Polynomial Regression Of Degree Research Articles

ML-Based Approach to Predict Carotid Arterial Blood Flow Dynamics

In the current study, a numerical model has been developed to simulate the blood flow characteristics in the human carotid artery. The data thus generated is analyzed to understand the blood flow variations and predict the flow characteristics using Machine Learning techniques. In developing the numerical model, the key features of the system, namely, the blood, is modeled as an incompressible Newtonian fluid, and the artery is an elastic pipe. This model is simulated using COMSOL software by varying the material properties of the artery. Univariate analysis was performed to gain insight into the features' behaviour and target variables. Subsequently, machine-learning regression models were trained using the data generated from the idealized human carotid artery. Furthermore, the validity of the data was ensured by comparing it with flow division ratios available in the literature. The evaluation of these models was conducted by calculating the Mean Absolute Error values for the test dataset, resulting in the following values: polynomial regressor (0.0106), hyper-tuned support vector regressor (0.0487), decision tree regressor (0.000), random forest regressor (0.0156), Adaboost (0.0508), gradient-boosting (0.0044), and XGboost (0.0043). A quantile loss function was employed to assess the prediction uncertainty. According to the theory of loss function, models with low loss values are considered good predictors. The prediction uncertainty was measured by applying quantile loss function, and it identified that the random forest regressor as the best predictor model for the data, followed by the polynomial regression of degree 3. Prediction intervals for the target variable were computed by leveraging the random forest quantile regressor model. Moreover, the developed polynomial model was utilized to investigate the presence of stenosis in the artery.

Determining the Best Model among Candidate Machine Learning Models for Chicago Suburb House Price Data

With the development of the society, there is an increasing need of predicting the house price in order to purchase properties economically. Myriads of machine learning methods have been developed to achieve this goal. In this paper, we narrow our focus on three types of machine learning methods-Linear Regression, Polynomial Regression, and Neural Network (Multilayer Perceptron Regressor) and train all of them on Chicago suburb house price dataset and employ all of them to make predictions. Performance is evaluated using mean squared error (MSE) and R square (R2) metrics. Preliminary experimental results demostrate that Neural Network outperforms the other candidate methods in all cases. Polynomial Regression of degree 2 outperforms Linear Regression in most cases but performs arbitrarily bad in worst cases. Therefore, Neural Network (MLP Regressor) is the best method for predicting Chicago suburb house price.

A Novel OFDM Format and a Machine Learning Based Dimming Control for LiFi

This paper proposes a new hybrid orthogonal frequency division multiplexing (OFDM) form termed as DC-biased pulse amplitude modulated optical OFDM (DPO-OFDM) by combining the ideas of the existing DC-biased optical OFDM (DCO-OFDM) and pulse amplitude modulated discrete multitone (PAM-DMT). The analysis indicates that the required DC-bias for DPO-OFDM-based light fidelity (LiFi) depends on the dimming level and the components of the DPO-OFDM. The bit error rate (BER) performance and dimming flexibility of the DPO-OFDM and existing OFDM schemes are evaluated using MATLAB tools. The results show that the proposed DPO-OFDM is power efficient and has a wide dimming range. Furthermore, a switching algorithm is introduced for LiFi, where the individual components of the hybrid OFDM are switched according to a target dimming level. Next, machine learning algorithms are used for the first time to find the appropriate proportions of the hybrid OFDM components. It is shown that polynomial regression of degree 4 can reliably predict the constellation size of the DCO-OFDM component of DPO-OFDM for a given constellation size of PAM-DMT. With the component switching and the machine learning algorithms, DPO-OFDM-based LiFi is power efficient at a wide dimming range.

