A Bivariate Semiparametric Control Chart Based on Order Statistics

In this article, a new bivariate semiparametric Shewhart‐type control chart is presented. The proposed chart is based on the bivariate statistic (X_(r), Y_(s)), where X_(r) and Y_(s) are the order statistics of the respective X and Y test samples. It is created by considering a straightforward generalization of the well‐known univariate median control chart and can be easily applied because it calls for the computation of two single order statistics. The false alarm rate and the in‐control run length are not affected by the marginal distributions of the monitored characteristics. However, its performance is typically affected by the dependence structure of the bivariate observations under study; therefore, the suggested chart may be characterized as a semiparametric control chart. An explicit expression for the operating characteristic function of the new control chart is obtained. Moreover, exact formulae are provided for the calculation of the alarm rate given that the characteristics under study follow specific bivariate distributions. In addition, tables and graphs are given for the implementation of the chart for some typical average run length values and false alarm rates. The performance of the suggested chart is compared with that of the traditional χ² chart as well as to the nonparametric SN² and SR² charts that are based on the multivariate form of the sign test and the Wilcoxon signed‐rank test, respectively. Finally, in order to demonstrate the applicability of our chart, a case study regarding a real‐world problem related to winery production is presented. Copyright © 2016 John Wiley & Sons, Ltd.     

An improved variable sample size and sampling interval S control chart

The standard deviation chart (S chart) is used to monitor process variability. This paper proposes an upper‐sided improved variable sample size and sampling interval (VSSI_t) S chart by improving the existing upper‐sided variable sample size and sampling interval (VSSI) S chart through the inclusion of an additional sampling interval. The optimal designs of the VSSI_t S chart together with the competing charts under consideration, such as the VSSI S and exponentially weighted moving average (EWMA) S charts, by minimizing the out‐of‐control average time to signal (ATS₁) and expected average time to signal (EATS₁) criteria, are performed using the MATLAB programs. The performances of the standard S, VSSI S, EWMA S, and VSSI_t S charts are compared, in terms of the ATS₁ and EATS₁ criteria, where the results show that the VSSI_t S chart surpasses the other charts in detecting moderate and large shifts, while the EWMA S is the best performing chart in detecting small shifts. An illustrative example is given to explain the implementation of the VSSI_t S chart.     

An enhanced CUSUM‐t chart for process mean

In this paper, we propose an auxiliary‐information–based (AIB) Crosier cumulative sum (CCUSUM) t chart for monitoring the process mean, namely, the AIB‐CCUSUM‐t chart. The run length characteristics of the proposed chart are computed using Monte Carlo simulation. The optimal parameters for the AIB‐CCUSUM‐t chart to detect specific mean shifts are computed. The fast initial response (FIR) feature is also attached with the proposed chart. It is found that the AIB‐CCUSUM‐t and FIR‐AIB‐CCUSUM‐t charts perform uniformly and substantially better than the CCUSUM‐t and FIR‐CCUSUM‐t charts, respectively. An example is presented to support the theory.     

Monitoring bivariate and trivariate mean vectors with a Shewhart chart

In this article, we propose the use of the mean chart to control multivariate processes. The basic idea is to control the mean vector of bivariate (X, Y) and trivariate (X, Y, Z) processes by alternating the charting statistic of the Shewhart chart. If the mean of X observations was the charting statistic to obtain the current sample point, then the mean of Y observations will be the charting statistic to obtain the next sample point (for the trivariate case, the mean of Z observations will be the charting statistic to obtain the sample point subsequent to the next one). As a Shewhart chart, the signal is given anytime a sample point is plotted beyond the control limits, independent of the charting statistic in use. A fair comparison between the proposed chart and the Hotelling chart is based on an equal number of measurements per sample. The Shewhart chart with alternated charting statistic (ACS) always outperforms the Hotelling chart, except for specific types of disturbances in quality characteristics highly correlated (ρ = 0.7). The ACS chart is substantially easier to operate and faster than the Hotelling chart in signaling changes in the mean vector of bivariate and trivariate processes. Even with fewer measurements per sample, the trivariate ACS chart outperforms the Hotelling chart.     

