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

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 $\left[0,\text{\hspace{0.17em}}T\right]$ is associated with each unit $i$ of the population. The population $U$ can be decomposed in $D$ disjoint domains ${U}_{d}$ of known sizes ${N}_{d},\text{\hspace{0.17em}}d=1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}D.$ Our goal is to estimate the mean curve of $Y$ in each domain:

${\mu }_{d}\left(t\right)=\frac{1}{{N}_{d}}\sum _{i\in {U}_{d}}\text{\hspace{0.17em}}{Y}_{i}\left(t\right),\text{ }t\in \left[0,\text{\hspace{0.17em}}T\right],\text{ }d=1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}D.\text{ }\text{ }\text{ }\text{ }\text{ }\left(2.1\right)$

In the population $U,$ we select a sample $s$ of size $n,$ based on a random sampling design $p\left(\cdot \right).$ Let ${\pi }_{i}=\text{Pr}\left(i\in s\right)$ the probability of inclusion of unit $i$ in sample $s$ and assumed to be positive for all units $i\in U.$ Let ${s}_{d}=s\cap {U}_{d}$ 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 ${\overline{X}}_{d}={\sum }_{i\in {U}_{d}}\text{\hspace{0.17em}}{X}_{i}/{N}_{d}$ on the domain $d=1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}D.$

In practice, the curves are not observed continuously, but only for a series of measurement instants $0={t}_{1}<{t}_{2}<\dots <{t}_{L}=T$ 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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