Cumulative Sum Statistic Research Articles

Arc length asymptotics for multivariate time series

AbstractThis paper quantifies the asymptotic behavior of sample arc lengths in a multivariate time series. Arc length is a natural measure of the fluctuations in a data series and can be used to quantify volatility. The idea is that processes with larger sample arc lengths exhibit larger fluctuations and hence suggest greater volatility. Here, a Gaussian functional central limit theorem for sample arc lengths is proven under finite second moment conditions. With equally spaced observations, the theory is shown to apply when the first differences of the series obey many of the popular stationary time series models in today's literature, including autoregressive moving‐average, generalized autoregressive conditional heteroscedastic, and stochastic volatility model classes. A cumulative sum statistic is introduced to identify series regimes of differing volatilities. Our applications consider log prices of asset series. Specifically, the results are used to detect nonstationary periods of stock prices. Copyright © 2014 John Wiley & Sons, Ltd.

Online Detection of Change on Information Streams in Wireless Sensor Network Modeled Using Gaussian Distribution

Wireless sensor network (WSN) is deployed to monitor certain physical quantities in a region. This monitoring problem could be stated as the problem of detecting a change in the parameters of a static or dynamic stochastic system. A moving window procedure is proposed to detect the systematic error, which occurs at an unknown time. It can detect the deviation in the mean of sensor measurements keeping variance as constant. The performance measures, such as the average run length (ARL) to detection delay and false alarms are computed for various window sizes. The performance comparison is done against traditional cumulative sum (CUSUM) method. The detection of change in mean using CUSUM is done with smaller delay compared to the proposed moving window detection procedure. In order to calculate CUSUM statistics, the number of measurements to keep in sensor memory increases with time. However, in the proposed moving window detection procedure, the number of stored measurements is limited by the size of the window. Therefore, it is advantageous to use the moving window procedure for change detection in sensor nodes that have very limited memory. A high probability of detection is achieved at the cost of larger window size and higher detection delay. However, we are able to achieve the maximum probability of detection even at a window size of 11.

累積和，2方向累積和，2重累積和統計量の理論と応用

累積和および2重累積和に基づく，形状制約および変化点仮説への総合的接近法を提案する．形状制約は用量反応解析において，パラメトリックモデルを仮定するのが難しい場合に有用なアプローチを与える．変化点仮説はいろいろな場面で時系列の変化を検出するために有用であり，形状制約と密接な関係がある．本論で取り上げる形状制約は単調性，および凸性であり，それぞれ段差，およびスロープ変化点モデルが対応する．提案される統計量は最大対比型と累積χ2型であり，よく知られた制約付き尤度比検定に比べ統計量の構造が簡単で，非正規モデルや2元表データ等，多様な問題への拡張が容易である．とくに，2元表データでは，行や列の水準に自然な順序が有る，無いといういろいろな場合が想定され，それらに応じて興味ある様々な手法が提案される．凸性検定に対して，新たに2重累積和に基づく方法が提案され，2階マルコフ性に基づく簡潔なアルゴリズムが展開される．最後に，3元表データへの拡張について言及される．

Using Cumulative Sum Statistics to Detect Inconsistencies in Unproctored Internet Testing

Unproctored Internet Testing (UIT) is becoming more popular in personnel recruitment and selection. A drawback of UIT is that cheating is easy and, therefore, a proctored test is often administered after an UIT procedure. To detect inconsistent test scores from UIT, a cumulative sum procedure (CUSUM) is proposed. This procedure is applied to empirical data from an adaptive computer-based test in a real personnel selection context. The usefulness of the CUSUM is illustrated, and the unique contribution of the CUSUM to existing procedures is discussed.

A CUSUM to Detect Person Misfit

This article extends the work by Armstrong and Shi on CUmulative SUM (CUSUM) person-fit methodology. The authors present new theoretical considerations concerning the use of CUSUM person-fit statistics based on likelihood ratios for the purpose of detecting cheating and random guessing by individual test takers. According to the Neyman–Pearson Lemma, the optimality of such statistics relies on how accurately normal and aberrant behaviors are modeled. General and specific models for cheating and random guessing are investigated. The detection rates of several statistics are compared using simulated data. Results showed that the likelihood-based CUSUM statistics that use the proposed models for aberrant behavior performed better than some of the more commonly used statistics, especially for cheating behavior.

