CUSUM Procedure Research Articles

On the ARL of Cusum in Multinomial Models

There is no known closed form expression for the average sample number, also known as average run length, of a multivariate cusum procedure N = min{M1, M2, · · · , Mm} for m ≥ 3, where Mi are univariate cusum procedures. The problem is generally considered to be hopelessly complicated for any model. In this paper, for the multinomial model we show, however, that there is a rather simple closed form expression for the ARL of N with an elementary proof. A bit surprisingly, we further show that the ARL of N is related to the ARLs of Mi the same way as the capacitance of a series network of capacitors is related to the capacitances of its own components.

Distribution-free image monitoring with application to battery coating process

This article presents a distribution-free image monitoring procedure for a manufacturing process, where a series of images are converted into a vector of two feature values extracted from singular value decomposition. Traditional image-based monitoring methods often make specific assumptions about marginal distributions and spatio-temporal dependence structures, which are often violated in real-world scenarios such as battery coating processes. To overcome this issue, we propose a distribution-free image monitoring procedure that detects a shift in the mean matrix of monitored images. Our method involves performing singular value decomposition of each image matrix in two ways to obtain two values, which are then combined into a bivariate vector. The bivariate vectors are monitored using a distribution-free multivariate CUSUM procedure, for which we determine control limits analytically, enabling convenient and easy implementation of the monitoring procedure. We demonstrate the effectiveness of our proposed procedure, as measured by average run lengths, using various simulated data and a real-data example from a battery coating process.

Detecting mean changes in experience sampling data in real time: A comparison of univariate and multivariate statistical process control methods.

Detecting early warning signals of developing mood disorders in continuously collected affective experience sampling (ESM) data would pave the way for timely intervention and prevention of a mood disorder from occurring or to mitigate its severity. However, there is an urgent need for online statistical methods tailored to the specifics of ESM data. Statistical process control (SPC) procedures, originally developed for monitoring industrial processes, seem promising tools. However, affective ESM data violate major assumptions of the SPC procedures: The observations are not independent across time, often skewed distributed, and characterized by missingness. Therefore, evaluating SPC performance on simulated data with typical ESM features is a crucial step. In this article, we didactically introduce six univariate and multivariate SPC procedures: Shewhart, Hotelling's T², EWMA, MEWMA, CUSUM and MCUSUM. Their behavior is illustrated on publicly available affective ESM data of a patient that relapsed into depression. To deal with the missingness, autocorrelation, and skewness in these data, we compute and monitor the day averages rather than the individual measurement occasions. Moreover, we apply all procedures on simulated data with typical affective ESM features, and evaluate their performance at detecting small to moderate mean changes. The simulation results indicate that the (M)EWMA and (M)CUSUM procedures clearly outperform the Shewhart and Hotelling's T² procedures and support using day averages rather than the original data. Based on these results, we provide some recommendations for optimizing SPC performance when monitoring ESM data as well as a wide range of directions for future research. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Structural Stability of the Relationship between Indirect Taxes and Government Expenditure in Nepal: Implications for Fiscal Policy

This study uses time series data of Nepal from fiscal years 1974/75 to 2017/18 to examine the dynamics of the relationship between indirect taxes and government spending. The study’s main objective is to examine the effect size of Value Added Tax (VAT), excise, and customs duties in predicting Nepal’s budget deficit. The data for the study purpose were obtained from the Government of Nepal, Ministry of Finance. The study used a retrospective methodology. The data were first checked for stationarity by employing the Augmented Dickey-Fuller Test, and then the link between the variables was determined using a multiple regression analysis based on ordinary least squares. The results showed that customs charges did not affect government spending in Nepal, while value-added and excise taxes had a statistically significant positive impact. The structural stability of the parameter was evaluated utilising the CUSUM and CUSUMSQ procedures. The results showed no evidence of misspecification or instability within the period predicted by the model. The residuals follow a normal distribution with no serial correlation, heteroscedasticity, or multicollinearity issues. The main implication of the study finding is that the government should exercise fiscal restraint and prioritise the collection of indirect taxes. Additionally, it recommends reducing customs taxes for new businesses and expanding the range of products subject to VAT.

