domain: info sde

text:
Gauthier Roussilhe
email: gauthier.roussilhe@student.rmit.edu.au

Anne-Laure Ligozat

Sophie Quinton

A long road ahead: a review of the state of knowledge of the environmental effects of digitization

Keywords: Sustainability, ICT, Environmental Footprint, Digitization

In this brief review, we report on the state of knowledge and its limitations regarding the environmental footprint of the digital sector. While the final energy consumption and the related carbon footprint of information and communication technologies (ICT) are well studied, other environmental factors and some phases of the life cycle are still poorly understood. New connected equipment and services currently being deployed make such assessments even more complex. In addition, more research is needed on the indirect effects of ICT (i.e. substitution, optimization and various rebound effects). Indeed, recent reports tend to ignore or minimize the negative effects of digitization. Because indirect effects depend on external factors such as regulations, prices, socio-cultural contexts, etc., extrapolations are extremely uncertain. Methodological elements have been proposed to perform such evaluations, but there is still a long way to go.

Introduction

While the environmental effects of digitization have been studied since the 1990s [START_REF] Hilty | Environmental informatics as a new discipline of applied computer science[END_REF], many questions are still open. The current environmental emergency and the need to transition rapidly towards more sustainable societies, combined with the ever increasing development, deployment and use of information and communication technologies (ICT), are making this topic particularly important today. This review intends to provide a broad overview of the current state of knowledge in the field. As it is often the case (see for example [START_REF] Horner | Known unknowns: Indirect energy effects of information and communication technology[END_REF] and [START_REF] Bieser | Assessing indirect environmental effects of information and communication technology (ICT): A systematic literature review[END_REF]), we will first discuss the direct environmental effects of ICT, namely those that result from the life cycle of the underlying equipment (enduser devices, networks and data centers). We will then address indirect effects, i.e. the (positive or negative) environmental effects of the digitization of other sectors and activities, e.g. through efficiency gains or rebound effects. We conclude with a discussion of gaps in knowledge and future research directions.

Estimating the direct effects of digitization

So far, most global assessments of the direct environmental effects of ICT have focused on electricity consumption (including manufacturing and use phases) and greenhouse gas (GHG) emissions (also called carbon footprint) -leaving out other major effects such as material or water footprint, but also pollutions. As a result, our review covers the same limited scope.

The most prominent publications on this topic are: Andrae & Edler [START_REF] Andrae | On global electricity usage of communication technology: trends to 2030[END_REF] which is known to be outdated and superseded by Andrae [START_REF] Andrae | New perspectives on internet electricity use in 2030[END_REF], Malmodin & Lundén [START_REF]The energy and carbon footprint of the global ICT and E&M sectors 2010-2015[END_REF] and Belkhir & Elmeligi [START_REF] Belkhir | Assessing ICT global emissions footprint: Trends to 2040 & recommendations[END_REF]. While they are based on the same breakdown of ICT equipment into data centers, telecommunication networks and end-user devices, these studies have different perimeters. By unifying the scopes considered in [START_REF] Andrae | On global electricity usage of communication technology: trends to 2030[END_REF], [START_REF]The energy and carbon footprint of the global ICT and E&M sectors 2010-2015[END_REF] and [START_REF] Belkhir | Assessing ICT global emissions footprint: Trends to 2040 & recommendations[END_REF], Freitag et al. [START_REF] Freitag | The real climate and transformative impact of ICT: A critique of estimates, trends, and regulations[END_REF] argue that GHG emissions from the ICT sector ranged from 2.1 to 3.9% of global emissions in 2020 (1.2 to 2.2 GtCO2e). Carbon footprint estimates and projections are computed using a mix of top-down views based on global numbers (worldwide data traffic, hardware shipments, etc.) and bottom-up estimates obtained mostly using life cycle assessments (LCAs).

The wide range of ICT's carbon footprint estimate, and most importantly the disagreement over the associated trend -Malmodin & Lundén [START_REF]The energy and carbon footprint of the global ICT and E&M sectors 2010-2015[END_REF] believe that it will stagnate in coming years, while Andrae [START_REF] Andrae | New perspectives on internet electricity use in 2030[END_REF] anticipates an increase -is explained by differences in the scope chosen, e.g. whether to include IoT, cryptocurrencies and other new services, in assumptions (about energy efficiency, ICT decarbonization etc.) but also in extrapolation methods: Malmodin & Lundén project trends based on equipment sales while Andrae & Edler's projections are based on the evolution of data traffic. While the former is too narrow in scope to anticipate future uses and services, the latter is known to be inherently flawed because increasing traffic does not necessarily lead to a similar increase in footprint due to improved equipment efficiency. In a recent paper, Pasek et al. [START_REF] Pasek | The world wide web of carbon: Toward a relational footprinting of information and communications technology's climate impacts[END_REF] identify six key factors that make estimating the carbon footprint of ICT and its future trends complex or even impossible: access to industry data, bottomup vs. top-down assessments, system boundaries (scope), geographic averaging, functional units (e.g. KWh/GB or KWh/subscriber), and energy efficiencies.

Data centers

There is an ongoing debate specifically about data centers, in particular regarding whether the observed and expected increase in data traffic will translate into an increase in data center electricity use1 . Recent work by Masanet et al. [START_REF] Masanet | Recalibrating global data center energyuse estimates[END_REF] estimates worldwide electricity consumption of data centers at 205 TWh in 2018, a modest 6% increase compared to 2010, despite a ten-fold increase in data traffic. In contrast, based on data sets from the European and German data center industry, Hintemann and Hinterholzer [START_REF] Hintemann | Energy consumption of data centers worldwide[END_REF] estimated worldwide energy consumption of data centers at around 400 TWh in 2020. The difference between these two figures can be explained by differences in the estimated number of hyperscale data centers (which are particularly efficient), and "differences in scope (e.g. including or excluding crypto mining), methodologies and assumptions" according to the International Energy Agency (IEA), which recently estimated that data center electricity consumption in 2021 was between 220 and 320 TWh [START_REF] Iea | Data Centres and Data Transmission Networks -Analysis[END_REF] (without including cryptocurrencies).

In addition to electricity, water consumption and related water stress have been pinpointed as a major issue for the data center industry [START_REF] Mytton | Data centre water consumption[END_REF][START_REF] Siddik | The environmental footprint of data centers in the United States[END_REF], as well as the renewal rate of IT equipment which is likely to increase its manufacturing footprint [START_REF] Gupta | Chasing carbon: The elusive environmental footprint of computing[END_REF].

Networks

Like for data centers, studies on networks are generally focused on electricity consumption as the use phase represents a significant part of their carbon footprint (around 80%) [START_REF] Malmodin | The electricity consumption and operational carbon emissions of ICT network operators 2010-2015[END_REF]. Malmodin and Lundén [START_REF] Malmodin | The electricity consumption and operational carbon emissions of ICT network operators 2010-2015[END_REF] assessed the carbon footprint of ICT networks, up to 169 MtCO2e and 242 TWh in 2015. By synthesizing most of the available studies on this subject, Coroamȃ [START_REF] Coroama | Investigating the inconsistencies among energy and energy intensity estimates of the internet. metrics and harmonising values[END_REF] estimates this consumption at 340 TWh in 2020. He points out the lack of coherence of the available results, e.g. a divergence factor of 5-26 between bottom-up and top-down studies. The IEA estimates that energy consumption of networks was between 260 and 340 TWh in 2021 [START_REF] Iea | Data Centres and Data Transmission Networks -Analysis[END_REF].

However, these studies do not account for manufacturing impacts that are yet to be better understood. Given that the carbon footprint of the 700,000 5G base stations installed in China was estimated at around 17±5 MtCO2e in 2020 [START_REF] Ding | Carbon emissions and mitigation potentials of 5G base station in China[END_REF] -accounting for manufacturing, transport, construction and use -with the use phase accounting for an estimated 9.8 MtCO2e, such impacts are clearly significant.

End-user devices

The environmental footprint of end-user devices is generally assessed through LCAs. The global footprint is then obtained in a bottom-up fashion by estimating the number of devices of various types that are manufactured (based on sales number), in use (based on estimates of the average lifespan of such devices) and discarded in a given year, and multiplying these numbers by the corresponding impact of each type of device. Clément et al. [START_REF] Clément | Sources of variation in life cycle assessments of smartphones and tablet computers[END_REF] performed a review of available LCAs of smartphones and tablets which showed the importance of the manufacturing phase, and specifically that of integrated circuits (ICs), in the environmental impact of devices. There are no up-to-date data for computers and displays (see [START_REF] Bhakar | Life cycle assessment of CRT, LCD and LED monitors[END_REF][START_REF] Epa | Life-cycle assessment of desktop computer displays: Summary of results[END_REF][START_REF] Teehan | Sources of variation in life cycle assessments of desktop computers[END_REF] for older studies).

