**5. Methodology and Results**

*Identification*

44 Practical Concepts of Quality Control

*Estimation:*

es the R2

*Checking:*

*Forecast:*

with this feature.

tributed (i.i.d. condition).

Identification is the most critical phase of the "Box and Jenkins" methodology, it is possible that several researchers to identify different models for the same series, using different crite‐ ria of choice (ACF, PACF, Akaike, etc..). Typically, the models should be parsimonious. The study analyzes the ACF and PACF, and attempts to identify the model. The process seeks to determine the order of (p,d,q), based on the behavior of the Autocorrelation Functions

After identifying the best model should then adjust and examine it. The adjusted models are compared using several criteria. One of the criteria is the of parsimony, in which it appears that the incorporation of coefficients additional improves the degree of adjustment (increas‐

freedom. One of ways to improve the degree of adjustment of this model to time series data

The inclusion of additional lags implies increasing the number of repressors, which leads to a reduction in the sum of squared residuals estimated. Currently, there are several criteria for selection of models that generate a trade-off between reductions in the sum of squared

Generally, when working with lagged variables are lost about the time series under study. Therefore, to compare alternative models (or competitors) should remain fixed number of

Aspiring to know the efficacy of the model found, takes place waste analysis. If the residuals are autocorrelated, then the dynamics of the series is not completely explained by the coeffi‐ cients of the fitted model. It should be excluded from the process of choosing the model(s)

An analysis of existence (or not) of serial autocorrelation of waste is made based on the func‐ tions of autocorrelation and partial autocorrelation of waste and their respective correlo‐ grams. It is noteworthy that, when estimating a model, it is desired that the error produced by it have characteristic "white noise" that is, this will be independent and identically dis‐

Predictions can be ex-ante, made to calculate future values of short-term variable in the study. Or, ex-post held to generate values within the sample period. The better these last, the more efficient the model estimated. We choose the best model throught the lower Mean Absolute Percentage Error (MAPE). It is a formal measure of the quality of forecasts ex-post. There‐ fore, the lower value of the MAPE is the best fit of forecasts of the model to time series data.

and reduces the sum of squared residuals) model, but you reduces the degrees of

(ACF) and Partial Autocorrelation (PACF), as well as their respective correlograms.

is to include lags additional in Cases AR (p), MA (q), ARMA (p, q) and ARIMA.

residuals and estimated a more parsimonious model.

information used for all models compared.

In this work we analyzed the Têxtil Oeste Ltda industry, whose Statistical Control of Proc‐ esses implantation happened in 1999. Here, we limited to analyze the control charts for con‐ tinuous variables as tools used for the control of the process. The conventional Shewhart control charts were used added of other appropriated models to transformations of autocor‐ relations data in data that are independent and usually distributed.

In thread's polypropylene process there are several outputs to consider critical. One of these outputs is the thread's resistance. In an effort to develop a control plan to assure quality of the appropriate surface, it was certain that the resistance has a main impact on surface quali‐ ty of the thread. So, to verify the quality of the thread, it's resistance should be controlled.

At once, the data used in this study is the daily data of the thread's polypropylene resistan‐ ces control.

These data are for the models identification and estimation and for the models predictive capacity analysis. Before control charts be applied, three fundamental assumptions must be met: The process is under control; the data are normally distributed; and the observations are independent.

Montgomey (2009) considers that the points out of control are stipulated reasonably well for the controls charts of Shewhart when the normality assumption is somewhat violated, but when observations aren't independent, control charts yield deceiving results. Many process‐ es don't produce independent observations. Alwan (1991) describes a method for control charting with autocorrelated data. The method involves fitting a time series curve and con‐ trol charting the residuals.

It was made a study that helped to verify where it is the largest instability of the process, so that we can make a better control of the system. It is suspected that the daily thread's resist‐ ance data aren't independent, and the result of a plot of these data, as showing in Figure 4, supports this belief.

The problem is to implement statistical control for a process that has autocorrelation (Dob‐ son, 1995). The Figure 4 shows us the great data variability. Calculations were performed to confirm the autocorrelation's suspected.

