Change point method with Rayleigh distribution

This paper intends to assess the performance between CUSUM and change point model of Rayleigh distribution by considering ARLs respectively. Finally we conclude that the performance of change point model is superior to CUSUM chart in the context of Rayleigh distribution.

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Introduction

The Statistical Process Control (SPC) aims to detect and diagnose situations where the process is gone out of statistical control. This type of problem involves two aspects namely process and statistical aspects. To have detailed outline of some of its statistical modeling aspects see Crowder et al. (1987) [1] . In fact the status of Statistical Quality Control may be described as one in which the process readings appear to follow a common statistical model. One model is that the process is in SQC, the successive process readings Xi's are independent and sampled from the same distribution. When the process goes out of control, it may behave in different ways. In general we can have two types of causes, those that affect single process readings and then disappear, and the second type of causes or sustainable causes. These causes will continue until they are identified and eliminated. In statistical terminology the isolated causes analogous to an out layer. The Shewart with an R or s-chart is an excellent tool for detecting these special causes, namely isolated or sustainable causes. However Shewart control char is less effective for detecting small changes in the process. Standard tools for detecting sustain changes are the CUSUM and EWMA chart. This paper focuses on another, less familiar; method aimed at detecting sustained changes is the Change Point formulation when the process averages follow non-Normal distribution.

Change point method

At first let us discuss Change Point Method when the process readings modeled by two Normal Distributions,

Here the number of observations 'n' is fixed in any traditional statistical settings, but this 'n' may increase unlimitedly in phase-II Statistical Process Control settings. Here both settings i.e. Phase-I and Phase-II will be discussed, with context indicating which of the two applies. In the case of in control distribution is N (μ1, σ1 2 ), the readings follows this distribution up to an epoch τ, the change point, at which point they shift to another Normal distribution different in mean i.e. μ1=μ2, in variability σ1 = σ2 or in both mean and variance. In change point method process readings leads to two statistical tasks namely testing task and estimating task. The testing task is to conform whether there has indeed being a change. If the change is there then the task of estimating τ, the time at which shift occurred. Sometimes we may have to estimate some or all of the parameters μ1, μ2, σ1 and σ2. In all discussions [see Hawkins, 2003] we work on change point has focused on shifts in mean only. i.e. μ1 ≠ μ2 but σ1 = σ2 = σ, throughout this paper this type of frame work is used here. From this change point method we have three scenarios based on the amount of process knowledge viz.  All parameters μ1, μ2, σ1 and σ2 are known exactly priory.  In control parameters μ and σ are known but μ2 is unknown.  All parameters μ1, μ2, σ1 and σ2 are unknown.  Throughout our discussions we concentrate on the third scenario namely all parameters are unknown under varying distributions.

Implementation of change point method

The formulae below mentioned indicate that the values are computationally burdensome. For the implementation of the method, construct two arrays of values of

( There is no need to store the running mean = , but it will be calculated 'on the fly'. The observation of the two new table entries can be calculated from the numerically stable recursions, when a new observation is added

The two sample statistic are calculated for every possible split point 1≤ j≤ n after finding (The customary variance estimator), using the value j leading to the maximum. The estimators are somewhat biased even though the maximum likelihood estimator are calculated by Hinkley (1970) [3] .

The maximizing j is that which maximizes jn E , so the searching step need only to evaluate Willsky and Jones (1976) [6] discussed the 'window' approach which is different from above method, in that observations more than w time periods into the past are not lost, since they are summarized in the window's leftmost S and W entry. The lost is the ability to split at these old instants. Appropriate values for the table size W may be in the 500 to 2000 range. It is very large as no interesting structure is lost, but small enough to compute for each new reading to less than 20000 operations.

Changepoint method with rayleigh distribution when the parameters are not know

Consider the model with none of the parameters known; we can test the presence of change point with another general Likelihood ration test. This test is a two sample t-test between before shift and after shift of the sequence, maximised across all possible change points; Worsely (1979) [7] . For a given change point 'j' where 1≤ j ≤ n-1, let

Be the mean of the (n-j) observations after shift. The residual sum of squares i.e. Vjn is given by

Here we assume that there is a single change point approach at epoch 'j', For a stable process, Tjn follows at t-distribution with n-2 degrees of freedom. An out of control signal is obtain as soon as

Exceeds some critical value, hn. Hawkins et al. (2003) [2] showed that, for any type I error, hn can be computed by using the Bonferroni inequality. They mentioned that the latter is conservative when a process measurement (n) is large. At same time, they provide empirical control limits, hn, α for different type I error, α, and various number of process observations. Also, for large n values (n≥1), they proposed the following approximation formula for computing hn, α

Where ln (.) is the natural log function.

