RESPONSE MODELING METHODOLOGY –

EMPIRICAL MODELING FOR ENGINEERING AND SCIENCE

 

Haim Shore

 

2005. World Scientific Publishing Co. Pte. Ltd.

 

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Table of Contents

 

Data Files (related to Numerical Examples in the Book)

 

Updated List of Refereed Publications

 

Comment: All papers authored by H. Shore, unless otherwise specified. Go to ABSTRACTS to see abstracts

 

(24) Shacham, M., Brauner, N., Shore, H., Benson-Karhi, D. Predicting Temperature-Dependent Properties by Correlations Based on Similarity of Molecular Structures – Application to Liquid Density

Journal: Engineering Chemistry Research

Year of Publication: 2008

 

(23) Comparison of Linear Predictors Obtained by Data Transformation, Generalized Linear Models (GLM) and Response Modeling Methodology (RMM)

Journal: Quality and Reliability Engineering International. 24(4). 389-399.

Year of Publication: 2008

 

(22) Shore H., Benson-karhi D., Brauner N., Shacham M. Prediction of Temperature-Dependent Properties by Correlations Based on Similarity of Molecular Structures

Journal: AIChE''07 Annual Meeting, Salt-Lake City, Nov. Proceedings (AIChE- American Institute of Chemical Engineers)

Year of Publication: 2007

 

(21) Benson-Karhi, D., Shore, H., Shacham, M. Modeling Temperature-Dependent Properties of Water via Response Modeling Methodology (RMM) and Comparison with Acceptable Models

Journal: Industrial & Engineering Chemistry Research. 46(10). 3446-3463.

Year of Publication: 2007

 

(20) Ladany, S., Shore, H. Profit Maximizing Warranty Period with Sales Expressed by A Demand Function.

Journal: Quality and Reliability Engineering International. 23(3). 291-301.

Year of Publication: 2007

 

(19) Shore, H., Benson-Karhi, D. Forecasting S-Shaped Diffusion Processes via Response Modeling Methodology

Journal: Journal of the Operational Research Society. 58(6). 720-729.

Year of Publication: 2007

 

(18) Comparison of Generalized Lambda Distribution (GLD) and Response Modeling Methodology (RMM) as general platforms for distribution fitting

Journal: Communications in Statistics (Theory& Methods). 36(15). 2805-2819.

Year of Publication: 2007

 

(17) Benson-Karhi, D., Shore, H., Shacham, M. Applying Response Modeling Methodology to Model Temperature-Dependnecy of Vapor Pressure.

Journal: Lecture Series on Computer and Computational Sciences. 4. 835-838.

Year of Publication: 2005

 

(16) Accurate RMM-based Approximations for the CDF of the Normal Distribution.

Journal: Communications in Statistics (Theory& Methods), 34(3). 507-513.

Year of Publication: 2005

 

(15) Response Modeling Methodology (RMM) - Maximum Likelihood Estimation Procedures

Journal: Computational Statistics and Data Analysis, 49, 1148-1172.

Year of Publication: 2005

 

(14) Non-Normal Populations in Quality Applications - A Revisited Perspective

Journal: Quality & Reliability Engineering International, 20(4), 375-382.

Year of Publication: 2004

 

(13) The Random Fatigue Life Model as a Special Case of the RMM Model- A Comment on Pascual (2004)

 Journal: Communications in Statistics (Simulation & Computation), 33(2), 537-539.

Year of Publication: 2004

 

(12) Response Modeling Methodology (RMM) - Validating Evidence from Engineering and the Sciences

Journal: Quality & Reliability Engineering International. 20, 61-79.

Year of Publication: 2004

 

(11) Response Modeling Methodology (RMM) - Current Distributions, Transformations and Approximations as Special Cases of the RMM Error Distribution

 Journal: Communications in Statistics (Theory & Methods), 33(7), 1491-1510.

