Hotelling T2 Chart Research Articles

Multivariate Statistical Process Monitoring and Control of Machining Process using Principal Components based Hotelling T2 Charts: A Machine Vision Approach

Multivariate Statistical Process Monitoring and Control of Machining Process using Principal Components based Hotelling T2 Charts: A Machine Vision Approach

MULTIVARIATE T2 CONTROL CHART BASED ON JAMES-STEIN AND SUCCESSIVE DIFFERENCE COVARIANCE MATRIX ESTIMATORS FOR INTRUSION DETECTION

The intrusion detection is a process to monitor the events taking place in a computer system or network and analyse the monitoring results to find signs of intrusion. The multivariate control chart, which is often used in the intrusion detection system, is Hotelling's T2. In this study, the Hotelling's T2 chart performance for intrusion detection is improved using the successive difference covariance matrix to estimate the covariance matrix and James-Stein estimator to estimate the mean vector. The control limits of the proposed chart are calculated using kernel density estimation. The performance of the proposed method, using T2 based on kernel density estimation control limit, outperforms the other control chart approaches in both training and testing dataset.

Integration of Machine Learning Techniques and Control Charts for Multivariate Processes

Using multivariate control chart instead of establishing univariate control chart for all variables in processes provides time and labor advantage. In addition, it is considered in the relations between variables. However, the statistical calculation of the measured values of all variables is seen as a single value in the control chart. Therefore, it is necessary to determine which variable(s) is the cause of the out of control signal. Effective corrective measures can only be developed when the causes of the fault(s) are determined correctly. The aim of the study is to determine the machine learning techniques that will accurately estimate the type of fault. With the Hotelling T2 chart, out of control signals are identified and the types of faults affected by the variables are defined. Various machine learning techniques are used to compare classification performances. The developed model was applied in the evaluation of the paint quality in a painting process. ANN was determined as the most successful techniques according to performance criteria. The novelty of the study is to classify the fault according to the types of faults, not the variables. Defining the faults according to its types will enable to take effective corrective actions quickly.

An Algorithm for Early Outbreak Detection in Multiple Data Streams

An Algorithm for Early Outbreak Detection in Multiple Data Streams

New sampling strategies to reduce the effect of autocorrelation on the synthetic T2 chart to monitor bivariate process

AbstractIn this paper, synthetic T2 chart is developed to monitor bivariate process with correlated variables and autocorrelated observations. The proposed chart is a combination of the Hotelling's T2 chart and the conforming run length chart. The operation and design of the chart are described when observations are autocorrelated and cross correlated. The first‐order vector autoregressive process VAR (1) is used to model the bivariate data from an autocorrelated process of interest. Using an average run length as performance measure criterion in the VAR (1) model, it is observed that autocorrelation seriously impact the performance of the synthetic T2 chart. To reduce the effect of autocorrelation on the performance of the synthetic T2 chart, the skip and mixed sampling strategies are implemented to form rational subgroups in the construction of synthetic T2 chart. The average run length performance of the synthetic T2 chart implementing these strategies is compared with that of the standard strategy of formation of rational subgroups. It is observed that implementing skip and mixed sampling strategies within rational subgroup improves the performance of the synthetic T2 chart.

Hotelling's T2 control charts with fixed and variable sample sizes for monitoring short production runs

AbstractShort production runs are common in enterprises that require a high degree of flexibility and variety in manufacturing processes. To date, past research on short production runs has little focus on the multivariate control charts. In view of this, fixed sample size (FSS) and variable sample size (VSS) Hotelling's T2 charts are designed to monitor the process mean when the production horizon is finite. Optimal parameters to minimize the out‐of‐control (1) truncated average run length (TARL) and (2) expected TARL (ETARL) are provided such that the in‐control TARL is equal to the number of inspections (say I). The numerical study considers the run length performances of the FSS and VSS T2 short‐run charts for both known and unknown shift sizes. The VSS T2 short‐run chart performs well in swiftly detecting various mean shifts in comparison with the FSS T2 short‐run chart. Additionally, the VSS T2 short‐run chart is superior to the FSS T2 short‐run chart, in terms of the truncated standard deviation of the run length, expected truncated standard deviation of the run length, probability that the chart signals an alarm within the I inspections, ie, P(I) and expected P(I). A case study on the impurity profile of a crystalline drug substance illustrates the implementation of the VSS T2 short‐run chart.

