Mean Squared Relative Error Research Articles

Parameter reduction, sensitivity studies, and correlation analyses applied to a mechanobiologically regulated bone cell population model of the bone metabolism.

When striving for reconstructing and predicting bone remodeling processes by means of mathematical models, cell population models have become a popular option. From a conceptual point of view, these models are able to take into account an arbitrary amount of regulatory mechanisms driving the development of bone cells and their activities. However, in most cases, the models include a large number of parameters; and most of those parameters cannot be measured, which certainly compromises the credibility of cell population models. Here, new insights are presented as to the potential improvement of this unsatisfactory situation. In particular, a previously published bone remodeling model was considered, and based on combination and merging of the original parameters, the total number of parameters could be reduced from 28 to 18, without impairing the model's versatility and significance. Furthermore, a comprehensive number of one- and two-variable sensitivity studies were performed, pointing out which parameters (alone and in combination with other parameters) influence the model predictions significantly - for that purpose, the mean squared relative error (MSRE) between simulations based on the original parameters and based on varied parameters was considered as failure measure. It has turned out that the model is significantly more sensitive to parameters which can be considered as phenomenological (such as differentiation, proliferation, and apoptosis rates) than to parameters which are directly related to specific processes (such as dissociation rate constants, and maximum concentrations of the involved factors). Using common correlation measures (such as Pearson, Spearman, and partial ranked correlation coefficients), correlation studies revealed that the correlations between most parameters and the MSRE are weak, while a few parameters exhibited moderate correlations. In conclusion, the results shown in this paper provide valuable insights concerning the design of new experiments allowing for measurement of the parameters which are most influential in the context of bone remodeling simulation.

Nonparametric relative recursive regression

Abstract In this paper, we propose the problem of estimating a regression function recursively based on the minimization of the Mean Squared Relative Error (MSRE), where outlier data are present and the response variable of the model is positive. We construct an alternative estimation of the regression function using a stochastic approximation method. The Bias, variance, and Mean Integrated Squared Error (MISE) are computed explicitly. The asymptotic normality of the proposed estimator is also proved. Moreover, we conduct a simulation to compare the performance of our proposed estimators with that of the two classical kernel regression estimators and then through a real Malaria dataset.

Raster Image Correlation Spectroscopy Performance Evaluation

Raster image correlation spectroscopy (RICS) is a fluorescence image analysis method for extracting the mobility, concentration, and stoichiometry of diffusing fluorescent molecules from confocal image stacks. The method works by calculating a spatial correlation function for each image and analyzing the average of those by model fitting. Rules of thumb exist for RICS image acquisitioning, yet a rigorous theoretical approach to predict the accuracy and precision of the recovered parameters has been lacking. We outline explicit expressions to reveal the dependence of RICS results on experimental parameters. In terms of imaging settings, we observed that a twofold decrease of the pixel size, e.g., from 100 to 50 nm, decreases the error on the translational diffusion constant (D) between three- and fivefold. For D = 1 μm2 s−1, a typical value for intracellular measurements, ∼25-fold lower mean-squared relative error was obtained when the optimal scan speed was used, although more drastic improvements were observed for other values of D. We proposed a slightly modified RICS calculation that allows correcting for the significant bias of the autocorrelation function at small (≪50 × 50 pixels) sizes of the region of interest. In terms of sample properties, at molecular brightness E = 100 kHz and higher, RICS data quality was sufficient using as little as 20 images, whereas the optimal number of frames for lower E scaled pro rata. RICS data quality was constant over the nM–μM concentration range. We developed a bootstrap-based confidence interval of D that outperformed the classical least-squares approach in terms of coverage probability of the true value of D. We validated the theory via in vitro experiments of enhanced green fluorescent protein at different buffer viscosities. Finally, we outline robust practical guidelines and provide free software to simulate the parameter effects on recovery of the diffusion coefficient.

