Newton Interpolation Research Articles

Principle, Methodology and Application for Data Cleaning techniques

Con-temporarily, information data has become the cornerstone of every company’s decision-making. In a vast flow of information, choosing the right data is the first step in developing successful predictions. After the determinations of the requirements, analysis purpose and prediction direction, outlier processing, missing value processing and repeated value processing are usually encountered. This paper introduces the limitations, advantages and disadvantages of different methods in application in detail. At the same time, this paper introduces some interpolation methods based on mathematical statistics, such as thermal interpolation, Lagrange interpolation and Newton interpolation. At the same time, it also provides the normal distribution processing method which is better in dealing with outlier problems, and the popular K-nearest neighbor algorithm. Finally, it illustrates the logic diagram of data cleaning in the data preparation stage. Overall, these results offer a guideline for selecting the appropriate treatment in the corresponding situation during data cleaning process.

Oscillatory, crossover behavior and chaos analysis of HIV-1 infection model using piece-wise Atangana–Baleanu fractional operator: Real data approach

There are many fatal diseases which are caused by virus. Different types of viruses cause different infections. One of them is HIV-1 infection which caused by retrovirus. HIV-1 infection is a hazardous disease that can lead to cancer, AIDS, and other serious illnesses. Several mathematical models have been proposed in the field and examined using various methods. In this manuscript, the newly suggested piece-wise (PW) Atangana–Baleanu (AB) fractional operator is used to examine HIV-1 infection. Some theorems related to the existence of the solution to the examined model are proved through fixed point results. The Ulam–Hyers (UH) stability and its different forms are presented for the proposed PW HIV-1 infection model. The considered model’s numerical results are attained via the Newton interpolation method. The results are graphically illustrated via MATLAB software to show the behavior of the considered model. Oscillatory and complex dynamics are obtained for some fractional orders and show the crossover behavior of the proposed model. The model’s simulation of infected class is fitted with the real data taken for six different countries. It proves the validity and accuracy of the suggested approach.

Study on indoor thermal comfort of different age groups in winter in a rural area of China’s hot-summer and cold-winter region

In this paper, both onsite measurements of thermal parameters and a subjective questionnaire of thermal comfort were conducted to investigate thermal comfort of different age groups during the winter season. This study was performed in a rural area of Tianmen, a city of China’s hot summer and cold winter region. Indoor environmental parameters were measured and a total of 1440 questionnaire samples were collected. The volunteers were divided into four groups, i.e., children, young people, middle-aged people and old people. The results indicated that in a cold environment with the temperature below 6 °C, old people were the most sensitive to cold and in a warm environment with the temperature above 19 °C, children were the most sensitive to heat. The neutral SET* (standard effective temperature) of children, young people, middle-aged people and old people in winter was 21.3 °C, 20.8 °C, 23.9 °C and 24.7 °C, which indicated discrepancies of SET* among different age groups. The neutral SET* model of different ages was predicted based on Newton interpolation. This study revealed differences in indoor thermal comfort among different age groups and could provide useful guidance for creating thermal comfortable environment for different ages.

ON COUPLED SYSTEM OF DRUG THERAPY VIA PIECEWISE EQUATIONS

This paper is devoted to establish some theoretical and computational results for a coupled system of drug therapy process. The considered problem is investigated by using the concept of piecewise modeling. We apply piecewise differentiation and integration to study the proposed model. We derive some adequate results for existence and uniqueness for solution to the proposed problems by using fixed point results. Further, we apply some numerical approach based on Newton interpolation formula for equally spaced arguments to find numerical results corresponding to our proposed model. We use some real values and two sets of fractional order to present the dynamics of the considered model graphically.

PIECEWISE FRACTIONAL-ORDER MODELING OF THE BREAST CANCER EPIDEMIOLOGY AFTER THE ATEZOLIZUMAB TREATMENT

In this paper, the therapeutic interactions are discussed with the aid of a mathematical model and piecewise differentiation technique. The model is based on the interaction between cell populations in therapeutic stress and cell populations in neoplastic transformation, referring specifically to triple-negative breast cancer (TNBC). The proposed computational approach provides an opportunity for the qualitative and parametric analysis of the clinical trials in question. The proposed mathematical model is based on the analysis of cell–cell interactions according to a logic that examines pathological stress and its influence on the dynamics of the cell population involved in tumor pathology. Detailed literature review and dynamical analysis of the proposed hypothesis are provided in this paper. The existence and non-negativity of the solutions are exploited, the hypothesis is stabilized; it is then further demonstrated with the aid of the piecewise derivative and the relevant application of the formula of Newton interpolation.

Modified Newton-residual interpolation for division of focal plane polarization image demosaicking.

