Third-order Polynomial Models Research Articles

Cubic response surface analysis: Investigating asymmetric and level-dependent congruence effects with third-order polynomial models.

Congruence hypotheses play a major role in many areas of psychology. They refer to, for example, the consequences of person-environment fit, similarity, or self-other agreement. For example, are people psychologically better adjusted when their self-view is in line with their reputation? A valid statistical approach that can be applied to investigate congruence hypotheses of this kind is quadratic Response Surface Analysis (RSA) in which a second-order polynomial model is fit to the data and appropriately interpreted. However, quadratic RSA does not allow researchers to investigate more precise expectations about a congruence effect. Do the data support an asymmetric congruence effect, in the sense that congruence leads to the highest (or lowest) outcome, but incongruence in one direction (e.g., self-view exceeds reputation) affects the outcome differently than incongruence in the other direction (e.g., self-view falls behind reputation)? Is there a level-dependent congruence effect, such that the amount of congruence is more strongly related to the outcome variable for some levels of the predictors (e.g., high self-view and reputation) than for others (e.g., low self-view and reputation)? Such complex congruence hypotheses have frequently been suggested in the literature, but they could not be investigated because an appropriate statistical approach has yet to be developed. Here, we present analytical strategies, based on third-order polynomial models, that enable users to investigate asymmetric and level-dependent congruence effects, respectively. To facilitate the correct application of the suggested approaches, we provide respective step-by-step guidelines, corresponding R syntax, and illustrative analyses using simulated and real data. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Stem profile of red oaks in a bottomland hardwood restoration plantation forest in the Arkansas Delta (USA)

Bottomland hardwoods are among the most diverse and productive forest ecosystems in the southeastern United States and are critically important for the provision of timber and non-timber ecosystem services. Red oaks, the dominant species in this group of forests, are of high ecological and economic value. Stem profile models are essential for accurately estimating the merchantable volume of oak trees, which is also closely indicative of total tree biomass and other ecosystem services given their allometric relationships. This study aims to develop and compare stem profiles among three red oak species in an 18-year old plantation forest using destructive sampling. Sixty trees randomly selected from an oak restoration plantation in the Arkansas Delta were felled for measuring the diameter-outside-bark (DOB) and diameter-inside-bark (DIB) at different stem heights. These sample composed of twenty trees from each of three species: cherry bark oak (CBO - Quercus pagoda Raf), Nuttall oak (NUT - Quercus texana Buckley), and Shumard oak (SHU - Quercus shumardii Buckl). Multiple models, including the segmented-profile model, form-class profile model, and second-and third-order polynomial models were fitted and compared. Results demonstrate that the form-class profile model was the best fitted for CBO and NUT, whereas the third-order polynomial model was the best for SHU. CBO tends to grow taller and has a higher wood density than NUT and SHU. These findings will inform restoration and management decisions of bottomland hardwood forests, especially red oaks in the region.

Optimization of Delamination and Thrust Force in the Drilling Process of Nanocomposites

A new design optimization technique is presented to improve the analytical performance of the drilling process of graphene oxide nano-composites. A detailed study was conducted for modeling-design-optimization of the drilling process using multiple nonlinear neuro-regression analyses for this goal. The data were slected from a literature study for this objective. The accuracy of the predictions of the nine potential functional structures presented for modeling the data was tested using a hybrid neuro-regression-based technique. Model selections to determine the objective functions were made by controlling the R2 values, limit values, and statistical results, respectively. The selected models were used in the optimization studies of delamination and thrust force values with four different optimization algorithms. The results show that the R2training and R2 training-adjust values give good results in the nine models as objective functions. However, R2testing values and statistical calculations were distinctive among all models. Furthermore, when the optimization results of the third-order polynomial and logarithmic models for both responses were compared to the reference study's results, it was observed that the current results were more closer to the test results.

Unified Hermite Polynomial Model and Its Application in Estimating Non-Gaussian Processes

AbstractNon-Gaussian processes beset many aspects of structural engineering analysis. To estimate non-Gaussian processes, various third-order Hermite polynomial models have been proposed and widely...

Modelling the growth of the brown frog (Rana dybowskii).

