Conventional Cumulative Sum Research Articles

A PCA-based algorithm for online false data injection and jamming attacks detection in cyber-physical systems

Cyber-physical systems face cyberattacks that hinder performance, increase cost, or even collapse the system. Thus, rapid and accurate attack detection is crucial for quick activation of defense mechanisms. Model-based attack detection approaches are known for their precision; however, the unavailability of the exact system model poses a challenge. In response, model-free approaches have gained increased attention as a practical alternative. In this paper, an online model-free algorithm for detecting false data injection and jamming attacks on CPSs is proposed. The method leverages principal component analysis to reconstruct the expected observations in a reduced dimension space, emphasizing the most effective principal components. Then, deciding on attacked or normal operation relies on analyzing either the Euclidean distance or the cosine similarity of the discrepancy between the expected and actual observations. The proposed metrics effectively expose subtle deviations from expected behavior, as any alteration in these components augments the distance between the observed and reconstructed values. The proposed method was compared with the conventional cumulative sum discriminator and Kalman-based algorithm, using an IEEE-14, IEEE-30, and IEEE-118 bus systems. The results demonstrate the superiority of the proposed algorithm in terms of various evaluation metrics, including F-score, precision, recall, and miss detection ratio.

BACKWARD CUSUM FOR TESTING AND MONITORING STRUCTURAL CHANGE WITH AN APPLICATION TO COVID-19 PANDEMIC DATA

It is well known that the conventional cumulative sum (CUSUM) test suffers from low power and large detection delay. In order to improve the power of the test, we propose two alternative statistics. The backward CUSUM detector considers the recursive residuals in reverse chronological order, whereas the stacked backward CUSUM detector sequentially cumulates a triangular array of backwardly cumulated residuals. A multivariate invariance principle for partial sums of recursive residuals is given, and the limiting distributions of the test statistics are derived under local alternatives. In the retrospective context, the local power of the tests is shown to be substantially higher than that of the conventional CUSUM test if a break occurs in the middle or at the end of the sample. When applied to monitoring schemes, the detection delay of the stacked backward CUSUM is found to be much shorter than that of the conventional monitoring CUSUM procedure. Furthermore, we propose an estimator of the break date based on the backward CUSUM detector and show that in monitoring exercises this estimator tends to outperform the usual maximum likelihood estimator. Finally, an application of the methodology to COVID-19 data is presented.

Fault detection strategy combining NARMAX model and Bhattacharyya distance for process monitoring

The complexity of modern chemical and petrochemical plants is becoming increasingly problematic in the recent years. At the same time, the demands to ensure safety and reliability of process operations rise. Early detection of abnormal event in complex real systems decrease maintenance cost and lead to guarantee the safety of human operators and environment. In the present work, a fault detection (FD) method which exploits the advantages of black-box modeling and statistical measure for fault detection in real chemical process as a distillation column is proposed. This technique is developed by applying the Nonlinear Auto-Regressive Moving Average with eXogenous input (NARMAX) model and Bhattacharyya distance (BD). In order to determine the NARMAX model, a real data set recorded during normal operations is used. Then, the BD is used to quantify on-line the dissimilarity between the current and reference probability distributions of the residual obtained from the NARMAX model for fault detection purposes. The ability of the proposed FD approach is demonstrated using real fault of separation unit. The obtained results indicate that the developed technique produces favorable performance compared to the conventional Cumulative Sum (CUSUM) test.

Real-Time Detection of Cyber-Physical False Data Injection Attacks on Power Systems

This article studies the online detection of false data injection attacks (FDIAs) and coordinated cyber-physical attacks (CCPAs) on power systems. By analyzing the hidden physical meaning of CCPAs, a cyber-physical FDIA with CCPA as a special case is proposed to establish the connection between FDIA and CCPA. Based on a discrete-time system dynamic model, an adaptive nonparametric cumulative sum (AN-CUSUM) detector is devised to deal with FDIAs and CCPAs simultaneously. The AN-CUSUM algorithm estimates the abnormal change of the state vector caused by attacks one step in advance and standardizes the decision statistics of conventional cumulative sum (CUSUM) to simplify the setting of thresholds. The proposed detector is robust to a wide range of time-varying system states and attack magnitudes. Moreover, a verification method is designed in the proposed detector to differentiate CCPAs and FDIAs according to unique characteristics of CCPAs, namely, the construction of CCPAs depends on the physical system parameter. Numerical results reveal that the proposed detector is more reliable to detect both FDIAs and CCPAs than existing CUSUM-based methods.

