Ratio Of Error Variance Research Articles

A Hybrid Model for Orthogonal Regression

AbstractLinear functional relationships are intended to be symmetric and therefore cannot generally be accurately estimated using ordinary least squares regression equations. Orthogonal regression (OR) models allow for errors in both Y and X and therefore can provide symmetric estimates of these relationships. The most well‐established OR model, the errors‐in‐variables (EIV) model, assumes that the observed scatter around the line is due entirely to errors of measurement in Y and X and that the ratio of the error variances is known. If most of the variance around the line is known to be due to the errors of measurement in Y and X, the EIV model can provide an unbiased maximum likelihood estimate for a functional relationship. However, if a substantial part of the variability around the line is due to natural variability, which is not attributable to errors of measurement in Y or X, the ratio of the measurement error variances is not well defined and the EIV model is not directly applicable. The main contribution of this report is the development of a hybrid model that provides plausible estimates for linear functional relationships in cases with substantial natural variability and substantial errors of measurement. An analysis of female and male differential test functioning between an essay test and an objective test used as parts of a licensure examination provides an illustration of the use of the hybrid model.

Parameter estimation in replicated linear functional relationship model in the presence of outliers

The relationship between two linear variables where both variables are observed with errors can be modeled using a linear functional relationship model. However, when there is no knowledge about the ratio of error variance, we proposed that one can use the replicated linear functional relationship model. The aim of this study is to compare the parameter estimates between unreplicated and replicated linear functional relationship model. The study also extends to examine the behavior of the estimators of the replicated linear functional relationship model in the presence of outliers. A simulation study is performed to investigate the performance of the model. In the absence of outlier, it is found that the value of the parameter estimates is almost similar for both models. Whereas in the presence of outliers, the parameter estimates of the replicated linear functional relationship model have a smaller mean square error as the number of observations increased. This suggests the superiority of the replicated model.

Quantile based estimation of biasing parameters in ridge regression model

Ridge regression is used to get precise estimates by introducing some bias when the problem of multicollinearity is present in the model. Biasing parameter plays a vital role in this bias-variance tradeoff. Several methods are available in literature for the estimation of biasing parameter. For high noise and multicollinearity, available methods do not perform well in terms of their mean square error. In this article, we proposed a new estimator and compared its performance with some other available estimators. Our suggested estimator is a class of estimators based on quantile of ratio of error variance to the canonical coefficients. Based on simulation study, our proposed estimator outperforms other estimators especially when multicollinearity and error variance are high. Finally, a real-life example is used to illustrate the application of the proposed estimator.

Improved PI neural network-based tension control for stranded wire helical springs manufacturing

During the winding process of stranded wire helical springs (SWHSs), uneven wire tension always results in high rejection rate and non-compliance service life of SWHSs. Combining the proportion integral neural network (PINN) with a simplified actuator model, this paper presents a new control scheme for the SWHS CNC machine to keep the wire tension uniform. The PINN is improved by introducing an error variance ratio, accounting for the interaction between wires, as a modifying factor in the second hidden layer. The actuator model is simplified based on the analysis of the dynamic characteristics of the actuator. The output value of the improved PINN is transferred into control voltage value by the simplified model. The tension of each wire is controlled by an improved PINN. In order to enhance the control performance, the network parameters are updated using the gradient-based back-propagation method. The validity and consistency of the improved PINN are verified by experiments. The results indicate that (1) the computation load is slight; (2) the rising time of the step response is within 1 s; (3) 89%-96% of tension deviation values of the wire 1 and wire 3 under different process parameters are within 10% of the reference tension value; (4) the standard deviation of the wire 2 with large disturbance is 8.24 N. Compared with other algorithms (incremental PI, multiple PIDNN, PI based particle swarm optimization), the control scheme based on the improved PINN has less computation load, faster response speed and better performance in the time-varying and nonlinear system with larger disturbance.