With the improvement of semiconductor processing technology, polarization sensors using division of focal plane have gradually become the mainstream method of polarization imaging. Similar to the color restoration method of the Bayer array sensor, the spatial information of polarized image is also recovered through the polarization demosaicking algorithm. In this paper, we propose a new modified Newton-residual interpolation polarization image demosaicking algorithm based on residual interpolation, which is suitable for a monochrome or color polarization filter array. First, we use the modified Newton interpolation method to generate edge-sensitive guiding images. Then, we carry out the improvement of the guide process during the residual interpolation by performing variance statistics on the local window image in the guiding process, so that the edges and flat image blocks have different guiding weights. Finally, we obtain edge-preserving results by applying these two improvements, which reduces the zipper effect and edge confusion. We compare the results of various algorithms on experimental data, demonstrating that our algorithm has impactful improvements in the evaluation metrics based on the ground-truth images.

In situ characterization of few-femtosecond laser pulses by learning from first-principles calculations

The field of ultrafast spectroscopy is based on lasers being able to produce pulses that are as short as a few femtoseconds. Due to their broad bandwidth, these ultrashort light transients are strongly affected by propagation through materials. Therefore, a careful characterization of their temporal profile is required before any application. We propose a scheme for their characterization in situ, ensuring that the pulse parameters are measured in the region where the interaction with the sample takes place. Our method is based on first-principles calculations for strong-field ionization of rare-gas atoms and autocorrelation. We introduce a machine-learning algorithm, called vector space Newton interpolation cage (VSNIC), that uses the results from the first-principles calculations as input and reconstructs from a strong-field autocorrelation pattern for an unknown pulse the pulse length and spectral width by narrow margins.

Investigation of robotic milling chatter stability prediction under different cutter orientations by an updated full-discretization method

Chatter can easily occur during robotic milling owing to the low rigidity of serial robots. Chatter affects the quality of the workpiece surface and damages the robot equipment, which is a bottleneck that hinders the application of robotic milling. An effective way to ensure milling stability is to rationally select the cutting parameters using high-precision stability lobe diagrams (SLDs). In this study, an updated full-discretization chatter stability prediction method based on the fourth-order Hermite and third-order Newton interpolation polynomial was proposed. A dynamical model of robotic milling was established by considering the regenerative effect, cutter structure modal coupling effect, and the influence of cutter orientations. Based on the established robotic milling dynamic model and the proposed full-discretization method, SLDs related to the axial depth of cut, spindle speed, lead angle, and tilt angle were generated. The dominant influencing factors of the robotic milling chatter stability under different robot postures and cutter orientations were analysed. The robotic milling experiment results show that the dynamic characteristics of the robot are different in each direction. When the milling cutter feeding direction is parallel to the x-axis direction of the global coordinate system, the regenerative effect is dominant. When the feed direction of the milling cutter is parallel to the y-axis of the global coordinate system, the cutter structure modal coupling effect is more obvious. This is possibly because the dynamic characteristics of the tool tip change during the milling process under different robot postures. However, the milling process becomes more stable with an increase in the lead or tilt angle. This is mainly because when the cutting depth is constant, the cutter–workpiece engagement area gradually decreases with an increase in the tool angle.

A Hybrid Interpolation Method for Fractional PDEs and Its Applications to Fractional Diffusion and Buckmaster Equations

This study presents a novel numerical method to solve PDEs with the fractional Caputo operator. In this method, we apply the Newton interpolation numerical scheme in Laplace space, and then, the solution is returned to real space through the inverse Laplace transform. The Newton polynomial provides good results as compared to the Lagrangian polynomial, which is used to construct the Adams–Bashforth method. This procedure is used to solve fractional Buckmaster and diffusion equations. Finally, a few numerical simulations are presented, ensuring that this strategy is highly stable and quickly converges to an exact solution.

Research on the maximum power point tracking method of photovoltaic based on Newton interpolation-assisted particle swarm algorithm

Abstract Solar energy has attracted a lot of attention because it is clean and has no pollution. However, due to the partially shaded condition, the photovoltaic array cannot work uniformly at the maximum power point, resulting in a large power loss. To improve this problem, the research of the maximum power point tracking (MPPT) algorithm is discussed by scholars. In this paper, an improved particle swarm optimization (PSO) algorithm is proposed to achieve the goal of MPPT, which uses Newton interpolation-assisted conventional PSO. After tracking to the maximum power point, the Newton interpolation method is used to calculate the maximum power point, reduce the number of iterations of the conventional PSO algorithm and reduce the steady-state oscillation. The simulation is carried out in MATLAB®/Simulink® and compared with conventional PSO. The results show that the improved PSO has better tracking accuracy and speed than the conventional algorithm, and the initial tracking speed is increased by >30%.