Sequential detection of transient signal by moving likelihood ratio statistic in an exponential family

ABSTRACT We first consider the sequential detection of transient signals by generalizing the moving average chart to exponential family and study the false detection probability (FDP) and power of detection (POD) in the steady state. Then windowed adjusted signed (or modified directed) likelihood ratio chart is studied by treating it as approximate normal random variable. In the multi-parameter exponential family, the detection of the transient change of one of the canonical parameters or a function of canonical parameters is considered by using the generalized adjusted signed likelihood ratio chart. Comparisons with window restricted CUSUM and Shiryayev-Roberts (S-R) procedures show that the generalized signed likelihood ratio chart performs quite well. Several important examples including the mean or variance change under normal model and a real-time example are used for illustration.

Quickest Change Detection With Non-Stationary Post-Change Observations

The problem of quickest detection of a change in the distribution of a sequence of independent observations is considered. The pre-change observations are assumed to be stationary with a known distribution, while the post-change observations are allowed to be non-stationary with some possible parametric uncertainty in their distributions. In particular, it is assumed that the cumulative Kullback-Leibler divergence between the post-change and the pre-change distributions grows in a certain manner with time after the change-point. For the case where the post-change distributions are known, a universal asymptotic lower bound on the delay is derived, as the false alarm rate goes to zero. Furthermore, a window-limited Cumulative Sum (CuSum) procedure is developed, and shown to achieve the lower bound asymptotically. For the case where the post-change distributions have parametric uncertainty, a window-limited (WL) generalized likelihood-ratio (GLR) CuSum procedure is developed and is shown to achieve the universal lower bound asymptotically. Extensions to the case with dependent observations are discussed. The analysis is validated through numerical results on synthetic data. The use of the WL-GLR-CuSum procedure in monitoring pandemics is also demonstrated.

Distribution-free multivariate time-series monitoring with analytically determined control limits

We consider the problem of detecting a shift in the mean of a multivariate time-series process with general marginal distributions and general cross- and auto-correlation structures. We propose a distribution-free monitoring procedure that does not need model fitting or trial-and-error calibration for control limits, which makes the procedure convenient to be implemented when a facility consists of many processes to be monitored. The main idea is to convert each observation vector into a one-dimensional T 2 quantity that captures cross-correlation. The T 2 quantities form a univariate auto-correlated process, and CUSUM statistics are constructed on the T 2 quantities. Then using the fact that the CUSUM statistics on the auto-correlated process behave as a reflected Brownian motion asymptotically under some conditions, the control limits of the CUSUM procedure are analytically determined by setting the first-passage time of the Brownian motion equal to a target in-control average run length. We compare the performance of our procedure with three competing procedures on simulated data with various cross- and auto-correlation and real data from a wafer etching process. The proposed procedure delivers actual in-control average run lengths close to the target and shows comparable or better performance in detecting a shift in mean than the competitors.

Sequential Change-Point Detection for Mutually Exciting Point Processes

We present a new CUSUM procedure for sequential change-point detection in self- and mutually-exciting point processes (specifically, Hawkes networks) using discrete events data. Hawkes networks have become a popular model in statistics and machine learning, primarily due to their capability in modeling irregularly observed data where the timing between events carries a lot of information. The problem of detecting abrupt changes in Hawkes networks arises from various applications, including neuroengineering, sensor networks, and social network monitoring. Despite this, there has not been an efficient online algorithm for detecting such changes from sequential data. To this end, we propose an online recursive implementation of the CUSUM statistic for Hawkes processes, which is computationally and memory-efficient and can be decentralized for distributed computing. We first prove theoretical properties of this new CUSUM procedure, then show the improved performance of this approach over existing methods, including the Shewhart procedure based on count data, the generalized likelihood ratio statistic, and the standard score statistic. This is demonstrated via simulation studies and an application to population code change-detection in neuroengineering.