Calculations were done to confirm the suspected autocorreation. The autocorrelation coeffi‐ cient for thread's resistance is defined as

$$r\_k = \frac{\sum\_{t=1}^{n-k} (\alpha\_t - \bar{\alpha})(\alpha\_{t+k} - \bar{\alpha})}{\sum\_{t=1}^n (\alpha\_t - \bar{\alpha})^2}, \ k = 0, 1, 2, \dots$$

where *k* = time periods ahead

*n* = total number of data

**Autocorrelations coefficients Thread's resistance** (Standard errors are white-noise estimates)


As we can see, the data are highly autocorrelated. The autocorrelation coefficients for lags 1-7 exceed two the standard errors. Before a control charts can be used, these data must be

To find an independent, normally distributed data set, Montgomery (2009) recommends to

The Box & Jenkins's methodology was used, to determine the parameters of the model (Box,

**Partial autocorrelations coefficients Thread's resistance** (Standard errors assume AR order of k-1)


model the structure and to develop the control charting of the residuals directly.

transformed to guarantee the independence of each observation.

24 +,025 ,0306 23 +,010 ,0306 22 +,027 ,0306 21 +,051 ,0307 20 +,045 ,0307 19 +,052 ,0307 18 +,028 ,0307 17 +,002 ,0307 16 +,093 ,0307 15 +,026 ,0308 14 +,056 ,0308 13 +,057 ,0308 12 +,015 ,0308 11 +,065 ,0308 10 +,088 ,0308 9 +,078 ,0308 8 +,059 ,0309 7 +,103 ,0309 6 +,112 ,0309 5 +,150 ,0309 4 +,108 ,0309 3 +,127 ,0309 2 +,128 ,0309 1 +,158 ,0310 Lag Corr. S.E.

**Figure 5.** Autocorrelations coefficients.

Jenkins and Reinsel, 2008).

24 +,004 ,0310 23 -,020 ,0310 22 +,002 ,0310 21 +,018 ,0310 20 +,017 ,0310 19 +,027 ,0310 18 +,004 ,0310 17 -,038 ,0310 16 +,065 ,0310 15 -,012 ,0310 14 +,020 ,0310 13 +,025 ,0310 12 -,037 ,0310 11 +,012 ,0310 10 +,036 ,0310 9 +,029 ,0310 8 -,002 ,0310 7 +,046 ,0310 6 +,055 ,0310 5 +,109 ,0310 4 +,068 ,0310 3 +,096 ,0310 2 +,106 ,0310 1 +,158 ,0310 Lag Corr. S.E.

**Figure 6.** Partial autocorrelations coefficients.

169,9 ,0000 169,2 ,0000 169,1 ,0000 168,3 ,0000 165,6 ,0000 163,5 ,0000 160,6 ,0000 159,8 ,0000 159,7 ,0000 150,6 ,0000 149,9 ,0000 146,6 ,0000 143,2 ,0000 142,9 ,0000 138,5 ,0000 130,4 ,0000 124,0 ,0000 120,3 ,0000 109,2 ,0000 96,06 ,0000 72,41 ,0000 60,15 ,0000 43,26 ,0000 26,15 ,0000 Q <sup>p</sup>

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47

**Figure 4.** Daily data.

The standard error at lag *k*, if *<sup>k</sup>* =1 is*Sek* <sup>=</sup> <sup>1</sup> *<sup>n</sup>* , and the standard error at lag *k*, if *k*1 is

$$S \, e\_k = \sqrt{\frac{1}{n} \left( 1 + 2 \sum\_{i=1}^{k-1} r\_i^2 \right)}$$

The autocorrelation coefficient for *k* =1 and *k* =2are:

$$r\_1 = \frac{160680,24}{971034} = 0,164$$

and

$$r\_2 = \frac{100113 \text{ 10 }}{971034} = 0,100$$

The standard error for *k* =1 and *k* =2are:

$$S \, e\_1 = \sqrt{\frac{1}{1041}} = 0,0310$$

and

$$Se\_2 = \sqrt{\frac{1}{1041} \text{L} + 2(0, 16)^2} \text{J} = 0.0317 \text{J}$$

The Figure 5 shows the autocorrelation coefficients and 2 standard errors for these coeffi‐ cients for up to 24 lags, and the Figure 6 shows the partial autocorrelation coefficients and the 2 standard errors for these coefficients for up to 24 lags.