A NUMERICAL SCHEME BASED ON TWO- AND THREE-STEP NEWTON INTERPOLATION POLYNOMIALS FOR FRACTAL–FRACTIONAL VARIABLE ORDERS CHAOTIC ATTRACTORS

The terms fractional differentiation and fractal differentiation are recently combined to form a new fractional differentiation operator. Several kernels, such as the power-law and the Mittag-Leffler function, are employed to investigate these novel operators. Both fractional-order and fractal dimensions are found in these developed operators. In the present investigation, we have applied these new operators to study certain chaotic attractors with Mittag-Leffler and power-law kernels. These chaotic models are solved using an effective numerical procedure called the Adams–Bashforth technique, based on two- and three-step Newton interpolation polynomials. In this analysis, the order of the fractional derivative is assumed to be the exponential, trigonometric, and hyperbolic forms of variables. The solutions obtained by the numerical procedure with two different kernels are portrayed in terms of chaotic plots.

A Predictive Mechanism Based on Newton Interpolation for Underwater Wireless Sensor Network

Despite their many applications and advantages inside the marine environment, the UWSN encounters many challenges that can affect the network performances, hence many issues has raised and has been discussed recently. In this paper, we have focused on the link failure and the probability of their occurs caused by the node's mobility, water current, the limited battery of the sensor, or even the presence of some interference. Our approach is based on the newton interpolation formula, using to evaluate the link quality between sensors before that a packet is sent and operates in a cross-layer fashion, the purpose is to detect and predict if a sensor node could move out from it sender’s range, or it can regain it current position, in addition we defined a predictive zone, where the sensors are evaluated before they are selected as forwarders, based on the well-known routing protocol, Vector-Based forwarding VBF, we have implemented our approach called CPN-VBF using the ns2-simulator, the simulation results has proved the merits of the proposed study.

Newton interpolation using $$\Re $$-Leja sequences

We investigate the computation and stability of hierarchical Newton interpolation formulas using \(\Re \)-Leja sequences. We carry out a thorough study, in particular from the angle of rounding errors (underflow/overflow), cost, and numerical stability. Furthermore, conditionings of Newton formulas are addressed and moderate estimates are derived.

Profitability prediction in thermal power enterprise based on the improved system dynamics model

This paper tries to improve the system dynamics model and to establish the model of the thermal power enterprise profitability by using Newton interpolation method to optimize the nonlinear function, MATLAB to fitting equation and data analysis; and through the VENSIM to present a simulation. The model is dissertated and validated by some thermal power enterprise's financial data. This model can provide enterprises with management decision-making reference.

Improvement of the WENO-NIP Scheme for Hyperbolic Conservation Laws

The WENO-NIP scheme was obtained by developing a class of L1-norm smoothness indicators based on Newton interpolation polynomial. It recovers the optimal convergence order in smooth regions regardless of critical points and achieves better resolution than the classical WENO-JS scheme. However, the WENO-NIP scheme produces severe spurious oscillations when solving 1D linear advection problems with discontinuities at long output times, and it is also very oscillatory near discontinuities for 1D Riemann problems. In this paper, we find that the spectral property of WENO-NIP exhibits the negative dissipation characteristic, and this is the reason why WENO-NIP is unstable near discontinuities. Using this knowledge, we develop a way of improving the WENO-NIP scheme by introducing an additional term to eliminate the negative dissipation interval. The proposed scheme, denoted as WENO-NIP+, maintains the same convergence property, as well as the same low-dissipation property, as the corresponding WENO-NIP scheme. Numerical examples confirm that the proposed scheme is much more stable near discontinuities for 1D linear advection problems with large output times and 1D Riemann problems than the WENO-NIP scheme. Furthermore, the new scheme is far less dissipative in the region with high-frequency waves. In addition, the improved WENO-NIP+ scheme can remove or at least greatly decrease the post-shock oscillations that are commonly produced by the WENO-NIP scheme when simulating 2D Euler equations with strong shocks.

A energy-conserving predictive preemptive multipath routing protocol for adhoc networks: a lifetime improvement