Accurate Tests and Intervals Based on Multivariate CUSUM Statistics

Suppose that we are monitoring incoming observations for a change in mean via a cumulative sum (CUSUM) test statistic. The usual nonparametric methods give first- and second-order approximations for the one- and two-sided cases. Withers and Nadarajah (Statistics and Probability Letters, 79, 2009, 689–697) and Withers and Nadarajah (Statistics and Probability Letters, 79, 2009, 2242–2250) showed how to improve the order of these approximations for linear statistics and for nonlinear statistics with non-zero asymptotic variances. Here, extensions are provided for nonlinear statistics with zero asymptotic variances and for multivariate statistics.

Multiple Attribute Control Charts with False Discovery Rate Control

The statistical cumulative sum (CUSUM) chart is a powerful tool for monitoring the attribute quality variable in manufacturing industry. In this article, we studied the multiplicity problem caused by simultaneously monitoring more than one attribute quality variable. Multiple binomial and Poisson CUSUM charts incorporating a multiple hypothesis testing technique known as false discovery rate control were proposed. The procedures for establishing the new control schemes were presented, and the performance of the new methods was evaluated using Monte Carlo simulation. The approximation methods for obtaining the p‐values of the CUSUM statistics for conducting the new control schemes were also provided and evaluated. The new methods were also illustrated with a real example. Copyright © 2011 John Wiley & Sons, Ltd.

CUSUM Hypothesis Tests and Alternative Response Probabilities for Finite Poisson Trials

This article develops a new cumulative sum statistic to identify aberrant behavior in a sequentially administered multiple-choice standardized examination. The examination responses can be described as finite Poisson trials, and the statistic can be used for other applications which fit this framework. The standardized examination setting uses a maximum likelihood estimate of examinee ability and an item response theory model. Aberrant and non aberrant probabilities are computed by an odds ratio analogous to risk adjusted CUSUM schemes. The significance level of a hypothesis test, where the null hypothesis is non-aberrant examinee behavior, is computed with Markov chains. A smoothing process is used to spread probabilities across the Markov states. The practicality of the approach to detect aberrant examinee behavior is demonstrated with results from both simulated and empirical data.

Epidemiological Surveillance of Birth Defects Compatible with Thalidomide Embryopathy in Brazil

The thalidomide tragedy of the 1960s resulted in thousands of children being born with severe limb reduction defects (LRD), among other malformations. In Brazil, there are still babies born with thalidomide embryopathy (TE) because of leprosy prevalence, availability of thalidomide, and deficiencies in the control of drug dispensation. Our objective was to implement a system of proactive surveillance to identify birth defects compatible with TE. Along one year, newborns with LRD were assessed in the Brazilian hospitals participating in the Latin-American Collaborative Study of Congenital Malformations (ECLAMC). A phenotype of LRD called thalidomide embryopathy phenotype (TEP) was established for surveillance. Children with TEP born between the years 2000–2008 were monitored, and during the 2007–2008 period we clinically investigated in greater detail all cases with TEP (proactive period). The period from 1982 to 1999 was defined as the baseline period for the cumulative sum statistics. The frequency of TEP during the surveillance period, at 3.10/10,000 births (CI 95%: 2.50–3.70), was significantly higher than that observed in the baseline period (1.92/10,000 births; CI 95%: 1.60–2.20), and not uniformly distributed across different Brazilian regions. During the proactive surveillance (2007–2008), two cases of suspected TE were identified, although the two mothers had denied the use of the drug during pregnancy. Our results suggest that TEP has probably increased in recent years, which coincides with the period of greater thalidomide availability. Our proactive surveillance identified two newborns with suspected TE, proving to be a sensitive tool to detect TE. The high frequency of leprosy and the large use of thalidomide reinforce the need for a continuous monitoring of TEP across Brazil.

Changepoints in the North Atlantic Tropical Cyclone Record

This article examines the North Atlantic tropical cyclone record for statistical discontinuities (changepoints). This is a controversial area and indeed, our end conclusions are opposite of those made in Dr. Kelvin Droegemeier’s July 28, 2009 Senate testimonial. The methods developed here should help rigorize the debate. Elaborating, we develop a level-α test for a changepoint in a categorical data sequence sampled from a multinomial distribution. The proposed test statistic is the maximum of correlated Pearson chi-square statistics. This test statistic is linked to cumulative sum statistics and its null hypothesis asymptotic distribution is derived in terms of the supremum of squared Brownian bridges. The methods are used to identify changes in the tropical cyclone record in the North Atlantic Basin over the period 1851–2008. We find changepoints in both the storm frequencies and their strengths (wind speeds). The changepoint in wind speed is not found with standard cumulative sum mean shift changepoint methods, hence providing a dataset where categorical probabilities shift but means do not. While some of the identified shifts can be attributed to changes in data collection techniques, the hotly debated changepoint in cyclone frequency circa 1995 also appears to be significant.

