A robust nonlinear identification algorithm using press statistic and forward regression

ABSTRACTGeneralized estimating equations (GEE) is one of the most commonly used methods for regression analysis of longitudinal data, especially with discrete outcomes. The GEE method accounts for the association among the responses of a subject through a working correlation matrix and its correct specification ensures efficient estimation of the regression parameters in the marginal mean regression model. This study proposes a predicted residual sum of squares (PRESS) statistic as a working correlation selection criterion in GEE. A simulation study is designed to assess the performance of the proposed GEE PRESS criterion and to compare its performance with its counterpart criteria in the literature. The results show that the GEE PRESS criterion has better performance than the weighted error sum of squares SC criterion in all cases but is surpassed in performance by the Gaussian pseudo-likelihood criterion. Lastly, the working correlation selection criteria are illustrated with data from the Coronary Artery Risk Development in Young Adults study.

Regression methods are employed in analyzing a Space Shuttle failure data set. A class of models based on transforms of cumulative time and cumulative failures is considered. It is used to predict both current as well as future log transformed failure intensity. This class includes several popular software reliability models including the exponential, logarithmic, and power models. It also includes models based on transforms of the time per failure, the time-varying analogue to the mean time between failures. Models are compared on the basis of their predictive performance as measured by the predicted residual sum of squares (PRESS) criterion. Time per failure is included as one of the independent variables in the models identified as having the lowest PRESS score, both when predicting current as well as future log failure intensity. Analyses are conducted using failures within a final subset of the observation interval. This subset is chosen through inspection of the plot of time per failure in terms of cumulative time. The impact of the choice of this subset is assessed by comparing results for analyses of that subset with analyses of the data in the complete observation interval.

The Box-Tidwell represents a commonly-used iterative approach in linear or nonlinear regression but is little used in reliability modeling. It provides a power transformation of the regressor variable in order to linearize the model, or (occasionally) defaults to a log transformation. Its main drawback is lack of convergence under certain circumstances which results in it recommending a log transformation inappropriately. The techniques developed in this paper significantly increase the Box-Tidwell's robustness and ensure a power transformation solution is consistently found. Used along with weighted least squares (WLS), the Box-Tidwell transformation represents a real alternative to maximum likelihood or graphical estimation. This paper takes the example of the power-law model used in reliability growth analysis and demonstrates the application and effectiveness of the robust Box-Tidwell. Extensive simulation modelling has shown it generally provides a better fit to the data than the alternative maximum likelihood estimates. It illustrates that maximum likelihood methods do not always provide the 'best' estimator in the sense of one that minimizes a suitable loss function. The comparative analysis was conducted using simulation of 10, 30, and 100 observations for the power-law model. Cross validation was conducted using the predicted residual sum of squares (PRESS) statistic. Contrary to expectation, the PRESS statistics shows that the parameter estimation by this methodology (called BTW) will provide the best fit to the data (in the sense of minimizing the sum of the squared errors), and not estimation by maximum likelihood methods. The BTW will provide the best interpolated predictions compared to the alternatives

SummaryPartial least square regression analysis was used to study the correlation between X variables (semolina quality, hydration level and mixing time) and Y variables, which were, in a first model, dough consistency during mixing, and, in a second model, dough properties after mixing (strength, elasticity, density) and leavening (maximum volume). The first model showed a predictive residual sum of squares (PRESS) of 2.98 and a predictive R2 (Q2) of 0.92, and highlighted the key role of hydration and mixing time on dough consistency. The second model had the best PRESS (8.25) and Q2 (0.94) values for dough volume and indicated that the volume increased with increasing mixing time until the dough consistency decreased of 20–30%. Dough volume was primarily affected by hydration. The model indicated that maximum volume after leavening, corresponding to optimum mixing time, was obtained with a soft and elastic dough, with a low‐density value.

