Univariate Functions Research Articles

Global optimization on non-convex two-way interaction truncated linear multivariate adaptive regression splines using mixed integer quadratic programming

Multivariate adaptive regression splines (MARS) has been adopted as a popular surrogate function to model the unknown relationships between input and output variables in complex systems. Searching optimal solutions on the complicated surrogate response surface of MARS serves as important tasks in various applications. In this paper, we present an efficient and effective approach to find a global optimal value of MARS models that incorporate two-way interaction terms which are products of truncated linear univariate functions (TITL-MARS). Specifically, with a MARS model consisting of linear and quadratic structures, we can reformulate the optimization problem on TITL-MARS into a mixed integer quadratic programming problem (TITL-MARS-OPT), which can be further solved in a more principled way. To illustrate the effectiveness of our proposed approach TITL-MARS-OPT, we come up with a genetic algorithm and a gradient descent algorithm to solve the original version of TITL-MARS, and we compared the performance of the proposed approach with the benchmark algorithms. Numerical experiments are conducted on a spectrum of examples with different levels of complexity, including 6 existing models and a real world application in the wind farm optimization. Our proposed approach can find a global optimal successfully and efficiently while the comparison algorithms fail to find a global optimal solution. In the end, Python code for TITL-MARS and TITL-MARS-OPT is made available on GitHub ( https://github.com/JuXinglong/TITL-MARS-OPT).

A preference for link operator functions can drive Boolean biological networks towards critical dynamics

Boolean modelling is a powerful framework to understand the operating principles of biological networks. The regulatory logic between biological entities in these networks is expressed as Boolean functions (BFs). There exist various types of BFs (and thus regulatory logic rules) which are meaningful in the biological context. In this contribution, we explore one such type, known as link operator functions (LOFs). We theoretically enumerate these functions and show that, among all BFs and even within the biologically consistent effective and unate functions (EUFs), the LOFs form a tiny subset. We then find that the AND-NOT LOFs are particularly abundant in reconstructed biological Boolean networks. By leveraging these facts, namely, the tiny representation of LOFs in the space of EUFs and their presence in the biological dataset, we show that the space of acceptable models can be shrunk by applying steady-state constraints to BFs, followed by the choice of LOFs which satisfy those constraints. Finally, we demonstrate that among a wide range of BFs, the LOFs drive biological network dynamics towards criticality.

Unconditional Bases for Homogeneous $$\alpha $$-Modulation Type Spaces

We construct orthonormal bases compatible with bi-variate homogeneous \(\alpha \)-modulation spaces and the associated spaces of Triebel–Lizorkin type. The construction is based on generating a separable \(\alpha \)-covering and using carefully selected tensor products of univariate brushlet functions with regard to this covering. We show that the associated systems form an unconditional bases for the homogeneous \(\alpha \)-spaces of Triebel–Lizorkin type.

Physics makes the difference: Bayesian optimization and active learning via augmented Gaussian process

Both experimental and computational methods for the exploration of structure, functionality, and properties of materials often necessitate the search across broad parameter spaces to discover optimal experimental conditions and regions of interest in the image space or parameter space of computational models. The direct grid search of the parameter space tends to be extremely time-consuming, leading to the development of strategies balancing exploration of unknown parameter spaces and exploitation towards required performance metrics. However, classical Bayesian optimization (BO) strategies based on the Gaussian process (GP) do not readily allow for the incorporation of the known physical behaviors or past knowledge. Here we explore a hybrid optimization/exploration algorithm created by augmenting the standard GP with a structured probabilistic model of the expected system’s behavior. This approach balances the flexibility of the non-parametric GP approach with a rigid structure of physical knowledge encoded into the parametric model. The fully Bayesian treatment of the latter allows additional control over the optimization via the selection of priors for the model parameters. The method is demonstrated for a noisy version of a standard univariate test function used to evaluate optimization algorithms and further extended to physical lattice models. This methodology is expected to be universally suitable for injecting prior knowledge in the form of physical models and past data in the BO framework.

The Use of a High-Pressure Water-Ice Jet for Removing Worn Paint Coating in Renovation Process.

