. 2

3 The Solution Proper analysis of data from such processes requires the use of multivariate statistical techniques. 3

4 Outline Multivariate SPC Multivariate control charts Multivariate capability analysis Data exploration and modeling Principal components analysis (PCA) Partial least squares (PLS) Neural network classifiers Design of experiments (DOE) Multivariate optimization 4

5 Example #1 Textile fiber Characteristic #1: tensile strength ± 1 Characteristic #2: diameter ±

6 Sample Data n = 100 6

7 X Individuals Chart - strength X Chart for strength CTR = UCL = LCL = Observation 7

8 X Individuals Chart - diameter X Chart for diameter CTR = 1.05 UCL = 1.06 LCL = Observation 8

9 frequency Capability Analysis - strength Process Capability for strength LSL = 114.0, Nominal = 115.0, USL = Normal Mean= Std. Dev.= DPM = Cp = 1.41 Pp = 1.40 Cpk = 1.38 Ppk = 1.36 K = strength 9

10 frequency Capability Analysis - diameter Process Capability for diameter LSL = 1.04, Nominal = 1.05, USL = 1.06 DPM = Normal Mean= Std. Dev.= Cp = 1.41 Pp = 1.36 Cpk = 1.39 Ppk = 1.35 K = diameter 10

11 strength Scatterplot Plot of strength vs diameter correlation = diameter 11

12 Multivariate Normal Distribution Multivariate Normal Distribution strength diameter 12

13 strength Control Ellipse Control Ellipse diameter 13

14 Multivariate Capability Determines joint probability of being within the specification limits on all characteristics Observed Estimated Estimated Variable Beyond Spec. Beyond Spec. DPM strength 0.0% % diameter 0.0% % Joint 0.0% %

15 Multivariate Capability Multivariate Normal Distribution DPM = strength diameter 15

16 diameter Capability Ellipse % Capability Ellipse MCP = strength 16

17 Mult. Capability Indices Defined to give the same DPM as in the univariate case. Capability Indices Index Estimate MCP 1.27 MCR DPM Z SQL

18 empirical data Test for Normality P-Values Shapiro-Wilk strength diameter Probability Plot strength diameter normal distribution 18

19 More than 2 Characteristics Calculate T-squared: T 2 i ( x x) 1 S i ( x i x) where S = sample covariance matrix x = vector of sample means 19

20 T-Squared T-Squared Chart 30 Multivariate Control Chart UCL = Observation 20

21 T-Squared Decomposition Subtracts the value of T-squared if each variable is removed. T-Squared Decomposition Relative Contribution to T-Squared Signal Observation T-Squared diameter strength Large values indicate that a variable has an important contribution. 21

22 rnormal(100,10,1) Control Ellipsoid Control Ellipsoid diameter strength 22

23 T-Squared Multivariate EWMA Chart Multivariate EWMA Control Chart UCL = 11.25, lambda = 0.2 Largest strength diameter Observation 23

24 Gen. Variance Generalized Variance Chart Plots the determinant of the variance-covariance matrix for data that is sampled in subgroups. (X 1.E-7) Generalized Variance Chart UCL = 3.281E-7 CL = E-8 LCL = Subgroup 24

25 Data Exploration and Modeling When 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. 25

26 Example #2 26

27 Matrix Plot MPG City MPG Highway Engine Size Horsepow er Fueltank Passengers Length Wheelbase Width U Turn Space Weight 27

28 Analysis Methods Predicting certain characteristics based on others (regression and ANOVA) Separating items into groups (classification) Detecting unusual items 28

29 Multiple Regression MPG City = *Engine Size *Horsepower *Passengers *Length *Wheelbase *Width *U Turn Space *Weight Standard T Parameter Estimate Error Statistic P-Value CONSTANT Engine Size Horsepower Passengers Length Wheelbase Width U Turn Space Weight R-squared = percent R-squared (adjusted for d.f.) = percent Standard Error of Est. = Mean absolute error =

30 Principal Components The goal of a principal components analysis (PCA) is to construct k linear combinations of the p variables X that contain the greatest variance. C1 a11x1 a12x 2... a1 p X p C2 a21x1 a22x 2... a2 p X p C k a X a X... k1 1 k 2 2 a kp X p 30

31 Eigenvalue Scree Plot Shows the number of significant components. 6 5 Scree Plot Component 31

32 Percentage Explained Principal Components Analysis Component Percent of Cumulative Number Eigenvalue Variance Percentage

