A Study on Piecewise Polynomial Smooth Approximation to the Plus Function

Imbalanced data learning is one of the most active and important fields in machine learning research. The existing class imbalance learning methods can make Support Vector Machines (SVMs) less sensitive to class imbalance; they still suffer from the disturbance of outliers and noise present in the datasets. A kind of Fuzzy Smooth Support Vector Machines (FSSVMs) are proposed based on the Smooth Support Vector Machine (SSVM) of O. L. Mangasarian. SSVM can be computed by the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) algorithm or the Newton‐Armijo algorithm easily. Two kinds of fuzzy memberships and three smooth functions can be chosen in the algorithms. The fuzzy memberships consider the contribution rate of each sample to the optimal separating hyperplane. The polynomial smooth functions can make the optimization problem more accurate at the inflection point. Those changes play the active effects on trials. The results of the experiments show that the FSSVMs can gain the better accuracy and the shorter time than the SSVMs and some of the other methods.

Smoothing methods, extensively used for solving important mathematical programming problems and applications, are applied here to generate and solve an unconstrained smooth reformulation of the support vector machine for pattern classification using a completely arbitrary kernel. We term such reformulation a smooth support vector machine (SSVM). A fast Newton–Armijo algorithm for solving the SSVM converges globally and quadratically. Numerical results and comparisons are given to demonstrate the effectiveness and speed of the algorithm. On six publicly available datasets, tenfold cross validation correctness of SSVM was the highest compared with four other methods as well as the fastest. On larger problems, SSVM was comparable or faster than SVMlight (T. Joachims, in Advances in Kernel Methods—Support Vector Learning, MIT Press: Cambridge, MA, 1999), SOR (O.L. Mangasarian and David R. Musicant, IEEE Transactions on Neural Networks, vol. 10, pp. 1032–1037, 1999) and SMO (J. Platt, in Advances in Kernel Methods—Support Vector Learning, MIT Press: Cambridge, MA, 1999). SSVM can also generate a highly nonlinear separating surface such as a checkerboard.

Human Development Index (HDI) is measuring achievements of human development based on basic components of quality of life. Human development index is low score if HDI is less than 60, moderate HDI between 60 to less than 70, high HDI between 70 to less than 80, and equal to 80 and more than 80 belong to high HDI. Smooth Support Vector Machine (SSVM) is a classification technique that is new. The algorithm used is Newton Armijo with linear kernel, polynomial kernel, and Radial Basis Function (RBF) kernel. The result of classification of human development index with SSVM method with linear kernel shows the prediction accuracy 84.77%, polynomial kernel 61.65%, and RBF kernel 100%. Radial Base Function Kernel (RBF) is the most accurate kernel in predicting human development index.

Support vector machine (SVM) is a supervised machine learning approach that was recognized as a statistical learning apotheosis for the small-sample database. SVM has shown its excellent learning and generalization ability and has been extensively employed in many areas. This paper presents a performance analysis of six types of SVMs for the diagnosis of the classical Wisconsin breast cancer problem from a statistical point of view. The classification performance of standard SVM (St-SVM) is analyzed and compared with those of the other modified classifiers such as proximal support vector machine (PSVM) classifiers, Lagrangian support vector machines (LSVM), finite Newton method for Lagrangian support vector machine (NSVM), Linear programming support vector machines (LPSVM), and smooth support vector machine (SSVM). The experimental results reveal that these SVM classifiers achieve very fast, simple, and efficient breast cancer diagnosis. The training results indicated that LSVM has the lowest accuracy of 95.6107 %, while St-SVM performed better than other methods for all performance indices (accuracy = 97.71 %) and is closely followed by LPSVM (accuracy = 97.3282). However, in the validation phase, the overall accuracies of LPSVM achieved 97.1429 %, which was superior to LSVM (95.4286 %), SSVM (96.5714 %), PSVM (96 %), NSVM (96.5714 %), and St-SVM (94.86 %). Value of ROC and MCC for LPSVM achieved 0.9938 and 0.9369, respectively, which outperformed other classifiers. The results strongly suggest that LPSVM can aid in the diagnosis of breast cancer.

Solving parametric piecewise polynomial systems

Over the last decades the finite element method, which was introduced by engineers in the 1960s, has become the perhaps most important numerical method for partial differential equations, particularly for equations of elliptic and parabolic types. This method is based on the variational form of the boundary value problem and approximates the exact solution by a piecewise polynomial function. It is more easily adapted to the geometry of the underlying domain than the finite difference method, and for symmetric positive definite elliptic problems it reduces to a finite linear system with a symmetric positive definite matrix. We first introduce this method in Sect. 5.1 for the case of a two-point boundary value problem and show a number of error estimates. In Sect. 5.2 we then formulate the method for a two-dimensional model problem. Here the piecewise polynomial approximations are defined on triangulations of the spatial domain, and in the following Sect. 5.3 we study such approximation in more detail. In Sect. 5.4 we show basic error estimates for the finite element method for the model problem, using piecewise linear approximating functions. All error bounds derived up to this point contain a norm of the unknown exact solution and are therefore often referred to as a priori error estimates. In Sect. 5.5 we show a so-called a posteriori error estimate in which the error bound is expressed in terms of the data of the problem and the computed solution. In Sect. 5.6 we analyze the effect of numerical integration, which is often used when the finite element equations are assembled in a computer program. In Sect. 5.7 we briefly describe a so-called mixed finite element method.KeywordsFinite Element MethodElliptic EquationQuadrature FormulaInterpolation ErrorPosteriori Error EstimateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Multi-objective reservoir operation during flood season considering spillway optimization