In the case of general change point formulation in which either or both of the parameters μ, σ may shift at the change point τ. Sullivan and Woodall (1996) [5] discuss this formulation and the resultant generalised likely hood ratio test. It provides a single diagnostic to detect shifts in either the mean or the variance or in both. This has the disadvantage of normality assumption. Furthermore, while bounds, approximations and extreme value results are known for the hull distribution of ,n max T , there is any hardly sample theory for Sullivan and Woodall statistic. Based on these reasons we will consider for the change point formulation to mean the formulation in the third scenario i.e. none of the process parameters is consider known exactly. In Phase I, with its static set of data X1, X2...Xn, traditional fixed sample statistical methods are appropriate. So, for example, it is appropriate to calculate ,n max T for the whole data set and test it against a suitable fractiles of the null distribution of the test statistic for that value of n. If the analysis indicates a lack of control in the Phase I data set, more data will be gathered after process adjustment until a clean data set is achieved. Phase II data are the process readings gathered subsequently unlike the fixed set of Phase I, they are from a never-ending stream.

As each new reading accrues, the SPC check is re-applied. For this purpose, fixed significance level control limits are not appropriate: rather, concern is with the run lengths, both in-and out-of-control. A convenient summary of the frequency of false alarms is the in-control average run length (ARL), which should be large, and self-contained GLR in which the maximised Likelyhood and the Likelyhood ratio are used both detection and estimation. In traditional methods viz. Shewart, CUSUM, EWMA charts, require a Phase I data set to have parameter estimates that can be used in the Phase II calculations. These methods require on to draw a connectional line below the Phase I data and separate the estimated data [Phase I] the SPC data [Phase II]. On contrast, in change point formulation one does not assume known parameters and hence does not require the estimates produced by a Phase I. Once the preliminaries are complete and the initial process stability has been achieved, the change point allows to go seamlessly into SPC in which, at each instant, all accumulated process readings are analysed and all data is used to test for the presence of a change point. When process remains in control, it also provides on ongoing stream of every improving estimates of the parameters. The change point model is a schematic approach in which each new observation Xn is added to the data set, the change point statistic is calculated for the sequence X1, X2...Xn. If

,n max T > n h . Where { n h is a suitably chosen sequence of control limits, then we conclude that there has been a change in mean. The important point is the choice of the control limits sequence { n h }. The ideal would be a sequence { n h }, such that the hazard or alarm rate [the conditional probability of a false alarm at any 'n', given that there was no previous alarm] was a constant α, as in case with the Shewart chart. When the hazard rate is constant the in-control ARL would be . This approach was used by Margavio et al. (1995) [4] in the context of an EWMA chart in context known parameters the false alarm changes point over time. Margavio et al. (1995) [4] derived control limits Sequence that would fix the false alarm for EWMA chart to a specified value. The present chapter attempts to obtained distribution of

,n max T . In fact obtaining distribution of ,n max T the sequence is far from being able to provide distributional theory sequence. Hence an attempt is made using simulation to tackle this problem. In fact the change point does not depend on the parameters estimation from Phase I so it is possible to star testing for a change point with the third process reading. Table 3.1 is. Obtained by simulation of 10 million sequence of length 200 using Rayleigh distribution. This table shows the control limits for α value of 0.05, 0.02, 0.01, 0.005, 0.002 and 0.001, corresponding in control ARLs of 20, 50, 100, 200, 500 and 1000 for different 'n' values ranging from 3 to 200. It can be noted that from table 3.1, as the 'α' values decreases the control limits  , n h decreases sharply initially, but then stabilizes. Similar type of behaviour can be observed even with Normal distribution; Hawkins (2003) [2] . It may not be reasonable that start testing at the third observation. However there are cases where the process shift occurs at third observation. In practice the particular should gather a modest number of observations to get an initial verification that the Rayleigh distribution was a reasonable fit. Then only the formal change point can be applied. In of this our main simulation is based on assumption of 9 readings without testing, with testing starting at the 10th observation, however 9 readings itself is hard to believe in a Rayleigh distribution for the Quality variable. Perhaps it is making reasonable compromise between the conflicting desires. The same 10 million sequence of length 200 are used to find out cut offs points up to n =200. The resulting control limits are presented in table 3.2. It is clear from the table that the ARL's are affected by the amount of history is gathered before the shift, with a faster response carrying with more history. These ARL's are also depends α and δ. It can also be observed that α is large or δ is large the ARL tends to be smaller, as one would anticipating. For different δ values in the range 0 to 3 we present resulting ARLs in figures 4.1 to 4.4 From the figures we note that the change point formulation is generalizing the optimal diagnostic for small shifts in the process average it is superior to the CUSUM chart with reference value k=0.25, where as in the case of medium size shifts change point model is better than k=0.5 CUSUM chart and when k=1 CUSUM chart performed well in all cases it is marginal only in the case of normal distribution, Hawkins (2003) [2] . Where as in the case of Rayleigh distribution change point model prove to be strongest diagnostic for detecting shifts in the process average when compare with the CUSUM chart. In other words while CUSUM chart it is never best, the change point method is always best in the case of Rayleigh distribution.