Online nonparametric monitoring of heterogeneous data streams with partial observations based on Thompson sampling

With the rapid advancement of sensor technology driven by Internet-of-Things-enabled applications, tremendous amounts of measurements of heterogeneous data streams are frequently acquired for online process monitoring. Such massive data, involving a large number of data streams with high sampling frequency, incur high costs on data collection, transmission, and analysis in practice. As a result, the resource constraint often restricts the data observability to only a subset of data streams at each data acquisition time, posing significant challenges in many online monitoring applications. Unfortunately, existing methods do not provide a general framework for monitoring heterogeneous data streams with partial observations. In this article, we propose a nonparametric monitoring and sampling algorithm to quickly detect abnormalities occurring to heterogeneous data streams. In particular, an approximation framework is incorporated with an antirank-based CUSUM procedure to collectively estimate the underlying status of all data streams based on partially observed data. Furthermore, an intelligent sampling strategy based on Thompson sampling is proposed to dynamically observe the informative data streams and balance between exploration and exploitation to facilitate quick anomaly detection. Theoretical justification of the proposed algorithm is also investigated. Both simulations and case studies are conducted to demonstrate the superiority of the proposed method.

General drawdown of general tax model in a time-homogeneous Markov framework

AbstractDrawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure), and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with more general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes [9, 20]. In this paper we further examine the general drawdown-related quantities in the (upward skip-free) time-homogeneous Markov process, and then in its (general) tax process by noticing the pathwise connection between general drawdown and the tax process.

Testing Randomness Online

The hypothesis of randomness is fundamental in statistical machine learning and in many areas of nonparametric statistics; it says that the observations are assumed to be independent and coming from the same unknown probability distribution. This hypothesis is close, in certain respects, to the hypothesis of exchangeability, which postulates that the distribution of the observations is invariant with respect to their permutations. This paper reviews known methods of testing the two hypotheses concentrating on the online mode of testing, when the observations arrive sequentially. All known online methods for testing these hypotheses are based on conformal martingales, which are defined and studied in detail. The paper emphasizes conceptual and practical aspects and states two kinds of results. Validity results limit the probability of a false alarm or the frequency of false alarms for various procedures based on conformal martingales, including conformal versions of the CUSUM and Shiryaev-Roberts procedures. Efficiency results establish connections between randomness, exchangeability, and conformal martingales.

Optimal Sequential Detection of Signals With Unknown Appearance and Disappearance Points in Time

The paper addresses a sequential changepoint detection problem, assuming that the duration of change may be finite and unknown. This problem is of importance for many applications, e.g., for signal and image processing where signals appear and disappear at unknown points in time or space. In contrast to the conventional optimality criterion in quickest change detection that requires minimization of the expected delay to detection for a given average run length to a false alarm, we focus on a reliable maximin change detection criterion of maximizing the minimal probability of detection in a given time (or space) window for a given local maximal probability of false alarm in the prescribed window. We show that the optimal detection procedure is a modified CUSUM procedure. We then compare operating characteristics of this optimal procedure with popular in engineering the Finite Moving Average (FMA) detection algorithm and the ordinary CUSUM procedure using Monte Carlo simulations, which show that typically the later algorithms have almost the same performance as the optimal one. At the same time, the FMA procedure has a substantial advantage -- independence to the intensity of the signal, which is usually unknown. Finally, the FMA algorithm is applied to detecting faint streaks of satellites in optical images.

