Daniel Simberloff
email: dsimberloff@utk.edu

Jean-Louis Martin

Piero Genovesi

Virginie Maris

David A Wardle

James Aronson

Franck Courchamp

Bella Galil

Emili Garcı ´a-Berthou

Michel Pascal

Petr Pys ˇek

Ronaldo Sousa

Eric Tabacchi

Montserrat Vila

Impacts of biological invasions: what's what and the way forward

Keywords: biosecurity, community and ecosystem impact, eradication, long-term management, societal perception

Study of the impacts of biological invasions, a pervasive component of global change, has generated remarkable understanding of the mechanisms and consequences of the spread of introduced populations. The growing field of invasion science, poised at a crossroads where ecology, social sciences, resource management, and public perception meet, is increasingly exposed to critical scrutiny from several perspectives. Although the rate of biological invasions, elucidation of their consequences, and knowledge about mitigation are growing rapidly, the very need for invasion science is disputed. Here, we highlight recent progress in understanding invasion impacts and management, and discuss the challenges that the discipline faces in its science and interactions with society.

Biological invasions: from asset to burden Biological invasions are a pervasive global change [START_REF]Global Change Ecosystems Research[END_REF]2], challenging the conservation of biodiversity and natural resources [3]. Recognition of this challenge fostered the growth of a new field (invasion science [START_REF] Richardson | Invasion science. The roads travelled and the roads ahead[END_REF]) dedicated to detecting, understanding, and mitigating invasion impacts (see Glossary). Invasion research has shown that the scope and complexity of consequences greatly exceed earlier perceptions. Research continues to spawn technological improvements to deal with impacts [START_REF] Asner | Invasive plants transform the three-dimensional structure of rain forests[END_REF][START_REF] Pys ˇek | Invasive species, environmental change and management, and health[END_REF], and invasion science underpins national and international regulatory frameworks protecting human health and economies [START_REF] Hulme | Grasping at the routes of biological invasions: a framework for integrating pathways into policy[END_REF].

Poised at a crossroads where ecology, social sciences, resource management, and economics meet, invasion science has been scrutinized from many perspectives, leading some to question the importance of invasion impacts and need for invasion science [START_REF] Davis | Don't judge species on their origins[END_REF]. The field has also been challenged in its ability to interact with society, even being tagged as xenophobic [START_REF] Young | Clarifying debates in invasion biology: a survey of invasion biologists[END_REF]. These criticisms have cultural and historical contexts.

Public perception of introduced populations is cultureand organism-dependent [START_REF] Coates | American Perceptions of Immigrant and Invasive Species: Strangers on the Land[END_REF]. Polynesians introduced rats for food, but others saw rats as a scourge on islands and Glossary Eradication: complete removal of all individuals of a distinct population, not contiguous with other populations. Extirpation: elimination of a local population, but with conspecifics remaining in contiguous populations or nearby. Impact: any significant change (increase or decrease) of an ecological property or process, regardless of perceived value to humans [16]. Introduced population: population that arrives at a site with intentional or accidental human assistance [START_REF] Richardson | Invasion science. The roads travelled and the roads ahead[END_REF]. Invasive population: introduced population that spreads and maintains itself without human assistance [START_REF] Richardson | Invasion science. The roads travelled and the roads ahead[END_REF]. Propagule pressure: the frequency with which a species is introduced to a site, combined with the number of individuals in each introduction event [START_REF] Simberloff | The role of propagule pressure in biological invasions[END_REF].

supported eradication programs. In Western societies, from the great explorations until the early 20th century, non-native species introduced by acclimatization societies were considered 'exotic' curiosities, often viewed as a resource [11]. Today, some still see many introduced populations as assets, because of aesthetic properties, popularity as ornamental plants and pets, or economic value. Certain non-natives, such as Eucalyptus in California, are so appreciated that they become cultural icons in their new ranges [START_REF] Nun ˜ez | Invasive species and the cultural keystone concept[END_REF]. However, as both intentional and unintentional introductions increased throughout the 20th century, biologists gathered mounting evidence of the threat that some introductions pose for native species and ecosystems and for human well-being. This led to the launching of modern invasion science during the mid-1980s [START_REF] Simberloff | Charles Elton: neither founder nor siren, but prophet[END_REF], a development that paralleled the rise of modern conservation biology and increasing public concern about biodiversity. Invasion scientists became increasingly predisposed against non-natives not because they originated elsewhere, but because the probability of negative impact by non-natives is far greater than for natives [14] and because the frequency of invasions has increased exponentially [START_REF]Delivering Alien Invasive Species Inventories for Europe (DAISIE) Handbook of Alien Species in Europe[END_REF]. This, rather than displaced xenophobia, is why origins matter to invasion scientisits.

Here, we aim to: (i) depict the growing understanding of invasion impacts; (ii) show how scientists have responded to the increased management burden this understanding imposes; (iii) discuss challenges posed by the interaction of the field with society; and (iv) suggest ways to advance the field and enhance its ability to respond to challenges.

A range of impacts: difficult to evaluate, uncertain, delayed, and pervasive Difficult to evaluate An ecological impact consists of any significant change in an ecological pattern or process [16]. Much popular literature and some scientific literature becloud invasion impacts and responses of species, communities, and ecosystems in two ways. First is the practice of designating native species as 'good' and introduced species as 'bad.' Species are neither and, furthermore, invasion pertains to the population level, not the species level. Different stakeholders can view an introduced population as 'harmful' or 'useful' [START_REF] Duffy | Why biodiversity is important to the functioning of real-world ecosystems[END_REF]. When the Japanese tiger prawn Marsupenaeus japonicus, which is native to the Red Sea, reached the Mediterranean through the Suez Canal, fishermen applauded, but it extirpated a native prawn (Melicertus kerathurus), epitomizing 'harmful' for conservationists [START_REF] Galil | Loss or gain? Invasive aliens and biodiversity in the Mediterranean Sea[END_REF]. Second, invasion impacts have been labeled 'good' or 'bad' depending on the effect on a particular ecosystem service (e.g., [START_REF] Schlaepfer | The potential conservation value of nonnative species[END_REF]). An impact is 'good' or 'bad' only from certain perspectives. Schlaepfer et al. [START_REF] Schlaepfer | The potential conservation value of nonnative species[END_REF] have pointed to introduced populations aiding conservation -for instance, introduced trees providing nesting habitat for birds. However, a non-native plant population perceived as beneficial can actually attract native birds for nesting but ultimately decrease their survival or reproduction [START_REF] Rodewald | Spreading messages about invasives[END_REF]. That nitrogen-fixing plants, disproportionately represented in non-native floras, enhance ecosystem nitrogen input, soil fertility, and productivity [21] is often seen as positive, but in oligotrophic systems it is commonly perceived as negative. Many cases of introduced populations aiding conservation involve negative effects on other species [22]. In several instances, an invader that has replaced natives substitutes as a resource for remaining natives. For example, introduced rats perform some pollination functions of the New Zealand birds that they helped eliminate [23].

