Curve Fitting Problem Research Articles

Robust baseline correction for Raman spectra by constrained Gaussian radial basis function fitting

Accurate baseline correction is a fundamental requirement for extracting meaningful spectral information and enabling precise quantitative analysis using Raman spectroscopy. Although numerous baseline correction techniques have been developed, they often require meticulous parameter adjustments and yield inconsistent results. To address these challenges, we have introduced a novel approach, namely constrained Gaussian radial basis function fitting (CGF). Our method involves solving a curve-fitting problem using Gaussian radial basis functions under specific constraints. To ensure stability and efficiency, we developed a linear programming algorithm for the proposed approach. We evaluated the performance of CGF using simulated Raman spectra and demonstrated its robustness across various scenarios, including changes in data length and noise levels. In contrast to standard methods, which frequently require complicated parameter adjustments and may exhibit varying errors, our approach provides a simple parameter search and consistently achieves low errors. We further assessed CGF using real Raman spectra, leading to enhanced accuracy in the quantitative analysis of the Raman spectra of chemical warfare agents. Our results emphasize the potential of CGF as a valuable tool for Raman spectroscopy data analysis, significantly advancing sophisticated analytical techniques.

Machine learning for determination of activity of water and activity coefficients of electrolytes in binary solutions

Activity of water and electrolytes in aqueous solutions is of utmost importance for multiple industrial applications. However, experimental determination of such values is time-consuming, while calculation of activity coefficients using numerical methods is challenging. By training neural networks models on literature data, one could predict activity of water and electrolytes easily, without requiring any experiment. In this paper, multiple descriptors (or features) are compared to predict activity coefficients of electrolytes and activity of water in electrolyte solutions. A neural network based on the Levenberg-Marquardt algorithm (LM-NN) showed high accuracy to calculate values, despite the small size of the training datasets. Both activity coefficients of electrolytes and activity of water in electrolyte solutions can be predicted accurately even on unseen data, using simple descriptors such as electrolyte concentration, ion sizes and charges. However, some discrepancies were observed due to the lack of representativeness of the training dataset. This could be solved by selecting training data sets that are similar (e.g. same group of the periodic table) to the unknown values, or by including available experimental data for the salt considered. The ability of the LM-NN to solve non-linear least square curve fitting problems makes it a good candidate to fit experimental activity coefficient data, with the advantage of simplicity as compared to e-NRTL or UNIQUAC methods. This method paves the way for accurate and quick determination of thermodynamic data for electrolyte solutions (and beyond) using machine learning, without necessitating large training datasets.

Adaptive tempered reversible jump algorithm for Bayesian curve fitting

Bayesian curve fitting plays an important role in inverse problems, and is often addressed using the reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. However, this algorithm can be computationally inefficient without appropriately tuned proposals. As a remedy, we present an adaptive RJMCMC algorithm for the curve fitting problems by extending the adaptive Metropolis sampler from a fixed-dimensional to a trans-dimensional case. In this presented algorithm, both the size and orientation of the proposal function can be automatically adjusted in the sampling process. Specifically, the curve fitting setting allows for the approximation of the posterior covariance of the a priori unknown function on a representative grid of points. This approximation facilitates the definition of efficient proposals. In addition, we introduce an auxiliary-tempered version of this algorithm via non-reversible parallel tempering. To evaluate the algorithms, we conduct numerical tests involving a series of controlled experiments. The results demonstrate that the adaptive algorithms exhibit significantly higher efficiency compared to the conventional ones. Even in cases where the posterior distribution is highly complex, leading to ineffective convergence in the auxiliary-tempered conventional RJMCMC, the proposed auxiliary-tempered adaptive RJMCMC performs satisfactorily. Furthermore, we present a realistic inverse example to test the algorithms. The successful application of the adaptive algorithm distinguishes it again from the conventional one that fails to converge effectively even after millions of iterations.

An analytical approach to evaluate the impact of age demographics in a pandemic.

