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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 TableCurve2D (a dedicated software for relational modeling),...
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... In fact, by allowing the data determine the final shape of the estimated model, RMM combines the unique advantages of the non-parametric approach ("no model specification required") without sacrificing the advantages associated with the parametric modeling approach. RMM has proved to be a versatile and effective modeling platform in diverse areas as distribution fitting (Shore, 2007 (Shore, 2008). Application of RMM to these areas has consistently shown that it may replace current models with negligible loss in accuracy. ...
Monitoring fetal growth via ultrasound requires modeling fetal biometry in terms of gestation age. In this study, we compare Response Modeling Methodology (RMM) to current models, in three stages. First, RMM is used to approximate 47 empirical mean models that have appeared in the literature, resulting in models with negligible loss in accuracy (unlike two other commonly applied models). Next, RMM models are fitted to sample averages of the Singaporean population and compared to a formerly published model. A similar simulation-based analysis is performed in the last stage for raw data of the British population. Altogether over seventy comparisons had been performed resulting in RMM consistently delivering better performing models.
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... This issue is relevant because an RMM model may provide good approximation to a given model and yet deliver poor fit to data generated from the approximated model. Former studies have shown that RMM delivers goodness-of-fit comparable to that of the data-generating models11,12 . To examine this issue in the ecological context, a simulation study was conducted, relating to the models specified inTable 1. ...
SPC monitoring of nonlinear relationships (profiles) has been the subject of much research recently. While attention is primarily given to the statistical aspects of the monitoring techniques, little effort has been devoted to developing a general modeling approach that would introduce "uniformity of practice" in modeling nonlinear profiles (analogously with the three-sigma limits of Shewhart control charts). In this article, we use Response Modeling Methodology (RMM) to demonstrate implementation of this approach to SPC monitoring of ecological relationships. Using ten ecological models that have appeared in the literature, it is first shown that RMM models can replace (approximate) current ecological models with negligible loss in accuracy. Computer simulation is then used to demonstrate that estimated RMM models and estimated data-generating ecological models achieve goodness-of-fit that is practically indistinguishable from one another. A regression-adjusted control scheme, based on control charts for the predicted median and for residuals variation, is developed and demonstrated for three types of "out of control" scenarios.
... 8). While it is simpler to use LAD* to estimate β p (since LAD* is calculated with each model fitting), PRLAD is the appropriate estimate for β p to monitor a future measurement (find more details in comment [2] below). However, it is more difficult to estimate, given the small number of absolute deviations available for its calculation. ...
An SPC-based methodology is developed to detect early deviations of fetal biometry from expected normal growth. Using Response Modeling Methodology (RMM), an initial fetal growth model is estimated in Phase I, based on four ultra-sound (US) measurements. Regression-adjusted SPC control scheme, based on a new median control chart and a control chart for residuals variation, monitor absolute deviations from the expected growth trajectory (as predicted by the growth model). Hadlock's boundary reference centiles, currently used by obstetricians as "specification limits", are also integrated in the monitoring scheme. As pregnancy progresses, newly collected data dynamically update the parameters of the median prediction model and of the associated control limits. Real longitudinal data from both normal pregnancies and those that resulted in adverse medical outcomes have been analyzed, using four fetal US biometric measures. Results show that non-smooth growth trajectory, expressed in exceptionally large absolute deviations from predicted median values, is a good precursor to possible prenatal adverse outcomes. Directions for future research are outlined.
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... The reason for selecting (8) for 'distribution fitting' [rather than (9)] is that it is the only RMM quantile function for which an explicit inverse function is available. The latter expresses the quantile Z in terms of the respective response quantile [refer to (18) below]. Existence of the inverse function is a prerequisite to our ability to formulate an ML function, from which ML estimates may be obtained. ...
... where φ is the standard normal pdf and z and g (z) in (20) are expressed in terms of the response values, y, via (18). From (20) the ML equation may be obtained, expressed in terms of the sample data and the unknown parameters. ...