CUTOFFS hn, α FOR SAMPLE SIZE ‘n’ AND HAZARD RATE ‘α’ STARTING AT SAMPLE 10	—	—	—	—	—	—
n	0.05	0.02	0.01	0.005	0.002	0.001
10	4.258586	5.021439	5.456973	6.027352	6.843208	7.513689
11	3.841319	4.546789	4.938576	5.464631	6.220676	6.844801
12	3.67287	4.333104	4.685837	5.168124	5.858124	6.42529
13	3.55255	4.180471	4.50531	4.956333	5.599158	6.12564
14	3.46231	4.065997	4.369914	4.79749	5.404933	5.900903
15	3.392123	3.976961	4.264606	4.673946	5.25387	5.726107
16	3.335973	3.905733	4.18036	4.57511	5.133019	5.58627
17	3.290033	3.847455	4.111431	4.494245	5.034141	5.471858
18	3.251749	3.79889	4.053991	4.426857	4.951743	5.376515
19	3.219355	3.757796	4.005387	4.369836	4.882021	5.29584
20	3.191589	3.722574	3.963727	4.320962	4.82226	5.22669
22	3.146469	3.665336	3.896029	4.24154	4.725147	5.114321
24	3.111375	3.620818	3.843375	4.179768	4.649616	5.026923
26	3.083301	3.585204	3.801252	4.13035	4.58919	4.957005
28	3.06033	3.556065	3.766788	4.089917	4.539751	4.899799
30	3.041189	3.531783	3.738068	4.056223	4.498552	4.852127
35	3.004885	3.48573	3.683598	3.992321	4.420416	4.761715
40	2.979259	3.453222	3.645149	3.947214	4.365261	4.697895
45	2.960204	3.429049	3.616559	3.913672	4.324248	4.65044
50	2.945479	3.410371	3.594466	3.887754	4.292556	4.613769
60	2.924211	3.38339	3.562555	3.850316	4.24678	4.560801
70	2.909588	3.364841	3.540616	3.824577	4.215308	4.524385
80	2.898918	3.351305	3.524606	3.805795	4.192342	4.497811
90	2.890788	3.340992	3.512408	3.791485	4.174845	4.477564
100	2.884388	3.332873	3.502806	3.78022	4.16107	4.461626
125	2.873094	3.318546	3.48586	3.76034	4.136761	4.433498
150	2.865721	3.309194	3.474798	3.747362	4.120893	4.415137
175	2.86053	3.302608	3.467009	3.738224	4.10972	4.402209
200	2.856677	3.29772	3.461228	3.731441	4.101426	4.392612