This paper addresses a methodological technique of leave-many-out cross-validation for choosing cutoff values in stepwise regression methods for simplifying the final regression model. A practical approach to choose cutoff values through cross-validation is to compute the minimum Predicted Residual Sum of Squares (PRESS). A leave-one-out cross-validation may overestimate the predictive model capabilities, for example see Shao (1993) and So et al (2000). Shao proves with asymptotic results and simulation that the model with the minimum value for the leave-one-out cross validation estimate of predictor errors is often over specified. That is, too many insignificant variables are contained in set ?i of the regression model. He recommended using a method that leaves out a subset of observations, called K-fold cross-validation. Leave-many-out procedures can be more adequate in order to obtain significant and optimal results. We describe various investigations for the assessment of performance of predictive regression models, including different values of K in K-fold cross-validation and selecting the best possible cutoff-values for automated model selection methods. We propose a resampling procedure by introducing alternative estimates of boosted cross-validated PRESS values for deciding the number of observations (l) to be omitted and number of folds/subsets (K) subsequently in K-fold cross-validation. Salahuddin and Hawkes (1991) used leave-one-out cross-validation to select equal cutoff values in stepwise regression which minimizes PRESS. We concentrate on applying K-fold cross-validation to choose unequal cutoff values that is F-to-enter and F-to-remove values which are then used for determining predictor variables in a regression model from the full data set. Our computer program for K-fold cross-validation can be efficiently used for choosing both equal and unequal cutoff values for automated model selection methods. Some previously analyzed data and Monte Carlo simulation are used to evaluate the proposed method against alternatives through a design experiment approach.

Traditional weight training programs feature alternating periods of exercise and recovery. In contrast, core stabilization tests used in wellness and fitness programs are performed continuously until volitional fatigue, which may lead to reduced adherence to fitness testing protocols. PURPOSE: To develop an interval prone bridge fitness test to assess core stability in healthy adults (ages 18-39 years). METHODS: Participants performed a prone bridge maneuver in alternating 15-sec work and 5-sec rest intervals with participants' RPE scores (0-10) recorded at the end of each work interval. The RPE score reported after 95 sec (RPE-95) was used to predict total interval prone bridge endurance time along with participants' self-reported level of physical activity (PA; sedentary = 0, low active = 1, active = 2, very active = 3). RESULTS: Multiple linear regression was employed to generate the following prediction equation (Total time (sec) = 300.0 - (23.4 × RPE-95) + (17.7 × PA); R =.86, SEE = 32.98 sec). Each predictor variable was statistically significant (RPE-95, p <.0001; PA, p = 0.006) and cross validation procedures using PRESS (predicted residual sum of squares) statistics revealed minimal shrinkage (Rp =.85 and SEEp = 32.89 sec). The mean and standard deviation (±SD) for the total duration of the interval prone bridge test and the RPE-95 data were 179.9 ± 65.2 sec and 6.3 ± 2.2, respectively. To assess test-retest reliability, a second test was completed about 48 hours after the first test. Reliability analyses (n = 45) yielded an acceptable test-retest intraclass reliability coefficient (ICC =.95, SEM = 12.7 sec) when comparing total interval prone bridge endurance times across days. CONCLUSION: The proposed interval prone bridge fitness test, and accompanying regression model, yields a relatively accurate estimate of total interval prone bridge test time in healthy men and women, using both RPE-95 and PA data.

The results obtained by implementing Principal Component Regression (PCR) according to three different criteria for choosing principal components (PCs), and those provided by Partial Least-Squares Regression (PSLR), in the determination of the active compound in a pharmaceutical preparation by near infrared diffuse reflectance spectroscopy are compared. The PCR-top down criterion used is commonly implemented in commercially available software: it selects consecutive PCs beginning with that possessing the largest eigenvalue. The other two criteria used do not assume the PCs with the largest eigenvalues to be the best predictors for the response variable; rather, the PCR-correlation criterion chooses only those PCs exhibiting the highest correlation with the response variable, and the PCR-best subset criterion selects those that provide the lowest predicted residual sum of squares ( PRESS) for an external prediction set. All the calibration methods tested exhibited a similar predictive ability (prediction errors ranged from 1.34% to 1.49%); however, the number of PCs used in the regression varied among them. The PLSR technique did not excel the methods based on selecting the best PCs for regression. Also, the PCR-correlation and PCR-best subset methods provided the same results and used fewer PCs than the PCR-top down method.