Response modeling methodology (RMM) is a general platform for modeling monotone convex relationships. Unique to RMM models is their ‘continuous convexity’ property, which allows the data to ‘select’ the final form of the model via the estimated parameters (analogously with the Box–Cox transformation). This renders RMM a versatile and effective platform for empirical modeling of random variation (‘distribution fitting’) and of systematic variation (‘relational modeling’). In this overview, we detail the motivation that led to the development of RMM, explain RMM core concepts, and introduce RMM basic model and variations. Allied maximum-likelihood estimation procedures are detailed, separately for models of random variation and for models of systematic variation. Numerical examples demonstrate RMM effectiveness in comparison to other current approaches. Current literature on RMM (about 25 publications), available software, and ongoing research are also addressed. WIREs Comp Stat 2011 3 357–372 DOI: 10.1002/wics.151
For further resources related to this article, please visit the WIREs website and also Wikipedia entry.
... The confidence intervals (in particular the 95% confidence interval) on the variable values and data point errors fit can be very useful indicators of the fit between the model and the data (Benson-Karhi et al., 2007). If all confidence intervals are smaller than the respective variable values, all of them are significant in the function fit; the best model is obtained with the lowest data point errors (Perry et al., 1999). ...
In this work, a computational method is proposed in order to choose the best interpolation function for experimental data. The optimal interpolation function follows the natural tendency of data, suffers a minimal influence by removing one data point, does not present over fitting and yields the best representation for large or rough data sets. High pressure Px data of binary mixtures containing ionic liquid were used as case studies: supercritical CO2+1-hexyl-3-methyl imidazolium hexafluorophosphate {[hmim][PF6]}, supercritical CO2+1-butyl-3-methyl imidazolium tetrafluoroborate {[bmim][BF4]} and supercritical CHF3+1-butyl-3-methyl imidazolium hexafluorophosphate {[bmim][PF6]}. The common practice of using the correlation coefficient to analyze the accuracy of the interpolation model is critically discussed. As a validation of the method, a thermodynamic consistency test is applied to the interpolated data. The results show that the mean values of the interpolation error for CO2+[hmim][PF6], CO2+[bmim][BF4] and CHF3+[bmim][PF6] are 0.47%, 0.33% and 0.13%, respectively. These values are within the reported experimental error, which has a range from 0.04% to 1.7%.
... However, very few attempts have been made to develop QSPRs to predict temperature-dependent properties for a wide range of temperature values (Godavarthy et al. 4 ). In this work, a framework for the prediction of temperature-dependent properties is established based on several recently developed techniques (Shacham et al., 5 Shore, 6 Shore et al., 7 Benson-Karhi et al., 8 and Cholakov et al. 9 ). ...
... When models for the target property are not available, empirical (regression) models for the predictive compounds must first be derived. One possibility of obtaining such models is via the new RMM (refer to Shore 6 for its introduction and to Shore et al. 7 and Benson-Karhi et al. 8 for a demonstrative application to modeling of physical properties). The main advantage of RMM-based models is that they typically deliver representation of the property's temperature-dependent variation, with adequate precision and a high level of stability, using only three parameters. ...
... where η is an appropriately relocated and rescaled T, namely, η ) 0 + 1 Τ ( 0 and 1 are real numbers), ε 1 and ε 2 are regressor and response errors, respectively, and M, R, and λ are the model's parameters. From this model, simplified two-, three-, and four-parameter models are derivable (Shore 6 ) that have been identified as most appropriate for modeling the temperature dependency of physical properties (Benson-Karhi et al. 8 ): ...
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 a linear relationship between property values of the target and the predictive compounds. Whenever necessary, response modeling methodology (RMM) can be employed to obtain a nonlinear regression model for representing property data of the predictive compounds. The application of the method is demonstrated under a variety of conditions by prediction of the temperature-dependent liquid-density variation of I-butene, toluene, n-hexane, and n-heneicosane. It is shown that straightforward application of the proposed method provides predictions with accuracy within the 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.
