Nondifferentiable Points Research Articles

On differentiability and mass distributions of typical bivariate copulas

Despite the fact that copulas are commonly considered as analytically smooth/regular objects, derivatives of copulas have to be handled with care. Triggered by a recently published result characterizing multivariate copulas via (d−1)-increasingness of their partial derivative we study the bivariate setting in detail and show that the set of non-differentiability points of a copula may be quite large. We first construct examples of copulas C whose first partial derivative ∂1C(x,y) is pathological in the sense that for almost every x∈(0,1) it does not exist on a dense subset of y∈(0,1), and then show that the family of these copulas is dense. Since in commonly considered subfamilies more regularity might be typical, we then focus on bivariate Extreme Value copulas (EVCs) and show that a topologically typical EVC is not absolutely continuous but has degenerated discrete component, implying that in this class typically ∂1C(x,y) exists in full (0,1)2.Considering that regularity of copulas is closely related to their mass distributions we then study mass distributions of topologically typical copulas and prove the surprising fact that topologically typical bivariate copulas are mutually completely dependent with full support. Furthermore, we use the characterization of EVCs in terms of their associated Pickands dependence measures ϑ on [0,1], show that regularity of ϑ carries over to the corresponding EVC and prove that the subfamily of all EVCs whose absolutely continuous, discrete and singular component has full support is dense in the class of all EVCs.

Predictive analysis of stochastic stock pattern utilizing fractional order dynamics and heteroscedastic with a radial neural network framework

Modelling of high-dimension chaotic, noisy, and non-stationary time series of complex fractal dynamics is a big challenge. In this research work, a novel approach of Leverage Convolution LC ARFIMA– GARCH model is presented for sequential learning of irregular, fractal dynamic patterns of stochastic time series dynamics. Different classical methods including the auto-regressive approach indecently are unable to capture fragile patterns and consequently convert some weak signals into random errors by creating white noise patterns. The Convolution Leverage provides an additional degree of freedom for critical noisy points and asymmetrical distribution dynamics to regulate the frequency of imbalance and skewed observation. The designed transformation in the ARFIMA model preserves the loss of information by shifting the skewed data distribution toward a normal pattern. The pattern of population parameters in the proposed leverage paradigm provides out of box approach to track and extract additional information on population variance patterns in the form of an additional degree of freedom to stabilize imbalance signals. The model can provide reliable performance for long-range dependence, particularly for mean reversion chaotic phase variation at critical non-differentiable points. The performance of the dynamic model is verified on chaotic real data of the Ireland Stock Market (ISEQ). The result statistics confirmed the optimized outcome with the addition of the GARCH heteroscedastic multimodal and radial basis neural network. The novel technique can help to address inherited challenges in the imbalance learning and extreme events of the time series modeling by monitoring the chaotic trajectory of fractal physical phenomena, particularly in, finance healthcare, climate, and intelligent sustainability.

A potential relation trigger method for entity-relation quintuple extraction in text with excessive entities

In the task of joint entity and relation extraction, the relationship between two entities is determined by some specific words in their source text. These words are viewed as potential triggers which are the evidence to explain the relationship but not marked clearly. However, the current models cannot make good use of the potential words to optimize components of entities and relations, but can only give separate results. These models aim to identify the type of relation between two entities mentioned in the source text by encoding the text and entities. Although some models can generate the weights for every single word by improving the attention mechanism, the weights will be influenced by the irrelevant words essentially, which is not needed in enhancing the influence of the triggers. We propose a joint entity-relation quintuple extraction framework based on the Potential Relation Trigger (PRT) method to get the highest probability of a word as the prompt in every time step and join the words together as relation hints. In specific, we leverage polarization mechanism in possibility calculation to avoid nondifferentiable points of the functions in our method when choosing. We find that their representation will improve the performance of the relation part with the exact range of the entities. Extensive experiments results demonstrate that the effectiveness of our proposed model achieves state-of-the-art performance on four RE benchmark datasets.

