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Change Point Analysis for Control Chart Setup, Monitoring, and
Process Capability Study
Shing I Chang and Jibing Yan
Department of Industrial and Manufacturing
Systems Engineering
Kansas State University
Manhattan, KS 66506
Abstract
The propod study investigates the possibility of applying wavelet decomposition to identify change points in a data t in various situations, such as during control chart tup, process monitoring, or process capability study stages. Wavelet analysis, often ud for signal processing, is first applied to decompo a stream of data in time domain into coefficients in various layers. Identifications would be
performed via patterns or energies in each layer. The propod framework offers a great potential for analysis of large data ts as most of today’s enterpris are looking forwards to applying data mining techniques to ek opportunities for improvement.
Keywords
Statistical Process Control (SPC), Change Point Analysis, Process Capability Study, Wavelet Analysis, Data Mining 1. Introduction
Statistical Process Control (SPC) has been widely ud to monitor either production or rvice process. Most of rearch in SPC focus on charting techniques. A pair of control charts, such as Mean (X-bar) and Range (R) charts, is applied on a quality characteristic to detect potential deviations from process mean and variance, usually referred to as process parameters. When process parameters are unknown, approximate 20 to 25 samples, each contains 4 to 5 obrvations, are often collected as a trial data t to estimate them. The most fundamental assumption is that a trial period, under which obrvations are collected, reprents “typical” operation of the process of interest. Therefore, process parameters remain constants during this period. This exemplary trial data t is then ud as a yardstick for future process monitoring. However, the problem is that this i
mportant assumption is rarely confirmed. The most common practice is just to check for tho points that plot outside the three sigma limits. If assignable caus which do not usually exhibit in the normal operations can be found to link to the points, the points would be scraped from the control limit calculation. Note that this procedure oversimplifies process assumptions that may deviate from process to process.
A related issue during the process monitoring stage is that, when a process shift occurs, we would like to know the moment of such an event. This is often referred to as the change-point problem. A good estimate of change point would provide a good estimate of the magnitude of process parameter shifts. Usually, the arch for a change point occurs right after control charts provide an out-of-control signal bad on the assumption of a single change-point. Therefore, it would be inappropriate to apply the change-point arching algorithm to trial data t collected during the tup stage becau it may contain multiple change points. Finally, had change points identified in a process characteristic data t, they would provide far more informative insight to the process of interest than the mere process capability indices along.
This study propos an integrated approach to identify change points in a data t no matter it is in the tup, process monitoring, or process capability study stages. Specifically, wavelet analysis often
ud for signal processing is propod to first decompo the data in time domain into coefficients in various layers. Identifications would be performed via patterns or energies in each layer. The propod framework offers a great potential for analysis of large data ts as most of today’s enterpris are looking forwards to applying data mining techniques to ek opportunities for improvement.
2. Background of Change Point Problems
The origion of change-point problem starts from the applied rearch in industrial area, and directly referred to the identification of the time and cau when a process shift took place. As early as the 1930s, Shewhart [1] mentioned the change-point problem. However, it is not until later in 1950s, Page [2], and Girshick and Rubin [3] propod veral methods relating it to Shewhart control charts, CUSUM charts, and GRSh test. Recent literature, connected change-point problems to control charting, aims at not only signaling process changes quicker but identifies the changing time and cau. For example, Samuel, Pignatiello and Calvin [4,5] considered the change of mean and variance in a normal process respectively with X chart, S chart, and R chart.
Considering the change-point problem from a more theoretical mathematic view at the end of 1950s,
Kolmogorov formulated the change-point problem into a general mathematical problem – the detection of disruption of stochastic homogeneity in any data t. It was then investigated intensively by many scientists such as Shiryaev [6,7], Chernoff and Zacks [8], and Hinkley [9,10]. During this period, most methods took a parametric approach. The constructed parametric change-point estimators depend on some a priori statistical information of the data, and are liable to the uncertainties of a priori information. Another school of study focud on nonparametric methods. The nonparametric estimators of change-point apply minimal a priori information, and the nonparametric methods are more robust. Bhattacharya and Frierson [11] put forward the first nonparametric test for the change-point problem with independent random variables. Darkhovsky and Brodsky [12] first propod to u the Kolmogorov-Smirnov type statistics and applied this method to change-point problems with dependent random variables.
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In this rearch, an alternative method – wavelet decomposition is propod to first transform obrvations into detailed channels and then arch for disruptions in the channels. The next ction provides basics of wavelet decomposition.
3. Background of Wavelet Decomposition
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Morlet [13] was the first to propo the wavelet transform for signal analysis. Daubechies [14] propod the construction of compactly supported orthonormal wavelet bas. Mallat [15, 16] realized that the orthonormal wavelet bas could be constructed systematically from a general framework named “Multiresolution analysis.” In wavelet reprentation, general functions are reprented in terms of simple, fixed building blocks at different scales and positions. The building blocks are the family of wavelets that are generated from a fixed function called “mother wavelet” by translation and scaling. Scaling operation is compressing and stretching of the mother wavelet and this is uful in catching the different frequency information of the function under analysis. The compression operation deals with the high frequency needs whereas the stretching operation deals with low frequency needs. Translating, as the name implies, is ud to obrve the time dependent information of the function under analysis. Thus a wavelet transform is capable of “zooming in” on the short-lived high frequency details and “zooming out” on long-lived low frequency details.