Year of Publication: 2004

 

(10) Response Modeling Methodology (RMM) - A New Approach to Model a Chemo-Response for a Monotone Convex/Concave Relationship

Journal: Computers and Chemical Engineering, 27(5), 715-726

Year of Publication: 2003

 

(9) Modeling a Response with Self-Generated and Externally-Generated Sources of Variation

Journal: Quality Engineering, 14(4), 563-578.

Year of Publication: 2002

 

(8) Shore, H., Brauner, N., Shacham, M. Modeling Physical and Thermodynamic Properties via Inverse Normalizing Transformations

Journal: Industrial and Engineering Chemistry Research, 41, 651-656.

Year of Publication: 2002

 

(7) Response Modeling Methodology (RMM) - Exploring the Implied Error Distribution

Journal: Communications in Statistics (Theory & Methods). 31(12). 2225-2249

Year of Publication: 2002

 

(6) Modeling a Non-Normal Response for Quality Improvement

Journal: International Journal of Production Research, V. 39(17), 4049-4063.

Year of Publication: 2001

 

(5) Three Approaches to Analyze Quality Data Originating in Non-Normal Populations

Journal: Quality Engineering, 13(2), 277-291.

Year of Publication: 2001

 

(4) Process Control for Non-Normal Populations Based on Inverse Normalizing Transformation

Journal: In Frontiers in Statistical Quality Control 6. Book. Lenz, H.-J., and Wilrich, P. –TH, Editors, Physica-Verlag ( a Springer-Verlag Company), 194- 206

Year of Publication: 2001

 

(3) Inverse Normalizing Transformations and an Extended Normalizing Transformation

Journal: In "Advances in Methodological and Applied Aspects of Probability and Statistics”". Book. N. Balakrishnan (Ed.). V.2 , pp. 131-140. Gordon and Breach Science Publishers.

Year of Publication: 2001

 

(2) General Control Charts for Attributes

Journal: IIE Transactions, 32, 1149-1160

Year of Publication: 2000

 

(1) General Control Charts for Variables

Journal: International Journal of Production Research, 38(8), 1875-1897.

Year of Publication: 2000


(B) ABSTRACTS

 

(24) Shacham, M., Brauner, N., Shore, H., Benson-Karhi, D. Predicting Temperature-Dependent Properties by Correlations Based on Similarity of Molecular Structures – Application to Liquid Density

2008. Engineering Chemistry Research

 

ABSTRACT: A novel method for predicting temperature-dependent properties is presented. The method involves the use of measured property values of predictive compounds that are structurally similar to the target compound, and molecular descriptor values. The quantitative structure-structure property relationship (QS2PR) is used to model the relationship between property values of the target and the predictive compounds. Whenever necessary, Response Modeling Methodology (RMM) is employed to develop a regression model for representing property data of the predictive compounds. The proposed method is demonstrated by prediction of temperature-dependent liquid-density variation of 1-butene, toluene, n-hexane and n-heneicosane. It is shown that straightforward application of the proposed method provides predictions with accuracy within experimental error level. An advantage of the proposed method over other similar prediction models is that it does not require measured property values of the target compound.

 

(23) Comparison of linear predictors obtained by data transformation, generalized linear models (GLM) and response modeling methodology (RMM).

2008. Quality and Reliability Engineering International

 

ABSTRACT: The data-transformation approach and Generalized Linear Modeling (GLM) both require specification of a transformation prior to deriving the linear predictor (LP). By contrast, Response Modeling Methodology (RMM) requires no such specifications. Furthermore, RMM effectively decouples modeling of the LP from modeling its relationship to the response. It may therefore be of interest to compare LPs obtained by the three approaches. Based on numerical quality problems that have appeared in the literature, these approaches are compared both in terms of the derived structure of the LPs and goodness-of-fit statistics. The relative advantages of RMM are discussed.