An Application of Infinite Programming on Desirability Functions

When assessing the quality of a system or product, it is necessary to take all responses into account and optimize them in a concurrent manner to find the factor levels that satisfy the overall system, process, or product properties to solve the robust design problem. This problem can be solved as a multi-response optimization problem. There are many methods suggested to solve this problem based on different disciplines like multi-objective optimization. In this study, we improve the theory of nondifferentiable desirability functions' optimization for which the so-called gradient-based methods are not useful. In this study, we propose an infinite programming approach for nondifferentiable desirability functions including more than one nondifferentiable point. We employed DNLP model of GAMS/BARON which is a nondifferentiable solver, however, the solution of more than one nondifferentiable point problem is resulted as infeasible. We also tested MATLAB/NOMAD which is a derivative-free solver for MINLP problems, however, MATLAB/NOMAD also did not succeed and could not solve this nondifferentiable problem. Lastly, we use a genetic algorithm that is implemented under MATLAB and it also cannot find a feasible solution. We use an example that is solved by different desirability functions approaches before and show that the desirability functions approach with more than one nondifferentiable point is a good alternative to the ones in the literature. We present the conclusion and future studies at the end of the paper.

Boundary Behavior of Conformal Maps on Domains with Corners

This paper investigates the behavior of conformal maps near corner points and other non-differentiable boundary points of planar domains. Conformal maps that preserve angles locally will exhibit singular behavior when approaching corners or other less smooth parts of the boundary. We classify types of isolated corner singularities and characterize the magnitude, derivatives, and integral properties of analytic functions near such points. Explicit mappings are constructed between model domains with cusps, wedges, slits, and logarithmic-type corner points. The behavior of the mapping functions is analyzed as the boundary coordinates approach the singular points. We establish several theorems describing the boundary limits, convergence, boundary correspondences, and boundary integrals of these conformal maps on domains with corners. The mapping properties provide insight into the effect of geometric singularities on analytic functions in application areas such as physics, fluid flow, and engineering problems involving complex mappings. The boundary behavior classifications developed here expand the mathematical understanding of conformal maps on domains with sharp corners or discontinuities.

Learning a surrogate Pekeris waveguide model

Underwater sound propagation models predict transmission loss (TL) given environmental parameters, such as seafloor sediment parameters. Also of interest is the reverse: Can seafloor parameters, such as sediment density, sound speed, and attenuation, be inferred from ocean acoustic data? This inverse problem can be addressed using information geometry tools for parameter identifiability analysis and model reduction, where we identify parameters that can be removed from a model without sacrificing accuracy. Critical to these methods for model reduction is the ability to evaluate derivatives of the model’s TL predictions with respect to model parameters. Automatic differentiation (AD) allows for rapid evaluation of a model’s derivatives but may face some challenges for implementation with sound propagation models, including models being written in “legacy code” and having non-differentiable points in the function space. Our solution is to train a surrogate machine learning (ML) model which can circumvent these challenges, and which we can apply AD to. We demonstrate this method using the Pekeris waveguide model of the ocean. We compare the surrogate and original model outputs and demonstrate the process of model reduction using a ML surrogate model. [Work supported by Office of Naval Research.]

Theoretically Scrutinizing Kinks on Efficient Frontiers and Computationally Reporting Nonexistence of the Tangent Portfolio for the Capital Asset Pricing Model by Parametric-Quadratic Programming

Researchers traditionally compute isolated points for an efficient frontier and assume a line which passes through a risk-free asset [Formula: see text] and is tangent to the frontier. The tangency plays pivotal roles for the capital asset pricing model (CAPM). However, the assumption may not hold in the presence of kinks (as non-differentiable points) on efficient frontiers. Kinks are detected by parametric-quadratic programming only and not by ordinary portfolio optimization. Up until now, there has been no research to theoretically scrutinize kink properties (especially implications to CAPM) and systematically quantify the nonexistence of the tangency. In such an area, this paper contributes to the literature. In theorems and corollaries, we prove the nonexistence of the tangency and substantiate that expected-return axis is composed of piecewisely connected intervals for which the tangency does not exist and intervals for which the tangency exists. Computationally, we reveal universal existence of kinks (e.g., 0.2 to 8.0 kinks for 5-stock to 1800-stock portfolio selections) and the tangency-nonexistence ratios as about 0.066.

Geodesic length and shifted weights in first-passage percolation

We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the ℓ 1 \ell ^1 distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produces singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments.

Application of Fractional Derivatives in Characterization of ECG graphs

In this paper, we have discussed the prediction of Left Ventricular Hypertrophy (LVH) and Right Ventricular Hypertrophy (RVH) of the heart from an Electrocardiogram (ECG) Graph using fractional order calculus. An ECG is a rough or unreachable curve that is continuous everywhere but non-differentiable at some points or all points where classical calculus fails. The purpose of this paper is to find left and right fractional derivatives at those non-differentiable points and then predict LVH and RVH by calculating the phase transition values (absolute difference of left and right fractional derivatives). Fractal dimension and Hurst exponents of V1, V2, V5, and V6 leads of the ECGs have been calculated for both problematic and normal ECGs. All such measures will help doctors to diagnose LVH and RVH from ECG in a more accurate manner as compared to the techniques which are under the classical order method.