... Since its development and introduction in the first years of the new millennium, RMM has been implemented in various fields of applied science, engineering and operations management. It has been shown to deliver good modeling capabilities, while preserving desirable " uniformity of practice " across widely divergent disciplines (for example, Shore (2003), Shore and , Ladany and Shore (2007) and Benson-Karhi, Shore, and Shacham (2007)). RMM includes, as special cases, numerous models that had been formerly derived within various scientific and engineering disciplines like physics, chemistry, chemical engineering, operations management and quality and reliability engineering. ...
Response Modeling Methodology (RMM) is a new approach for empirical
modeling of systematic variation and of random variation. Applied to various fields of science, engineering and operations management, RMM has been shown to deliver good modeling capabilities while preserving desirable “uniformity of practice” across widely divergent disciplines. In this paper, RMM is briefly outlined, and its basic philosophy, relative to other approaches, is discussed. A detailed numerical example demonstrates application of RMM for distribution fitting and compares the results to fitting by generalized lambda distribution. Initial results are described from an ongoing research that statistically compares goodness-of-fit obtained from fitting several families of distributions to a sample of commonly applied distributions. The results suggest that it is possible to rank widely used families of distributions in terms of their capability to serve as general platforms for distribution fitting.
Pure-compound property data are at present available only for a small fraction of the compounds, pertaining to such diverse areas as chemistry and chemical engineering, environmental engineering and environmental impact assessment, hazard and operability analysis. Therefore, methods for reliable prediction of property data are needed. In particular, prediction of temperature-dependent properties (like vapor pressure, vapor and liquid density, viscosity or specific heat) poses a special challenge because of the limited amount of experimental data available at widely varying temperature ranges. Current methods used to predict temperature-dependent properties can be classified into "group contribution" methods, methods based on the "corresponding-states principle" (for an extensive review of these methods see Poling et al., 2001), and "asymptotic behavior" correlations (see, for example, Marano and Holder, 1997). Reliable methods for predicting temperature-dependent properties have not yet been established. Even for vapor pressure, probably the most extensively investigated temperature-dependent property, prediction errors often reach several tens of percent (Poling et al., 2001). Furthermore, to predict properties of a target compound, many of these methods require experimental data about the target compound (critical properties, for example), and as such cannot be used for prediction of properties for compounds not yet synthesized. In recent years, there has been increasing interest in using, for prediction of constant properties, molecular descriptors integrated into Quantitative Structure Property Relationship (QSPR). However, very few attempts have been made to model, for prediction purposes, temperature-dependent properties (excluding vapor pressure; relate to Dearden et al., 2003). In this work, we apply several new techniques that we have developed recently (Shacham et al., 2004, Shore, 2005, Benson-Karhi et al., 2007, Cholakov et al., 2007) for prediction of temperature-dependent properties. To predict a particular property, a structure-structure correlation is first identified, using only molecular descriptors of the predictive compounds. The latter are compounds "similar" to the target compound, and for which experimental data for the target property are available. The QS2PR method of Shacham et al. (2004) is used to identify such similar compounds and to derive a linear structure-structure correlation. Typically, 2-4 predictive compounds are included in the QS2PR model. If the target compound and the potential predictive compounds belong to the same homologous series, use of the short-cut QS2PR method of Cholakov et al. (2007) can be considered. Two different methods have been developed to derive, for a given target compound, a model for the temperature-dependency of the target property. The first method is based on the availability of such models for the predictive compounds (e.g., Antoine equation parameters