Often, impacts usually seen as negative from the ecosystem perspective are perceived as positive by some societal segments. Invasion by Pinaceae throughout the southern hemisphere [START_REF] Richardson | Pinus: a model group for unlocking the secrets of alien plant invasions?[END_REF] usually reduces litter quality, impairs decomposition, and depletes many soil species [START_REF] Dehlin | Tree seedling performance and belowground properties in stands of invasive and native tree species[END_REF], but the fast-growing trees support timber industries. Many invasive flammable grasses modify fire regimes and drive ecosystems to an alternative stable state [26] but may benefit livestock and so ranchers. Hence, the full range of ecological, economic, and sociological consequences should be considered when an invasion impact is evaluated.

Uncertain but real and often delayed

When a species is proposed for introduction or a recent introduction is detected, invasion science suggests cause for concern. In Europe, for example, even though only 11% of over 10 000 non-native populations are known so far to cause measurable ecological impacts [START_REF] Vila | How well do we understand the impacts of alien species on ecosystem services? A pan-European, cross-taxa assessment[END_REF], this results in many problems. Among established aquatic species introduced to six European countries, 69% have recognized ecological impacts [START_REF] Garcı ´a-Berthou | Introduction pathways and establishment rates of invasive aquatic species in Europe[END_REF]. These percentages are underestimates, because many impacts are subtle or in inaccessible habitat and so are characterized only after intensive study. Native species can also suddenly spread into new habitats, but the risk of ecologically harmful impacts is greatly elevated for non-natives: by a factor of 40 for plants of the USA, for example [14]. Furthermore, molecular methods increasingly reveal that what had been regarded as 'invasion' might instead result from new genotypes from distant sources [START_REF] Lavergne | Increased genetic variation and evolutionary potential drive the success of an invasive grass[END_REF]. Concern is also heightened because many introduced populations remain innocuous for extended periods before spreading and becoming invasive [START_REF] Crooks | Lag times[END_REF][START_REF] Essl | Socioeconomic legacy yields an invasion debt[END_REF]. For instance, Brazilian pepper remained restricted in Florida for a century before rapidly expanding across a wide area [START_REF] Crooks | Lag times[END_REF], whereas plants introduced to Europe might take 150-400 years to reach their fullest areal extent [START_REF] Gasso | Spreading to a limit: the time required for a neophyte to reach its maximum age[END_REF].

Pervasive from the population to the community and ecosystem levels Some population effects are obvious. Introduced terrestrial predators are often seen eating endemic island prey; although population consequences require careful study, we are unsurprised to learn these are dire. The same applies to introductions of predatory fish to lakes, which have caused both local extirpations and global extinctions of native fish and amphibians [START_REF] Vitule | Introduction of non-native freshwater fish can certainly be bad[END_REF]. The growing roster of well-studied cases of such population impacts is enormous (e.g., [START_REF]Encyclopedia of Biological Invasions[END_REF]).

A recent finding is that certain extremely consequential impacts, particularly at the ecosystem level, are not readily detected. An example is the multiple effects of introduced nitrogen-fixers on ecosystem functions [35]. Although some impacts affecting entire communities and ecosystems have been recognized since at least the 1980s [35], more have become apparent as invasion science has undergone a shift from a primary focus on impacts on particular species (e.g., endemic island birds devastated by introduced predators) to cumulative impacts on ecosystems [21].

Increasing emphasis on community and ecosystem impacts has revealed important, sometimes unsuspected, effects of introductions from all major trophic groups. For instance, many invasive plants transform ecosystems both above-and belowground, particularly when they differ in functional traits from native flora and when those traits drive ecosystem processes [START_REF] Wardle | Terrestrial ecosystem responses to species gains and losses[END_REF]. Although invasive plants frequently have traits more associated with rapid resource acquisition than those of natives [START_REF] Bardgett | Aboveground-Belowground Interactions: Biotic Interactions, Ecosystem Properties and Global Change[END_REF], literature syntheses and meta-analyses often find no large or consistent overall differences between native and introduced plant populations in terms of functional traits [38] or effects on belowground processes [START_REF] Ehrenfeld | Effects of exotic plant invasions on soil nutrient cycling processes[END_REF][START_REF] Liao | Altered ecosystem carbon and nitrogen cycles by plant invasion: a meta-analysis[END_REF]. This is because invasive plant effects depend not only on the types of invader, but also on characteristics of the invaded ecosystem [16].

A growing number of studies shows introduced consumers, such as herbivores, decomposers, and predators (Table 1), transforming community composition and ecosystem properties through trophic cascades and changed nutrient cycling. Some invasive consumers remove or add physical structures, altering erosion regimes and changing habitat suitability for other species [START_REF] Simberloff | How common are invasion-induced ecosystem impacts?[END_REF]. Belowground, several invasive consumers also radically transform ecosystems (Table 1).

A range of actions: prevention, eradication, and longterm management

From caution to prevention The litany of negative, far-reaching impacts of invasions suggests that proposed introductions warrant great caution. Greater efforts are needed to screen pathways and vectors that bring unintended introductions and to detect invaders quickly. By the time impacts are noted, irreversible changes might have occurred [START_REF] Vila | Ecological impacts of invasive alien plants: a meta-analysis of their effects on species, communities and ecosystems[END_REF] or palliative measures might be too costly or impossible [START_REF] Rejma ´nek | When is eradication of exotic pest plants a realistic goal? In Turning the Tide: the Eradication of Invasive Species[END_REF]. Guiding principles on invasive species adopted by the Convention on Biological Diversity (2002) reflect these findings: prevention is the priority response; early detection, rapid response, and possible eradication should follow when prevention fails. Long-term management is the last option. Statistics confirm the validity of this approach (Box 1, Figure 1). Hemlock replacement by species producing higher quality litter that stimulates nutrient cycling [START_REF] Lovett | Forest ecosystem responses to exotic pests and pathogens in eastern North America[END_REF] Black-tailed deer (Odocoileus hemionus columbianus and red deer (Cervus elaphus)

Replacement of understory plants by plants producing poor-quality litter, altering nutrient cycling and the soil food web [START_REF] Wardle | Impacts of introduced browsing mammals in New Zealand forests on decomposer communities, soil biodiversity and ecosystem properties[END_REF] Decline in understory vegetation dramatically reducing arthropods and songbirds [64] North American beaver (Castor canadensis)

