On ergodic two-armed bandits

A device has two arms with unknown deterministic payoffs and the aim is to asymptotically identify the best one without spending too much time on the other. The Narendra algorithm offers a stochastic procedure to this end. We show under weak ergodic assumptions on these deterministic payoffs that the procedure eventually chooses the best arm (i.e., with greatest Cesaro limit) with probability one for appropriate step sequences of the algorithm. In the case of i.i.d. payoffs, this implies a "quenched" version of the "annealed" result of Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454] by the law of iterated logarithm, thus generalizing it. More precisely, if $(\eta_{\ell,i})_{i\in \mathbb {N}}\in\{0,1\}^{\mathbb {N}}$, $\ell\in\{A,B\}$, are the deterministic reward sequences we would get if we played at time $i$, we obtain infallibility with the same assumption on nonincreasing step sequences on the payoffs as in Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454], replacing the i.i.d. assumption by the hypothesis that the empirical averages $\sum_{i=1}^n\eta_{A,i}/n$ and $\sum_{i=1}^n\eta_{B,i}/n$ converge, as $n$ tends to infinity, respectively, to $\theta_A$ and $\theta_B$, with rate at least $1/(\log n)^{1+\varepsilon}$, for some $\varepsilon >0$. We also show a fallibility result, that is, convergence with positive probability to the choice of the wrong arm, which implies the corresponding result of Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454] in the i.i.d. case.Comment: Published in at http://dx.doi.org/10.1214/10-AAP751 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Semiparametric estimation of a two-component mixture of linear regressions in which one component is known

A new estimation method for the two-component mixture model introduced in \cite{Van13} is proposed. This model consists of a two-component mixture of linear regressions in which one component is entirely known while the proportion, the slope, the intercept and the error distribution of the other component are unknown. In spite of good performance for datasets of reasonable size, the method proposed in \cite{Van13} suffers from a serious drawback when the sample size becomes large as it is based on the optimization of a contrast function whose pointwise computation requires O(n^2) operations. The range of applicability of the method derived in this work is substantially larger as it relies on a method-of-moments estimator free of tuning parameters whose computation requires O(n) operations. From a theoretical perspective, the asymptotic normality of both the estimator of the Euclidean parameter vector and of the semiparametric estimator of the c.d.f.\ of the error is proved under weak conditions not involving zero-symmetry assumptions. In addition, an approximate confidence band for the c.d.f.\ of the error can be computed using a weighted bootstrap whose asymptotic validity is proved. The finite-sample performance of the resulting estimation procedure is studied under various scenarios through Monte Carlo experiments. The proposed method is illustrated on three real datasets of size $n=150$, 51 and 176,343, respectively. Two extensions of the considered model are discussed in the final section: a model with an additional scale parameter for the first component, and a model with more than one explanatory variable.Comment: 43 pages, 4 figures, 5 table

Semiparametric estimation of a two-component mixture model

Suppose that univariate data are drawn from a mixture of two distributions that are equal up to a shift parameter. Such a model is known to be nonidentifiable from a nonparametric viewpoint. However, if we assume that the unknown mixed distribution is symmetric, we obtain the identifiability of this model, which is then defined by four unknown parameters: the mixing proportion, two location parameters and the cumulative distribution function of the symmetric mixed distribution. We propose estimators for these four parameters when no training data is available. Our estimators are shown to be strongly consistent under mild regularity assumptions and their convergence rates are studied. Their finite-sample properties are illustrated by a Monte Carlo study and our method is applied to real data.Comment: Published at http://dx.doi.org/10.1214/009053606000000353 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Semiparametric topographical mixture models with symmetric errors

Motivated by the analysis of a Positron Emission Tomography (PET) imaging data considered in Bowen et al. (2012), we introduce a semiparametric topographical mixture model able to capture the characteristics of dichotomous shifted response-type experiments. We propose a local estimation procedure, based on the symmetry of the local noise, for the proportion and locations functions involved in the proposed model. We establish under mild conditions the minimax properties and asymptotic normality of our estimators when Monte Carlo simulations are conducted to examine their finite sample performance. Finally a statistical analysis of the PET imaging data in Bowen et al. (2012) is illustrated for the proposed method.Comment: 19 figure

Impact of climate change on the occurrence of late Holocene glacial lake outburst floods in Patagonia : a sediment perspective

