Page 24 - Contributed Paper Session (CPS) - Volume 2

CPS1408 Caston S. et al.
ℎ
where denotes the parameter, () represents the basis function
ℎ

with the dimension of the basis being denoted by q. The parameter estimates
of equation (1) are obtained by minimising the function given in equation (3).

| () = ∑ (y , − ∑ ( ) ), (3)
,

=1 =1

where ρτ is the pinball loss function. The AQR models are given in equation
(4).

()Φ( ) [ , − {∑ , ( ) + ∑ ∑ ( ) ( )}] = ()Θ( ) , . (4)

=1 =1 =1

Selection of variables is done using the least absolute shrinkage and
selection operator (Lasso) for hierarchical interactions method developed by
Bien et al. [1] and implemented in the R package “hierNet” ([2]). The objective
is to include an interaction where both variables are included in the model.
The restriction known as the strong hierarchy constraint is discussed in detail
in Ben and Tibshirani [2] and Lim and Hastie [9].

2.2 Forecast error measures
There are several error measures for probabilistic forecasting which include
among others the continuous rank probability score, the logarithmic score and
the quantile loss that is also known as the pinball loss. In this paper we use the
pinball loss function which is relatively easy to compute and interpret ([6]). The
pinball loss function is given as
( − ) if >

( , ) = { (1 − )( − ) if > , (5)

where is the quantile forecast and is the observed value of hourly

electricity demand.

2.3 Prediction intervals
For each of the models, = 1, … , we compute the prediction interval
,
widths (PIWs) which we shall abbreviate as = 1, … , , = 1, … , as
,
follows:
PIW = UL − LL , (6)

where ULij and LLij are the upper and lower limits of the prediction interval,
respectively. The analysis for determining the model which yields narrower
PIW is done in this study using box and whisker plots, together with the
probability density plots.

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