Change in watershed hydrology and nutrient cycling, transforming forests into meadows [START_REF] Anderson | The effects of invasive North American beavers on riparian plant communities in Cape Horn, Chile: do exotic beavers engineer differently in sub-Antarctic ecosystems?[END_REF] Rabbitfish (Siganus spp.) Reduction of habitat complexity and species richness, and alteration of food webs [START_REF] Sala | Alien marine fishes deplete algal biomass in the eastern Mediterranean[END_REF] Introduced predators Yellow crazy ant (Anoplolepis gracilipes) Dramatic reduction of red crab population, increasing tree seedling density, and reduction of litter decomposition [START_REF] O'dowd | Invasional meltdown: do invasive ants facilitate secondary invasions? In Ant Ecology[END_REF] Ship rat (Rattus rattus), Norway rat (Rattus norvegicus), and Arctic fox (Vulpes lagopus)

Predation on seabirds thwarting nutrient transfer from ocean to land. Rats change belowground community, nutrient cycling, and decomposition

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Foxes change soil fertility and transform grasslands to shrub-and forb-dominated ecosystems [START_REF] Croll | Introduced predators transform subarctic islands from grassland to tundra[END_REF] Rainbow trout (Oncorhynchus mykiss) Usurping terrestrial insects falling into streams, causing native char to shift to foraging for insects feeding on bottom algae, increasing algal biomass, decreasing insect emergence and spider populations [START_REF] Baxter | Fish invasion restructures stream and forest food webs by interrupting reciprocal prey subsidies[END_REF] Nile perch (Lates niloticus) Driving over 150 native fish species to extinction, including many phytoplanktivores and detritivores, favoring increased algal blooms and submersed vegetation, and massively increased prawn populations; perch fisheries attracted more humans, further exacerbating eutrophication [START_REF] Pringle | Nile Perch[END_REF] Belowground invasions Root pathogenic fungi and oomycetes (notably Armillaria and Phytophthora spp.)

Causing massive tree death in Australia and California with wide-ranging impacts aboveand belowground [START_REF] Loo | Ecological impacts of non-indigenous invasive fungi as forest pathogens[END_REF] Earthworms In deglaciated parts of North America lacking native worms, causing loss of organic matter, nutrient mineralization, enhanced plant invasion, loss of rare native species, and altered soil invertebrate communities [START_REF] Hendrix | Pandora's Box contained bait: the global problem of introduced earthworms[END_REF] Predatory flatworm (Arthurdendyus triangulatus)

In British islands and Faroe islands, depleting lumbricid earthworm populations, reducing soil porosity and drainage, increasing waterlogging, increasing domination by Juncus, and reducing mole density

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Invasive bivalves Providing shelter and substrate, altering sediment chemistry, grain size, and organic matter content by sediment reworking, and increasing light penetration by filter feeding [START_REF] Sousa | Non-indigenous invasive bivalves as ecosystem engineers[END_REF] Sphaeroma quoyanum (isopod) In California, creating galleries that reduce sediment stability and increase erosion, ultimately converting saltmarshes to mudflats [START_REF] Talley | Habitat utilization and alteration by the burrowing isopod, Sphaeroma quoyanum, in California salt marshes[END_REF] Prevention can occur at different stages, such as constricting pathways, intercepting movements at borders, and assessing risk for intentional imports. Improved ballast water treatment exemplifies pathway constriction. Mid-ocean ballast-water exchange for ships heading to freshwater ports can reduce freshwater zooplankton concentration in ship tanks by 99% [44]. Interception programs can reduce propagule pressure of potential invaders. In Europe between 1995 and 2004, over 80% of the 302 intercepted non-native insect species were not established in Europe [START_REF] Roques | Tentative analysis of the interceptions of non-indigenous organisms in Europe 1995-2004[END_REF]. New Zealand has intercepted at least 27 non-native mosquito species, including important disease vectors [START_REF] Derraik | A survey of the mosquito (Diptera: Culicidae) fauna of the Auckland Zoological Park[END_REF]. Stringent biosecurity can bring huge economic benefits (Box 1). For instance, the Australian plant quarantine program provides savings by screening out putative invaders. Even after accounting for lost revenue from the few non-weeds that might be excluded, screening could save the Australian economy US$1.67 billion over 50 years [START_REF] Keller | Risk assessment for invasive species produces net bioeconomic benefits[END_REF].

From early action to eradication Whatever the prevention effort, some species enter any jurisdiction; it is important to detect them and respond quickly as such actions are decisive in preventing invasions [START_REF] Jarrad | Improved design method for biosecurity surveillance and early detection of non-indigenous rats[END_REF][START_REF] Kaiser | Spatial economic analysis of early detection and rapid response strategies for an invasive species[END_REF]. Early detection can be improved by innovative tools, such as monitoring for environmental DNA [START_REF] Ficetola | Species detection using environmental DNA from water samples[END_REF]. Molecular approaches are increasingly used to monitor invasions in vulnerable environments [START_REF] Chown | DNA barcoding and the documentation of alien species establishment on sub-Antarctic Marion Island[END_REF]. Early detection allows for cost-effective removal. For introduced plants in New Zealand, early extirpation costs on average 40 times less than later attempts to extirpate widely established populations [START_REF] Harris | Estimating the Benefit of Early Control of all Newly Naturalised Plants[END_REF]. The contrast in timing of management between invasions of the Mediterranean and California by the alga Caulerpa taxifolia is telling (Box 1).

Prompt removal is also ecologically less risky than later interventions. Eradicating well-established invaders can cause surprises, such as release of another, previously suppressed non-native [53]. Thus, substantial research is required about the ecosystem role of a longstanding invader before eradication is attempted. This precaution does not apply to populations detected early, which will not have established strong interspecific relations within the invaded community.

Eradication technologies have improved dramatically. Of more than 1000 attempted eradications, 86% succeeded, including several long-standing invasions [START_REF] Genovesi | Are we turning the tide? Eradications in times of crisis: how the global community is responding to biological invasions[END_REF]. Avoidance of non-target effects has also improved [53]. Eradication can be cheaper than long-term management. For instance, 1 year of removing coypu (Myocastor coypus) in Italy would cost more than twice as much as the entire successful British eradication campaign [START_REF] Panzacchi | Population control of coypu Myocastor coypus in Italy compared to eradication in UK: a cost-benefit analysis[END_REF]. Eradication increasingly helps threatened species recover. It has improved the

Box 1. Costs and benefits: lessons from statistics

Reducing propagule pressure pays: biosecurity in New Zealand and Australia Stringent biosecurity based on risk assessment, as applied in New Zealand and Australia, has significantly reduced the number of invasions. Europe and New Zealand had similar invasion rates of non-native mammals through the 19th century, but no invasions occurred in New Zealand after public perceptions shifted and biosecurity policies were adopted (Figure I).