**Figure 5.** Autocorrelations coefficients.

**Real observations**

Daily data

The Figure 5 shows the autocorrelation coefficients and 2 standard errors for these coeffi‐ cients for up to 24 lags, and the Figure 6 shows the partial autocorrelation coefficients and

*<sup>n</sup>* , and the standard error at lag *k*, if *k*1 is


500

550

600

650

700

750

800

850

Resistance

**Figure 4.** Daily data.

*<sup>n</sup>* (1 <sup>+</sup> <sup>2</sup>∑ *i*=1 *k*−1 *ri* 2 )

<sup>971034</sup> =0,16

<sup>971034</sup> =0,10

<sup>1041</sup> =0,0310

<sup>1041</sup> <sup>1</sup> <sup>+</sup> 2(0,16)

The standard error for *k* =1 and *k* =2are:

*Sek* <sup>=</sup> <sup>1</sup>

and

*<sup>r</sup>*<sup>1</sup> <sup>=</sup> 160680,24

*<sup>r</sup>*<sup>2</sup> <sup>=</sup> 100113,10

*Se*<sup>1</sup> <sup>=</sup> <sup>1</sup>

*Se*<sup>2</sup> <sup>=</sup> <sup>1</sup>

and

500

The standard error at lag *k*, if *<sup>k</sup>* =1 is*Sek* <sup>=</sup> <sup>1</sup>

The autocorrelation coefficient for *k* =1 and *k* =2are:

<sup>2</sup> =0,0317

the 2 standard errors for these coefficients for up to 24 lags.

550

600

650

700

750

800

850

46 Practical Concepts of Quality Control

As we can see, the data are highly autocorrelated. The autocorrelation coefficients for lags 1-7 exceed two the standard errors. Before a control charts can be used, these data must be transformed to guarantee the independence of each observation.

To find an independent, normally distributed data set, Montgomery (2009) recommends to model the structure and to develop the control charting of the residuals directly.

The Box & Jenkins's methodology was used, to determine the parameters of the model (Box, Jenkins and Reinsel, 2008).

**Figure 6.** Partial autocorrelations coefficients.

**Partial autocorrelation coefficients for transformed data** Thread's resistence: ARIMA (1,1,1) Residuals (Standard errors assume AR order of k-1)


<sup>2</sup> =5,991. As the calculation qui-square value was *χ* <sup>2</sup> =

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49

Figure 8 and 9 show that the defined data is independent from an observation to another

5,0415, and it is smaller than the critical value, the data are considered as normal. Now the

**Lower Limit Upper Limit Obs Exp (Obs-Exp)²/ Exp** 649,0708 743,3123 744 696,1916 3,2831 648,9230 743,4601 672 696,1916 0,8406 648,7757 743,6074 702 696,1916 0,0485 648,6288 743,7543 690 696,1916 0,0551 648,4823 743,9008 720 696,1916 0,8142 Total 5,0415

observation. And the table 1 shows the Chi-square test to verify the normality.

24 -,010 ,0310 23 -,032 ,0310 22 -,007 ,0310 21 +,006 ,0310 20 +,002 ,0310 19 +,008 ,0310 18 -,019 ,0310 17 -,052 ,0310 16 +,047 ,0310 15 -,036 ,0310 14 -,002 ,0310 13 -,003 ,0310 12 -,062 ,0310 11 -,007 ,0310 10 +,008 ,0310 9 -,005 ,0310 8 -,035 ,0310 7 +,012 ,0310 6 +,016 ,0310 5 +,058 ,0310 4 +,003 ,0310 3 +,024 ,0310 2 +,018 ,0310 1 -,001 ,0310 Lag Corr. S.E.

For two degrees of freedom, *χ*0,05

**Table 1.** Test the Chi-square.

**Figure 9.** Partial autocorrelations coefficients for transformed data.

behavior of the productive process can be verified.

Figure 10 shows (*<sup>X</sup>*¯) and (*<sup>S</sup>*) charts for the real data.