Large Occurrence Patterns of New Zealand Deep Earthquakes: Characterization by Use of a Switching Poisson Model

The paper studies the statistical properties of deep earthquakes around North Island, New Zealand. We first evaluate the catalogue coverage and completeness of deep events according to cusum (cumulative sum) statistics and earlier literature. The epicentral, depth, and magnitude distributions of deep earthquakes are then discussed. It is worth noting that strong grouping effects are observed in the epicentral distribution of these deep earthquakes. Also, although the spatial distribution of deep earthquakes does not change, their occurrence frequencies vary from time to time, active in one period, relatively quiescent in another. The depth distribution of deep earthquakes also hardly changes except for events with focal depth less than 100 km. On the basis of spatial concentration we partition deep earthquakes into several groups—the Taupo-Bay of Plenty group, the Taranaki group, and the Cook Strait group. Second-order moment analysis via the two-point correlation function reveals only very small-scale clustering of deep earthquakes, presumably limited to some hot spots only. We also suggest that some models usually used for shallow earthquakes fit deep earthquakes unsatisfactorily. Instead, we propose a switching Poisson model for the occurrence patterns of deep earthquakes. The goodness-of-fit test suggests that the time-varying activity is well characterized by a switching Poisson model. Furthermore, detailed analysis carried out on each deep group by use of switching Poisson models reveals similar time-varying behavior in occurrence frequencies in each group.

Mean shift testing in correlated data

Several tests for detecting mean shifts at an unknown time in stationary time series have been proposed, including cumulative sum (CUSUM), Gaussian likelihood ratio (LR), maximum of F(Fmax) and extreme value statistics. This article reviews these tests, connects them with theoretical results, and compares their finite sample performance via simulation. We propose an adjusted CUSUM statistic which is closely related to the LR test and which links all tests. We find that tests based on CUSUMing estimated one-step-ahead prediction residuals from a fitted autoregressive moving average perform well in general and that the LR and Fmax tests (which induce substantial computational complexities) offer only a slight increase in power over the adjusted CUSUM test. We also conclude that CUSUM procedures work slightly better when the changepoint time is located near the centre of the data, but the adjusted CUSUM methods are preferable when the changepoint lies closer to the beginning or end of the data record. Finally, an application is presented to demonstrate the importance of the choice of method.

Early warning CUSUM plans for surveillance of negative binomial daily disease counts

Automated public health surveillance of disease counts for rapid outbreak, epidemic or bioterrorism detection using conventional control chart methods can be hampered by over-dispersion and background (‘in-control’) mean counts that vary over time. An adaptive cumulative sum (CUSUM) plan is developed for signalling unusually high incidence in prospectively monitored time series of over-dispersed daily disease counts with a non-homogeneous mean. Negative binomial transitional regression is used to prospectively model background counts and provide ‘one-step-ahead’ forecasts of the next day's count. A CUSUM plan then accumulates departures of observed counts from an offset (reference value) that is dynamically updated using the modelled forecasts. The CUSUM signals whenever the accumulated departures exceed a threshold. The amount of memory of past observations retained by the CUSUM plan is determined by the offset value; a smaller offset retains more memory and is efficient at detecting smaller shifts. Our approach optimises early outbreak detection by dynamically adjusting the offset value. We demonstrate the practical application of the ‘optimal’ CUSUM plans to daily counts of laboratory-notified influenza and Ross River virus diagnoses, with particular emphasis on the steady-state situation (i.e. changes that occur after the CUSUM statistic has run through several in-control counts).

TWO-PASS CUSUM TO IDENTIFY AGE-CLUSTER OUTBREAKS

The paper introduces a two-pass adaptive cumulative sum (CUSUM) statistic to identify age clusters (age grouping) that significantly contribute to epidemics or unusually high counts. If epidemiologists know that an epidemic is confined to a narrow age group, then this information not only makes it clear where to target the epidemiological effort but also helps them decide whether to respond. It is much easier to control an epidemic that starts in a narrow age range of the population, such as pre-school children, than an epidemic that is not confined demographically or geographically.

Model‐Free CUSUM Methods for Person Fit

This article demonstrates the use of a new class of model‐free cumulative sum (CUSUM) statistics to detect person fit given the responses to a linear test. The fundamental statistic being accumulated is the likelihood ratio of two probabilities. The detection performance of this CUSUM scheme is compared to other model‐free person‐fit statistics found in the literature as well as an adaptation of another CUSUM approach. The study used both simulated responses and real response data from a large‐scale standardized admission test.