33 Components Table of Component Weights Component Component 1 2 Engine Size Horsepower Passengers Length Wheelbase Width U Turn Space Weight First component *Engine Size *Horsepower *Passengers *Length *Wheelbase *Width *U Turn Space *Weight Second component *Engine Size *Horsepower *Passengers *Length *Wheelbase *Width *U Turn Space *Weight 33

34 C_2 Interpretation Plot of C_2 vs C_ Type Compact Large Midsize Small Sporty Van C_1 34

35 Principal Component Regression MPG City = *size *unsportiness Standard T Parameter Estimate Error Statistic P-Value CONSTANT size unsportiness R-squared = percent R-squared (adjusted for d.f.) = percent Standard Error of Est. = Mean absolute error =

36 Partial Least Squares (PLS) Similar to PCA, except that it finds components that minimize the variance in both the X s and the Y s. May be used with many X variables, even exceeding n. 36

37 Percent variation Component Extraction Starts with number of components equal to the minimum of p and (n-1). Model Comparison Plot X Y Number of components 37

38 Engine Size Horsepower Passengers Length Wheelbase Width U Turn Space Weight Stnd. coefficient Coefficient Plot PLS Coefficient Plot MPG City MPG Highway Fueltank

39 Model in Original Units MPG City = *Engine Size *Horsepower *Passengers *Length *Wheelbase *Width *U Turn Space *Weight 39

40 Classification Principal 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. 40

41 unsportiness 6 Types of Automobiles Plot of unsportiness vs size Type Compact Large Midsize Small Sporty Van size 41

42 Neural Networks Input layer Pattern layer Summation layer Output layer (2 variables) (93 cases) (6 neurons) (6 groups) 42

43 Bayesian Classifier Begins with prior probabilities for membership in each group Uses a Parzen-like density estimator of the density function for each group g j ( X ) 1 n j n j i 1 exp X X 2 i 2 43

44 Options The prior probabilities may be determined in several ways. A training set is usually used to find a good value for. 44

45 Output Number of cases in training set: 93 Number of cases in validation set: 0 Spacing parameter used: (optimized by jackknifing during training) Training Set Percent Correctly Type Members Classified Compact Large Midsize Small Sporty Van Total

46 unsportiness Classification Regions Classification Plot sigma = Type Compact Large Midsize Small Sporty Van size 46

47 unsportiness Changing Sigma Classification Plot sigma = 0.3 Type Compact Large Midsize Small Sporty Van size 47

48 unsportiness Overlay Plot Classification Plot sigma = 0.3 Type Compact Large Midsize Small Sporty Van size 48

49 unsportiness Outlier Detection 5 Control Ellipse size 49

50 unsportiness Cluster Analysis Cluster Scatterplot Method of k-means,squared Euclidean Cluster Centroids size 50

51 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. 51

52 Example #3 Myers and Montgomery (2002) describe an experiment on a chemical process: Response variable Conversion percentage Goal 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% 52

53 Experiment run time temperature catalyst conversion activity (minutes ) (degrees C ) (percent )

54 Step #1: Model Conversion Standardized Pareto Chart for conversion AC C:catalyst CC B:temperature BB BC AA AB A:time Standardized effect 54

55 catalyst Step #2: Optimize Conversion Goal: maximize conversion Optimum value = Factor Low High Optimum time temperature catalyst Contours of Estimated Response Surface temperature= time conversion

56 Step #3: Model Activity Standardized Pareto Chart for activity A:time C:catalyst AA AB B:temperature BC BB CC AC Standardized effect 56

57 catalyst Step #4: Optimize Activity Goal: maintain activity at 57.5 Optimum value = 57.5 Factor Low High Optimum time temperature catalyst Contours of Estimated Response Surface temperature=210.0 activity time 57

58 Desirability, d Step #5: Select Desirability Fcns. Maximize Desirability Function for Maximization s = 0.2 s = s = 1 s = s = 8 0 Low Predicted response High 58

59 Des irability, d Desirability Function Hit Target 1 Desirability Function for Hitting Target s = 0.1 t = 0.1 s = 1 t = 1 s = 5 t = Low Target Predicted response High 59

60 Combined Desirability m I I I 1/ I d 1d 2 d m j j D m where m = # of factors and 0 I j 5. D ranges from 0 to 1. 60

61 Example Optimum value = Factor Low High Optimum time temperature catalyst Weights Weights Response Low High Goal First Second Impact conversion Maximize activity Response Optimum conversion activity

62 catalyst Desirability Contours Contours of Estimated Response Surface temperature= time Desirability

63 Desirability Desirability Surface Estimated Response Surface temperature= time catalyst 63

64 catalyst Overlaid Contours Overlay Plot temperature=210.0 conversion activity time 64

65 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. 65

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