Monitoring and Identification of Influential Process Characteristics in the Presence of Autocorrelation
Chapter One
1.1 AIM AND OBJECTIVES OF THE STUDY
The aim of this research work is to develop a methodology for implementing the multivariate statistical control to processes that contain significant autocorrelation and non normal in order to improve industrial process decision.
The objectives are to
- Construct traditional Hotelling’s T 2 chart to monitor the process using the original data containing the autocorrelation and non-normality
- Subjecting the original data to VAR(1) model in order to remove the autocorrelation and transform the data to satisfy the normality assumption, and then construct Hotelling’s T2 to monitor the residuals of the VAR(1) model.
- Adopt the MYT decomposition approach to identify the influential variables among the five process characteristics under
CHAPTER TWO
LITERATURE REVIEW
INTRODUCTION
There exist two phases in Statistical Process Control (SPC), namely, phase 1 and phase 2. Phase 1 is considered as a retrospective phase, and it constitutes set of individual observations obtained from an in-control process whereby control limits are determined, which involve estimation of the unknown statistic(s), with the aim of achieving observations from an in-control process. Individual observation from such in-control process is then used as a reference data in
phase 2. The phase I control limits for the Hotelling’s T 2 control chart is as given below.
UCL and LCL are the upper control limit, lower control limit respectively.
However, phase 2 is mainly used to monitoring future observations. The phase II control limits for the T 2 control chart is as given below.
UL LCL
(2.2)
where
F , p,m p
is a 1
percentile of F distribution with parameters p and m
degree of freedom , m is the size of base sample, p is the number of variables,
X i is the vector of individual observations and is the chosen level of significance. This research work will focus on phase 1 and phase 2 of control chart.
It is well known that, an efficient control chart must continue to sample as long as the process is in control and must give signal (out-of-control) to stop the sampling as fast as possible whenever the situation arises. Some of the widely used univariate control chart based on performance includes: Shewart chart, Cumulative Sum chart (CUSUM), Exponentially Weighted Moving Average chart (EWMA) to mention but few, which has their corresponding multivariate control chart such as Hotelling’s , Multivariate Cumulative Sum chart (MCUSUM) and
Multivariate Exponential Weighted Moving Average chart (MEWMA) respectively.
UNIVARIATE CONTROL CHART
In some statistical control applications the process would have one particular observation known as quality or process characteristics. Univariate Control Charts involve monitoring individual process or quality characteristic with the aid of control chart, since such univariate control chart can only monitor one quality characteristic in a single chart, this chart is not limited to one quality characteristic alone, two or more quality characteristics can also be monitored but the users would have to look at each quality characteristic separately, that is independently, by doing this, any correlation among the quality characteristics would be considered unimportant and ignored.
SHEWART CONTROL CHART
The Shewart control chart is the most known chart, whose name emerges from Walter Shewart who established them in his pioneering work in 1931. Shewart control chart is used to detect assignable causes in a process. It uses X-bar chart to monitor and control the mean of a process while R and S chart are used to monitor and control the variability of the process.
Assuming is a sample of n independent, identically and normally distributed random
variables with mean and standard deviation , (both known). Then the average of this sample is distributed as a normal variable with mean µ and standard deviation then the control
limits of the X-bar chart are as follows.
UC LCL=
(2.3)
Where UCL and LCL are the upper and lower control limits respectively. Limitation of the shewart charts includes its inability to consider the weight of the previous observations (it is memory less). Although, it is well known for its sensitivity in detecting large shift in the process mean, but very insensitive to small shift in the process mean. Once a shewart chart signals, it implies the presence of assignable cause that demands investigation. Champ and Woodall (1987),Reynolds et al (1988), Reynolds et al (1989),Prabhu et al (1994) to mention but a few had studied the Shewart chart and improved on it.
CHAPTER THREE
METHODOLOGY
MONITORING MULTIVARIATE TIME SERIES
Nowadays, statistical process control (SPC) applications, involve more than one quality characteristic to be monitored. Monitoring these quality characteristics individually might not be correct enough since there might be correlation among the characteristics therefore, simultaneous monitoring of such characteristics is important since correlation among the characteristics will be considered in the process. In multivariate statistical process control applications, several variables are of interest, there are three main multivariate control charts usually used which
comprises the Hotelling’s T 2 control chart, Multivariate Exponentially-Weighted Moving
Average (MEWMA) and Multivariate Cumulative Sum (MCUSUM) control charts. For the
purpose of this research, we consider Hotelling’s T 2 control chart technique for monitoring
simultaneously several correlated process characteristics. Hotelling’s T 2 chart is the multivariate extension of univariate control chart.
However, in real life situation, quality or process characteristics are collected or measured overtime it is of importance to consider the effect of autocorrelation and partial autocorrelation. We suggest vector autoregressive model Var(1), when there is existence of autocorrelation in the data. We are going to confirm the correlation among the process characteristics and then the existence of autocorrelation will be investigated among the process characteristics. By the means of correlogram, we confirmed the existence of autocorrelation in the original data, since the correlogram is visual verification of autocorrelation which gives you an idea as to the order of autocorrelation as well as whether there is existence of autocorrelation in the regression equation.
CHAPTER FOUR
RESULTS AND DISCUSSION
RESULTS
The data for this research was collected from Ashaka Cement factory in Gombe State; the data contain 84 data point, embedded with autocorrelation and partial autocorrelation and non- normality.