Nonparametric monitoring of multivariate data via KNN learning

Process monitoring of multivariate quality attributes is important in many industrial applications, in which rich historical data are often available thanks to modern sensing technologies. While multivariate statistical process control (SPC) has been receiving increasing attention, existing methods are often inadequate as they are sensitive to the parametric model assumptions of multivariate data. In this paper, we propose a novel, nonparametric k-nearest neighbours empirical cumulative sum (KNN-ECUSUM) control chart that is a machine-learning-based black-box control chart for monitoring multivariate data by utilising extensive historical data under both in-control and out-of-control scenarios. Our proposed method utilises the k-nearest neighbours (KNN) algorithm for dimension reduction to transform multivariate data into univariate data and then applies the CUSUM procedure to monitor the change on the empirical distribution of the transformed univariate data. Extensive simulation studies and a real industrial example based on a disk monitoring system demonstrate the robustness and effectiveness of our proposed method.

Monitoring parameter changes in models with a trend

We develop a CUSUM-type monitoring procedure based on the ordinary least squares residuals for detecting structural changes in models with a trend. A proper boundary function is designed to control the size. We derive the limiting null distribution and the consistency of the procedure under the alternative. In addition, we derive the asymptotic distribution of the delay time for the CUSUM procedure as well as the fluctuation procedure proposed by Qi etal. (2016). The simulation and empirical results indicate that although neither procedure is uniformly superior to the other, the CUSUM test is more suitable for an early break.

Change-point detection for Lévy processes

Since the work of Page in the 1950s, the problem of detecting an abrupt change in the distribution of stochastic processes has received a great deal of attention. In particular, a deep connection has been established between Lorden’s minimax approach to change-point detection and the widely used CUSUM procedure, first for discrete-time processes, and subsequently for some of their continuous-time counterparts. However, results for processes with jumps are still scarce, while the practical importance of such processes has escalated since the turn of the century. In this work, we consider the problem of detecting a change in the distribution of continuous-time processes with independent and stationary increments, that is, Levy processes, and our main result shows that CUSUM is indeed optimal in Lorden’s sense. This is the most natural continuous-time analogue of the seminal work of Moustakides [Ann. Statist. 14 (1986) 1379–1387] for sequentially observed random variables that are assumed to be i.i.d. before and after the change-point. From a practical perspective, the approach we adopt is appealing as it consists in approximating the continuous-time problem by a suitable sequence of change-point problems with equispaced sampling points, and for which a CUSUM procedure is shown to be optimal.

Comparison of statistical algorithms for syndromic surveillance aberration detection

ObjectiveTo investigate whether alternative statistical approaches can improve daily aberration detection using syndromic surveillance in England.IntroductionSyndromic surveillance involves monitoring big health datasets to provide early warning of threats to public health. Public health authorities use statistical detection algorithms to interrogate these datasets for aberrations that are indicative of emerging threats. The algorithm currently in use at Public Health England (PHE) for syndromic surveillance is the ‘rising activity, multi-level mixed effects, indicator emphasis’ (RAMMIE) method (Morbey et al, 2015), which fits a mixed model to counts of syndromes on a daily basis. This research checks whether the RAMMIE method works across a range of public health scenarios and how it compares to alternative methods.MethodsFor this purpose, we compare RAMMIE to the improved quasi-Poisson regression-based approach (Noufaily et al, 2013), currently implemented at PHE for weekly infectious disease laboratory surveillance, and to the Early Aberration Reporting System (EARS) method (Rossi et al, 1999), which is used for syndromic surveillance aberration detection in many other countries. We model syndromic datasets, capturing real data aspects such as long-term trends, seasonality, public holidays, and day-of-the-week effects, with or without added outbreaks. Then, we compute the sensitivity and specificity to compare how well each of the algorithms detects synthetic outbreaks to provide recommendations for the most suitable statistical methods to use during different public health scenarios.ResultsPreliminary results suggest all methods provide high sensitivity and specificity, with the (Noufaily et al, 2013) approach having the highest sensitivity and specificity. We showed that for syndromes with long-term increasing trends, RAMMIE required modificaiton to prevent excess false alarms. Also, our study suggests further work is needed to fully account for public holidays and day-of-the-week effects.ConclusionsOur study will provide recommendations for which algorithm is most effective for PHE's syndromic surveillance for a range of different syndromes. Furthermore our work to generate standardised synthetic syndromic datasets and a range of outbreaks can be used for future evaluations in England and elsewhere.ReferencesNoufaily, A., Enki, D. G., Farrington, C. P., Garthwaite, P., Andrews, N. and Charlett, A. (2013). An Improved Algorithm for Outbreak Detection in Multiple Surveillance Systems. Statistics in Medicine, 32(7), 1206-1222.Morbey, R. A., Elliot, A. J., Charlett, A., Verlander, A. Q, Andrews, N. and Smith, G. (2013). The application of a novel ‘rising activity, multi-level mixed effects, indicator emphasis’ (RAMMIE) method for syndromic surveillance in England, Bioinformatics, 31(22), 3660-3665.Rossi, G, Lampugnani, L, Marchi, M. (1999), An approximate CUSUM procedure for surveillance of health events. Statistics in Medicine, 18, 2111–2122