Rapid response versus procrastination for an invasive Pacific alga

In both Europe and California, the invasive Pacific alga, Caulerpa taxifolia, was quickly detected in small patches. In California, a US$7 000 000 eradication effort mounted within 6 months of discovery succeeded in 2 years [START_REF] Woodfield | taxifolia from Agua Hedionda Lagoon and Huntington Harbour, California, Southern California Caulerpa Action Team 97[END_REF]. In the Mediterranean, procrastination for several years allowed the species to spread to thousands of hectares off the coasts of Spain, France, Monaco, Italy, Croatia, and Tunisia [97], and it is now ineradicable with current technology. Red List conservation status of 11 bird, five mammal, and one amphibian species [START_REF] Mcgeoch | Global indicators of biological invasion: species numbers, biodiversity impact and policy responses[END_REF]. However, reinvasion risk can remain high, necessitating ongoing monitoring and occasional intervention. Such costly programs might not be deemed high priorities unless they are part of broader programs aimed at maintaining biodiversity and sustainable resource use (Box 2).

From long-term management to restoration For cases where eradication fails or is not attempted, longterm management has improved, with more ambitious targets than a mere decade ago. New technologies demonstrate that long-term management of invaders is neither futile nor necessarily damaging to non-targets. For example, a synthetic larval pheromone attracting anadromous adult sea lampreys is replacing barriers and lampricides in parts of the Great Lakes [START_REF] Fine | Isolation and biological activity of the multi-component sea lamprey migratory pheromone and new information in its potency[END_REF]. A 'Super Sucker' vacuum device enabled removal of two invasive algae from Hawaiian coral reefs [START_REF] Pala | Invasion biologists suck it up[END_REF]. 'BioBullets,' minute beads containing atomized potassium chloride coated with non-ionic surfactant, helped remove zebra mussels from UK water facilities [START_REF] Aldridge | Microencapsulated BioBullets for the control of biofouling zebra mussels[END_REF]. When non-native populations have long been present, management is more complicated because costs tend to be greater, probability of success lower, and stakeholders might favor maintaining the invader. Examples of the complexities of their management include ship rats that affect seabirds of Mediterranean islands [START_REF] Martin | Black rats, island characteristics and colonial nesting birds in the Mediterranean: current consequences of an ancient introduction[END_REF] and some agricultural weeds [START_REF] Williamson | Measuring the impact of plant invaders in Britain[END_REF]. Decisions on managing longstanding populations must be case-specific and entail the best information on invasion impact, likelihood of success and recovery, management methods, and possible non-target impacts.

Removing or reducing an invader often does not suffice to re-establish native communities and ecosystems, and so active restoration can be crucial [START_REF] Gaertner | Insights into invasion and restoration ecology: time to collaborate towards a holistic approach to tackle biological invasions[END_REF]. The many advances in restoration science are beyond the scope of this paper; Box 2 provides an example integrating invasion management with restoration.

Challenges posed by how society perceives invasions

An unresolved issue for managing invasions is articulating biodiversity conservation with other social concerns. Such concerns depend on how the public perceives a phenomenon, and perceptions are shaped by multiple factors of which scientific knowledge is only one. This underscores the need for invasion scientists to transfer knowledge effectively.

Perception of non-native species: a multifaceted process Perceptions by society of introduced species can change. For example, a century ago, citizens and government agencies collaborated to introduce deer to the Haida Gwaii archipelago (British Columbia, Canada) as a source of meat [START_REF] Golumbia | History and current status of introduced vertebrates on Haida Gwaii[END_REF]. Today, government agencies consider deer a problem, and the local community increasingly deplores the impact of deer on vegetation [START_REF] Golumbia | History and current status of introduced vertebrates on Haida Gwaii[END_REF]64].

For the public to perceive an impact, a visible effect by a visible invader is usually critical. Impacts belowground or underwater are not as easily recognized as those aboveground. Landscapes overwhelmed by kudzu vines are striking, whereas disruption of soil organisms and processes by introduced earthworms did not attract significant attention until research revealed they can transform North American forests [START_REF] Hendrix | Pandora's Box contained bait: the global problem of introduced earthworms[END_REF]. Similarly, direct threats to endangered endemic or charismatic species attract attention, whereas gradual changes in abundance and distribution of common species or ecological properties tend to pass unnoticed. Invasion scientists need to alert the public and policymakers to subtle or non-obvious impacts.

The appearance and reputation of the invader also matter, often independently of its impact. Plans to kill introduced ungulates (e.g., feral domestic animals such as horses in the USA, camels in Australia, or deer on islands) or charismatic species such as mute swans in North America or gray squirrels in Europe often encounter opposition from the public and from animal defenders, whereas campaigns against invertebrates face no such resistance.

Communicating with society: a need for clarification New concepts, sometimes produced by science, can also influence public perception. For example, the suggestion that local biodiversity can be maintained or even enriched in the face of invasions [START_REF] Sax | Species invasions and extinction: the future of native biodiversity on islands[END_REF] imparts a positive message that can lead to invasions being seen more as opportunities than as problems. So does the proposition that 'novel ecosystems' including non-native species can provide ecosystem services equivalent to those of nativedominated ecosystems [START_REF] Hobbs | Novel ecosystems: implications for conservation and restoration[END_REF]. The public does not necessarily perceive that maintenance of local biodiversity by virtue of invasions is often at the expense of global biodiversity (e.g., Box 1, Hawaiian avifauna) or that criteria for designating an ecosystem 'novel' have not

Box 2. Carefully planned restoration improves invasion management: The Zena Forest Restoration Program

Zena Forest (Willamette Valley, Oregon) exemplifies restoration in the Pacific Northwest, where oak woodlands occupy interior valleys between the Coast and Cascade mountain ranges. Oregon white oak (Quercus garryana) is fire-adapted and grows in pure stands or with Douglas fir (Pseudotsuga menziesii), ponderosa pine (Pinus ponderosa), bigleaf maple (Acer macrophyllum), and other trees depending on site conditions and past timber exploitation. Oregon white oak is not shade-tolerant and, without fire, is overtopped by native Douglas fir and grand fir (Abies grandis), as well as by various introduced plants [100].

Part of Zena Forest (120 ha) was purchased in 2009 by Willamette University for restoration research and education. To restore oak savanna, most competing Douglas firs are removed or killed (and left as snags). All non-native cherry (Prunus spp.) are removed, along with other invaders (e.g., blackberry (Rubus sp.), scotch broom (Cytisus scoparius) and English hawthorn (Crateagus laevigata)). One goal is to identify 'legacy' oaks and clear competing trees, so that the oaks become wide crowned and have more wildlife value owing to larger acorn crops and development of cavities for small mammals and birds [100].