Maize is a feed crop with high nutritive value. However, the production environment for maize is becoming increasingly severe due to labor shortage or the introduction of organic stock farming. The aim of this study is to establish a monitoring system to support the production management of a maize field. This study examines the method of forecasting maize yield, feed quality, time required to acquire information on the field, and the platform used for remote sensing. Our aim is to evaluate the maize nutritive component at the time of harvest, using remote sensing. Calibration models to estimate the concentrations of various nutritive components such as moisture, total digestible nutrients (TDN), crude protein (CP), and organic cell wall (OCW) were developed using a dataset consisting of observed concentrations and spectral data acquired from a hyperspectral imaging sensor carried in the industrial unmanned helicopter. Prior to obtaining the calibration models, the samples were divided in half as calibration (n = 30) and validation (n = 30) datasets. Calibration models were obtained using partial least squares regression (PLSR) to estimate the maize nutritive composition. The estimation accuracy of the calibration models was examined by applying cross validation, and the evaluation index (EI) method was applied for evaluating the practicality of the calibration model. The EI evaluates the standard deviation of prediction (SDP), and can provide relative evaluation among different components. Five ranks are used: (A) very high (EI, <12.4%); (B) high (EI, 12.524.9%); (C) low (EI, 25.037.4%); (D) very low (EI, 25.037.4%); and (E) estimation impossible (EI, >50.0%). A calibration model has practical accuracy when its EI rank is A, B, or C at less than 37.4%. And the coefficient of determination (R2) of the observed and estimated values for calibration and validation was calculated for each model.In the results of the calibration models, the number of latent variables for estimate nutritive components was four for CP, three for moisture, TDN, and OCW. According to the relationship between the observed and estimated values obtained by the calibration model, most samples in the scatter plot of all nutritive compositions were distributed along the 45 line. The predicted residual sum of squares (PRESS) was 0.89 (moisture),0.82 (TDN), 0.71 (CP),and 0.75 (OCW), respectively. The R2 values of the calibration for all nutritive compositions were high (R2, >0.70); also, the R2 values of the estimation tests for all nutritive compositions were high (R2, >0.79). In addition, all models were ranked B by the EI test, confirming their practicality. In conclusion, this sensing system is a useful technique for estimating the nutritive composition of a maize field.

We conduct an exploratory data analysis of a specific system test data set that includes concomitant variables measuring aspects of test effort as well as the failure count and time variables considered in standard software reliability analyses. We consider a family of Poisson process models for which failure rates may depend on time as well as on the test effort variables. We analyze failure rates computed from this test data using least squares parameter estimation and model selection via the predicted residual sum of squares (PRESS). >

An automatic nonlinear predictive model-construction algorithm is introduced based on forward regression and the predicted-residual-sums-of-squares (PRESS) statistic. The proposed algorithm is based on the fundamental concept of evaluating a model's generalisation capability through crossvalidation. This is achieved by using the PRESS statistic as a cost function to optimise model structure. In particular, the proposed algorithm is developed with the aim of achieving computational efficiency, such that the computational effort, which would usually be extensive in the computation of the PRESS statistic, is reduced or minimised. The computation of PRESS is simplified by avoiding a matrix inversion through the use of the orthogonalisation procedure inherent in forward regression, and is further reduced significantly by the introduction of a forward-recursive formula. Based on the properties of the PRESS statistic, the proposed algorithm can achieve a fully automated procedure without resort to any other validation data set for iterative model evaluation. Numerical examples are used to demonstrate the efficacy of the algorithm.

The analytic function for the amplified spontaneous emission spectrum of InP/InGaAsP multi-quantum wells is studied by spectrum fitting. Three fitting functions, Lorentz, Gaussian and Sech line shape functions are chosen, and the analytical expressions for the above three functions are obtained with Levenberg-Marquardt algorithm, respectively. The center wavelength of Lorentz line shape function spectrum fitting is 1548.707 nm with 66.23 nm of full-width half maximum (FWHM), -0.00036484 mW power compensation, 0.98294 of R-square and 4.7674310-6 of residual sum of squares; the center wavelength of Gaussian line shape function spectrum fitting is 1548.651 nm with 61.42 nm of FWHM, 0.00212 mW power compensation, 0.99191 of R-square and 2.2650510-6 of residual sum of squares; the center wavelength of Sech line shape function spectrum fitting is 1548.787 nm with 36.99 nm of FWHM, 0.00222 mW power compensation, 0.98128 of R-square and 5.2433110-6 of residual sum of squares. It can be seen that Gaussian line shape function spectrum fitting has the highest R-square and smallest residual sum of squares, and the residual squares of data are symmetrically distributed among 0.0001. Gaussian line shape function spectrum fitting has higher fitting degree. It is demonstrated that InP/InGaAsP multi-quantum wells is a kind of active layer quantum well structure semiconductor material, whose amplified spontaneous emission spectrum line shape belongs to inhomogeneous broadening due to the effect of lattice defects, the corresponding line shape function is Gaussian line shape function, and the amplified spontaneous emission spectrum line shape function can be used for designing the optical passive devices.

