Local linear regression with adaptive orthogonal fitting for the wind power application

Short-term forecasting of wind generation requires a model of the function for the conversion of me-teorological variables (mainly wind speed) to power production. Such a power curve is nonlinear and bounded, in addition to being nonstationary. Local linear regression is an appealing nonparametric ap-proach for power curve estimation, for which the model coefficients can be tracked with recursive Least Squares (LS) methods. This may lead to an inaccurate estimate of the true power curve, owing to the assumption that a noise component is present on the response variable axis only. Therefore, this assump-tion is relaxed here, by describing a local linear regression with orthogonal fit. Local linear coefficients are defined as those which minimize a weighted Total Least Squares (TLS) criterion. An adaptive es-timation method is introduced in order to accommodate nonstationarity. This has the additional benefit of lowering the computational costs of updating local coefficients every time new observations become available. The estimation method is based on tracking the left-most eigenvector of the augmented covari-ance matrix. A robustification of the estimation method is also proposed. Simulations on semi-artificial datasets (for which the true power curve is available) underline the properties of the proposed regression and related estimation methods. An important result is the significantly higher ability of local polynomia

A unified wavelet-based modelling framework for non-linear system identification: the WANARX model structure

A new unified modelling framework based on the superposition of additive submodels, functional components, and wavelet decompositions is proposed for non-linear system identification. A non-linear model, which is often represented using a multivariate non-linear function, is initially decomposed into a number of functional components via the wellknown analysis of variance (ANOVA) expression, which can be viewed as a special form of the NARX (non-linear autoregressive with exogenous inputs) model for representing dynamic input–output systems. By expanding each functional component using wavelet decompositions including the regular lattice frame decomposition, wavelet series and multiresolution wavelet decompositions, the multivariate non-linear model can then be converted into a linear-in-theparameters problem, which can be solved using least-squares type methods. An efficient model structure determination approach based upon a forward orthogonal least squares (OLS) algorithm, which involves a stepwise orthogonalization of the regressors and a forward selection of the relevant model terms based on the error reduction ratio (ERR), is employed to solve the linear-in-the-parameters problem in the present study. The new modelling structure is referred to as a wavelet-based ANOVA decomposition of the NARX model or simply WANARX model, and can be applied to represent high-order and high dimensional non-linear systems

Wavelet Neural Networks: A Practical Guide

Wavelet networks (WNs) are a new class of networks which have been used with great success in a wide range of application. However a general accepted framework for applying WNs is missing from the literature. In this study, we present a complete statistical model identification framework in order to apply WNs in various applications. The following subjects were thorough examined: the structure of a WN, training methods, initialization algorithms, variable significance and variable selection algorithms, model selection methods and finally methods to construct confidence and prediction intervals. In addition the complexity of each algorithm is discussed. Our proposed framework was tested in two simulated cases, in one chaotic time series described by the Mackey-Glass equation and in three real datasets described by daily temperatures in Berlin, daily wind speeds in New York and breast cancer classification. Our results have shown that the proposed algorithms produce stable and robust results indicating that our proposed framework can be applied in various applications

The wavelet-NARMAX representation : a hybrid model structure combining polynomial models with multiresolution wavelet decompositions

A new hybrid model structure combing polynomial models with multiresolution wavelet decompositions is introduced for nonlinear system identification. Polynomial models play an important role in approximation theory, and have been extensively used in linear and nonlinear system identification. Wavelet decompositions, in which the basis functions have the property of localization in both time and frequency, outperform many other approximation schemes and offer a flexible solution for approximating arbitrary functions. Although wavelet representations can approximate even severe nonlinearities in a given signal very well, the advantage of these representations can be lost when wavelets are used to capture linear or low-order nonlinear behaviour in a signal. In order to sufficiently utilise the global property of polynomials and the local property of wavelet representations simultaneously, in this study polynomial models and wavelet decompositions are combined together in a parallel structure to represent nonlinear input-output systems. As a special form of the NARMAX model, this hybrid model structure will be referred to as the WAvelet-NARMAX model, or simply WANARMAX. Generally, such a WANARMAX representation for an input-output system might involve a large number of basis functions and therefore a great number of model terms. Experience reveals that only a small number of these model terms are significant to the system output. A new fast orthogonal least squares algorithm, called the matching pursuit orthogonal least squares (MPOLS) algorithm, is also introduced in this study to determine which terms should be included in the final model

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

Very-short-term probabilistic wind power forecasts by sparse vector autoregression

A spatio-temporal method for producing very-short-term parametric probabilistic wind power forecasts at a large number of locations is presented. Smart grids containing tens, or hundreds, of wind generators require skilled very-short-term forecasts to operate effectively, and spatial information is highly desirable. In addition, probabilistic forecasts are widely regarded as necessary for optimal power system management as they quantify the uncertainty associated with point forecasts. Here we work within a parametric framework based on the logit-normal distribution and forecast its parameters. The location parameter for multiple wind farms is modelled as a vector-valued spatio-temporal process, and the scale parameter is tracked by modified exponential smoothing. A state-of-the-art technique for fitting sparse vector autoregressive models is employed to model the location parameter and demonstrates numerical advantages over conventional vector autoregressive models. The proposed method is tested on a dataset of 5 minute mean wind power generation at 22 wind farms in Australia. 5-minute-ahead forecasts are produced and evaluated in terms of point and probabilistic forecast skill scores and calibration. Conventional autoregressive and vector autoregressive models serve as benchmarks