Multivariate process capability using principal component analysis in the presence of measurement errors

Various different definitions of multivariate process capability indices have been proposed in the literature. Most of the research works related to multivariate process capability indices assume no gauge measurement errors. However, in industrial applications, despite the use of highly advanced measuring instruments, account needs to be taken of gauge imprecision. In this paper we are going to examine the effects of measurement errors on multivariate process capability indices computed using the principal components analysis. We show that measurement errors alter the results of a multivariate process capability analysis, resulting in either a decrease or an increase in the capability of the process. In order to achieve accurate process capability assessments, we propose a method useful for overcoming the effects of gauge measurement errors.     

Efficient $$\hbox {SMC}^2$$ schemes for stochastic kinetic models

Fitting stochastic kinetic models represented by Markov jump processes within the Bayesian paradigm is complicated by the intractability of the observed-data likelihood. There has therefore been considerable attention given to the design of pseudo-marginal Markov chain Monte Carlo algorithms for such models. However, these methods are typically computationally intensive, often require careful tuning and must be restarted from scratch upon receipt of new observations. Sequential Monte Carlo (SMC) methods on the other hand aim to efficiently reuse posterior samples at each time point. Despite their appeal, applying SMC schemes in scenarios with both dynamic states and static parameters is made difficult by the problem of particle degeneracy. A principled approach for overcoming this problem is to move each parameter particle through a Metropolis-Hastings kernel that leaves the target invariant. This rejuvenation step is key to a recently proposed \(\hbox {SMC}^2\) algorithm, which can be seen as the pseudo-marginal analogue of an idealised scheme known as iterated batch importance sampling. Computing the parameter weights in \(\hbox {SMC}^2\) requires running a particle filter over dynamic states to unbiasedly estimate the intractable observed-data likelihood up to the current time point. In this paper, we propose to use an auxiliary particle filter inside the \(\hbox {SMC}^2\) scheme. Our method uses two recently proposed constructs for sampling conditioned jump processes, and we find that the resulting inference schemes typically require fewer state particles than when using a simple bootstrap filter. Using two applications, we compare the performance of the proposed approach with various competing methods, including two global MCMC schemes.     

Estimating total and specific process capability indices in three-stage processes with measurement errors

In multistage processes, two kinds of process capability indices (specific and total process capability indices) are defined in each stage. The total process capability index calculates the capability of each stage when it is affected by the previous stages and the specific process capability index calculates the capability of the stage when the effects of the previous stages are eliminated. Process capability indices in multistage processes are proposed under the assumption of no measurement errors. However, sometimes this assumption may be violated and leads to misleading interpretations. In this paper, the effects of measurement errors on the specific and total process capability indices in the second and third stages of the three-stage processes are statistically analysed. In addition, the effects of the measurement errors on the bias and the mean square error value of the total and specific process capability indices in the second and third stages are studied. Finally, the effects of the measurement errors on the specific and total process capability indices are shown through a numerical example.     

Evaluation of process capability in multivariate nonlinear profiles

ABSTRACT

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.     

Statistical analysis of process capability indices with measurement errors: The case ofC p

Process capability indices (PCIs) have been widely used in manufacturing industries to previde a quantitative measure of process potential and performance. While some efforts have been dedicated in the literature to the statistical properties of PCIs estimators, scarce attention has been given to the evaluation of these properties when sample data are affected by measurement errors. In this work we deal with the problem of measurement errors effects on the performance of PCIs. The analysis is illustrated with reference toC _p , i.e. the simplest and most common measure suggested to evaluate process capability. The authors would like to thank two anonymous referees for their comments and suggestion that were useful in the preparation and improvement of this paper. This work was partially supported by a MURST research grant.     

Multivariate Capability Indices: Distributional and Inferential Properties

Process capability indices have been widely used in the manufacturing industry for measuring process reproduction capability according to manufacturing specifications. Properties of the univariate processes have been investigated extensively, but are comparatively neglected for multivariate processes where multiple dependent characteristics are involved in quality measurement. In this paper, we consider two commonly used multivariate capability indices MC_p and MC_pm, to evaluate multivariate process capability. We investigate the statistical properties of the estimated MC_p and obtain the lower confidence bound for MC_p. We also consider testing MC_p, and provide critical values for testing if a multivariate process meets the preset capability requirement. In addition, an approximate confidence interval for MC_pm is derived. A simulation study is conducted to ascertain the accuracy of the approximation. Three examples are presented to illustrate the applicability of the obtained results.     