A model‐free approach to reduce the effect of autocorrelation on statistical process control charts

AbstractThe presence of autocorrelation in time series data significantly affects the performance of statistical process control charts. For process data collected in real time, consecutive observations result in serial dependence that violates the assumption of sample independence. This paper focuses on the improvement of statistical process control (SPC) charts of individual observations when they are designed to monitor stationary processes in the presence of data autocorrelation. Three widely used SPC tools, Shewhart chart, Hotelling's T2 chart, and PCA‐based SPC charts, are chosen to demonstrate the proposed approach. There has been a modified version of SPC chart shown to be effective in accommodating the autocorrelation behavior, relying on a model‐based estimation of chart parameters. Process model estimation, however, is a challenging task, especially for complex systems. In this paper, a model‐free monitoring approach is proposed for SPC charts of individual observations, aimed at reducing the effect of autocorrelation on chart performance. It relies on a skipping strategy that creates sample bins with little or no autocorrelation. A single skipping chart and a combined skipping chart (CSC) are established, and a parameter sensitivity study is performed through simulations. In the case studies, the advantages of the CSC are noted as being a model‐free approach and having a performance consistent with that of the benchmark modified chart.

A statistical monitoring strategy for a pulp and paper industry

The current article features the endeavour to develop a monitoring strategy for paper and pulp industry. The complexity involved in production of better quality pulp warrants the close monitoring of the process characteristics associated with the process. Partial least square regression (PLSR) which is a multivariate statistical projection-based technique is used for building the process representation which aids in the simultaneous monitoring of multiple correlated characteristics. Individual Hotelling T2 chart has been developed to monitor each grade of paper being produced. However this leads to implementation of different control charts for different grades of paper adding to the dexterity of the methodology. In order to solve the above problem methodology hence developed is further extended to develop a combined control chart to monitor all the grades of paper simultaneously. Furthermore a comparative study of the competence of the developed unified control chart with respect to the individual charts is done.

Vibrational spectroscopy and multivariate control charts: A new strategy for monitoring the stability of captopril in the pharmaceutical industry

The quality control of pharmaceutical production has focused on controlling impurity formation and product degradation. High temperatures, humidity, and the presence of hygroscopic excipients affect the stability of captopril. These conditions accelerate the degradation of captopril and favor the formation of captopril disulfide. In this study, multivariate control charts were combined with vibrational spectroscopy to monitor the stability of captopril. Fifty spectra of different batches of normal captopril were recorded in the mid-infrared (MIR) and near-infrared (NIR) spectral regions, and they were used in the training phase of multivariate charts. The validation phase consisted of 20 lots of normal captopril and 10 lots past their expiration dates. A stability study was performed by subjecting one batch of captopril to a temperature of 40±2°C and relative humidity of 75±5% for a period of six months in a climatic chamber. The captopril samples were collected weekly from the climatic chamber during this period, for a total of 24 spectra. Principal component analysis was performed on the MIR and NIR datasets, and the scores of the principal components were used in the construction of Hotelling's T2 chart, whereas the squared prediction error (SPE) chart was built with the residuals of the model. In the validation phase, the SPE chart adequately identified expired lots in these two infrared regions. A strategy based on the successive projections algorithm in combination with a multivariate cumulative sum chart was developed to improve the sensitivity of Hotelling's T2 charts in the MIR region in the monitoring step of the stability study. Our results indicate that vibrational spectroscopy combined with control charts is a useful tool for monitoring changes in the stability pattern of captopril.

Economic design based on a multivariate Bayesian VSI chart with a dual control limit

ABSTRACTIn this paper, a multivariate Bayesian variable sampling interval (VSI) control chart for the economic design and optimization of statistical parameters is designed. Based on the VSI sampling strategy of a multivariate Bayesian control chart with dual control limits, the optimal expected cost function is constructed. The proposed model allows the determination of the scheme parameters that minimize the expected cost per time of the process. The effectiveness of the Bayesian VSI chart is estimated through economic comparisons with the Bayesian fixed sampling interval and the Hotelling's T2 chart. This study is an in-depth study on a Bayesian multivariate control chart with variable parameter. Furthermore, it is shown that significant cost improvement may be realized through the new model.