The paper presents the results of investigations into the possibility of using ahigh-pressure water-ice jet as a new method for removing a worn-out paint coating from the surface of metal parts (including those found in means of transportation) and for preparing the base surface for the application of renovation paint coating. Experimental investigations were carried out in four stages, on flat specimens, sized S × H = 75 × 115 mm, cut from sheet metal made of various materials such as steel X5CrNi18-10, PA2 aluminium alloy and PMMA polymethyl methacrylate (plastic). In the first stage, the surfaces of the samples were subjected to observation of surface morphology under a scanning electron microscope, and surface topography (ST) measurements were made on a profilographometer. Two ST parameters were analysed in detail: the maximum height of surface roughness Sz and the arithmetic mean surface roughness Sa. Next, paint coatings were applied to the specimens as a base. In the third stage, the paint coating applied was removed by means of a high-pressure water-ice jet (HPWIJ) by changing the values of the technological parameters, i.e., water jet pressure pw, dry ice mass flow rate , distance between the sprinkler head outlet and the surface being treated (the so-called working jet length) l2 and spray angle κ for the following constants: the number of TS = 4 holes, water hole diameter φ = 1.2 mm and sprinkler head length Lk = 200 mm. Afterwards, the surface morphology was observed again and the surface topography of the specimen was investigated by measuring selected 3D parameters of the ST structure, Sz and Sa. The results of investigations into the influence of selected HPWIJ treatment parameters on the surface QF removal efficiency obtained are also presented. Univariate regression functions were developed for the mean stripping efficiency based on the following: dry ice mass flow rate , working jet length l2 and spray angle κ. Based on these functions, the values of optimal parameters were determined that allow the maximum efficiency of the process to be obtained. A 95% confidence region for the regression function was also developed. The results demonstrated that HPWIJ treatment does not interfere with the geometric structure of the base material, and they confirmed the possibility of using this treatment as an efficient method of removing a worn paint layer from bases made of various metal and plastic materials, and preparing it for applying a new layer during renovation.

Hyper-Gaussian regularized Whittaker–Kotel’nikov–Shannon sampling series

The reconstruction of a bandlimited function from its finite sample data is fundamental in signal analysis. It is well known that oversampling of a bandlimited function leads to exponential convergence in its reconstruction. A simple and efficient Gaussian regularized Shannon sampling formula has been proposed in G. W. Wei [Quasi wavelets and quasi interpolating wavelets, Chem. Phys. Lett. 296 (1998) 215–222] with such an exponential convergence ability. We show that all hyper-Gaussian regularized formulas share this desired property. The analysis is built on estimates on the Fourier transform of the hyper-Gaussian functions. We also establish error bounds for the reconstruction of derivatives of a univariate bandlimited function, and for multivariate bandlimited functions.

Multivariate Frequency Analysis for Streamflow Drought Having Different Time Resolution Using Archimedean Copula Functions

This study explored return periods of the streamflow deficit characteristics by threshold levels with different time resolutions (daily, weekly, and monthly) using univariate and Copula functions, reflecting hydrological regimes like rapid recoveries from drought and anthropogenic dam operation by monthly basis in South Korea. As a result, the daily-based threshold had a large variability in severities and durations. This is because threshold levels include a large variability of daily streamflow with a small time interval. Joint return periods of duration under extreme droughts were similar to those from univariate frequency analysis. The severities in these extreme droughts were also around the average level, but the duration was relatively long, leading to an increase in the joint return period. This result helps to effectively manage water resource preparing for the assessment on the drought risk with a long duration.

Definitional schemes for primitive recursive and computable functions

The unary primitive recursive functions can be defined in terms of a finite set of initial functions together with a finite set of unary and binary operations that are primitive recursive in their inputs. We reduce arity considerations, by show that two fixed unary operations suffice, and a single initial function can be chosen arbitrarily. The method works for many other classes of functions, including the unary partial computable functions. For this class of partial functions we also show that a single unary operation (together with any finite set of initial functions) will never suffice.

The Generalized HR $q$-Derivative and Its Application to Quaternion Least Mean Square Algorithm

Quaternion adaptive filters have been widely used in processing three-dimensional and four-dimensional signals. To improve the performance of quaternion adaptive filtering algorithms, this letter first proposes a novel derivation rule named generalized HR (GHR) <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative based on the concept of <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative and quaternion GHR derivative. Then, the product rules of GHR <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative are deduced, and the results of GHR <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative regarding some common univariate and multivariable functions are given. In addition, based on the proposed GHR <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative, the cost function of the quaternion least mean square (QLMS) algorithm is minimized to generate the <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-QLMS algorithm. Finally, an example of Lorentz chaotic time-series prediction verifies the effectiveness of the proposed <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-QLMS algorithm.

Multivariate fractal interpolation functions: some approximation aspects and an associated fractal interpolation operator

In the classical (non-fractal) setting, the natural kinship between theories of interpolation and approximation is well explored. In contrast to this, in the context of fractal interpolation, the interrelation between interpolation and approximation is subtle, and this duality is relatively obscure. The notion of α-fractal functions provides a proper foundation for the approximation-theoretic facet of univariate fractal interpolation functions (FIFs). However, no comparable approximation-theoretic aspects of FIFs have been developed for functions of several variables. The current article intends to open the door for intriguing interactions between approximation theory and multivariate FIFs. To this end, in the first part of this article, we develop a general framework for constructing multivariate FIFs, which is amenable to provide a multivariate analogue of an α-fractal function. Multivariate α-fractal functions provide a parameterized family of fractal approximants associated with a given multivariate continuous function. Some elementary aspects of the multivariate fractal (not necessarily linear) interpolation operator that sends a continuous function defined on a hyperrectangle to its fractal analogue are studied. As in the univariate setting, the notion of α-fractal functions serves as a basis for fractalizing various results in multivariate approximation theory, including that of multivariate splines. For our part, we provide some approximation classes of multivariate fractal functions and prove a few results on the constrained fractal approximation of real-valued continuous functions of several variables.

