行政院國家科學委員會專題研究計畫 期中進度報告
碎形理論在品質管理上之研究(1/2)
計畫類別： 個別型計畫
計畫編號： NSC94-2213-E-011-040-
執行期間： 94 年 08 月 01 日至 95 年 07 月 31 日 執行單位： 國立臺灣科技大學工業管理系
計畫主持人： 周碩彥
計畫參與人員： 張凱翔、周士豪、徐啟桓
報告類型： 精簡報告
報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢
中 華 民 國 95 年 7 月 5 日
NSC 942213E011040
94 08 01 95 07 31
! "
E-Mail: [email protected]
Abstract
Statistical process control (SPC) for short- term auto-correlated processes has received a great deal of attention in recent years. Approaches for dealing with such processes exist. However, no approaches are available for dealing with long-term auto-correlated processes such as fractional Brownian motion (fBm), which has been proved a suitable model to describe many processes.
Therefore, it is necessary to bring up an idea to help manager to monitor that.
Since fBm process has some characteristic properties like self-similarity, increment stationary, variance law and so forth, this research combines the autoregressive T2 control chart with fBm and designs a guideline for monitoring that by transforming the observations according to its properties. Finally, we will explain the effects of the Hurst exponent; the parameter controls the scaling and degree of long-range dependent of fBm process, with sensitivity analysis and numerical example.
Keyword: statistical process control, control chart, autocorrelation, auto-correlated process, fractal, fractional Brownian motion, Hurst exponent, Hurst law
1. Motivation and Objective
Control charts are effective tools in many industries to monitor processes with the objective
of improving process quality and productivity.
Traditionally, the statistical properties of control charts have been evaluated under the assumption that observations on the process output at different times are independent. However, the independence assumption is often violated for processes of interest in many applications because the observations from those exhibit autocorrelation that may be the result of dynamics that are inherent to the process.
There are two types of auto-correlated processes: short-term dependent processes and long-term dependent processes. Fractional Brownian motion (fBm) is one of the latter that was put forward by Mandelbrot after Hurst’s findings, fractal theory. It has been successfully used to model the variate of many natural phenomena because it considers the dependence between each variate of natural phenomena. In addition to natural phenomena, it has been proved that fBm is the suitable model to describe many artificial processes, such as stock trend line of finance, the pressure signal in time at a certain height of industrial airlift reactor, and so forth.
With improvements in measurement and data collection technology, short-term auto-correlated processes like Markov or short time ARMA processes are often used to model the data sampled at higher rates, which leads to data autocorrelation every now and then. However, with a variety of technology applied in engineering field, the properties of observations taken from the process are complex and singular increasingly. That results in the inadequacy of the independence assumption
and the process modeled by short-term auto-correlated processes.
For example, when there is significant autocorrelation in observations from a process, the in-control average run length (ARL) under the independence assumption will be decreased, and that will lead to an average false alarm rate much higher than expected or desired if the auto- correlation is positive. In addition to the effect on independence assumption, if the autocorrelation of processes decays too slowly, the observations modeled by short-term auto-correlated processes may result in a large number of parameters, potentially obfuscating connection between the monitored process and its model. Hence providing an applicable model to describe and designing a methodology to monitor those is an important topic.
Recently, statistical process control (SPC) for short-term auto-correlated processes has received a great deal of attention. Approaches for dealing with such processes exist. But so far, no approaches are available for dealing with long- term auto-correlated processes such as fBm, which has been proved a suitable model to describe certain processes. Therefore, it is necessary to bring up an idea to help manager to monitor that.
Since fBm process has some characteristic properties like self-similarity, increment stationary, variance law and so on, this paper designs a guideline for monitoring that by transforming the observations according to its properties.
2. The Guideline for Monitoring fBm Process
After Mandelbrot and Ness’s reporting [2], many scientists, engineers and statisticians have used fractional Brownian motion to re-model their existing phenomena and problems in their own fields. Since fBm has been used in applications such as hydrology, economics, astronomy, electronics, geophysics, finance, chemistry, and so forth, and the use of SPC method is so universal that many researchers use it to detect data in all kinds of fields, it is necessary to design a guideline to monitor observations taken from fBm process.
It is difficult to monitor fBm because it is a multivariate process and has lots of characteristic properties. In order to apply the autoregressive T2 control chart in the detection of fBm process, we have to transform that into an unvaried process first. It is necessary to find the value of Hurst exponent, H of the fBm, so as to alter fBm process by using the self-similarity and variance law properties. In addition, this research develops a relation between the H value and the autocorrelation function of the transformed fBm data, and uses that to decide the dimension of the multivariate vector of T2 approach.
3. Result and Discussion
Since the objective of this research is to develop a method to detect fBm process, we assume the data sampled follow fBm process, and the time delay between samples is not needed. That means Tt2 can be formed for t p p= , +1,p+2,p+3,K instead of being restricted within t= p p p, 2 ,3 K
t2
T the statistic of the autoregressive T2 control chart
p the dimension of the vector, the parameter of chi-square distribution
In order to monitor the fBm process by SPC method, we have to transform the observations to remove the variation of its variance. According to the properties of fBm, we can use the Statistical Self-similarity and variance law to do that.