Performance analysis of hotelling T2 under multivariate inspection errors

In this paper, we analysed the effect of multivariate normal inspection errors on the performance of the Hotelling T2 for individual observations during Phase II. The computed and charted statistic based on contaminated measurements may lead to inferior performance in the false alarm rate, power of detection and average run length (ARL). In this paper, we derive explicitly how the multivariate inspection errors are related to the Hotelling T2 statistic. Performance of the multivariate statistical process control approach in terms of in-control and out-of-control ARL is reported with respect to different cases/structures of the inspection errors. Given fixed sizes of measurement error covariances, both the false alarm rate and detection time are dramatically affected by just 10% measurement error. High correlations of the measurement errors induce larger variances and covariances of the observed process characteristics, leading to inferior performance of the T2 chart. This research finding has advantages that one can predict how the Hotelling T2 chart will be inferior with respect to the measurement structure, i.e. %GRR. The lemmas provided in this research yield an explicit expression for numerical analysis of the inferior performance of the T2 chart.

Image Analysis and High Dimensional Control Chart for Inspection of Sausage Color Homogeneity and Uniformity

Abstract Sausage color usually influences consumers’ selection due to the perceptions of quality. Extensive studies have applied image processing to capture the characteristics of food products according to the high-dimensional nature of the resultant images. However, the color homogeneity (i. e. “within pack” variation) and uniformity (i. e. “between-pack” variation) have rarely been studied. Therefore, this paper proposes a new framework to detect both variations using images. In addition, a new approach has been developed to deal with high-dimension data involving colorimetric characteristics, namely L *, a *, b *, hue (h) and chroma (C *). These high-dimensional data are transformed to represent color homogeneity and uniformity. Hotelling T 2 chart is used to detect color abnormalities. Our approach indicates that the out-of-control items can be identified with the control chart signals. Nonetheless, the out-of-control signals alone are inadequate for determination of the possible causes. Then, the proposed analysis framework was subsequently applied to identify possible causes that contributed to the process deviations. Furthermore, prior to the experiments with sausages, the image inspection device was tested for gauge repeatability and reproducibility.

The economic and economic-statistical designs of the Hotelling's T2 chart based on the expected average run length

ABSTRACTExisting economic and economic-statistical designs require practitioners to specify the Mahalanobis Distance Shift Size (MDSS) as an exact value. However, practitioners may find it difficult to specify this distance. This article proposes the economic and economic-statistical designs of the Hotelling's T2 chart, where practitioners do not have to specify the MDSS. Adopting optimal design parameters based on the wrong MDSS results in a significant increase in cost. In comparison, adopting the optimal design parameters based on the proposed methodology results in a slight increase in cost. This article also studies the effects of different input parameters and statistical constraints.

The robust economic statistical design of the Hotelling's T2 chart

ABSTRACTEconomic statistical designs aim at minimizing the cost of process monitoring when a specific scenario or a set of estimated process and cost parameters is given. But, in practice the process may be affected by more than one scenario which may lead to severe cost penalties if the wrong design is used. Here, we investigate the robust economic statistical design (RESD) of the T2 chart in an attempt to reduce these cost penalties when there are multiple scenarios. Our method is to employ the genetic algorithm (GA) optimization method to minimize the total expected monitoring cost across all distinct scenarios. We illustrate the effectiveness of the method using two numerical examples. Simulation studies indicate that robust economic statistical designs should be encouraged in practice.

Framework-based on SQC and DMAIC to monitor die casting process parameters

The article attempts to reduce process variation and improve the product quality by using both off-line and online quality control techniques. This task may be achieved by developing an integrated model based on statistical quality control (SQC) and Six Sigma-based D-M-A-I-C (define-measure-analyse-improve-control). The goal of study was achieved by using various quality control tools viz. brainstorming sessions, histograms, failure mode effect analysis (FMEA), cause and effect diagram, control charts, principle component analysis and response surface methodology. Principle component analysis (PCA) was employed for building of the process representation and Hotelling T2 chart based on the principle component score was used for detection of abnormal behaviour. Further response surface methodology was used to optimise the process parameters. There was a reduction in the overall process variability by employment of optimised process parameters. The suggested integrated model seems to be effective for process monitoring of the proposed case study and may be used to monitor other similar processes as well.

A Distribution-Free Multivariate Phase I Location Control Chart for Subgrouped Data from Elliptical Distributions

In quality control, a proper Phase I analysis is essential to the success of Phase II monitoring. A literature review reveals no distribution-free Phase I multivariate techniques in existence. This research develops a Phase I location control chart for multivariate elliptical processes. The resulting in-control reference sample can then be used to estimate the parameters for Phase II monitoring. Using Monte Carlo simulation, the proposed method is compared with the Hotelling's T2 Phase I chart. Although Hotelling's T2 chart is preferred when the data are multivariate normal, the proposed method is shown to perform significantly better under nonnormality. This article has supplementary material online.