Well-controlled development leads to uniform body size and a better growth rate; therefore, the ability to determine the growth rate of frogs and their period of sexual maturity is essential for producing healthy, high-quality descendant frogs. To establish a working model that can best predict the growth performance of frogs, the present study examined the growth of one-year-old and two-year-old brown frogs (Rana dybowskii) from metamorphosis to hibernation (18 weeks) and out-hibernation to hibernation (20 weeks) under the same environmental conditions. Brown frog growth was studied and mathematically modelled using various nonlinear, linear, and polynomial functions. The model input values were statistically evaluated using parameters such as the Akaike’s information criterion. The body weight/size ratio (Kwl) and Fulton’s condition factor (K) were used to compare the weight and size of groups of frogs during the growth period. The results showed that the third- and fourth-order polynomial models provided the most consistent predictions of body weight for age 1 and age 2 brown frogs, respectively. Both the Gompertz and third-order polynomial models yielded similarly adequate results for the body size of age 1 brown frogs, while the Janoschek model produced a similarly adequate result for the body size of age 2 brown frogs. The Brody and Janoschek models yielded the highest and lowest estimates of asymptotic weight, respectively, for the body weights of all frogs. The Kwl value of all frogs increased from 0.40 to 3.18. The K value of age 1 frogs decreased from 23.81 to 9.45 in the first four weeks. The K value of age 2 frogs remained close to 10. Graphically, a sigmoidal trend was observed for body weight and body size with increasing age. The results of this study will be useful not only for amphibian research but also for frog farming management strategies and decisions.

Dielectric properties of chili powder in the development of radio frequency and microwave pasteurisation

ABSTRACTFood products are dielectric materials. The ability to store and dissipate electromagnetic energy of food in dielectric heating is determined by its dielectric properties. To develop pasteurisation protocols based on radio frequency (RF) and microwave (MW) heating of chili powder, dielectric properties of chili powder were measured using an open-ended coaxial-line probe and an impedance analyzer over a frequency range from 8 to 3000 MHz. Penetration depth of chili powder was calculated at five selected frequencies. Results showed that the dielectric properties of chili powder decreased with increase in frequency from 8 to 3000 MHz, but increased with increase in moisture content from 4 to 15% (w. b.) and with increase in temperature from 25 to 85°C. Dielectric constant and loss factor of different density samples showed little difference at low temperature, but large difference at high temperature. RF waves have larger penetration depth in chili powder than MW, and therefore could result in better heating uniformity especially for thicker samples during pasteurisation process. Geometry of sample and heating runaway phenomena may also affect the application of RF pasteurisation for foods. The relationship between moisture content, temperature, and dielectric properties of chili powder at 13.56, 27.12, 40.68, 915 and 2450 MHz can be described by third-order polynomial models with high coefficient of determination (R2>0.89), which can be used to predict temperature- and moisture-dependent dielectric constant and loss factor of chili powder with high accuracy.

Experimental Determination and Modeling of the Solubility of Sodium Chloride in Subcritical Water from (568 to 598) K and (10 to 25) MPa

The solubility of sodium chloride was investigated by using a continuous flow method over the temperature and pressure ranges of (568 to 598) K and (10 to 25) MPa, respectively. The results showed that the solubility of sodium chloride increased with increasing water density. In general, over the low-density ranges, the increase of solubility is not significant. Whereas, over the high-density ranges, the solubility greatly increases with density. The experimental solubility data were also correlated with seven empirical and semiempirical models (empirical, enthalpy, Cp-, Flory–Huggins, ionization, second-order polynomial, and third-order polynomial models). The correlated results indicated that the second-order polynomial model provided the best fit. The solubility data of sodium chloride in sub-, near-, and supercritical water from this work and literature were collected with a view to evaluating the correlative and predictive capability of these models over a wide range. The enthalpy model gave the best...

Antioxidant activity optimisation of Spirulina platensis C-phycocyanin obtained by freeze-thaw, microwave-assisted and ultrasound-assisted extraction methods

In vitro antioxidant activities of C-phycocyanin extracts obtained from Spirulina platensis by freeze-thaw, microwave-assisted extraction (MAE) and ultrasound-assisted extraction (UAE) methods were evaluated according to 1,1-diphenyl-2-picryl-hydrazyl radical scavenging capacity. Full factorial design was used to optimise the extraction time and medium pH using multiple linear regression analysis. The third-order polynomial models with high R2 values (0.925, 0.946 and 0.858) were sufficiently fitted to the experimental antiradical activities of C-phycocyanin extracted by the three extraction methods. The optimised results showed that the lowest IC50 value for C-phycocyanin extracted by freeze-thaw extraction method (IC50 = 0.031 mg/ml) was obtained after 4 h of extraction at pH=8.00. While these values for MAE (IC50 = 0.021 mg/ml) and UAE (IC50 = 0.036 mg/ml) methods were 22.5 s at pH=8.00, and 6.71 min at pH=7.57, respectively. The models adequacy was confirmed by extracting the C-phycocyanin under the o...