Artificial neural network applied forecast on a parametric study of Calophyllum inophyllum methyl ester-diesel engine out responses

This experimental work presents a parametric investigation of Calophyllum inophyllum methyl ester (CIME)-diesel engine operations and artificial neural network (ANN) applied forecast of the engine out responses. The engine tests were performed for five test fuels from idle to full load conditions with the stipulated increment of 25% of the load for every run at three selected injection timings (21°, 23° and 25° CA bTDC) for 220bar, 260bar and 300bar injection pressures. The experimental outcomes indicated that twenty percentage blend of the biodiesel in neat diesel (CIME20) showed the highest brake thermal efficiency (BTE) among the CIME-diesel operations for 300bar injection pressure at 23° CA bTDC injection timing whereas BTE for the test fuels reduced at advanced and retarded injection timings at full load. CO, UBHC, dry soot and engine out O2 emissions were reduced at the advanced injection timing whereas NO and CO2 emissions increased. Using steady state experimental data, separate ANN models were proposed to forecast performance (BTE, BSEC, EGT) and emission (CO, CO2, UBHC, NO, dry soot and engine out O2) parameters with percentage load, blend percentage, injection pressure and injection timing as selected input control variables. The proposed ANN models indicated an impressive agreement as correlation coefficient (R) and mean absolute percentage error (MAPE) values perceived in the range of 0.99879–0.99993 and 0.87–4.62% respectively with remarkably lower root mean squared errors. Besides, lower values of mean squared relative error (MSRE) and noteworthy Nash-Sutcliffe coefficient of Efficiency (NSE) indices reasonably demonstrated robustness of the proposed models. Moreover, observed values of forecasting uncertainty Theil U2 indicated more effective outcomes for a credible model forecasting ability.

Estimating Spatial Precipitation Using Regression Kriging and Artificial Neural Network Residual Kriging (RKNNRK) Hybrid Approach

A hybrid model, combining regression kriging and neural network residual kriging (RKNNRK), is developed for determining spatial precipitation distribution. The RKNNRK model is compared with current spatial interpolation models, including simple kriging (SK), ordinary kriging (OK), universal kriging (UK), regression kriging (RK) and neural network residual kriging (NNRK). Results show that hybrid models, including RK, NNRK and RKNNRK, performed better than SK, OK and UK, based on the coefficient of efficiency (CE), coefficient of determination (r2), index of agreement (d), mean squared relative error (MSRE), mean absolute error (MAE), root-mean-square error (RMSE), and mean squared error (MSE). Of the six spatial interpolation models, the RKNNRK model was the most accurate, and the NNRK model was the second best.

Mixed gas separation study for the hydrogen recovery from H 2/CO/N 2/CO 2 post combustion mixtures using a Matrimid membrane

In this work, the membrane separation of hydrogen from binary, ternary and quaternary mixtures of H 2, N 2, CO and CO 2 is presented. Hydrogen permeability through a polyimide Matrimid 5218 membrane was experimentally obtained using the constant pressure technique. The influence of the feed gas composition, temperature (30–100 °C), pressure range (up to 6 bar), and flow rates was experimentally analyzed. As expected, the pure gas permeability of H 2 was only slightly dependant on pressure and had an average value of 17.7 × 10 −14 m 3(STP) m m −2 s −1 kPa −1 at 30 °C. Hydrogen permeability was not affected by the presence of nitrogen and carbon monoxide, and as a result the mixed gas selectivities for the H 2/N 2/CO mixtures are very close to the selectivities calculated from pure gas permeation data. On the contrary, a strong dependency of the hydrogen permeability on CO 2 concentration was observed even at low concentrations of CO 2. A reduction of 42% of the hydrogen permeability coefficient was obtained when a mixture of 10/90 vol.% H 2/CO 2 was used as feed gas. Accordingly H 2/CO 2 selectivity decayed from a value of 4.2 calculated from pure gas permeabilities to 2.7 when permeation data were obtained in mixed gas experiments. The preferential sorption of CO 2 on the Langmuir sites of the excess free volume portion of the polymer allowed explaining and quantifying this phenomenon. The “dual-mode sorption, partial immobilization” model was used to describe H 2 and CO 2 permeation behavior of pure, binary, ternary and quaternary mixtures. The model sorption parameters for N 2 and CO 2 in the polymer Matrimid 5218 were obtained from the literature meanwhile those for H 2 and for CO were unknown and resulted from the fitting of the experimental data to the proposed model. Satisfactory agreement between predicted permeability results and experimental data with a correlation coefficient ( R) higher than 0.95 and mean squared relative error (MSRE) lower than 0.01 was attained. Thus, this work reports useful knowledge related to the intrinsic material properties, considering gas mixtures of industrial interest and essential when other membrane configurations like hollow fibers, mixed matrix membranes or polymer blends are proposed.

