Cumulative Sum Procedure Research Articles

Self-Starting Cusum Charts for Location and Scale

In some quality control problems, it is not known what the exact process mean and standard deviation are under control but it is desired to determine whether there have been drifts from the conditions obtained at the process start-up. This situation is not well-covered by standard cumulative sum procedures, which generally assume known process parameters. This paper uses the running mean and standard deviation of all observations made on the process since start-up as substitutes for the unknown true values of the process mean and standard deviation. Using some theoretical properties of independence of residuals, two pairs of cusums are set up: one testing for constancy of location of the process, and the other for constancy of the spread. While the process is under control, both these cusum pairs are of approximately normal N(O, 1) quantities (and therefore are well understood), but if the location, the spread or both change, then non-centrality is introduced into one or both of the location and scale cusum pairs, and it drifts out-of-control. It is shown that the procedure performs well in detecting changes in the process, even in comparison with the often utopian situation in which the process mean and variance are known exactly prior to the start of the cusum. precisely if large numbers of spurious signals are to be avoided. While it has usually been assumed in discussion of cusums that the process specification is known exactly, there are many circumstances in which it is to some extent uncertain, and under these conditions the assumption of known mean and variance is inappropriate. A case of particular interest to the author arises in the control of assaying in a chemical laboratory for bias and precision. A good way of doing this is (Mandel 1964) by using reference materials. A reference material is a batch of the same sort of material as is assayed in the laboratory (obtained for example from a previous consignment), portions of which are assayed regularly along with the production material. A depar- ture of the assays of the reference material from their previous mean or standard deviation indicates a change in bias or precision. Note that the control required is simply for any change in the values obtained for the reference material. It is immaterial what the true assay of the reference material (the ideal target process mean) actually is, and the quality control system should not be predicated on the assumption of an exact knowledge of the process mean. This indeterminancy presents

The cusum test of homogeneity with an application in spontaneous abortion epidemiology.

We test the hypothesis of homogeneity over time in the rate of an event with a modification of Page's cumulative sum (cusum) procedure. The modification draws on an indifference zone approach to testing that is well suited to the epidemiologic study of retrospective data. We provide procedures for evaluating the significance level of the largest observed cusum value occurring in a sequence of fixed length. We draw inferences unconditionally with a known baseline rate of the event, and conditionally on observed margins with unknown baseline rate. Data collected over a five year period from one New York City hospital serve to illustrate the modified cusum test in a study of fluctuations in the proportions of chromosomally normal, trisomic, X-monosomic and triploid spontaneous abortions.

Characterization of profiles of salivary progesterone concentrations during the luteal phase of fertile and subfertile women.

Salivary progesterone concentrations were measured in daily samples collected between 08.00 and 09.00 h throughout the menstrual cycle of women with a history of fertility. The luteal-phase salivary progesterone profiles in these normally menstruating, healthy women were characterized using a computer program based on a cumulative sum procedure. This method of statistical analysis led to the development of a 'progesterone boundary diagram', the inner and outer domains of which distinguished between the profiles of salivary progesterone considered compatible with fertility, and those observed in subfertile women attending an infertility clinic.

Cumulative sum (cusum) control schemes

Cumulative Sum (CUSUM) quality control schemes are becoming widely used i n industry because they are powerful, versatile, and easy to use. They cumulate recent process data to quickly detect out-of-control situations. They also serve as a powerful diagnostic tool. There are now more than 10,000 CUSUM control schemes in use daily in Du Pont. This paper describes design and implementation procedures for CUSUM control schemes with emphasison properties that are recorded as counts. The paper will describe recent developments which make CUSUM procedures more useful and more powerful. The recent developments described are: Fast Initial Response (FIR) CUSUM which gives extra sensitivity to out-of-control situations at start up or after a(possiblyin effective)control action. A Combined She whart-CUSUM which combines the key features of CUSUM schemes and She whart Schemes by adding She whart Control Limits to a CUSUM scheme. ROBUST CUSUM schemes which are no tunduly influenced by a few outliers or fliers occurrin...

