# Sample-based estimation of mean electricity consumption curves for small domains

Section 6. Conclusions and outlooks

In this article, we proposed four approaches for estimating mean curves by sampling for small domains. The first two consist of projecting curves in a finite space and using the usual methods for estimating total real variables for each base vector in the projection space. In this case, we use either unit-level linear mixed models or linear regression. The last two approaches consist of predicting each curve of the unsampled units using a non-parametric model and aggregating those predictions to determine the estimated mean curves for each domain. The models used to build the predictions are regression trees adapted to functional data build using the Courbotree approach of Stéphan and Cogordan (2009) or random forests adapted to functional data built by aggregating random Courbotree trees. For each approach, we also proposed a process for approximating the variance of mean curve estimators based on a bootstrap.

Our tests showed that the linear mixed models gave the best results and, for this particular data set, made it possible to divide the error committed by approximately seven in relation to the Horvitz-Thompson estimators. The regression trees come next, followed by the linear functional regressions.

This work can be extended in various ways. In particular, we feel that the approach based on the aggregation of non-parametric estimates of curves using regression trees or random forests is promising. An interesting possibility for improvement could be the use of more relevant distances than the Euclidean distance in the split criteria that builds our regression trees. We could thus use the Mahalanobis distance, the Manhattan distance, or a “dynamic time warping” distance.

Another possibility could be to build this split criterion by applying the Euclidian distance not on the discretized curves, but on a transformation of those curves, by projection in a wavelet base, or on non-linear summaries, such as variational autoencoders from deep learning models (see, for example, LeCun, Bengio and Hinton, 2015).

We can also question the choice of depth of the regression tree, the minimum size of the leaves and the number of trees in the forest. The criteria usually used in non-parametric statistics to answer this question are usually based on the principle of cross-validation. However, our objective here is not to determine the best possible prediction for each population unit, but a prediction that gives the best estimate of the mean curve by domain, which is not necessarily the same thing. It would therefore be best to adapt the cross-validation criteria to reflect our objective.

Finally, we note that the introduction of random effects in the linear models results in improved prediction, which leads us to think that there are characteristics in the domains that are not explained solely by the auxiliary information. It could therefore be relevant to adapt the functional regression trees to include the random effects. One solution, for example, would be to extend the algorithm from Hajjem, Bellavance and Larocque (2014), based on an EM algorithm as part of the functional data.

## Acknowledgements

The authors thank Hervé Cardot for the fruitful discussions and the associate editor and two referees for their remarks and comments, which helped greatly improve this article.

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