Accurate diagnosis of Alzheimer's disease (AD) and Mild Cognitive Impairment (MCI) was identified on an early stage is essential in the healthcare industry to stop degeneration. The Smooth Support Vector Machine (SSVM) model, Principal Component Analysis (PCA), feature extraction, and Magnetic Resonance Imaging (MRI) image prepossessing are the components for the diagnosis of AD is proposed in this research at early stage. To assist in the classifier's training, we proposed a novel Improved Weighed Quantum Lion Optimization (IWQLO). The SSVM parameters are specifically proposed to be optimized using a new Switching delayed Lion Optimization (SLO) algorithm. The IWQLO-SSVM approach was effectively used to classify AD and MCI utilizing MRI scans of the Alzheimer's disease Neuroimaging Initiative (ADNI) database and Outcome and Assessment Information Set (OASIS) database. For six example scenarios, the classification accuracy of our proposed method is acceptable. Testing show that the proposed approach improves the performance measures such as accuracy, precision, specificity, sensitivity and recall for detecting the early stage AD diagnosis.

This paper presents an event-driven simulation methodology for injection-locked oscillators. The proposed method adopts a phase-domain macromodel based on a perturbation projection vector (PPV), which expresses the oscillator's phase response using a nonlinear, time-varying differential equation. By expressing the PPV using a piecewise polynomial function, the PPV model can be simulated in an event-driven fashion on a digital logic simulator like SystemVerilog. While this piecewise polynomial approximation introduces a trade-off between the simulation accuracy and speed, our results show that the simulation accuracy quickly reaches its optimal value as the piecewise interval number increases even with first-order polynomials. The experimental results with an LC oscillator, ring oscillator, and burst-mode CDR models operating with both periodic and aperiodic inputs demonstrate that the proposed models achieve speeds up to 2 ~ 30× that of transistor-level Spectre simulations with the same accuracy.

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In practice, we are interested in approximating a function f(x) near x = a by a polynomial of degree M. Other issues involved in approximation are how good the approximation is and over what interval can we expect the approximation to be good. In this paper we present approximation that is realized as piecewise polynomial function where in each interval ƒ(x) is approximated by a Taylor polynomial of low order. The piecewise polynomial approximation can be effectively implemented using Farrow structure.

We design a new, fast algorithm for agnostically learning univariate probability distributions whose densities are well-approximated by piecewise polynomial functions. Let ƒ be the density function of an arbitrary univariate distribution, and suppose that ƒ is OPT-close in Li- distance to an unknown piecewise polynomial function with t interval pieces and degree d. For any γ > 0, our algorithm draws n = Õγ(t(d + 1)/ ∊2) samples from ƒ, runs in time Õ(n), and with probability at least 9/10 outputs an Ογ (t)-piecewise degree-d hypothesis H that is (3 + γ) · OPT + ∊ close to f. Our approximation factor almost matches the best known information-theoretic (but computationally inefficient) upper bound of 3.Our general algorithm yields (nearly) sample- optimal and nearly-linear time estimators for a wide range of structured distribution families over both continuous and discrete domains in a unified way. For most of our applications, these are the first sample-optimal and nearly-linear time estimators in the literature. As a consequence, our work resolves the sample and computational complexities of a broad class of inference tasks via a single “meta-algorithm”. Moreover, we demonstrate that our algorithm performs very well in experiments.Our algorithm consists of three levels: (i) At the top level, we employ an iterative greedy algorithm for finding a good partition of the real line into the pieces of a piecewise polynomial. (ii) For each piece, we show that the sub-problem of finding a good polynomial fit on the current interval can be solved efficiently with a separation oracle method. (iii) We reduce the task of finding a separating hyperplane to a combinatorial problem and design a nearly-linear algorithm for this problem. Combining these three procedures gives a density estimation algorithm with the claimed guarantees.

Purpose – The purpose of this paper is to present an original mixed least squares method for solving problems in dynamic elasticity. Design/methodology/approach – The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation. The proposed theoretical formulation is given the general three-dimensional case and is tested numerically on the solution of one-dimensional wave equations. Findings – To test the performance of the method, it has been implemented in an original computer code, using object-oriented logic and written from scratch. Two one-dimensional problems from the mathematical physics, with well-known exact analytical solutions, have been solved. The numerical examples include a forced vibrating spring, fixed at its both ends and a rod, vibrating under its own weight, when one of its ends is fixed and the other is traction-free. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values. Originality/value – The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications to two and three-dimensional problems in dynamic elasticity.

The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F -spaces N 0 p , n {\mathbf {N}}_0^{p,n} (introduced for this purpose in [2]). We obtain two o -results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in N 0 p , n {\mathbf {N}}_0^{p,n} we define a derivative D n , σ f {D^{n,\sigma }}f in L σ , σ = ( n + p − 1 ) − 1 {L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}} , which agrees with D n f {D^n}f for smooth f , and prove several properties of D n , σ {D^{n,\sigma }} .

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

Recently, smooth rank function was proposed for matrix completion problem. The main idea of this method was based on a continuous and differentiable approximation of the rank function. In this paper, a new approach for solving low-rank matrix recovery problem based on smooth function is proposed. It not only uses a smooth function to approximate the rank function, but also approximates the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm with a continuous and differentiable function. In addition, gradient decreasing approach can be used to solve the minimization problem. Finally, experimental results show that this new algorithm provides accurate results in almost all of our testing scenarios with a reasonable running time. Especially, it has higher approximation performance than other methods.