Concurrent object regression

Modern-day problems in statistics often face the challenge of exploring and analyzing complex non-Euclidean object data that do not conform to vector space structures or operations. Examples of such data objects include covariance matrices, graph Laplacians of networks, and univariate probability distribution functions. In the current contribution a new concurrent regression model is proposed to characterize the time-varying relation between an object in a general metric space (as a response) and a vector in Rp (as a predictor), where concepts from Fréchet regression is employed. Concurrent regression has been a well-developed area of research for Euclidean predictors and responses, with many important applications for longitudinal studies and functional data. However, there is no such model available so far for general object data as responses. We develop generalized versions of both global least squares regression and locally weighted least squares smoothing in the context of concurrent regression for responses which are situated in general metric spaces and propose estimators that can accommodate sparse and/or irregular designs. Consistency results are demonstrated for sample estimates of appropriate population targets along with the corresponding rates of convergence. The proposed models are illustrated with human mortality data and resting state functional Magnetic Resonance Imaging data (fMRI) as responses.

Association between mixed dietary B vitamin intake and insulin resistance in US middle-aged and older adults without diabetes: The Bayesian kernel machine regression approach.

In daily life, the intake of dietary nutrients is mixed. However, evidence for the association between mixed dietary B vitamin intake and insulin resistance is limited. In this study, we estimated the joint effect of intake of various dietary B vitamins on insulin resistance. This cross-sectional study used data from the National Health and Nutrition Examination Survey 2011-2018. We included 1,628 middle-aged and 1,058 older adults without diabetes. Multivariable logistic regression and Bayesian kernel machine regression models were constructed. In the multivariable logistic regression, when all B vitamins were included in the model, the ORs (95% CIs) of insulin resistance were 3.06 (1.00-9.37) and 0.42 (0.19- 0.93) for the highest quartile of vitamin B-1 and B-12 intake in the middle-aged group when the lowest quartile was the reference. In the older group, no significant association was observed. In the Bayesian kernel machine regression analysis, a negative trend was noted between mixed B vitamin intake and insulin resistance in both examined groups. The univariate exposure-response function indicated that vitamin B-12 intake was negatively associated with insulin resistance in the middle-aged group, and that vitamin B-6 and dietary folate equivalent intakes were negatively associated with insulin resistance in older group. The bivariate exposure-response function indicated a potential interaction effect between dietary intake of vitamin B-12 and those of vitamin B-1, B-2, niacin, and dietary folate equivalent on insulin resistance in older people. Our results suggest that mixed dietary B vitamin intake tends to decrease the OR of insulin resistance both in middle-aged and older people.

Approximation by linear combinations of translates of a single function

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution with a single function, and prove upper bounds of the $L^p$-approximation convergence rate by these methods, when $n \to \infty$, for $1 \leq p \leq \infty$. We also generalize these results to classes of multivariate functions defined the convolution with the tensor product of a single function. In the case $p=2$, for this class, we also prove a lower bound of the quantity characterizing best approximation of by arbitrary linear combinations of $n$ translates of arbitrary function.

Classification in high dimension with an Alternative Feature Augmentation via Nonparametrics and Selection (AFANS)

We propose in this article a new high-dimensional classification method. It is called AFANS and relies on the penalized logistic regression model. It consists of transforming the data using univariate conditional probability functions. The aim is to put more emphasis on the significant features and hence to allow a better classification rule. We estimate the different univariate conditional probability functions using a simple procedure based on the Gini index. In addition to that, we first establish under mild conditions that the misclassification excess risk bound of the sparse logistic regression, with the LASSO penalty, is rate-optimal and then use this to derive the misclassification excess risk bound of AFANS. Finally, a detailed empirical study, to illustrate the advantages of our approach with respect to several competitors, is conducted on both simulated and real data sets.

Predicting Logarithm of Real Numbers to any Real Number Base

This paper aims to propose an algorithm that can predict the values of logarithmic function for any domain and for any bases with minimum possible error using basic mathematical operations. It talks about a univariate quadratic function that overlaps with the graph of the common logarithmic function to some extent, and using this resemblance to fullest advantage. It proposes a formula and certain algorithms based on the same formula.

Approximation factor of the piecewise linear functions in Mamdani fuzzy system and its realization process1

Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.