Why KDAC? A general activation function for knowledge discovery

Deep learning oriented named entity recognition (DNER) has gradually become the paradigm of knowledge discovery, which greatly promotes domain intelligence. However, the activation function of DNER fails to treat gradient vanishing, no negative output or non-differentiable existence, which may impede the exploration of knowledge due to the omission and incomplete representation of the latent semantic. To break through the dilemma, we present a novel activation function termed KDAC. Detailly, KDAC is an aggregation function with multiple conversion modes. The backbone is the interaction between exponent and linearity, and the both ends are extended through adaptive linear divergence, which can surmount the gradient vanishing and no negative output. Crucially, the non-differentiable points can be alerted and eliminated by an approximate smoothing algorithm. KDAC has a series of brilliant properties, such as nonlinear, stable near-linear transformation and derivative, as well as dynamic style, etc. We perform experiments based on BERT-BiLSTM-CNN-CRF model on six benchmark datasets containing different domain knowledge, such as Weibo, Clinical, E-commerce, Resume, HAZOP and People's daily. The evaluation results show that KDAC is advanced and effective, and can provide more generalized activation to stimulate the performance of DNER. We hope that KDAC can be exploited as a promising activation function to devote itself to the construction of knowledge.

Type-V intermittency from Markov binary block visibility graph perspective

In this work, type-V intermittency is studied from Markov binary block visibility graphs perspective. We considered a piecewise quadratic Poincare map that is a simple model to exhibit this type of intermittency. The mechanism of type-V intermittency is collision of a stable fixed point with a point of discontinuity of the Poincare map. We study the behavior of a dynamical system in the vicinity of the discontinuous or non-differentiable points (NDP) using networks language. Numerical results showed that there is a logarithmic scaling law logε for the average laminar length of the type-V intermittency. We also described their properties based on statistical tools such as the length between reinjection points and the average laminar length. For further investigation, we verified the degree distribution of the complex network generated by type-V intermittency time series and finally, predicted the behavior of type-V intermittency by the proposed theoretical degree distributions.

Comments on 'A weighting analogue to pair matching in propensity score analysis' by L. Li and T. Greene.

Li and Greene (A weighting analogue to pair matching in propensity score analysis. Int J Biostat 2013;9:215-34) propose that estimates derived by the matching weight (MW) estimator are similar to those derived by the one-to-one propensity score matching estimator. The MW estimator has some useful properties, however, some regularity conditions need to be confirmed to derive an asymptotic distribution since the MW has a non-differentiable point. In this letter, we confirm the asymptotic distribution of the MW estimator and the sufficient conditions to achieve it.

Non-differentiable saddle points and sub-optimal local minima exist for deep ReLU networks

Whether sub-optimal local minima and saddle points exist in the highly non-convex loss landscape of deep neural networks has a great impact on the performance of optimization algorithms. Theoretically, we study in this paper the existence of non-differentiable sub-optimal local minima and saddle points for deep ReLU networks with arbitrary depth. We prove that there always exist non-differentiable saddle points in the loss surface of deep ReLU networks with squared loss or cross-entropy loss under reasonable assumptions. We also prove that deep ReLU networks with cross-entropy loss will have non-differentiable sub-optimal local minima if some outermost samples do not belong to a certain class. Experimental results on real and synthetic datasets verify our theoretical findings.

Criteria Determining the Number of Estimate Points of Moment Method for Performance Functions Including Non-differentiable Points

Moment method is famous for its calculation efficiency. Moments of the performance function are obtained by point estimates based on Hermite Integration. The reliability index and probability of failure can be obtained using existing standardization function or existing distribution systems. The precision of moment method increases monotonously with the increasing number of estimating points. The performance functions with non-differentiable points are frequently encountered when elasto-plastic behavior dominates for a structure under loading. Thus the accuracy and efficiency of moment method should be addressed for performance functions with non-differentiable points. In this paper, an example of structural performance functions is presented to show the existence of non-differentiable points. Two example performance functions are used to verify the accuracy and efficiency of moment method. The MCS method is employed to obtain the accurate results of probability of failure. The relative errors of probability of failure from moment method are obtained through a comparison with the results from MCS method. The fluctuation of the relative errors with the number of estimate points is also presented. The required number of estimate points for performance functions with non-differentiable points is discussed through the above two examples. Finally, criteria to determine the estimate points for performance functions with non-differentiable points are proposed and verified the applicability.