Fuzzy modeling of vegetable straw cover crop productivity at different nitrogen doses

The Direct Planting System (DPS), also referred to as direct seeding or no-tillage farming, is distinguished by the permanent occupation of the agricultural land with plant crops besides the maintenance of the straw under the soil surface. However, for its effectiveness, it is necessary to adopt suitable cover crops for each region. Sorghum has a high tolerance to the dry season and a high volume of dry matter production, which are important characteristics for adoption as a cover crop. Millet, besides having a high resistance to periods of water stress, has also a high concentration of biomass production. In this context, the present work aimed to analyze, through the use of mathematical models, the effects of nitrogen in sorghum and millet crops, adopted as vegetation cover crops, under different levels of nitrogen in relation to the dry matter production variables. The comparison of the fuzzy (Gaussian and triangular) and polynomial regression models was performed using statistical tests and verification of the determination coefficient. Analytically, the best statistical parameters for triangular fuzzy modeling were verified in comparison to the Gaussian fuzzy model and polynomial regression of degree two. However, in relation to the behavior of the mathematical modeling curve, the Gaussian model proved to be best in keeping with the reality of the agronomic phenomena, mainly due to the smoothing of the curve over the data set.

SEASONS IN STUTTGART: DEVELOPING A GOOGLE EARTH ENGINE TOOL FOR HEAT ISLAND MAPPING

Abstract. This study generates a process in GEE (Google Earth Engine) for SUHI (Surface Urban Heat islands) identification derived from TIRS (Thermal Infrared Sensor) and OLI (Operational Land Imager) sensors of Landsat 8 imagery in the area of Stuttgart, Germany. By comparing the temperature images in winter and summer seasons through a regression model, a relation between the Surface Cover (SC), the Terrain Shape (DEM) and the LST (Land Surface Temperature) is established. A Python code is developed for modelling the data and displaying the results linked to GEE. Three different models are used to establish the relationship between different variables (Temperature, Height, Wind etc.). Accuracy/goodness of fit of these models are measured using R-squared and standard error. Results shows that polynomial regression of 3rd order degree fits best to the dataset used in this study. Moreover, it is found that temperature values are not perfect for this study, as Landsat 8 have been acquired at 10’o clock in the morning (local time), whereas night time acquisition (which was not available for Stuttgart, Germany) would be best suited for the study. The results indicate that urban areas and meadow (open areas without vegetation) get the bigger values of temperature. Terrain Shape with respect to height indicates that the bigger the height, the lower the temperature in most of the regions. This project provides insight into the development of applications using a web-based platform and leads to a fast and accurate result for identifying the SUHI effect. It can contribute to the necessity of planning more vegetation areas in order to reduce hot temperature values in Stuttgart.

Half-Blind Calibration for the Efficient Compensation of Parasitic Cross-Sensitivities in Nonlinear Multisensor Systems

We extend the method of half-blind calibration (HBC) to sensor systems with a nonlinear response. HBC ensures the efficient compensation of undesired cross-sensitivities in sensor systems despite a reduced calibration effort. During HBC, one applies load combinations covering the ranges of the measurands of primary interest of the system and the undesired disturbances. However, only the measurand values and the output signals of the individual sensors of the system are recorded. In the proposed nonlinear HBC, these data are fitted using multiple polynomial regression, treating the sensor data as predictor variables and each measurand of interest as the dependent variable. For each such measurand, the extracted polynomial function of the sensor signals predicts cross-sensitivity-compensated values. The method is demonstrated using a silicon-based magnetic sensor system, where stress and temperature constitute disturbances. During calibration, controlled values of the out-of-plane component of the magnetic induction are applied, while stress and temperature are varied without accurate knowledge of their values. Output signals of the Hall sensor and of co-integrated stress and temperature sensors provide the sensor signals. All respond nonlinearly to stress and temperature. Using a polynomial regression of degree 4, the system achieves an accuracy of the measured magnetic induction of $136~\mu \text{T}$ in the ranges of ±21 mT, −40 °C to 125 °C, and up to −80 MPa of in-plane stress. This performance was achieved without quantitative knowledge of the disturbances during calibration. Nonlinear HBC has the potential to considerably simplify calibration equipment and procedures.