The Chi-square test was executed, to verify the normality:

**Figure 7.** Residuals of thread's resistance.

The Figures 8 and 9 show that the obtained model is adapted to the resistance data. The au‐ tocorrelation coefficients were calculated for the transformed data defined for the model ARIMA (1,1,1), to validate that the autocorrelation has been removed from the data.

**Figure 8.** Autocorrelation coefficients.

**Figure 9.** Partial autocorrelations coefficients for transformed data.

Figure 8 and 9 show that the defined data is independent from an observation to another observation. And the table 1 shows the Chi-square test to verify the normality.

For two degrees of freedom, *χ*0,05 <sup>2</sup> =5,991. As the calculation qui-square value was *χ* <sup>2</sup> = 5,0415, and it is smaller than the critical value, the data are considered as normal. Now the behavior of the productive process can be verified.

The Chi-square test was executed, to verify the normality:


**Table 1.** Test the Chi-square.

**Thread's resistance ARIMA (1,1,1) Residuals**

Daily data

The Figures 8 and 9 show that the obtained model is adapted to the resistance data. The au‐ tocorrelation coefficients were calculated for the transformed data defined for the model

ARIMA (1,1,1), to validate that the autocorrelation has been removed from the data.

**Autocorrelation coefficients for transformed data** Thread' resistance: ARIMA (1,1,1) Residuals (Standard errors are white-noise estimates)




19,83 ,7063 19,72 ,6586 18,73 ,6619 18,52 ,6158 18,42 ,5597 18,42 ,4947 18,38 ,4308 17,85 ,3986 14,08 ,5929 11,82 ,6929 10,19 ,7479 10,16 ,6806 10,12 ,6052 6,34 ,8497 6,29 ,7905 6,17 ,7231 6,16 ,6296 5,10 ,6476 4,88 ,5593 4,60 ,4664 ,96 ,9161 ,95 ,8137 ,34 ,8450 ,00 ,9738 Q p



0

50

100

150

Resistance


**Figure 7.** Residuals of thread's resistance.

24 -,010 ,0306 23 -,031 ,0306 22 -,014 ,0307 21 +,010 ,0307 20 -,001 ,0307 19 +,006 ,0307 18 -,022 ,0307 17 -,060 ,0307 16 +,046 ,0308 15 -,039 ,0308 14 -,005 ,0308 13 -,006 ,0308 12 -,060 ,0308 11 -,007 ,0308 10 +,011 ,0308 9 -,003 ,0309 8 -,032 ,0309 7 +,015 ,0309 6 +,016 ,0309 5 +,059 ,0309 4 +,003 ,0309 3 +,024 ,0309 2 +,018 ,0310 1 -,001 ,0310 Lag Corr. S.E.

**Figure 8.** Autocorrelation coefficients.



0

50

100

150

48 Practical Concepts of Quality Control

Figure 10 shows (*<sup>X</sup>*¯) and (*<sup>S</sup>*) charts for the real data.

Figure 11 shows (*<sup>X</sup>*¯) and (*<sup>S</sup>*) charts for transformed data. Verifications revealed that the sys‐ tem had been drained during this time period and actions were taken to correct the prob‐ lem. The problem was in the first observations, which were ignored, the normality condition

Applications of Control Charts Arima for Autocorrelated Data

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51

Through the Figure 11 we can observe that the control charts for the same data, indicate that the residual values are practically inside of control limits for the average. According to War‐ dell, Moskowitz and Plant (1994) it is entirely possible in traditional control charts, the points are out of the limits because of the systematic or the common causes and not because

According to Reid and Sanders (2002), there are several types of statistical quality control (SQC) techniques. One category of SQC techniques consists of descriptive statistics tools such as the mean, range, and standard deviation. These tools are used to describe quality characteristics and relationships. Another category of SQC techniques consists of statistical process control (SPC) methods that are used to monitor changes in the production process. To understand SPC methods you must understand the differences between common and as‐