On Cumulative Sum Procedures and the Sprt with Applications

SUMMARY Both one-sided and two-sided cumulative sum procedures of Page (1954) are shown to be closely related to a sequential probability ratio test (SPRT). The generating functions (Laplace transforms) of cumulative sum procedures are found in terms of the generating functions of a certain SPRT. An example is discussed and some applications are given.

On the Markov Chain Approach to the Two-Sided CUSUM Procedure

A Markov chain representation is given for the two-sided, possibly asymmetric, cumulative sum (CUSUM) procedure of Ewan and Kemp (1960) for integer-valued random variables. The approach taken by Brook and Evans (1972) for the one-sided CUSUM procedure to determine exact and approximate expressions for the run length distribution, its moments, and its percentage points is extended to the two-sided CUSUM procedure. The number of states included in the Markov chain is minimized in order to make the methods as efftcient as possible.

On the Markov Chain Approach to the Two-Sided CUSUM Procedure

A Markov chain representation is given for the two-sided, possibly asymmetric, cumulative sum (CUSUM) procedure of Ewan and Kemp (1960) for integer-valued random variables. The approach taken by Brook and Evans (1972) for the one-sided CUSUM procedure to determine exact and approximate expressions for the run length distribution, its moments, and its percentage points is extended to the two-sided CUSUM procedure. The number of states included in the Markov chain is minimized in order to make the methods as efftcient as possible.

A Note on Page's Two-Sided Cumulative Sum Procedure

A symmetric version of Page's (1954) two-sided cumulative sum procedure is discussed and a formula for the average run length given. Asymptotic normality of run length under suitable normalization is established.

United States Department of Agriculture Cusum Acceptance Sampling Procedures

A set of USDA acceptance sampling plans based on cumulative sum (cusum) procedures is described. The plans, for inspection by attributes, are indexed by AQL's and apply to continuous production grouped in lots or batches. They are simple to apply and provide better discrimination than single sampling plans. The plans differ from published cusum procedures in that starting values are used and the cusum value is not reset to zero after failure.

A note on Page's two-sided cumulative sum procedure

SUMMARY A symmetric version of Page's (1954) two-sided cumulative sum procedure is discussed and a formula for the average run length given. Asymptotic normality of run length under suitable normalization is established.

Some first passage problems related to cusum procedures

Let X 1, X 2,… be independent random variables, and set W n = max(0, W n-1 + X n ), W 0 = 0, n ⩾ 1. The so-called cusum (cumulative sum) procedure uses the first passage time T( h) = inf{ n ⩾ 1: W n ⩾ h}for detecting changes in the mean μ of the process. It is shown that lim h →∞ μET( h)/ h = 1 if μ > 0. Also, a cusum procedure for detecting changes in the normal mean is derived when the variance is unknown. An asymptotic approximation to the average run length is given.

Cumulative sum technique and its application to the analysis of peristimulus time histograms

The cumulative sum procedure introduced by Hurst (1950) involves subtraction of a control reference level from a series of datum points and adding the differences consecutively. The resultant curve shows trends indicating changed levels of points. The techniques has been applied to peristimulus time histograms in order to reveal small changes in the probability of spike occurrences normally obscured by random fluctuations. The combined process of cumulative sum and PSTH cuts the overall number of trials needed in real time and reduces the uncertainty of decision making during the experiment.

Wald's approximations to the average run length in cusum procedures

Wald's approximation to the ARL(average run length in cusum) (cumulative sum) procedures are given for an exponential family of densities. From these approximations it is shown that Page's (1954) cusum procedure is (in a sense) identical with a cusum procedure defined in terms of likelihood ratios. Moreover, these approximations are improved by estimating the excess over the boundary and their closeness is examined by numerical comparisons with some exact results. Some examples are also given.

Process Monitoring with Variable Element Sizes

Cumulative sum procedures are well established for monitoring continuous processes to detect changes in level of one or more process parameters. The methods outlined below extend the use of cusums to situations where the sample elements vary in magnitude. Corresponding Shewhart procedures are also presented and some estimation problems are considered.