 

(22) Shore H., Benson-karhi D., Brauner, N., Shacham M. Prediction of Temperature-Dependent Properties by Correlations Based on Similarity of Molecular Structures

2007. AIChE''07 Annual Meeting, Salt-Lake City, Nov. Proceedings (AIChE - American Institute of Chemical Engineers)

 

(21) Benson-Karhi, D., Shore, H., Shacham, M. Modeling Temperature-Dependent Properties of Water via Response Modeling Methodology (RMM) and Comparison with Acceptable Models

2007. Industrial & Engineering Chemistry Research

 

ABSTRACT: Response Modeling Methodology (RMM) is a new empirical modeling methodology, recently developed. In this paper a new structured procedure to compare relational models in terms of goodness-of-fit and stability is developed and applied to evaluate three types of models: Those obtained by TableCureve2Dâ (a dedicated software for relational modeling) , acceptable models recommended by DIPPRâ (a widely used database for constant and temperature-dependent physical properties) and models derived by RMM. For four pure substances, oxygen, argon, nitrogen and water, model comparison had been conducted over fourteen temperature-dependent physical and thermodynamic properties. Summary tables of ranking the various models are provided. Detailed results are reported in this paper for water. The three variations of the RMM model (2-, 3- and 4-parameter models) compare favorably with others both in terms of goodness-of-fit and stability. The unique desirable properties of RMM models are discussed.

 

(20) Ladany, S., Shore, H. Profit Maximizing Warranty Period with Sales Expressed by a Demand Function

2007. Quality and Reliability Engineering International

 

ABSTRACT: The problem of determining the optimal warranty period, assumed to coincide with the manufacturer's LSL for the life-time of the product, is addressed. It is assumed that the quantity sold depends via a Cobb-Douglas-type demand function on the sale-price and on the warranty period, and that both the cost incurred for a non-conforming item and the sale-price increase with the warranty period. A general solution is derived using RMM (Response Modeling Methodology) and a new approximation for the standard normal CDF. The general solution is compared to exact optimal solutions derived under various distributional scenarios. Relative to the exact optimal solutions, RMM-based solutions are accurate to at least the first three significant digits. Some exact results are derived for the uniform and the exponential distributions.

 

(19) Shore, H., Benson-Karhi, D. Forecasting S-shaped Diffusion Processes via Response Modeling Methodology

2007. Journal of the Operational Research Society

 

ABSTRACT: Diffusion processes abound in various areas of corporate activities, such as the time-dependent behavior of cumulative demand of a new product, or the adoption rate of a technological innovation. In most cases, the proportion of the population that has adopted the new product by time t behaves like an S-shaped curve, which resembles the sigmoid curve typical to many known statistical distribution functions. This analogy has motivated the common use of the latter for forecasting purposes. Recently, a new methodology for empirical modeling has been developed, denoted Response Modeling Methodology (RMM). The error distribution of the RMM model has been shown to model well variously shaped distribution functions, and may therefore be adequate to forecast sigmoid-curve processes. In particular, RMM may be applied to forecast S-shaped diffusion processes. In this paper, forty-seven data sets, assembled from published sources by Meade and Islam (1998), are used to compare the accuracy and the stability of RMM-generated forecasts, relative to current commonly applied models. Results show that in most comparisons RMM forecasts outperform those based on any individually selected distributional model.

 

(18) Comparison of Generalized Lambda Distribution (GLD) and Response Modeling Methodology (RMM) as General Platforms for Distribution Fitting

2007. Communications in Statistics (Theory& Methods)

 

ABSTRACT: Distribution fitting is widely practiced in all branches of engineering and applied science. Yet only few studies have examined the relative capability of various parameter-rich families of distributions to represent a wide spectrum of diversely-shaped distributions. In this paper, two such families of distributions, Generalized Lambda Distribution (GLD) and Response Modeling Methodology (RMM), are compared. For a sample of some commonly used distributions, each family is fitted to each distribution, using two methods: Fitting by minimization of the L2 norm (minimizing density function distance) and non-linear regression applied to a sample of exact quantile values (minimizing quantile function distance). The resultant goodness-of-fit is assessed by four criteria: the optimized value of the L2 norm and three additional criteria, relating to quantile function matching. Results show that RMM is uniformly better than GLD. An additional study includes Shore’s quantile function (QF) and again RMM is the best performer, followed by Shore’s QF and then GLD.