Wind Speed and Direction on Water Application Uniformity of a Mechanical Lateral-Move Irrigation System

Abstract. In order to develop models for representation of Christiansen’s Uniformity (CU) and Distribution Uniformity (DU) as a function of wind speed, 32 in-field tests evaluating a mechanical lateral-move irrigation system, used in rice production, were carried out in southern Rio Grande do Sul, Brazil. These tests were used to generate two third-order polynomial models for estimation of CU and DU, which were then validated based on a cross-validation approach. The generated models had their accuracy quantified by means of the following statistical measures: determination coefficient (R2), reliability and performance index (c), root mean square error (RMSE), and Nash-Sutcliffe coefficient (CNS). Wind direction had no significant influence on CU and DU. The CU values estimated from the cross-validation method were compared to those observed, resulting in R2 = 0.44, c = 0.53, RMSE = 1.82%, and CNS = 0.43. Likewise, DU values estimated from the cross-validation method were compared to the observed values, culminating in R2, c, RMSE, and CNS equal to 0.41%, 0.51%, 2.81% and 0.40%, respectively. The models developed in this study can be useful as a support tool for decision making when applying mechanical lateral-move irrigation systems, allowing estimation of CU and DU values with satisfactory precision for wind speeds less than 5.5 m s-1. Keywords: In-field tests, Rice, Sprinkler irrigation.

Dielectric Properties of Raw Milk as Functions of Protein Content and Temperature

To understand the effect of protein content on dielectric properties (dielectric constant e ′ and dielectric loss factor e ″) of raw milk and to provide information for developing protein content detector being suitable for routine laboratory analysis or real-time quality monitoring, the values of e ′ and e ″ of raw cow’s milk with the protein content of 3.21–7.12 % were measured over the frequency range from 10 to 4500 MHz at temperatures from 25 to 75 °C by using a vector network analyzer and an open-ended coaxial-line probe. The results showed that the e ′ decreased with increasing either frequency or temperature. e ″ decreased linearly with frequency in a log-log plot at low frequency end and had minimums at about 2000–3500 MHz. The minimums increased with temperature. Below about 600 MHz, e ″ increased with increasing temperature and decreased above 1000 MHz. e ′ increased linearly with an increase of protein content below about 150 MHz and decreased linearly above 600 MHz, and e ″ increased linearly with increasing protein content over full investigated frequency range. The developed third-order polynomial models could describe the dielectric properties of raw cow’s milk as functions of protein content and temperature at a given frequency exactly. If the dielectric properties and temperature of milk can be obtained, its protein content could be sensed.

Dielectric Properties of Ground Hazelnuts at Different Frequencies, Temperatures, and Moisture Contents

<abstract><title><italic>Abstract.</italic></title> To develop advanced drying methods using microwave or radio frequency energy, the dielectric constant (ε′) and loss factor (ε″) of ground hazelnuts at moisture contents between 4.6% and 20.3% wet basis (w.b.) were determined from 10 to 4500 MHz at 20°C to 60°C using an open-ended coaxial probe and a network analyzer. The results showed that both ε′ and ε″ decreased with increasing frequency but increased with increasing moisture content and temperature. Both ε′ and ε″ decreased more at the low frequency end of the range than at the high frequency end, especially for high moisture levels. They increased almost linearly with increasing temperature. Third-order polynomial models could be used to describe ε′ and ε″ as functions of moisture content and temperature at selected frequencies. Each model provided a good fit to the experimental data at a significance level of 0.0001 and with a coefficient of determination greater than 0.990. The power penetration depth decreased with increasing frequency, moisture content, and temperature. Large penetration depths at frequencies below 100 MHz may provide large-scale drying.

Temperature-Dependent Development of the Five Psyllophagous Ladybird Predators of Agonoscena pistaciae (Hemiptera: Psyllidae)

Abstract The temperature-dependent development of five psyllophagous ladybirds, Adalia bipunctata (L.), Coccinella undecimpunctata aegyptica (Reiche), Exochomus nigripennis (Erichson), Hippodamia variegata (Goeze), and Oenopia conglobata contaminata (Menetries) (Coleoptera: Coccinellidae), was studied on the common pistachio psylla, Agonoscena pistaciae Burckhardt and Lauterer (Hemiptera: Psyllidae), in the laboratory. The lower developmental thresholds for total development (egg to adult) of the predators ranged from 10.9 (O. c. contaminata) to 14°C (C. u. aegyptica). The lowest and highest degree-day (DD) requirement values for total development of the studied cocinellids were estimated for C. u. aegyptica (166.67 DD) and E. nigripennis (344.83 DD), respectively. The optimum temperature for development of the studied ladybirds, calculated using third-order polynomial and Lactin models, was in the range of 27.5–30°C. This study showed that all the five psyllophagous ladybirds successfully complete their development in a wide range of temperatures from 17.5 to 30°C, indicating their high potential for use in biological control programs against A. pistaciae.

