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A Smooth Nonparametric Approach to Multivariate Process Capability
Abstract
The determination of the capability of a stable process with a multivariate quality characteristic using standard methods usually requires the assumption that the quality characteristic of interest follows a multivariate normal distribution. Unfortunately, multivariate normality is a difficult assumption to assess. Further, departures from this assumption can result in erroneous conclusions. In this article, I propose assessing the capability of a process using a nonparametric estimator. This estimator is based on a kernel estimate of an integral of a multivariate density. Bandwidth selection for this method is based on a smoothed bootstrap estimate of the mean squared error of the estimator. I also address the issue of constructing approximate confidence intervals. An example is presented that applies the proposed method to bivariate nonnormal process data. The performance of the resulting estimator is then compared to the sample proportion and a normal parametric estimate in a simulation study.
... According to Polansky [24], presence of a point near the boundary of the rectangular specification region indicates that a significant amount of tail area of the process distribution may overlap the boundary leading towards poor process capability value. This is actually the case here, as can be seen in Figures 1 and 2. ...
... However, the hybrid bootstrap confidence interval forp(H, S) is [0, 0.1372], which is fairly wide and accrding to Polansky [24], this happens due to small sample size. ...
... 9. The MPCIs defined by Ciupke [10] and Polansky [24] do not require any distributional assumption and they both considered the process to be of poor capability level. ...
Process Capability Index (PCI) is a very popular tool for assessing performance of processes (often involving a single quality characteristic). Multivariate process capability indices (MPCI) are comparatively new to the literature and hence often involve some difficulties in practical applications. One such hurdle is multivariate normality assumption of the underlying distribution of the quality characteristics. While such assumption gives some computational as well as theoretical advantage in formulating MPCIs, often data encountered in practice do not follow multivariate normal distribution due to several reasons. Consequently, the computed values of the MPCIs may give misleading results. In the present article, we have made performance analysis of some MPCIs in the light of a dataset which has been widely used in the literature, particularly in the context of MPCIs. Most of these MPCIs were already applied to the said data in the literature and our objective is to make their comparative performance analysis. In this context, the data, though actually non-normal, is often concluded as multivariate normal by several researchers. Therefore, while making the performance analysis of the MPCIs, this aspect has also been incorporated to put emphasis on the importance of distributional assumption in a multivariate process capability analysis.
... Among them, Chan et al. (1991) and Chen (1994) have assumed multivariate normality of the data and have concluded that the process is capable. On the contrary, Polansky (2001) has addressed the problem from non-parametric view point and has shown using kernel estimation procedure that the process is actually poor in terms of capability. According to Polansky (2001), such incapability of the process is due to a point that is near the boundary of the specification set. ...
... On the contrary, Polansky (2001) has addressed the problem from non-parametric view point and has shown using kernel estimation procedure that the process is actually poor in terms of capability. According to Polansky (2001), such incapability of the process is due to a point that is near the boundary of the specification set. This is evident from Figure 2, featuring the plot of the transformed data as well. ...
... This is evident from Figure 2, featuring the plot of the transformed data as well. Thus, our view regarding the capability of the process is concurred by Polansky's (2001) observation. This is because in all the cases except the case of Polansky (2001), the underlying distribution of the quality characteristics was assumed to be multivariate normal, while actually this is not the case. ...
In manufacturing industries, it is often seen that the bilateral specification limits corresponding to a particular quality characteristic are not symmetric with respect to the stipulated target. A unified superstructure Cp′′(u,v) of univariate process capability indices was specially designed for processes with asymmetric specification limits. However, as in most of the practical situations a process consists of a number of inter-related quality characteristics, subsequently, a multivariate analogue of Cp′′(u,v), which is called CM(u,v), was developed. In the present paper, we study some properties of CM(u,v) like threshold value and compatibility with the asymmetry in loss function. We also discuss estimation procedures for plug-in estimators of some of the member indices of CM(u,v). Finally, the superstructure is applied to a numerical example to supplement the theory developed in this article.
... Finally, the capability assessments for multivariate processes with non-normal process distributions have been studied by Abbasi and Niaki [2], Ahmad et al. [1], Polansky [53] and so on. ...
... The MPCIs discussed in the present section can also be used for this purpose. In particular, Polansky [53] used the same data and concluded that the performance of the process is not satisfactory, which supports the observations made by Chatterjee and Chakraborty [14]. Moreover,the approach of transforming the data to multivariate normality and then applying C M (u, v) is easier to execute as compared to using Polansky's [53] MPCI. ...
... In particular, Polansky [53] used the same data and concluded that the performance of the process is not satisfactory, which supports the observations made by Chatterjee and Chakraborty [14]. Moreover,the approach of transforming the data to multivariate normality and then applying C M (u, v) is easier to execute as compared to using Polansky's [53] MPCI. ...
In the context of statistical quality control, process capability index (PCI) is one of the widely accepted approaches for assessing the performance of a process with respect to the pre-assigned specification limits. The quality characteristic under consideration can have differnt types of specification limits like bilateral, unilateral, circular and so on. Use of single PCI for all the situations could be misleading. Hence appropriate PCIs need to be chosen based on the characteristics of the specification limits. Similar situations may arise for multivariate characteristics as well. In the present chapter, we have discussed about some of the PCIs for different specification limits including some PCIs for multivariate characteristics. A few numerical examples are given to suppliment our theoretical discussion.
... A critical concept in the area of statistical process control is the assessment of manufacturing process capability. Process capability is seen as the ability of the manufacturing process to consistently produce items within the given specifications of quality characteristics, i.e., within the tolerances for Design Parameters (DPs) and Process Variables (PVs) (Polansky, 2001;Ding et al., 2005;Montgomery, 2005). Therefore, process capability is related to process fallout rate, which is defined as the probability of DP and PV measurements to be outside of design tolerances. ...
... Kernel density estimation is a popular technique for smooth estimation of the density, consisting of a chosen kernel function (such as Gaussian, Epanechnikov) and bandwidth matrix (width of the kernel function) to estimate the density (Silverman, 1986;Scott, 1992). This paper follows the approach as implemented by Polansky (2001) using optimum bandwidth selection based on research conducted by Wand and Jones (1995). Figure 3 illustrates a case of non-normal process representation us- ing Kernel Density Estimate (KDE)-based confidence regions with the mean set at design nominal. ...
... Wang et al. (2000) reviewed major multivariate process capability indices under the assumption of normality. Polansky (2001) proposed a non-parametric, distribution-free estimator to compute multivariate process capability that is calculated based on the KDE of the probability density function (p.d.f.). This methodology is also used to estimate the probability of failures to generate control charts (Polansky, 2005). ...
This paper introduces a methodology for functional capability analysis and optimal process adjustment for products with failures that occur when design parameters and process variables are within tolerance limits (in-specs). The proposed methodology defines a multivariate functional capability space (FC-Space) using a mathematical morphology operation, the Minkowski sum, in order to represent a unified model with (i) multidimensional design tolerance space; (ii) in-specs failure region(s); and, (iii) non-parametric, multivariate process measurements represented as Kernel Density Estimates (KDEs). The defined FC-Space allows the determination of a desired process fallout rate in the case of products with field failures that occur within design tolerances (in-specs). The outlined process adjustment approach identifies the optimum position of the process mean in order to minimize the overlap between the KDEs and in-specs failure regions, i.e., achieve the minimum possible process fallout rate for current process variation. The FC-Space-based process adjustment methodology is illustrated using a case study from the electronics industry where the in-specs failure region is identified based on warranty information analysis.
... The classical approach is to transform the data appropriately so that the transformed data is normally distributed. The works of Polansky (1998Polansky ( , 2000Polansky ( , 2001 fall in this category. A more recent approach is to use PCIs specifically defined for non-Gaussian data. ...