Regression between earthquake magnitudes having errors with known variances

Recent publications on the regression between earthquake magnitudes assume that both magnitudes are affected by error and that only the ratio of error variances is known. If X and Y represent observed magnitudes, and x and y represent the corresponding theoretical values, the problem is to find the a and b of the best-fit line $y = a x + b$ . This problem has a closed solution only for homoscedastic errors (their variances are all equal for each of the two variables). The published solution was derived using a method that cannot provide a sum of squares of residuals. Therefore, it is not possible to compare the goodness of fit for different pairs of magnitudes. Furthermore, the method does not provide expressions for the x and y. The least-squares method introduced here does not have these drawbacks. The two methods of solution result in the same equations for a and b. General properties of a discussed in the literature but not proved, or proved for particular cases, are derived here. A comparison of different expressions for the variances of a and b is provided. The paper also considers the statistical aspects of the ongoing debate regarding the prediction of y given X. Analysis of actual data from the literature shows that a new approach produces an average improvement of less than 0.1 magnitude units over the standard approach when applied to $M_{w}$ vs. $m_{b}$ and $M_{w}$ vs. $M_{S}$ regressions. This improvement is minor, within the typical error of $M_{w}$ . Moreover, a test subset of 100 predicted magnitudes shows that the new approach results in magnitudes closer to the theoretically true magnitudes for only 65 % of them. For the remaining 35 %, the standard approach produces closer values. Therefore, the new approach does not always give the most accurate magnitude estimates.

A Case Study to Select an Optimal Split-Plot Design for a Mixture-Process Experiment Based on Multiple Objectives

ABSTRACTWith increasingly constrained budgets, it is now becoming more desirable to get more information from each experiment and to have an intentional strategy for selecting designs for split-plot experiments that balance multiple competing objectives. Lu and Anderson-Cook (2014) developed a decision-making process for selecting an optimal split-plot design (SPD) for flexible objectives/criteria based on a Pareto front. The method allows exploration of all contending non-inferior choices with their trade-offs to enable an informed and justifiable decision based on understanding the potential impact of subjective aspects. This article considers a case study of a mixture-process experiment that seeks an SPD with a good balance of precise model coefficient estimates as measured by D-efficiency and low experimental cost, which is a function of both the time required to run the experiment as well as the financial cost. The D-efficiency is a function of the whole plot-to-subplot error variance ratio, a quantity that is typically not known a priori when the choice of a design must be made. The Pareto front approach is applied and graphical tools are used to quantify the trade-offs between criteria and robustness of design performance to different user-selected preferences for the criteria. A substantially different pattern of design performance robustness to the uncertainty of the specified variance ratio is demonstrated compared to non-mixture experiments.

Reply to "Comment on 'General Orthogonal Regression Relations between Body-Wave and Moment Magnitudes' by Ranjit Das, H. R. Wason, and M. L. Sharma" by Paolo Gasperini and Barbara Lolli

Reply| March 01, 2014 Reply to “Comment on ‘General Orthogonal Regression Relations between Body‐Wave and Moment Magnitudes’ by Ranjit Das, H. R. Wason, and M. L. Sharma” by Paolo Gasperini and Barbara Lolli Ranjit Das; Ranjit Das Earthquake Engineering Department, Indian Institute of Technology Roorkee, Roorkee 247667, Indiaranjit244614@gmail.comwason2009@gmail.comsharmamukat@gmail.com Search for other works by this author on: GSW Google Scholar H. R. Wason; H. R. Wason Earthquake Engineering Department, Indian Institute of Technology Roorkee, Roorkee 247667, Indiaranjit244614@gmail.comwason2009@gmail.comsharmamukat@gmail.com Search for other works by this author on: GSW Google Scholar M. L. Sharma M. L. Sharma Earthquake Engineering Department, Indian Institute of Technology Roorkee, Roorkee 247667, Indiaranjit244614@gmail.comwason2009@gmail.comsharmamukat@gmail.com Search for other works by this author on: GSW Google Scholar Author and Article Information Ranjit Das Earthquake Engineering Department, Indian Institute of Technology Roorkee, Roorkee 247667, Indiaranjit244614@gmail.comwason2009@gmail.comsharmamukat@gmail.com H. R. Wason Earthquake Engineering Department, Indian Institute of Technology Roorkee, Roorkee 247667, Indiaranjit244614@gmail.comwason2009@gmail.comsharmamukat@gmail.com M. L. Sharma Earthquake Engineering Department, Indian Institute of Technology Roorkee, Roorkee 247667, Indiaranjit244614@gmail.comwason2009@gmail.comsharmamukat@gmail.com Publisher: Seismological Society of America First Online: 14 Jul 2017 Online ISSN: 1938-2057 Print ISSN: 0895-0695 © 2014 by the Seismological Society of America Seismological Research Letters (2014) 85 (2): 352–353. https://doi.org/10.1785/0220130145 Article history First Online: 14 Jul 2017 Connected Content Commentary: Comment on “General Orthogonal Regression Relations between Body‐Wave and Moment Magnitudes” by Ranjit Das, H. R. Wason, and M. L. Sharma Cite View This Citation Add to Citation Manager Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Get Permissions Search Site Citation Ranjit Das, H. R. Wason, M. L. Sharma; Reply to “Comment on ‘General Orthogonal Regression Relations between Body‐Wave and Moment Magnitudes’ by Ranjit Das, H. R. Wason, and M. L. Sharma” by Paolo Gasperini and Barbara Lolli. Seismological Research Letters 2014;; 85 (2): 352–353. doi: https://doi.org/10.1785/0220130145 Download citation file: Ris (Zotero) Refmanager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentBy SocietySeismological Research Letters Search Advanced Search The authors’ pointwise response to the comments raised by Paolo Gasperini and Barbara Lolli is given below. For the sake of brevity, we will refer hereafter to their comments as G&L. First, Das et al. (2012) derived the general orthogonal regression (GOR) relation between mb and Mw following the minimization procedure as described by Fuller (1987) for error variance ratio (η) value equal to one. In order to obtain an estimate of the dependent variable (Mw) from the derived GOR line for any observed value of the independent variable (mb,obs),... You do not have access to this content, please speak to your institutional administrator if you feel you should have access.