The time required to identify and confirm risk factors for new diseases and to design an appropriate treatment strategy is one of the most significant obstacles medical professionals face. Traditionally, this approach entails several clinical studies that may last several years, during which time strict preventative measures must be in place to contain the epidemic and limit the number of fatalities. Analytical tools may be used to direct and accelerate this process. This study introduces a six-state compartmental model to explain and assess the impact of age demographics by designing a dynamic, explainable analytics model of the SARS-CoV-2 coronavirus. An age-stratified mathematical model taking the form of a deterministic system of ordinary differential equations divides the population into different age groups to better understand and assess the impact of age on mortality. It also provides a more accurate and effective interpretation of the disease evolution, specifically in terms of the cumulative numbers of infected cases and deaths. The proposed Kermack-Mckendrick model is incorporated into a non-linear least-squares optimization curve-fitting problem whose optimized parameters are numerically obtained using the Levenberg-Marquard algorithm. The curve-fitting model's efficiency is proved by testing the age-stratified model's performance on three U.S. states: Connecticut, North Dakota, and South Dakota. Our results confirm that splitting the population into different age groups leads to better fitting and forecasting results overall as compared to those achieved by the traditional method, i.e., without age groups. By using comprehensive models that account for age, gender, and ethnicity, regional public health authorities may be able to avoid future epidemics from inflicting more fatalities and establish a public health policy that reduces the burden on the elderly population.

Generative Wind Power Curve Modeling via Machine Vision: A Deep Convolutional Network Method With Data-Synthesis-Informed-Training

This study develops a novel data-synthesis-informed-training U-net (DITU-net) based method to automate the wind power curve (WPC) modeling without data pre-processing. The proposed data-synthesis-informed-training (DIT) process has following steps. First, different from traditional studies regarding the WPC modeling as a curve fitting problem, we renovate the WPC modeling formulation from a machine vision aspect. To develop sufficiently diversified training samples, we synthesize supervisory control and data acquisition (SCADA) WPC data based on a set of S-shape functions representing WPCs. These synthesized SCADA data and WPC functions are visualized as images, named the synthesized SCADA WPC and synthesized neat WPC, and paired as training samples. A deep generative model based on U-net is developed to approximate the projection recovering the synthesized neat WPC from the synthesized SCADA WPC. The developed U-net based model is applied into observed SCADA data and can successfully generate the neat WPC. Moreover, a pixel mapping and correction process is developed to derive a mathematical form depicting the neat WPC generated previously. The proposed DITU-net only needs to train once and does not require any data preprocessing in applications. Numerical experiments based on 76 WTs are conducted to validate the superiority of the proposed method via benchmarking against classical WPC modeling methods.

Three-Parameter P-S-N Curve Fitting Based on Improved Maximum Likelihood Estimation Method

The P-S-N curve is a vital tool for dealing with fatigue life analysis, and its fitting under the condition of small samples is always concerned. In the view that the three parameters of the P-S-N curve equation can better describe the relationship between stress and fatigue life in the middle- and long-life range, this paper proposes an improved maximum likelihood method (IMLM). The backward statistical inference method (BSIM) recently proposed has been proven to be a good solution to the two-parameter P-S-N curve fitting problem under the condition of small samples. Because of the addition of an unknown parameter, the problem exists in the search for the optimal solution to the three-parameter P-S-N curve fitting. Considering that the maximum likelihood estimation is a commonly used P-S-N curve fitting method, and the rationality of its search for the optimal solution is better than that of BSIM, a new method combining BSIM and the maximum likelihood estimation is proposed. In addition to the BSIM advantage of expanding the sample information, the IMLM also has the advantage of more reasonable optimal solution search criteria, which improves the disadvantage of BSIM in parameter search. Finally, through the simulation tests and the fatigue test, the P-S-N curve fitting was carried out by using the traditional group method (GM), BSIM, and IMLM, respectively. The results show that the IMLM has the highest fitting accuracy. A test arrangement method is proposed accordingly.