The process capability index (PCI) has been one of the most useful indicators for evaluating the capability of a manufacturing process. Since PCI is based on sample observations, it is essentially an estimated value. Hence, it is natural to think of a confidence interval (CI) of the PCI. In this paper, bootstrap confidence intervals and highest posterior density (HPD) credible intervals of non-normal PCIs, C N pmk , CNpm, C N pk and CNp are studied through simulation when the underlying distribution is two parameter logistic-exponential (LE). First, maximum likelihood method is used to estimate the model parameters and then three boot-strap confidence intervals (BCIs) are considered for obtaining CIs of non-normal PCIs, C N pmk , CNpm, C N pk and CNp. Next, Bayesian estimation is studied with respect to symmetric (squared error) loss function using gamma priors for the model parameters. In order to assess the performance of BCIs and HPD credible intervals of C N pmk , CNpm, C N pk and CNp with respect to average width, coverage probabilities and relative coverage, Monte Carlo simulations are conducted. Finally, a real data set, related to weight of the rubber edge of the speaker driver has been analyzed for illustrative purpose.
... Nonetheless, how to tackle the issue of asymmetric tolerances is still not clear as the asymmetric tolerance allocation problem is rarely addressed in the literature [24]. One possible approach to tackle unknown pdf and asymmetric tolerances is to generate random samples through the computer simulation [25], [26], which is still an area to be explored. However, these data-driven studies for asymmetric tolerance allocation are rarely addressed in the literature, as they are generally not necessary for traditional manufacturing processes. ...
Tolerance allocation is a design tool that is proven crucial for enhancing the cost effectiveness and productivity of manufacturing systems. The growing implementation of additive manufacturing (AM) with its unique characteristics requires novel tolerance allocation methodologies to be developed. More specifically, many of the assumptions in traditional tolerance allocation methods such as normality assumption, a priori known probability density function of data, and symmetricity of tolerances cannot be seamlessly applied to AM processes. Furthermore, as the obtained dimensional errors of components in AM processes are significantly affected by the decisions made during the manufacturing stage (e.g., selected process, material, layer thickness, and build direction), the manufacturing parameters need to be jointly considered for allocating feasible and optimum tolerances during the product design phase. In this paper, a methodology for joint dimensional tolerance allocation and manufacturing of assemblies fabricated by AM processes is proposed based on the asymmetric distribution of errors and considering assembly requirements, namely, specification and confidence level. The bootstrap statistical technique is used to estimate the unknown population's statistics. A cyclic optimization approach is adopted to tackle the formulated problem. The numerical examples are provided to illustrate the effectiveness of the proposed method.
... where ( ) is the inverse cumulative distribution of the standard normal random variable and is the multivariate normal probability of multivariate data that falls within a multivariate specification limits. [11] used a nonparametric approach of multivariate kernel-density estimation to find P NC and computed capability index using Eq. (1). ...
In many cases, the quality of a manufactured product is determined by more than one characteristic and often, these quality characteristics are correlated. A number of methods for dealing with quality evaluation of multivariate processes have been proposed in the literature. However, some of these studies do not consider correlation among quality characteristics. In this paper, two new approaches for estimating the proportion of non-conformance for correlated multivariate quality characteristics with nominal specifications are proposed: (i) the modified tolerance region approach and (ii) the target distance approach. In the first approach, the p number of correlated variables are analysed based on the projected shadow of the p-dimensional hyper ellipsoid so that the ability to visualise the tolerance region and the process region for p > 2 is preserved. In the second approach, the correlated variables are combined and a new variable called the target distance is introduced. The proportion of non-conformance results estimated using both methods were used to compute the multi-variate capability index and the total expected quality cost. This study also suggest modification to the NMCp index as proposed in Pan and Lee (2010) such that the process capability for p > 2 can be measured correctly. The application of both approaches is demonstrated using two examples and it is shown that both methods i.e. the modified tolerance region and the target distance methods are capable of estimating the capability of multivariate processes.
... This leads to the fallacious assumption of normality while the deviation is significantly observable. In these cases, the normality violation is not a rare occurrence in process control; especially when the number of samples is not large enough (Polansky, 2001). Schilling & Nelson (1976) investigated the effects of non-normality on the control limits of the X-bar control chart. ...
The growing demand for statistical process monitoring has led to the vast utilization of multivariate control charts. Complicated structure of the measured variables associated with highly correlated characteristics, has given rise to daily increasing urge for reliable substitutes of conventional methods. In this regard, projection methods have been developed to address the issue of high correlation among characteristics by transforming them to an uncorrelated set of variables. Principal component analysis (PCA)-based control charts are widely used to overcome the issue of correlation among measured variables by defining linear transformations of the existing variables to a new uncorrelated space. Newly transformed variables explain different amount of variations in the measured variables with the first PC explaining the highest amount, the second PC explains the second highest one, and so on. PCA, also gives the opportunity of dimension reduction to the researcher, in cost of losing a part of information extracted from observed variation, yet using all the original measured variables. In spite of the mentioned strength of the PCA based methods, the underlying assumption of observations to be normally distributed, has limited the applicability of PCA-based schemes, as the normality assumption is widely violated in real practices. With this in regard, a distribution-free method to establish the limits of PCA-based control charts can be a good modification to keep the scheme reliable when the normality assumption is not met. The proposed method presented in this paper is based on support vector machines (SVM) as a substitute for conventional methods to construct control limits for PCA-based control charts. As SVM uses real-world observations of the process, no distributional assumption is required to construct control limits. Extensive simulation experiments are conducted using normal and non-normal datasets to compare the performance of the proposed method with those of the conventional and some non-parametric methods existing in the literature. The results show a relatively good performance of the proposed method compared to others in terms of the average and the standard deviation of run lengths.
... Di Bucchianico et al. [21] develop a nonparametric procedure for constructing multivariate tolerance regions that relies on the choice of a class of sets to which the tolerance region belongs (for example, ellipsoids or hyperrectangles). Polansky [22] also uses the tolerance regions approach to evaluate the capability of a manufacturing process. ...
Process capability is determined by comparing the actual performance of a process with required specifications. Several indices have been proposed to report the capability of a process in the univariate case. When performance is tracked in several dimensions, an extension of these indices is required. We present several multivariate capability indices, and extend the idea of multivariate tolerance regions for assessing the capability of a process.
Keywords:
process capability;
specification limits;
multivariate statistical process control;
Mahalanobis T2 charts;
multivariate tolerance regions
... Di Bucchianico et al. [21] develop a nonparametric procedure for constructing multivariate tolerance regions that relies on the choice of a class of sets to which the tolerance region belongs (for example, ellipsoids or hyperrectangles). Polansky [22] also uses the tolerance regions approach to evaluate the capability of a manufacturing process. ...
Process capability is determined by comparing the actual performance of a process with required specifications. Several indices have been proposed to report the capability of a process in the univariate case. When performance is tracked in several dimensions, an extension of these indices is required. We present several multivariate capability indices, and extend the idea of multivariate tolerance regions for assessing the capability of a process.
... Calculation of proposed capability index boils down to calculation of the process yield. To calculate the process yield, it necessitates to apply a curve fitting method to approximate the quality characteristic distribution, ( ). f x Polansky [32,33] used nonparametric approach (basically kernel density estimation) to estimate process yield for both univariate as well as multivariate quality characteristics. Ciarlini et al. [9] used bootstrap methodology to estimate tail probabilities even in regions not supported by data, with accuracy independent of the sample variance also when data are not nearly normal. ...
