Multivariate Coefficient Of Variation Research Articles

Inference for all variants of the multivariate coefficient of variation in factorial designs

AbstractThe multivariate coefficient of variation (MCV) is an attractive and easy‐to‐interpret effect size for the dispersion in multivariate data. Recently, the first inference methods for the MCV were proposed for general factorial designs. However, the inference methods are primarily derived for one special MCV variant while there are several reasonable proposals. Moreover, when rejecting a global null hypothesis, a more in‐depth analysis is of interest to find the significant contrasts of MCV. This paper concerns extending the nonparametric permutation procedure to the other MCV variants and a max‐type test for post hoc analysis. To improve the small sample performance of the latter, we suggest a novel bootstrap strategy and prove its asymptotic validity. The actual performance of all proposed tests is compared in an extensive simulation study and illustrated by real data analysis. All methods are implemented in the R package GFDmcv, available on CRAN.

Optimal designs of variable sample size control chart for monitoring the multivariate coefficient of variation in short production runs

Short production runs are common in industries that require high flexibility and variety in manufacturing. Only a limited amount of researches on multivariate control charts in short production runs exists in the literature. This article proposes two one-sided adaptive charts for monitoring the multivariate coefficient of variation (MCV) in short production runs by varying the sample size (called the VSS MCV SPR charts). The two one-sided adaptive charts are the upper-sided and lower-sided VSS MCV SPR charts for monitoring increasing and decreasing shifts, respectively, in the process MCV. The optimal parameters of the VSS MCV SPR charts in minimizing the out-of-control truncated average run length (TARL) value, for various shift sizes, are presented. The proposed VSS MCV SPR charts outperform their non-adaptive short production runs Shewhart MCV counterparts (called the SH MCV SPR charts), as the former have smaller out-of-control TARL value than the latter, for the same shift size. The sensitivity of the VSS MCV SPR charts, in terms of the probability of signaling an alarm within I inspections is also studied, where the VSS MCV SPR charts surpass their corresponding SH MCV SPR charts. An example for implementing the VSS MCV SPR charts is provided.

Efficient CUSUM control charts for monitoring the multivariate coefficient of variation

Monitoring the coefficient of variation (CV) is of interest when there is a constant proportion between the varying standard deviation and the varying mean in the process. In a multivariate process, monitoring the multivariate CV (MCV) is useful to assess the relative variability of the process. With the aim to improve the existing MCV charts’ performance, this paper develops two one-sided cumulative sum (CUSUM) charts for monitoring the MCV. Based on the Markov chain model, some run length (RL) measures are evaluated and used to show the superiority of the proposed charts over the competing MCV charts under the deterministic shift. Moreover, when the shift size is unknown in practice, the performance of the proposed CUSUM MCV charts is evaluated. Finally, an example is provided to show the implementation of the proposed CUSUM MCV charts.

Monitoring multivariate coefficient of variation with individual observations

AbstractMonitoring the multivariate coefficient of variation (MCV) is conducted when the multivariate process variation changes proportionally with the mean vector because under this situation the traditional multivariate charts for the mean and variance cannot work correctly. All existing MCV charts in the literature are implemented based on sample MCVs computed from samples of size larger than unity each. However, there are situations when forming samples (of ) is not feasible. To address the aforementioned limitation of existing MCV charts, in this paper, an EWMA chart based on individual observations for monitoring the MCV is developed. Numerical studies based on Monte Carlo simulations are conducted to investigate the run length properties of the proposed chart in detecting either increases or decreases in the MCV, for two and three quality characteristics and various parameter values. A real bivariate dataset on the diameter and length of dowel pins is adopted to illustrate the implementation of the proposed chart.

Monitoring the multivariate coefficient of variation in presence of autocorrelation with variable parameters control charts

ABSTRACT The coefficient of variation is a very important process parameter in many processes. A few control charts have been considered so far for monitoring its multivariate counterpart, i.e., the multivariate coefficient of variation (MCV). In addition, autocorrelation is very likely to occur in processes with high sampling frequency. Hence, designing suitable control charts and investigating the effect of autocorrelation on these charts is necessary. However, no control chart has been developed so far for the coefficient of variation that is capable of accounting for autocorrelation in either univariate or multivariate cases. This paper fills the gap by developing multivariate Shewhart-type control charts to monitor MCV with different autocorrelation structures for the observations: vector autoregressive, vector moving average, and vector mixed autoregressive and moving average. In addition, we add variable parameters adaptive features to the Shewhart-type scheme, in order to improve its performance. We develop a Markov chain model to get the statistical performance measures; then, we perform extensive numerical analyses to evaluate the effect of autocorrelation on adaptive and non-adaptive charts in the presence of downward and upward MCV shifts. Finally, we present an illustrative example from a healthcare process to show the implementation of this scheme in real practice.