Mobile device are widely used today in MANETs, due to their rich functionality. However, route failure may occur due to lower receive signal strength, mobility, congestion and device failures. Also, the battery life of these devices is very limited and deploying high resource consuming applications such as streaming on these mobile devices, is a challenging task. It is extremely important to efficiently use of the contained resources on these devices when they participate in a mobile ad hoc network. In this paper, we propose an optimization of the cross-layer networking mechanism. Our work focuses on MAC and routing layers of the OSI stack. We propose a cooperation of the routing layer with the MAC layer power-control technique to decrease the energy consumption in transmission within adhoc networks. We propose an energy-conserving multipath routing protocol for adhoc networks lifetime improvement protocol called E-PPAOMDV (Energy aware Predictive Preemptive AOMDV). This protocol is based on new metric to preserves the residual energy of nodes and balances the consumed energy to increase the network lifetime. Also, we propose a mechanism based on Newton interpolation, to distinguish between both situations, failures due to congestion or mobility, and consequently avoiding unnecessary route repair process. The E-PPAOMDV was implemented using NS-2. The simulation results demonstrate the merits of our proposed E-PPAOMDV. Our proposal improves the performance of mobile ad hoc networks by extending the lifetime of the network and decreasing the average consumed energy with approximately 1 to 3%, while the average end-to-end delay is reduced by 33%, normalized routing load by 20 to 27%; also, increasing the packet delivery ratio with approximately 2-10% and the throughput with 5% when compared with AOMR-LM.

Determination of the drying temperature of the medicinal plant prickly capers - “Capparis spinosa L.” fruits by the method of mathematical modeling

In this paper, the optimal drying temperature of the medicinal plant prickly capers - “Capparis spinosa L.” was studied and mathematical models were determined by comparing experimental data with data calculated using the Lagrange and Newton interpolation formula and the choice of regression analysis. The regression equation has the form of a polynomial and a polynomial level. The mathematical model in which the level of the polynomial was 6 and higher was almost indistinguishable from the models in the form of a polynomial which level was 4 or 5. When determining the optimal drying temperature of the fruit of the medicinal plant prickly capers - “Capparis spinosa L.” by mathematical modeling, it is shown that graphs with a polynomial degree of n=3 and higher have the same values as graphs in which the degree of the polynomial is almost equal to n=2. The coincidence of the theoretical results with the experimental data obtained in determining the optimal drying temperature for the fruit of the medicinal plant prickly capers - “Capparis spinosa L.”, shows the adequacy of the mathematical model. It was found that the optimal drying temperature of the medicinal plant prickly capers - “Capparis spinosa L.” is 333 – 343 K and this is confirmed by mathematical modeling.

Inverse central ordering for the Newton interpolation formula

An inverse central ordering of the nodes is proposed for the Newton interpolation formula. This ordering may improve the stability for certain distributions of nodes. For equidistant nodes, an upper bound of the conditioning is provided. This bound is close to the bound of the conditioning in the Lagrange interpolation formula, whose conditioning is the lowest. This ordering is related to a pivoting strategy of a matrix elimination procedure called Neville elimination. The results are illustrated with examples.

An Adaptive Strategy Based on Repetitive Predictive Control for Improving Adaptability of LCL-Type Grid-Connected Inverters Under Weak Grid

For current-controlled grid-connected inverters with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LCL</i> filter, the essence of grid voltage full feedback is to improve the output impedance of grid-connected inverters to infinity, which can eliminate the influences of grid voltage distortion and the non-negligible grid impedance. However, existing full feedforward strategies are complex and control delay cannot be compensated precisely, which leads to the instability of system when the full feedforward of the voltage at the point of common coupling under the weak grid is adopted. This article first proposes a simplified full feedforward strategy based on grid-side inductor volt-age differential feedback active damping (GIVD-AD). Second, repetitive prediction (RP) is used to compensate for the control delay. Then, a virtual impedance branch is added and an adaptive algorithm for modifying virtual impedance based on the difference between predicted grid voltages by Newton interpolation prediction (NIP) and RP is proposed. With the proposed method, grid-connected inverters can work stably when grid impedance changes suddenly and exhibit strong rejection ability against grid-voltage harmonics. Finally, simulative and experimental results from a 3-kW single-phase grid-connected inverter are provided to prove the effectiveness of the proposed strategy.

Improved variable scale-convex-peak method for weak signal detection

Based on the rectification method of detectable domain and elimination method of blind domain, the variable scale-convex-peak method for the identification frequency of weak signal is improved. For the initial phase of weak signal to be detected, the analytic expression of chaotic threshold is obtained by using the stochastic Melnikov method. The influence of phase on chaotic threshold is clarified, which leads to that the frequency of weak signal can be identified. Specifically, within a period, the initial phase can be divided into the detectable domain and the blind domain. In the detectable domain, when the frequency identification deviates, the rectification method of detectable domain is introduced. Specifically, the Newton interpolation method is used to give the calculation formula to rectify the deviation, which improves the accuracy of frequency identification. In the blind domain, the elimination method of blind domain is given by constructing the detection equations. The feasibility of the improved variable scale-convex-peak method is verified by numerical simulation and circuit experiment. Furthermore, by comparing the identified frequency value with the theoretical fault frequency value, the fault location of the wheelset bearing of the high-speed train is determined. The improved variable scale-convex-peak method is applied when the initial phase is in the frequency detectable domain or the blind domain, respectively.