A generalized process capability index, defined as the ratio of proportion of specification conformance (or, process yield) to proportion of desired (or, natural) conformance, has been developed. Almost all the process capabilities defined in the literature are directly or indirectly associated with this generalized index. Normal as well as non-normal and continuous as well as discrete random variables could be covered by this new index. It can also be assessed under either unilateral or bilateral specifications. We deal with the proposed index in case of normal, exponential and Poisson processes. Under each distributional assumption, point estimators for the proposed index are suggested and compared through simulation study. Two real-world applications have been discussed using the proposed index.
... Avec un resserrement constant des tolérances allouées dans la fabrication de produits, les designers doivent avoir de meilleurs indicateurs pour bienétablir, du premier coup, les paramètres de conception par rapport Article publié par EDP Sciences Déviations spatiales réelles de l'élément i μx i , μy i Erreurs systématiques estimées de l'élément i σÉcart type estimé d'une variable aléatoire aux variables du procédé utilisé [4]. Pour répondreà cette demande, nombreuses recherches ont démontré de nouvelles méthodes très efficaces pour modéliser les variations provenant des différentesétapes de fabrication d'une pièce [5,6], et pour surveiller et contrôler statistiquement un procédé possédant plusieurs variables [7,8]. ...
This paper proposes a new methodology to exploit the capability of manufacturing processes in the statistical calculation of the tolerance of localization of a set of features according to the standards ISO 1101 and ASME Y14.5. The number of features that form the pattern studied, the systematic and random errors in the manufacturing process will all be retained and included in the approach. An explicit mathematical model is developed to identify the statistical distribution functions for different types of localization tolerances. From these distributions, we present a methodology to estimate the values of tolerance that can meet a compliance threshold for a pre-established value, and vice versa. The article also presents a series of charts that are usable in an industrial context. Many examples are illustrated and a case study is presented.
... Tools such as statistical process control SPC, functional dimensioning and tolerancing, statistical tolerancing [2], and capability indices [3] have been developed to ease the designer's task in estimating the possible geometrical variations of an assembly and, at the same time, obtain more robust control and quality improvement. With a constant tightening of tolerances allocated in product manufacturing, designers need better indicators to establish design parameters in relation to process variables, from the outset [4]. To meet this demand, many studies have recently demonstrated new methods for monitoring and controlling a process with multiple statistical variables [5][6][7]. ...
This paper considers a way of measuring a process capability index in order to obtain the geometric tolerance of a pattern of position elements according to the ASME Y14.5 standard. The number of elements present in the pattern, as well as its material condition (least LMC or maximum MMC), are taken into consideration during the analysis. An explicit mathematical model will be developed to identify the distribution functions (PDF and CDF) of defects on the location and diameter. Using these distributions and the Hasofer-Lind index, we will arrive at a new definition of process capability-meaning the value of tolerances that can meet the threshold of x% compliance. Finally, our method is validated using a variety of typical case studies. [DOI:10.1115/1.4005797]
... 1 the indices which measure the ratio of a tolerance region to a process region, such as the research proposed by Wang et al. (2000), Shahriari et al. (1995), and Taam et al. (1993) 2 the indices based on proportion of conforming items, such as the research proposed by Polansky (2001), Pal (1999), and Chen (1994) 3 the indices based on principal components analysis (PCA), for instance the research proposed by Wang and Chen (1999) 4 the indices based on the extension of univariate PCI such as those introduced by Holmes and Mergen (1999) and Chen et al. (2003). ...
While most of the methods developed for computing process capability indices (PCI) concentrate on cases with normally or continuous non-normally distributed quality characteristics, computing this measure for processes with mixed distributed data has not been investigated so far. In this paper, a new method is proposed for computing (PCI) for mixed binary-normal quality characteristics. In the proposed method, first a mixed binary-normal distribution is considered to be fitted on the available data. Having estimated the unknown parameters of the fitted distribution using maximum likelihood estimation and genetic algorithm, the proportion of the conforming items of thecorresponding distribution is estimated by Monte Carlo simulation runs. Finally, the PCI is computed based on the relationship of PCI and proportion of conforming items. The performance of the proposed method is evaluated using simulation studies as well as a case study in a plastic injection moulding process.
... Di Bucchianico et al. [29] developed a nonparametric procedure for constructing multivariate tolerance regions that relies on the choice of a class of sets to which the tolerance region belongs (for example ellipsoids or hyper-rectangles). Polansky [30] also used the tolerance regions approach proposed in [26] to evaluate the capability of a manufacturing process. ...
The semiconductor industry ranges from the design and production of semiconductors on silicon wafers to automatic placement robots that insert semiconductor devices on hybrid microcircuits.Wafers consist of electronic circuits or chips that are characterized by electrical and mechanical characteristics. Process modeling and simulations provide predictions of geometries and material properties of semiconductor devices and wafer structures and help design and improve manufacturing processes such as photolithography, etching, deposition, and ion implantation. In this paper, we focus on three application areas of industrial statistics to the semiconductor industry. These are: (1) process capability indices, (2) process monitoring, and (3) multivariate statistical process control. We refer to two case studies that set a context and provide examples to the presented techniques. Copyright © 2012 John Wiley & Sons, Ltd.
... In202122232425262728293031 multivariate capability indices have been developed and presented for assessing process capability. Some of multivariate process capability indices are defined based on the ratio of a tolerance region to a process region, such as the method proposed by Shahriari et al. [25], while some authors have used principal components analysis or the probability of producing nonconforming items, such as Wang and Chen [32] and Polansky [33], respectively. A general framework for research topics about profile monitoring could be found in a review article which is presented by Woodall [34]. ...
In some situations, the quality of a process or product is characterized by a linear regression model between two or more variables which is called a linear regression profile. Moreover, in some cases, several correlated quality characteristics can be modeled as a set of linear functions of one explanatory variable which is typically referred to as multivariate simple linear profiles structure. On the other hand, process capability index is an important concept in statistical process control and measures the ability of the process to provide products that meet certain specifications. Little work, however, is done to evaluate the capability of a process with profile quality characteristic. This paper proposes three new methods for measuring process capability in multivariate simple linear profiles. Performance of the proposed methods is evaluated through simulation studies. In addition, the applicability of the proposed methods is illustrated using a real case of calibration application.
... See, for instance, the indices introduced by Wang et al. [14], Shahriari et al. [15] and Taam et al. [16]. The second category computes the indices based on the probability of nonconforming items such as studies by Pal [17], Chen [18] and Polansky [19] . In the third category using principle component analysis both normal and non-normal cases have been studied for multivariate process capability indices; e.g., see Wang and Chen [20]. ...
Most of the researches developed for single response and multi response optimization problems are based on the normality assumption of responses, while this assumption does not necessarily hold in real situations. In the real world processes, each product can contain correlated responses which follow different distributions. For instance, multivariate non-normal responses, multi-attribute responses or in some cases mixed continuous-discrete responses. In this paper a new approach is presented based on multivariate process capability index and NORTA inverse transformation for multi response optimization problem with mixed continuous-discrete responses. In the proposed approach, assuming distribution function of the responses is known in advance based on historical data; first we transform the multivariate mixed continuous-discrete responses using NORTA inverse transformation to obtain multivariate normal distributed responses. Then the multivariate process capability index is computed in each treatment. Finally, for determining the optimum treatment, the geometric mean value of multivariate Process Capability Index (PCI) is computed for each factor level and the most capable levels are selected as the optimum setting. The performance of the proposed method is verified through a real case study in a plastic molding process as well as simulation studies with numerical examples.
... Capability indices are typically subject to distributional assumptions. To avoid this problem, Polansky (2001) has used a nonparametric approach. Our paper, instead of using a traditional capability index, uses P(D), the probability of a defective. ...