Reply to ‘Comment on “Magnitude conversion problem using general orthogonal regression” by H. R. Wason, Ranjit Das and M. L. Sharma’ by Paolo Gasperini and Barbara Lolli

We completely reject the conclusions of P. Gasperini and Barbara Lolli made in the concluding remarks of their comments on the methodology and results of our paper (Wason et al. 2012). Based on reasoning and comparisons made between true values on the general orthogonal regression (GOR) line and the estimates of the variable given by the two procedures, we would like to reiterate that the proposed methodology is sound and provides improved estimates of the true points lying on the GOR line compared to the commonly used procedure for magnitude conversion. Hence, the use of the proposed method based on GOR in Wason et al. (2012) is suggested for magnitude conversion of data sets having error variance ratio of magnitudes (η) less than or equal to 2.5. However, for η > 2.5, the commonly used procedure of getting the unbiased estimates using observed values of the predictor variable in the GOR relation shall yield better estimates.

Magnitude conversion problem using general orthogonal regression

SUMMARY In this study, a procedure for the application of general orthogonal regression (GOR) towards conversion of different magnitude types is described. Through minimization of the squares of orthogonal residuals, GOR relation is obtained in terms of the abscissas (Mx*) of the projected points corresponding to the observed data pairs (Mx, obs, My, obs). In many studies, Mx* is replaced by Mx, obs in the GOR relation for convenience of obtaining the estimates of a preferred magnitude type for given magnitude values. Such forms of GOR, however, lead to biased estimates of the dependent variable. To represent the GOR relation correctly in terms of Mx, obs, a linear relation has been obtained between Mx* and Mx, obs using given points and the corresponding projected points on the GOR line. Based on events data for the whole globe during the period 1976–2007, GOR relations have been derived for conversion of mb to Mw,mb to Ms,mb to Me and Ms to Mw following the proposed procedure and using specific error variance ratio (η) values. The superiority of the GOR relations obtained following the proposed procedure over the commonly used forms has been shown by computing the absolute average difference and standard deviation between the observed and the estimated values using events data not used in the derivation. It is observed that the proposed GOR relations yield better estimates compared to the commonly used GOR forms. This procedure has been further tested for a wide range of η values between 0.1 and 7.0. The procedure proposed in this study can be used for the purpose of catalogue homogenization where GOR relations are applicable for conversion of different magnitude types.

Magnitude conversion to unified moment magnitude using orthogonal regression relation

Homogenization of earthquake catalog being a pre-requisite for seismic hazard assessment requires region based magnitude conversion relationships. Linear Standard Regression (SR) relations fail when both the magnitudes have measurement errors. To accomplish homogenization, techniques like Orthogonal Standard Regression (OSR) are thus used. In this paper a technique is proposed for using such OSR for preparation of homogenized earthquake catalog in moment magnitude Mw.For derivation of orthogonal regression relation between mb and Mw, a data set consisting of 171 events with observed body wave magnitudes (mb,obs) and moment magnitude (Mw,obs) values has been taken from ISC and GCMT databases for Northeast India and adjoining region for the period 1978–2006. Firstly, an OSR relation given below has been developed using mb,obs and Mw,obs values corresponding to 150 events from this data set. Mw=1.3(±0.004)mb,proxy-1.4(±0.130), where mb,proxy are body wave magnitude values of the points on the OSR line given by the orthogonality criterion, for observed (mb,obs, Mw,obs) points. A linear relation is then developed between these 150 mb,obs values and corresponding mb,proxy values given by the OSR line using orthogonality criterion. The relation obtained is mb,proxy=0.878(±0.03)mb,obs+0.653(±0.15).The accuracy of the above procedure has been checked with the rest of the data i.e., 21 events values. The improvement in the correlation coefficient value between mb,obs and Mw estimated using the proposed procedure compared to the correlation coefficient value between mb,obs and Mw,obs shows the advantage of OSR relationship for homogenization.The OSR procedure developed in this study can be used to homogenize any catalog containing various magnitudes (e.g., ML, mb, MS) with measurement errors, by their conversion to unified moment magnitude Mw. The proposed procedure also remains valid in case the magnitudes have measurement errors of different orders, i.e. the error variance ratio is different from unity.