Analysis of cointegrated models with measurement errors

We study the asymptotic properties of the reduced-rank estimator of error correction models of vector processes observed with measurement errors. Although it is well known that there is no asymptotic measurement error bias when predictor variables are integrated processes in regression models [Phillips BCB, Durlauf SN. Multiple time series regression with integrated processes. Rev Econom Stud. 1986;53:473–495], we systematically investigate the effects of the measurement errors (in the dependent variables as well as in the predictor variables) on the estimation of not only cointegrating vectors but also the speed of the adjustment matrix. Furthermore, we present the asymptotic properties of the estimators. We also obtain the asymptotic distribution of the likelihood ratio test for the cointegrating ranks. We investigate the effects of the measurement errors on estimation and test through a Monte Carlo simulation study.     

Skewness Reduction Approach in Multi-Attribute Process Monitoring

Since the product quality of many industrial processes depends upon more than one dependent variable or attribute, they are either multivariate or multi-attribute in nature. Although multivariate statistical process control is receiving increased attention in the literature, little work has been done to deal with multi-attribute processes. In this article, we develop a new methodology to monitor multi-attribute processes. To do this, first we transform multi-attribute data in a way that their marginal probability distributions have almost zero skewness. Then, we estimate the transformed covariance matrix and apply the well-known T ² control chart. In order to illustrate the proposed method and evaluate its performance, we use two simulation experiments and compare the results with the ones from both MNP chart and the χ² control chart.     

Inference on C pk for autocorrelated data in the presence of random measurement errors

The present paper examines the properties of the C _pk estimator when observations are autocorrelated and affected by measurement errors. The underlying reason for this choice of subject matter is that in industrial applications, process data are often autocorrelated, especially when sampling frequency is not particularly low, and even with the most advanced measuring instruments, gauge imprecision needs to be taken into consideration. In the case of a first-order stationary autoregressive process, we compare the statistical properties of the estimator in the error case with those of the estimator in the error-free case. Results indicate that the presence of gauge measurement errors leads the estimator to behave differently depending on the entity of error variability.     

A simulation study and evaluation of multivariate forecast based control charts applied to ARMA processes

Much research had been performed in the area of control charting techniques for monitoring autocorrelated processes, especially regarding forecast based monitoring schemes. Forecast based monitoring schemes involve fitting an appropriate time-series model to the process, generating one step ahead forecast errors, and monitoring the forecast errors with traditional control charts. Another method introduced into the literature involves using multivariate control charts to monitor the ARMA derived one-step-ahead (OSA) and two-step-ahead (TSA) forecast errors. This article provides a broad simulation study and evaluation of the suggested multivariate approaches in regards to various ARMA(1,1) and AR(1) processes, and a comparison to their univariate counterparts.     

Multivariate nonparametric methods in two-way balanced designs: performances and limitations in small samples

Investigations of multivariate population are pretty common in applied researches, and the two-way crossed factorial design is a common design used at the exploratory phase in industrial applications. When assumptions such as multivariate normality and covariance homogeneity are violated, the conventional wisdom is to resort to nonparametric tests for hypotheses testing. In this paper we compare the performances, and in particular the power, of some nonparametric and semi-parametric methods that have been developed in recent years. Specifically, we examined resampling methods and robust versions of classical multivariate analysis of variance (MANOVA) tests. In a simulation study, we generate data sets with different configurations of factor''s effect, number of replicates, number of response variables under null hypothesis, and number of response variables under alternative hypothesis. The objective is to elicit practical advice and guides to practitioners regarding the sensitivity of the tests in the various configurations, the tradeoff between power and type I error, the strategic impact of increasing number of response variables, and the favourable performance of one test when the alternative is sparse. A real case study from an industrial engineering experiment in thermoformed packaging production is used to compare and illustrate the application of the various methods.     

The non parametric regression estimate with dependent measurement errors

ABSTRACT

Non parametric regression estimation with measurement errors data has received great attention, and deconvolution local polynomial estimators can be used to deal with the problem that the errors are independent of other variables in the literature. In this article, the copula method is applied to tackle the case that the errors may depend on covariates, and the asymptotic properties of the resulting estimators are derived. Two simulations are conducted to illustrate the performance of the proposed estimators.     

Control chart for monitoring multivariate COM-Poisson attributes

Statistical process control of multi-attribute count data has received much attention with modern data-acquisition equipment and online computers. The multivariate Poisson distribution is often used to monitor multivariate attributes count data. However, little work has been done so far on under- or over-dispersed multivariate count data, which is common in many industrial processes, with positive or negative correlation. In this study, a Shewhart-type multivariate control chart is constructed to monitor such kind of data, namely the multivariate COM-Poisson (MCP) chart, based on the MCP distribution. The performance of the MCP chart is evaluated by the average run length in simulation. The proposed chart generalizes some existing multivariate attribute charts as its special cases. A real-life bivariate process and a simulated trivariate Poisson process are used to illustrate the application of the MCP chart.     