Algorithms for Change Detection With Sparse Signals

We consider the change detection problem where the pre-change observation vectors are purely noise and the post-change observation vectors are noise-corrupted compressive measurements of sparse signals with a common support, measured using a sensing matrix. In general, post-change probability density function (pdf) of the observations depends on parameters such as the support and variances of the sparse signal. When these parameters are unknown, we propose two approaches. In the first approach, we approximate the post-change pdf based on the known parameters such as mutual coherence of the sensing matrix and bounds on the signal variances. In the second approach, we parameterize the post-change pdf with an unknown parameter and try to estimate this parameter using two different methods, stochastic gradient descent and generalized likelihood ratio. In both these approaches, we employ the conventional cumulative sum (CUSUM) algorithm with various decision statistics such as the energy of the observations, correlation values with columns of the sensing matrix and the maximum value of such correlations. We analytically characterize the worst case detection delay and average run length to false alarm performance of pdf-approximation based approach. We also numerically study the performance and offer insights on the relevance of various decision statistics in different signal to noise ratio (SNR) regimes. We also address the problem of designing sensing matrices with small mutual coherence by using designs from quantum information theory. One such design using equi-angular lines has an additional structure which allows exact characterization of the post-change pdf of the correlation values, even when the support set of the sparse signal is unknown. We apply our detection algorithms with sensing matrix designed from equi-angular lines to a massive random access problem and show their superior performance over conventional Gold codes.

A new dual CUSUM mean chart

AbstractThe conventional cumulative sum (CUSUM) chart is usually designed based on a known shift size. In usual practice, shift size is often unknown and can be assumed to vary within an interval. With such a range of shift size, the dual CUSUM (DCUSUM) chart provides more sensitivity than the CUSUM chart. In this paper, we propose dual Crosier CUSUM (DCCUSUM) charts with and without fast initial response features to efficiently monitor the infrequent changes in the mean of a normally distributed process. Monte Carlo simulations are used to compute the run length characteristics of one‐sided and two‐sided DCCUSUM charts. These run length characteristics are compared with those of the CUSUM, Crosier CUSUM, Shewhart‐CUSUM, and DCUSUM charts in terms of the integral relative average run length. It turns out that the proposed chart shows better performance when detecting a range of mean shift sizes. A real dataset is considered to illustrate the implementation of existing and proposed charts.

Statistical Process Control Using a Dynamic Sampling Scheme

This article considers statistical process control (SPC) of univariate processes, and tries to make two contributions to the univariate SPC problem. First, we propose a continuously variable sampling scheme, based on a quantitative measure of the likelihood of a process distributional shift at each observation time point, provided by the p-value of the conventional cumulative sum (CUSUM) charting statistic. For convenience of the design and implementation, the variable sampling scheme is described by a parametric function in the flexible Box–Cox transformation family. Second, the resulting CUSUM chart using the variable sampling scheme is combined with an adaptive estimation procedure for determining its reference value, to effectively protect against a range of unknown shifts. Numerical studies show that it performs well in various cases. A real data example from a chemical process illustrates the application and implementation of our proposed method. This article has supplementary materials online.

Increasing the Sensitivity of Cumulative Sum Charts for Location

The cumulative sum (CUSUM) chart is a very effective control charting procedure used for the quick detection of small‐sized and moderate‐sized changes. It can detect small process shifts missed by the Shewhart‐type control chart, which is sensitive mainly to large shifts. To further enhance the sensitivity of the CUSUM control chart at detecting very small process disturbances, this article presents CUSUM control charts based on well‐structured sampling procedures, double ranked set sampling, median‐double ranked set sampling, and double‐median ranked set sampling. These sampling techniques significantly improve the overall performance of the CUSUM chart over the entire process mean shift range, without increasing the false alarm rate. The newly developed control schemes do not only dominate most of the existing charts but are also easy to design and implement as illustrated through an application example of real datasets. The control schemes used for comparison in this study include the conventional CUSUM chart, a fast initial response CUSUM chart, a 2‐CUSUM chart, a 3‐CUSUM chart, a runs rules‐based CUSUM chart, the enhanced adaptive CUSUM chart, the CUSUM chart based on ranked set sampling (RSS), and the single CUSUM and combined Shewhart–CUSUM charts based on median RSS. Copyright © 2014 John Wiley & Sons, Ltd.