QUALITY OF CLONAL SEEDLINGS OF Tectona grandis Linn F. ROOTED IN DIFFERENT CONCENTRATIONS OF INDOLEBUTYRIC ACID

The general objective of this paper is to evaluate the effect of different IBA concentrations on greenhouse, shade and full sun survival, root vigor and seedling quality in Tectona grandis. To do so, mini-cuttings were prepared containing two pairs of leaves reduced to 25% of their original size (7.0 ± 1.0 cm) in length and bevel cut at the base. They were submitted to five treatments with different concentrations of indolebutyric acid (0, 500, 1000, 1500 e 2000 mg L-1), remaining in greenhouse, house of shade and in full sun for 30, 20 and 15 days, respectively. Survival was evaluated in greenhouse, house of shade and full sun; radicial vigor, height, diameter of the neck, ratio between shoot height and diameter, root dry mass, aerial, and total mass. The experimental design was in randomized blocks, with five replications of 15 plants and five treatments. Data were submitted to the normality test, Bartlett test and polynomial regression of the second degree. As a result, it was verified that the use of IBA did not influence root vigor and survival, however, it provided seedlings with a higher dry mass, being the best the 2000 mg L-1 treatment.

МЕТОДИКА АВТОМАТИЗИРОВАННОГО МОНИТОРИНГА ЛИНЕЙНЫХ ОБЪЕКТОВ НЕФТЕГАЗОВОГО КОМПЛЕКСА С БЕСПИЛОТНОГО ВОЗДУШНОГО СУДНА

Introduction. The aim of the work is to create a methodology for automated monitoring of linear objects in the oil and gas complex based on the developed mathematical model of flame recognition from a video protocol obtained from a free aircraft. Materials and methods. In a typical solution for recognizing a flame from a free aircraft, an infrared camera is used which effectively solves the detection problem, but has a high cost and is an additional attachments, which increases the weight of the apparatus and reduces its flight capabilities. In existing systems with video camera, the operator makes a decision about the presence of combustion, receiving information from the monitor of the control panel in real time. Theoretical basis. To improve this principle, the algorithm for recognizing the flame in automatic mode, implemented in the program “Video detector flame 2.0 FD”, its difference in the analysis of the video protocol on-board. To implement the algorithm developed, the device of the Russian production Supercam S350 was selected. The flight conditions for the implementation of the methodology were chosen on the basis of the limitations of the underlying mathematical model, while the altitude was 40 m. The condition of sufficiency of the study of one point from two different angles with an overlap of 10 % is accepted in the work. The operational speed of the aircraft during monitoring is 50 km/h. Results and its discussion. To ensure the uniformity of capture of the frames of the underlying surface, polynomial regression of the third degree is determined depending on the true speed in the range from 20 to 70 km/h. The operational characteristics of the selected aircraft and the boundary conditions of the developed mathematical model are generalized and summarized in the methodology of automated monitoring of linear objects of the oil and gas complex. Conclusions. The proposed algorithm uses a digital video camera of standard resolution, performing the detection task without specialized equipment. This significantly reduces the cost of the device and opens the possibility of broad integration into existing free aircraft due to cross-platform software.

Correlation among extinction efficiency and other parameters in an aggregate dust model

We study the extinction properties of highly porous Ballistic Cluster-Cluster Aggregate dust aggregates in a wide range of complex refractive indices (, ) and wavelengths ( m). An attempt has been made for the first time to investigate the correlation among extinction efficiency (), composition of dust aggregates (), wavelength of radiation (λ) and size parameter of the monomers (x). If k is fixed at any value between 0.001 and 1.0, increases with increase of n from 1.4 to 2.0. and n are correlated via linear regression when the cluster size is small, whereas the correlation is quadratic at moderate and higher sizes of the cluster. This feature is observed at all wavelengths (ultraviolet to optical to infrared). We also find that the variation of with n is very small when λ is high. When n is fixed at any value between 1.4 and 2.0, it is observed that and k are correlated via a polynomial regression equation (of degree 1, 2, 3 or 4), where the degree of the equation depends on the cluster size, n and λ. The correlation is linear for small size and quadratic/cubic/quartic for moderate and higher sizes. We have also found that and x are correlated via a polynomial regression (of degree 3, 4 or 5) for all values of n. The degree of regression is found to be n and k-dependent. The set of relations obtained from our work can be used to model interstellar extinction for dust aggregates in a wide range of wavelengths and complex refractive indices.

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.