A multivariate adaptive control chart for simultaneously monitoring of the process parameters

There have been some advances in multivariate control charts in recent years. This paper presents a new simultaneous scheme for monitoring both the mean and variability of a multivariate normal process in a single chart, which is developed by improving and modifying another recently proposed scheme. We not only propose a new control scheme but also make it adaptive by varying all control chart parameters. Our scheme, for the first time, considers the process variability in two forms: “covariance matrix” and “multivariate coefficient of variation (MCV)”. This scheme, again for the first time, considers simultaneous monitoring of the MCV with another process parameter (in our case, the mean vector). In addition, we develop a Markov chain model to compute the average run length and average time to signal values. We conduct extensive numerical analyses to measure the performance of the proposed scheme in two scenarios of process variability. At last, we present a numerical example by using a real dataset from a healthcare process to illustrate how the scheme can be implemented in practice.

Abnormal vibration detection of wind turbine based on temporal convolution network and multivariate coefficient of variation

A working wind turbine generates a large amount of multivariate time-series data, which contain abundant operation state information and can predict impending anomalies. The anomaly detection of the wind turbine nacelle that houses all of the generating components in a turbine have been challenging due to its inherent complexities, systematic oscillations and noise. To address these problems, this paper proposes an unsupervised time-series anomaly detection approach, which combines deep learning with multi-parameter relative variability detection. A normal behavior model (NBM) of nacelle vibration is firstly built upon training normal historical data of the supervisory control and data acquisition (SCADA) system in the high-resolution domain. To better capture the temporal characteristics and frequency information of vibration signals, the vibration spectrum vector is integrated with the multivariate time-series data as inputs and the spectrum-embedded temporal convolutional network (SETCN) is then used to extract latent features. The anomalies are detected through a multi-variate coefficient of variation (MCV) based anomaly assessment index (AAI) of relative variability among vibration residuals and environment parameters of the nacelle. The approach considers the time-series characteristics of input data and preserves the spatio-temporal correlation between variables. Validations using data collected from real-world wind farms demonstrate the effectiveness of the proposed approach.

Evaluation of Variable Parameter MCV Control Chart in Downward Process Shifts

Previous studies revealed that the coefficient of variation (CV) is important in ensuring process quality, especially for monitoring a process where its process mean and variance are highly correlated. The fact that almost all industrial process monitoring involves a minimum of two or more related quality characteristics being monitored simultaneously. The existing adaptive charts for monitoring the multivariate CV are focused on detecting upward process shifts. The downward process monitoring is crucial since it shows process improvement. Very little research works are available on the downward adaptive multivariate CV chart. This makes it difficult for the quality engineer who wishes to implement the adaptive multivariate CV chart on the downward process monitoring. Therefore, this paper filled the research gap by proposing a downward variable parameter chart for the multivariate coefficient of variation. The performance measures of the proposed charts are derived based on the Markov-chain approach. Numerical comparisons between the proposed and existing charts have been made, in terms of the average time to signal criterion. The findings reveal that the proposed charts outperform the existing charts for detecting small and moderate downward process shifts

An Optimal Adaptive Variable Sample Size Scheme for the Multivariate Coefficient of Variation

Development of an efficient process monitoring system has always received great attention. Previous studies revealed that the coefficient of variation (CV) is important in ensuring process quality, especially for monitoring a process where its process mean and variance are highly correlated. The fact that almost all industrial process monitoring involves a minimum of two or more related quality characteristics being monitored simultaneously, this paper incorporates the salient feature of the adaptive sample size VSS scheme into the standard multivariate CV (MCV) chart, called the VSS MCV chart. A Markov chain model is developed for the derivation of the chart’s performance measures, i.e the average run length (ARL), the standard deviation of the run length (SDRL), the average sample size (ASS), the average number of observations to signal (ANOS) and the expected average run length (EARL). The numerical comparison shows that the proposed chart prevails over the existing standard MCV chart for detecting small and moderate upward and downward MCV shifts.