 

(17) Benson-Karhi, D., Shore, H., Shacham, M. Applying Response Modeling Methodology to Model Temperature-Dependnecy of Vapor Pressure.

2005. Lecture Series on Computer and Computational Sciences

 

ABSTRACT: In modeling chemical properties, theory-based relationships often represent available data accurately. However, when goodness-of-fit is not satisfactory, empirical modeling is called for. Recently, a new empirical modeling methodology has been developed, denoted Response Modeling Methodology (RMM). The new approach is intended to model monotone convex relationships. In this paper we apply RMM to model the temperature dependence of vapor pressure. The resulting models are compared to well-known and widely used property correlation equations. These include the “Acceptable Models”, currently recommended by DIPPR[1] and models recommended by “Table Curve”, a dedicated empirical modeling software. Recently developed methodologies for data-based comparison of models are employed to select the best model. Results show that RMM can represent satisfactorily curves of different shapes. Further research would extend the results introduced to other combinations of chemical properties and substances.  

 

(16) Accurate RMM-based Approximations for the CDF of the Normal Distribution

2005. Communications in Statistics (Theory& Methods)

ABSTRACT: A variation of the RMM error distribution, used to model the exponential distribution, has recently been applied to derive a three-parameter approximation for the standard normal CDF, with a maximum absolute error of order (10)-5. In this short communication, a simple modification enhances the accuracy to the order of (10)-6. Another RMM-based approximation, based on the original RMM error distribution, achieves an absolute maximum error of (10)-7. The simplicity of the new non-polynomial approximations qualifies them to be conveniently integrated into stochastic optimization (models like inventory models) or to be used in applications. That modeling of the exponential distribution via the RMM model could produce such highly accurate approximation for the standard normal CDF seems to lend further validity to the RMM model.

 

(15) Response Modeling Methodology (RMM)- Maximum Likelihood Estimation Procedures

 2005. Computational Statistics and Data Analysis

ABSTRACT: Response modeling methodology (RMM) is a new approach for empirical modeling. ML estimation procedures for the RMM model are developed. For relational modeling, the RMM model is estimated in two phases. In the first phase, the structure of the linear predictor (LP) is determined and its parameters estimated. This is accomplished by combining canonical correlation analysis with linear regression analysis. The former procedure is used to estimate coefficients in a Taylor series approximation to an unspecified response transformation. Canonical scores are then used in the latter procedure as response values in order to estimate coefficients of the LP. In the second phase, the parameters of the RMM model are estimated via ML, given the LP estimated earlier. For modeling random variation, it is assumed that the LP is constant and a new simple percentile-based estimating procedure is developed. The new estimation procedures are demonstrated for some published data.

(14) Non-Normal Populations in Quality Applications - A Revisited Perspective

2004. Quality & Reliability Engineering International

 

ABSTRACT: Much research effort has recently been focused on methods to deal with non-normal populations. While for weak non-normality the normal approximation is a useful choice (as in Shewhart control charts), moderate to strong skewness requires alternative approaches. In this short communication, we discuss the properties required from such approaches, and revisit two new ones. The first approach, for attributes data, assumes that the mean, the variance and the skewness measure can be calculated. These are then incorporated in a modified normal approximation, which preserves these moments. Extension of the Shewhart chart to skewed attribute distributions (e.g. the geometric distribution) is thus achieved. The other approach, for variables data, fit a member of a four-parameter family of distributions. However, unlike similar approaches, sample estimates of at most the second degree are employed in the fitting procedure. This has been shown to result in a better representation of the underlying (unknown) distribution than methods based on four-moment matching. Some numerical comparisons are given.