The Effect of Autocorrelation on the Hotelling T2 Control Chart

One of the basic assumptions for traditional univariate and multivariate control charts is that the data are independent in time. For the latter, in many cases, the data are serially dependent (autocorrelated) and cross‐correlated because of, for example, frequent sampling and process dynamics. It is well known that the autocorrelation affects the false alarm rate and the shift‐detection ability of the traditional univariate control charts. However, how the false alarm rate and the shift‐detection ability of the Hotelling T2 control chart are affected by various autocorrelation and cross‐correlation structures for different magnitudes of shifts in the process mean is not fully explored in the literature. In this article, the performance of the Hotelling T2 control chart for different shift sizes and various autocorrelation and cross‐correlation structures are compared based on the average run length using simulated data. Three different approaches in constructing the Hotelling T2 chart are studied for two different estimates of the covariance matrix: (i) ignoring the autocorrelation and using the raw data with theoretical upper control limits; (ii) ignoring the autocorrelation and using the raw data with adjusted control limits calculated through Monte Carlo simulations; and (iii) constructing the control chart for the residuals from a multivariate time series model fitted to the raw data. To limit the complexity, we use a first‐order vector autoregressive process and focus mainly on bivariate data. © 2014 The Authors. Quality and Reliability Engineering International Published by John Wiley & Sons Ltd.

Sensitivity Analyses of the Economic and Economic Statistical Designs of the Multivariate Synthetic T2 Chart

This article performs a sensitivity analyses of the synthetic T2 chart using fractional factorial design, which integrates the interaction effects. We are interested in the effects of the input parameters on the optimal cost, chart's parameters, and average run lengths. We also look at the input parameters responsible for the increase in cost and improvement in statistical performance under statistical constraints, and investigate how the input parameters influence the binding effect of the statistical constraints. The sensitivity analyses of the synthetic T2 chart are compared with that of the Hotelling's T2 chart, and parameters responsible for the cost advantage of the synthetic T2 chart are identified.

Principal component analysis for the early detection of mastitis and lameness in dairy cows

This investigation analysed the applicability of principal component analysis (PCA), a latent variable method, for the early detection of mastitis and lameness. Data used were recorded on the Karkendamm dairy research farm between August 2008 and December 2010. For mastitis and lameness detection, data of 338 and 315 cows in their first 200d in milk were analysed, respectively. Mastitis as well as lameness were specified according to veterinary treatments. Diseases were defined as disease blocks. The different definitions used (two for mastitis, three for lameness) varied solely in the sequence length of the blocks. Only the days before the treatment were included in the blocks. Milk electrical conductivity, milk yield and feeding patterns (feed intake, number of feeding visits and time at the trough) were used for recognition of mastitis. Pedometer activity and feeding patterns were utilised for lameness detection. To develop and verify the PCA model, the mastitis and the lameness datasets were divided into training and test datasets. PCA extracted uncorrelated principle components (PC) by linear transformations of the raw data so that the first few PCs captured most of the variations in the original dataset. For process monitoring and disease detection, these resulting PCs were applied to the Hotelling's T 2 chart and to the residual control chart. The results show that block sensitivity of mastitis detection ranged from 77·4 to 83·3%, whilst specificity was around 76·7%. The error rates were around 98·9%. For lameness detection, the block sensitivity ranged from 73·8 to 87·8% while the obtained specificities were between 54·8 and 61·9%. The error rates varied from 87·8 to 89·2%. In conclusion, PCA seems to be not yet transferable into practical usage. Results could probably be improved if different traits and more informative sensor data are included in the analysis.

Model-based Multivariate Monitoring Charts for Autocorrelated Processes

Autocorrelation or nonstationarity may seriously impact the performance of conventional Hotelling's T2 charts. We suggest modeling processes with multivariate autoregressive integrated moving average time series models and propose two model-based monitoring charts. One monitors the predicted value and provides information about the need for mean adjustments. The other is a Hotelling's T2 control chart applied to the residuals. The average run length performance of the residual-based Hotelling's T2 chart is compared with the observed data-based Hotelling's T2 chart for a group of first-order vector autoregressive models. We show that the new chart in most cases performs well. Copyright © 2013 John Wiley & Sons, Ltd.