Curve fitting and jump detection on nonparametric regression with missing data

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.

Background: Nowadays, forecasting models based on nonparametric models have been developed in many branches of science such as mathematics and economics. However, the relatively complicated structure of these models has made them less practical in medical sciences. Objectives: In this study, we investigated the application of a nonparametric regression model to predict the psychological symptoms appearing six months after mild traumatic brain injury. We also made a comparison between the performance of nonparametric regression and artificial neural network models in predicting the psychological symptoms. Methods: In a six-month period during 2015 - 2017, information of 100 mild traumatic patients was included in a prospective cohort study. The data were then divided randomly into two groups of training (n = 50) and testing (n = 50) for 100 times. In the training group, the focus was on 100 artificial neural network and nonparametric regression models. However, in the testing group, a comparison was made between the values obtained using the two final models. To compare the models, the ROC curve and the accuracy rate were finally applied. Results: According to the obtained results, the nonparametric regression model showed an accuracy rate of 91.25% while the neural network model had an accuracy rate of 85.34%. In the experimental set for both neural network and nonparametric regression models, the area under the ROC curve appeared to be 81.51% and 85.73 %, respectively. Conclusions: The nonparametric regression model appeared to be more powerful than the neural network model in predicting psychological symptoms.

The aim of this study was to investigate the curve fitting and model selection problem of the torque-velocity relationship of elbow flexors and extensors in untrained females. The second goal was to determine the optimal models in different function classes and the best, among the optimal ones. Lastly, test the best models to predict the torque were tested. Using the polynomials (second - fourth degree) and Boltzmann sigmoid functions, and a different presentation of data points (averages, a point cloud, etc.), we determined the optimal models by both error criteria: minimum residual sum of squares and minimum of the maximal absolute residue. To assess the best models, we applied Akaike and Bayesian information criteria, Hausdorff distance and the minimum of the smallest maximal absolute residue and the predictive torque-velocity relationships of the best models with torque values, calculated beyond the experimental velocity interval. The application of different error and model selection criteria showed that the best models in the majority of cases were polynomials of fourth degree, with some exceptions from second and third degree. The criteria values for the optimal Boltzmann sigmoids were very close to those of the best polynomial models. However, the predicted torque-velocity relationships had physiological behavior only in Boltzmann's sigmoid functions, and their parameters had a clear interpretation. The results obtained suggest that the Boltzmann sigmoid functions are suitable for modeling and predicting of the torque-velocity relationship of elbow flexors and extensors in untrained females, as compared to polynomials, and their curves are physiologically relevant.

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.

Impact of simultaneous omission of a variable and an observation on a linear regression equation

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.

When experimental data are submitted to analysis of variance, the assumption of data homoscedasticity (variance homogeneity among treatments), associated to the adopted mathematical model must be satisfied. This verification is necessary to ensure the correct test for the analysis. In some cases, when data homoscedascity is not observed, errors may invalidate the analysis. An alternative to overcome this difficulty is the application of the specific residue analysis, which consists of the decomposition of the residual sum of squares in its components, in order to adequately test the correspondent orthogonal contrasts of interest between treatment means. Although the decomposition of the residual sum of squares is a seldom used procedure, it is useful for a better understanding of the residual mean square nature and to validate the tests to be applied. The objective of this review is to illustrate the specific residue application as a valid and adequate alternative to analyze data from experiments following completely randomized and randomized complete block designs in the presence of heteroscedasticity.

The objective of this study was to evaluate alternatives for modeling the age of dam at calving (AOD) effect on the weaning weight of Charolais-Zebu crossbred calves. Data from 56,965 calves were analyzed, using statistical models considering the fixed effects of the contemporary groups, sire and dam genetic groups, and AOD. The AOD effect was fitted to models using annual age classes, and ordinary quadratic to quintic-ordered polynomials (OP) or segmented polynomials (SP) with two, three, four, six and twelve evenly spaced intervals. In the case of segmented polynomials, general linear and quadratic effects and only one quadratic additional term from each knot were considered. The AOD effects were nested within sex of calf in all cases. According to the fitting criteria, the F-test for the reduction of residual sum of squares, coefficient of determination, residual sum of squares and mean of squared residuals, the three interval segmented polynomial (two knots) fitted to the data as well as the more complex polynomials.