for vapor pressure calculations). In this case, the property value for the target compound can be calculated (point by point) at any specified temperature. To this aim, the calculated property values for the predictive compounds (at a specified temperature) are introduced into the structure-structure correlation to obtain the corresponding property value for the target compound. For some properties (vapor pressure, for example), it will be advantageous, when using this method, to set the property value and then calculate the matching temperature as this yields more accurate predictions In cases where models for the target property are not available, empirical (regression) models for the predictive compounds must first be derived. One possibility to derive such models is via the new Response Modeling Methodology (RMM; refer to Shore, 2005, for its introduction, and to Benson-Karhi et al., 2007, for a demonstrative application to modeling water properties). The main advantage of RMM-based models is that they typically deliver representation of the property's temperature-dependent variation, with adequate precision and high level of stability, using only two or three parameters. Thus, the same model can be used over a wide spectrum of properties, and for data available at various temperature ranges. The parameters of the RMM model, derived for the predictive compounds, can then be inserted into the structure-structure correlation to obtain RMM parameter values for the target compound. The proposed methods were applied to various properties of several hydrocarbons, and to some oxygen containing organic compounds. The properties tested included liquid density, vapor pressure, heat of vaporization, solid, liquid and ideal gas heat capacity, second virial coefficient, liquid and vapor viscosity, liquid and vapor thermal conductivity and surface tension. Highly accurate predictions were generally obtained. Detailed results of the evaluation of the proposed method will be presented in the conference.
References
1. Benson-Karhi, D., Shore, H., Shacham, M., "Modeling Temperature-Dependent Properties of Water via Response Modeling Methodology (RMM) and Comparison with Acceptable Models", Ind. Eng. Chem. Res., Web Release Date: April, 5th, 2007; (Correlation) DOI: 10.1021/ie061252x
2. Cholakov, G. St., Stateva, R. P., Shacham, M., Brauner, N., " Prediction of Properties in Homologous Series with a Shortcut QS2PR Method", AIChE J , 53(1), 150-159 (2007)
3. Dearden, J. C. ?Quantitative Structure?Property Relationships for Prediction of Boiling Point, Vapor Pressure, and Melting Point?, Environmental Toxicology and Chemistry, 22( 8), 1696?1709 (2003).
4. Marano, J.J., Holder, G.D., "General Equations for Correlating the Thermo-physical Properties of n-Paraffins, n-Olefins and other Homologous Series. 2. Asymptotic Behavior Correlations for PVT Properties", Ind. Eng. Chem. Res., 36, 1887-1894 (1997).
5. Poling, B.E., Prausnitz, J. M., O'Connel, J. P., Properties of Gases and Liquids, 5th Ed., McGraw-Hill, New York (2001).
6. Shacham, M., Brauner, N., Cholakov, G. St., Stateva R. P., ?Property Prediction by Correlations Based on Similarity of Molecular Structures?, AIChE J., 50(10), 2481-2492 (2004).
7. Shore, H., Response Modeling Methodology - Empirical Modeling for Engineering and Science. World Scientific Publishing Co. Ltd., Singapore (2005).
In a recent paper, temperature-dependent properties of water were modeled via response modeling methodology (RMM), and the resultant models were compared to models obtained by TableCurve2D (a dedicated software for relational modeling), and to “Acceptable models”, recommended by DIPPR (a widely used database for constant and temperature-dependent physical properties). In this paper, we extend the comparison to oxygen, argon and nitrogen. Model comparison has been conducted for 10 temperature-dependent physical and thermodynamic properties. Detailed results are reported in this paper. Summary tables, which rank the various models in terms of goodness-of-fit and stability over all properties, are provided. The three variations of the RMM model (two-, three-, and four-parameter models) compare favorably with other models, often with more parameters, in terms of both goodness-of-fit and stability. The unique desirable properties of RMM models are discussed.
The data-transformation approach and generalized linear modeling 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 in terms of both the derived structure of the LPs and goodness-of-fit statistics. The relative advantages of RMM are discussed. Copyright © 2007 John Wiley & Sons, Ltd.
