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A robust least square approach for forecasting models: an application to Brazil’s natural gas demand

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Abstract

The robust least square method has been introduced in the literature as a new parameter estimation technique to deal with the presence of data uncertainties. In this paper we propose to use the robust least square method combined with log-linear Cobb–Douglas model as an alternative for developing forecast models. We first extend the robust least square method to the case which allows uncertainties only in some columns of the data matrix as well as to include weighting matrices on the past data observations and on the uncertainties. Afterwards we compare the robust and ordinary least square methods for the yearly estimate for the natural gas demand in Brazil, considering the total demand as well as the industrial and power sectors demand. Regarding the power sector case, a further contribution of the paper is to analyze the impact of the reservoirs’ levels over the demand of natural gas by thermoelectric power plants in an energy mix dominated by hydropower. Although both methods, the robust and the ordinary least square, presented similar results, the robust approach gave a slightly better result and presented reasonable long-run elasticities related to the demand of natural in the country, indicating that can it be a good alternative to overcome the difficulties associated with the use of short time series and unreliable data on the forecast of energy consumption in emerging markets.

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Authors and Affiliations

  1. Department of Telecommunications and Control Engineering, EPUSP-Polytechnic School of the University of São Paulo, São Paulo, Brazil

    Oswaldo L. V. Costa

  2. Department of Production Engineering, EPUSP, São Paulo, Brazil

    Celma de Oliveira Ribeiro, Linda Lee Ho & Erik Eduardo Rego

  3. Graduate Energy Program of the Institute of Energy and Environment at the University of São Paulo, São Paulo, Brazil

    Virginia Parente & Javier Toro

Authors
  1. Oswaldo L. V. Costa
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  2. Celma de Oliveira Ribeiro
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  3. Linda Lee Ho
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  4. Erik Eduardo Rego
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  5. Virginia Parente
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  6. Javier Toro
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Correspondence to Oswaldo L. V. Costa.

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Research supported in part by Project Fapesp/Shell Research Centre for Gas Innovation, Grant 2014/50279-4. The first, second, third and fourth authors received financial support from CNPq (Brazilian National Research Council), Grants 304091/2014-6, 307126/2018-8, 304670/2014-6 and 303865/2017-2 respectively.

Appendix

Appendix

1.1 Second-order cone programming (SOCP)

A SCOP is defined as follows:

$$\begin{aligned}&\min _{\in \mathbb {R}^n} \mathbf c ^\prime \mathbf x \nonumber \\&\text {subject to}\,\,\,\,\Vert \mathbf A _i\mathbf x +\mathbf b _i\Vert \le \mathbf e _i^\prime \mathbf x + \mathbf d _i, i=1,\ldots ,\kappa \end{aligned}$$
(16)

where \(\mathbf c ,\mathbf e _i\in \mathbb {R}^n\), \(\mathbf A _i \in \mathbb {R}^{n_i\times n}\) and \(\mathbf d _i\in \mathbb {R}\). As pointed out in [28] SOCP includes several important standard classes of convex optimization problems, such as linear programming (LP), quadratic programming (QP) and quadratically constrained quadratic program (QCQP). Several very efficient primal-dual interior-point methods for SOCP have been developed in the last few years, which makes this class of problem very attractive from the numerical point of view.

1.2 Proof of Theorem 1

The goal of this subsection is to show (5), following the same approach as in Theorem 3.1 in [12]. First we notice that

$$\begin{aligned} \Vert \mathbf{W }\big (\mathbf A _1 \mathbf x _1 + (\mathbf A _2+\varDelta \mathbf A _2) \mathbf x _2&- (\mathbf b +\varDelta \mathbf b )\big )\Vert = \Vert \mathbf{W }(\mathbf{Ax } - \mathbf b )+ \mathbf{W } \begin{bmatrix} \varDelta \mathbf A _2&\varDelta \mathbf b \end{bmatrix} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert \nonumber \\&\le \Vert \mathbf{W }(\mathbf{Ax } - \mathbf b )\Vert + \Vert \mathbf{W } \begin{bmatrix} \varDelta \mathbf A _2&\varDelta \mathbf b \end{bmatrix} {\varvec{\varGamma }^{-1}}{\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert \nonumber \\&\le \Vert \mathbf{W }(\mathbf{Ax } - \mathbf b )\Vert + \Vert \mathbf{W } \begin{bmatrix} \varDelta \mathbf A _2&\varDelta \mathbf b \end{bmatrix} {\varvec{\varGamma }^{-1}} \Vert \Vert {\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert \nonumber \\&\le \Vert \mathbf{W }(\mathbf{Ax } - \mathbf b )\Vert + \rho \Vert {\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert \end{aligned}$$
(17)