Oxygen permeation of Ba xSr 1 − xCo 0.8Fe 0.2O 3 − δ perovskite-type membrane: Experimental and modeling

Ba x Sr 1 − x Co 0 .8 Fe 0 .2 O 0.3 − δ ( x = 0.2, 0.5, 0.8) dense membranes were prepared by a combined EDTA and citrate complexing method. In our previous works, effects of sintering temperature, sintering dwell time and pressing pressure on microstructure and theoretical densities of the membranes were examined and finally the best corresponding values were reported as 1100 °C, 8–9 h and 200–250 MPa, respectively. In the present work, effects of temperature (650–950 °C), feed flow rate (100–200 cm³/min), sweep gas flow rate (40–80 cm³/min) and membrane thickness (4–5 mm) on oxygen permeation behavior of the Ba x Sr 1 − x Co 0 .8 Fe 0 .2 O 0.3 − δ membranes were investigated. Also, a mathematical model based on Nernst–Planck equation was developed to predict oxygen permeation through the perovskite-type membranes. Both bulk diffusion and surface reactions were incorporated into the model. It was observed that surface reactions are not elementary and a correction term should be introduced into the model to compensate this effect. Also, using a dimensionless Reynolds number, effect of feed flow rate on oxygen flux was taken into account. With aids of these modifications, it was realized that, there is a reasonable agreement between predicted results and experimental data with correlation coefficient (R) of higher than 0.960 and mean squared relative error (MSRE) of lower than 0.022 for all the membranes. Oxygen vacancy bulk diffusion coefficient (D v), surface exchange rate constants (k f and k r), contribution of each resistance to oxygen permeation and characteristic thickness (L c) of the BSCF membranes were also estimated.

Prediction of ternary gas permeation through synthesized PDMS membranes by using Principal Component Analysis (PCA) and fuzzy logic (FL)

Implementation of try and error method for membrane preparation procedures is a time and cost consuming technique. This study tries to present a novel idea to make membrane preparation procedure heuristic. Applying this method, helps researchers to predict performance of a membrane prior to its preparation. At first, a number of membranes are prepared and characterized. Then, their measured separation properties are used for prediction of performance of a membrane before its preparation. Furthermore, after preparation of each new membrane, its relevant data can be added to the data bank of the model to improve its capability for the next predictions. Therefore, this model will be improved step by step, after each new preparation. Fuzzy logic-based (FL) model and Principal Component Analysis (PCA) were employed to predict permeability of C 3H 8, CH 4 and H 2 in ternary gas mixtures using a membrane gas separation module. Based on Placket–Burman (P–B) design, eight different polydimethylsiloxane (PDMS) membranes were synthesized using different preparation conditions, solvent concentration, crosslinker concentration, catalyst concentration, type (or boiling point) of solvent, stirring time and synthesis times in ambient temperature and in an oven at 80 °C, and their gas permeation properties were investigated. In an innovating procedure, effects of operating conditions, including feed temperature, pressure, flow rate, C 3H 8 and H 2 concentration, as well as preparation conditions on the permeability of gasses through the synthesized membranes were investigated. Basically, in order to develop a FL model to predict permeability of gasses through all the synthesized membranes, synthesis and operating conditions should be considered, simultaneously, and this extends dimensionality of the problem. In all engineering problems, as the number of variables increases, the corresponding data matrix extends. To overcome the problem, PCA method was randomly used for seven of the prepared membranes from P–B design, and it was shown that the first four principal components could describe almost all of the variation in the data matrix. This means that the dimensionality of the problem reduced from 12 to 4. Using the first four principal components, a Sugeno type FL inference system was trained and applied to predict permeability of gasses. FL modeling results showed that there is an excellent agreement between the experimental data and the predicted values, with mean squared relative error (MSRE) of less than 0.0095. The developed model was used for the 8th membrane and its ability to predict separation parameters of this membrane was confirmed.

