Resistance to learning and the evolution of cooperation

In many evolutionary algorithms, crossover is the main operator used in generating new individuals from old ones. However, the usual mechanism for generating offsprings in spatially structured evolutionary games has to date been clonation. Here we study the effect of incorporating crossover on these models. Our framework is the spatial Continuous Prisoner's Dilemma. For this evolutionary game, it has been reported that occasional errors (mutations) in the clonal process can explain the emergence of cooperation from a non-cooperative initial state. First, we show that this only occurs for particular regimes of low costs of cooperation. Then, we display how crossover gets greater the range of scenarios where cooperative mutants can invade selfish populations. In a social context, where crossover involves a general rule of gradual learning, our results show that the less that is learnt in a single step, the larger the degree of global cooperation finally attained. In general, the effect of step-by-step learning can be more efficient for the evolution of cooperation than a full blast one

Rewarding cooperation in social dilemmas

One of the most direct human mechanisms of promoting cooperation is rewarding it. We study the effect of sharing a reward among cooperators in the most stringent form of social dilemma. Thus, individuals confront a new dilemma: on the one hand, they may be inclined to choose the shared reward despite the possibility of being exploited by defectors; on the other hand, if too many players do that, cooperators will obtain a poor reward and defectors will outperform them. By appropriately tuning the amount to be shared we can cast a vast variety of scenarios, including traditional ones in the study of cooperation as well as more complex situations where unexpected behavior can occur. We provide a complete classification of the equilibria of the nplayer game as well as of the evolutionary dynamics. Beyond, we extend our analysis to a general class of public good games where competition among individuals with the same strategy exists

Prediction Bands for Functional Data Based on Depth Measures

We propose a new methodology for predicting a partially observed curve from a functional data sample. The novelty of our approach relies on the selection of sample curves which form tight bands that preserve the shape of the curve to predict, making this a deep datum. The involved subsampling problem is dealt by algorithms specially designed to be used in conjunction with two different tools for computing central regions for functional data. From this merge, we obtain prediction bands for the unobserved part of the curve in question. We test our algorithms by forecasting the Spanish electricity demand and imputing missing daily temperatures. The results are consistent with our simulation that show that we can predict at the far horizon.Supported by the Spanish Ministerio de Educación,Cultura y Deporte under grant FPU15/00625. Partially supported by the Spanish Ministerio de Economía y Competitividad under grant ECO2015-66593-P

Localization processes for functional data analysis

We propose an alternative to k-nearest neighbors for functional data whereby the approximating neighboring curves are piecewise functions built from a functional sample. Using a locally defined distance function that satisfies stabilization criteria, we establish pointwise and global approximation results in function spaces when the number of data curves is large. We exploit this feature to develop the asymptotic theory when a finite number of curves is observed at time-points given by an i.i.d. sample whose cardinality increases up to infinity. We use these results to investigate the problem of estimating unobserved segments of a partially observed functional data sample as well as to study the problem of functional classification and outlier detection. For such problems our methods are competitive with and sometimes superior to benchmark predictions in the field. The R package localFDA provides routines for computing the localization processes and the estimators proposed in this article

On projection methods for functional time series forecasting

Two nonparametric methods are presented for forecasting functional time series (FTS). The FTS we observe is a curve at a discrete-time point. We address both one-step-ahead forecasting and dynamic updating. Dynamic updating is a forward prediction of the unobserved segment of the most recent curve. Among the two proposed methods, the first one is a straightforward adaptation to FTS of the k-nearest neighbors methods for univariate time series forecasting. The second one is based on a selection of curves, termed the curve envelope, that aims to be representative in shape and magnitude of the most recent functional observation, either a whole curve or the observed part of a partially observed curve. In a similar fashion to k-nearest neighbors and other projection methods successfully used for time series forecasting, we "project" the k-nearest neighbors and the curves in the envelope for forecasting. In doing so, we keep track of the next period evolution of the curves. The methods are applied to simulated data, daily electricity demand, and NOx emissions and provide competitive results with and often superior to several benchmark predictions. The approach offers a model-free alternative to statistical methods based on FTS modeling to study the cyclic or seasonal behavior of many FTS.The authors acknowledge insightful comments and suggestions from two reviewers and the Associate Editor. Antonio Elías is supported by the Spanish Ministerio de Educación, Cultura y Deporte under grant FPU15/00625 and the research stay grant EST17/00841. Antonio Elías and Raúl Jiménez are partially supported by the Spanish Ministerio de Economía y Competitividad under grant ECO2015-66593-P and PID2019-109196GB-I00/AEI/10.13039/501100011033. Funding for open access charge: Universidad de Málaga / CBUA. Part of this article was conducted during a stay at Australian National University. Antonio Elías is grateful to Han Lin Shang for his hospitality and insightful and constructive discussions

A Depth for Censured Functional Data

Censured functional data are becoming more recurrent in applications. In those cases, the existing depth measure are useless. In this paper, an approach for measuring depths of censured functional data is presented. Its performance for finite samples is tested by simulation, showing that the new depth agrees with a integrated depth for uncensured functional data.Antonio Elías is supported by the Spanish Ministerio de Educación, Cultura y Deporte under grant FPU15/00625. Antonio Elías and Raúl Jiménez are partially supported by the Spanish Ministerio de Economía y Competitividad under grant ECO2015-66593-P