Services sector and economic growth in India

This study examines the long-run equilibrium and short-run dynamic relationship between services sector and Gross Domestic Product (GDP) and between services and nonservices sectors in India. The model is estimated using the optimal single-equation and the maximum-likelihood system estimators. All the estimators consistently suggest the cointegrating relationship between services sector and GDP as well as between services and nonservices sectors. The estimates of long-run elasticity parameters are statistically significant and dimensionally consistent across the estimators. The conventional Cumulative Sum (CUSUM) and the new CUSUM and Moving Sum (MOSUM) tests suggest the stability of the equilibrium residuals and reinforce the cointegrating relationship between the model series. The error correction model provides some support for unidirectional Granger-causality from services sector to GDP. The impulse response and variance decomposition analyses instead suggest the bidirectional causality between services sector and GDP and between services and nonservices sectors. The stable growth of services sector is essentially crucial to absorb the adverse effects of exogenous weather shocks in agriculture and industry and provide resilience to the economy.

Design and Application of Risk Adjusted Cumulative Sum for Strength Monitoring of Ready Mixed Concrete

The cumulative sum procedure is an effective statistical process control tool that can be used to monitor the quality of ready mixed concrete during its production process. In this paper, an attempt has been made to design and apply a new cumulative sum procedure for the ready mixed concrete industry, which takes care of the risks involved in and associated with the production of concrete. This new procedure can be termed as risk adjusted cumulative sum. The 28-day characteristic cube compressive strengths of the various grades of concrete and detailed information regarding the production process and the risks associated with production were collected from the operational ready mixed concrete plants in and around Ahmedabad and Delhi, two important cities in India. The risks were quantified using a likelihood-based scoring method. Finally, a risk adjusted cumulative sum model was developed by imposing the weighted score of the estimated risks on the conventional cumulative sum plot. This model is a more effective and realistic tool for monitoring the strength of ready mixed concrete.

A comparison of formulas for the design of cumulative sum control charts

AbstractA conventional cumulative sum control chart has a V‐mask where each arm makes an angle θ with the horizontal and has a lead distance d. These parameters are usually related to h (=dA tan(ϑ), A is a scaling factor) and k(=A tan(ϑ)). Two systems of formulas are examined for deriving h and k for selected values of the in control and out of control average run lengths. This is the first time an exhaustive comparison has been made. Copyright © 2009 John Wiley & Sons, Ltd.

Adaptive CUSUM procedures with Markovian mean estimation

This paper proposes using a Markovian-type mean estimating procedure in the conventional cumulative sum (CUSUM) control scheme to update its reference value in an adaptive way. This generalizes a class of Markovian adaptive CUSUM (ACUSUM) schemes to achieve the aim of providing an overall good performance over a range of future expected but unknown mean shifts. A two-dimensional Markov chain model is developed to analyze the run length performance of the new scheme. A comparison of run length performance of the proposed ACUSUM scheme and other control charts is shown favorable to the former.

A Robust-Likelihood Cumulative Sum Chart

In practice, the cumulative sum (CUSUM) control chart is often used to detect small shifts in the mean of a normally distributed process, but it performs poorly for thick-tailed processes and for large shifts. This article provides a robust-likelihood cumulative sum (RLCUSUM) chart that discounts outliers and yet has the ability to detect large shifts quickly. The new chart is motivated by the likelihood underpinnings of the CUSUM. It is based on the likelihood of a variate constructed to ensure robust performance of the resulting chart. The new chart is compared with the conventional CUSUM in terms of average run length, with the finding that the RLCUSUM is much better than the conventional CUSUM, especially for large shifts. For small shifts, the two charts are essentially equivalent. We study the properties of the conditional limiting distribution. A final application of the RLCUSUM in assessing livestock disease shows that the method is applicable in process control.