Distribution-free tests for polynomial regression based on simplicial depth

A general approach for developing distribution-free tests for general linear models based on simplicial depth is presented. In most relevant cases, the test statistic is a degenerated U-statistic so that the spectral decomposition of the conditional expectation of the kernel function is needed to derive the asymptotic distribution. A general formula for this conditional expectation is derived. Then it is shown how this general formula can be specified for polynomial regression. Based on the specified form, the spectral decomposition and thus the asymptotic distribution is derived for polynomial regression of arbitrary degree. The power of the new test is compared via simulation with other tests. An application on cubic regression demonstrates the applicability of the new tests and in particular their outlier robustness.

Optimal designs which are efficient for lack of fit tests

Linear regression models are among the models most used in practice, although the practitioners are often not sure whether their assumed linear regression model is at least approximately true. In such situations, only designs for which the linear model can be checked are accepted in practice. For important linear regression models such as polynomial regression, optimal designs do not have this property. To get practically attractive designs, we suggest the following strategy. One part of the design points is used to allow one to carry out a lack of fit test with good power for practically interesting alternatives. The rest of the design points are determined in such a way that the whole design is optimal for inference on the unknown parameter in case the lack of fit test does not reject the linear regression model. To solve this problem, we introduce efficient lack of fit designs. Then we explicitly determine the ek-optimal design in the class of efficient lack of fit designs for polynomial regression of degree k−1.

Efficient lack of fit designs that are optimal to estimate the highest coefficient of a polynomial

Polynomial regression models are often used in practice although practitioners are seldom sure whether their assumed linear regression model is at least approximately true. Therefore, designs are most interesting in practice with which the linear model can be checked. Optimal designs for polynomial regression, however, have not this property. To get practically attractive designs, Bischoff and Miller [2003. Optimal designs efficient for lack of fit tests. Preprint.] have suggested the following strategy. One part of the design points are used to be able to carry out a lack of fit test with a good power for practically interesting alternatives. The rest of the design points are determined in such a way that the whole design is optimal for inference on the unknown parameter in case the lack of fit test does not reject the linear regression model. To solve this problem, efficient lack of fit designs were introduced in Bischoff and Miller [2003. Optimal designs efficient for lack of fit tests. Preprint.]. In this class of designs we explicitly determine e k -optimal designs for polynomial regression of degree k - 1 in this paper. More exactly, for k ⩽ 4 we determine all e k -optimal designs for polynomial regression of degree k - 1 . For k > 4 we determine the e k -optimal designs for polynomial regression of degree k - 1 if we restrict ourselves to designs that deliver highly efficient lack of fit tests.

A Bayesian Approach for Predicting With Polynomial Regression of Unknown Degree

This article compares three methods for computing the posterior probabilities of the possible orders in polynomial regression models. These posterior probabilities are used for forecasting using Bayesian model averaging. It is shown that Bayesian model averaging provides a closer relationship between the theoretical coverage of the high-density predictive interval (HDPI) and the observed coverage than those corresponding to selecting the best model. The performance of the different procedures is illustrated with simulations and some known engineering data.

Robust designs for polynomial regression by maximizing a minimum of D- and D1-efficiencies

In the common polynomial regression of degree m we determine the design which maximizes the minimum of the $D$-efficiency in the model of degree $m$ and the $D_1$-efficiencies in the models of degree $m-j,\dots, m +k$ ($j, k\ge 0$ given). The resulting designs allow an efficient estimation of the parameters in the chosen regression and have reasonable efficiencies for checking the goodness-of-fit of the assumed model of degree $m$ by testing the highest coefficients in the polynomials of degree $m-j,\dots, m +k$ . Our approach is based on a combination of the theory of canonical moments and general equivalence theory for minimax optimality criteria. The optimal designs can be explicitly characterized by evaluating certain associated orthogonal polynomials.