One-sided variable sampling interval EWMA control charts for monitoring the multivariate coefficient of variation in the presence of measurement errors

Online monitoring of the multivariate coefficient of variation (MCV) can be of interest in many real situations in which the dispersion of a multivariate process is meant to remain constant with regard to its position. To this aim, several control charts have been recently proposed in the literature. In this paper, the new one-sided adaptive charts to monitor the MCV are proposed. The chart applies a variable sampling interval (VSI) strategy on an exponentially weighted moving average (EWMA) scheme, aiming at benefiting from the known advantages of both approaches. Formulas to optimally determine the parameters of the chart are derived and presented. The proposed chart is shown to outperform competing charts under most circumstances. The presence of the measurement errors is considered to understand their effects on the charts. An illustrative example is also included.

Gradient Statistics Aware Power Control for Over-the-Air Federated Learning

Federated learning (FL) is a promising technique that enables many edge devices to train a machine learning model collaboratively in wireless networks. By exploiting the superposition nature of wireless waveforms, over-the-air computation (AirComp) can accelerate model aggregation and hence facilitate communication-efficient FL. Due to channel fading, power control is crucial in AirComp. Prior works assume that the signals to be aggregated from each device, i.e., local gradients have identical statistics. In FL, however, gradient statistics vary over both training iterations and feature dimensions, and are unknown in advance. This paper studies the power control problem for over-the-air FL by taking gradient statistics into account. The goal is to minimize the aggregation error by optimizing the transmit power at each device subject to average power constraints. We obtain the optimal policy in closed form when gradient statistics are given. Notably, we show that the optimal transmit power is continuous and monotonically decreases with the squared multivariate coefficient of variation (SMCV) of gradient vectors. We then propose a method to estimate gradient statistics with negligible communication cost. Experimental results demonstrate that the proposed gradient-statistics-aware power control achieves higher test accuracy than the existing schemes for a wide range of scenarios.

Monitoring multivariate coefficient of variation with upward Shewhart and EWMA charts in the presence of measurement errors using the linear covariate error model

AbstractIn practice, measurement errors exist and ignoring their presence may lead to erroneous conclusions in the actual performance of control charts. The implementation of the existing multivariate coefficient of variation (MCV) charts ignores the presence of measurement errors. To address this concern, the performances of the upward Shewhart‐MCV and exponentially weighted moving average MCV charts for detecting increasing MCV shifts, using a linear covariate error model, are investigated. Explicit mathematical expressions are derived to compute the limits and average run lengths of the charts in the presence of measurement errors. Finally, an illustrative example using a real‐life dataset is presented to demonstrate the charts’ implementation.

Variable sampling interval EWMA chart for multivariate coefficient of variation

A control chart for monitoring the multivariate coefficient of variation (MCV) is used when the focus is on monitoring the ratio of relative multivariate variability to the mean of a multivariate process. The MCV chart is useful in process monitoring when practitioners are not interested in the consistency of the mean vector or covariance matrix. This study proposes a one-sided upward variable sampling interval (VSI) exponentially weighted moving average (EWMA) chart to detect increasing shifts in MCV squared and shows the derivation of formulae to evaluate the performance of the VSI EWMA- chart by using the Markov chain approach. Comparative investigations show that the proposed chart outperforms existing MCV charts in detecting shifts in process MCV. A numerical example that uses real data reveals that the proposed chart performs well in actual applications.

Multivariate coefficient of variation charts with measurement errors

The effects of measurement errors on the performances of multivariate charts have not been extensively investigated compared with univariate charts. More specifically, control charts for monitoring the multivariate coefficient of variation (MCV) in the existing literature are not investigated under the assumption that measurement errors are present. This research proposes two one-sided MCV charts in the presence of measurement errors, one for detecting decreasing MCV shifts and another for detecting increasing MCV shifts. The distributional properties of the sample MCV in the presence of measurement errors are derived. Additionally, the formulae to compute the control limits and performance measures of the proposed charts are derived. Furthermore, a step-by-step procedure to implement the MCV charts with measurement errors is presented. Finally, an application of the proposed charts is illustrated using real data.

One-sided Downward Control Chart for Monitoring the Multivariate Coefficient of Variation with VSSI Strategy

In recent years, control charts monitoring the coefficient of variation (CV), denoted as the ratio of the variance to the mean, is attracting significant attention due to its ability to monitor processes in which the process mean and process variance are not independent of each other. However, very few studies have been done on charts to monitor downward process shifts, which is important since downward process shifts show process improvement. In view of the importance of today’s competitive manufacturing environment, this paper proposes a one-sided chart to monitor the downward multivariate CV (MCV) with variable sample size and sampling interval (VSSI), i.e. the VSSI D MCV chart. This paper monitors the MCV as most industrial processes simultaneously monitor at least two or more quality characteristics, while the VSSI feature is incorporated, as it is shown that this feature brings about a significant improvement of the chart. A Markov chain approach was adopted for designing a performance measure of the proposed chart. The numerical comparison revealed that the proposed chart outperformed existing MCV charts. The implementation of the VSSI D MCV chart is illustrated with an example.