Electrochemical Impedance Spectroscopy as an Alternative to Determine Dielectric Constant of Potatoes at Various Moisture Contents

The dielectric (DE) properties, specifically the DE constant (ε') and loss factor (ε''), were measured for vacuum-dried and freeze-dried potato samples at a microwave frequency of 2.45 GHz over a range of different moisture contents (MCs) using a DE probe and also a 2-probe electrochemical impedance spectroscopy (EIS). Third-order polynomial models (ε' = f₁(MC); and ε'' = f₂(MC)) at room temperature were developed for regression analysis. Additionally, at various temperatures (T), biphasic 3rd-order polynomial models (ε' = f₁(MC, T); and ε'' = f₂(MC, T)) were obtained to determine ε' and ε'' as a function of MC and T using measured data. The vacuum-dried potato sample showed a good fitness of ε' and ε'' (R² = 0.95 and 0.96, respectively) to the regression model with the range of MCs from 18% to 80% (w/w), while the freeze-dried potato sample showed a good fitness of ε' and ε'' to the 1st-phase regression model with MC < 50% w/w (R² = 0.95 and 0.96, respectively) and the 2nd-phase regression model with MC > 50% w/w (R² = 0.94 to 0.96). EIS measurements were also used to obtain correlation impedances for ε' and ε'' determined by the DE probe method. The resulted regression analysis meets the demands for simple, rapid, and accurate assessment for transient values of ε' and ε'' of food products during dehydration/drying processes. The EIS method was verified to be a successful alternative to direct measurements of ε' and ε''.

Conditions of Slope-Rotatability for Third-Order Polynomial Regression Models

In experimental design for response surface analysis, it is sometimes of interest to estimate the difference of responses at two points. If differences at points close together are involved, the design that reliably estimates the slope of the response surface is important. In particular, Hader and Park (1978) suggested the concept of slope-rotatability and studied slope rotatable central composite designs. Until now, many response surface designs including central composite designs have been suggested for fitting second order response surface models. However, we often need to fit third-order polynomial regression models. In this article, we suggest extended central composite designs (ECCDs) to fit third-order models and find the necessary and sufficient conditions for slope-rotatability over all directions in the third-order polynomial models.

On the Order of Polynomial Regression Models for Manoeuvring Forces

The paper is investigating certain aspects of polynomial regression models used for representation of hydrodynamic forces acting on a ship performing normal (moderate) manoeuvres. A unique polynomial model of increased order developed by Oltmann, Sharma and Wolf at the end of 70s on the basis of high-precision model tests was taken as a primary one. The model was used to generate sample sets of responses which were further approximated with simpler third-order polynomial models presented here as secondary regressions. The structure of the secondary models was chosen using the stepwise forward regression procedure and the coefficients were further estimated by the least-square algorithm. The resulting model turned out to be much more economical in terms of the number of regressors and estimated parameters and does not include regressors of higher order and acceleration-velocity cross-coupling terms present in the primary model. At the same time, comparisons of the force/moment responses, zigzag manoeuvre time histories and of the spiral curves demonstrated that no tangible loss of accuracy can be traced. The conclusion is made that no terms of the order higher than three should enter polynomial models used in ship manoeuvring. Copyright ©2012 IFAC

Evaluation of the Boltzmann equation as an alternative model in the selection of the high-yield subsample within the framework of the compositional nutrient diagnosis system