Glacial Lake Outburst Floods (GLOFs) constitute one of the most notorious and destructive geohazards worldwide. They have occurred throughout recorded history and form a constant threat for local communities near glacierized regions. Although the recent deglaciation resulted in an increase in glacial lakes, both in size and number, worldwide, little is known about the possible relationship between climate variability and GLOF occurrence. Yet, it is generally assumed that GLOF frequency is currently increasing due to global glacier retreat. This is mainly due to a lack of continuous long-term flood records. Current records of GLOF occurrence, which are based on instrumental and historical data, are intrinsically limited to the last centuries. Consequently, the effect of climate change on GLOF occurrence and the likely evolution of GLOFs under future climate change scenarios remains unclear. However, a comprehensive understanding of the link between climate change, glacier variability, and GLOF occurrence is fundamental for future GLOF predictions and to improve flood hazard assessments. As in many other glacierized regions, GLOFs are a well-known phenomenon in the Patagonian Andes. They are particularly pronounced in the Baker region of Chilean Patagonia (47–48 °S), where repeated GLOFs occurred from the abrupt drainage of ice-dammed Cachet 2 Lake between April 2008 and November 2020. During these events, water from Cachet 2 Lake spills into Colonia River, a tributary of Baker River, and increases both river discharge and sediment suspended concentrations. Colonia GLOFs are able to block the regular Baker River flow and result in the inundation of large areas upstream of the Colonia-Baker confluence, such as the Valle Grande floodplain. Downstream, the Baker River triples in discharge and large amounts of sediment are transported, and ultimately deposited, in fjords. The repeated Baker River GLOFs during the 21st century and the location of the Baker River, which drains most of the eastern side of the Northern Patagonian Icefield (NPI) and therefore integrates meltwater from several lake-river systems, makes the Baker region ideally suited to investigate GLOFs and to study the impact of climate change on GLOF occurrence. To examine how GLOFs are recorded in fjord sediments, this study mapped the bathymetry of the head of Martínez Channel, i.e. the fjord in which the Baker River discharges, using multibeam echosounding. Results show that the subaquatic delta of Baker River is deeply incised by sinuous channels. The presence of sediment waves and coarser sediment within these channels suggest recent channel activity by turbidity currents. The latter is confirmed by sediment records collected at the head of the fjord, which reveal the presence of turbidites intercalated within silty background sediments, particularly on the delta plain in front of the main submarine channel. Although the turbidity currents are most likely generated by elevated river discharge and the associated relatively high suspended sediment loads, most turbidites are not related to GLOFs. Instead, they seem to represent other extreme discharge events, such as extreme precipitation or rain-on-snow events. By comparing geochemical and sedimentological results obtained on the sediment cores to the recent GLOF history of Baker River, we show that the recent 21st century Cachet 2 GLOF deposits can be distinguished from background sediments by their finer grain size and lower organic carbon content, reflecting the increased input of glacial sediments during GLOFs. In addition, the results obtained on the fjord sediment cores demonstrate that the 21st century GLOFs from Cachet 2 Lake, which occurred less than one year apart, are not recorded as individual layers but as units richer in sediment of glacial origin. This suggests that it is not possible to reconstruct GLOF frequency nor magnitude solely based on fjord sediments. Although 21 GLOFs from Cachet 2 Lake occurred between 2008 and 2017, the deposits with the clearest GLOF signature represent the initial events, implying that more glacial sediment was released during those first GLOFs, possibly due to lake-bed erosion. Consequently, it appears that sediment availability plays a more important role than flood magnitude in controlling GLOF deposit properties. Although GLOF frequency and magnitude cannot be accurately reconstructed using fjord sediments, high accumulation rates at the head of Martínez Channel highlight the potential of fjord sediment archives to establish pre-historical GLOF records at high temporal resolution. In addition, the bathymetric imagery and the sediment records obtained at the head of Martínez Channel show that site selection and multi-coring are fundamental to reconstruct the Baker River GLOF history, as fjord heads are dynamic sedimentary environments with rapidly migrating channels. Ideal locations to reconstruct GLOFs are found on the delta slope, away from any submarine channel influence. GLOF deposits are best identified close to the river mouth, as background sediments become progressively finer and less organic, thus more similar to GLOF deposits, with increasing distance from the lip of the Baker River delta. Given the unique context of the Baker River system, where a significant portion of the watershed is vegetated and where the fine and organic-poor signature of GLOF deposits clearly contrasts with the slightly coarser and organic background sediments, our results may only be applicable to fjord sediments from temperate regions. Distinguishing GLOF deposits from background sediments would likely be more challenging in high latitude fjords. Sediments deposited in floodplains constitute another faithful recorder of Baker River GLOFs. In the Valle Grande floodplain, which is located immediately upstream of the Colonia-Baker confluence, GLOFs are registered as organic-poor deposits intercalated within organic-rich background sediments. In contrast to marine archives, the sediments of the Valle Grande floodplain have lower accumulation rates, and can therefore be used to determine changes in GLOF occurrence on longer timescales (late Holocene). Based on four radiocarbon-dated sediment cores collected in the Valle Grande floodplain, our results show that high-magnitude GLOFs occurred intermittently in the upper Baker River watershed over the past 2.75 kyr. Two periods of increased flood activity occurred between approximately 2.57 and 2.17 cal kyr BP, and from 0.75 to 0 cal kyr BP. Comparison with independent proxy records of glacier variability reveals that these two periods of increased flood frequency match with Neoglacial advances. These advances seem to result from lower-than-average temperatures and wetter conditions. Based on these results, we suggest that there is a strong, yet indirect, link between climate variability and GLOF occurrence. We hypothesize that, on multi-millennial timescales, high-magnitude GLOFs from eastern NPI glaciers are more frequent at times when glaciers are larger and thicker, as such glaciers most likely form larger and stronger ice dams, which in turn are able to retain larger lakes. Our results therefore suggest that the probability that high-magnitude GLOFs occur decreases as glaciers thin and retreat. Conversely, the frequency of lower magnitude GLOFs tends to increase during glacier recession because of the rapid growth of glacial lakes and formation of new lakes. Although isolated cases of new lakes formed behind large glaciers could still produce large GLOFs locally, the likelihood of large lake drainage and therefore high-magnitude GLOF occurrence decreases. This study supports the use of sediment-based GLOF records in other GLOF-prone regions for proper flood hazard assessment. A broader knowledge of the impact of climate change on GLOF occurrence can help to prevent further development in flood-prone regions and will reduce the vulnerability of communities to floods. Long-term paleoflood records can be of great importance for integrating spatial planning and planned infrastructure projects, such as hydroelectric dams, particularly since electricity demand is increasing with economic growth