The efficiency of run rules schemes for the multivariate coefficient of variation in short runs process

In real industries, multivariate process monitoring is crucial as there are many instances that involve at least two quality variables to be monitored simultaneously. The short runs process is commonly seen in production after the industry moved toward flexible manufacturing. Monitoring the coefficient of variation (CV) is useful in a wide variety of scientific areas. In a view of the importance in monitoring the CV and the fact that most real-life data in process monitoring are multivariate in nature, this paper proposes to monitor the multivariate CV in short runs process by means of run rules (RR) control charts. A Markov chain model is established for designing the proposed charts. The statistical performances of the RR multivariate CV (MCV) and Shewhart MCV (SH MCV) charts are compared in terms of the truncated average run length and the expected truncated average run length. The results show that the proposed charts surpass the SH MCV chart for detecting small and moderate multivariate CV shifts. The implementation of the RR MCV chart in the short runs process is illustrated with an example using a real dataset.

Variable sampling interval Shewhart control charts for monitoring the multivariate coefficient of variation

AbstractIn many industrial manufacturing processes, the ratio of the variance to the mean of a quantity of interest is an important characteristic to ensure the quality of the processes. This ratio is called the coefficient of variation (CV). A lot of control charts have been designed for monitoring the CV of univariate quantity in the literature. However, the CV control charts for multivariate quantity have not received much attention yet. In this paper, we investigate a variable sampling interval (VSI) Shewhart control chart for monitoring multivariate CV. The time between two consecutive samples is allowed to vary according to the previous value of the multivariate CV, which will help the chart to detect the process shifts faster. The comparison with the fixed sampling interval Shewhart chart is implemented to highlight the advantage of the VSI method. Finally, an illustrative example is demonstrated on real data.

A proposed variable parameter control chart for monitoring the multivariate coefficient of variation

AbstractAn efficient process monitoring system is important for achieving sustainable manufacturing. The control charting technique is one of the most effective techniques to monitor process quality. In certain processes where the process mean and variance are not independent of one another, the coefficient of variation (CV), which measures the ratio of the standard deviation to the mean, should be monitored. In line with industrial settings, where at least two or more variables are monitored simultaneously in most processes, this paper proposes a variable parameter (VP) chart to monitor the multivariate CV (MCV). Formulae and algorithms to optimize the various performance measures are discussed. The proposed VP MCV chart is designed based on a Markov chain approach. The performance comparison shows that the proposed VP MCV chart prevails over the existing MCV charts, in terms of the average time to signal (ATS), standard deviation of the time of signal (SDTS), and expected average time to signal (EATS) criteria. An example is presented to illustrate the implementation of the proposed VP MCV chart.

The efficiency of run rules schemes for the multivariate coefficient of variation: a Markov chain approach

ABSTRACT Control charts are one of the important tools to monitor quality. The coefficient of variation (CV) is a common measure of dispersion in many real-life applications. Recently, CV control charts are proposed to monitor processes which do not have a constant mean and a standard deviation which changes with the mean. These processes cannot be monitored by standard control charts which monitor the mean and/or standard deviation. This research proposes the monitoring of the multivariate coefficient of variation (MCV) by means of run rules (RR MCV) control charts, which is not available in the existing literature. The design of these charts is obtained using a Markov-chain approach. The proposed charts are simple to implement. The performance of the RR MCV and Shewhart MCV (SH MCV) charts are compared in terms of the average run length (ARL) and the expected average run length (EARL). An example is illustrated based on a real dataset. The findings revealed that the performance of the proposed charts surpasses the SH MCV chart for detecting small and moderate MCV shifts.

One-sided synthetic control charts for monitoring the multivariate coefficient of variation

ABSTRACTShewhart's type control charts for monitoring the Multivariate Coefficient of Variation (MCV) have recently been proposed in order to monitor the relative variability compared with the mean. These approaches are known to be rather slow in the detection of small or moderate process shifts. In this paper, in order to improve the detection efficiency, two one-sided Synthetic charts for the MCV are proposed. A Markov chain method is used to evaluate the statistical performance of the proposed charts. Furthermore, computational experiments reveal that the proposed control charts outperform the Shewhart MCV control chart in terms of the average run length to detect an out-of-control state. Finally, the implementation of the proposed chart is illustrated with an example using steel sleeves data.