that the present thesis method is able to obtain – at least for the simulated time series used – more or less acceptable scenario forecasts. Regardless, the method still needs to be improved to reduce the underestimation of the forecast uncertainty. In the meantime, the other presented method should be preferred in cases in which an accurate description of the uncertainty is needed, but scenario forecasts are not strictly required.
Notice as a final remark, that the evaluation of the scenario forecasts using the pointwise evaluation metrics (i.e. the average pinball-loss and modified reliability deviation) is considered adequate for the experiment presented in the current section. Nonetheless, evaluation values for multivariate dis-tributions (as e.g., the energy score [174]) are considered to be better at evaluating scenario forecasts, especially when comparing different scenario forecasting techniques. This is to be taken into consideration in future re-lated works.
2.5 Novelty and Remaining Questions
The current chapter describes several methods whose novelty can be sum-marized as follows:
1. The possibility of training quantile regressions without the need of us-ing Equation (1.16), thus openus-ing the possibility of trainus-ing quantile regressions with complex data mining techniques (i.e. ANN) with-out much effort, since their training algorithms remain unchanged (cf.
Section 2.2.1)
2. The obtainment of interval forecasts based on the new type of quantile regressions (cf. Section 2.2.2)
3. The straightforward obtainment of non-parametric distribution fore-casts (cf. Section 2.2.3)
2 Probabilistic Forecasting
4. The feasibility of obtaining parametric distribution forecasts (i.e.
parametric probabilistic forecasts) without requiring a distribution assumption (cf. Section 2.2.4)
5. The creation of scenario forecasts without the need of explicitly de-scribing the correlation structure between neighboring time series val-ues (cf. Section 2.2.5)
Additionally, the acceptable results obtained in the experiments demon-strate that the developed methods offer possible solutions to the first three open questions in Section 1.4 and thus fulfill the first three objectives of the current thesis (cf. Section 1.5). Nonetheless, a series of new question can also be raised, for instance:
• How do different values of the parameter th and different distance measures (cf. Equation (2.1)) influence the quality of the NNQF-based quantile regressions?
• How do quantile regressions based on the NNQF fare in comparison to “true" quantile regressions trained with more complex data mining techniques, as e.g., ANNs?
• Can the results of quantile regressions with different probabilities (trained with the same data mining technique) be improved by letting them choose their features independently of one another?
• Does the use of a non-linear interpolation (cf. Equation (2.10)) in-fluences the quality of the non-parametric CDF forecasts and of all forecasts based on them?
• Does the selection of differentFˆ˜s(y|x) andFˆ˜e(y|x) (cf. Equation (2.11)) functions influence the quality of the non-parametric CDF forecasts and of all forecasts based on them?
2.5 Novelty and Remaining Questions
• What improvements can be done to the scenario forecast method to avoid underestimating the forecast uncertainty?
• How can the implementation of the scenario forecast approach be op-timized, for it to allow the creation of scenario forecasts based on more complex data mining techniques (e.g., ANNs) in an acceptable amount of time?
Questions as the ones previously described are to be tackled in future re-lated works.
3 Hierarchical Probabilistic Forecasting
3.1 Overview
The current chapter presents a concept for estimating the joint distribution of two correlated future time series values, as well as for obtaining coherent hierarchical probabilistic forecasts based only on several joint distribution forecasts. A coherent hierarchical forecast is a collection of forecasts for all time series in a hierarchy in which the forecasts of each time series actually represent the sum of those forming it. Since time series are normally cast independently of one another, the coherency of the hierarchical fore-casts is not assured. Furthermore, additional problems arise when trying to estimate coherent hierarchical probabilistic forecasts; since estimating the CDF of the sum of several correlated values requires the estimation of a mul-tivariate CDF, which is not trivial to compute. Therefore, the current chapter offers an alternative based on the sequential estimation of joint distribution functions. An example of a time series hierarchy is given in Figure 3.1. The time series in the third level in Figure 3.1 represent, for example, loads of various households; with those in the second level representing loads at the substation level, and the time series at the top being the aggregation of all substations.
The present chapter is divided in three main sections. The first presents a straightforward method for obtaining joint distribution forecasts, the sec-ond describes an approach for obtaining coherent probabilistic hierarchi-cal forecasts, and the last describes the approaches’ novelty and some
re-3 Hierarchical Probabilistic Forecasting
Level 1
Level 2
Level 3
Level 1 Time Series 1 (L1TS1)
L2TS1 L2TS2 L2TS3
L3TS1 L3TS2 L3TS3 L3TS4 L3TS5
Figure 3.1: Example of a time series hierarchy
maining open questions. Furthermore, a series of examples are also pre-sented. The data used for these examples stems from three correlated sim-ulated load time series, {PLi[k]; k = 1, . . . , K; i = 1, 2, 3}, and their sum {PLS[k]; k = 1, . . . , K}. These time series represent three years of hourly measurements, i.e. K = 26280 and are obtained using the same benchmark generator as in [3, 156], as well as some additional modifications; more in-formation thereof can be found in Appendix B.1. The used time series are assumed to represent the load of three really similar households, as well as the load measured at a substation they are all connected to. For the sake of illustration, Figure 3.2 shows a segment of the used time series.
30000 3100 3200 3300 3400 3500
0.2 0.4 0.6 0.8
PPPP
P P P P
Figure 3.2: Simulated load time series used in the examples of the present chapter. Please refer to Appendix B.1 for more information about their obtainment.