Optimization of electrode system for the aircraft time-of-flight ion-marking airspeed and angle of attack sensor

A technique of optimizing the electrode system of the differential recorder for the aircraft time-of-flight ion-marking airspeed and aircraft angle of attack sensor by the criterion of ion-marking recording error variance minimum is considered. The technique is based on analyzing the function of spatial distribution for the ion-marking and recorder electrode interaction field potential and determining a point for which this function reaches the maximum slope. The relations making it possible to synthesize the sensor electrode system by the complex criterion as a ratio of error variance to transformation channel response are obtained.

Graphical solutions for structural regression assist errors-in-variables modelling

Structural regression attempts to reveal an underlying relationship by compensating for errors in the variables. Ordinary least-squares regression has an entirely different purpose and provides a relationship between error-included variables. Structural model solutions, also known as the errors-in-variables and measurement–error solutions, use various inputs such as the error–variance ratio and x-error variance. This paper proposes that more accurate structural line gradient (coefficient) solutions will result from using the several solutions together as a system of equations. The known data scatter, as measured by the correlation coefficient, should always be used in choosing legitimate combinations of x- and y-error terms. However, this is difficult using equations. Chart solutions are presented to assist users to understand the structural regression process, to observe the correlation coefficient constraint, to assess the impact of their error estimates and, therefore, to provide better quality estimates of the structural regression gradient.

Robust Estimators of Errors-in-Variables Models: Part 1

It is well known that consistent estimators of errors-in-variables models require knowledge of the ratio of error variances. What is not well known is that a Joint Least Squares estimator is robust to a wide misspecification of that ratio. Through a series of Monte Carlo experiments we show that an easy-to-implement estimator produces estimates that are nearly unbiased for a wide range of the ratio of error variances. These MC analyses encompass linear and nonlinear specifications and also a system on nonlinear equations where all the variables are measured with errors.

Common Slope Tests for Bivariate Errors-in-Variables Models

Summary Likelihood ratio tests are derived for bivariate normal structural relationships in the presence of group structure. These tests may also be applied to less restrictive models where only errors are assumed to be normally distributed. Tests for a common slope amongst those from several datasets are derived for three different cases – when the assumed ratio of error variances is the same across datasets and either known or unknown,and when the standardised major axis model is used. Estimation of the slope in the case where the ratio of error variances is unknown could be considered as a maximum likelihood grouping method. The derivations are accompanied by some small sample simulations,and the tests are applied to data arising from work on seed allometry.

Testing alternative methods of analysing differences in fish condition, with an example using the mosquitofish Gambusia holbrooki

In this study we have examined two different methods, the ANCOVA and a weighted procedure, in order to highlight the problems associated with the use of ANCOVA when the assumption of homogeneity of error variances among data sets is wrongly assumed. When error variances are unequal, ANCOVA procedure may lead to estimates of common regression coefficients with larger standard error values than those obtained when a weighted method is employed. Moreover, the ANCOVA test for homogeneity of regression coefficients seems to be differently affected when error variances are not all equal. Specifically, in the case of unequal sample sizes, when large error variance is combined with the small group and the larger: smaller error variance ratio was more than 2, ANCOVA tests for homogeneity of regression coefficients could be substantially affected leading to a Type I error. Although the ANCOVA approach is a simple and robust method in testing for equality of regression coefficients and related analysis, its application should be preceded by tests to determine homogeneity of error variances to ensure its correct use and avoid erroneous interpretations of the data analysed. The weighted method could be considered as a valuable alternative method to ANCOVA when homogeneity of error variances and regression coefficients cannot be assumed. Data for Gambusia holbrooki from Lake Banyoles and Alt Emporda salt marshes (Catalonia, Spain) are used to illustrate these procedures.