A comparison of CUSUM, EWMA, and temporal scan statistics for detection of increases in poisson rates

AbstractVarious control chart methods have been used in healthcare and public health surveillance to detect increases in the rates of diseases or their symptoms. Although the observations in many health surveillance applications are often discrete, few efforts have been made to ex‐ plore the behavior of detection methods in discrete distributions. Joner et al. (Statist. Med. 2008; 27:2555–2575) investigated and compared the performance of the scan statistic methods with the cumulative sum (CUSUM) charts under a Bernoulli distribution. In this paper we compare the performance of three detection methods: temporal scan statistic, CUSUM, and exponential weighted moving average (EWMA) when the observations follow the Poisson distribution. A simulation study showed that the Poisson CUSUM and EWMA charts generally outperformed the Poisson scan statistic methods. In comparisons between CUSUM and EWMA, the CUSUM charts were superior in dealing with a large shift with a later change in time. However, the EWMA charts outperformed the CUSUM charts in situations with a small shift and an early change in time. The methods were also compared with thyroid cancer using a real data set. Copyright © 2009 John Wiley & Sons, Ltd.

A Parametric Cumulative Sum Statistic for Person Fit

This article develops a new cumulative sum (CUSUM) statistic to detect aberrant item response behavior. Shifts in behavior are modeled with quadratic functions and a series of likelihood ratio tests are used to detect aberrancy. The new CUSUM statistic is compared against another CUSUM approach as well as traditional person-fit statistics. A simulation study demonstrates the advantage of the proposed method. Also, the person-fit methods are applied to real response data from the administration of a high-stakes exam. The use of CUSUM charts to help visually identify types of aberrant behavior is demonstrated.

A CUSUM-based method for monitoring simple linear profiles

In most statistical process control applications, the quality of a process or product is characterized by univariate or multivariate quality characteristics and monitored by the corresponding univariate and multivariate control charts, respectively. However, sometimes, the quality of a process or a product is better characterized by a relationship between a response variable and one or more explanatory variables. This relationship, which can be linear, nonlinear, or even a complicated model, is referred to as a profile. So far, several methods have been proposed for monitoring simple linear profiles. In this paper, a new method based on cumulative sum statistics is proposed to enhance monitoring of linear profiles in phase II. The performance of the proposed method is evaluated by average run length criterion. A comprehensive comparison is also conducted between the performance of the proposed method and the existing methods for monitoring simple linear profiles. The results show that the proposed method performs satisfactorily. In addition, the effects of reference value, sample size, and corrected sum of squares of explanatory variables on the performance of the proposed method are investigated.

False Discovery Rate-Adjusted Charting Schemes for Multistage Process Monitoring and Fault Identification

Most statistical process control research focuses on single-stage processes. This article considers the problem of multistage process monitoring and fault identification. This problem is formulated as a multiple hypotheses testing problem; however, as the number of stages increases, the detection power of multiple hypotheses testing methods that seek to control the type I error rate decreases dramatically. To maintain the detection power, we use a false discovery rate (FDR) control approach, which is widely used in microarray research. Two multistage process monitoring and fault identification schemes—an FDR-adjusted Shewhart chart and an FDR-adjusted cumulative sum (CUSUM) chart—are established. To apply the FDR approach, the distribution of the CUSUM statistics are obtained based on Markov chain theory and Brownian motion with drift models. The detection and fault identification power of the new schemes are evaluated by the Monte Carlo method. The results indicate that the novel FDR-adjusted approaches are better at identifying the faulty stage than the conventional type I error rate control approach, especially when multiple out-of-control stages are present.

Distribution-free cumulative sum control charts using bootstrap-based control limits

This paper deals with phase II, univariate, statistical process control when a set of in-control data is available, and when both the in-control and out-of-control distributions of the process are unknown. Existing process control techniques typically require substantial knowledge about the in-control and out-of-control distributions of the process, which is often difficult to obtain in practice. We propose (a) using a sequence of control limits for the cumulative sum (CUSUM) control charts, where the control limits are determined by the conditional distribution of the CUSUM statistic given the last time it was zero, and (b) estimating the control limits by bootstrap. Traditionally, the CUSUM control chart uses a single control limit, which is obtained under the assumption that the in-control and out-of-control distributions of the process are Normal. When the normality assumption is not valid, which is often true in applications, the actual in-control average run length, defined to be the expected time duration before the control chart signals a process change, is quite different from the nominal in-control average run length. This limitation is mostly eliminated in the proposed procedure, which is distribution-free and robust against different choices of the in-control and out-of-control distributions.