Looking more precisely at key components such as ICs, Bardon et al [START_REF] Bardon | DTCO including sustainability: Power-performance-area-costenvironmental score (PPACE) analysis for logic technologies[END_REF] and [START_REF] Pirson | The Environmental Footprint of IC Production: Meta-Analysis and Historical Trends[END_REF] estimate that the environmental footprint of more advanced technological nodes (below 10nm) is increasing, possibly due to new manufacturing processes. Beyond these familiar devices, a whole new range of devices are appearing with the advent of the Internet of Things (IoT). Pirson and Bol showed that the carbon footprint of IoT edge devices with different hardware profiles can vary up to a factor of 158× [START_REF] Pirson | Assessing the embodied carbon footprint of IoT edge devices with a bottom-up life-cycle approach[END_REF]. Many other devices are poorly accounted for. For instance, electronic devices (ICs, sensors, etc.) used in other sectors, such as the car industry, are understudied, although they are at the core of new mobility solutions. Kemp et al. [START_REF] Kemp | Life cycle greenhouse gas impacts of a connected and automated SUV and van[END_REF] or Sudhakar et al. [START_REF] Sudhakar | Data centers on wheels: Emissions from computing onboard autonomous vehicles[END_REF] estimate that electronic equipment could represent an important part of the life cycle GHG emissions of an autonomous electric vehicle.

Challenges due to new digital technologies

Emerging digital technologies such as blockchain or Artifical Intelligence (AI) represent new challenges for assessing the environmental effects of ICT. For instance, as the use of AI is becoming ubiquitous, the environmental impact of training natural language processing (NLP) models has become a source of concern -see for example Strubell et al. [START_REF] Strubell | Energy and policy considerations for modern deep learning research[END_REF] and Patterson et al [START_REF] Patterson | The carbon footprint of machine learning training will plateau, then shrink[END_REF][START_REF] Patterson | Carbon emissions and large neural network training[END_REF]. While these studies roughly follow the division between data centers, networks and end-user devices, the terms used and the areas of focus differ. AI/ML applications look at the training phase generally done in data centers but rarely integrate the use phase on various devices, called in this case the inference phase. Furthermore, the manufacturing phase is not always taken into account.

Gupta et al. [START_REF] Gupta | Chasing carbon: The elusive environmental footprint of computing[END_REF] emphasize the increasing part of manufacturing in the life cycle of computing and AI in particular while Wu et al. [START_REF] Wu | Sustainable AI: Environmental implications, challenges and opportunities[END_REF], Kaack et al. [START_REF] Kaack | Aligning artificial intelligence with climate change mitigation[END_REF] and Ligozat et al. [START_REF] Ligozat | Unraveling the hidden environmental impacts of AI solutions for environment life cycle assessment of ai solutions[END_REF] advocate for more extensive evaluations of AI systems, taking into account the whole life cycle of equipment, the different phases of AI, and its indirect effects. New digital technologies and their uses influence the expansion of digital infrastructures and could increase the absolute global environmental footprint, regardless of the efficiency of the equipment that supports them. These effects must therefore be studied in a broader framework than that of an isolated technology deployment.

Estimating the indirect effects of digitization

Different classifications have been proposed to describe the indirect environmental effects of digitization. In 2006, Hilty et al. [START_REF] Hilty | The relevance of information and communication technologies for environmental sustainability -a prospective simulation study[END_REF] proposed a classification between 1st order effects related to the life cycle of a product or service, 2nd order effects related to the efficiency and substitution effects of a service, and 3rd order effects related to behavioral and structural changes brought about by a service. Hilty and Aebischer [START_REF] Hilty | ICT for sustainability: An emerging research field[END_REF] proposed a new version of this classification in 2014 with the LES model, with the L standing for life-cycle impact, the E for enabling impact (benefits of using ICT services) and S for structural impact (socioeconomic impacts of ICT). Horner et al. [START_REF] Horner | Known unknowns: Indirect energy effects of information and communication technology[END_REF] summarize the different categorizations that have been proposed in the past: indirect effect of a single service (efficiency, substitution, direct rebound), indirect effect of complementary services (indirect rebound), economy-wide and society-wide indirect effects. For the sake of simplicity, we use here the first classification used by Hilty et al. by separating the enabling effects, i.e. the benefits resulting from the use of an ICT service, and the rebound effects, whether they are direct, indirect or socio-economic wide.

While enabling effects are well understood in the form of efficiency or substitution, rebound effects require further explanation. A rebound effect is identified as a positive impact (less resources consumed, more efficient process) on a service or product which leads to an increased demand on the latter or beyond. Coroamȃ and Mattern [START_REF] Coroamȃ | Digital reboundwhy digitalization will not redeem us our environmental sins[END_REF] identify several types of rebound effect that could apply to digital technologies: a direct rebound effect (optimization of a product leads to an increased consumption), a backfire (when the rebound effect is more impactful that the original solution), indirect rebounds (when the rebound happens out of the original scope), time rebound (linked to the cost of time-saving technologies), and macro-level rebounds (economy/society-wide). Moreover, Bieser and Hilty notice that structural effects (macro-level rebounds) are mostly ignored in the literature. [START_REF] Bieser | Assessing indirect environmental effects of information and communication technology (ICT): A systematic literature review[END_REF]

Enabling effects

To the best of our knowledge, no scientific publication so far has provided an assessment of the global (current or potential) enabling effects of digitization for GHG emissions mitigation2 . Still, two claims have been widely shared amongst stakeholders and tech companies since 2015: that digitization could reduce global GHG emissions by up to 20% (12 GtCO2e) in 2030 according to GeSI [START_REF] Gesi | SMARTer2030 -ICT solutions for 21st century challenges[END_REF], a partnership of companies from the ICT sector; and that mobile communication technologies enabled the avoidance of 2.1 GtCO2e in 2018 according to GSMA [START_REF] Gsma | The enablement effect[END_REF], the association representing the interests of mobile operators worldwide. Both reports show major methodological flaws and their results should be used with extreme caution, as we explain below.

The GeSI report proceeds by extrapolation from industrial case studies and conversations with experts. For a given digital solution, the authors select a few GHG abatement levers and corresponding impacts, weigh them by an adoption rate (different for OECD and non-OECD countries) and compare the result to a baseline scenario based on IPCC and WRI projections. As an example, they estimate a potential of avoided emissions for mobility at 3.6 GtCO2e by 2030, based on the hypothesis that digitization could reduce car production by -15%, fuel consumed by -30%, and all freight (maritime, road, air) between -20 and -30%. Similarly, GSMA's estimates are obtained by multiplying, for each case studied, an avoided emission factor by the number of smartphones in use (or machineto-machine connections) to obtain a global estimate. The largest source of avoided emissions relates to accommodation sharing (AirBnB, Couchsurfing), up to 221 MtCO2e avoided in 2018. This result is based on an avoided emission factor of 56.2 kgCO2e per smartphone, obtained through the answers of 6,000 smartphone users to an online survey and a single study commissioned by AirBnB in 2014 comparing GHG emissions of shared accommodation vs. hotel. Of the 2.1 Gt of potential emissions avoided, around 10% are thus obtained by a global extrapolation of the responses of this sample of 6.000 people.

The complexity of performing macro-studies, pointed out in [START_REF] Malmodin | Considerations for macro-level studies of ICT's enabling potential[END_REF], is also addressed by Malmodin and Coroamȃ [START_REF] Malmodin | Assessing ict's enabling effect through case study extrapolation-the example of smart metering[END_REF] who pinpoint shortcomings due to extrapolations from case studies -in particular, the fact that the representativeness of these case studies is critical but rarely discussed. For instance, a pilot project or a case study works somewhere under specific conditions, this result cannot be extrapolated until you manage to reproduce the same conditions: a smart thermostat works better in an well insulated house, a bike-sharing app is more effective once you develop a biking infrastructure and so on. Furthermore, there is no guarantee that indirect effects, whether positive or negative, are maintained over time, so it is better to keep assumptions and a baseline scenario conservative.

Other issues that appear even at smaller scales, which are discussed by Coroamȃ et al. [START_REF] Coroamȃ | A methodology for assessing the environmental effects induced by ict services: Part I: Single services[END_REF], Bergmak et al. [START_REF] Bergmark | A methodology for assessing the environmental effects induced by ict services: Part II: Multiple services and companies[END_REF] and Rasoldier et al. [START_REF] Rasoldier | How realistic are claims about the benefits of using digital technologies for GHG emissions mitigation?[END_REF] include: ignoring life-cycle impacts (i.e. the environmental footprint of the ICT service/product under study), and structural impacts (rebound effects, society-wide effects); and overlooking the fact that the success or failure of an ICT service depends on factors outside of the ICT sector (policies, price, culture, etc.). Methodologies to address this are proposed from the perspective of a single service in [START_REF] Coroamȃ | A methodology for assessing the environmental effects induced by ict services: Part I: Single services[END_REF] and [START_REF] Rasoldier | How realistic are claims about the benefits of using digital technologies for GHG emissions mitigation?[END_REF], and from the perspective of multiples services and companies in [START_REF] Bergmark | A methodology for assessing the environmental effects induced by ict services: Part II: Multiple services and companies[END_REF].