Young oak stands are thinned to 100-125 trees per ha. Invasive brush species are mechanically removed or treated with herbicides. Heavy mowing and prescribed burning reduce fuel loads and open areas for native grasses and wildflowers. Following reintroduced burning, seeds of native grasses and forbs are sown to restore native prairie. This is important because two butterfly species endangered in the Willamette Valley need these plants for food and reproduction. Ongoing monitoring and evaluation make this a valuable pilot site for similar restoration in the region.

been provided, and quantification of services such ecosystems provide has barely begun [START_REF] Mascaro | Novel forests maintain ecosystem processes after the decline of native tree species[END_REF].

To conceptualize their research objects and to communicate with the public and land managers, invasion scientists, as do other scientists [START_REF] Nagel | The Structure of Science -Problems in the Logic of Scientific Explanation[END_REF], developed metaphors. They have been accused of relying on a strongly normative, loaded vocabulary, borrowing military images [70] or worse, using xenophobic and racist expressions [71]. However, use of terms such as 'invasion' to emphasize the spread of many phenomena considered harmful is commonplace (cf. metaphors used in public health and economics). In invasion science, as in other contexts, researchers talk about 'invasions' because what is observed really is reminiscent of armies moving. Media reports on biological invasions often attract readers' attention with military metaphors. However, invasion scientists, indeed all scientists, should beware of value-laden vocabulary to avoid the risk of sterile ideological arguments.

The charge of xenophobia, fear of strangers because of their origins, is of a different nature. As stated, populations of non-native species are not problematic because they are not native per se, but because they are more likely than natives to cause ecological damage. The recommendation to prevent introductions and to beware of newly established populations is consistent with the precautionary principle, which has complex ethical and social implications. It comes with incentives to kill organisms solely because of the 'potential' problems that they could pose. These can be sentient beings for which one might have ethical concerns [START_REF] Perry | Improving interactions between animal rights groups and conservation biologists[END_REF]. It reduces people's freedom to keep non-native plants and animals. It also limits economic activities that can release invaders (e.g., horticulture or the pet trade). These costs are the price to be paid to avoid further devastating impacts on ecosystems and human well-being and increasing global homogenization of ecosystems. The wish to maintain the global diversity of native communities and ecosystems has nothing to do with xenophobia. On the contrary, it stems from principles similar to those that defend the right for every human society to retain its cultural distinctiveness, as proclaimed by the Council of Europe [73] and UNESCO [74].

The way forward

From the obvious to the subtle and pervasive As a cross-disciplinary field linking with many other fields [START_REF] Pys ˇek | Invasive species, environmental change and management, and health[END_REF], invasion science will continue to provide important insights into areas as diverse as conservation biology, evolutionary biology, population ecology, ecosystem ecology, global change science, and restoration ecology. However, above all, invasion science will lead to improved understanding of how adding a single species to a community can greatly modify biodiversity and ecosystem functioning.

Invasion science must develop better metrics for quantifying and categorizing impacts [16,[START_REF] Vila | How well do we understand the impacts of alien species on ecosystem services? A pan-European, cross-taxa assessment[END_REF] to improve prioritization of management and risk assessments. Economists require such quantified information for valuing impacts and courses of action in the cost-benefit analyses of individual species that are a pillar of invasion economics [75]. Attempts to mitigate invasion impacts on ecosystems and services that they provide will have to account for the entire range of impacts and complexity of interspecific interactions. This will require invasion science to focus not only on loss of charismatic species globally, or changes in populations locally, but also on the consequences of invasions for regional diversity. These often pertain to the distribution and abundance of obscure community members, to changes in trophic networks that can affect ecosystem functioning, and to concepts such as diversity homogenization at the landscape scale. Few studies have focused on these subjects despite their potential importance to biodiversity conservation. This comprehensive approach remains the core of understanding and mitigating the consequences of biotic homogenization [START_REF] Ville ´ger | Homogenization patterns of the world's freshwater fish faunas[END_REF].

From principles to applications: bridging the gaps Biosecurity policies and strategies must be updated regularly to reflect new findings [START_REF] Pys ˇek | Invasive species, environmental change and management, and health[END_REF]. Policies on invasions are in fact solidly based on science. The Convention on Biological Diversity (decision VI/23) incorporated findings of invasion science to define its guiding principles on invaders, ranking prevention, early detection and rapid response, and eradication as noted above. The same principles were again stressed by the G8 Environment in the Charter of Syracuse of 2009 and by leaders of the most influential world conservation organizations [START_REF] Lambertini | Invasives: a major conservation threat[END_REF], who called for strengthening, not weakening, the struggle against invasions.

However, whereas key principles on how to respond to invasions are well advanced, their application remains limited. For management, the key challenge is to bridge the gap between growing scientific understanding of impacts and management action. This will require better integration of ecological perspectives and knowledge with socioeconomic considerations and human perceptions of invasions. For instance, most cost-benefit analyses simply attempt to detail costs and benefits of particular invasions post facto. Therefore, they do not encompass prevention, the preferred management strategy [75]; neither do they address the ongoing costs and benefits of different possible policy decisions regarding invasions in general (Courtois P. Mulier C., and Salles, J-M. personal communication). Economists cannot advance in this direction without information from invasion scientists on the gamut of impacts and possible strategies to deal with them. With such knowledge, economists could then inform the public about economic consequences of various courses of action, just as ecologists inform them about ecological consequences, but it is the public and its policymakers who must define societal objectives and decide how to achieve them.

From colliding worldviews to practical solutions Some controversies about invasion management are rooted in divergent ethical frameworks. Three main worldviews collide in a 'triangular affair' [START_REF] Callicott | Animal Liberation: a triangular affair[END_REF]. Ecocentrists focus on ecological entities or processes. They readily admit that invasions should be avoided and that, when it is feasible, invaders should be eradicated to protect biodiversity. Anthropocentrists believe only humans deserve direct moral consideration. They do not worry about ecological impacts of invasions unless these also drive economic or social damages. Zoocentrists accord equal moral consideration to every sentient being. They oppose sacrificing the interests of individual animals for the sake of human interests or biodiversity per se and have often opposed eradication plans. This triangle can be 'squared' by adding a fourth corner, surely under-represented, of biocentrists, who care for every individual living being, sentient or not. It would be vain for scientists to try to settle such a debate, but anyone engaged in invasion management should pay attention to these underlying ethical issues [START_REF] Perry | Improving interactions between animal rights groups and conservation biologists[END_REF].