 

(13) The Random Fatigue Life Model as a Special Case of the RMM Model- A Comment on Pascual (2003)

2004. Communications in Statistics (Simulation & Computation)

 

(12) Response Modeling Methodology (RMM)- Validating Evidence from Engineering and the Sciences

2004. Quality & Reliability Engineering International

 

ABSTRACT: Modeling a response in terms of the factors that affect it is often required in quality applications. While the normal scenario is commonly assumed in such modeling efforts, leading to the application of linear regression analysis, there are cases when the assumptions underlying this scenario are not valid and alternative approaches need to be pursued, like the normalization of the data or generalized linear modeling. Recently, a new response modeling methodology (RMM) has been introduced, which seems to be a natural generalization of various current scientific and engineering mainstream models, where a monotone convex (concave) relationship between the response and the affecting factor (or a linear combination of factors) may be assumed. The purpose of this paper is to provide the quality practitioner with a survey of these models and demonstrate how they can be derived as special cases of the new RMM. A major implication of this survey is that RMM can be considered a valid approach for quality engineering modeling and, thus, may be conveniently applied where theory-based models are not available or the goodness-of-fit of current empirically-derived models is unsatisfactory. A numerical example demonstrates the application of the new RMM to software reliability-growth modeling. The behavior of the new model when the systematic variation vanishes (there is only random variation) is also briefly explored.

 

(11) Response Modeling Methodology (RMM)- Current Distributions, Transformations and Approximations as Special Cases of the RMM Error Distribution

2004. Communications in Statistics (Theory & Methods)

 

ABSTRACT: Recently a new Response Modeling Methodology (RMM) has been introduced, which models the relationship between a response and the affecting factors, assuming only that this relationship is monotone convex. It has been demonstrated that many current relational models, developed over the years in various branches of engineering and the sciences, are in fact special cases of the RMM model. In this paper, we proceed to demonstrate that the RMM error distribution delivers as special cases some well-known statistical distributions, or related transformations and approximations. This establishes the RMM error distribution as a highly versatile platform for modeling random variation. New accurate non-polynomial approximations to the CDF of the normal distribution, with error less than 0.00002, and to the Poisson quantile function, demonstrate this versatility.

 

(10) Response Modeling Methodology (RMM) - A New Approach to Model a Chemo-Response for a Monotone Convex/Concave Relationship

2003. Computers and Chemical Engineering

 

ABSTRACT: Modeling the variation of a response in terms of the variation transmitted to it by a related factor (or factors) comprises the bulk of the scientific and engineering research effort. Often, we may reasonably assume that the relationship between the response and the affecting factor (or a linear combination of factors) is monotone convex (concave). To model such a relationship, a new Response Modeling Methodology (RMM) has recently been introduced and shown to represent a natural generalization of many theoretical or empirically-derived models, developed over the years in various scientific and engineering disciplines. In particular, many models of chemistry and chemical engineering fall into this category. In this paper, we demonstrate application of the new methodology to the modeling of a chemical response, and discuss extension to non-monotone relationships. It is shown that some well-known and widely used property correlation equations are indeed special cases of the new model. The allied estimation procedures are applied to some published data sets, related to temperature dependence of vapor pressure and solid heat capacity. We assess the effectiveness of the new approach relative to current models.

 

(9) Modeling a Response with Self-Generated and Externally-Generated Sources of Variation

2002. Quality Engineering

 

ABSTRACT: Attempts to model the variation of a random response in terms of the factors that affect it and its own self-generated variability constitute the bulk of the scientific and engineering research effort. This is particularly valid for quality engineering, where modeling of a response is often required to solve quality problems or to improve quality. While models that are derived from established domain-specific theories are commonly used in various disciplines, a pragmatic approach may be conceived that assembles under a single general model features that are shared by models developed in disparate and unrelated disciplines. In this paper, we develop a new approach compatible with this concept. On the basis of recently developed inverse normalizing transformations, the new model provides the quantile-relationship between a response (the dependent variable) and the factors that affect it (the independent variables), assuming only that this relationship is either uniformly convex or concave. Furthermore, two independent sources of variation, one internal and one external, are assumed to account for the observed response variation. We demonstrate the validity of the new approach by showing that the new model is a generalization of models that are currently in use in three disparate engineering disciplines: hardware reliability, software reliability, and chemical engineering. Employing some previously published data sets, the modeling competence of the new approach is demonstrated.