In the light of all these issues, it appears clearly that the existing grey literature focuses on positive effects, while ignoring or minimizing possible negative outcomes.

Rebound effects

Rebound effects are particularly hard to track as they can apply from the micro to the macro scale. Coroamȃ and Mattern [START_REF] Coroamȃ | Digital reboundwhy digitalization will not redeem us our environmental sins[END_REF] note that "the mechanisms behind rebound effects in general, and thus of digital rebound as well, are essentially non-technical in nature. Their roots reside in economics and in human behavior." Following this observation, Lange et al. [START_REF] Lange | Digitalization and energy consumption. does ICT reduce energy demand?[END_REF] study if ICT reduces energy demand, including direct and indirect effects. Their conclusion points out two energy-reducing effects (increase of energy efficiency and sectoral change / tertiarization) and two energy-increasing effects (direct effects and economic growth).

At a smaller scale, it is possible to observe rebound effects. For instance, based on English travel patterns from 2005 to 2019, Caldarola and Sorrell [START_REF] Caldarola | Do teleworkers travel less? evidence from the English national travel survey[END_REF] estimate that while teleworkers take fewer trips than non-teleworkers, they travel farther, suggesting that teleworking does not reduce travel. On a similar topic, Ward et al. [START_REF] Ward | Air pollution, greenhouse gas, and traffic externality benefits and costs of shifting private vehicle travel to ridesourcing services[END_REF] study the effects of switching from private vehicle travel to transportation networks companies (TNCs) such as Uber or Lyft. Based on a Monte Carlo simulation of 100,000 passenger trips in six U.S. cities, they estimate that shifting from private vehicle travel to TNCs reduces air pollutants by 50-60% on average, but increases fuel consumption and greenhouse gases emissions by 20% due to deadheading, and external costs by 60% due to congestion, crashes and noise. Rebound effects can help us understand where digitization cannot be a solution in today's society, but studying them could also help figure out in which social, economic, technological, political conditions digitization could actually work.

Studies that focus on enabling effects generally avoid studying rebound effects because of their complexity and because they depend on behavioral or structural changes. For instance, the authors of the GeSI report [START_REF] Gesi | SMARTer2030 -ICT solutions for 21st century challenges[END_REF] discussed previously also estimate a rebound effect of 1.37 GtCO2e that is not deducted from the 12 GtCO2e total of potential avoided emissions due to the fact that "the science behind rebound is generally tricky and a matter of debate." This argument is however debatable since enabling effects such as the energy savings obtained by a smart meter (taken into account in the GeSI or GSMA estimates) are equally complex: a smart meter does not in itself guarantee energy savings, it is the potential changes in behavior enabled by the visualization of consumption data that could lead to avoided GHG emissions. There has thus been so far a double standard when it comes to modeling indirect effects. This needs to change if we want to estimate potential net effects, integrating both enabling effects and rebound effects.

Gaps in knowledge and future research directions

In its last report [START_REF] Pathak | Technical summary[END_REF], the IPCC Working Group III mentions: "At present, the understanding of both the direct and indirect impacts of digitalisation on energy use, carbon emissions and potential mitigation is limited." Our brief review of the current state of knowledge is in line with this statement.

Table 1 lists the gaps in knowledge that we have identified in Section 2 regarding the direct environmental effects of digitization. Data from manufacturers, whether American, European or Asian, are not available for public research, resulting in the environmental factors related to manufacturing being understudied. End of life (EoL) is even less understood because of the lack of traceability of end-of-life equipment and the geographical dispersion of formal and informal treatment of e-waste, while significant pollution can take place at the local level. Besides, because global studies have focused on the ICT sector, there are no environmental assessments of ICT products that go directly to other sectors (e.g. chips for the automotive industry). Finally, providing a reliable estimate of the environmental footprint of a digital service is still a complex exercise, especially with new services being promoted every year (metaverse, etc.).

Indirect effects of digitization are even more complicated to assess. Global studies from the industry mostly focused on positive impacts, presenting only partial and unreliable estimates. Research about indirect effects of digitization raises many methodological questions that are still open. We have summarized in Table 2 the gaps in knowledge and recommendations for future work discussed in Section 3.

While ICT are at the core of most critical infrastructures and human activities, the role they can play to mitigate GHG emissions remains unclear. More research is needed to identify under which multi-faceted conditions (taking into account territorial specificities) a digital solution can have positive effects, and under which it has negative effects, meaning it is better not to deploy it or even to "un-digitize" it depending on the situation. Last but not least, one should acknowledge that our times require urgent action and that answers to many questions raised in this paper may not get scientifically sound answers quickly enough. One crucial question that remains is then: what do we need to know to make decisions? Table 1: Summary of the gaps in knowledge regarding direct effects of digitization, based on [START_REF]The energy and carbon footprint of the global ICT and E&M sectors 2010-2015[END_REF], [START_REF] Belkhir | Assessing ICT global emissions footprint: Trends to 2040 & recommendations[END_REF], [START_REF] Andrae | New perspectives on internet electricity use in 2030[END_REF] and [START_REF] Freitag | The real climate and transformative impact of ICT: A critique of estimates, trends, and regulations[END_REF]. [START_REF] Bieser | Assessing indirect environmental effects of information and communication technology (ICT): A systematic literature review[END_REF], [START_REF] Coroamȃ | A methodology for assessing the environmental effects induced by ict services: Part I: Single services[END_REF], [START_REF] Rasoldier | How realistic are claims about the benefits of using digital technologies for GHG emissions mitigation?[END_REF], and [START_REF] Coroamȃ | Digital reboundwhy digitalization will not redeem us our environmental sins[END_REF].

Extrapolation method

Global extrapolation from small scale leading to misestimates Use of a conservative extrapolation factor / refer to a present effect / study based on random sampling scheme / avoid global extrapolations or limit the scale (R)

Rebound effects

Mostly ignored Need for future assessments of rebound effect including more than just ICT service usage (R)

  Time perspective including efficiency gains, induced demand, etc. (R) External social, economic, environmental factors inclusion Mostly ignored or implicit hypotheses External conditions in which an ICT solution can have and cannot have a positive impact (G/R) Input data Small-case and uncertainty due to lack of availability Better open data policies and governance / limited extrapolation of uncertain data (R)

Table 2 :

 2 Summary of the gaps in knowledge and methodological recommendations regarding indirect effects of digitization, based on

		Historical focus	Gaps of knowledge
	Life cycle stages	Use phase	Raw material extraction, manufacturing, maintenance, EoL
	Environmental impact	Final energy consumption, GHG	4 studied out of 19 recommended by the European Union Product
	indicators	emissions	Environmental Footprint
	Infrastructures	Data center, networks, end-user	Edge computing, IoT / IIoT, telecom satellites (no data), network
		devices	deployment, etc.
	Services	Data transfer (esp. streaming)	IA / ML life cycle, AR / VR related services, blockchains (esp.
			manufacturing), etc.
	Geographical areas	North America, Europe	Asia for manufacturing, EoL (worldwide)
	Sectors	ICT / E&M sector	All sectors except ICT / E&M sector

Note that this leaves out other life cycle impacts such as energy consumption due to manufacturing.

Note that a few scientific studies do exist at the macro (but not global) scale, e.g. Malmodin et al.[START_REF] Malmodin | Considerations for macro-level studies of ICT's enabling potential[END_REF].

* This article gives a clear view on what can be expected from the rebound effects of digital technologies, both from a positive and negative point of view.