Statistical Performance of a Control Chart for Individual Observations Monitoring the Ratio of Two Normal Variables

Statistical Process Control monitoring of the ratio Z of two normal variables X and Y has received too little attention in quality control literature. Several applications dealing with monitoring the ratio Z can be found in the industrial sector, when quality control of products consisting of several raw materials calls for monitoring their proportions (ratios) within a product. Tables about the statistical performance of these charts are still not available. This paper investigates the statistical performance of a Phase II Shewhart control chart monitoring the ratio of two normal variables in the case of individual observations. The obtained results show that the performance of the proposed chart is a function of the distribution parameters of the two normal variables. In particular, the Shewhart chart monitoring the ratio Z outperforms the (p = 2) multivariate T² control chart when a process shift affects the in‐control mean of X or, alternatively, of Y and the correlation among X and Y is high and when the in‐control means of X and Y shift contemporarily to opposite directions. The sensitivity of the proposed chart to a shift of the in‐control dispersion has been investigated, too. We also show that the standardization of the two variables before computing their ratio is not a good practice due to a significant loss in the chart's statistical performance. An illustrative example from the food industry details the implementation of the ratio control chart. Copyright © 2013 John Wiley & Sons, Ltd.     

Attribute Charts for Monitoring the Mean Vector of Bivariate Processes

This article proposes two Shewhart charts, denoted np_xy and np_w charts, which use attribute inspection to control the mean vector (μ_x; μ_y)′ of bivariate processes. The units of the sample are classified as first‐class, second‐class, or third‐class units, according to discriminate limits and the values of their two quality characteristics, X and Y. When the np_xy chart is in use, the monitoring statistic is M = N₁ + N₂, where N₁ and N₂ are the number of sample units with a second‐class and third‐class classification, respectively. When the np_w chart is in use, the monitoring statistic is W = N₁ + 2N₂. We assume that the quality characteristics X and Y follow a bivariate normal distribution and that the assignable cause shifts the mean vector without changing the covariance matrix. In general, the synthetic np_xy and np_w charts require twice larger samples to outperform the T² chart. Copyright © 2014 John Wiley & Sons, Ltd.     

Abundance of elliptic dynamics on conservative three-flows

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C ¹-residual (dense G _δ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-setting counterpart of Newhouse's Theorem 1.3 (S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 (1977), pp. 1061–1087). Our result follows from two theorems, the first one says that if Λ is a hyperbolic invariant set for some class C ¹ zero divergence vector field X on M, then either X is Anosov, or else Λ has empty interior. The second one says that, if X is not Anosov, then for any open set U ? M there exists Y arbitrarily close to X such that Y ^t has an elliptical closed orbit through U.     

Control charts for monitoring the ratio of two Poisson rates

In this paper, we are concerning in monitoring the ratio ρ of two Poisson rates by control charts. Let X and Y be two independent Poisson random variables with means λ₁ and λ₂=λ₁/ρ, respectively. The study considers that only individual observations X_i and Y_i are available at each sampling time i. The performance in detecting shifts on the ratio ρ using several statistics, some based on normalized transformations, is evaluated by an extensive simulation study. Two types of control charts, Shewhart and exponentially weighted moving average (EWMA), are considered. The one-sided control chart with upper control limit (UCL) is applied so that we are focusing on detecting when the ratio ρ shifted to higher rate in this paper. The results pointed out that EWMA control chart is a better alternative. Some guidelines indicating which statistics yield best performance are proposed for the practitioners.     

An introduction to chaotic and random time series analysis

Chaos refers to the paradoxical evolution of a deterministic system in a way that is disordered—to the point that the time dependence of the physical variables appears stochastic. A need for data analysis procedures to detect, model, and separate chaotic and random processes has arisen from this recently understood paradigm. Many special techniques have been designed for chaotic data; the unification of these with conventional time series analysis is a developing field. This tutorial uses examples to explain the origin of chaotic behavior and the relation of chaos to randomness. Two powerful mathematical results are described: (1) a representation theorem guarantees the existence of a specific time-domain model for chaos and addresses the relation between chaotic, random, and strictly deterministic processes, and (2) a theorem assures that information on the behavior of a physical system in its complete state space can be extracted from time-series data on a single observable. These theorems form the basis of a practical data analysis scheme, as follows: given N observations of a variable Y, i.e., {Y_n, n = 1,2,3, …, N}, define X = A * Y and maximize, with respect to the parameters of A, a function H(X) that measures degree of chaos. This maximization is carried out by minimizing the dimension covered by the data in the M-dimensional space (X_n, X_n+1, X_n+2, …, X_n+M?1). The resulting dimension D either (1) increases continuously with M or (2) levels off and remains constant (= D_max) beyond a certain point. In case (1) or if D_max is quite large X is random; if case (2) holds and D_max is small, we have chaos. The inverse of A found in this procedure is an estimate of the filter in the moving average model for Y.     