 

(8) Modeling Physical and Thermodynamic Properties via Inverse Normalizing Transformations

2002.  Shore, H., Brauner, N., Shacham, M. Industrial and Engineering Chemistry Research.

 

ABSTRACT: The Inverse Normalizing Transformation (INT) represents a generalization of the inverse of the Box-Cox transformation. It is shown that several well-known and widely used property correlation equations, such as the Antoine, the truncated Riedel, the Rackett and the Guggenheim equations can be derived from the INT. Its use is demonstrated for modeling temperature dependence of vapor pressure, solid and liquid heat capacity, vapor and liquid viscosity and surface tension data. It is shown that the INT can represent satisfactorily curves of different shapes and, as such, its use can be beneficial in modeling temperature dependence of various physical and thermodynamic properties.

 

(7) Response Modeling Methodology (RMM)- Exploring the Implied Error Distribution

2002. Communications in Statistics (Theory & Methods)

 

ABSTRACT: Modeling efforts in engineering and the sciences often attempt to describe the relationship between a response and some external affecting factor (or a linear combination of factors), where the modeled relationship is known to be monotone convex (or concave). Recently, a new general model had been developed (in the framework of a new response modeling methodology, RMM), which was demonstrated to be a natural generalization of current mainstream models in many scientific and engineering disciplines, like physics, chemistry, chemical engineering, electric engineering or reliability engineering (hardware and software). The error structure of this model comprises two normal error components, which together define the error distribution associated with the response. In this paper, we derive this error distribution and investigate its properties. We show that widely-used theoretical distributions may be well represented by the new distribution, surprisingly not to be found in textbooks.

 

(6) Modeling a Non-Normal Response for Quality Improvement

2001. International Journal of Production Research

 

ABSTRACT: Efforts to improve quality require that the factors affecting it be identified. This allows either removal of root-causes for low quality, or finding optimal settings for the investigated product or process. When the common assumptions of the normal scenario are not met two alternative approaches are commonly pursued: Normalization of data and the use of generalized linear models (GLM). Recently, a third alternative has been developed, that models a response subject to self-generated random variation and externally-generated systematic and random variation. It is assumed that the relationship between the response and the externally generated variation is uniformly convex (or concave). A unique feature of the new model is that both its structure and the parameters' values are determined solely by the data on hand (no theory-based arguments are required). Here, we compare the effectiveness of the new methodology relative to current approaches when applied to response modeling in quality improvement efforts. We do this by using published data sets which have been formerly analyzed within the framework of either the normalizing approach or the GLM approach (or both). The relative merits of the new methodology are demonstrated and discussed.

 

(5) Three Approaches to Analyze Quality Data Originating in Non-Normal Populations

2001. Quality Engineering

 

ABSTRACT: Quality procedures used by practitioners often assume underlying normality. When this assumption is unsubstantiated by empirical evidence and the exact distribution is unknown, various approaches are commonly pursued, like normalizing the available data or fitting a member from a known three- or four-parameter family of distributions (like Pearson). These approaches and others, while theoretically valid, often pose for the quality practitioner difficulties in implementation that detract appreciably from their attractiveness. In recent years, three alternative approaches to analyze data from non-normal populations have been developed, and their usefulness demonstrated (via refereed publications). The purpose of this paper is to expose the quality practitioner to these new approaches, and demonstrate their merits, relative to current approaches, with the expressed intent to enhance their implementation within the quality discipline. For each of the new methodologies, the motivation for developing it is first given, followed by a brief exposition of the allied analysis-routines and a review of published references, where the approach has been implemented to solve practical problems. A detailed numerical example demonstrates an application to a quality-related problem.

 

(4) Process Control for Non-Normal Populations Based on Inverse Normalizing Transformation

2001. In Frontiers in Statistical Quality Control 6. Book. Lenz, H.-J., and Wilrich, P. –TH, Editors, Physica-Verlag (Springer-Verlag Company), 194- 206.