When forest stock volume is quantitative estimated using SPOT-5, QuickBird and ALOS optical data with linear regression model, the optimal ratio of remote sensing band is chosen from the above three types of optical remote sensing data respectively, which is a significant part. In this study, the experiments are taken in Zhazuo Forest of Xiuwen County of Guizhou Province. Comprehensive utilization of the three optical data, the selected ratio of band is confirmed according with characteristics of the forest region. Optimization of the ratio of band remote sensing method used is the criteria of mean residual sum of square called RMSq. In this paper the multicollinearity which commonly exsits between ratio of the original band is analyzed and studied to get rid of its unfavorable influence in this paper. By means of the criteria of mean residual sum of square, the ratio of remote sensing band which determines the impact of forest stock volume estimation is confirmed finally. Conclusions are as follows: Compared with the selected band, multiple-correlation has been greatly reduced. The optimal ratio of remote sensing band such as SP4, SP2-3/2+3, SP 1-4/1+4, SP1*3/2 has an important role on the interpretation of forest stock volume estimation.

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.

Electromyography (EMG) and mechanomyography (MMG) have been used to directly evaluate muscle function through the electromechanical aspect of muscle contraction. The purpose of this study was to establish new absolute indices to describe muscle contraction performance during dynamic exercise by combining EMG and displacement MMG (dMMG) measured simultaneously using our previously developed MMG/EMG hybrid transducer system. Study participants were eight healthy male non-athletes (controls) and eight male athletes. EMG and dMMG of the vastus medialis were measured for 30 s during four cycles of recumbent bicycle pedaling (30, 60, 90, and 120 W) and on passive joint movement. Total powers were calculated based on the time domain waveforms of both signals. Muscle contraction performance was verified with the slope of regression line (SRL) and the residual sum of squares (RSS) obtained from EMG and dMMG correlation. EMG and dMMG has increased with the work rate. Force and EMG were similar between groups, but dMMG showed a significant difference with load increase. Athletes had significantly higher SRL and significantly lower RSS than controls. The average value divided by SRL and RSS was higher in athletes than in controls. The indices presented by the combined approach of EMG and dMMG showed a clear contrast between the investigated groups and may be parameters that reflect muscle contraction performance during dynamic exercise.

Dynamic experiments were carried out on the adsorption of copper (II) ions in a laboratory packed bed of activated carbon. Both pulse and step change inputs were used. Measurements were reproducible and replicate measurements furnished nearly identical parameters for a heterogeneous model of the activated carbon bed assuming Langmuir adsorption isotherm. The initial concentration of copper (II) ion was 3 ppm and the average particle size of activated carbon was 1.5 mm. The adsorption experiments in the shallow beds of activated carbon, 3 and 5 cm bed height had little effect on the parameters. Hydraulic loading, 25 and 50 ml min-1, affected dispersion and the volumetric mass transfer coefficient, but unexpectedly also changed the effective diffusivity. This apparently comes from the incorporation of errors due to instrumental noise and model approximations in the parameters. A 5 min pulse input has a low residual sum of square (RSS) with all of set of starting point while a 10 min pulse has a low RSS using some set of starting point. A step up input has a high RSS that may arise from a large number of data. These affected the Langmuir adsorption isotherm constant and diffusivity as expected from their concentration dependency. Step change and pulse simulations using the model agreed well with the experiments.

For checking on heteroscedasticity in regression models, a unified approach is proposed to constructing test statistics in parametric and nonparametric regression models. For nonparametric regression, the test is not affected sensitively by the choice of smoothing parameters which are involved in estimation of the non parametric regression function. The limiting null distribution of the test statistic remains the same in a wide range of the smoothing parameters. When the covariate is one-dimensional, the tests are, under some conditions, asymptotically distribution-free. In the high-dimensional cases, the validity of bootstrap approximations is investigated. It is shown that a variant of the wild bootstrap is consistent while the classical bootstrap is not in the general case, but is applicable if some extra assumption on conditional variance of the squared error is imposed. A simulation study is performed to provide evidence of how the tests work and compare with tests that have appeared in the literature. The approach mayreadily be extended to handle partial linear, and linear autoregressive models.