Design and implementation of qth quantile‐unbiased tr‐chart for monitoring times between events

The times between events control charts have been proposed in literature for statistical monitoring of high‐yield processes by observing the waiting times up to r ^th (r ≥ 1  ) non‐conforming items or defects. The average run length (ARL) is the most widely used performance measure to evaluate the chart's performance, but in recent years, it has been subjected to criticisms. Because the run length distribution is highly skewed and hence, the ARL is not necessarily a typical value of the run length. Thus, evaluation of the control chart based on ARL alone could be misleading. In this paper, the quantiles of run length distribution are considered, instead of ARL, to design the t_r ‐chart. Further, we eliminate the bias in q ^th quantile function of the t_r ‐chart for both the known and unknown parameter case. In particular, the MRL‐unbiased t_r ‐chart is discussed in detail and compared with the ARL‐unbiased t_r ‐chart. It is found that the MRL‐unbiased t_r ‐chart outperforms than the corresponding ARL‐unbiased chart in unknown parameter case. It is also found that the proposed chart requires less phase I observations than that of the earlier studies has been suggested.     

On the spectral representation method in simulation

Two models X_n(t) and Y_n(t) are considered for generating samples of stationary band-limited Gaussian processes. The models are based on the spectral representation method and consist of a superposition of n harmonics. The harmonics of X_n(t) have random phase and amplitude while the harmonics of Y_n(t) only have random phase. It is shown that the two models are equal in the second-moment sense. However, Y_n(t) has stronger ergodic properties than X_n(t). On the other hand, X_n(t) is a Gaussian process for any value of n while Y_n(t) is asymptotically Gaussian as n approaches infinity. It is demonstrated that the rejection of X_n(t), because of its weak ergodic property, or of model Y_n(t), because of its nonGaussian distribution, it is not generally justified. One special case in which Y_n(t) should not be used is that of Gaussian processes with power concentrated at a few discrete frequencies.     

Monitoring of Time‐Between‐Events with a Generalized Group Runs Control Chart

Control charting methods for time between events (TBE) is important in both manufacturing and nonmanufacturing fields. With the aim to enhance the speed for detecting shifts in the mean TBE, this paper proposes a generalized group runs TBE chart to monitor the mean TBE of a homogenous Poisson failure process. The proposed chart combines a TBE subchart and a generalized group conforming run length subchart. The zero‐state and steady‐state performances of the proposed chart were evaluated by applying a Markov chain method. Overall, it is found that the proposed chart outperforms the existing TBE charts, such as the T, T_r, EWMA‐T, Synth‐T_r, and GR‐T_r charts. Copyright © 2015 John Wiley & Sons, Ltd.     

A New Synthetic Exponentially Weighted Moving Average Control Chart for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely recognized as a potentially powerful process monitoring tool of the statistical process control because of their excellent speed in detecting small to moderate shifts in the process parameters. Recently, new EWMA and synthetic control charts have been proposed based on the best linear unbiased estimator of the scale parameter using ordered ranked set sampling (ORSS) scheme, named EWMA‐ORSS and synthetic‐ORSS charts, respectively. In this paper, we extend the work and propose a new synthetic EWMA (SynEWMA) control chart for monitoring the process dispersion using ORSS, named SynEWMA‐ORSS chart. The SynEWMA‐ORSS chart is an integration of the EWMA‐ORSS chart and the conforming run length chart. Extensive Monte Carlo simulations are used to estimate the run length performances of the proposed control chart. A comprehensive comparison of the run length performances of the proposed and the existing powerful control charts reveals that the SynEWMA‐ORSS chart outperforms the synthetic‐R, synthetic‐S, synthetic‐D, synthetic‐ORSS, CUSUM‐R, CUSUM‐S, CUSUM‐ln S², EWMA‐ln S² and EWMA‐ORSS charts when detecting small shifts in the process dispersion. A similar trend is observed when the proposed control chart is constructed under imperfect rankings. An application to a real data is also provided to demonstrate the implementation and application of the proposed control chart. Copyright © 2014 John Wiley & Sons, Ltd.     