Christine Reynier

Catherine Bernard

Mrs Dalloway

Virginia Woolf

Texte

« Le Texte

Jonathan Bloom

V S Pritchett

William Trevor

et la réflexion politique dans Aaron's

For possibly, if fiction is, as we suggest, in difficulties, it may be because nobody grasps her firmly and defines her severely. She has had no rules drawn up for her, very little thinking done on her behalf. [. . .] But this part of his duty, if it is his duty, Mr. Forster expressly disowns. He is not going to theorise about fiction except incidentally. (Woolf 1994, 601) In this review of E.M. Forster's Aspects of the Novel, 'The Art of Fiction', 1 Virginia Woolf calls for a theory of fiction that does not exist yet in 1927 and that Forster, according to her, fails to provide. It can be argued that in the innumerable reviews and essays she herself wrote from the beginning to the end of her career and that have been somewhat underrated by critics for a long while, 2 Woolf provides such a theory even if she challenges the very definition of the term 'theory' by embedding it in a genre, the modern essay, which 'admits [. . .] of sudden boldness and metaphor' (Woolf 1994, 218) rather than of abstraction. 3 In those texts (now partly collected by Andrew McNeillie), while reviewing books, passing judgment or making general statements about literature, she gradually and very indirectly comes to define the ideal art of writing while championing, like most creative writers who are also critics, her own art of writing. And although she posits herself as a critic and reader of other writers' fiction, the reader she embodies may also appear as the ideal reader of her own fiction. Woolf's ambiguous position is therefore challenging from the start and it is her theory of writing and reading, as it is scattered in her essays, that I would like to examine here, knowing full well that the scope of her essays is so vast that my attempt is bound to be fragmentary and will only come as a modest complement to the already existing studies of Woolf's essays. 4 What sort of a theoretician is Woolf? What does her theory of fiction mainly consist in? These are the questions I shall address by soliciting Woolf's essays and comparing them with her contemporaries', namely E.M. Forster, Ford Madox Ford, Joseph Conrad, D.H. Lawrence or T.S. Eliot's, thus grounding them in their cultural context while showing how ground-breaking they are in the light of her successors'.

1

Woolf is well-known for condemning in her diary 'the damned egotistical self, which ruins Joyce and Richardson' (Woolf 1953, January 26, 1920). Similarly, in her essays, she voices her admiration for Greek literature, 'the impersonal literature' (Woolf 1994, 39) which is read and appreciated because 'we have either no sense or a very weak one of the personality of the Greek dramatists' (Woolf 1994, 274), as well as for writers who, unlike Hemingway, refrain from revealing their sex in their writings: '[t]he great writers lay no stress upon sex one way or the other' (Woolf 1994, 454), she writes in 'An Essay in Criticism'. Her call for a selfless, ungendered 5 and anonymous literature, that she will reiterate down to her last unfinished essay, 'Anon', sounds very much like T.S. Eliot's theory of impersonality, minus the religious rhetoric of asceticism, and is close to E.M. Forster's conception of fiction as 'unsigned' (Forster 1979, 84): 'all literature tends towards a condition of anonymity, and [. . .], so far as words are creative, a signature merely distracts us from their true significance. [. . .] The signature, the name, belongs to the surface personality, and pertains to the world of information, it is a ticket, not the spirit of life' (Forster 1979, 82, 87).

However, in other essays, especially 'Personalities', Woolf seems to contradict herself when she argues in favour of personality and recognises that Shakespeare is much read because readers can find the man in his writings even if his work is not exactly a form of personal literature: 'most people have precisely that personal feeling for him which I think they have not for Aeschylus' (Woolf 1972, 275); she goes on showing that Shakespeare is one of '[t]hese great artists who manage to infuse the whole of themselves into their works, yet contrive to universalise their identity so that, though we feel Shakespeare everywhere about, we cannot catch him at the moment in any particular spot' (Woolf 1972, 275). Similarly in Thomas Brown's prose, the signature of the author is always traced on the page and 'the figure of the author' 6 (Woolf 1988, 156) is never far away: 'Somewhere, everywhere, now hidden, now apparent in whatever is written down is the form of a human being' (Woolf 1988, 156); and yet autobiographical facts are discounted. Personality is here redefined as a form of universality -or https://journals.openedition.org/ebc/9416 3/11 commonality -rather than the expression of an individuality and comes close to the impersonal. In the end, both impersonality and personality can be at work in a literary text. 7 Woolf comes up with a complex theory of impersonal personality that encompasses both T.S. Eliot's theory of impersonality and opposing theories and can be said to span the developments of theories ranging from Roland Barthes's 'Death of the Author' to his own later re-inscription of the author in Roland Barthes par Roland Barthes, from structuralism to post-structuralism and even deconstruction, if we think of Jacques Derrida's concept of signature. 8 Woolf seems to sum up beforehand years of debate on the question of authorship and authority and delineates a space in-between personality and impersonality, an ambivalent and paradoxical space that implicitly establishes paradox as one of the main tenets of modernist fiction.

If for the early Barthes, the birth of the reader signals the death of the author, 9 for Woolf, author and reader are not seen in antithetical and exclusive terms. Much like E.M. Forster who acknowledges that there is, between reader and author, a 'sense of cooperation' that turns into 'an infection', a transformation of the one by the other, 10 Woolf believes in the 'happy alliance' (Woolf 1994, 215) of writer and reader that both inhabit the ambivalent personal/impersonal space. 11 Not only should the reader be the writer's 'fellow-worker and accomplice' (Woolf 1972, 2) but there should also be a total empathy between them: 'Do not dictate to your author; try to become him' (Woolf 1972, 2); 12 and their roles should be interchangeable: 'Perhaps the quickest way to understand the elements of what a novelist is doing is not to read, but to write' (Woolf 1972, 2). Such a conception of writing and reading suggests some form of exchange and circulation between writer and reader, which is confirmed by what Woolf writes about Thomas Brown, one of her favourite writers: Browne, she remarks, is an amateur writer who is not paid for the books he writes: 'He is free since it is the offering of his own bounty to give us as little or as much as he chooses ' (Woolf 1988,159). Writing is seen as a gift the writer makes to the reader and this generates in turn the reader's generosity: 'one of the invariable properties of beauty [beautiful writing] is that it leaves in the mind a desire to impart. Some offering we must make ' (Woolf 1988, 159). Woolf's (ideal) reader is not only receptive to fiction, a passive receptacle without any preconceptions, 13 a 'non-preoccupied mind' (Ford 271) or 'virgin mind' (Ford 273) like Ford Madox Ford's cabman's mind modelled on Tolstoy's peasant's mind, 14 he is also ready to give; a similar give-and-take logic of writing and reading has been explored by Derrida in La Dissémination 15 or by Lucette Finas who writes: 'The reader's gift to the text rests on the text's gift to the reader'. 16 If the beauty of Thomas Browne's prose is a gift, the prose of his predecessors, however different, is also one. When reading Hakluyt's book in 'Reading', the narrator's 'attention wanders ' (Woolf 1988, 148), carried along by the rhythm of the sentences and the visions of the Elizabethan age they conjure up. Like Barthes's reader, the narrator reads looking up, 17 or as Woolf writes in 'How Should One Read a Book?': 'how great a part the art of not reading plays in the art of reading ' (1988, 393). Yet the narrator's attention only wanders to explore further the unknown territories evoked in the narrative, broadening its scope, developing its potentialities while the text yields in return a rich, inexhaustible treasure to the reader: 'And so, as you read on across the broad pages with as many slips and somnolences as you like, the illusion rises and holds you of banks slipping by on either side, and glades opening out, of white towers revealed, of gilt domes and ivory minarets. It is, indeed, an atmosphere, not only soft and fine, but rich, too, with more than one can grasp at any single reading' (Woolf 1988,149). Reading Hakluyt's book further illustrates Woolf's conception of reading and writing as a gift. 'Multiplication du lu par le lisant . . .', Lucette Finas will write (Finas 14). Something which, according to Woolf, is only possible with one category of writers, the laymen who, like Chaucer, have her favour because 'we are left to stray and stare and make out a meaning for ourselves' (Woolf 1994, 32) whereas 'the priests', like Wordsworth, Coleridge and Shelley 'take you by the hand and lead you straight to the mystery' (Woolf 1994, 31). The laymen guarantee a freedom of interpretation based on 1972,248) rather than their representational power or, more exactly, from their connotative rather than denotative power, an opposition that Barthes, following the linguist Hjelmlev, was to make in S/Z in 1970 and that Woolf made as early as April 20, 1937 in a radio broadcast entitled 'Craftsmanship'. A tongue-in-cheek defence of the sign language which, with one gable or one star, as in the Michelin or Baedeker's guidebooks, can come to the point more clearly and concisely than words. This ironic plea clearly amounts to enhancing the power of words 'not to express one simple statement but a thousand possibilities' (Woolf 1972, 246), 'without the writer's will; often against his will' (248); and reading them 'allow [s] the sunken meanings to remain sunken, suggested, not stated; lapsing and flowing into each other like reeds on the bed of a river' (248). Words are shown to exceed the writer's intentionality and disseminate meaning, thus preventing reading from turning into a hermeneutic quest. 18 Reading would then be better defined as a moment of openness, and the two terms, openness and moment, are important here. Openness first: reading as a moment of openness is best synthesised in 'Reading' where the beginning of the essay presents itself as a 'reading scene'. The narrator is sitting in an armchair by the window reading while enjoying 'the view across country [. . .] from the window' so that 'it seemed as if what [he] read was laid upon the landscape, not printed' (Woolf 1988, 142). Together with outer space, time past and time present enter the space of the book; the gardener mowing the lawn 'was part of the book' as well as 'Keats and Pope behind him, and then Dryden and Sir Thomas Browne' (Woolf 1988, 142). Through the chronotope of the book, reading is seen here as a moment of silent conversation both between the space of the book and outer space and between the time of famous writers of the past and the time of present common people. The position of the narrator, sitting in the library of a large old house opening onto a garden, can be seen as a metaphor of reading as a moment of openness to the other, whether dead or alive, famous or not, and a moment of conjunction between fiction and reality. Far from adopting the paradoxical posture of Proust who associates reading with a closed space, 19 Woolf associates the reader with a space in-between the private and the public spheres and conceives of reading as a form of opening and contact, a generous disposition and an encounter with alterity. Woolf could say with Derrida: 'lire un livre, c'est-à-dire [. . .] inviter la parole d'un autre chez vous' (Derrida 1994, 256). Furthermore, in this scene, the narrator-reader metaphorically welcomes famous and obscure voices just as Woolf the reviewer accepted to review classics as well as 'the rubbish-heap' of literature (Woolf 1972, 5) and Woolf the writer wrote biographies of the famous as well as the forgotten, Elizabeth Barrett Browning or Roger Fry, servants or dogs. 20 The model reader also functions as the model writer and writing is presented as requiring the same generous disposition as reading.