Piecewise structure of Lyapunov functions and densely checked decrease conditions for hybrid systems

We propose a class of locally Lipschitz functions with piecewise structure for use as Lyapunov functions for hybrid dynamical systems. Subject to some regularity of the dynamics, we show that Lyapunov inequalities can be checked only on a dense set and thus we avoid checking them at points of nondifferentiability of the Lyapunov function. Connections to other classes of locally Lipschitz or piecewise regular functions are also discussed, and applications to hybrid dynamical systems are included.

Distance functions with dense singular sets

We characterize the denseness of the singular set of the distance function from a -hypersurface in terms of an inner ball condition and we address the problem of the existence of viscosity solutions of the Eikonal equation whose singular set (i.e. set of non-differentiability points) is not no-where dense.

Optimal control of a non-smooth quasilinear elliptic equation

<p style='text-indent:20px;'>This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a <inline-formula><tex-math id="M1">\begin{document}$ PC^1 $\end{document}</tex-math></inline-formula> function with countably many points of nondifferentiability, we show that both conditions are equivalent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples.</p>

A strong order 3/4 method for SDEs with discontinuous drift coefficient

Abstract In this paper we study strong approximation of the solution of a scalar stochastic differential equation (SDE) at the final time in the case when the drift coefficient may have discontinuities in space. Recently, it has been shown in Müller-Gronbach & Yaroslavtseva (2020, On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient. Ann. Inst. Henri Poincaré Probab. Stat., 56, 1162–1178) that for scalar SDEs with a piecewise Lipschitz drift coefficient, and a Lipschitz diffusion coefficient that is nonzero at the discontinuity points of the drift coefficient the classical Euler–Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\in [1,\infty )$. Up to now this was the best $L_p$-error rate available in the literature for equations of that type. In the present paper we construct a method based on finitely many evaluations of the driving Brownian motion that even achieves an $L_p$-error rate of at least $3/4$ for all $p\in [1,\infty )$ under additional piecewise smoothness assumptions on the coefficients. This is the first higher order method known in the literature for SDEs of that type. We add that the $L_p$-error rate $3/4$ cannot be improved in general. To obtain the upper error bound for our new method, we prove in particular that a quasi-Milstein scheme achieves an $L_p$-error rate of at least $3/4$ in the case of coefficients that are both Lipschitz continuous and piecewise differentiable with Lipschitz continuous derivatives, which is of interest in itself. The latter error rates are obtained via a detailed analysis of the average size of increments of the time-continuous quasi-Milstein scheme over time intervals in which the scheme crosses a point of nondifferentiability of the coefficients.

Prediction of Ventricular Hypertrophy of Heart Using Fractional Calculus

In this paper we have studied Left and Right ventricular hypertrophy of heart from ECGs using fractional calculus. An ECG is a rough or unreachable curve which is continuous everywhere but nondifferentiable at some points or all points where classical calculus fails. Our purpose of this paper is to find left and right fractional derivatives at those non-differentiable points and then predict LVH and RVH by calculating the phase transition values (absolute difference of left and right fractional derivatives). Our investigation shows that for LVH patients the phase transition values at the non-differentiable points of V1, V2, V5 and V6 leads are higher than those for normal ECG. For RVH ECG the phase transition values at S are smaller than those of R in V1 leads which are just opposite to the case of normal ECGs. Fractal dimension and Hurst Exponents of V1, V2, V5 and V6 leads of the ECGs have been calculated for both problematic and normal ECGs. All such measures may help doctors to diagnose LVH and RVH from ECG more accurately than the technique they use.

On some universal Morse–Sard type theorems

The classical Morse–Sard theorem claims that for a mapping v:Rn→Rm+1 of class Ck the measure of critical values v(Zv,m) is zero under condition k≥n−m. Here the critical set, or m-critical set is defined as Zv,m={x∈Rn:rank∇v(x)≤m}. Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the Ck category.Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v:Rn→Rd belongs to the Hölder class Ck,α, 0≤α≤1, then for every q>m the identityHμ(Zv,m∩v−1(y))=0 holds for Hq-almost all y∈Rd, where μ=n−m−(k+α)(q−m).Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa.The result is new even for the classical Ck-case (when α=0); similar result is established for the Sobolev classes of mappings Wpk(Rn,Rd) with minimal integrability assumptions p=max⁡(1,n/k), i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces.The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).