Refractive index and density of Na-, Rb- and mixed Na,Rb-aluminogermanate glasses

Refractive index and density measurements are given for the ternary systemsX M₂O :Y Al₂O₃: (1 -X -Y)GeO₂ where M = sodium and rubidium for Na/(Na + Rb) = 0, 0.3, 0.5, 0.7 and 1.0; (1 -X -Y) = 0.7, 0.8 and 0.9 andY/X = 0, 0.3 and 1.0. The values of the molar refractivity,R, are related to the structure of the glass. At low alkali concentrations, the decrease inR with increasingX is related to the closer packing of the network. At high alkali concentrations the changes inR appear to be controlled by the introduction of non-bridging oxygens into the network. Alumina additions to the germania network strongly increaseR and this appears to be related to change in the network packing brought about by the conversion of Ge⁴⁺ from octahedral to tetrahedral coordination.     

New EWMA control charts for monitoring process dispersion using auxiliary information

The exponentially weighted moving average (EWMA) control chart is a well‐known statistical process monitoring tool because of its exceptional pace in catching infrequent variations in the process parameter(s). In this paper, we propose new EWMA charts using the auxiliary information for efficiently monitoring the process dispersion, named the auxiliary‐information–based (AIB) EWMA (AIB‐EWMA) charts. These AIB‐EWMA charts are based on the regression estimators that require information on the quality characteristic under study as well as on any related auxiliary characteristic. Extensive Monte Carlo simulation are used to compute and study the run length profiles of the AIB‐EWMA charts. The proposed charts are comprehensively compared with a recent powerful EWMA chart—which has been shown to be better than the existing EWMA charts—and an existing AIB‐Shewhart chart. It turns out that the proposed charts perform uniformly better than the existing charts. An illustrative example is also given to explain the implementation and working of the AIB‐EWMA charts.     

Internal symmetry groups for perfect elastoplasticity under axial‐torsional loadings

Abstract

In the material modeling of experimental axial‐torsional strain control tests, the hoop and radial strains are always unknown, a priori, and hence can not be viewed as inputs. This greatly complicates constitutive model analyses because the resulting differential equations become highly nonlinear. To tackle this problem, we demonstrate two new formulations. By using the two‐integrating factors idea we derive two Lie type systems in the product space M ¹⁺¹?M ¹⁺¹. The Lie algebra is the direct sum so(1, 1)?so(1, 1), and correspondingly the symmetry group is the direct product SO_o (1, 1) ?SO_o (1, 1). Then, by using the one‐integrating factor idea we convert the nonlinear constitutive equations to a Lie type system X=A(X, t)X with A?sl(2, 1, R), a Lie algebra of the special orthochronous pseudo‐linear group SL(2, 1, R). The underlying space is a cone in the pseudo‐Riemann manifold. Consistent numerical methods are also developed according to these Lie symmetries.     

Book Reviews–10-Year Index 1959–1968

The dependence structure of a stationary time series {X_t } is usually discussed in terms of the observed variables X_t . Let F(x) be the common marginal c.d.f. of the X_t We propose a model involving only the transformed series {Z_t }, where Z_t = F(X_t ). The conditional distribution of Z_t given Z _t–1, Z _t–2, … is expressed in terms of Z_t , Z _t1, … and parameters β₁, β₂, …. The case where Z_t depends only on Z _t–1 is considered in detail. Estimators for the β's are found, and an example using a normal Markov process is given.     

Chitosan-TPP nanoparticles stabilized by poloxamer for controlling the release and enhancing the bioavailability of doxazosin mesylate: in vitro,and in vivo evaluation

Objective: Control the release and enhance the bioavailability of chitosan-doxazosin mesylate nanoparticles (DM-NPs).

