Interpretable Modeling for Short- and Medium-Term Electricity Demand Forecasting

We consider the problem of short- and medium-term electricity demand forecasting by using past demand and daily weather forecast information. Conventionally, many researchers have directly applied regression analysis. However, interpreting the effect of weather on the demand is difficult with the existing methods. In this study, we build a statistical model that resolves this interpretation issue. A varying coefficient model with basis expansion is used to capture the nonlinear structure of the weather effect. This approach results in an interpretable model when the regression coefficients are nonnegative. To estimate the nonnegative regression coefficients, we employ nonnegative least squares. Three real data analyses show the practicality of our proposed statistical modeling. Two of them demonstrate good forecast accuracy and interpretability of our proposed method. In the third example, we investigate the effect of COVID-19 on electricity demand. The interpretation would help make strategies for energy-saving interventions and demand response.

Efficiently transforming from values of a function on a sparse grid to basis coefficients

In many contexts it is necessary to determine coefficients of a basis expansion of a function ${f}\left(x_1, \ldots, x_D\right) $ from values of the function at points on a sparse grid. Knowing the coefficients, one has an interpolant or a surrogate. For example, such coefficients are used in uncertainty quantification. In this chapter, we present an efficient method for computing the coefficients. It uses basis functions that, like the familiar piecewise linear hierarchical functions, are zero at points in previous levels. They are linear combinations of any, e.g. global, nested basis functions $\varphi_{i_k}^{\left(k\right)}\left(x_k\right)$. Most importantly, the transformation from function values to basis coefficients is done, exploiting the nesting, by evaluating sums sequentially. When the number of functions in level $\ell_k$ equals $\ell_k$ (i.e. when the level index is increased by one, only one point (function) is added) and the basis function indices satisfy ${\left\lVert\mathbf{i}-\mathbf{1}\right\lVert_1 \le b}$, the cost of the transformation scales as $\mathcal{O}\left(D \left[\frac{b}{D+1} + 1\right] N_\mathrm{sparse}\right)$, where $N_\mathrm{sparse}$ is the number of points on the sparse grid. We compare the cost of doing the transformation with sequential sums to the cost of other methods in the literature.

New Modeling Approaches Based on Varimax Rotation of Functional Principal Components

Functional Principal Component Analysis (FPCA) is an important dimension reduction technique to interpret the main modes of functional data variation in terms of a small set of uncorrelated variables. The principal components can not always be simply interpreted and rotation is one of the main solutions to improve the interpretation. In this paper, two new functional Varimax rotation approaches are introduced. They are based on the equivalence between FPCA of basis expansion of the sample curves and Principal Component Analysis (PCA) of a transformation of the matrix of basis coefficients. The first approach consists of a rotation of the eigenvectors that preserves the orthogonality between the eigenfunctions but the rotated principal component scores are not uncorrelated. The second approach is based on rotation of the loadings of the standardized principal component scores that provides uncorrelated rotated scores but non-orthogonal eigenfunctions. A simulation study and an application with data from the curves of infections by COVID-19 pandemic in Spain are developed to study the performance of these methods by comparing the results with other existing approaches.

Adaptive Filtering Based on Legendre Polynomials

Abstract:<div><br><div><pre><p>In system identification scenarios, classical adaptive filters, such as the recursive least squares (RLS) algorithm, predict the system impulse response. If a tracking delay is acceptable, interpolating estimators capable of providing more accurate estimates of time-varying impulse responses can be used; channel estimation in communications is an example of such applications. The basis expansion model (BEM) approach is known to be efficient for non-adaptive (block) channel estimation in communications. In this paper, we combine the BEM approach with the sliding-window RLS (SRLS) algorithm and propose a new family of adaptive filters. Specifically, we use the Legendre polynomials, thus the name the SRLS-L adaptive filter. The identification performance of the SRLS-L algorithm is evaluated analytically and via simulation. The analysis shows significant improvement in the estimation accuracy compared to the SRLS algorithm and a good match between the theoretical and simulation results. The performance is further investigated in application to the self-interference cancellation in full-duplex underwater acoustic communications, where a high estimation accuracy is required. A field experiment conducted in a lake shows significant improvement in the cancellation performance compared to the classical SRLS algorithm.</p> </pre></div></div>