9

Such openness (to the other) ensures a fully responsive reading, i.e. the ability to be surprised by the work of art; in Woolf's words, it turns reading into an 'intensification of life' (Woolf 1972, 41), a moment of 'delight' (Woolf 1972, 39) when the reader is surprised or 'bewildered'. 21 In 'On Being Ill', Woolf shows surprise has a double origin. The person who is sick is evoked lying in bed 'disinterested and able [. . .] to look, for example, at the sky' (321) and discover 'a strangely overcoming spectacle' which 'has been going on all the time without [his] knowing it' (321). The recumbent posture becomes the image of the reader's 'disinterested contemplation' 22 and rediscovery of the ordinary world, i.e. of his experience of the process of defamiliarisation operated by fiction. In a metaphorical way, Woolf defines what Viktor Schklovsky was to call, at about the same time, ostranenie or defamiliarisation. What she herself calls, in 'A Sketch of the Past', 'the shock-receiving capacity' (Woolf 1978, 83) of the writer, which is here shown to be transferable to the reader. And Attridge will write: 'the act of breaking down the familiar is also the act of welcoming the other' (Attridge 2004, 26). Incomprehensibility has an enormous power over us in illness [. . .] In health meaning has encroached upon sound. Our intelligence domineers over our senses. But in illness, with the police off duty, we creep beneath some obscure poem by Mallarmé or Donne [. . .] and the words give out their scent, and ripple like leaves, and chequer us with light and shadow, and then, if at last we grasp the meaning, it is all the richer for having travelled slowly up with all the bloom upon its wings. (Woolf 1994, 324) Illness thus sharpens the reader's senses or, more exactly, is turned into a metaphor of the writer's attempt at drawing the reader's attention to words as signifiers, to the musicality, sensuality or visual qualities of language -another form of defamiliarisation -something the poetic words 'the words give out their scent, and ripple like leaves, and chequer us with light and shadow', perform while pointing at it. Woolf's desire for defamiliarisation together with her insistence on the aesthetic power of words is reminiscent of the preface to 'The Nigger of the Narcissus' where Conrad defines the writer's task as to 'make [us] hear, to make [us] feel, to make [us] see!' what we had previously been insensitive to and asks for 'unswerving devotion to the perfect blending of form and substance', the only way of activating the power of 'the old, old words, worn thin, defaced by ages of careless usage' (Conrad XLIX) . It also partly echoes Ford's words about what constitutes 'the sole real province of all the arts', i.e. 'the beauty of music -that is to say, of music without much meaning, but of very great power to stir the emotions' (Ford 2002, 154). 23 On the whole reading, for Woolf, is synonymous with discovery, surprise and aesthetic involvement. Such a theory of reading, dissecting and foregrounding pleasure, paves the way for Sontag and Barthes's erotics of reading; 24 it also turns reading into a moment of being, the event of language and not in language, a Deleuzian event. 25 Taking into account Woolf's conception of reading and writing as inextricably bound together and almost interchangeable, and as mainly generous, open dispositions, we could finally define her art of fiction as eminently ethical, as offering the possibility, in Attridge's words: 'to read inventively, to respond to the inventiveness of the work in an inventive way and thus affirm and prolong its inventiveness. I would call this a responsible reading, a reading that attempts to do justice to the alterity, singularity, and inventiveness of the literary work; and it's here that I locate the ethical' (Attridge 2003, 33). And Attridge adds: 'the peculiar pleasure of the literary response (over and above all the pleasure to be gained from new information, sensuous patterning, stirring of memory, and so on) lies in this apprehension of otherness and in the demands it makes' (Attridge 2003, 35). 26 Woolf's anatomy of fiction, as delineated in her various essays, is a complex network that we have just began exploring; it seems to posit her in some ways as a follower of the Eliotian line which has constituted the critical orthodoxy for a long while. Yet we have seen that she departs from it in more ways than one and often comes close to E.M. Forster's views on literature. 27 However she cannot be said to be an adept of Art for Art's sake, a line E.M. Forster claims to defend. 28 She exceeds by far the boundaries of a self-contained fiction by clearly adopting in her essays an ethical position, involving openness and responsibility as well as aesthetic commitment.

Whether she evades an Art for Art's sake conception of fiction in other ways remains to be seen. I shall argue that in the essays dealing with literature which we are concerned with, Woolf voices her belief in other forms of literary commitment, but in an extremely submerged way, not comparable with the straightforward manner she adopts in her often commented so-called feminist and political essays, A Room of One's Own and Three Guineas. 'Craftsmanship' is one of the most telling essays in that respect. There she explains that 'Words, English words, are full of echoes, of memories, of associations' (Woolf 1972, 248). Dissemination, the 'power of suggestion' (248) of words therefore appears not simply as a formal choice in favour of open meaning in an 248) -and with the country's history: 'They have been out and about, on people's lips, in their houses, in the streets, in the fields, for so many centuries' (248), the history of the people, of everyman and everywoman. Words are the living palimpsests and memory of English history. Woolf points here not only to a phenomenon that critics, in the wake of T.S. Eliot will study under the banner of intertextuality, but also to the grounding of words and fiction in history, a historical context that precludes the autonomy of fiction.

In the same essay, Woolf adds: 'they [words] are so stored with meanings, with memories, [. . .] they have contracted so many famous marriages' (248) and she expatiates on this metaphor, further personifying words by referring to the necessity they feel to change: 'it is their nature to change' ( 251), what she also calls their 'falling in love, and mating together': 'Royal words mate with commoners. English words marry French words, German words, Indian words, Negro words, if they have a fancy' (250). The evolution of language is here linked with historical circumstances: the invasion of England, the wars she fought, and the reality of Empire. History has altered language and necessarily weighs on the writing of fiction. What Woolf also insists on in this passage is the mating of royal words with commoners, of uneducated words with educated ones as well as of Indian or Negro words with English ones. Language is depicted as a democratic space where social hierarchies are blurred ('They are highly democratic, too; they believe that one word is as good as another', 250) and where different cultures enter into a dialogue with each other. These statements are first of all an indirect way of taking into account the origin of words, the social class that coined them, the nation they were born in, i.e. their socio-cultural background, dismissing once more a conception of language and the fiction that uses it as autonomous; they also extol the virtues of impurity and point out the dialogic forces at work in language, implicitly stating the political nature of language, its submerged but nonetheless real capacity to defend a pluralistic and democratic ideal. In a very light, humorous and metaphorical manner, Woolf unmasks the unconscious of language which precludes its autonomy, something that Bakhtine and his successors will examine at length. If language enables the writer to make aesthetic choices, it also, as 'a social and historical construct' (Lecercle 1999, 223), constrains the writer in a double movement that Lecercle identifies as its 'enabling constraints' (Lecercle 1999, 116). 29 If we read her essays closely, we can see that Woolf engages in a conversation with her contemporaries while laying down the problematics of her own fiction and what has come to be called modernist fiction, presenting the latter as a complex literary phenomenon, both experimental, self-contained and conditioned by its context, capable of commitment, thus thinking through the critical orthodoxy that has long prevailed and only been recently reappraised. 30 While defining fiction in her essays and retaining the essayist's tone and the writer's style, for the reader's greatest pleasure, Woolf elaborates a form of criticism that should suffice to establish her as one of the major critics of the century; she defines a form of criticism which we could call, borrowing the term Derrida uses about reading, 31 dischronic criticism, a form of criticism which is haunted by previous and contemporary theories, but also, through a logic of invagination, by theories that were still to come. And paraphrasing Derrida's statement about Joyce, we could say about Woolf: 'Elle nous a tous lus et pillés, celle-là'. 32 In other words, her essays preclude all literary categorisation and point instead to a constant circulation between what critics have chosen to call modernism and post-modernism. Gualtieri focuses on the form of the essay, its place within the history of the genre and the generic blurrings Woolf experiments with; she is also interested in Woolf's concern for women's points of view, especially in A Room of One's Own and in her historiographic concerns, particularly in Three Guineas. In her most valuable study, Melba Cuddy-Keane scrutinizes the process of reading in Woolf's essays, first locating them in their cultural context, then focusing on some essays and underlining their pedagogic nature, the dialogic relation between reader and writer as well as 'the ideal of a classless, democratic, but intellectual readership' they promote (Cuddy-Keane 2).