since \(\Vert \mathbf{W } [ \varDelta \mathbf A _2 \,\,\varDelta \mathbf b ] {\varvec{\varGamma }^{-1}} \Vert \le \rho \). Define now \(\begin{bmatrix} \varDelta \mathbf A _2^*&\varDelta \mathbf b ^* \end{bmatrix}\) as

$$\begin{aligned} \begin{bmatrix} \varDelta \mathbf A _2^*&\varDelta \mathbf b ^* \end{bmatrix} = \rho \mathbf v \frac{1}{\Vert {\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert }\begin{bmatrix}{} \mathbf x _2^\prime&-\mathbf 1 \end{bmatrix} \varvec{\varGamma } \end{aligned}$$
(18)

where \(\mathbf v \) is defined, for the case in which \(\mathbf{Ax } \ne \mathbf b \), as

$$\begin{aligned} \mathbf v = \frac{1}{\Vert \mathbf{W }(\mathbf{Ax }-\mathbf b )\Vert } (\mathbf{Ax }-\mathbf b ) \end{aligned}$$
(19)

and for the case in which \(\mathbf{Ax } = \mathbf b \), as \(\mathbf v = \mathbf e \) where \(\mathbf e \) is a vector such that \(\Vert \mathbf{W }{} \mathbf e \Vert =1\). From (18) we have that

$$\begin{aligned} \Vert \mathbf{W }\begin{bmatrix} \varDelta \mathbf A _2^*&\varDelta \mathbf b ^* \end{bmatrix} {\varvec{\varGamma }^{-1}} \Vert = \rho \Vert \mathbf{W }{} \mathbf v \Vert \frac{1}{\Vert {\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert }\Vert \begin{bmatrix}{} \mathbf x _2^\prime&-\mathbf 1 \end{bmatrix} {\varvec{\varGamma }}\Vert = \rho \end{aligned}$$

since \(\Vert \mathbf{W }{} \mathbf v \Vert =1\). Moreover, for the case in which \(\mathbf{Ax } \ne \mathbf b \), we have from (18), (19) that

$$\begin{aligned}&\Vert \mathbf{W }(\mathbf{Ax } - \mathbf b )+ \mathbf{W } \begin{bmatrix} \varDelta \mathbf A _2^*&\varDelta \mathbf b ^* \end{bmatrix} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix}\Vert \nonumber \\&\quad =\Vert \mathbf{W }(\mathbf{Ax } - \mathbf b )+ \rho \frac{1}{\Vert \mathbf{W }(\mathbf{Ax }-\mathbf b )\Vert } \mathbf{W }(\mathbf{Ax }-\mathbf b ) \frac{1}{\Vert {\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert }\begin{bmatrix}{} \mathbf x _2^\prime&-\mathbf 1 \end{bmatrix} \varGamma \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert \nonumber \\&\quad =\Vert \mathbf{W }(\mathbf{Ax } - \mathbf b )\Vert + \rho \Vert {\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert \end{aligned}$$
(20)

and similarly for the case in which \(\mathbf{Ax }=\mathbf b \) we have that

$$\begin{aligned} \Vert \mathbf{W }(\mathbf{Ax } - \mathbf b )+ \mathbf{W } \begin{bmatrix} \varDelta \mathbf A _2^*&\varDelta \mathbf b ^* \end{bmatrix} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix}\Vert&= \Vert \rho \mathbf{W } \mathbf e \frac{1}{\Vert {\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert }\begin{bmatrix}{} \mathbf x _2^\prime&-\mathbf 1 \end{bmatrix} \varvec{\varGamma } \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert \nonumber \\&=\rho \Vert \mathbf{W } \mathbf e \Vert \Vert {\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert = \rho \Vert {\varvec{\varGamma }} \begin{bmatrix}{} \mathbf x _2\\ -\mathbf 1 \end{bmatrix} \Vert . \end{aligned}$$
(21)

By combining (17), (20) and (21) we obtain (5).

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Costa, O.L.V., de Oliveira Ribeiro, C., Ho, L.L. et al. A robust least square approach for forecasting models: an application to Brazil’s natural gas demand. Energy Syst 11, 1111–1135 (2020). https://doi.org/10.1007/s12667-019-00351-1

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