A functional time series is the realization of a stochastic process where each observation is a continuous function defined on a finite interval. These processes are commonly found in electricity markets and are gaining more importance as more market data become available and markets head toward continuous-time marginal pricing approaches. Forecasting these time series requires models that operate with continuous functions. This paper proposes a new functional forecasting method that attempts to generalize the standard seasonal ARMAX time series model to the L ² Hilbert space. The structure of the proposed model is a linear regression where functional parameters operate on functional variables. The variables can be lagged values of the series (autoregressive terms), past observed innovations (moving average terms), or exogenous variables. In this approach, the functional parameters used are integral operators whose kernels are modeled as linear combinations of sigmoid functions. The parameters of each sigmoid are optimized using a Quasi-Newton algorithm that minimizes the sum of squared errors. This novel approach allows us to estimate the moving average terms in functional time series models. The new model is tested by forecasting the daily price profile of the Spanish and German electricity markets and it is compared to other functional reference models.

Debris cones are a very common landform in temperate high mountains. They are the most representative examples of the periglacial and nival processes. This work studies the dynamic behavior of two debris cones (Cone A and Cone B) in the Picos de Europa, in the north of the Iberian Peninsula. Their evolution was measured uninterruptedly throughout each August for 10 years (2009–2018) using the Terrestrial Laser Scanning (TLS) technique. The observations and calculations of the two debris cones were treated independently, but both showed the same behavior. Therefore, if these results are extrapolated to other debris cones in similar environments (temperate high mountain), they should show behavior similar to that of the two debris cones analyzed. Material falls onto the cones from the walls, and transfer of sediments follows linear trajectories according to the maximum slope. In order to understand the linear evolution of the two debris cones, profiles were created along the maximum slope lines of the Digital Elevation Model (DEM) of 2009, and these profile lines were extrapolated to the remaining years of measurement. In order to determine volumetric surface behavior in the DEMs, each year for the period 2009–2018 was compared. In addition, the statistical predictive value for position (Z) in year 2018 was calculated for the same planimetric position (X,Y) throughout the profiles of maximum slopes. To do so, the real field data from 2009–2017 were interpolated and used to form a sample of curves. These curves are interpreted as the realization of a functional random variable that can be predicted using statistical techniques. The predictive curve obtained was compared with the 2018 field data. The results of both coordinates (Z), the real field data, and the statistical data are coherent within the margin of error of the data collection.

In this paper, we provide improvements in the additive constant of the current best asymptotic upper bound for the maximum number of halving lines for planar sets of n points, where n is an even number. We also improve this current best upper bound for small values of n, namely, $106\le n\le 336$. To obtain this enhancements, we provide lower bounds for the sum of the squares of the degrees of the vertices of a graph related to the halving lines.