Significance: Improve DM bioavailability for the treatment of benign prostatic hyperplasia and hypertension.

Methods: Plackett–Burman design was utilized to screen the variables affecting the quality of DM-NPs prepared by ionic gelation method. The investigated variables were initial drug load (X₁), chitosan percentage (X₂), tripolyphosphate sodium (TPP) percentage (X₃), poloxamer percentage (X₄), homogenization speed (X₅), homogenization time (X₆) and TPP addition rate (X₇). The prepared DM-loaded NPs have been fully evaluated for particle size (Y₁), Zeta potential (Y₂), production yield (Y₃), entrapment efficiency (Y₄), loading capacity (Y₅), initial burst (Y₆), and cumulative drug release (Y₇). Finally, DM pharmacokinetic has been investigated on healthy albino male rabbits by means of non-compartmental analysis.

Results: The combination of variables showed variability of Y₁, Y₂, and Y₃ equal to 122–710?nm, 3.49–23.63?mV, and 47.31–92.96%, respectively. While Y₄ and Y₅, reached 99.87%, and 8.53%, respectively. The prepared NPs revealed that X₂, X₃, and X₄ are the variables that play the important role in controlling the release behavior of DM from the NPs. The in vivo pharmacokinetic results indicated the enhancement in bioavailability of DM by 7 folds compared to drug suspension and the mean residence time prolonged to 23.72?h compared to 4.7?h of drug suspension.

Conclusion: The study proved that controlling the release of DM from NPs enhance its bioavailability and improve the compliance of patients with hypertension or benign prostatic hyperplasia.     

New Synthetic Control Charts for Monitoring Process Mean and Process Dispersion

A statistical quality control chart is widely recognized as a potentially powerful tool that is frequently used in many manufacturing and service industries to monitor the quality of the product or manufacturing processes. In this paper, we propose new synthetic control charts for monitoring the process mean and the process dispersion. The proposed synthetic charts are based on ranked set sampling (RSS), median RSS (MRSS), and ordered RSS (ORSS) schemes, named synthetic‐RSS, synthetic‐MRSS, and synthetic‐ORSS charts, respectively. Average run lengths are used to evaluate the performances of the control charts. It is found that the synthetic‐RSS and synthetic‐MRSS mean charts perform uniformly better than the Shewhart mean chart based on simple random sampling (Shewhart‐SRS), synthetic‐SRS, double sampling‐SRS, Shewhart‐RSS, and Shewhart‐MRSS mean charts. The proposed synthetic charts generally outperform the exponentially weighted moving average (EWMA) chart based on SRS in the detection of large mean shifts. We also compare the performance of the synthetic‐ORSS dispersion chart with the existing powerful dispersion charts. It turns out that the synthetic‐ORSS chart also performs uniformly better than the Shewhart‐R, Shewhart‐S, synthetic‐R, synthetic‐S, synthetic‐D, cumulative sum (CUSUM) ln S², CUSUM‐R, CUSUM‐S, EWMA‐ln S², and change point CUSUM charts for detecting increases in the process dispersion. A similar trend is observed when the proposed synthetic charts are constructed under imperfect RSS schemes. Illustrative examples are used to demonstrate the implementation of the proposed synthetic charts. Copyright © 2014 John Wiley & Sons, Ltd.     

A sensitive non-parametric EWMA control chart

In practice, we may not always have normally distributed quality characteristics of interest. This leads to the need for non-parametric techniques which are not dependent on the assumptions about the parent distribution. This study develops a non-parametric exponentially weighted moving average (EWMA) chart (namely the NPSS_EWMA chart) for an improved monitoring of process location. The proposal is based on the use of sign statistics on a moving pattern in an EWMA setup. The design structure of the proposed chart is developed and its performance is evaluated in terms of different properties including average run length (ARL), standard deviation run length (SDRL), percentiles, relative ARL (RARL), extra quadratic loss (EQL), and performance comparison index (PCI). The proposal is compared with recently developed non-parametric counterparts namely NPS_EWMA, NPAS_EWMA, and NPS_CUSUM charts. It is observed that the design structure of the proposed NPSS_EWMA chart outshines the existing counterparts. An application example is also included in the study for practical demonstration.