Multiscale regression on unknown manifolds

<abstract><p>We consider the regression problem of estimating functions on $ \mathbb{R}^D $ but supported on a $ d $-dimensional manifold $ \mathcal{M} ~~\subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $ \mathcal{M} $ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $ \mathbb{R}^D $. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.</p></abstract>

Fault Type Classification of 150 kV Transmission Line using Wavelet Multi-Resolution Analysis Method

To identify the fault, wavelet method is used in solving complex protection problems. This study uses a new approach, namely the wavelet multi resolution analysis method with its application where multi resolution analysis works to analyze signals at different frequencies with the same resolution. In this study, the classification of fault types that occur in the 150 kV transmission line quickly and accurately is carried out using the wavelet multi resolution analysis method. This research is included in applied research and was designed using computer simulation software, namely ATP and MATLAB. The data transmission system used is the Maninjau Hydroelectric Power Plant transmission line to Pauh Limo Substation. The modeled transmission system is given 1-phase to ground, 2-phase to ground, 2- phase, 3-phase and lightning faults. To determine the accuracy of this classification, the fault is varied according to the distance and impedance of the disturbance. From the analysis of the simulation results and calculations, based on the wavelet multi resolution analysis method used in fault classifying, the average value of the approximation coefficient used to classify the type of fault is obtained. Based on the results of the study, it can be said that all types of fault analyzed in this study have met the classification requirements using the wavelet multi resolution analysis method

Fault Type Classification of 150 kV Transmission Line using Wavelet Multi-Resolution Analysis Method

To identify the fault, wavelet method is used in solving complex protection problems. This study uses a new approach, namely the wavelet multi resolution analysis method with its application where multi resolution analysis works to analyze signals at different frequencies with the same resolution. In this study, the classification of fault types that occur in the 150 kV transmission line quickly and accurately is carried out using the wavelet multi resolution analysis method. This research is included in applied research and was designed using computer simulation software, namely ATP and MATLAB. The data transmission system used is the Maninjau Hydroelectric Power Plant transmission line to Pauh Limo Substation. The modeled transmission system is given 1-phase to ground, 2-phase to ground, 2- phase, 3-phase and lightning faults. To determine the accuracy of this classification, the fault is varied according to the distance and impedance of the disturbance. From the analysis of the simulation results and calculations, based on the wavelet multi resolution analysis method used in fault classifying, the average value of the approximation coefficient used to classify the type of fault is obtained. Based on the results of the study, it can be said that all types of fault analyzed in this study have met the classification requirements using the wavelet multi resolution analysis method

The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform

This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. Log-periodicity is associated with the deep symmetry of discrete scale invariance in physical systems. Furthermore, there is a close connection between such chirping techniques, and other useful applications such as wavelet decomposition for multi-resolution analysis. Motivated to uncover additional properties, we propose an alternative, simpler parameterisation of the original Altes waveforms. From this, it becomes apparent that we have a flexible family of hyperbolic chirps suitable for the detection of accelerating time-series oscillations. The proposed formalism reveals the original chirps to be a set of admissible wavelets with desirable properties of regularity, infinite vanishing moments and time-frequency localisation. As they are self-similar, these “Altes chirplets” allow efficient implementation of the scale-invariant hyperbolic chirplet transform (HCT), whose basis functions form hyperbolic curves in the time-frequency plane. Compared with the rectangular time-frequency tilings of both the conventional wavelet transform and the short-time Fourier transform, the HCT can better facilitate the detection of chirping signals, which are often the signature of critical failure in complex systems. A synthetic example is presented to illustrate this useful application of the HCT.

Multi-resolution wavelet basis for solving steady forced Korteweg–de Vries model

A steady forced Korteweg–de Vries (fKdV) model which includes gravity, capillary, and pressure distributions is solved numerically using the wavelet Galerkin method. The anti-derivatives of Daubechies wavelets are developed as the basis of the solution subspaces for the mixed boundary condition type. Accuracy of numerical solutions can be improved by increasing the number of wavelet levels in the multi-resolution analysis. The theoretical result of convergence rate is also shown. The problem can be viewed as gravity-capillary wave flows over an applied pressure distribution. The flow regime can be characterized by subcritical, supercritical, and critical flows depending on the value of the Froude number. Trapped depression and elevation waves are found over the pressure distribution. For a near-critical flow regime, a generalized solitary wave with ripples is presented. This shows a capillary effect in balance to gravity and the pressure force on the free surface.