When output random variables are a function (known as a transfer function) of input random variables, Monte Carlo simulation has often been used to examine the sensitivity of the outputs to changes to the inputs. An important and commonly used measure of the outputs is their process capability (the probability that an output is within specification limits). In this paper, we show how to efficiently conduct extensive analysis of the sensitivity of the process capability of outputs to changes to inputs. Specifically, we show how a single set of simulation replications can be used to efficiently estimate the process capability as a function of each input random variable's values, its parameters, and truncation of its values at chosen limits. The approach is extremely flexible; the effects of changes to the distributional form of an input variable alone or in combination with the previously mentioned changes are easily evaluated.
... The percentile method is a particular bootstrap method used to obtain the results in the last column of tables 2 and 4. See Efron (1982) for more information on the bootstrap method. Precedents for using bootstrap confidence intervals for multivariate capability include Littig et al. (1992) and Polansky (2001). Using bias corrected bootstrap intervals (also used in tables 2 and 4) is a method to fairly centre the bootstrap interval around the average (Gunter 1992) if its initial values are biased. ...
Recent literature has proposed multivariate capability indices, but does not suggest a method for measuring quality characteristics in a way that links production irregularities directly to their causes. Our objective is to present a new approach to multivariate capability indices that uses process-oriented basis representation (POBREP) which allows the computing of cause-related index values. The proposed method focuses on independent process-oriented multivariate data by employing regression coefficients as data. These coefficients measure the amount of the characteristic patterns induced by particular problems or incidents that can occur in the system. Two examples from the electronics industry (the chip capacitor process and solder paste process) use simulated data and Monte Carlo integration to demonstrate the new process-oriented capability method. A reduction of estimation error was realized when using process-oriented capability. For the chip capacitor problem, capability error is 24–54% when using ordinary multivariate data. However, when using process-oriented data the error is less than 3%. Capability is difficult to compute from sample data in the solder paste example without the process-oriented approach. Future research should propose a multivariate capability measure for dependent process-oriented data.
... The standard assumption behind majority of SPC and MSPC methods mentioned above is that the process variables follow a Gaussian distribution (Rose 1991), a questionable assumption in several industrial processes (Polansky 2001), and in particular, highly automated processes (Chinnam and Kolarik 1992). (Schilling and Nelson 1976) and many other researchers have investigated the effects of non-normality on the control limits and charting performance. ...
It is important to monitor manufacturing processes in order to improve product quality and reduce production cost. Statistical Process Control (SPC) is the most commonly used method for process monitoring, in particular making distinctions between variations attributed to normal process variability to those caused by ‘special causes’. Most SPC and multivariate SPC (MSPC) methods are parametric in that they make assumptions about the distributional properties and autocorrelation structure of in-control process parameters, and, if satisfied, are effective in managing false alarms/-positives and false-negatives. However, when processes do not satisfy these assumptions, the effectiveness of SPC methods is compromised. Several non-parametric control charts based on sequential ranks of data depth measures have been proposed in the literature, but their development and implementation have been rather slow in industrial process control. Several non-parametric control charts based on machine learning principles have also been proposed in the literature to overcome some of these limitations. However, unlike conventional SPC methods, these non-parametric methods require event data from each out-of-control process state for effective model building. The paper presents a new non-parametric multivariate control chart based on kernel distance that overcomes these limitations by employing the notion of one-class classification based on support vector principles. The chart is non-parametric in that it makes no assumptions regarding the data probability density and only requires ‘normal’ or in-control data for effective representation of an in-control process. It does, however, make an explicit provision to incorporate any available data from out-of-control process states. Experimental evaluation on a variety of benchmarking datasets suggests that the proposed chart is effective for process monitoring.
Process Capability Indices (PCIs) are major tools in Geometric Dimensioning & Tolerancing (GD&T) for quantifying the production quality, monitoring production or prioritizing projects. Initially, PCIs were constructed for studying each characteristic of the process independently. Then, they have been extended to analyze several dependent characteristics simultaneously. Nowadays, with the increasing complexity of the production parts, for example in aircraft engines, the conformity of one part may rely on the conformity of hundreds of characteristics. Moreover, those characteristics being dependent, it may be misleading to make decisions only based on univariate PCIs. However, classical multivariate PCIs in the literature do not allow treating such amount of data efficiently, unless assuming Gaussian distribution, which is not always true. Regarding those issues, we advocate for PCIs based on some transformation of the conformity rates. This presents the advantage of being free from distributional assumptions, such as the Gaussian distribution. In addition, it has direct interpretation, allowing it to compare different processes. To estimate the PCIs of parts with hundreds of characteristics, we propose to use Vine Copulas. This is a very flexible class of models, which gives precise estimation even in high dimension. From an industrial perspective, the computation of the estimator can be costly. To answer this point, we explain how to compute a lower bound of the proposed PCI, which is faster to calculate. We illustrate our method adaptability with simulations under Gaussian and non-Gaussian distributions. We apply it to compare the production of Fan Blades of two different factories.
En las compañías manufactureras, es indispensable conocer la capacidad que tienen los procesos de cumplir especificaciones o metas relacionadas con la eficiencia operativa, ya sea al planear las condiciones de calidad en manufactura o al momento de evaluar la gestión a través de los sistemas de gestión integrados. En las décadas recientes, ha surgido el concepto de capacidad del proceso o desempeño del proceso, que proporciona una estimación cuantitativa de qué tan conforme es un proceso. Este trabajo ilustra una metodología para calcular un indicador de capacidad de proceso multivariado validado en una compañía productora de bebidas gaseosas, el cual resume el comportamiento del sistema de gestión integrado y orienta a los administradores de procesos a tomar decisiones estratégicas sobre el control y la mejora de los procesos con base en la identificación de variables claves de procesos pertenecientes a los diferentes sistemas de gestión, basados históricamente con valores variables analizadas de manera univariada, recurriendo a análisis densos y sin percepción de las correlaciones posibles entre los diferentes factores de los sistemas integrados de gestión de calidad. La metodología está basada en el análisis de la base de datos correspondiente a los resultados de los indicadores de gestión de los diferentes sistemas de calidad, obtenidos históricamente y almacenados en los sistemas de información de la compañía. Estos datos se trataron como variables aleatorias distribuidas normalmente y agrupadas matemáticamente como variables con comportamientos distribuidos con chi – cuadrado, estableciendo metas o valores nominales de resultados de los sistemas de calidad. De estos cálculos resultaron valores apropiados a un desarrollo estable del sistema de calidad, logrando disminuir la dispersión a través del cálculo del indicador de capacidad y reflejando la maduración del sistema integral de gestión.
We present a two-phase methodology based on the concept of depth to measure the capability of processes characterized by the functional relationship of multivariate nonlinear profile data, treated as multivariate functional observations. In the first phase, the modified tolerance region is estimated using a historical data set, while in the second, a current process is assessed using the proposed three-component vector, where the first component measures the volume ratio between the current process region and the modified tolerance region; the second measures the probability that the median of the current process is within the modified tolerance region, and the third measures the probability that the current process region is inside the modified tolerance region. To facilitate interpretation, a single index is derived from this capability vector. A simulation study is carried out to assess the performance of the proposed method. An real example illustrates the applicability of this approach.
In some complicated cases, a multivariate process/product can be characterized better by a combined technical specification vector. A combined technical specification vector is referred to as the case that the quality of a process/product is evaluated by different continuous and discrete random variables at the same time. In this case that there is also the covariance condition, the vector could not follow a single joint distribution. This research approaching kernel method and support vector data description (SVDD) proposes a new multivariate cumulative scheme named MK‐CUSUM to monitor the random vector of a process/product with combined technical specifications. The numerical analysis indicates that the proposed scheme is capable of detecting out‐of‐control conditions effectively when the process experiences different shifts in the random mean vector. The comparative performance report addresses that the proposed scheme is superior compared to the models of literature.