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—	CUTOFFS h FOR SAMPLE SIZE ‘n’ AND HAZARD RATE ‘α’ STARTING AT SAMPLE 3 n, α	—	—	—	—	—	—
	n	0.05	0.02	0.01	0.005	0.002	0.001
	3	26.4517	61.65674	118.3213	239.6466	611.6192	1158.239
	4	8.148804	11.68613	16.77718	23.96427	39.21047	55.75856
	5	5.891949	6.953319	8.9649	11.48376	16.82633	21.60409
	6	5.104095	5.451373	6.673182	8.110302	11.4155	14.03367
	7	4.719309	4.747417	5.63837	6.646386	9.192542	11.05301
	8	4.497061	4.348107	5.063179	5.851018	8.02302	9.523984
	9	4.355226	4.094638	4.702189	5.358792	7.314035	8.612264
	10	4.258586	3.921439	4.456973	5.027352	6.843208	8.013689
	11	4.189662	3.796806	4.280913	4.790654	6.510132	7.593683
	12	4.138848	3.703635	4.149243	4.614154	6.263356	7.284337
	13	4.100447	3.631928	4.047656	4.478133	6.073976	7.047961
	14	4.070882	3.575467	3.967333	4.370557	5.924579	6.862067
	15	4.0478	3.530193	3.90256	4.283685	5.804082	6.712457
	16	4.029593	3.493347	3.849476	4.212323	5.705113	6.589751
	17	4.015129	3.462993	3.805383	4.15286	5.622586	6.487513
	18	4.003588	3.437734	3.768344	4.102711	5.552882	6.401188
	19	3.994364	3.416538	3.73693	4.059982	5.493362	6.327467
	20	3.987002	3.398627	3.710068	4.02325	5.442055	6.263887
	22	3.976538	3.370409	3.666892	3.963679	5.358423	6.160113
	24	3.970215	3.349692	3.634158	3.917862	5.293551	6.079398
	26	3.966777	3.334327	3.608908	3.881917	5.242126	6.01518
	28	3.965401	3.322883	3.589178	3.853269	5.200642	5.963144
	30	3.965522	3.31438	3.573619	3.83015	5.166698	5.92034
	35	3.970065	3.301886	3.547307	3.789134	5.104778	5.841425
	40	3.97808	3.297231	3.532486	3.763587	5.064119	5.78858
	45	3.987793	3.297093	3.524515	3.747407	5.036458	5.751721
	50	3.998296	3.299678	3.52085	3.737231	5.017231	5.725286
	60	4.019917	3.309271	3.521035	3.727685	4.994256	5.691687
	70	4.041075	3.321638	3.526596	3.726261	4.983276	5.673205
	80	4.061185	3.33498	3.53474	3.729106	4.97892	5.663231
	90	4.080104	3.348487	3.544119	3.7343	4.978461	5.658384
	100	4.09785	3.361779	3.554036	3.74081	4.980391	5.656776
	125	4.137614	3.393147	3.57909	3.759533	4.990664	5.660564
	150	4.171928	3.421475	3.602962	3.778972	4.004237	5.669669
	175	4.201978	3.446974	3.625098	3.797784	4.01868	5.680807
	200	4.228651	3.470029	3.645496	3.815569	4.033051	5.692607

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Conclusions

In this paper control limits or cut off points [  , n h ] are obtained by simulation of 10 million sequence of length 200 with the process readings start at 3 and 10 are obtained in the context of Rayleigh distribution. I.e. when the phenomenon under consideration follows Rayleigh distribution respectively. This paper also provides a comparative study between change point model and CUSUM chart by considering ARLs in the context of Rayleigh distribution respectively. We found that the performance of change point model is superior to CUSUM chart in the context of Rayleigh distribution, it is found that the change point model is slightly worse than CUSUM chart in detecting small shifts, which is similar case with the normal distribution Hawkins et al. (2003) [2] . It would be very interesting to study the behaviour of change point model and comparing its performance with CUSUM chart in the context of non-normal distributions i.e. when the process average follows non-normal distribution.

Figures and tables from the paper

α	start	0	0.5	0.6	0.75	1.25	1.5	1.75	2	2.25	2.5	3
0.02	10	3.289152	20.7541	36.5089	102.6066	144.3342	42.71179	21.04804	13.24886	9.659798	7.452006	5.09
	25	5.624377	20.6541	35.10036	96.95557	134.0896	32.6479	11.04804	10.98858	6.847978	5.47006	2.839615
	50	2.070861	20.3541	31.56612	89.44565	124.332	27.14786	8.980432	9.288579	5.729783	4.450598	11.84368
	100	33.30872	21.2541	28.33235	84.35545	113.312	22.50896	7.080432	5.385795	3.747978	2.77006	1.258894
	250	0.638146	24.7541	22.50036	68.95557	103.3196	18.71959	7.004315	3.245795	2.659978	2.750598	21.96378
0.01	10	4.00041	24.3428	42.7633	121.0012	170.1435	50.04397	24.68882	15.50775	11.23847	8.639493	0.977985
	25	3.635815	24.4428	38.50814	110.9017	160.4053	42.06973	15.38824	11.57749	8.634698	5.64493	6.287495
	50	4.439894	23.7428	33.53981	102.4963	149.1053	32.72732	8.623937	10.27493	7.596984	4.6993	2.13432

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1: The Arl of the Change Point Procedure When Shift Occurs At the ‘Start’ Position with Size ‘ δ ’

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