5 In France, Catherine Bernard has written on 'Virginia Woolf essayiste ou l'écriture sans pédigrée' and Frédéric Regard has published a stimulating study of three major essays, 'Mr Bennett and Mrs Brown', A Room of One's Own and Three Guineas in La Force du féminin; see also my analysis of Woolf's essays on biography (Reynier 2003).

6 Woolf will develop this in her theory of androgyny in A Room of One's Own which enables her to escape the binary construction of gender.

7 Now the title of a book by Maurice Couturier.

8 Similarly Ford defines the ideal author as 'sedulous to avoid letting his personality appear in the course of his book. On the other hand, his whole book [. . .] is merely an expression of his personality' (Ford 1995, 265). Lisa Low, in her analysis of Woolf's essays, writes that 'Woolf expresses a lifelong quest to remove the personal element from her work, not because by impersonality she will transcend what is common, but because by being impersonal she will become common' (Rosenberg & Dubino 265) and concludes that in her late essay 'personalities', Woolf is not so clearly in favour of impersonality, preferring to weigh the pros and cons of personality. Such a dissociation of impersonality and personality seems to me contradictory with the notion of 'common' which includes every man, every woman and therefore, the 'I'.

9 See especially Signéponge.

10 Roland Barthes writes: 'la naissance du lecteur doit se payer de la mort de l'Auteur' (Barthes 1984, 69).

  Umberto Eco will call the openness of the text, what E.M. Forster calls 'Expansion. [. . .] Not completion'(Forster 1927, 116), what D.H. Lawrence refers to as its absence of fixedness when he writes: 'Give me nothing fixed, set, static' (Lawrence 76).Such active reading, for Woolf, derives from the suggestive power of words (Woolf

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what

  However illness keeps the reader's mind blurred and makes it difficult for him to understand what he is reading. But this disadvantage is turned upside down by Woolf who writes:

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  Bibliographieopen text. It is also connected with history, both with literary history -as in the example Woolf takes of 'the splendid word 'incarnadine' [. . .] who can use it without remembering also 'multitudinous seas'?' (

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E.M. Forster writes: 'What is so wonderful about great literature is that it transforms the man who reads it towards the condition of the man who wrote, and brings to birth in us also the creative impulse'(Forster 1979, 84).

Michael Kaufmann, in his analysis of The Common Reader, argues that Woolf's desire to give up the ego tends towards a communal 'we', 'a communal structure of artists and reader-critics' (Rosenberg & Dubino 149), and adds that Woolf 'anticipates reader-response criticism by several decades' (Rosenberg & Dubino 149). Beyond the essays included in The Common Reader, this can be extended to most of Woolf's essays.

A considerably revised version of this lecture given in 1926 at Hayes Court, a girls' school, is published in(Woolf 1988, 388-400); the wording if not the substance is different.

In 'How Should One Read a Book', Woolf defines her ideal reader thus: 'Most commonly we come to books with blurred and divided minds, asking of fiction that it shall be true, of poetry that it shall be false, of biography that it shall be flattering, of history that it shall enforce our own prejudices. If we could banish all such preconceptions when we read, that would be an admirable beginning'(Woolf 1972, 2).

Ford writes: 'Now the one sensible thing in the long drivel of nonsense with which Tolstoi misled this dull world was the remark that art should be addressed to the peasant' (270) and adds that 'in Occidental Europe the non-preoccupied mind-which is the same thing as the peasant intelligence-is to be found scattered throughout every grade of society' (271) but is certainly not an English gentleman, the latter being too much concerned with conventions. The cabman is the very opposite and embodies the ideal reader's 'non-preoccupied mind'. 'On Impressionism', Second article, Poetry and Drama 2.6 (December 1914).

See especially 'La Pharmacie de Platon' about reading and writing being one and about what he calls 'adding' or 'supplement': 'ajouter n'est pas ici autre chose que donner à lire. [. . .] Le supplément de lecture ou d'écriture doit être rigoureusement prescrit mais par la nécessité d'un jeu, signe auquel il faut accorder le système de tous ses pouvoirs'(Derrida 1972, 80).

This is my translation. Finas writes: 'Le don du lecteur au texte suppose le don du texte au lecteur' (Finas 14).

Barthes writes about 'lire en levant la tête' (Barthes

1984, 33).19 'Dissemination' would then synthesise more adequately than 'polysemy' what Woolf describes in this essay. Derrida defines dissemination in the following way: 'S'il n'y a donc pas d'unité thématique ou de sens total à se réapproprier au-delà des instances textuelles, [. . .] le texte n'est plus l'expression ou la représentation [. . .] de quelquevérité qui viendrait se diffracter ou se rassembler dans une littérature polysémique. C'est à ce concept herméneutique de polysémie qu'il faudrait substituer celui de dissémination'(Derrida 1972, 319).

E.M. Forster compares Woolf and Sterne's capacity for bewilderment(Forster 1974, 12), a term Woolf herself uses about reading Chekhov in 'The Russian Point of View'. For a discussion of this term, see my 'Virginia Wolf's Ethics of the Short Story, Etudes Anglaises, forthcoming.

A Kantian concept Fry, Bell or Woolf resorted to so as to evoke the contemplation of a work of art or, in Woolf's case, the reading of a book. On that subject, see Froula, especially 1-32.

'Les Jeunes and Des Imagistes', first published in Outlook 33 (9 May 1914) 653. Ford admired this quality in Imagiste poetry which had found a new form for the narrative of emotion whereas the novel was still groping for one, according to him.

Barthes, in 1973, makes a distinction between le '[t]exte de plaisir: celui qui contente' and the '[t]exte de jouissance: celui qui met en état de perte [. . .] met en crise son rapport au langage'(Barthes 1973, 25-26). As for Sontag, she argues in favour of an art that would make us 'recover our senses' and, paraphrasing Conrad, she writes: 'We must learn to see more, to hear more, to feel more.[. . .] In place of a hermeneutics, we need an erotics of art' (Sontag 312).

See on that subject Lecercle who writes that Woolf's moments of being are 'the best illustrations I know of the Deleuzian event'(Lecercle 2002, 151).

Attridge here is close to what Derrida writes about deconstructive reading as the experience of the irreducibility of alterity; see (Derrida 1987).

See (Reynier 2005, 13-24).

In 'Art for Art's Sake' Forster defines a work of art as 'a self-contained entity' (89) possessing an 'internal order' or harmony, a 'form' of its own(Forster 1979, 94), a late conception of art which somewhat jars with his more ethical earlier practice of fictional writing.

4 See especially Leila Brosnan who challenges the dividing line between Woolf's non-fiction and fiction and draws attention to her essays and journalism 'as a body of writing that develops and reveals its own self-determining aesthetic dimensions as well as an associated historical environment' (4) and notices that 'the study of the essay and Woolf's essays in particular, is still relatively undeveloped' (10). In her discussion of Woolf's essays, she 'initiate[s] a dialogue between critical approaches' (8) rather than try to see how Woolf anticipated them.; she is interested mainly in the essay as a specific genre. The collection of articles published by C. Rosenberg and J. Dubino discusses some of Woolf's essays in a perceptive, although sometimes indirect way; when some points they discuss will surface, reference will be made to them. Elena , vol. 4: 1925-1928, ed. Andrew McNeillie, London: Hogarth P., 1994, 388-400. 'Is Fiction an Art?', The Essays of Virginia Woolf, vol. 4: 1925-1928, ed. Andrew McNeillie, London: Hogarth P., 1994, 457-65. DOI : 10.1111/0044-0124.00468 W , Virginia, 'On Being Ill', The Essays of Virginia Woolf, vol. 4: 1925-1928, ed. Andrew McNeillie, London: Hogarth P., 1994, 317-29. DOI : 10.1111/j.1741-6248.1993.00199.x W , Virginia, 'On Not Knowing Greek', The Essays of Virginia Woolf, vol. 4: 1925-1928, ed. Andrew McNeillie, London: Hogarth Press, 1994, 38-53. 'Reading', The Essays of Virginia Woolf, vol. 3: 1919-1924, ed. Andrew McNeillie, New York: HBJ, 1988, 141-61. W , Virginia, 'The Art if Fiction', The Essays of Virginia Woolf, vol. 4: 1925-1928, ed. Andrew McNeillie, London: Hogarth P., 1994, 599-603. W , Virginia, 'The Modern Essay', The Essays of Virginia Woolf, vol. 4: 1925-1928, ed. Andrew McNeillie, London: Hogarth P., 1994, 216-27. W , Virginia, 'The Pastons and Chaucer', The Essays of Virginia Woolf, vol. 4: 1925-1928, ed. Andrew McNeillie, London: Hogarth P., 1994, 20-38. W , Virginia, 'The Patron and the Crocus', The Essays of Virginia Woolf, vol. 4: 1925-1928, ed. Andrew McNeillie, London: Hogarth Press, 1994, 212-15.