Since X t( t) X t( ) 1H ( (X t r t) X t( ))
+ ∆ − ∆ r + ∆ −
(for any r>0)
and the increment [ (X t+ ∆ −t) X t( )] ~ (0,N σH2∆t2H), set
Xt 1H B tH( )
=t
H( )
B t the observation from fBm process Xt the fBm data after transforming
and it becomes the unvaried auto-correlated
Hurst exponent to transform the process. Chapter 2 has introduced the most widely used method, R/S analysis:
lo g ( )
lo g ( 2)
T T
R S
H = T
H Hurst exponent, 0<H < 1
R the range of the partial sums of deviations of T
the time series from its sample mean S the sample standard deviation T
T the sampling period
Next, we can find the covariance of the transformed data and use T2 chart to monitor that.
The auto-covariance is
( ,t t k) [( t )( t k )] k
Cov X X+ =E X −µ X+ −µ =r .
We know that E B t k[ H( + −) B tH( )]2=σH2k2H , the variance law
[ H( ) H( )] H H
E B t k B t σ k
⇒ + − = ,
and E B[ H(0+ + −t k) BH(0)]2=E B t k[ H( + )]2 =σH2(t k+ )2H
where
2 0
2 1/ 2 ( 1/ 2)
2
[ (|1 | | | ) 1 ]
2 ( 1/ 2)
H H
H d
H H
σ =Γ σ+ ∫−∞ −τ − − −τ − τ +
[13]
[ ( ), ( )] 1[| |2 | |2 | | ]2
2
H H H
H H
Cov B t B s = t + s − −t s
then we get
1 1
[ ( ), ( )]
( )
1 1
=E[ ( ) ( )]
( )
1 1
[ ( )] [ ( )]
( )
=E[ 1 ( ) ( )]
( )
1 1
{ [ ( )]}{ [ ( )]}
( )
= 1 { [ ( ) ( )] [ (
( )
H H
H H
H H
H H
H H
H H
H H
H H
H H
H H
H H H
H H
Cov B t k B t
t k t
B t k B t
t k t
E B t k E B t
t k t
B t k B t t k t
E B t k E B t
t k t
E B t k B t E B t t k t
+ +
+ ⋅ −
+ + +
+ −
+ + +
+ −
+
2 2 2
2 2 2
)] [ ( )]}
= 1 [ ( ), ( )]
( )
1 1
= { | | | | }
( ) 2
1 1
= [ | | | | ]
( ) 2
H
H H
H H
H H H
H H
H H H
H H
k E B t Cov B t k B t
t k t
t t k t k t
t k t
t t k k
t k t
+ + +
⋅ + + − + −
+
⋅ + + −
+
k= 1 1[2 | |2 | | ]2
2
( )
H H H
H H t t k k
t k t
γ + ⋅ + + −
γk the auto-covariance of X t µ the mean of the fBm process
µ0 the mean of the fBm process when it is in-control
α the parameter of chi-square distribution t the sampling instant
k time lag
σ the variance of the observations
H2
σ a positive constant, the variance of the fBm process
After that, we can design the T2 control chart by using these equations and parameters we have developed.
4. Continuing Work
There are more and more applications of the SPC method. In addition to the industrial field, scientist applied it to finance field, even the efficiency of employees has been measured by that. But the model adopted to fit is not certainly accuracy. It is intuitive that fBm can be a good model to fit those.
The method for monitoring fBm process in this research will be useful in these applications.
Now we successfully develop a method to transform the observations to remove the variation of its variance and find the covariance of the
transformed data for monitoring the fBm process.
The concept we use is to combine the autoregressive T2 control chart with fractional Brownian motion. Most important, this research is going to derive the autocorrelation function of the distribution converged from fBm process according to the self-similarity and variance law properties. And we want to find the relationship between Hurst exponent and the dimension of the multivariate vector using in T2 approach.
In the future, we will carefully define more notations and assumptions in the model and provide a procedure for monitoring fBm process.
And then we will analyze the sensitivity to find out the effect and the relationship between the Hurst exponent H and other parameters. Finally, it is important to illustrate the guideline by a numerical example using Matlab to generate fBm process by the technique of Cholesky that has been chosen as one of the fundamental fBm synthesizers. [15]
Besides, a comparison with the existing method for monitoring auto-correlated process may be made by Monte Carlo simulation. And then we will clearly know the advantages of every method. Moreover, the Hurst exponent may be used to estimate the standard deviation of the process while it is usually estimated by the range and variance of the samples. If we find the relationship between the two, maybe we can develop a control chart designed for fBm process based on original observations without transformed. And its advantage is easily understood. Since a lot of research techniques of fractional Brownian motion exit, it is possible to find a way to synthesize an fBm process in terms of its parameters and properties to fit the one want to detect, and then computes the residuals. The residuals will be regarded as random variables if it is an unbiased estimation. Hence we can plot those residuals on the standard control chart and monitor them. Those subjects and directions may be able to be extended from this research in the future.
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