 

ABSTRACT: When the process distribution is non-normal traditional Shewhart charts may not be applicable. A common practice in such cases is to normalize the process data by applying the Box-Cox power transformation. The general effectiveness of this transformation implies that an inverse normalizing transformation (INT) may also be effective, namely: a power transformation of the standard normal quantile may deliver good representation for the original process data without the need to transform them. Using the Box-Cox transformation as a departure point, three INTs have recently been developed and their effectiveness demonstrated. In this paper we adapt one of these transformations to develop a new scheme for process control when the underlying process distribution is non-normal. The adopted transformation has three parameters, and these are identified by matching of the median, the mean and the mean of the log (of the original monitoring statistic). Given the low mean-squared-errors (MSEs) associated with sample estimates of these parameters, the new transformation may be used to derive control limits for non-normal populations without resorting to estimates of third and fourth moments, notoriously known for their high MSEs. Implementation of the new approach to monitor processes with non-normal distributions is demonstrated.

 

(3) Inverse Normalizing Transformations and an Extended Normalizing Transformation

2001. In Advances in Methodological and Applied Aspects of Probability and Statistics. Book. N. Balakrishnan (Ed.). V.2 , pp. 131-140. Gordon and Breach Science Publishers.

 

ABSTRACT: When available data are non-normal, a common practice is to normalize them by applying the Box-Cox power transformation. The general effectiveness of this transformation implies that an inverse normalizing transformation, namely: a power transformation of the standard normal quantile, may effectively deliver a general representation for many of the commonly applied theoretical statistical distributions. Employing as a departure point the Box-Cox transformation, we develop in this paper several inverse normalizing transformations, and define criteria for their effectiveness and for the adequacy of the allied normalizing transformation (when such exists). In terms of these criteria, the new transformations are compared to the inverse Box-Cox transformation, and estimation procedures for their parameters are developed. A certain three-parameter inverse normalizing transformation, where the standard normal quantile appears only once, serves to derive a new normalizing transformation. The latter turns out to be an extension of the traditional Box-Cox transformation. For a sample of commonly used distributions, the new normalizing transformation is compared to the Box-Cox transformation and is shown to achieve a better normalizing effect. An application to preventive maintenance is numerically demonstrated.

 

(2) General Control Charts for Attributes

2000. IIE Transactions

 

ABSTRACT: Traditional Shewhart-type control charts ignore the skewness of the plotted statistic. Occasionally, the skewness is too large to be ignored, and in such cases the classical Shewhart chart ceases to deliver satisfactory performance. In this paper, we develop a general framework for constructing Shewhart-like control charts for attributes based on fitting a quantile function that preserves all first three moments of the plotted statistic. Furthermore, these moments enter explicitly into the formulae for calculating the limits. To enhance the accuracy of these limits the value of the skewness measure used in the calculations is inflated by 44%. This inflation rate delivers accurate control limits for diversely-shaped attribute distributions like the binomial, the Poisson, the geometric and the negative binomial. A new control chart for the M/M/S queueing model is developed and its performance evaluated.

 

(1) General Control Charts for Variables

2000. International Journal of Production Research

 

ABSTRACT: When the distribution of the monitoring statistic used in statistical process control is non-normal traditional Shewhart charts may not be applicable. A common practice in such cases is to normalize the data, using the Box-Cox power transformation. In this paper, we develop an inverse normalizing transformation (INT), namely, a transformation that expresses the original process variable in terms of the standard normal variable. The new INT is used to develop a general methodology for constructing process control schemes for either normal or non-normal environments. Simplified versions of the new INT result in transformations with a reduced number of parameters, allowing fitting procedures which require only low-degree moments (second degree at most). The new procedures are incorporated in some suggested SPC schemes which are numerically demonstrated. A simple approximation for the CDF of the standard normal distribution, with a maximum error (for z>0) of  ±0.00002, is a by-product of the new transformations.  

 



[1] Design Institute for Physical Properties