Common causes of variation are based on random causes that cannot be identified. A certain amount of common or normal variation occurs in every process due to differences in materi‐ als, workers, machines, and other factors. Assignable causes of variation, on the other hand, are variations that can be identified and eliminated. An important part of statistical process control (SPC) is monitoring the production process to make sure that the only variations in the process are those due to common or normal causes. Under these conditions we say that a production process is in a state of control. You should also understand the different types of quality control charts that are used to monitor the production process: x-bar charts, R-range

In this chapter we show how to use the techniques of quality control for autocorrelated data. Thus, the data collected were analyzed simultaneously, the continuous variables, to find a possible reason for lack of control in the final stages of production. We presented methods for using the techniques of statistical quality control for correlated observations. It is the au‐ tocorrelation data, is modeled by the continuous variables ARIMA. With the residuals ob‐

The traditional Shewhart control charts can be used for process control, even when the as‐ sumptions of independent observations are transgressed, by removing the autocorrelation with a time series models. For applying those techniques, the thread's resistance stayed in control state for the average. The result was a decrease in the variation of surface quality of the polypropylene thread that is produced, while simultaneously it increased the surface

was verified, and the control charts *<sup>X</sup>*¯ and S were replotted.

of occurrence of special causes.

signable causes of variation.

charts, p-charts, and c-charts, Reid and Sanders (2002).

tained in the models, we applied the Shewhart control charts.

**6. Conclusion**

quality average.

**Figure 10.** *X*¯ and *S* charts for real data.

Through the illustration 10 we can notice the sequence of observations and limits of the tra‐ ditional Shewhart charts, where several points were out of the control limits, indicating that the process is apparently out of control. In fact, before the transformation of the data, we found the data really correlated what took us to model for a process ARIMA (Wardell, Mos‐ kowitz and Plante, 1994).

**Figure 11.** (*X*¯) and (*S*) charts for transformed data.

Figure 11 shows (*<sup>X</sup>*¯) and (*<sup>S</sup>*) charts for transformed data. Verifications revealed that the sys‐ tem had been drained during this time period and actions were taken to correct the prob‐ lem. The problem was in the first observations, which were ignored, the normality condition was verified, and the control charts *<sup>X</sup>*¯ and S were replotted.

Through the Figure 11 we can observe that the control charts for the same data, indicate that the residual values are practically inside of control limits for the average. According to War‐ dell, Moskowitz and Plant (1994) it is entirely possible in traditional control charts, the points are out of the limits because of the systematic or the common causes and not because of occurrence of special causes.

### **6. Conclusion**

**X-BAR** Mean:681,795 ( *681,795* ) Proc. sigma:26,5389 ( *26,5389* ) n:35,86

Samples

**Std.Dev.** Mean: 26,3327 ( *26,3327* ) Sigma:3,16669 ( *3,16669* ) n:35,862

Samples

**X-BAR** Mean:,026976 ( *,026976* ) Proc. sigma:27,2476 ( *27,2476* ) n:35,83

Samples

**Std.Dev.** Mean: 27,0360 ( *27,0360* ) Sigma:3,25285 ( *3,25285* ) n:35,828

Samples

1 5 11 16 21 26

1 5 11 16 21 26

1 5 11 16 21 26

1 5 11 16 21 26

Through the illustration 10 we can notice the sequence of observations and limits of the tra‐ ditional Shewhart charts, where several points were out of the control limits, indicating that the process is apparently out of control. In fact, before the transformation of the data, we found the data really correlated what took us to model for a process ARIMA (Wardell, Mos‐

Histogram of Means

No of obs

No of obs

Histogram of Means

No of obs

No of obs

**Figure 11.** (*X*¯) and (*S*) charts for transformed data.

0 4 8 12

Histogram of Std.Dv's

0 4 8 12 16

0 4 8 12

Histogram of Std.Dv's

0 10 20 30

**X-Bar chart:** VAR3

50 Practical Concepts of Quality Control

 **S chart:** VAR3

 5,0 10,0 15,0 20,0 25,0 30,0 35,0 40,0 45,0

**Figure 10.** *X*¯ and *S* charts for real data.

kowitz and Plante, 1994).

