Recently, there have been several papers that discuss the extension of the Pinball loss Support Vector Machine (Pin-SVM) model, originally proposed by Huang et al.,[1][2]. Pin-SVM classifier deals with the pinball loss function, which has been defined in terms of the parameter $\tau$. The parameter $\tau$ can take values in $[ -1,1]$. The existing Pin-SVM model requires to solve the same optimization problem for all values of $\tau$ in $[ -1,1]$. In this paper, we improve the existing Pin-SVM model for the binary classification task. At first, we note that there is major difficulty in Pin-SVM model (Huang et al. [1]) for $ -1 \leq \tau < 0$. Specifically, we show that the Pin-SVM model requires the solution of different optimization problem for $ -1 \leq \tau < 0$. We further propose a unified model termed as Unified Pin-SVM which results in a QPP valid for all $-1\leq \tau \leq 1$ and hence more convenient to use. The proposed Unified Pin-SVM model can obtain a significant improvement in accuracy over the existing Pin-SVM model which has also been empirically justified by extensive numerical experiments with real-world datasets.

More recently there has been an interest in density forecasts for the hourly prices, motivated by considerations of risk management. See [1,2] for extensive reviews. In this paper, we provide a new formulation with a focus upon price spreads, and specifically, we forecast the density functions for the intraday spreads in the day-ahead prices. The optimal operation of storage facilities, e.g., batteries and electric vehicles, or load shifting programmes, e.g., demand-side management, over daily cycles depends upon these spreads if they are operated as merchants, arbitraging buying and selling from the wholesale market. Furthermore, if the risk is a consideration, analysis of the mean differences in price levels would be inadequate, and we therefore directly estimate the density functions of all hourly spreads in prices at the day-ahead stage. These forecasts ahead of the day-ahead auctions would be needed to help traders decide whether they want to be buyers or sellers at each hour and thereby optimise their bids and offers. Our specification, estimation and forecasting of these arbitrage spreads are new and computationally-intensive. Based upon day-ahead forecasts for the drivers of electricity prices, such as demand, wind and solar production, gas and coal prices, forecasts for electricity price levels have been proposed from various methods, e.g., [3-6] and some for price densities [1,7], but apparently no methods have been developed specifically for forecasting intraday spread densities. Until recently storage assets, such as pumped hydro storage, would regularly store energy at night and discharge at the daily peak demand periods, which were quite predictable. However with the penetration of wind and especially solar generating facilities, the peak and trough hours in prices move around the day and in sunny locations with substantial solar energy, e.g., California, the lowest prices may often be in the middle of the day [8].

The estimation of f τ( x) is difficult but, more informative than estimation of only mean regression f ( x). The estimation of f τ( x) for different values of τ can briefly describe the different characteristics of the conditional distribution of y/x . In many real world problems, the estimation of mean regression f ( x) is not required or enough, rather they require the estimation of quantile f τ(x). The study of quantile regression problem has initially been started in 1978 by Koenkar and Bassett[1]. Later, it has been briefly discussed and described by Koenker in his book (Koenker, [2]). Koenkar and Bassett [1] proposed the pinball loss function for the estimation of the quantile function f τ(x). For a given quantile τ (0, 1), the pinball loss function was an asymmetric loss function suitable for quantile estimation. It was given by P τ( u) null τu if u 0, (τ 1)u otherwise.

In this paper, we propose a novel asymmetric $\epsilon$-insensitive pinball loss function for quantile estimation. There exists some pinball loss functions which attempt to incorporate the $\epsilon$-insensitive zone approach in it but, they fail to extend the $\epsilon$-insensitive approach for quantile estimation in true sense. The proposed asymmetric $\epsilon$-insensitive pinball loss function can make an asymmetric $\epsilon$- insensitive zone of fixed width around the data and divide it using $\tau$ value for the estimation of the $\tau$th quantile. The use of the proposed asymmetric $\epsilon$-insensitive pinball loss function in Support Vector Quantile Regression (SVQR) model improves its prediction ability significantly. It also brings the sparsity back in SVQR model. Further, the numerical results obtained by several experiments carried on artificial and real world datasets empirically show the efficacy of the proposed `$\epsilon$-Support Vector Quantile Regression' ($\epsilon$-SVQR) model over other existing SVQR models.