Sensitivity of Drosophila Heat Shock Transcription Factor to Low pH

The heat shock transcription factor (HSF) mediates the induction of heat shock gene expression. The activation of HSF involves heat shock-induced trimerization, binding to its cognate DNA sites, and the acquisition of transcriptional competence. In this study, the oligomeric properties of Drosophila HSF were analyzed by equilibrium analytical ultracentrifugation and gel filtration chromatography. Previous findings showed that trimerization of purified Drosophila HSF was directly sensitive to heat and oxidation (1). Here we report that low pH, in the physiological range, also directly induces HSF trimerization and DNA binding in vitro. Furthermore, the induction of HSF trimerization by low pH is synergistic with the actions of heat and oxidation. Since heat or chemical stress leads to a moderate decrease of intracellular pH, we suggest that intracellular acidification may contribute to activating the heat shock response in vivo.

D-optimal designs for weighted polynomial regression

Using an Equivalence Theorem and the theory of Sturm-Liouville systems, we determine the D-optimal designs for some classes of weighted polynomial regression of degree d on the interval [−1, 1]. We also show that the number of the optimal support points for such models is d + 1. and that the optimal support points can be computed through a specific eigenvalue of a certain band matrix. Some results of Karlin and Studden ( Ann. Math. Statist. 37 (1996), 783–815) turn out to be special cases of our results.

E-optimal designs for polynomial regression without intercept

We give all E-optimal designs for the mean parameter vector in polynomial regression of degree d without intercept in one real variable. The derivation is based on interplays between E-optimal design problems in the present and in certain heteroscedastic polynomial setups with intercept. Thereby the optimal supports are found to be related to the alternation points of the Chebyshev polynomials of the first kind, but the structure of optimal designs essentially depends on the regression degree being odd or even. In the former case the E-optimal designs are precisely the (infinitely many) scalar optimal designs, where the scalar parameter system refers to the Chebyshev coefficients, while for even d there is exactly one optimal design. In both cases explicit formulae for the corresponding optimal weights are obtained. Remarks on extending the results to E-optimality for subparameters of the mean vector (in heteroscdastic setups) are also given.

On a mixture of the D- and D1-optimality criterion in polynomial regression

We consider a mixture of the D1-optimality criterion (minimizing the variance of the estimate for the highest coefficient) and the D-optimality criterion (minimizing the volume of the ellipsoid of concentration for the unknown parameter vector) in the polynomial regression model of degree n ⊆ N. The mixture is defined as a weighted product of both optimality criteria and explicit solutions are given for the proposed criterion. The derived designs have excellent efficiencies compared to the G-, D-, and D1-optimal design. This is illustrated in some examples for polynomial regression of lower degree. The optimal designs are calculated using the theory of canonical moments. Further applications are given in the field of model robust designs for polynomial regression models where only an upper bound for the degree of the polynomial regression model is known by the experimenter before the experiments are carried out.

Saturability of hepatic iron deposits in genetic hemochromatosis

The relationship of pretreatment serum ferritin and hepatic iron concentration to body iron removed by venesections was evaluated in 33 patients with genetic hemochromatosis. The median values of the three variables considered were 1,950 micrograms/L (range = 255 to 10,000), 1,175 micrograms/100 mg dry weight (range = 270 to 4,310) and 10 gm (range = 2 to 41), respectively. At basal liver biopsy 18 patients had cirrhosis, 6 patients had fibrosis and 9 patients had a normal pattern; siderosis was degree 3 in 6 patients and degree 4 in 27 patients. The results of fitting a polynomial regression of second degree showed that the curve of serum ferritin on iron removed was a straight line (R2 = 0.79, with a significant coefficient of linearity, p less than 0.01, and a nonsignificant coefficient of curvature), whereas that of hepatic iron concentration on iron removed showed a curvature (R2 = 0.62, with significant coefficient of linearity and curvature, p less than 0.01) and reached a plateau. The sigmoid model fit the curve of hepatic iron concentration on iron removed (R2 = 0.61), which suggested a saturation of hepatic iron storage capability; the asymptote corresponded to a hepatic iron concentration of about 2,000 micrograms/100 mg. In alcoholic patients (17 cases) the location of the sigmoid was greater than in nonalcoholic patients. Our results suggest that iron deposition occurs in the liver before other organs are involved and that with massive iron overload hepatic deposits reach saturation, after which hepatic iron concentration does not always reflect the amount of total stores. Alcohol consumption could slow the saturation of hepatic iron deposits.