Electric load forecasting has been extensively studied during the past century. While many models and their variants have been proposed and tested in load forecasting literature, most of the existing case studies have been conducted using the data collected under normal operating conditions. A recent case study shows that four representative load forecasting models easily fail under data integrity attacks. To address this challenge, we propose three robust load forecasting models including two variants of the iteratively re-weighted least squares regression models and an $L_{1}$ regression model. Numerical experiments indicate the dominating performance of the three proposed robust regression models, especially $L_{1}$ regression, compared to other representative load forecasting models.

Robust Regression Models for Load Forecasting

We analytically investigate univariate C1 continuous cubic L1 interpolating splines calculated by minimizing an L1 spline functional based on the second derivative on 5-point windows. Specifically, we link geometric properties of the data points in the windows with linearity, convexity and oscillation properties of the resulting L1 spline. These analytical results provide the basis for a computationally efficient algorithm for calculation of L1 splines on 5-point windows.

Univariate Cubic L1 Interpolating Splines: Analytical Results for Linearity, Convexity and Oscillation on 5-PointWindows

$$L^1$$ splines have been under development for interpolation and approximation of irregular geometric data We investigate the advantages in terms of shape preservation and computational efficiency of calculating univariate cubic $$L^1$$ spline fits using a steepest-descent algorithm to minimize a global data-fitting functional under a constraint implemented by a local analysis-based interpolating-spline algorithm on 5-node windows Comparison of these locally calculated $$L^1$$ spline fits with globally calculated $$L^1$$ spline fits previously reported in the literature indicates that the locally calculated spline fits preserve shape on the average slightly better than the globally calculated spline fits and are computationally more efficient because the locally-calculated-spline-fit algorithm can be parallelized

/pdf/approximation-of-irregular-geometric-data-by-locally-50uy8ye32w.pdf

Approximation of Irregular Geometric Data by Locally Calculated Univariate Cubic L^1 Spline Fits

Principal Component Analysis (PCA) is widely used for identifying the major components of statistically distributed point clouds. Robust versions of PCA, often based in part on the l1 norm (rather than the l2 norm), are increasingly used, especially for point clouds with many outliers. Neither standard PCA nor robust PCAs can provide, without additional assumptions, reliable information for outlier-rich point clouds and for distributions with several main directions (spokes). We carry out a fundamental and complete reformulation of the PCA approach in a framework based exclusively on the l1 norm and heavy-tailed distributions. The l1 Major Component Detection and Analysis (l1 MCDA) that we propose can determine the main directions and the radial extent of 2D data from single or multiple superimposed Gaussian or heavy-tailed distributions without and with patterned artificial outliers (clutter). In nearly all cases in the computational results, 2D l1 MCDA has accuracy superior to that of standard PCA and of two robust PCAs, namely, the projection-pursuit method of Croux and Ruiz-Gazen and the l1 factorization method of Ke and Kanade. (Standard PCA is, of course, superior to l1 MCDA for Gaussian-distributed point clouds.) The computing time of l1 MCDA is competitive with the computing times of the two robust PCAs.

/pdf/l1-major-component-detection-and-analysis-l1-mcda-3mt8ky1ff8.pdf

ℓ1 Major Component Detection and Analysis (ℓ1 MCDA): Foundations in Two Dimensions

On shape-preserving capability of cubic L1 spline fits

We compare univariate L1 interpolating splines calculated on 5-point windows, on 7-point windows and on global data sets using four different spline functionals, namely, ones based on the second derivative, the first derivative, the function value and the antiderivative. Computational results indicate that second-derivative-based 5-point-window L1 splines preserve shape as well as or better than the other types of L1 splines. To calculate second-derivative-based 5-point-window L1 splines, we introduce an analysis-based, parallelizable algorithm. This algorithm is orders of magnitude faster than the previously widely used primal affine algorithm.

https://www.mdpi.com/1999-4893/3/3/311/pdf

Univariate Cubic L1 Interpolating Splines: Spline Functional, Window Size and Analysis-based Algorithm

In this paper, univariate cubic L 1 interpolating splines based on the first derivative and on 5-point windows are introduced. Analytical results for minimizing the local spline functional on 5-point windows are presented and, based on these results, an efficient algorithm for calculating the spline coefficients is set up. It is shown that cubic L 1 splines based on the first derivative and on 5-point windows preserve linearity of the original data and avoid extraneous oscillation. Computational examples, including comparison with first-derivative-based cubic L 1 splines calculated by a primal affine algorithm and with second-derivative-based cubic L 1 splines, show the advantages of the first-derivative-based cubic L 1 splines calculated by the new algorithm.

Univariate cubic L1 interpolating splines based on the first derivative and on 5-point windows: analysis, algorithm and shape-preserving properties

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Univariate Cubic L1 Interpolating Splines: Spline Functional, Window Size and Analysis-based Algorithm

Univariate cubic L1 interpolating splines based on the first derivative and on 5-point windows: analysis, algorithm and shape-preserving properties