Using nonlinear functional relationship regression to fit fisheries models

Functional relationship regression refers to that class of statistical model where a functional relationship is assumed to exist between two arithmetic variables, but the two arithmetic variables can only be viewed with measurement error and (or) natural variability. The goal is to estimate the underlying functional relationship when observing only the variables containing error or natural variability. While statistical details need to be worked out, some general approaches to this class of problem can be recommended. First of all, nonlinear least squares can provide maximum likelihood estimates when the error variance ratio, λ;, is assumed known. Furthermore, the usual estimates of standard errors from nonlinear least squares, while theoretically flawed for this class of model, appear to provide usable estimates for many practical problems. Next, a K-sample F test, for testing the equality of nonlinear functional relationship regression curves, is proposed. Finally, computer memory-saving algorithms are suggested for situations where sample sizes are large. Methods proposed here are applied to three types of functional curves commonly estimated in fisheries biology: a stock-recruitment curve, allometry, and the von Bertalanffy growth curve.

Using nonlinear functional relationship regression to fit fisheries models

Functional relationship regression refers to that class of statistical model where a functional relationship is assumed to exist between two arithmetic variables, but the two arithmetic variables can only be viewed with measurement error and (or) natural variability. The goal is to estimate the underlying functional relationship when observing only the variables containing error or natural variability. While statistical details need to be worked out, some general approaches to this class of problem can be recommended. First of all, nonlinear least squares can provide maximum likelihood estimates when the error variance ratio, λ;, is assumed known. Furthermore, the usual estimates of standard errors from nonlinear least squares, while theoretically flawed for this class of model, appear to provide usable estimates for many practical problems. Next, a K-sample F test, for testing the equality of nonlinear functional relationship regression curves, is proposed. Finally, computer memory-saving algorithms are suggested for situations where sample sizes are large. Methods proposed here are applied to three types of functional curves commonly estimated in fisheries biology: a stock-recruitment curve, allometry, and the von Bertalanffy growth curve.

A statistical framework for multivariate latent variable regression methods based on maximum likelihood

A statistical framework is developed to contrast methods used for parameter estimation for a latent variable multivariate regression (LVMR) model. This model involves two sets of variables, X and Y, both with multiple variables and sharing a common latent structure with additive random errors. The methods contrasted are partial least squares (PLS) regression, principal component regression (PCR), reduced rank regression (RRR) and canonical co-ordinate regression (CCR). The framework is based on a constrained maximum likelihood analysis of the model under assumptions of multivariate normality. The constraint is that the estimates of the latent variables are restricted to be linear functions of the X variables, which is the form of the estimates for the methods being contrasted. The resulting framework is a continuum regression that goes from RRR to PCR depending on the ratio of error variances in the X and Y spaces. PLS does not arise as a member of the continuum; however, the method does offer some insight into why PLS would work well in practice. The constrained maximum likelihood result is also compared with the unconstrained maximum likelihood analysis to investigate the impact of the constraint. The results are illustrated on a simulated example. Copyright © 1999 John Wiley & Sons, Ltd.

Saddlepoint approximation in the linear structural relationship model

It is shown that the joint maximum likelihood estimator of slope and intercept of the regression line in the classical (known error-variance ratio) linear structural relationship model can be represented as a solution of a two- dimensional M- equation. Therefore, it is possible to use a general saddlepoint approximation for multidimensional M- equations. Under normality assumptions we express the solution of the implicit multivariate “centering equation” in an explicit form. This allows a considerable saving of computing time. By integrating out numerically an unwanted variable one is also able to find the saddlepoint approximation for the slope- estimator. Numerical examples illustrate the efficiency of the approximation.

Bootstrap Confidence Regions for Functional Relationships in Errors-in- Variables Models

We suggest bootstrap methods for constructing confidence bands (and intervals) for an unknown linear functional relationship in an errors-invariables model. It is assumed that the ratio of error variances is known to lie within an interval $\Lambda = \lbrack\lambda_1, \lambda_2\rbrack$. A confidence band is constructed for the range of possible linear relationships when $\lambda \in \Lambda$. Meaningful results are obtained even in the extreme case $\Lambda = \lbrack 0, \infty\rbrack$, which corresponds to no assumption being made about $\Lambda$. The bootstrap bands have several interesting features, which include the following: (i) the bands do not shrink to a line as $n \rightarrow \infty$, unless $\Lambda$ is a singleton (i.e., $\lambda_1 = \lambda_2)$; (ii) percentile-$t$ versions of the bands enjoy only first-order coverage accuracy, not the second-order accuracy normally found in simpler statistical problems.