One way to proceed towards consensus and social agreement would be to acknowledge the legitimacy of these different ethical commitments and to try to overcome theoretical disagreements through collective search for practical solutions to problems related to specific invasions [START_REF] Maris | From adaptive management to adjustive management: a pragmatic account of biodiversity values[END_REF]. Indeed, what is shared by the three (or four) corners of the affair is a kind of absolutism regarding moral principles. However, real people in real life usually compromise among these poles depending on the specifics of situations. Moral concerns are more a collective elaboration of norms from the ground up than purely deductive applications of theoretical principles. By honestly and respectfully engaging in this debate, scientists and managers might not only influence public perceptions but also test and improve their own moral positions.

Understanding invasions in a changing world Finally, two major challenges must be overcome related to how scientists and the public perceive introduced populations, their consequences, and their management. First is the need to shift attention further from dominant focus on the properties of invading organisms to how anthropogenic changes in ecosystems facilitate many invasions (e.g., [START_REF] Gaertner | Insights into invasion and restoration ecology: time to collaborate towards a holistic approach to tackle biological invasions[END_REF][START_REF] Macdougall | Are invasive species the drivers or passengers of change in degraded ecosystems?[END_REF] but see [81]). Such a shift can lead to new ways to prevent invasions or to mitigate consequences of ongoing ones, for example through grazing or water management policies. Second is the need to convey information to the public about the range, scope, and consequences of less obvious effects of invasions, such as those affecting ecosystem processes. Scientists are also well positioned to elucidate the complexities of invasions and to explore realistic management options. Although most invasion scientists endorse a normative commitment towards biodiversity, their proper role as scientists, in terms of public discourse, is to educate citizens in a way that informs debate within society about how to think about and manage invasions.

Figure 1 .

 1 Figure 1. Management strategy against invasive species. The optimal strategy evolves with time since introduction, with management efficiency decreasing and management costs increasing with time since introduction.

Figure I .

 I Figure I. Numbers of non-native mammal species established in Europe and New Zealand over eight centuries (compiled by P. Genovesi and M. Clout).

Table 1 .

 1 Examples of ecosystem and community transformations by invasive consumer populations

	Species	Transformation

Acknowledgments

The workshop that spawned this paper was sponsored by CEFE-CNRS, the University of Montpellier (UM2) and the Re ´gion Languedoc Roussillon. P.P. was supported by grant no. P504/11/1028 (Czech Science Foundation), long-term research development project no. RVO 67985939 (Academy of Sciences of the Czech Republic), institutional resources of Ministry of Education, Youth and Sports of the Czech Republic, and acknowledges the support by Praemium Academiae award from the Academy of Sciences of the Czech Republic. D.A.W. was supported by a Wallenberg Scholars award, E.G.B. by the Spanish Ministry of Science (projects CGL2009-12877-C02-01 and CSD2009-00065), M.V. by the Spanish Ministry of Science projects CGL2009-7515 and CSD2008-00040 and the Junta de Andalucı ´a RNM-4031, and F.C. by the French ANR (2009 PEXT 010 01). We thank three anonymous reviewers for constructive comments, Jean-Michel Salles, Pierre Courtois, and Chloe ´Mulier for discussion, and Be ´renge `re Merlot for assistance with references.


Cécilia Damon

Antoine Liutkus

Alexandre Gramfort

Slim Essid

NON-NEGATIVE MATRIX FACTORIZATION FOR SINGLE-CHANNEL EEG ARTIFACT REJECTION

Keywords: EEG, artifact removal, nonnegative matrix factorisation, source separation, Gaussian model

New applications of Electroencephalographic recording (EEG) pose new challenges in terms of artifact removal. In our work we target applications where the EEG is to be captured by a single electrode and a number of additional lightweight sensors are allowed. Thus, this paper introduces a new method for artifact removal for single-channel EEG recordings using nonnegative matrix factorisation (NMF) in a Gaussian source separation framework. We focus the study on ocular artifacts and show that by properly exploiting prior information on the latter, through the analysis of electrooculographic recordings, our artifact removal results on single-channel EEG are comparable to the results obtained with the classic multi-channel Independent Component Analysis technique.

INTRODUCTION

Electroencephalographic (EEG) recordings capture a mixture of endogenous brain activities and extraneous environmental and physiological artifacts such as power grid noise, eye movements, heart beat or muscle activities. These artifacts make it difficult to analyze and interpret EEG data since they tend to overlap with the cerebral signals of interest, which are generally of lower energy than the artifacts [START_REF] O G Lins | Ocular artifacts in EEG and event-related potentials I: Scalp topography[END_REF]. Consequently researchers (in the neuroscience and biomedical engineering communities) have devised EEG artifact removal techniques that have proven effective for particular sources of artifacts such as eye movements or heart beats [START_REF] Croft | Removal of ocular artifact from the EEG: a review[END_REF][START_REF] T-P | Removing electroencephalographic artifacts by blind source separation[END_REF]. However many such techniques can be considered as being mainly geared towards experimental (especially medical) setups. Indeed, the sources of artifacts are generally intentionally limited, especially by requiring that the subject wearing the EEG device remain as steady as possible, and the processing occurs offline, possibly requiring a user's intervention to complete the artifact decontamination process. The latter is in particular true for one of the most popular approaches which relies on Independent Component Analysis (ICA) [START_REF] Jung | Extended ica removes artifacts from electroencephalographic recordings[END_REF].

Nevertheless new applications of EEG recording, for instance brain-computer interfaces or human-activity monitor-ing, pose new challenges in terms of artifact removal as they call for fully automatic techniques, that would be additionally amenable to real-time processing. A few proposals have been made along this line in previous works where two types of methods can be distinguished. The first exploit various statistical properties of the artifacts in the time-and frequencydomains [START_REF] Levan | A system for automatic artifact removal in ictal scalp EEG based on independent component analysis and Bayesian classification[END_REF][START_REF] Mammone | Automatic artifact rejection from multichannel scalp EEG by wavelet ica[END_REF], the second rely on prior knowledge about either the artifacts [START_REF] James | Temporally constrained ICA: an application to artifact rejection in electromagnetic brain signal analysis[END_REF][START_REF] Molla | Multivariate emd based approach to eog artifacts separation from EEG[END_REF] or the signal of interest [START_REF] Yuan Zou | Automatic EEG artifact removal based on ica and hierarchical clustering[END_REF]. The former are specific to standard artifacts whereas the latter can integrate different kinds of prior information.

A common major drawback to such techniques, which is actually common to all ICA-based approaches, is the fact that they can only handle overdetermined mixtures, which entails using many EEG electrodes (at least as many as the sources of artifacts plus one for the useful information). This is clearly prohibitive for the general public applications that we are interested in, where the EEG setup is to be maintained as light as possible and ideally be limited to a single electrode, while allowing the use of other types of lightweight sensors, for example Electromyographic (EMG), Electrocardiographic (ECG), or inertial measurement sensors.