A note on the tau-additive measures

In this note, we point out that some results shown in Jónás et al. (2022) [2] are not valid. In that paper, a class of unary functions called tau function was introduced. By using the tau functions, the authors introduced a class of monotone measures called tau-additive measure and asserted that a tau function is either concave or convex and, as a consequence, a tau-additive measure is either submodular or supermodular. We present two examples to show that all these assertions are false.

Recovery of continuous functions of two variables from their Fourier coefficients known with error

In this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes $W^{\psi}_{2,p}$, $1 \leq p &lt; \infty$, consist of functions that are given in terms of generalized smoothness $\psi$. Namely, we consider the two-dimensional case which complements the recent results from [Res. Math. 2020, 28 (2), 24-34] for the classes $W^{\psi}_p$ of univariate functions.&#x0D; As to available information, we are given the noisy Fourier coefficients $y^{\delta}_{i,j} = y_{i,j} + \delta \xi_{i,j}$, $\delta \in (0,1)$, $i,j = 1,2, \dots$, of functions with respect to certain orthonormal system $\{ \varphi_{i,j} \}_{i,j=1}^{\infty}$, where the noise level is small in the sense of the norm of the space $l_p$, $1 \leq p &lt; \infty$, of double sequences $\xi=( \xi_{i,j} )_{i,j=1}^{\infty}$ of real numbers. As a recovery method, we use the so-called $\Lambda$-method of summation given by certain two-dimensional triangular numerical matrix $\Lambda = \{ \lambda_{i,j}^n \}_{i,j=1}^n$, where $n$ is a natural number associated with the sequence $\psi$ that define smoothness of the investigated functions. The recovery error is estimated in the norm of the space $C ([0,1]^2)$ of continuous on $[0,1]^2$ functions.&#x0D; We showed, that for $1\leq p &lt; \infty$, under the respective assumptions on the smoothness parameter $\psi$ and the elements of the matrix $\Lambda$, it holds \[ \Delta( W^{\psi}_{2,p}, \Lambda, l_p)= \sup\limits_{ y \in W^{\psi}_{2,p} } \sup\limits_{\| \xi \|_{l_p} \leq 1} \Big\| y - \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \lambda_{i,j}^n ( y_{i,j} + \delta \xi_{i,j}) \varphi_{i,j} \Big\|_{C ([0,1]^2)} \ll \frac{ n^{\beta + 1 - 1/{p}}}{\psi(n)}.\]

Using Hamiltonian Monte Carlo via Stan to estimate crop input response functions with stochastic plateaus

Bayesian analysis provides a principled way to quantify uncertainty and incorporate data and prior knowledge into parameter estimates. This paper makes the latest in Bayesian methods easier for others to use with a focus on a stochastic plateau crop production function. Previous studies have used Bayesian techniques to model stochastic plateau functions. No published work provides computation guidelines to estimate stochastic plateau functions using Bayesian approaches using computationally efficient algorithms such as Hamiltonian Monte Carlo (HMC). HMC is the default choice in Stan software, but the learning curve to master the software is steep. We provide HMC estimation instructions for a univariate stochastic plateau function using the R environment's brms and RStan . We also compare HMC estimation results with those using Gibbs sampling via Just Another Gibbs Sampler (JAGS) software using the R Interface R2jags . The programs are relatively compact and are included in the paper. Simulation results from HMC are consistent across different priors and are similar to those generated by a Gibbs sampler. This paper serves as a tutorial on the use of HMC for estimating plateau-type production functions. The code, and method, can be extended to other nonlinear models. • Stochastic plateau crop response functions (SPCRFs) are common in agronomic trials. • No study provides a Bayesian computation to aid estimation of SPCRFs. • Study makes it easier for others to estimate SPCRFs using modern Bayesian methods. • HMC is the default choice in Stan but the learning curve to master it is steep. • Modeling procedures and Stan codes are demonstrated in detail to be used easily.

Ear morphology and morphometry as potential forensic tools for identification of the Hausa, Igbo and Yoruba populations of Nigeria

BackgroundThe human external ear is unique in every individual in terms of shape, size and dimension making it suitable in forensic anthropology for sex estimation and personal identification purposes. The study aimed to evaluate sexual dimorphism and ethnic specificity of the external ear in major Nigerian ethnic populations.ResultsThere was variation in the morphological features of the external ear of the sampled subjects. The external ear features vary in the right and left ears in both sexes of the ethnic groups. All variables were statistically significant (p < 0.05) except ear width. Univariate discriminant function gave sex prediction accuracies between 56.4 and 57.3% for left and right ears, respectively. Population-specific sex prediction accuracy using stepwise discriminant analysis of left ear variables ranged 58–69.7% and 57.5–74.2% for right ear.ConclusionThe ear parameters showed potential for sex estimation, but cannot be solely relied upon for personal identification.