T2 CONTROL CHART TO PROCESSES WITH CROSS-AUTOCORRELATION

Requeijo, José Gomes1

Souza, Adriano Mendonça2

1UNIDEMI, Faculty of Science and Technology, New University of Lisbon, Portugal,

+351212948567, jfgr@fct.unl.pt

2Department of Statistics, Federal University of Santa Maria, Brasil,

+555532208486, amsouza@smail.ufsm.br

Abstract:

Today's society is characterized by consumers becoming more educated and demanding for products and

services they use. Wherever, organizations are structured to respond the explicit or implicit needs of

consumers, which lead to increased competitiveness of organizations. This competitiveness is reflected in the

ongoing quest to provide high quality products and excellence process development. The monitoring of

productive processes is essential to respond adequately to these objectives and therefore the implementation

of statistical techniques suited to different situations is crucial, as is the case of statistical control charts

introduced in the 1920s by Walter Shewhart. The Statistical Process Control (SPC), initially applied to a

single variable, proved inadequate in studying simultaneously several quality characteristics of the same

product. T2 Charts, introduced in 1985 by Alt, are the appropriate statistical techniques to the simultaneous

control of the averages of several product characteristics, advantages over the univariate study, once on the

other hand it prevents the production of a large number of documents and leads to more accurate analysis

by considering the correlation among variables. As for the univariate study, the multivariate control is based

on the assumptions of Normality and data independence of all characteristic. When the assumption of

independence of at least one characteristic is violated, the construction of T2 charts directly on the data is an

inadequate solution, being necessary to isolate the autocorrelation for each variable and construct T2 charts

based on residuals/forecast errors. Another relevant question, that has not deserved much attention by the

scientific community, is related with possibility of cross-autorrelation existence among variables, i.e., the

autocorrelation occurring in a variable is not exclusive of the influence of that variable but its effect may be

transmitted for part or all variables. This paper presents a methodology for implementation of multivariate

statistical control, considering the cross-autocorrelation, both for Phase I (preliminary or retrospective) and

Phase II (monitoring) statistical control. In Phase I, it seems that the process is in-control and estimates the

process parameters, the mean vector and covariance matrix. In addition, the multivariate process capability

using specific indicators will be study. One suggestion is the use of capability indices CpM, PV, and LI.

Verified the stability and capability of the process, Phase II begins consisting of monitoring the process in

real time based on estimates performed in Phase I. When there is significant autocorrelation at least in one

of the variables, it is necessary to model the process in order to contemplate cross-autocorrelation effect

acting in the variables. Variables are estimated by a VAR(p) model, considering only the significant

parameters. An example application for Phase I of multivariate statistical process control is presented, as

well as multivariate process capability.

Keywords: Multivariate Statistical Process Control (MSPC); Multivariate Process Capability;

Autocorrelation; Model VAR(p).

1. Introduction The multivariate statistical process control (MSPC) deserved attention of many researchers, such as Alt

(1985), Doganaksoy et al. (1991), Mason and Young (1998), Hayter and Tsui (1994), Lowry and

Montgomery (1995), Mason et al. (1997), Nedumaran and Pignatiello (1998), Egyptian clover et al (2007),

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822

Montgomery (2008 ), Pereira and Requeijo (2008) and Young-il and Adams (2009), among others. Due to

the further article publication by Alt (1985), its great coverage is considered by many one of the greatest

contributions to the study of this issue, being an indispensable reference to be checked.

In many cases the assumptions of normality and independence are not always verified, so that traditional

developments seem to be inadequate. To address these possible situations, it is suggested the data

transformation when distributions do not follow the Normal distribution and modeling of the process if there

is autocorrelation of quality characteristics.

As for the univariate study, multivariate control is based on the assumptions of Normality and independence

of data of all characteristic. When the assumption of independence of at least one characteristic is violated,

the construction of T2 charts directly on the data is an inadequate solution, being necessary to isolate the

autocorrelation for each variable and construct T2 charts based on residuals/forecast errors. Another

relevant question that has not deserved much attention by the scientific community is related to the

possibility of cross-autorrelation existence among variables, i.e., the autocorrelation occurring in a variable

is exclusive of the influence of that variable but its effect may be transmitted for part or all variables. This

paper presents a methodology for implementation of multivariate statistical control considering the cross-

autocorrelation, both for Phase I (preliminary or retrospective) and for Phase II (monitoring) statistical

control.

Phase I culminates with the multivariate study of process capability. If this proves ability to produce

according to technical specifications defined in the stage of design and product/process development is

initiated the Phase II, which consists of monitoring the process in real time. In Phase II, the main focus is the

identification of possible special causes of variation and subsequent corrective actions to be implemented in

the process. About multivariate analysis of process capability, are referred the studies of Taam et al. (1993),

Chen (1994), Shahriari et al. (1995), Wang et al. (2000) and Kotz and Johnson (2002).

2. Methodology After collecting data for analysis, the existence of autocorrelation among observations is verified and since it

is working on multivariate data the cross-correlation among variables is investigated.

Determined the existence of correlation between variables and the autocorrelation, it is estimated the model

of autoregression vectors (VAR), so that only the significant coefficients are present in the model, thus

enabling the estimation of residuals with blank noise characteristics, i.e. with zero mean, constant variance

and non-auto-correlated.

For the implementation of multivariate statistical control of the process mean vector, when there is

significant autocorrelation of one or more characteristics of quality is suggested modeling the process

through a VAR (p) in detriment of integrated autoregressive models and mobile means (ARIMA), so that to

be contemplated the possibility of cross-autocorrelation between variables under study. This study will be

based on developments of Box, Jenkins and Reinsel (2008), Morettin (2006), Souza et al. (2004), Enders

(1995), Maddala (1992), Alexander (2005) and Bueno (2008).

