Sample-based estimation of mean electricity consumption curves for small domains
Section 2. Notations and framework

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Let a population of interest $U$ of size $N.$ A (demand) curve $Y_i\left(t\right)$ measured for each instant $t$ belonging to an interval of time $\left[\mathrm{0,}T\right]$ is associated with each unit $i$ of the population. The population $U$ can be decomposed in $D$ disjoint domains $U_d$ of known sizes $N_d\mathrm{,}d\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}D.$ Our goal is to estimate the mean curve of $Y$ in each domain:

$${\mu }_d\left(t\right)=\frac{1}{N_d}{\displaystyle \sum _{i\in U_d}}{Y}_i\left(t\right)\mathrm{,}t\in \left[\mathrm{0,}T\right]\mathrm{,}d\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}D\mathrm{.}(2.1)$$

In the population $U,$ we select a sample $s$ of size $n,$ based on a random sampling design $p(\cdot ).$ Let ${\pi }_i\mathrm{<mo>=</mo>}\text{Pr}\left(i\in s\right)$ the probability of inclusion of unit $i$ in sample $s$ and assumed to be positive for all units $i\in U.$ Let $s_d\mathrm{<mo>=</mo>}s\cap U_d$ the portion of $s$ belonging to domain $U_d$ of random size $n_d,$ which can be equal to 0 for one or more domains.

We assume that we also have a dimensional vector $p$ of auxiliary variables (non-dependent on time) $X_i$ that will be assumed to be known for each individual $i$ in the population and with a known average ${\overline{X}}_d\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U_d}}{X}_i/N_d$ on the domain $d\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}D.$

In practice, the curves are not observed continuously, but only for a series of measurement instants $\mathrm{0\; <mo>=</mo>}{t}_1\mathrm{<mo><</mo>}{t}_2\mathrm{<mo><</mo>}\dots \mathrm{<mo><</mo>}{t}_L\mathrm{<mo>=</mo>}T$ that are also assumed to be equidistant and identical for all individuals. It is also assumed that there are no missing values.

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