Distribution system operators that seek operational excellence are supposed to design and implement capable processes throughout their value stream and assess how well their real services match the design specifications. Process capability analysis is a critical stage in systematic and sophisticated quality engineering approaches such as six sigma, which are based on quantitative studies. The capability of a distribution network to maintain the power quality and voltage level within an acceptable range is one of the most important aspects of design and utilisation of electricity distribution network. Capability analysis has been successfully used in the previous studies to assess the quality of conformance in the power distribution network. However, these studies were limited to a voltage level and other aspects of quality such as total harmonic distortion were neglected. In addition, parametric methods were not available to overcome the problem of multimodality in voltage distribution, especially in the presence of distributed generation. In the current study, for the first time, a parametric approach based on decomposing complex distributions to a few Gaussian distributions by artificial bee colony optimisation were used for evaluating the capability of the distribution network to maintain voltage level and power quality indices in predefined ranges.
Conventional process capability indices (PCIs) are usually determined under the assumption that quality characteristics follow a normal distribution. If a process does not follow a normal distribution, however, conventional PCIs do not provide valid measures of process capability, especially in terms of the number of nonconforming parts. Methods to remedy the shortcoming of the conventional PCIs when the population is skewed can be divided into five categories; to use normalizing transformations like Box–Cox power transformation or Johnson transformation, to fit an empirical distribution or a known three- or four-parameter distribution such as Burr or Pearson distribution to the original data and use the quantiles of the fitted distribution, to modify the standard definition of PCIs in order to increase their robustness to skewness, to construct PCIs with estimate of the process yield, and to develop heuristic methods adequate for skewed distributions. This article provides a compact survey and brief comments on the skewed PCIs.
Keywords:
process capability index;
nonnormality;
skewed population;
normalizing transformation;
quantile
Conventional process capability indices (PCIs) are usually determined under the assumption that quality characteristics follow a normal distribution. If a process does not follow a normal distribution, however, conventional PCIs do not provide valid measures of process capability, especially in terms of the number of nonconforming parts. Methods to remedy the shortcoming of the conventional PCIs when the population is skewed can be divided into five categories; to use normalizing transformations like Box?Cox power transformation or Johnson transformation, to fit an empirical distribution or a known three- or four-parameter distribution such as Burr or Pearson distribution to the original data and use the quantiles of the fitted distribution, to modify the standard definition of PCIs in order to increase their robustness to skewness, to construct PCIs with estimate of the process yield, and to develop heuristic methods adequate for skewed distributions. This article provides a compact survey and brief comments on the skewed PCIs.
A multivariate exponentially weighted moving average (MEWMA) control chart is proposed for detecting process shifts during the phase II monitoring of simple linear profiles (SLPs) in the presence of within-profile autocorrelation. The proposed control chart is called MEWMA-SLP. Furthermore, two process capability indices are proposed for evaluating the capability of in-control SLP processes, and their utilization is demonstrated through examples. Intensive simulations reveal that the MEWMA-SLP chart is more sensitive than existing control charts in detecting profile shifts.
The capability of a manufacturing process refers to the ability of the process to produce items that are within set specifications. The determination of the capability of a stable manufacturing process with a multivariate quality characteristic using standard methods usually requires the assumption that the quality characteristic of interest follows a multivariate normal distribution, an assumption that is difficult to assess in practice. Departures from this assumption can result in erroneous conclusions, which can be costly to the manufacturer. In this paper, we propose assessing the capability of a process using a nonparametric Bayesian framework. This framework relies on using mixtures of Dirichlet processes to elicit a prior on the multivariate distribution of the quality characteristic. The posterior distribution of the distribution of the quality characteristic can then be used to induce a posterior distribution on the process fallout rate, the mean distance between the quality characteristic and a specified target value, or on an associated process capability index. The methodology is demonstrated through three examples using real-world data. Particular emphasis in these examples is given toward specifying the parameter values required to specify the prior distribution and on interpretation of the results.
These days Shewhart control chart for evaluating stability of the process is widely used in various field. But it must follow strict assumption of distribution. In real-life problems, this assumption is often violated when many quality characteristics follow non-normal distribution. Moreover, it is more serious in multivariate quality characteristics. To overcome this problem, many researchers have studied the non-parametric control charts. Recently, SVDD (Support Vector Data Description) control chart based on RBF (Radial Basis Function) Kernel, which is called K-chart, determines description of data region on in-control process and is used in various field. But it is important to select kernel parameter or etc. in order to apply the K-chart and they must be predetermined. For this, many researchers use grid search for optimizing parameters. But it has some problems such as selecting search range, calculating cost and time, etc. In this paper, we research the efficiency of selecting parameter regions as data structure vary via simulation study and propose a new method for determining parameters so that it can be easily used and discuss a robust choice of parameters for various data structures. In addition, we apply it on the real example and evaluate its performance.
The capability of a manufacturing process is a measure of the ability of the process to consistently produce items within engineering specifications. Most early attempts to assess the capability of a process were based on the assumption that the quality characteristic of interest follows a parametric distribution, usually normal. Several studies of process data indicate that many quality characteristics follow nonnormal distributions.Further, theoretical and empirical studies of measures of process capability indicate that violations of the assumption relating to the distribution of the quality characteristic can produce serious errors in conclusions about the capability of the process. This article reviews methods that have been developed to assess process capability within a nonparametric framework. Such methods require few specific assumptions about the distribution of the quality characteristic. The methods reviewed in this article include those based on nonparametric tolerance limits, nonparametric estimates of the process yield, and indices based on the percentiles of the process distribution.Keywords:extreme value theory;kernel estimate;natural tolerance;nonparametric;process capability;process fallout rate;process yield
Process capability indices measure the ability of a process to provide products that meet certain specifications. Few references deal with the capability of a process characterized by a functional relationship between a response variable and one or more explanatory variables, which is called profile. Specifically, there is not any reference analysing the capability of processes characterized by multivariate nonlinear profiles. In this paper, we propose a method to measure the capability of these processes, based on principal components for multivariate functional data and the concept of functional depth. A simulation study is conducted to assess the performance of the proposed method. An example from the sugar production illustrates the applicability of this approach.
Standard measurement-system analysis criteria assume the gauge measures a single variable. In automotive-body manufacturing, measurement systems take data for many quality characteristics, yet manufacturers evaluate each response independently. To support using these data as a multivariate response, this paper develops multivariate extensions of gauge-approval criteria precision to tolerance ratio, percent R&R, and signal-to-noise ratio. These criteria use the volume of constant-density contours to characterize variability, the role of the standard deviation in single-variable models. This paper contains a MANOVA method using expected mean squares for estimating the variance-covariance matrices for one-factor, two-factor, and three-factor gauge studies. The paper demonstrates how to fit the MANOVA model and estimate the multivariate criteria using automotive body panel gauge-study data.
Quantifying the "capability" of a manufacturing process is an important
initial step in any quality improvement program. Capability is usually defined
in dictionaries as "the ability to carry out a task, to achieve an objective".
Process capability indices(PCIs) is defined as a combination of materials,
methods, equipments and people engaged in producing a measurable output. PCIs
which establish the relationships between the actual process performance and
the manufacturing specifications, have been a focus of research in quality
assurance and process capability analysis. Capability indices that qualify
process potential and process performance are practical tools for successful
quality improvement activities and quality program implementation. As a matter
of fact, all processes have inherent statistical variability, which can be
identified, evaluated and reduced by statistical methods. Generalized Process
Capability Index, defined as the ratio of proportion of specification
conformance (or, process yield) to proportion of desired (or, natural)
conformance. We review the process capability indices in case of normal,
non-normal, discrete and multivariate process distributions and discuss the
inferential aspects of some of these process capability indices. Relations
among the process capability indices have also been illustrated with examples.