Essoham Ali
email: essoham.ali@univ-ubs.fr

Kim-Hung Pho
email: phokimhung@tdtu.edu.vn

A novel model for count data: Zero-Inflated Probit Bell model with application to ecological data

Keywords: Estimation, Simulation, Probit, Zero-inflated Bell, Ecological data

A novel model for count data is proposed in this work. This new model is briefly called the Zero-Inflated Probit Bell (ZIPBell) model. The ZIPBell model can be used to simulate and analyze traditional count data and can be used well in the simulation and analysis of count data with the phenomenon that the frequency for zero occupies a very high percentage. In the scope of this study, we first provide the general formula, functions or related equations for the ZIPBell model. The results of asymptotic inferences for the ZIPBell model are also given in this work. Maximum likelihood estimator (MLE) is used to check its performance of the proposed model. According to the numerical illustrations via several simulation researches and the analysis of a practical data set, it has been seen that the MLE method has offered correct and reliable inferences. Finally, the inferences and conclusions of this study are mentioned.

Introduction

In scientific research, the topic of regression models is always interested and noticed by scientists and researchers. As we all know that, regression models can be applied in almost all areas of life and has many interesting practical applications. It can be practically applied in applied statistics, applied mathematics, economics, finance, medicine, education, engineering, social sciences and so on. Because the application of the regression model is so great, it is used a lot in the researches of scientists. Therefore in this study we are also interested in regression models.

When it comes to regression models, the first thing we are concerned in is the data for the outcome variable. Depending on the type of data, there are suitable regression models to choose for scientific research. Specifically, in this work we are interested in count data.

Count data can be understood simply that it will receive values that are natural numbers for the response variable, and this data is very common and exceedingly often encountered a lot in real data. Applications on using regression models for analysis related to count data can be found in [START_REF] Truong | Zero-inflated poisson regression models: Applications in the sciences and social sciences[END_REF].

For count data, conventional regression models comprise Poisson, Negative Binomial (NB) and Binomial are regularly performed as the first attention in regression analyses. If in that situation the count data occupies a very large number of zeros in the data set, it is often referred to as zero-inflated (ZI) count data. For the ZI count data set, the regression models as mentioned above often do not fit well enough in the analysis (see [START_REF] Lukusa | Review of zero-inflated models with missing data[END_REF]. For such data sets, the ZI count models commonly activate better than conventional regression models. The widesread and crucial ZI count models such as: ZI Poisson (ZIP), ZI Negative Binomial (ZINB), and ZI Binomial (ZIB). These models were first introduced and proposed in the researches of [START_REF] Lambert | Zero-inflated Poisson regression, with an application to defects in manufacturing[END_REF], [START_REF] Ridout | A score test for testing a zero-inflated Poisson regression model against zero-inflated negative binomial alternatives[END_REF][START_REF] Hall | Zero-inflated Poisson and binomial regression with random effects: a case study[END_REF], respectively. Moreover, a list of ubiquitous ZI models can be referred in [START_REF] Lukusa | Review of zero-inflated models with missing data[END_REF].

Recently as we know a novel model that can be employed and executed to ZI count data is ZI Bell (ZIBell) model, which was first introduced by [START_REF] Lemonte | Zero-inflated Bell regression models for count data[END_REF]. Some interesting studies on the ZIBell model can be seen in: [START_REF] Abolhassani | Zero-inflated Bell scan: A more flexible spatial scan statistic[END_REF] provided other perspectives on how to construct a ZIBell model. In addition, [START_REF] Algamal | Performance of the Ridge and Liu Estimators in the zero-inflated Bell Regression Model[END_REF] conducted a comparison of the working performance of the Liu and Ridge estimators in this model. Recently, [START_REF] Ali | Statistical Inference in a Zero-Inflated Bell Regression Model[END_REF] presented the asymptotic properties of the ZIBell model.

The multivariate ZIBell model as well as its inferences and applications can be found in [START_REF] Lemonte | Multivariate zero-inflated Bell distribution and its inference and applications[END_REF], and so on.

Indeed in some cases when doing regression analysis for real data sets, the ZIBell model may be a better fit for the data set than the more widespread ZI counting models such as ZIP, ZINB and ZIB. No need to say where far away, in the study of [START_REF] Lemonte | Zero-inflated Bell regression models for count data[END_REF] have showed that the count of the infected blood cell (IBC) data set would fit the ZIBell model better than some other ZI count models. Therefore we would like to study and learn the problems related to the ZIBell model in this work.

The ZIBell Model and Estimating Equations

The probability mass function (PMF) of the ZIBell distribution is offerred by (see [START_REF] Ali | Statistical Inference in a Zero-Inflated Bell Regression Model[END_REF]:

P (Y = y) =        π + (1 -π) exp 1 -e W (µ) , y = 0 (1 -π) exp 1 -e W (µ) W (µ) y B y y! , y > 0 (1)
where µ > 0 and π ∈ (0, 1). Note also that the original for Eqn. (1) mentioned in the study of [START_REF] Lemonte | Zero-inflated Bell regression models for count data[END_REF]. If Y follows a ZIBell distribution with the two parameters µ and π, then we desinagate that Y ∼ ZIBell(µ, π). In addition, we should know that W (.) is the Lambert function, and B y are the Bell numbers with

B n = 1 e ∞ k=0 k n k! . We let log(µ i ) = β ⊤ x i and logit(π i ) = τ ⊤ s i ,
where β = (β 1 , β 2 , ..., β p ) T , and τ = (τ 1 , τ 2 , ..., τ q ) T are parameters need to be evaluated,

x T i = [x 1 , x 2 , ...,
x n ] T , and s T i = [s 1 , s 2 , ..., s n ] T . The log-likelihood of θ = (β ⊤ , τ ⊤ ) ⊤ of the ZIBell model is given by (see, [START_REF] Lemonte | Zero-inflated Bell regression models for count data[END_REF])

n i=1 -log 1 + e τ ⊤ s i + J i log e τ ⊤ s i + exp 1 -e W (e β ⊤ x i ) +(1 -J i ) Y i log W (e β ⊤ x i ) -e W (e β ⊤ x i ) ,
where J i = 1 {Z i =0} . We should note that the MLE technique can be used optimize ℓ(θ) to get θ where the estimate of θ = (β ⊤ , τ ⊤ ) ⊤ .

3 The Proposed Model: ZI Probit Bell (ZIPBell) model

Research motivation

Note that the ZIBell model is created by performing two processes to generate the zeros. A process is usually modeled using the Bell distribution to produce a count (counting zeros), some of which may be zero. The remaining process is frequently modeled with a binary distribution in order to generate zeros. The logit model is often viewed as the easiest of the family of binary models because it is built through exponential functions. It is therefore not too difficult to understand that the logit model is frequently used in processes that use the binary model.

Besides, as we all know that the probit model is established and built on the basis of the normal distribution and this distribution is always considered an important distribution with many good and beautiful properties in statistics. For that reason, of the two steps mentioned above, one uses a binary model. We aim to use the probit model (see [START_REF] Pho | Pearson chi-squared and unweighted residual sum of square tests of fit for a probit model[END_REF] to create a new model, which is called for short: the ZI Probit Bell (ZIPBell) model. The proposed and the ZIBell models have numerous important meanings and applications in practice. Therefore, it is interesting and makes a lot of sense to propose this new regression model in the family of ZI count models.

Besides, the proposal of a new regression model helps researchers, readers or science lovers have more tools and more choices to analyze data. Because data is more and more available, both in quantity and in different forms. Hence, this proposal is very important and makes a lot of sense in the current situation and reality. Everything from formulating the general formula for the proposed model, the necessary equations for the estimation, the asymptotic properties, numerical analyzes for the proposed model are fully provided in the next subsection.

