indice de capacidad multivariada

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Introduction

Univariate Capability Indices

Multivariate Capability Indices

Principal Component Analysis

Preliminary results

Conclusions

Multivariate analysis to measure the capability

of a process

Roberto Jose Herrera Acosta.

Universidad del Atlntico

FACULTAD DE INGENIERIA

Barranquilla 2013

Roberto Jose Herrera Acosta Multivariate analysis
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Introduction




Preliminary results

Conclusions

Content

1 Introduction

2 Univariate Capability Indices

3 Multivariate Capability Indices

4 Principal Component Analysis

5 Preliminary results

6 Conclusions

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Introduction




Preliminary results

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Introduction

After estimating the parameters of a process, the next step

is to determine the process capability

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Introduction




Preliminary results

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Introduction



In multivariate quality control there is no consensus on the

methodology required to measure the capability of a

process


I d i
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Introduction




Preliminary results

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Introduction



In multivariate quality control there is no consensus on the

methodology required to measure the capability of a

process

In this paper, two schemes are usually presented tocalculate the capability of a multivariate process in the field


I t d ti
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Preliminary results

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Content

1 Introduction





6 Conclusions


Introduction
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Preliminary results

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Capacity Indices

Univariate

It is estimated as the ratio of the allowable variability and thevariability observed.

Cp=

LES LEI6

Ca=1 | m|d


Introduction
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Preliminary results

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features

In this scheme univariate variable quality characteristic is

assumed normally distributed.

A Process mean is centered in the region of tolerance,the index of capabilityCp=1, indicating that the 0,27%ofthe units are nonconforming.

Conversely if the process mean is far from the center, it is

likely that the process is producing a significant percentageof nonconforming units.


Introduction
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Preliminary results

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Index centering capability

Univariate

Cpk=min

LES

3

, LEI

3


Introduction
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Preliminary results

Conclusions

Index centering capability

Univariate

Cpk=min

LES 3

, LEI

3

features

The capability index has a sensitivity to the changes in the

magnitude of the variance of the process, the centering

and the specification limits.

Depending on the above changes, the percentage of

nonconformity of the process, despite having the same

Cpk, can be significantly different.

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Content

1 Introduction





6 Conclusions

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Introduction to Multivariate capability index

Multivariate capability

Each of the vvariables has some technical specifications,where the vector 0 contains the target values for each of

the characteristics of quality measures in the process.

The objective is to use the data X, the mean vector and

covariance matrix , or distribution process, comparedwith engineering specifications to reach an acceptabledefinition of capability.


Introduction
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Preliminary results

Conclusions

features

Multivariate capability

In the univariate case, the location and dispersion of the

normal curve defines a values range, as well as tolerancelimits define an interval.

As univariate capability indices provide a comparison

between these lengths.

In the multivariate case the comparison is more complex,assuming multivariate normal distribution, the intervals are

similar to elliptical or ellipsoidal contours.


Introduction

U i i C bili I di
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Preliminary results

Conclusions

featuresMultivariate capability

In two dimensions, the ranges of tolerance form a

rectangular tolerance region, in three or more dimensions

define a hypercube.

For more complex specifications, tolerance regions may

have complicated shapes.

Multivariate

The comparison of various forms, locations, sizes and

orientations that arise from the statistical distribution, as well as

process specifications, leads to quite different definitions of

capability in the multivariate domain.


Introduction

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Proposals multivariate capability indices

Proposals

The reason for the region of tolerance to the region of the

process.

The proportion of nonconforming products.

other approaches that use loss functions.

Using principal components.

In this paper we discuss two methods: based on theproportion of nonconforming products and the method by

principal components

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Preliminary results

Conclusions

The proportion of nonconforming products

Multivariate

Boyles(1994)propose the following overall capability index,

Cpkj=1

31

vj=1

2

3Cpkj 1+1

2

whereCpkjdenotes theCpkvalue of thejthcharacteristic forj=1,2, ..., vandv is the number of characteristics.


IntroductionUnivariate Capability Indices
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The proportion of nonconforming products

Multivariate

The new index,CTp

k, may be viewed as a generalization of the

single characteristic yield index, Cpk, considered by Boyles

(1994).

CTpk=1

3

1

1

2

1 Cdr

Cdp +

1

2

1 +Cdr

Cdp whereCdr=

(T)d andCdp=

d


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Content

1 Introduction

2





6 Conclusions


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PCA

As for the principal component analysis(PCA)the objective isto obtain a reduced dimension or construct new linearly

independent variables as shown in Wang (1998). If X is a

matrix of rankr, then its singular value decomposition is:

X=U(r)D(r)Vt(r) (1)

whereV(r) andU(r) are matrices whose columns are theorthonormalized vectors associated with nonzero eigenvalues

ofXtX,XXt andD(r) is the diagonal matrix diag

(12...

r)


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p y



Preliminary results

Conclusions


PCA

The main standards are obtained by applying the singular value

decomposition to the matrix ZtZ wherezij is the corresponding

standard element ofX. It decomposition defines thecoordinates ofnproduct on the th principal axis as:

PCA

=Zv =p

j=1

vjZj

where vectorV is the th column vectorV(r)


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Preliminary results

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PCA

the coordinates of thepvariables overth principal axis aredefined:


=Ztu =

ni=1

uiZi


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PCA

PCA

The variability associated with each principal component

defines the ratio of each eigenvalue to the sum of eigenvaluesas shown:

PCA

p=1

= p (2)


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Preliminary results

Conclusions


PCA

To get the multivariate capability index is calculated a value of

Zby the standardized principal component transformation, the

value ofZis obtained from the probability measure 1 P.WherePis defined as:

pi=1

P0,1[ai


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Preliminary results

Conclusions


PCA

To get the multivariate capability index is calculated by:

MCpk= 1

31()

PCA

whereis the probability that a process produce a conformingproduct



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PCA

Another proposal is Wang and Chen(1998)

MCp=

vi=1

Cp,PCi

1v

PCwhereCp,PCiis univariate capability index for the j th

component andvis the number of selected eigenvalues.



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Content

1 Introduction





6 Conclusions



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Cuadro:Comparison table multivariate capability indices

1 Chen PCA0.9973 % 0,73 1,04/1,21= 0,860.9900 % 1,07 1,12/1,21= 0,920.9500 %

1,33

1,54/1,21= 1,27

Results

For 95%the method of Chen(2001) and the indices of themethodsPCAhave values greater than unity.

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Preliminary results

Conclusions

Conclusions

The notion of capacity in the multivariate context is not

uniquely defined.The method Chen(2001) an the principal component

application allows to evaluate the probability measure of

the simultaneous implementation of the specifications and

produce an index of capacity subject to the values ofz

related to the probability measures of compliance.





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p y


Preliminary results

Conclusions

END OF PRESENTATION

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