A Markov Chain Model for the Adaptive CUSUM Control Chart

In practice, when the magnitude of a future mean shift is unknown, it is always desired to design a control chart to perform reasonably well over a range of shifts rather than to optimize the performance at detecting a particular level of shifts. Compared with the conventional cumulative sum (CUSUM) control chart designed based on a prespecified mean shift, the adaptive CUSUM (ACUSUM) chart proposed by Sparks (2000) can detect a broader range of mean shifts. This paper develops a two-dimensional Markov chain model to analyze the performance of ACUSUM charts. Moreover, a more general operating model is suggested for the current ACUSUM chart to simplify its implementation.

Dual CUSUM control schemes for detecting a range of mean shifts

Conventional quality control procedures, such as the CUmulative SUM (CUSUM) and exponentially weighted moving average charts are usually designed based on a mean shift with a given size. In practice, the exact value of the shift size is often unknown and can only be reasonably assumed to vary within a certain range. Such a range of shifts deteriorates the performance of existing control charts. In this paper, a quality control scheme, a Dual CUSUM (DCUSUM), is applied that combines two CUSUM charts to detect the range of shifts. The out-of-control signal is triggered if either one of the CUSUM statistics goes out of the DCUSUM control limits. In particular, a design procedure for the DCUSUM charts is developed and an analytical formula for the Average Run Length (ARL) calculation is obtained via the Markov chain method. The proposed DCUSUM charts are compared with the conventional CUSUM and combined Shewhart-CUSUM charts. Based on a proposed criterion, the integrated relative ARL, the proposed schemes show better performance in detecting a range of mean shifts.

Method for detecting spectral changes in the frequency domain

The use of frequency response estimates for change detection is considered. The class of faults considered affects the spectral properties of the system. Spectral changes are detected as deviations in the frequency response, through the choice of a suitable residual. Taking into consideration the statistical nature of a finite-time estimator, a χ2 detector is then designed for the residual. The performance of the fixed sample size detector is compared with a conventional cumulative sum algorithm, and found to be better. A DC motor is used as an implementation example to illustrate the performance and properties of the detector.

A Group of Weighted Moving Averages: An efficient plan for monitoring a range of process changes

It is common to monitor quality characteristics of process outputs or inputs in production processes. Early detection of deviations from target is an important part of the intervention strategy for keeping the process on target. This paper proposes the use of moving averages with smaller sample sizes to provide an early signal of large location changes from target, while moving averages with larger sample sizes are used to give an early signal of smaller location changes from target. If the averaging process gives more recent observations greater weight then it responds to changes quicker than the simple average. Therefore, this paper illustrates that the joint application of a group of weighted moving averages of variable sample sizes is an efficient monitoring plan for early detection of both large and small deviations from target. These plans are compared to the conventional cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) plans using simulation studies and two applied examples.

Weighted moving averages: an efficient plan for monitoring specific location shifts

Exponentially weighted moving averages (EWMA) or Cumulative sum (CUSUM) monitoring plans are used in industry for early detection of deviations from target for process measurements. This paper illustrates the advantage of using weighted moving averages (WMA) in this regard, particular when the size of the shift is known in advance. WMA plans are compared to the conventional CUSUM and the EWMA procedures using a simulation study.

A Group of Moving Averages Control Plan for Signaling Varying Location Shifts

Control plans consisting of a group of moving averages (GMA plans) of various sizes are proposed for monitoring processes. Since moving averages of different sizes retain several levels of memory of past observations, these plans have good average run length (ARL) properties over a range of location shifts. The ARLs of GMA plans are compared with the conventional cumulative sum and the exponentially weighted moving average procedures.

CUSUM Charts for Signalling Varying Location Shifts

The conventional cumulative sum (CUSUM) with k = 0.5 is often used as the default CUSUM statistic when future shifts are unknown. In this paper, CUSUM procedures are designed to be efficient at signalling a range of future expected but unknown location shifts. Two approaches are advocated. The first uses three simultaneous conventional CUSUM statistics with different resetting boundaries. This results in a procedure that has, on average, several levels of memory, and thus signals a broader range of location shifts more efficiently than the conventional CUSUM with k = 0.5. The second uses an adaptive CUSUM statistic that continually adjusts its form to be efficient for signalling a one-step-ahead forecast in deviation from its target value. Average run length (ARL) is used to compare the relative performance of procedures. Several applications are used to illustrate procedures.