Estimation of body mass from skeletal metrics can reveal important insights into the paleobiology of archeological or fossil remains. The standard approach constructs predictive equations from postcrania, but studies have questioned the reliability of traditional measures. Here, we examine several skeletal features to assess their accuracy in predicting body mass. Antemortem mass measurements were compared with common skeletal dimensions from the same animals postmortem, using 115 rhesus macaques (male: n = 43; female: n = 72). Individuals were divided into training (n = 58) and test samples (n = 57) to build and assess Ordinary Least Squares or multivariate regressions by residual sum of squares (RSS) and AIC weights. A leave-one-out approach was implemented to formulate the best fit multivariate models, which were compared against a univariate and a previously published catarrhine body-mass estimation model. Femur circumference represented the best univariate model. The best model overall was composed of four variables (femur, tibia and fibula circumference and humerus length). By RSS and AICw, models built from rhesus macaque data (RSS = 26.91, AIC = -20.66) better predicted body mass than did the catarrhine model (RSS = 65.47, AIC = 20.24). Body mass in rhesus macaques is best predicted by a 4-variable equation composed of humerus length and hind limb midshaft circumferences. Comparison of models built from the macaque versus the catarrhine data highlight the importance of taxonomic specificity in predicting body mass. This paper provides a valuable dataset of combined somatic and skeletal data in a primate, which can be used to build body mass equations for fragmentary fossil evidence.

The present article addresses the problem of inference in a multiscale computational model of pulmonary arterial and venous blood circulation. The model is a computationally expensive simulator which, given specific parameter values, solves a system of nonlinear partial differential equations and returns predicted pressure and flow values at different locations in the arterial and venous blood vessels. The standard approach in parameter calibration for computer code is to emulate the simulator using a Gaussian Process prior. In the present work, we take a different approach and emulate the objective function itself, i.e. the residual sum of squares between the simulations and the observed data. The Efficient Global Optimization (EGO) algorithm [2] is used to minimize the residual sum of squares. A generalization of the EGO algorithm that can handle hidden constraints is described. We demonstrate that this modified emulator achieves a reduction in the computational costs of inference by two orders of magnitude.

Agricultural settlement is one important policy goal adopted in Libya, and many other developing countries in Africa and the Middle East, to improve the level of living in rural areas. These countries share several general characteristics. Their agricultural production is low. They have a high percentage of nomadic and seminomadic people in their rural populations. They face a high rate of migration from rural to urban areas. Development under such conditions not only requires providing the right quantities and qualities of physical inputs, but also the selection of the most promising settlers as well. Therefore, this study of the "Gebel Al Akh dar" settlement scheme was designed to indicate the major sources of income of farmers on the project as well as to identify the major influences on their farm output. Hence, an attempt was made to identify factors which are essential for farm success in an unfavorable agro-climate environment. Specifically, the objectives of the study were: (1) to explain the variation in net farm income on the project, (2) to investigate the social and the economic conditions of the settlement scheme, and (3) to study the economic conditions of the settlers by investigating their levels and sources of farm income. Data for the study were collected by personal interview from a random sample of 120 farms. The framework of analysis used in testing the hypothesized relations were multiple regression, "t" tests on differences between mean incomes, and simple analysis of variance. Factor analysis was used in constructing two indices of modernization which, in turn, were regressed on selected socio economic variables. Sources of differences in incomes for settlers by project areas were examined using "t" tests on mean differences and analysis of variance tests. A stepwise multiple regression format with the "mean residual sum of squares" criterion (MSE) was used to identify the significant and relevant variables included in the production function analysis. The analysis suggested some alterations are needed in policy decisions with respect to farm inputs, selection and training of settlers, and research programs. For example, hired labor was found to be more rewarding than the use of tractors and other inputs on the scheme. This suggests that the machinery-labor mix of the project needs to be reconsidered. Also, the traditional attitudes of settlers, found to be elements of success in farming, direct the attention to new criteria for settler selection. Finally, since management ability was the most important variable on both dry and irrigated farms, training programs are needed that focus on local agricultural problems while considering abilities of settlers and the region's social and economic conditions.

SummaryAn exposition of the missing plot technique often applied in analysis of variance is given in very general terms.Non‐orthogonality mostly implies heavy computations. If the scheme of observations is almost orthogonal this technique, however, supplies in a simple way unbiassed and efficient estimates of the expectation values which occur in a linear hypothesis underlying an analysis of variance. Moreover the correct residual sum of squares required for a test or a confidence interval estimation is obtained without difficulty.A correct test of an effect or an interaction will be provided by two estimates, the first under the null‐hypothesis, the second under the alternative hypothesis. In the case of non‐orthogonality this may imply two separate applications of the discussed technique. The difference between the two residual sums of squares will be used for the numerator of a valid F‐criterion.The technique is illustrated by an example.