Finally we also consider the process capability indices using conditional
ordering and transforming multivariate data to univariate one using the concept
of structural function.
Most multivariate process capability indices proposed by other researchers provide quality measure in losses due to variability and being off-target. Some suggested loss function models that estimate losses due to rejection and quality costs but failed to consider the correlation between the multivariate quality characteristics. In this paper, we propose a novel new approach for estimating the total expected quality cost that provides quality cost measure not only in losses due to variability and being off-target but also losses due to rejection. The proposed model also estimates the quality costs for correlated quality characteristics with nominal specifications and provides the estimated costs due to individual characteristics; it helps to identify those process quality characteristics which have high variation when compared with their specification spread. The model will also improve the estimate of the rejection cost suggested by earlier researchers as it identifies the reject region as the region outside of the modified tolerance region. We present four different correlated bivariate normal process scenarios where the proposed method can be applicable and demonstrate that our approach provides a robust tool in estimating the risk of the product being rejected and performs as well as existing methods in estimating the total quality cost.
ultivariate process capability indices (MPCI), as an important means of statistical process control (SPC), can be used to ensure the high reliability of semiconductors manufacturing process. However, the reasonable sampling number is an important factor when considering MPCI values. As general, the large sample number requires much effort and time, or even cannot be achieved. In this paper, we evaluated the impact of different sample size on the calculations of multivariate process capability indices using simulation and analyses. After getting enough data and choosing disparate sample numbers, corresponding multivariate process capability indices can be obtained, which demonstrate the relationship between sampling numbers and calculation results. The conclusions have critical guiding significance for manufacturing semiconductors with high reliability requirement.
In some quality control applications, quality of a product or a process can be characterized by a profile defined as a functional relationship between a response variable and one or more explanatory variables. Many researchers have contributed to the development of linear and nonlinear profiles to monitor a process or product. However, less work has been devoted to the development of process capability indices in profile monitoring to evaluate process performance with respect to specification limits. This paper presents a process capability analysis when the quality characteristic of interest is represented by a linear profile. Simulation analyses along with a real case study in leather industry are used to evaluate the performance of the proposed method. Results indicate satisfactory performance.
This paper proposes multivariate process capability indices (PCIs) for skewed populations using rand modified process region approaches. The proposed methods are based on the multivariate version of a weighted standard deviation method which adjusts the variance-covariance matrix of quality characteristics and approximates the probability density function using several multivariate Journal distributions with the adjusted variance-covariance matrix. Performance of the proposed PCIs is investigated using Monte Carlo simulation, and finite sample properties of the estimators are studied by means of relative bias and mean square error.
Process capability indices (PCIs) are used in statistical process control to evaluate the capability of the processes in satisfying the customer's needs. In the past two decades varieties of PCI are introduced by researchers to analyze the process capability with univariate or multivariate quality characteristics. To the best of our knowledge, most famous multivariate capability indices are proposed when the quality characteristics have both upper and lower specification limits. These indices are incapable to assess the multivariate processes capability with unilateral specification. In this article, we propose a new multivariate PCI to analyze the processes with one or more unilateral specification limits. This new index also accounts for all problems in the best PCIs of the literature. The performance of the proposed index is evaluated by real cases under different situations. The results show that the proposed index performs satisfactorily in all cases considered. Copyright © 2012 John Wiley & Sons, Ltd.
Folded normal distribution arises when we try to find out the distribution of absolute values of a function of a normal variable. The properties and uses of univariate and bivariate folded normal distribution have been studied by various researchers. We study here the properties of multivariate folded normal distribution and indicate some areas of applications.
With the advent of modern technology, manufacturing processes have become very sophisticated; a single quality characteristic can no longer reflect a product's quality. In order to establish performance measures for evaluating the capability of a multivariate manufacturing process, several new multivariate capability (NMC) indices, such as NMCp and NMCpm, have been developed over the past few years. However, the sample size determination for multivariate process capability indices has not been thoroughly considered in previous studies. Generally, the larger the sample size, the more accurate an estimation will be. However, too large a sample size may result in excessive costs. Hence, the trade-off between sample size and precision in estimation is a critical issue. In this paper, the lower confidence limits of NMCp and NMCpm indices are used to determine the appropriate sample size. Moreover, a procedure for conducting the multivariate process capability study is provided. Finally, two numerical examples are given to demonstrate that the proper determination of sample size for multivariate process indices can achieve a good balance between sampling costs and estimation precision.
In the chapter, the term “manufacturing process” refers to the set of relevant engineering processes together with all the other factors—such as raw material and manufacturing environment—that enter the process of translating product design into the finished product. The interaction between the product design and the manufacturing process cannot be ignored. These issues highlight the challenge of quantifying the capability of industrial processes. This chapter provides an overview of the progress that has been made so far in meeting this challenge. As is frequently the case in scientific investigations, the first attempts at quantifying the capability of industrial processes were made in the context of a highly simplified model—the model of statistical control in one dimension with the univariate normal distribution lurking in the background. A large part of the process capability literature is concerned with this model and/or its variations and extensions, and this literature has found useful applications in industry. The chapter provides an overview of the literature from the point of view of applications and/or theoretical developments. It includes estimation and testing for the various process control indices. The chapter considers the Bayesian approach to quantifying process capability. The remarkable progress in Markov chain Monte Carlo techniques has opened new vistas for the application of the Bayesian paradigm.
This paper focuses on the monitoring techniques applied in multivariate processes when the underlying distribution of the quality characteristics departs from normality. For most conventional control charts, such as Hotelling's T 2 charts, the design of the control limits is commonly based on the assumption that the quality characteristics follow a multivariate normal distribution. However, this may not be reasonable in many real-world problems. This paper addresses this issue and proposes a monitoring approach motivated by statistical learning theory, which has been applied successfully in the field of pattern recognition. The developed multivariate control chart is based on the kernel distance, which is a measure of the distance between the 'kernel centre' and the incoming new sample to be monitored. The kernel distance can be calculated using support vector methods. This chart makes use of information extracted from in-control preliminary samples. A case study demonstrates that the kernel-distance-based chart can perform better than conventional charts when the underlying distribution of the quality characteristics is not multivariate normal.
In this paper a comparison between Mingoti and Glória's (2003) and Niverthi and Dey's (2000) multivariate capability indexes is presented. Monte Carlo simulation is used for the comparison and some confidence intervals were generated for the true capability index by using bootstrap methodology.Neste artigo é apresentada uma comparação entre os índices de capacidade multivariados de Mingoti e Glória (2003) e Niverthi e Dey (2000). O método de simulação de Monte Carlo é utilizado na comparação, e intervalos de confiança para o verdadeiro valor do índice de capacidade do processo são construídos através da metodologia Bootstrap.
We analyze, from both theoretical and practical point of view, the use of the smoothed bootstrap in the estimation of a functional T(f) of the underlying density. We consider a plug-in approach based on the use of an estimator of type T(fˆ n ) where fˆ n is a nonparametric (kernel) estimator of f. First, we obtain a result of asymptotic validity for the smoothed bootstrap in this case. Second, we present a simulation study analyzing the performance of the smoothed bootstrap versus the ordinary non-smoothed version of this method.