In this study, we adopted Mallat’s multiresolution approach using Daubechies’s mother wavelets. A signal S (considered as obrvations over time) was first decompod into approximate and detailed channels, each carries specialized information that contributes to reconstruction back to S. It has been demonstrated in many applications that the quality of such reconstructions is remarkable. Furth
ermore, not only that the storage quantity of the summation of all channels is smaller than S so that compression is possible, but also each channel carries distinguished traits about S that would rve as “signatures” of S. Motivated by this property of wavelet transform, we study the possibility of applying wavelet analysis for change point problems.
4. A Propod Framework for Wavelet Analysis on Change Point Problems
As alluded earlier, solving a change point problem is important to three stages of process control: tup stage, monitoring stage, and capability study stage. Most literatures related to change point problems focus on the process monitoring stage. However, it is the other two stages, solutions and evaluations on process stability enable decision makers to obtain the complete picture. An earlier analysis during control chart tup stage would help find if a process exhibits stationary behavior or what types of control charts are suitable.
Investigation on multiple change point problem has great potential to compliment capability study. Kotz and Johnson [17] provided the definition of process capability indices PCI as a “single-number asssments of ability to meet specification limits on quality characteristic(s) of interest.” Over a period time, two identical PCI numbers may
provide different meaning. For example, one number is obtained from a very stationary operation with an “average” performance while the other process is mixed with excellent process performance and bad deviation from target. Both process would be classified under a PCI index with the same “grade.” However, the cour of action to achieve continuous improvement would be very different. This information would not be provided by PCI along. The propod method is an attempt to provide more informative analysis on a process over a period a time. Through clustering of similar distributions of a process period, decision makers can study the variation and bias of process of interest.
From the point view of time ries, i.e., considering obrvations from a process over time as a time ries or a signal S, it can also be reprented by a function )(t f . The basic assumption of wavelet analysis for process asssment is that the channels from a multiresolution decomposition of S would provide some “clues” of changes in process parameters, i.e., mean and variance.
For discrete wavelet analysis, the multiresolution reprentation of a signal )(t f is the following:
)()()(00t f A t f D t f m m m m +=
∑−∞= (1)
If 0m =J , the wavelet ries approximation to a signal )(t f is expresd in terms of the signals:
)(...)()()()(11t f D t f D t f D t f A t f J J J ++++≈− (2)
The terms in this approximation sum constitute a decomposition of the signal into signal components )(t f A J ,)(t f D J , …, and )(1t f D which are also known as channels. J A is the coefficient vector for the appoximate channel which reprents the general profile of the signal S while D i , i=1, 2, …, J are the coefficient vectors for detailed channels. The detailed channel D i with larger i provides more flutuation information about the signal.
With this multiresolution structure in mind, our initial attempt is to perform a wavelet transform on a ries of data from a process with an univariate quality characteristic. Suppo there is a process shift in mean, variance or both. We would like to pick up dinstinct sigatures in the wavelet channels that would tell one from another. During the tup stage, quality engineers usually do not know the process paramters. Therefore, the task is to identify if the process of interest is stable or not and where the data under stable process conditions is clustered according to the wavelet signature. The data under steady state is then ud to estimate the process parameters and build control charts.
During the process monitoring stage, wavelet decompostion can be applied after a control chart prov
ides an out-of-control signature. The distinct signature is then ud to locate along the time line where the shift has occurred. Finally, during process capability study stage, the wavelet signatures may appear multiple times at various locations. This information would help access process capability by analysis of frequecy, time between occurance, and signature clustering patterns.
At this point, a crutial question is prompted, i.e., what is the information that the wavelet channels can provide but the original signal S can not? After all, S contains the entire information of the time ries. Why is it necessary to decompo this time ries? At the first glance of the time ries plot, e.g., Figure 1, it is easy to “e” the pattern changes in the original signal. However, this task is very difficult for computer algorithm to “e” a pattern change from raw data. The analogy is that it is very easy for a human to distinguish dogs from cats but it would take tremendous efforts to develop a computer algorithm to perform the same task. With the wavelet decomposition, the information in the wavelet channels is more readliy ud by computer algorithms. The next ction provides an example to illustrate wavelet decompostion of a time ries and the possible u of the wavelet channels for decision making.
5. Wavelet Analysis on Simulated Data Sets
In this example, the simulated data t contained 1000 obrvations as shown in Figure 1. The first 500 random reprented the in-control process mean 10=µ and standard deviation 1.0=σ, i. e.,)1.0,10(~2N X while process shifts occurred at 501st obrvation. The last 500 numbers had the shift magnitudes of mean shift σδ1=and variance shift σρ2=, i. e.,)2.0,1.10(~2N X
.
01002003004005006007008009001000
9.49.6
9.8
10
10.2
10.4
10.6
10.8
11
Figure 1. A data t contains 1000 obrvations from two populations
We applied Daubechies 8 mother wavelets to decompo the data to the 9th level as shown in Figu
re 2. Note that some wavelet-decompod channels contained the “signatures” of change point. For example, at level 5 (D5) there is a sharp spike around point 500 corresponding to the mean shift. And at level 1 and 2 (D1 and D2) the vibration magnitudes become bigger around point 500, which is also corresponding to the variance change. This result suggests that a process obrvation data t can be treated as a signal and decompod into hierarchical channels, which carry mean shifts and variance changes in different layers and locations.
Acknowledgement
This rearch is partially funded by Office of Naval Rearch (Grant N00014-01-1-0917) and Advanced Manufacturing Institute of Kansas State University.
Approximation A9
Detail D1
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Detail D2
Detail D3
Detail D4
玉器知识Detail D5
Detail D6
Detail D7
Detail D8Detail D9走自己的路让别人说去
Figure 2. A Wavelet Decomposition with Daubechies 8 Mother wavelets
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