**X-Bar chart:** RESIDUOS

 **S chart:** RESIDUOS

 5,0 10,0 15,0 20,0 25,0 30,0 35,0 40,0 45,0 50,0

 -20 -15 -10 -5 0 5 10 15 20

668,500 681,795 695,090

16,8492 26,3327 35,8493


17,2941 27,0360 36,8113 According to Reid and Sanders (2002), there are several types of statistical quality control (SQC) techniques. One category of SQC techniques consists of descriptive statistics tools such as the mean, range, and standard deviation. These tools are used to describe quality characteristics and relationships. Another category of SQC techniques consists of statistical process control (SPC) methods that are used to monitor changes in the production process. To understand SPC methods you must understand the differences between common and as‐ signable causes of variation.

Common causes of variation are based on random causes that cannot be identified. A certain amount of common or normal variation occurs in every process due to differences in materi‐ als, workers, machines, and other factors. Assignable causes of variation, on the other hand, are variations that can be identified and eliminated. An important part of statistical process control (SPC) is monitoring the production process to make sure that the only variations in the process are those due to common or normal causes. Under these conditions we say that a production process is in a state of control. You should also understand the different types of quality control charts that are used to monitor the production process: x-bar charts, R-range charts, p-charts, and c-charts, Reid and Sanders (2002).

In this chapter we show how to use the techniques of quality control for autocorrelated data. Thus, the data collected were analyzed simultaneously, the continuous variables, to find a possible reason for lack of control in the final stages of production. We presented methods for using the techniques of statistical quality control for correlated observations. It is the au‐ tocorrelation data, is modeled by the continuous variables ARIMA. With the residuals ob‐ tained in the models, we applied the Shewhart control charts.

The traditional Shewhart control charts can be used for process control, even when the as‐ sumptions of independent observations are transgressed, by removing the autocorrelation with a time series models. For applying those techniques, the thread's resistance stayed in control state for the average. The result was a decrease in the variation of surface quality of the polypropylene thread that is produced, while simultaneously it increased the surface quality average.

Many companies, because they believe in the advantages that can be obtained from the prac‐ tice of SQC, invest many resources in the implementation, especially of the conventional control charts, called Shewhart charts. Since it is not necessary to a thorough knowledge of statistics, is more favorable to the deployment of these graphs by the companies, but not al‐ ways the results are as expected. There is a concern with the correlation of data.

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[6] Cochrane, J. H. (2005). Time Series for Macroeconomics and Finance. Chicago:

[7] Dobson, B. (1995). Control charting dependent data: a case study. *Quality Engineer‐*

[8] Jiang, W., Tsui, K., & Woodall, W. (2000). A new SPC monitoring method: the ARMA

[9] Fischer, S. (1982). Séries Univariantes de Tempo-Metodologia Box and Jenkins. Porto

[10] Juran, J. M. (1993). Made in U.S.A.: A renaissance in quality. *Harvard Business Review*,

[11] Loureço Filho, R. C. B. (1964). Controle Estatístico de Qualidade. Rio de Janeiro: Liv‐

[12] Montgomery, D. C. (2009). Introduction to Statistical Quality Control. Vol. 1, 6 Ed.,

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[15] Reid, R. D., & Sanders, N. R. (2002). Operations Management. 3rd Edition. John Wil‐

[16] Russo, S., Rodrigues, P. M. M., & Camargo, M. E. (2006). Aplicação de Séries Tempo‐ rais na Série Teor de Umidade da Areia de Fundição da Indústria Fundimisa. *Revista Gestão Industrial*, 1808-0448, 2(1), 35-45, Universidade Tecnológica Federal do Paraná-UTFPR, Campus Ponta Grossa- Paraná- Brasil, D.O.I.: 10.3895/

[17] Shewhart, W. A. (1931). Economic control of quality of the manufactured product.

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In this context, the text presented throughout this chapter can serve as a reference to the in‐ dustries that face difficulties in deploying statistical quality control. However, one must be careful with the type of variables to analyze what is being proposed, which allows us to con‐ clude that this proposed combination of techniques for time series with control charts, claim to be complete and extended to cover all possible difficulties we can find. In the classic mod‐ el of monitoring, there is no such information to identify an non conform item, in the end of teh proces, no one knows how to do for the same does not happen, because the variables used in the previous process are autocorrelated.