Process Capability Indices, such as Cp and Cpk have been widely used as statistical tools to assess the manufacturing process performance. These indices provide numerical measures on process precision, process accuracy and process performance. Extensive researches have been done on the univariate process capability indices in the last two decades. However, process capability indices for processes with multiple quality characteristics have received little attention, comparatively. The multivariate process capability indices, which are used for evaluation of processes with correlated quality characteristics such as weight, height and width, could be investigated, in depth. In this research the concept of process capability and its relevant indices in univariate and multivariate cases are discussed. Based on the previous works, a new multivariate process capability vector (NMPCV) is introduced. This vector is based on a modification of the traditional multivariate process capability indices. The value and the power of this new index are evaluated using field and simulated data. The results of this research show that NMPCV is a better tool for judging the process accuracy, the process precision and the process performance. When the characteristics are highly correlated, the results are much better.
Probability bounds can be derived for distributions whose covariance matrices are ordered with respect to Löwner partial ordering, a relation that is based on whether the difference between two matrices is positive definite. One example is Anderson’s Theorem. This paper develops a probability bound that follows from Anderson’s Theorem that is useful in the assessment of multivariate process capability. A statistical hypothesis test is also derived that allows one to test the null hypothesis that a given process is capable versus the alternative hypothesis that it is not capable on the basis of a sample of observed quality characteristic vectors from the process. It is argued that the proposed methodology is viable outside the multivariate normal model, where the p-value for the test can be computed using the bootstrap. The methods are demonstrated using example data, and the performance of the bootstrap approach is studied empirically using computer simulations.
The main usefulness of a capability index is to relate the actual variability of the process with the admissible one. This admissible variability is, in turn, related with the nonconforming proportion. Hence, the capability index should be closely related to the nonconforming proportion. In univariate and centered processes, the classical C
p
index explicitly admits this interpretation. For instance, if C
p
= 0.5, the standard deviation should be reduced to 50% to attain C
p
= 1. However, for noncentered processes and multivariate processes, there is a lack of capability indices that admit such an interpretation. This article fills this gap in the literature and proposes univariate and multivariate capability indices that have a direct interpretation of how much the variability of the process should increase or decrease to attain a unitary index. Some numerical examples are used to compare the proposed indices with the existing ones, showing the advantages of the proposals.
Traditional multivariate quality control charts assume that quality characteristics follow a multivariate normal distribution. However, in many industrial applications the process distribution is not known, implying the need to construct a flexible control chart appropriate for real applications. A promising approach is to use support vector machines in statistical process control. This paper focuses on the application of the ‘kernel-distance-based multivariate control chart’, also known as the ‘k-chart’, to a real industrial process, and its assessment by comparing it to Hotelling’s T2 control chart, based on the number of out-of-control observations and on the Average Run Length. The industrial application showed that the k-chart is sensitive to small shifts in mean vector and outperforms the T2 control chart in terms of Average Run Length.
A measure of process capability for the multivariate normal case is proposed. This measure takes into account both proximity to the target and the variation observed in the process. The result is analogous to the univariate measure of process capability referred to as Cpm. Some statistical properties associated with the measure are examined. Multivariate specification limits and their creation are also discussed.
Consider the problem of estimating θ=θ(P) based on datax
n
from an unknown distributionP. Given a family of estimatorsT
n, β
of θ(P), the goal is to choose β among β∈I so that the resulting estimator is as good as possible. Typically, β can be regarded as a tuning or smoothing parameter, and proper choice of β is essential for good performance ofT
n, β
. In this paper, we discuss the theory of β being chosen by the bootstrap. Specifically, the bootstrap estimate of β,\(\hat \beta _n\), is chosen to minimize an empirical bootstrap estimate of risk. A general theory is presented to establish the consistency and weak convergence properties of these estimators. Confidence intervals for θ(P) based on\(T_{n,\hat \beta _n }\), are also asymptotically valid. Several applications of the theory are presented, including optimal choice of trimming proportion, bandwidth selection in density estimation and optimal combinations of estimates.
Exact analytic expressions for the bootstrap mean and variance of any L-estimator are obtained, thus eliminating the error due to bootstrap resampling. The expressions follow from the direct calculation of the bootstrap mean vector and covariance matrix of the whole set of order statistics. By using these expressions, recommendations can be made about the appropriateness of bootstrap estimation under given conditions.
For interval estimation of a proportion, coverage probabilities tend to be too large for "exact" confidence intervals based on inverting the binomial test and too small for the interval based on inverting the Wald large-sample normal test (i.e., sample proportion ± z-score × estimated standard error). Wilson's suggestion of inverting the related score test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes. The 95% score interval has similar behavior as the adjusted Wald interval obtained after adding two "successes" and two "failures" to the sample. In elementary courses, with the score and adjusted Wald methods it is unnecessary to provide students with awkward sample size guidelines.
We propose a multivariate process capability index (PCI) over a general tolerance zone which includes ellipsoidal and rectangular solid ones as special cases. Our multivariate PCI appears to be a natural generalization of the PCI C p for a univariate process to a multivariate process. Computing aspects of the proposed multivariate PCI are discussed in detail, especially for a bivariate normal process. It is noted that its distributional and inferential aspects are difficult to deal with. Resampling methods and a Monte Carlo procedure are suggested to overcome this difficulty. Some examples with a set of real data are presented to illustrate and examine the proposed multivariate PCI.
The question of smoothing when using the non-parametric version of the bootstrap for estimation of population functionals is reconsidered. In general, there is no global preference for procedures based on a smoothed version of the empirical distribution rather than the empirical distribution itself. In the majority of problems smoothing influences only the second order properties of the estimator, while requiring greater computation and choice of a suitable amount of smoothing. There are problems, however, where smoothing may affect the rate of convergence of the estimator. We discuss an example of such a problem and consider issues relating to empirical choice of whether to smooth, and by how much. A procedure based on the bootstrap for choice of bandwidth is suggested and illustrated.
Quality characteristics analyzed in statistical process control (SPC) often are required to be normally distributed. This is true in many types of control charts and acceptance sampling plans, as well as in process capability studies. If a characteristic is not normally distributed, but normal-based techniques are used, serious errors can result. One approach to solving this problem is to transform the non-normal data to normality using the Johnson system of distributions. In this paper, we use the sample quantile ratio, in conjunction with the Shapiro-Wilk test of normality, to find a suitable transformation for non-normal data. Examples of fitting non-normal SPC data are presented and discussed. The effect of the Johnson transformation on an SPC procedure involving an estimator for the population standard deviation is studied using non-normal data and Johnson-transformed data.
The determination of the capability of a stable process using the standard process capability indices requires that the quality characteristic of interest be normally distributed. Departures from normality can result in erroneous conclusions when using these indices. In this paper we propose assessing the capability of a process using a nonparametric estimator. This estimator is based on kernel estimation of the distribution function. Bandwidth selection for this method can be based either on a normal reference distribution or on a nonparametric estimate. An example is presented that applies this proposed method to non-normal process data. The performance of the resulting estimator is then compared with the sample proportion and a normal-based estimate in a simulation study. © 1998 John Wiley & Sons, Ltd.
It is widely believed that the number of resamples required for bootstrap variance estimation is relatively small An argument based on the unconditional coefficient of variation of the Monte Carlo approximation, suggests that as few as 25 resamples will give reasonable results. In this article we argue that the number of resamples should, in fact, be determined by the conditional coefficient of variation, involving only resampling variability. Our conditional analysis is founded on a belief that Monte Carlo error should not be allowed to determine the conclusions of a statistical analysis and indicates that approximately 800 resamples are required for this purpose. The argument can be generalized to the multivariate setting and a simple formula is given for determining a lower bound on the number of resamples required to approximate an m-dimensional bootstrap variance-covariance matrix.