Modeling snowflake plasmas in RFX-mod2: A test bed for SOL and edge physics characterization

The definition of low-current snowflake plasma equilibria in RFX-mod2 experiment is carried out in terms of both plasma conditions and engineering requirements. The snowflake magnetic configuration is obtained by solving a least-squares curve fitting problem with properly defined linear contraints. Positive and negative triangularity snowflake equilibria are defined and compared with a standard single-null plasma. The magnetic field topology is characterized in terms of local magnetic shear and connection length revealing all the potential advantages of a wider low poloidal field region. The RFX-mod2 improved diagnostic system and its magnet flexibility would allow a detailed characterization of the scrape off layer and plasma edge physics in snowflake configurations.

Novel model prediction time-to-event analysis: data validation and estimation of 200 million cases in the global COVID-19 epidemic

ObjectivesAssessment of recuperation and death times of a population inflicted by an epidemic has only been feasible through studying a sample of individuals via time-to-event analysis, which requires identified participants....

A Polynomial Fitting Problem: The Orthogonal Distances Method

The classical curve-fitting problem to relate two variables, x and y, deals with polynomials. Generally, this problem is solved by the least squares method (LS), where the minimization function considers the vertical errors from the data points to the fitting curve. Another curve-fitting method is total least squares (TLS), which takes into account errors in both x and y variables. A further method is the orthogonal distances method (OD), which minimizes the sum of the squares of orthogonal distances from the data points to the fitting curve. In this work, we develop the OD method for the polynomial fitting of degree n and compare the TLS and OD methods. The results show that TLS and OD methods are not equivalent in general; however, both methods get the same estimates when a polynomial of degree 1 without an independent coefficient is considered. As examples, we consider the calibration curve-fitting problem of a R-type thermocouple by polynomials of degrees 1 to 4, with and without an independent coefficient, using the LS, TLS and OD methods.

Fractal Perturbation of the Nadaraya–Watson Estimator

One of the main tasks in the problems of machine learning and curve fitting is to develop suitable models for given data sets. It requires to generate a function to approximate the data arising from some unknown function. The class of kernel regression estimators is one of main types of nonparametric curve estimations. On the other hand, fractal theory provides new technologies for making complicated irregular curves in many practical problems. In this paper, we are going to investigate fractal curve-fitting problems with the help of kernel regression estimators. For a given data set that arises from an unknown function m, one of the well-known kernel regression estimators, the Nadaraya–Watson estimator m^, is applied. We consider the case that m is Hölder-continuous of exponent β with 0<β≤1, and the graph of m is irregular. An estimation for the expectation of |m^−m|2 is established. Then a fractal perturbation f[m^] corresponding to m^ is constructed to fit the given data. The expectations of |f[m^]−m^|2 and |f[m^]−m|2 are also estimated.

Atomic Formulation of the Boolean Curve Fitting Problem

Boolean curve fitting is the process of finding a Boolean function that takes given values at certain points in its Boolean domain. The problem boils down to solving a set of ‘big’ Boolean equations that may or may not be consistent. The usual formulation of the Boolean curve fitting problem is quite complicated, indeed. In this paper, we formulate the Boolean curve fitting problem using the technique of atomic decomposition of Boolean equations. This converts the problem into a set of independent switching equations. We present the solution of these switching equations and express the solution in very simple and compact forms. We also present the consistency and uniqueness conditions for this problem again in very compact forms. A few illustrative examples are given. These examples clearly pinpoint the simplicity gained by the Boolean-equation solving step within the overall Boolean-fitting procedure. The method presented here can be applied to the design of Boolean functions for cryptographic systems.

ScatterHough: Automatic Lane Detection from Noisy LiDAR Data

Lane detection plays an essential role in autonomous driving. Using LiDAR data instead of RGB images makes lane detection a simple straight line, and curve fitting problem works for realtime applications even under poor weather or lighting conditions. Handling scatter distributed noisy data is a crucial step to reduce lane detection error from LiDAR data. Classic Hough Transform (HT) only allows points in a straight line to vote on the corresponding parameters, which is not suitable for data in scatter form. In this paper, a Scatter Hough algorithm is proposed for better lane detection on scatter data. Two additional operations, neighbor voting and neighbor vote-reduction, are introduced to HT to make points in the same curve vote and consider their neighbors’ voting result as well. The evaluation of the proposed method shows that this method can adaptively fit both straight lines and curves with high accuracy, compared with benchmark and state-of-the-art methods.