Testing process capability based on Cpm in the presence of random measurement errors

Process capability indices have been widely used in the manufacturing industry providing numerical measures on process performance. The index C_p provides measures on process precision (or product consistency). The index C_pm, sometimes called the Taguchi index, meditates on process centring ability and process loss. Most research work related to C_p and C_pm assumes no gauge measurement errors. This assumption insufficiently reflects real situations even with highly advanced measuring instruments. Conclusions drawn from process capability analysis are therefore unreliable and misleading. In this paper, we conduct sensitivity investigation on process capability C_p and C_pm in the presence of gauge measurement errors. Due to the randomness of variations in the data, we consider capability testing for C_p and C_pm to obtain lower confidence bounds and critical values for true process capability when gauge measurement errors are unavoidable. The results show that the estimator with sample data contaminated by the measurement errors severely underestimates the true capability, resulting in imperceptible smaller test power. To obtain the true process capability, adjusted confidence bounds and critical values are presented to practitioners for their factory applications.     

Consistent estimation of regression coefficients in ultrastructural measurement error model using stochastic prior information

The problem of consistent estimation of regression coefficients in a multivariate linear ultrastructural measurement error model is considered in this article when some additional information on regression coefficients is available a priori. Such additional information is expressible in the form of stochastic linear restrictions. Utilizing stochastic restrictions given a priori, some methodologies are presented to obtain the consistent estimators of regression coefficients under two types of additional information separately, viz., covariance matrix of measurement errors and reliability matrix associated with explanatory variables. The measurement errors are assumed to be not necessarily normally distributed. The asymptotic properties of the proposed estimators are derived and analyzed analytically as well as numerically through a Monte Carlo simulation experiment.     

Sample size determination for estimating multivariate process capability indices based on lower confidence limits

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 NMC _p and NMC _pm , 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 NMC _p and NMC _pm 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.     

CAPABILITY MEASURES FOR m-DEPENDENT STATIONARY PROCESSES

Process capability indices, providing numerical measures on process potential and process performance, have received substantial research attention. Most research assumes that the process is normally distributed and the process data are independent. In real-world applications such as chemical, soft drinks, or tobacco/cigaratte manufacturing processes, process data are often auto-correlated. In this paper, we consider the capability indices Cp, Cpk, Cpm, Cpmk for strictly m-dependent stationary processes. We investigate the statistical properties of their natural estimators. We derive the asymptotic distributions, and establish confidence intervals so that capability testing can be performed.     

CUSUM control schemes for monitoring the covariance matrix of multivariate time series

Modified cumulative sum (CUSUM) control charts and CUSUM schemes for residuals are suggested to detect changes in the covariance matrix of multivariate time series. Several properties of these schemes are derived when the in-control process is a stationary Gaussian process. A Monte Carlo study reveals that the proposed approaches show similar or even better performance than the schemes based on the multivariate exponentially weighted moving average (MEWMA) recursion. We illustrate how the control procedures can be applied to monitor the covariance structure of developed stock market indices.     

The impact of integrated measurement errors on modeling long-run macroeconomic time series

ABSTRACT

Data spanning long time periods, such as that over 1860–2012 for the UK, seem likely to have substantial errors of measurement that may even be integrated of order one, but which are probably cointegrated for cognate variables. We analyze and simulate the impacts of such measurement errors on parameter estimates and tests in a bivariate cointegrated system with trends and location shifts which reflect the many major turbulent events that have occurred historically. When trends or shifts therein are large, cointegration analysis is not much affected by such measurement errors, leading to conventional stationary attenuation biases dependent on the measurement error variance, unlike the outcome when there are no offsetting shifts or trends.     

A combined mixed-s-skip sampling strategy to reduce the effect of autocorrelation on the X̄ scheme with and without measurement errors

In order to reduce the effect of autocorrelation on the X¯ monitoring scheme, a new sampling strategy is proposed to form rational subgroup samples of size n. It requires sampling to be done such that: (i) observations from two consecutive samples are merged, and (ii) some consecutive observations are skipped before sampling. This technique which is a generalized version of the mixed samples strategy is shown to yield a better reduction of the negative effect of autocorrelation when monitoring the mean of processes with and without measurement errors. For processes subjected to a combined effect of autocorrelation and measurement errors, the proposed sampling technique, together with multiple measurement strategy, yields an uniformly better zero-state run-length performance than its two main existing competitors for any autocorrelation level. However, in steady-state mode, it yields the best performance only when the monitoring process is subject to a high level of autocorrelation, for any given level of measurement errors. A real life example is used to illustrate the implementation of the proposed sampling strategy.KEYWORDS: Autocorrelation, measurement errors, mixed samples strategy, multiple measurements, skipping sampling strategy, steady-state, zero-state