Calculation of the yield cutoff value, as a means for establishing the distinction between the high- and low-yield subsamples, and subsequent estimation of the corresponding crop reference values, is an important step within the framework of the compositional nutrient diagnosis system. In the present work, the former was conducted through a new approach, viz., the modeling of the cumulative variance ratio function vs. yield relationship with a sigmoidal (Boltzmann) equation. The study was conducted with a database comprised of 78, 5–9-year-old ‘Valencia’ orange trees ( Citrus sinensis L. Osbeck), grafted on ‘Cleopatra’ tangerine rootstocks ( Citrus reshni, Hort. ex. Tan.), and grown in Carabobo State (Venezuela). The performance of the proposed model was compared to that of the conventional cubic-polynomial equation by means of the root mean square of the percentage deviations (RMSPD). This estimator was preferred over the commonly used coefficient of determination, because the high values of the latter were inconsistent with the lack of fit of the third-order polynomial models observed in the corresponding scatter plots. The performance of the Boltzmann models was clearly superior in all cases, leading to its exclusive selection for further compositional nutrient diagnostic (CND) calculations. The proposed approach was used for the estimation of the yield cutoff value, reference norms (as row-centered log-ratios, V X * ), and optimum foliar concentrations ( C X * ) for N, P, K, Ca, and Mg in the aforementioned crop. A yield cutoff value of 114.10 kg tree −1, corresponding to a reference χ 2-value of 4.95, was derived from this approach. CND norms were V N * = 0.472 ± 0.226 , V P * = − 2.306 ± 0.199 , V K * = − 0.360 ± 0.209 , V Ca * = 0.431 ± 0.300 , V Mg * = − 2.158 ± 0.477 , and V Rd * = 3.920 ± 0.138 . Optimum nutrient concentrations, expressed in g kg −1, were C N * = 29.57 ± 3.59 , C P * = 1.94 ± 0.69 , C K * = 13.04 ± 2.78 , C Ca * = 30.20 ± 10.95 , and C Mg * = 2.31 ± 1.02 . These values, though of preliminary nature, may render more accurate nutritional diagnoses in regional applications, particularly if the age and rootstock of the trees under evaluation match those of the trees from which the norms were derived.

Optimization of Rubber Wood Drying by Response Surface Method and Multiple Contour Plots

By adopting the central-composite experiment design, the response surface methodology was used to optimize operating conditions of rubber wood drying. The independent variables are initial moisture content of rubber wood, and three drying environment parameters namely, temperature, relative humidity, and air velocity. The investigating responses are final moisture content, drying time, and energy consumption. The restriction of the optimization is the designated final moisture content, which is not greater than 16%. The third-order polynomial models with transformed responses were developed from experiment data to generate 3-D response surfaces and contour plots. The analysis of variance (ANOVA) was performed to identify the significant parameters affecting the rubber wood drying. Drying temperature and holding relative humidity are those two influential operating parameters that significantly control the final moisture of rubber wood and affect the drying time and energy. The multiple contour plots of drying responses show that the optimum operating regions are located mainly at high temperature drying zone. The high temperature drying practice can save energy and drying time by 44 and 25% respectively, in comparison to the conventional temperature drying.

Grinding performance and physical properties of wheat and barley straws, corn stover and switchgrass

Abstract Wheat and barley straws, corn stover and switchgrass at two moisture contents were ground using a hammer mill with three different screen sizes (3.2, 1.6 and 0.8 mm ). Energy required for grinding these materials was measured. Among the four materials, switchgrass had the highest specific energy consumption ( 27.6 kW h t −1 ), and corn stover had the least specific energy consumption ( 11.0 kW h t −1 ) at 3.2 mm screen size. Physical properties of grinds such as moisture content, geometric mean diameter of grind particles, particle size distribution, and bulk and particle densities were determined. Second- or third-order polynomial models were developed relating bulk and particle densities of grinds to geometric mean diameter within the range of 0.18– 1.43 mm . Switchgrass had the highest calorific value and the lowest ash content among the biomass species tested.

Minimax designs for estimating the slope of a third-order response surface in a hypercubic region

The design criterion considered is minimization of the variance of the estimated slope of a response surface maximized over all points in the factor space. Optimal designs under the minimax criterion, within some classes of commonly used designs, are derived for third-order polynomial models over hypercubic regions.

Optimization in photochemical reaction detection : Application to high-performance liquid chromatography—photolysis—electrochemical detection

Chemometric techniques have been applied to optimization of response in photochemical reaction detection. Multivariate optimization using a factorial experimental design was employed to assess the effects of mobile phase pH and UV irradiation time on the amperometric response obtained by high-performance liquid chromatography—photolysis—electrochemical detection for four cardiovascular drugs. The procedure is more representative of the dynamic analytical conditions encountered in photochemical reaction detection than previous methods employing univariate optimization or off-line static methods. Stepwise multiple linear regression with backward elimination was used to fit the experimental data to empirical reduced third-order polynomial models. Plots of response were used to determine optimum conditions and visualize the interactions between pH and irradiation time. The design and construction of a low-cost on-line photochemical reactor for high-performance liquid chromatography is also discussed.