Piecewise Linear Coefficients Research Articles

Classification of abnormalities in mammograms by new asymmetric fractal features

In this paper we use fractal method for detection and diagnosis of abnormalities in mammograms. We have used 168 images that were carefully selected by a radiologist and their abnormalities were also confirmed by biopsy. These images included asymmetric lesions, architectural distortion, normal tissue and mass lesion where in case of mass lesion they included circumscribed benign, ill-defined and spiculated malignant masses. At first, by using wavelet transform and piecewise linear coefficient mapping, image enhancement were done. Secondly detection of lesions was done by fractal method as a ROI. Since in investigation of breast cancer, it is important that fibroglandular tissues in both breasts be symmetric and for each asymmetric density, evaluation for malignancy is necessary, we define new fractal features based on extracting asymmetric information from lesions. The fractal features were evaluated on 5 data sets using SVM classifier which enabled to achieve high accuracy in classification of mammograms and diagnostic results. We have also investigated the performance of image enhancement in classification of each data set which shows different effects of enhancement on different lesion types.

Calculating Lyapunov Value for the Logistic Equation with Rapidly Oscillating Delay

We consider the local dynamic of the logistic equation with rapidly oscillating timeperiodic piecewise constant or piecewise linear coefficient of delay. It was shown that the averaged equation is a logistic equation with two delays in first case and logistic equation with distributed delay in second case. The criterion of equilibrium point stability was obtained in both cases. Dynamical properties of the original equation were considered in the critical case of equilibrium point of averaged equation stability problem. It was shown, that local dynamic in the critical case is defined by Lyapunov value whose sign depends on the parameters of the problem.

Perpetual American options in a diffusion model with piecewise-linear coefficients

Abstract We derive closed form solutions to the discounted optimal stopping problems related to the pricing of the perpetual American standard put and call options in an extension of the Black–Merton–Scholes model with piecewise-constant dividend and volatility rates. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. We present explicit algorithms to determine the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options.

Perpetual American Options in a Diffusion Model with Piecewise-Linear Coefficients

We derive closed form solutions to the discounted optimal stopping problems related to the pricing of the perpetual American standard put and call options in a diffusion model with piecewise-linear coefficients. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. We present explicit algorithms to determine the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options.

Polynomial Differential Equations with Piecewise Linear Coefficients

Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the multiplicity is the same when the coefficients are polynomial functions of degree n, or piecewise linear functions with n segments.

Piecewise linear regularized solution paths

We consider the generic regularized optimization problem β̂(λ)=arg minβ L(y, Xβ)+λJ(β). Efron, Hastie, Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407–499] have shown that for the LASSO—that is, if L is squared error loss and J(β)=‖β‖1 is the ℓ1 norm of β—the optimal coefficient path is piecewise linear, that is, ∂β̂(λ)/∂λ is piecewise constant. We derive a general characterization of the properties of (loss L, penalty J) pairs which give piecewise linear coefficient paths. Such pairs allow for efficient generation of the full regularized coefficient paths. We investigate the nature of efficient path following algorithms which arise. We use our results to suggest robust versions of the LASSO for regression and classification, and to develop new, efficient algorithms for existing problems in the literature, including Mammen and van de Geer’s locally adaptive regression splines.