REPRODUCING KERNEL HILBERT SPACES OF FRACTAL INTERPOLATION FUNCTIONS FOR CURVE FITTING PROBLEMS

In this paper, we show that the spaces of some types of fractal interpolation functions are reproducing kernel Hilbert spaces with two different types of inner products. Then we apply these results to curve fitting problems. We establish the fractal interpolation functions that are in reproducing kernel Hilbert spaces and that minimize the regularized empirical error.

Iterative Least Square Optimization for the Weights of NURBS Curve

NURBS curves have been widely applied in the field of data points approximation, and their fitting accuracy can be improved by adjusting the values of their weights. When applying the NURBS curve, it is difficult to obtain the optimal weights values due to the nonlinearity of the curve fitting problem with NURBS. In this paper, a weights iterative optimization method for NURBS curve fitting is proposed, where the geometric property of weight has been adopted to iteratively obtain the adjusting values of the weights with the least square method. The effectiveness and convergence of the proposed method are demonstrated by numerical experiments. The results show that the proposed method can obtain higher fitting accuracy than other iterative optimization methods. Meanwhile, it has the merits of data noise robustness, high accuracy with small-scale knots, and flexibility. Hence, the proposed method is suitable for applications including noisy data approximation and skinned surface generation.

Damped Nyquist Plot for the Phase and Gain Optimization of Power System Stabilizers

This paper presents a method to optimize the tuning of a power system stabilizer (PSS) with the objective of simultaneously damping electromechanical poles of different nature. The damped Nyquist plot (DNP) is used to determine the PSS phase and gain compensation requirements to place each pole at a predefined location in the complex plane. A non–linear least–squares curve fitting problem is solved to find the set of PSS parameters that leads to the optimum trade–off for all dominant poles based on a single eigenvalue calculation per operation scenario. The PSS tuning is performed on a two–bus synthetic power system model whose results are validated against a three-machine IEEE benchmark model results. The advantages of using the DNP method together with the synthetic system to perform a PSS tuning in practice are emphasized.

On Selecting Composite Functions Based on Polynomials for Responses Describing Extreme Magnitudes of Structures

The main aim of this work was to investigate a numerical error in determining limit state functions, which describe the extreme magnitudes of steel structures with respect to random variables. It was assisted here by the global version of the response function method (RFM). Various approximations of trial points generated on the basis of several hundred selected reference composite functions based on polynomials were analyzed. The final goal was to find some criterion—between approximation and input data—for the selection of the response function leading to relative a posteriori errors less than 1%. Unlike the classical problem of curve fitting, the accuracy of the final values of probabilistic moments was verified here as they can be used in further reliability calculations. The use of the criterion and the associated way of selecting the response function was demonstrated on the example of steel diagrid grillages. It resulted in quite high correctness in comparison with extended FEM tests.

A Novel Statistical Optimization Algorithm for Estimating Perfusion Curves in Susceptibility Contrast-Enhanced MRI.

Dynamic susceptibility contrast-enhanced magnetic resonance imaging is an important tool for evaluating intravascular indicator dynamics, which in turn is valuable for understanding brain physiology and pathophysiology. This procedure usually involves fitting a gamma-variate function to observed concentration-time curves in order to eliminate undesired effects of recirculation and the leakage of contrast agents. Several conventional curve-fitting approaches are routinely applied. The nonlinear optimization methods typically are computationally expensive and require reliable initial values to guarantee success, whereas a logarithmic linear least-squares (LL-LS) method is more stable and efficient, and does not suffer from the initial-value problem, but it can show degraded performance, especially when a few data or outliers are present. In this paper, we demonstrate, that the original perfusion curve-fitting problem can be transformed into a gamma-distribution-fitting problem by treating the concentration-time curves as a random sample from a gamma distribution with time as the random variable. A robust maximum-likelihood estimation (MLE) algorithm can then be readily adopted to solve this problem. The performance of the proposed method is compared with the nonlinear Levenberg-Marquardt (L-M) method and the LL-LS method using both synthetic and real data. The results show that the performance of the proposed approach is far superior to those of the other two methods, while keeping the advantages of the LL-LS method, such as easy implementation, low computational load, and dispensing with the need to guess the initial values. We argue that the proposed method represents an attractive alternative option for assessing intravascular indicator dynamics in clinical applications. Moreover, we also provide valuable suggestions on how to select valid data points and set the initial values in the two traditional approaches (LL-LS and nonlinear L-M methods) to achieve more reliable estimations.