Therefore attempts at single-channel EEG artifact removals have been made [START_REF] James | Extracting multisource brain activity from a single electromagnetic channel[END_REF]. Our work falls in this category.

We propose a novel automatic artifact removal scheme, inspired by the current state-of-the-art in underdetermined audio source separation [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation. Audio, Speech, and Language Processing[END_REF][START_REF] Févotte | Notes on nonnegative tensor factorization of the spectrogram for audio source separation : statistical insights and towards self-clustering of the spatial cues[END_REF][START_REF] Liutkus | Gaussian processes for underdetermined source separation[END_REF], exploiting nonnegative matrix/tensor factorization (NMF) in a probabilistic Gaussian framework.

While NMF has already been used for EEG-feature extraction [START_REF] Lee | Group nonnegative matrix factorization for EEG classification[END_REF], its use for EEG artifact removal within a Gaussian source separation framework is to our knowledge completely novel. Our method is further described in Section 2 before an experimental validation is proposed in Section 3 where our approach is compared to the classic ICA method on real EEG recordings. While these experiments focus on artifacts related to eye movements and blinks (exploiting Electrooculographic (EOG) recordings) being a strong source of contamination, it is important to note that the advantage of our method is its potential to handle many concurrent sources of artifacts simultaneously even when a single EEG electrode is used for recording, provided that some auxiliary signals describing the temporal activations of those sources of artifacts can be captured, which is often easily realized.

It is worth mentioning that our approach is advantageous compared to regression techniques used in numerous works on EEG artifact rejection [START_REF] Gratton | A new method for off-line removal of ocular artifact[END_REF] which required the use of "clean" recordings of sources of artifacts. In particular when treating ocular artifacts, the EOG recordings used to estimate the artifact signals tend to be contaminated by the EEG signals which are then subtracted from the brain signals of interest, hence causing a serious difficulty for such regression techniques.

NMF-BASED ARTIFCAT REJECTION

We hereafter explain how nonnegative matrix factorisation is used in order to perform EEG artifact rejection following a probabilistic blind source separation paradigm. We first expose the theoretical foundations and the general model underlying the separation procedure adopted, then we describe how the latter is specifically applied to our particular data configuration, that is single-channel EEG accompanied with auxiliary signals describing the sources of artifacts.

Probabilistic model and general separation procedure

For the sake of generality, we suppose that there are I observable time-series x (t, i), each x (•, i) corresponding to one of the EEG sensors. For a given sensor i, we assume that each x (•, i) is the sum of J underlying signals ỹ (•, i, 1) , . . . , ỹ (•, i, J) which are called latent components in this study. Hence, we simply have :

∀ (t, i) , x (t, i) = J j=1 ỹ (t, i, j) . (1)
Thus, model (1) can be understood as stating that the observed I time-series can actually be decomposed as the sum of J latent sets of I time-series. The objective of the processing under study then becomes to extract those J sets. That way, each x (•, i) of the observed EEG data will be decomposed into its J constituent {ỹ (•, i, j)} j=1,...,J latent components. In this paper, a distinction is made between stochastic processes such as ỹ (•, i, j) or x (•, i) and their realizations ỹ (•, i, j) and x (•, i) which are written in bold type.

In the following, we will make use of a Time-Frequency (TF) representation of the signals considered. More specifically, x (•, •, i) will denote the Short Term Fourier Transform (STFT) of the mixture x (•, i), so that x (f, n, i) ∈ C is its spectrum at frequency bin f for frame index n. Similarly, y (f, n, i, j) denotes the STFT of the i th channel of latent component j at TF bin (f, n). All signals are supposed to have the same number F of frequency indices and the same number N of frames.

Following [START_REF] Liutkus | Gaussian processes for underdetermined source separation[END_REF], all {ỹ (•, i, j)} i,j are here supposed to be independent and Locally Stationary Gaussian Processes (LSGP, see [START_REF] Liutkus | Gaussian processes for underdetermined source separation[END_REF]). The main attractive feature of this model is that computations can be readily performed in the STFT representation. Indeed it can be be shown [START_REF] Benaroya | Audio source separation with a single sensor[END_REF][START_REF] Cemgil | Prior structures for Time-Frequency energy distributions[END_REF][START_REF] Liutkus | Gaussian processes for underdetermined source separation[END_REF] that Minimum Mean Squared Error (MMSE) estimates of the latent components are then given through WIENER filtering by : ŷ (f, n, i, j) = P (f, n, i, j)

J j=1 P (f, n, i, j) x (f, n, i) (2)
where P (f, n, i, j) > 0 is the Power Spectral Density (PSD) of the i th channel of component j at TF bin (f, n).

This separation procedure is very simple to implement. One just has to compute the STFTs of the observed EEG signals x (•, i), and then compute (2) for all (f, n, i, j). Separated time series are then simply recovered through an inverse STFT procedure, which consists in inverse Fourier transforms followed by overlap-add procedures.

The main issue with the proposed model is of course the important number F N IJ of its parameters P (f, n, i, j). Still, one of its most interesting features is the way prior knowledge can be taken into account in order to strongly reduce the number of its parameters. Indeed, several facts can be considered for modeling P :

• Even if the I different channels of each latent component are modeled as independent, their Power Spectral Densities may be strongly related. In this study, we will assume that :

P (f, n, i, j) = Q ij P (f, n, j) , (3)
with Q ij ≥ 0. Equation (3) amounts to writing that all channels of a given component j share the same PSD P (f, n, j) up to a nonnegative scaling factor Q ij .

• For one given latent component j, we may assume that its PSD P (f, n, j) as it appears in (3) is further structured so that it simply consists of one given spectral template W j that is modulated by a time-varying activation gain H j . This can be written :

P (f, n, j) = W j H ⊤ j f,n , (4)
where W j and H j are F × 1 and N × 1 nonnegative vectors.

As can be seen, these two simplifications lead to a reduction of the number of parameters from F N IJ to (F + N + I) J, which is quite remarkable. We are then left with the problem of modeling the PSD of observed signals x (•, i) using a Nonnegative Tensor Factorization (NTF [START_REF] Cichocki | Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation[END_REF]), which simplifies to Nonnegative Matrix Factorisation (NMF) in the single-channel case. Hence, though we here focus on the latter situation, the method that we present can be also used for multi-sensor data.

