A Better Alternative to Piecewise Linear Time Series Segmentation

Time series are difficult to monitor, summarize and predict. Segmentation organizes time series into few intervals having uniform characteristics (flatness, linearity, modality, monotonicity and so on). For scalability, we require fast linear time algorithms. The popular piecewise linear model can determine where the data goes up or down and at what rate. Unfortunately, when the data does not follow a linear model, the computation of the local slope creates overfitting. We propose an adaptive time series model where the polynomial degree of each interval vary (constant, linear and so on). Given a number of regressors, the cost of each interval is its polynomial degree: constant intervals cost 1 regressor, linear intervals cost 2 regressors, and so on. Our goal is to minimize the Euclidean (l_2) error for a given model complexity. Experimentally, we investigate the model where intervals can be either constant or linear. Over synthetic random walks, historical stock market prices, and electrocardiograms, the adaptive model provides a more accurate segmentation than the piecewise linear model without increasing the cross-validation error or the running time, while providing a richer vocabulary to applications. Implementation issues, such as numerical stability and real-world performance, are discussed.Comment: to appear in SIAM Data Mining 200

Intrusion Detection Systems Using Adaptive Regression Splines

Past few years have witnessed a growing recognition of intelligent techniques for the construction of efficient and reliable intrusion detection systems. Due to increasing incidents of cyber attacks, building effective intrusion detection systems (IDS) are essential for protecting information systems security, and yet it remains an elusive goal and a great challenge. In this paper, we report a performance analysis between Multivariate Adaptive Regression Splines (MARS), neural networks and support vector machines. The MARS procedure builds flexible regression models by fitting separate splines to distinct intervals of the predictor variables. A brief comparison of different neural network learning algorithms is also given

An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching

Adaptive transient solution of nonuniform multiconductor transmission lines using wavelets

Abstract—This paper presents a highly adaptive algorithm for the transient simulation of nonuniform interconnects loaded with arbitrary nonlinear and dynamic terminations. The discretization of the governing equations is obtained through a weak formula-tion using biorthogonal wavelet bases as trial and test functions. It is shown how the multiresolution properties of wavelets lead to very sparse approximations of the voltages and currents in typical transient analyzes. A simple yet effective time–space adaptive al-gorithm capable of selecting the minimal number of unknowns at each time iteration is described. Numerical results show the high degree of adaptivity of the proposed scheme. Index Terms—Electromagnetic (EM) transient analysis, multi-conductor transmission lines (TLs), wavelet transforms. I

Einstein equations in the null quasi-spherical gauge III: numerical algorithms

We describe numerical techniques used in the construction of our 4th order evolution for the full Einstein equations, and assess the accuracy of representative solutions. The code is based on a null gauge with a quasi-spherical radial coordinate, and simulates the interaction of a single black hole with gravitational radiation. Techniques used include spherical harmonic representations, convolution spline interpolation and filtering, and an RK4 "method of lines" evolution. For sample initial data of "intermediate" size (gravitational field with 19% of the black hole mass), the code is accurate to 1 part in 10^5, until null time z=55 when the coordinate condition breaks down.Comment: Latex, 38 pages, 29 figures (360Kb compressed

Condition number analysis and preconditioning of the finite cell method

The (Isogeometric) Finite Cell Method - in which a domain is immersed in a structured background mesh - suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling relation between the condition number of (I)FCM system matrices and the smallest cell volume fraction. Ill-conditioning stems either from basis functions being small on cells with small volume fractions, or from basis functions being nearly linearly dependent on such cells. Based on these two sources of ill-conditioning, an algebraic preconditioning technique is developed, which is referred to as Symmetric Incomplete Permuted Inverse Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the SIPIC preconditioner in improving (I)FCM condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the Poisson problem and for two- and three-dimensional problems in linear elasticity, in which Nitche's method is applied in either the normal or tangential direction. The accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method

Quantile regression with varying coefficients

Quantile regression provides a framework for modeling statistical quantities of interest other than the conditional mean. The regression methodology is well developed for linear models, but less so for nonparametric models. We consider conditional quantiles with varying coefficients and propose a methodology for their estimation and assessment using polynomial splines. The proposed estimators are easy to compute via standard quantile regression algorithms and a stepwise knot selection algorithm. The proposed Rao-score-type test that assesses the model against a linear model is also easy to implement. We provide asymptotic results on the convergence of the estimators and the null distribution of the test statistic. Empirical results are also provided, including an application of the methodology to forced expiratory volume (FEV) data.Comment: Published at http://dx.doi.org/10.1214/009053606000000966 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org