StrokeStrip

When creating freeform drawings, artists routinely employ clusters of overdrawn strokes to convey intended, aggregate curves. The ability to algorithmically fit these intended curves to their corresponding clusters is central to many applications that use artist drawings as inputs. However, while human observers effortlessly envision the intended curves given stroke clusters as input, existing fitting algorithms lack robustness and frequently fail when presented with input stroke clusters with non-trivial geometry or topology. We present StrokeStrip , a new and robust method for fitting intended curves to vector-format stroke clusters. Our method generates fitting outputs consistent with viewer expectations across a vast range of input stroke cluster configurations. We observe that viewers perceive stroke clusters as continuous, varying-width strips whose paths are described by the intended curves. An arc length parameterization of these strips defines a natural mapping from a strip to its path. We recast the curve fitting problem as one of parameterizing the cluster strokes using a joint 1D parameterization that is the restriction of the natural arc length parameterization of this strip to the strokes in the cluster. We simultaneously compute the joint cluster parameterization and implicitly reconstruct the a priori unknown strip geometry by solving a variational problem using a discrete-continuous optimization framework. We use this parameterization to compute parametric aggregate curves whose shape reflects the geometric properties of the cluster strokes at the corresponding isovalues. We demonstrate StrokeStrip outputs to be significantly better aligned with observer preferences compared to those of prior art; in a perceptual study, viewers preferred our fitting outputs by a factor of 12:1 compared to alternatives. We further validate our algorithmic choices via a range of ablation studies; extend our framework to raster data; and illustrate applications that benefit from the parameterizations produced.

Extended Bezier Model-Based Human Target Localization Algorithm by Doppler Radar

This letter proposes a Doppler through-wall radar (TWR) positioning method based on the Hough transform of the extended Bezier model. First, the Hough transform based on the extended Bezier model is used to extract the interested target components from received echoes and estimate their instantaneous frequency (IF). Next, the real-time position of the target is estimated according to the Doppler processing. To improve the adaptability of the algorithm to different target trajectories, especially to solve the problem of asymmetric curve fitting, the proposed extended Bezier model can approach the practical IF curve by dynamically adjusting two parameters, which can not only keep the relatively low computational complexity but also significantly improve the positioning accuracy. Moreover, the proposed algorithm effectively solves the problem of “frequency ambiguity.” The experimental results demonstrate and validate that the proposed algorithm is effective in the field of human sensing applications of Doppler radar.

Perception, Planning and Control for Self-Driving System Based on On-board Sensors

The autonomous vehicle can recognize and understand environment, self-control, and achieve the driving level of the human driver. To create this kind of system, the following work was carried out: (a) a real time lane detection system is proposed, based on vision system functions, using webcam camera; (b) In order to detect curve lanes, deep learning is applied for lane detection, based on fully Convolutional Neural Network (CNN); (c) The cubic spline interpolation method is used for path generation, based on Global Positioning System (GPS) data, where distance between two adjacent path generation points is same. Compared with the connection method of the cubic polynomial fitting algorithm, the curve fitted path by the cubic spline interpolation method is smoother and more satisfied with the vehicle motion pattern; (d) Based on frenet coordinate frame, optimize trajectory planning method is used to plan safe, easy and comfort trajectory. Compared to the Cartesian coordinate frame, frenet coordinate frame simplifies the solution of road curve fitting problems, especially in the case of complex road environment; (e) Fuzzy sliding mode control method is proposed for vehicle steering control. Finally, the simulation and experiment results are used to prove the effectiveness of the proposed methods.