Below are the results of various approaches to the Hotelling’s T 2 control chart
HOTELLING’S T2 CONTROL CHART CONSTRUCTED WITH ORIGINAL DATA
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATION
SUMMARY
The summary will be categorized into two stages: Stage I focusing on the identification of out-of-control, and stage II focusing on the interpretation of out-of-control
Stage I:The data used for quality control are increasingly multivariate and sampled in time at a high sampling rate. Hence the data are typically autocorrelated especially if sampled quickly relative to the dynamics of the system being monitored, however, ignoring such autocorrelation can greatly impair the performance of a (multivariate) control chart.
We investigated the autocorrelation, partial autocorrelation and normality assumption in data, and we adopted different approach in applying Hotelling’s T 2 ;
- Hotelling’s T 2control chart was constructed using the original data, and then the control chart showed that the process is in control, knowing well that the data is autocorrelated and failed the normality
- Hotelling’s T 2control chart was constructed using residuals from the VAR(1) model, then the control chart showed that the process is out of control, knowing well that the data from the residuals is free from autocorrelation and partial
- Hotelling’s T 2control chart was constructed using residuals from the VAR(1) model and transformed to normal, then control chart showed that the process is out of control, knowing well that the data from the residuals is free from autocorrelation, partial autocorrelation and satisfied the normality
By comparing the Hotelling’s T 2 control chart using the original data with the Hotelling’s
T 2 control chart using the residuals, it is revealed that the effect of ignoring the verification of auto-correlation and normality can impair the performance of the Hotelling’s T 2 control chart.
This is an undesirable situation, leaving the process unchecked, bearing in mind that the process is in-control when in fact, some points in the process have actually gone out-of-control as seen in this work will definitely lead to production of deformed products, likewise the investigation of possible cause of the problem may not even arise, and production of substandard product will recur. But if a process is been tested for failure of some statistical assumption, and appropriate model or transformation is applied, then a better industrial decision will be achieved.
Stage II: MYT decomposition method which partitions the overall Hotelling’s T 2 into orthogonal component gives important information on how to detect influential variable. However, even with the reduced computation scheme, the numerous computations involved in the process of identification especially when the number of variable is large, is discouraging. Despite this disadvantage the MYT approach has a wonderful way of evaluating the contribution of individual variables and their joint contributions. This work illustrate the decomposing of T 2 statistic for five variables which involved 12 decomposition and 80 number of unique term required to construct the decomposition chart are evaluated. And this aided the identification of the outrageous variables in the process.
CONCLUSIONS
From this research we were able to deduce the effect of ignoring two of the most important assumption of the multivariate statistical process control. The control chart for residuals detected two out of control points , likewise, the normality transformation effect brought the out of control points to just only one point, but the control chart using the original data could not detect this points, the likely reason for this insensitivity could be attributed to presence of autocorrelation and failure of normality in the original data, since it was confirmed (see the appendix) that the original data contain autocorrelation, in the same vein, the reason for control chart for residuals to be able to identify the two out of control can be justify by the ability of the fitted model to remove the autocorrelation through a time series model, as it can be vividly seen (see the appendices) that there is no autocorrelation and autocorrelation in the residual data. Thereby the assumption of independence is achieved in the fitted model alongside the normality assumption mentioned earlier. This research lies in the application of control charts of fitted model (VAR) in detriment of control charts for traditional multivariate control chart, wherein some mistakes of observing a process as a stable process, when in fact there are assignable causes of variation in the process. (False alarm), and also seeing a process as an unstable process, when in fact it is stable (False alarm). This can be attributed to inappropriate approach to parameters estimation and incorrect analysis about the process capability. Wastage of resources (time, energy) that may be required to recoup the process or products as a result of non- interference of the process on time, when in fact the process needed to be adjusted.
More so, we proceeded to identify the influential variables, in this research we adopted MYT approached and we integrated it with a graphical modification for easy and vivid identification of the influential variables. The MYT decomposition is a very reliable method of indentifying influential variable in a process, it easy and simple to estimate the decomposition term more especially when the number of variable are small say two or three, but tends to be complex when variable increase say four, five, and so on. We were able to illustrate the invariant property for five variables and show how variables that deviate from the characteristics of the underline process could be detected, the graphical approach aid easy and vivid understanding and identification of any variable which may have contributed to the out of control points. In order to indulge in any corrective action in the process, appropriate identification of combination of process characteristic responsible for the signal of the control chart is paramount. Therefore the proposed contribution chart is easy, vivid unambiguous approach to see the contribution of each and every variable alongside their interrelationship with other variables.
RECOMMENDATION
Whenever a multivariate statistical process is to be monitored, it is important to check the underlying assumptions of independent and normality of the observations so as not to drive towards wrong decision
CONTRIBUTION TO KNOWLEDGE
This research work succeeded in comparing Hotelling’s T 2 control chart with and without autocorrelation, partial autocorrelation and non normality, which will enhance industrial decision. We also derived and simplified MYT decomposition for five variables, as this derivation and simplification will help quality engineer and others to develop decomposition model for larger variables. Finally this derived decomposition will be applicable in some other related studies that require interpretation of out of control process.
FUTURE WORK
Area such as non parametric applications in statistical process control can be adopted in the plight of failure of normality assumption. And the result in this thesis can be compared. Likewise A computer program can be written to simplify the decomposition of Larger variables
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