Maximum likelihood (ML) and minimum variance unbiased (MVU) estimators of the proportion nonconforming in univariate and bivariate normal random samples are compared for the case where the moments of the distribution are assumed to be unknown and each variable has lower and upper specification limits. Both types of estimator have skewed distributions when the proportion nonconforming and sample size are small, and the MVU estimator has a substantial probability of being zero in these situations. Using Pitman's closeness criterion, the ML estimators are nearly always superior to the MVU estimator for the cases considered. Using the MSE criterion, the MVU estimator is superior to the ML estimator when the distribution of the ML estimator is quite skewed. After transforming the estimators to symmetry, the ML estimator has smaller MSE than the MVU estimator.
Classical evaluation of process capability indices (PCI) supposes that the process is normally distributed with mean μ and standard deviation σ. If the process is not normal, then this evaluation may yield strange results. In order to evaluate non-normal process capability indices, we propose to use relations between process capability indices and the proportion of nonconforming items and to estimate this proportion using a fitting method based on Burr's distributions.
Truncated data is a common occurrence in many industrial processes. This is particularly true in the situation of a supplier that has several customers, each with different specifications on the same product. Each customer will typically only observe a ..
In this article, we present a method for establishing an upper bound on the fraction nonconforming estimator for any unimodally distributed process. This bound can be used as a worst-case estimate of process capability. Our method does not use curve-fitting techniques or assume normality of the process distribution. Hence, the technique is more intuitive and simpler and because normality is not required, it may be applied with confidence on a much larger class of distributions. The fraction nonconforming bounds established using this technique can be combined with parametric estimates to provide a sequence of upper bounds on the fraction nonconforming and hence characterize the capability of the process.
In this paper we examine the behaviour of bivariate generalizations of the process capability indices Cp, CPU and CPL. It is shown that when the characteristics of interest have a bivariate normal distribution the distribution of is related to the bivariate χ distribution and that of is related to the bivariate noncentral t distribution. Since the process depends on two variables, it seems reasonable to require that BOTH variables conform with the definition of capability; that is, the process will be considered not capable if either of the indices is below a prescribed value. Tests of hypotheses concerning this requirement on the indices, CP1 and CP2 , are developed. Critical points, based on the bivariate χ distribution, for testing these hypotheses are tabulated.
We propose a widely applicable method for choosing the smoothing parameters for nonparametric density estimators. It has come to be realized in recent years (e.g., see Hall and Marron 1987; Scott and Terrell 1987) that cross-validation methods for finding reasonable smoothing parameters from raw data are of very limited practical value. Their sampling variability is simply too large. The alternative discussed here, the maximal smoothing principle, suggests that we consider using the most smoothing that is consistent with the estimated scale of our data. This greatly generalizes and exploits a phenomenon noted in Terrell and Scott (1985), that measures of scale tend to place upper bounds on the smoothing parameters that minimize asymptotic mean integrated squared error of density estimates such as histograms and frequency polygons. The method avoids the extreme sampling variability of cross-validation by using ordinary scale estimators such as the standard deviation and interquartile range, which have order n −1 variability; cross-validated parameters have orders of variability such as n −1/5. The disadvantage is that maximal smoothing parameters are conservative, rather than asymptotically optimal. Because they tend to lose information, they should be used in conjunction with other data displays that retain more of the features of the original sample. On the other hand, such conservative methods are widely valued by statisticians because they discourage naive overinterpretation of one's data. Maximal smoothing parameters are here derived for histograms and kernel methods, using not only the standard deviation but several more resistant methods of scale estimation. The method is then applied to density estimation on the half-line, on finite intervals, and in several variables.
The basic kernel density estimator in one dimension has a single smoothing parameter, usually referred to as the bandwidth. For higher dimensions, however, there are several options for smoothing parameterization of the kernel estimator. For the bivariate case, there can be between one and three independent smoothing parameters in the estimator, which leads to a flexibility versus complexity trade-off when using this estimator in practice. In this article the performances of the different possible smoothing parameterizations are compared, using both the asymptotic and exact mean integrated squared error. Our results show that it is important to have independent smoothing parameters for each of the coordinate directions. Although this is enough for many situations, for densities with high amounts of curvature in directions different to those of the coordinate axes, substantial gains can be made by allowing the kernel mass to have arbitrary orientations. The "sphering" approaches to choosing this orientation are shown to be detrimental in general, however.
This chapter deals with statistical methods that, in some way, avoid mathematical difficulties that one would be facing using traditional approaches. The traditional approach of mathematical statistics is based on analytic expressions, or formulas, so avoiding these might seem itself a formidable task, especially in view of the chapters that so far have been covered. It should be pointed out that we have no objection of using mathematical formulas—in fact, some of these are pleasant to use. However, in many cases, such formulas are simply not available or too complicated to use.
We develop a unified framework within which many commonly used bootstrap critical points and confidence intervals may be discussed and compared. In all, seven different bootstrap methods are examined, each being usable in both parametric and nonparametric contexts. Emphasis is on the way in which the methods cope with first- and second-order departures from normality. Percentile-$t$ and accelerated bias-correction emerge as the most promising of existing techniques. Certain other methods are shown to lead to serious errors in coverage and position of critical point. An alternative approach, based on "shortest" bootstrap confidence intervals, is developed. We also make several more technical contributions. In particular, we confirm Efron's conjecture that accelerated bias-correction is second-order correct in a variety of multivariate circumstances, and give a simple interpretation of the acceleration constant.
We discuss the following problem given a random sample X = (X
1, X
2,…, X
n) from an unknown probability distribution F, estimate the sampling distribution of some prespecified random variable R(X, F), on the basis of the observed data x. (Standard jackknife theory gives an approximate mean and variance in the case R(X, F) = \(\theta \left( {\hat F} \right) - \theta \left( F \right)\), θ some parameter of interest.) A general method, called the “bootstrap”, is introduced, and shown to work satisfactorily on a variety of estimation problems. The jackknife is shown to be a linear approximation method for the bootstrap. The exposition proceeds by a series of examples: variance of the sample median, error rates in a linear discriminant analysis, ratio estimation, estimating regression parameters, etc.
Fitting SPC Data Using a Sample Quantile Ratio
- Y.-M Chou
- A M Polansky
Chou, Y.-M., and Polansky, A. M. (1996), " Fitting SPC Data Using a Sample Quantile Ratio, " in Proceedings of the Section on Quality and Productivity, American Statistical Association, pp. 9–16.
Process Capability Calculations for Non-normal Dis-tributions
- J A Clements
Clements, J. A. (1989), " Process Capability Calculations for Non-normal Dis-tributions, " Quality Progress, 22, 95–100.
Practical Nonparametric Statistics Small Samples and Non-normal Capability
- W J Conover
- Wiley
- G K Constable
- J R Hobbs
Conover, W. J. (1980), Practical Nonparametric Statistics (2nd ed.), New York: Wiley. Constable, G. K., and Hobbs, J. R. (1992), " Small Samples and Non-normal Capability, " in ASQC Technical Conference Transactions, Milwaukee, W I: American Society for Quality Control, pp. 37–43.
Local Polynomial Modelling and its Applica-tions Using Johnson Curves to Describe Non-normal Pro-cess Data
- J Fan
- I Gijbels
Fan, J., and Gijbels, I. (1996), Local Polynomial Modelling and its Applica-tions, London: Chapman and Hall. Farnum, N. R. (1996), " Using Johnson Curves to Describe Non-normal Pro-cess Data, " Quality Engineering, 9, 329–336.
Joint Con dence Regions for Correlated Multidimensional Environmental Control Variables
- G H Otto
- D W Scott
Otto, G. H., and Scott, D. W. (1996), " Joint Con dence Regions for Correlated Multidimensional Environmental Control Variables, " in Proceedings of the Section on Statistics and the Environment, American Statistical Association, pp. 71–76.