After obtaining a new data set represented by blank noise residuals originated from the modeling VAR (p)

and meeting the requirements of control charts for independence and normality is applied the multivariate

control charts.

Like the univariate statistical process control (SPC), the multivariate statistical process control (MSPC) is

also divided into two different phases, Phase I and Phase II. In Phase I is verified the process stability and in

MSPC’s Phase II future data are monitored. The traditional MSPC can only be implemented if the joint

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823

distribution of p variables is a multivariate normal distribution ( )ΣΣΣΣµµµµ ,N p with mean vector µµµµ = (µ1, µ2, ...,

µp)’and covariance matrix ΣΣΣΣ. It is also assumed that variables considered in isolation are independent and

identically distributed according to normal distributions.

2. Case Study

The application refers to the multivariate statistical process control represented by the analysis of three

quality characteristics, the cooling temperatures related to the tile firing process of a Brazilian ceramics

company. 92 hourly measurements were carried out simultaneously to these three temperatures named as

"RESF1", "RESF2" and "RESF3."

Two approaches are presented to compare the results. The first one (approach 1) consists of verifying the

existence of autocorrelation in the process and applying the methodology of VAR modeling to determine the

residues and analyzed them by the Hotelling T2 chart. The second one (approach 2) consists of implementing

the T2 charts directly to the original data series related to the three quality characteristics, with no concern to

check the existence of autocorrelation in the process.

3.1 Approach 1

Based on 92 observations was carried out the MSPC’s Phase 1 following the methodology of residues

estimation in order to address the cross-autocorrelation between variables by VAR estimation methodology.

In this context, it was verified by the FACE and FACPE that the process has significant auto-correlation, so

that its modeling should be made. It was found that the process can be adjusted by a VAR (1) model, where

the coefficients showed in equation (1) were estimated, having been considered in modeling only statistically

significant coefficients.

++++

⋅⋅⋅⋅

++++

−−−−

====

−−−−

−−−−

−−−−

t

t

t

,t

,t

,t

t

t

t

ε

ε

ε

X

X

X

, ,

, ,

,

,

,

,

X

X

X

3

2

1

13

12

11

3

2

1

46210525500

06233020160

0084900

53811

46666

626121

where "X1" "X2" and "X3" represent, respectively, the quality characteristics "RESF1", "RESF2" and

"RESF3", 0ΦΦΦΦ is the vector of coefficients defining the means of the process, 1ΦΦΦΦ is the matrix of

autoregressive coefficients and tε is the vector of residues (blank noise).

−−−−

====ΦΦΦΦ

53811

46666

626121

0

,

,

,

,

====ΦΦΦΦ

46210525500

06233020160

0084900

1

,,

,,

,

, ====tε

t

t

t

ε

ε

ε

3

2

1

Subsequently, it was verified through residues analysis by means of the estimate correlation sample function

(FACE) and partial autocorrelation sample function (FACPE) that residues are not auto-correlated and

follow the normal distribution. Based on the residues determined for each time t was constructed its T2 chart

presented in Figure 1.

Proceedings of the 41st International Conference on Computers & Industrial Engineering

824

T 2

CHART

Phase 1

0

3

6

9

12

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91

Observation n.

T 2

T ² LCL UCL

Figure 1. T2 control chart of residues

The analysis of residues’ T2 chart shows that the process on three temperatures under study is stable, ie,

under statistical control, since no statistical value is the upper limit of control. It is also verified that at the

time 53 ( 1611253 .T ==== value) the point is near the upper limit of control ( 24112 .UCL

T==== ), although still with

lower value.

3.2 Approach 2

With the original values from the 92 observations was constructed the T2 chart, which is presented in Figure

2.

T 2

CHART

Phase 1

0

2

4

6

8

10

12

14

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91

Observation n.

T 2

T ² LIC LSC

Figure 2: T2 control chart of original daa

The analysis of original data’s T2 chart wrongly reveals the existence of four special causes of variation, with

upper limit statistical values of control (items 3, 6, 53 and 54).

Comparison of Approach 1 and Approach 2

Comparing T2 control charts of residues and original data (Figure 1 and Figure 2) is verified that the effect of

ignoring the verification of auto-correlation in the process leads to a drastic increase of false alarms. This is

Proceedings of the 41st International Conference on Computers & Industrial Engineering

825

an undesirable situation, since the investigation of possible causes leading to eventual (false) anomalies in

the process will not lead to the identification of any problem.

3. Conclusions

This paper aims to develop a methodology for implementation of multivariate statistical control to processes

that exhibit significant autocorrelation. Overall, the main conclusion from this research lies in the application

of T2 control charts for auto-correlated data in detriment of control charts for traditional multivariate.

The multivariate SPC is more appropriate than the univariate when is necessary to simultaneously control

several variables. The multivariate study allows detecting possible changes in the correlation of variables

with less information for the analysis (i.e. a single graph is sufficient). However, T2 charts applied directly to

the original data, ignoring the process autocorrelation, show many signs of special causes when actually,

these signs are natural in the process. After modeling the process through VAR (p) models, the application of

T2control charts to residues is revealed to be an effective method to the statistical control of auto-correlated

processes. These models have advantages over standard models of ARIMA modeling for beholding the cross

autocorrelation of different variables under study.

Some of these mistakes can be enumerated by not being considered autocorrelation, such as:

i) Considering a stable process, when actually there are special causes of variation, which corresponds to an

unstable process, ii) considering a process out of statistical control, when it is really statistically stable

(increase of false alarms), iii) incorrectly estimating parameters of the process, iv) performing inaccurate

and/or incorrect analyses about process capability and v) loss of resources by undue interference in the

production process in order to solve problems that really do not exist (false alarms).

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