NMF decomposition

A fact which is now acknowledged in the litterature [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF][START_REF] Liutkus | Gaussian processes for underdetermined source separation[END_REF] is that learning a NMF model such as (4) through maximum likelihood estimation is equivalent to minimizing the Itakura-Saito (IS) divergence1 between the power spectrogram of the observations and the model :

Ŵ , Ĥ, Q = argmin W,H,Q f,n,i d IS   |x (f, n, i)| 2 J j=1 W f j H nj Q ij   , (5)
where W j and H j have been gathered as the J columns of matrices W and H of respective dimensions F × J and N × J. Very efficient iterative algorithms exist which permit to estimate W , H and Q that minimize [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF]. The reader is referred to studies such as [START_REF] Févotte | Algorithms for nonnegative matrix factorization with the beta-divergence[END_REF][START_REF] Liutkus | Informed source separation through spectrogram coding and data embedding[END_REF][START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation. Audio, Speech, and Language Processing[END_REF] for the details of those algorithms.

The NMF problem may also be solved using other cost functions such as the Euclidean distance (i.e. l 2 norm) and the Kullback-Leibler divergence commonly used in the NMF litterature [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF]. In our experiments (presented in Section 3) the use of the Itakura-Saito cost has been assessed by comparison to results obtained based on Euclidean cost functions.

The main issue with this approach lies in the fact that the latent components obtained that way, even if they correctly sum up to the observed mixtures, do not necessarily correspond to the latent components we are looking for. To address this issue, we propose an approach based on supervised initialization of the model parameters.

Informed NMF initialization

One important feature of the current work is the use of auxiliary signals characterizing the sources of artifacts to aid their rejection in single-channel EEG analysis. This is achieved by initializing the learning process with results of the NMF decomposition of such auxiliary signals.

In contrast, single-channel source separation is not possible with ICA which is constrained to be used in (over)determined settings. Still, ICA can equally benefit from the initialization of components to be extracted with the available auxiliary signals, which is actually what we do when we compare our method to ICA. However, the latter requires the addition of a new EEG channel for each new source of artifact so that the minimum number of EEG channels that need to be used to handle L sources of artifacts must be L + 1 (one channel per artifact component plus one for EEG useful information). Using NMF source separation, this restriction is removed and we are able to integrate multiple known artifact sources in the learning process with a single EEG channel.

We proceed as follows. As a first step, we perform a NMF decomposition on each reference source providing K init spectral components W init and activation gains H init . Then we use these activations, H init , to initialize the NMF decomposition of the EEG data with K components such that K > K init . This strategy has proven more effective than the alternative consisting in using W init instead, as it better copes with the variability of the spectral patterns characterizing the artifacts (captured in W init ) across different sensors. The remaining K -K init components of the EEG NMF decomposition are randomly initialized. Once this NMF model is learned, the artifact and decontaminated EEG signals can be easily reconstructed through WIENER filtering.

EXPERIMENTS AND RESULTS

We now present results on real EEG signals corrupted by ocular artifacts. We mainly aim at proving the efficiency of NMF in removing ocular artifact in single channel EEG analysis comparing to 2 channel-based ICA source separation.

Validation procedure

Constrained FastICA and NMF source separation methods are here used with a given EOG reference channel to automatically remove ocular artifacts. The EEG data are public2 and were acquired at Martinos Center for Biomedical Imaging on a single subject with a 60 EEG-electrode cap combined with a 306-channel MEG Neuromag Vectorview system in a magnetically shielded room. Simultaneously, ocular movements were recorded with a vertical EOG channel. The experiment consists of audio and visual stimuli. The data was recorded at a rate of 600 Hz. We worked on 2 particular EEG channels located to the front of the cap being highly contaminated by eye movements.

As described in Section 2.3, for both methods initialization is performed with the given artifact reference signal to guide the source learning process. For FastICA, this merely consists in initializing a component of the mixing matrix with the EOG signal and the other components are generated randomly. For NMF, the initialization is done as described in 2.3.

The validation procedure includes two steps : a training step during which the hyperparameters of each source separation method are learned on one half of the dataset and a test step during which the best hyperparameter is tested on the other half of the dataset. Both centered EEG and EOG data are split in two sets to perform the initialization. Five models with 100 iterations each have been learned on both EEG-and EOG-based NMF models and only those with the smallest cost-function value have been selected.

While we necessarily estimated only 2 components for ICA, we have been able to test a range of hyperparameters for NMF by varying the number of components assessed to the artifact on the one hand (i.e. 2, 4, 6, 8), and to EEG sources on the other hand (i.e. 4, 6, 8, 16).

To compare ICA and NMF results, we used two similarity measures, the mutual information and the correlation, and defined two criteria. The first criterion assesses the similarity between the estimated artifact and EOG signals while the second assesses the EEG denoising quality by computing the ratio of the similarity measure between the noise-suppressed and EOG signal to the similarity measure between the original and EOG signals. Below, we present the results of this comparison.

Results

We only show the NMF results obtained with the Itakura-Saito cost function (that corresponds to maximum likelihood inference in our Gaussian framework). Our tests with NMF using the l 2 norm have completely failed. This can be explained by a nice feature of the Itakura-Saito cost, that is its scale invariance, which makes it more robust to the varying dynamics of EEG and EOG signals. In order of appearance, the blue signal is the original signal, the red signal is the denoised EEG signal and the last signals include the EOG signal in black dotted line and the estimated artifact signal in red solid line.

Both methods have succeeded in extracting the eye movement artifact signal and thus in denoising the EEG signal. The bar graph and the signals visual inspection prove that the NMF approach with one channel is as efficient as the FastICA method with two channels.

CONCLUSION

In this study we showed how NMF can address the problem of EEG artifact removal even in underdetermined settings, namely when the number of EEG channels is less than the number of brain sources and noisy signals (heart beat, eye blinks, etc.). Our experimental results on real data are promising for modern applications of EEG with lightweight devices more and more widely available on the market. Future work will consider many concurrent sources of artifacts including less-studied ones occurring when the subjects are in motion.

ACKNOWLEDGMENT

This work was supported by the DGA and the DGCIS under the MEEGAPERF project.

Fig. 1 .

 1 Fig. 1. FastICA signal decomposition on two frontal EEG channels.

Fig. 2 .

 2 Fig. 2. NMF signal decomposition on a single EEG channel with eight EEG components and four artifact components. In order of appearance, the blue signal is the original signal, the red signal is the denoised signal and the last signals include the EOG signal in black dotted line and the estimated artifact signal in red solid line.

Fig. 3 .

 3 Fig. 3. Mutual information-based and correlation-based denoising measures for each method, ICA in red bars and NMF in green bars. Denoising measures are averaged over channels for ICA.

The Itakura-Saito divergence between two nonnegative scalars a and b is defined as d IS (a | b) = a b -log a b -1.

http://martinos.org/mne/