Repeated SPRT charts for monitoring INAR(1) processes

AbstractPoisson integer valued autoregressive (INAR) models have been proposed for modeling correlated count data. Poisson lognormal (PLN) INAR models extend their use to overdispersed contexts. In this paper, we will propose the use of a repeated Sequential Probability Ratio Test (SPRT) procedure to detect change in first‐order INAR and PLN INAR models. We consider change in the mean, the autocorrelation parameter, and the overdispersion parameter. Simulation results show the repeated SPRT procedure performs favorably relative to previously proposed CUSUM procedures that are based on either the observations themselves or residuals of the observations from predicted values. A dataset on invasive insect species is used to illustrate the repeated SPRT procedure.

Joint Monitoring of Process Location and Dispersion Based on CUSUM Procedure and Generalized Likelihood Ratio in the Presence of Measurement Errors

In this paper, we propose a new methodology based on the combination of cumulative sum procedure and generalized likelihood ratio statistic for joint monitoring of the process location and dispersion. Then, we explore the effect of measurement errors on detecting ability of the proposed control chart when (i) the variance of measurement error is constant (ii) the variance of measurement error increases linearly as the level of the process mean increases. We also utilize multiple measurements on each sample point in order to decrease the adverse effects of measurement errors on the performance of the proposed control charts. Two numerical examples based on simulation studies are given to evaluate the ability of the proposed methods in terms of average run length, median run length, standard deviation of run length, and the first and third quantile points of the run length distribution (Q1 and Q3). Finally, a real life example is given to illustrate the application of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.

Detecting at‐Most‐m Changes in Linear Regression Models

In this article, we provide a new procedure to test for at‐most‐ changes in the time‐dependent regression model , that is, β1 = β2 = ⋯ = βT under the no‐change null hypothesis against the alternative if and β(j) ≠ β(ℓ) for some with . Our procedure is based on weighted sums of the residuals, incorporating the possibility of changes. The weak limit of the proposed test statistic is the sum of two double‐exponential random variables. A small Monte Carlo simulation illustrates the applicability of the limit results in case of small and moderate sample sizes. We compare the new method to the cumulative sum control chart (CUSUM) and standardized (weighted) CUSUM procedures and obtain the power curves of the test statistics under the alternative. We apply our method to find changes in the unconditional four‐factor capital asset pricing model.

Estimation of Change-point and Post-change Parameters after Adaptive Sequential CUSUM Test in an Exponential Family

In this paper, we consider an adaptive sequential CUSUM procedure in an exponential family where the change-point and post-change parameters are estimated adaptively. It is shown that the adaptive CUSUM procedure is efficient at the first order. The conditional biases of the estimation for the change-point and post-change parameter are studied. Comparison with the classical CUSUM procedure in the normal case is made. Nile river flow and average global temperature data sets are used for demonstration.