Ternary gas permeation through a synthesized PDMS membrane: Experimental and modeling

A single layer PDMS membrane was synthesized and gas permeation behavior of ternary gas mixtures consisting of C 3H 8, CH 4 and H 2 through it was investigated. Permeability, solubility and diffusivity of gases were measured at different upstream pressures, feed temperatures and feed compositions. The results showed that, increasing pressure and temperature decreases permeability of more condensable gas, C 3H 8, while increases that of lighter gases, H 2 and CH 4. Solubility of all components in the gas mixture increased with increasing pressure and decreasing temperature. Diffusivity of all components increased with increasing temperature. With increasing pressure that of the lighter gases increased, while diffusivity of the more condensable gas decreased. Increasing C 3H 8 concentration in the feed improved permeation properties of all components. A mathematical model was also developed based on Fick's law of diffusion to predict permeation behavior of ternary gas mixtures. The developed model is able to predict the permeability, solubility, diffusivity and activation energy of all components as well as the permeate composition using feed stream data. It was realized that, there is a reasonable agreement between predicted results and experimental data with correlation coefficient ( R) of higher than 0.95 and mean squared relative error (MSRE) of lower than 0.2.

Coupling a mathematical and a fuzzy logic-based model for prediction of zinc ions separation from wastewater using electrodialysis

Abstract This paper presents experimental data, a fuzzy logic (FL) model, a mathematical model (MM) and a coupled MM–FL model for a laboratory scale electrodialysis (ED) cell. The aim was to predict separation percent (SP) of zinc ions as a function of concentration, temperature, flow rate and voltage. At first, a Sugeno type FL inference system was applied to model zinc ions separation from wastewater using ED. FL modeling results showed that there is an excellent agreement between the experimental data and the predicted values, with mean squared relative error (MSRE) of less than 0.01. Then, the results of a previously developed MM were presented. The MM related SP to hydrodynamic dimension of the ED cell and operation conditions via two distinct parameters. This ability favored the MM for scale-up applications. However, based on MSRE of the MM (about 24), it could not obviously predict the experimental data as well as FL. Hence, as a final step, the MM was coupled with FL to achieve benefits of both. It was found out that the developed coupled model (MM–FL) is able to predict SP of zinc ions at all operating condition and almost every dimension to a high degree of accuracy (MSRE = 0.05).

On the Role of Semielasticity in Statistics

Assun¸cão showed that the lower bound for the mean squared relative error (MSRE) of an unbiased estimator can be expressed in terms of elasticity, thus providing a meaningful interpretation of the MSRE lower bound. Where the MSE is preferred to MSRE, however, the idea of elasticity does not apply and must be replaced by semielasticity. In this article, we restate the Fisher information measure, the Cramér-Rao lower bound, and the identification problem in terms of semielasticity.

Performance of nonparametric species richness estimators in a high diversity plant community

Abstract. The efficiency of four nonparametric species richness estimators — first‐order Jackknife, second‐order Jackknife, Chao2 and Bootstrap — was tested using simulated quadrat sampling of two field data sets (a sandy ‘Dune’ and adjacent ‘Swale’) in high diversity shrublands (kwongan) in south‐western Australia. The data sets each comprised > 100 perennial plant species and > 10 000 individuals, and the explicit (x‐y co‐ordinate) location of every individual. We applied two simulated sampling strategies to these data sets based on sampling quadrats of unit sizes 1/400th and 1/100th of total plot area. For each site and sampling strategy we obtained 250 independent sample curves, of 250 quadrats each, and compared the estimators’ performances by using three indices of bias and precision: MRE (mean relative error), MSRE (mean squared relative error) and OVER (percentage overestimation). The analysis presented here is unique in providing sample estimates derived from a complete, field‐based population census for a high diversity plant community. In general the true reference value was approached faster for a comparable area sampled for the smaller quadrat size and for the swale field data set, which was characterized by smaller plant size and higher plant density. Nevertheless, at least 15–30% of the total area needed to be sampled before reasonable estimates of St (total species richness) were obtained. In most field surveys, typically less than 1% of the total study domain is likely to be sampled, and at this sampling intensity underestimation is a problem. Results showed that the second‐order Jackknife approached the actual value of St more quickly than the other estimators. All four estimators were better than Sobs (observed number of species). However, the behaviour of the tested estimators was not as good as expected, and even with large sample size (number of quadrats sampled) all of them failed to provide reliable estimates. First‐ and second‐order Jackknives were positively biased whereas Chao2 and Bootstrap were negatively biased. The observed limitations in the estimators’ performance suggests that there is still scope for new tools to be developed by statisticians to assist in the estimation of species richness from sample data, especially in communities with high species richness.