Alien Species and the Water Framework Directive - Questionnaire Results

Alien species constitute a major pressure in aquatic environments, both ecologically and economically. This recognition has initiated a debate on the role of alien species in ecological status classifications. We distributed a questionnaire to review how EU Member States (MSs) deal with alien species in their national status assessments under the Water Framework Directive (WFD). The questionnaire was filled and returned by 23 EU MSs and Norway. Analysis of the questionnaire returns and referred methods revealed the existence of a wide range of approaches: (1) The majority of MSs do not take alien species explicitly into account for classification under the WFD. This implies that the biological methods that were developed for assessing anthropogenic pressures are assumed to be able to pick up pressures by alien species, but this remains to be verified. Some of these MSs do acknowledge the need for a more direct or supplementary account of alien species, but fail to do so because of the lack of information on the impact and distribution of alien species within their territory. (2) Few MSs assess ecological status using the pressure-based tools, and subsequently downgraded the status in case of presence of predefined high-impact alien species. The main critiques to this approach are that it puts too much weight on alien species, compared to other pressures, and that it may results in downgrading of many water bodies, even in the absence of any perspectives for remediation of the problem. (3) Others account for alien species by attributing aliens a different score than native species in a metric, or by including alien species in comparisons with alien-free reference communities. Such approaches may blur the impact assessment of other pressures, and may obscure the magnitude of the alien species problem. (4) Almost all MSs support the idea of a supplementary biopollution index. Such an index would uncouple alien species and anthropogenic pressure assessments, and allow for a correct appraisal of the problem without affecting the WFD classification. A pan-European index may not be feasible because of different trade-offs between practicability and accuracy across MSs. A better harmonization of the views on alien species and water body classification within Europe is desirable, but a challenging task. It requires an agreed interpretation and usage of alien species related terms, and an increased compatibility and completeness of national and regional alien species lists. These topics have been added for discussion to the 2010-2012 mandate of the WFD Working Group on Ecological Status.JRC.DDG.H.5-Rural, water and ecosystem resource