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

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

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

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