The general formula

Notice that the variables and two parameters π and µ in this section are similarly defined as in the previous part. The general formula of the ZIPBell model has the following form:

P (Y = y|x, s) = πI(y = 0) + (1 -π) exp 1 -e W (µ) W (µ) y B y y! , (2)
for y = 0, 1, 2, . . . , W (.) is the Lambert function, and B y are the Bell numbers.

The log-likelihood function

According to (1) then the ZIPBell model has the general likelihood function form:

L n (θ) = n i=1 π i + (1 -π i ) exp(1 -e W (µ) ) I(Y i =0) (1 -π i ) exp(1 -e W (µ) ) W (µ) Y i .B y Y i ! I(Y i >0) (3)
Performing the natural logarithm L(θ) in (3), one has the associated log-likelihood function of the ZIPBell model can then be expressed as follows:

log L n (θ) = n i=1 I(Y i = 0) log π + (1 -π) exp(1 -e W (µ) ) (4) + n i=1 I(Y i > 0) Y i log(W (µ)) + log (1 -π) -e W (µ) + 1 + log(B y ) -log(Y i !)
For the sake of simplicity as in the study by [START_REF] Lemonte | Zero-inflated Bell regression models for count data[END_REF], formula (5) can be rewritten as follows:

log L n (θ) = n i=1 I(Y i = 0) log π + (1 -π) exp(1 -e W (µ) ) (5) + n i=1 I(Y i > 0) Y i log(W (µ)) + log (1 -π) -e W (µ)
Note that the formula (5) is the general form for the ZIPBell model. In the case of the probability of ZI is modeled by the probit model as follows:

log(µ i ) = β ⊤ x i and probit(π i ) = F (τ ⊤ s i )
where F is the C.D.F of N (0, 1), and f is the first derivative of F . Basing on (5) the associated log-likelihood of θ = (β ⊤ , τ ⊤ ) ⊤ is given by

ℓ(θ) = n i=1 J i log F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) +(1 -J i ) Y i log[W (e β ⊤ x i )] + log[1 -F (τ ⊤ s i )] -e W (e β ⊤ x i ) . (6)
and

P (Y = 0|x, s) = π + (1 -π) exp(1 -e W (µ) ) = F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i )
3.4 Estimating Score Function (ESF)

Following to ℓ(θ) = n i=1 ℓ i (θ)
, where ℓ(θ) is given in ( 6), the ESF is offered as follows:

U F,n (θ) = 1 √ n ∂ℓ(θ) ∂θ = 1 √ n      ∂ℓ(θ) ∂τ ∂ℓ(θ) ∂β      = 1 √ n n i=1 S i (θ). ( 7
) S i (θ) = ∂ℓ i (θ)/∂θ = S T i1 (θ) , S T i2 (θ) T , S i1 (θ) = ∂ℓ i (θ) ∂τ , S i2 (θ) = ∂ℓ i (θ) ∂β i = 1, . . . , n. S i1 (θ) = f (τ T X i ) 1 -F (τ ⊤ s i ) . I(Y i = 0) -F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) , (8)
S i2 (θ) = D.e W (e β ⊤ x i ) .F (τ ⊤ s i ). I(Y i = 0) -F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) (9) + D W (e β ⊤ x i ) . Y i -1 -F (τ ⊤ s i ) W (e β ⊤ x i ).e W (e β ⊤ x i ) with D = ∂ ∂β [W (e β ⊤ x i )]. The detailed transformations of S i (θ) is offered in Appendix.
We need to know that, the way of expressing S i1 (θ) in ( 8) and S i2 (θ) in ( 9) can help us in inferring the asymptotic properties of the estimator θ. Besides, we can easily see that

E[U F,n (θ)] = 0, U F,n (

Asymptotic Properties

Before proceeding to study the asymptotic properties of θ F , some necessary conditions are needed to carry out this work. These conditions are similar to the study of [START_REF] Pho | Zero-inflated probit Bernoulli model: a new model for binary data[END_REF], they are briefly expressed as (K1) and (K2). Recall that:

U F,n (θ) = 1 √ n n i=1 S i (θ), S i (θ) = ∂ℓ i (θ)/∂θ = S T i1 (θ) , S T i2 (θ) T , S i1 (θ) = ∂ℓ i (θ) ∂τ , S i2 (θ) = ∂ℓ i (θ) ∂β i = 1, . . . , n.
where S i1 (θ) and S i2 (θ) are given in ( 8) and ( 9), respectively.

(K1) For the sake of simplicity, with any column vector a, we denote a ⊗2 = aa T , and

E [(S 1 (θ)) ⊗2
] is positive definite in a neighborhood of the true θ.

(K2) The first derivatives of U F,n (θ) w.r.t θ exist almost surely in a neighborhood of true θ.

Besides, note that in such a neighborhood, the first derivatives are uniformly bounded above by a function of (Y, X, Z), in the case of the expectations exist.

For the convenience of the following inferences, we call:

G F,n (θ) = - 1 √ n ∂U F,n (θ) ∂θ T = - 1 n n i=1 ∂S i (θ) ∂θ T .
Because every component of G F,n (θ) is a mean of i.i.d random variables, we have:

E[G F,n (θ)] = 1 n n i=1 E E -∂S i (θ) ∂θ T Y i , X i , Z i = E E -∂S 1 (θ) ∂θ T Y 1 , X 1 , Z 1 = E -∂S 1 (θ) ∂θ T = G F (θ).
By performing the weak law of large numbers (WLLN), it's not too hard to see that

G F,n (θ) p → G F (θ).
We represent θ F as the root of U F,n (θ) = 0. Theorem 1 provides the asymptotic properties of θ F and the proof for this theorem is discussed in detail in Appendix.

Theorem 1 Under (K1) and (K2)

(a) θ p -→ θ as n-→∞. (b) √ n( θ-θ) d -→ N (0, ∆ F ), where ∆ F = G -1 F (θ)Q F (θ)[G -1 F (θ)] T and Q F (θ) = E [(S 1 (θ)) ⊗2 ].
Because it's a fact that the variance of the score function is the inverse of the Fisher information matrix, thus G F (θ) = Q F (θ) hence a simple formula for ∆ F can be rewritten as: ∆ F = G -1 F (θ). The transformations to obtain expressions G F (θ) and Q F (θ) are similar in the study of [START_REF] Pho | Zero-inflated probit Bernoulli model: a new model for binary data[END_REF]. Due to the simplicity of scientific research and this research objective focusing on simulation studies and analyzing real data sets, the transformations to obtain G F (θ) and Q F (θ) are omitted in this work.

Numerical study

In this section, we assess finite-sample properties of the MLE with the first case that is both X and S are a univariate and the second case is both X and S are vectors of covariates.

Case (i) : Both X and S are a univariate. We simulate data from a ZIPBell model defined by :

µ(X) = exp(β 0 + β 1 X 1 + β 1 X 2 ) and probit(π i ) = τ 0 + τ 1 X 1 + τ 2 S 2
where X 1 , X 2 and S 2 are independently drawn from normal N (0, 1), uniform U(1, 3) and exponential E(1) distributions respectively. Parameters β and τ are chosen as β = (-0.25, -0.9, 0.3) ⊤ and τ = (-0.55, 0.7, 0) ⊤ . Case (ii) : Both X and S are vectors of covariates. Simulation design is essentially similar.

µ(X) = exp(β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X 4 + β 5 X 5 ) and probit(π i ) = τ 0 + τ 1 S 1 + τ 2 S 2 + τ 3 S 3 + τ 4 S 4 + τ 5 S 5 . X 1 , • • • , X 5 are simulated independently from normal N (0, 1), uniform U[1, 3], binomial
B(1, 0.5), normal N (-2, 1), and binomial B(2, 0.5). Covariates S 2 , • • • , S 5 are drawn independently from a normal N (0, 1), normal N (-1, 1), binomial B(1, 0.5) distributions respectively. Linear predictors are allowed to share some common terms by letting S 1 = X 2 and S 2 = X 5 . Parameters β and τ are taken as β = (0.5, -0.9, 0.3, 0.8, 0.1, -0.4) ⊤ and τ = (1.5, -0.9, -.2, 0.7, 0, 0) ⊤ . Note that the parameter τ is chosen in order to obtain the desired zero rate which is 63%.

For the two cases, our simulation results are based on N = 1000 simulated samples. We consider the following sample sizes: n = 200, 500, 1000 and 2000. For each estimator, we report the average bias, empirical standard deviation (SD), average standard error (SE), empirical root mean square error (RMSE) and empirical coverage probability (CP) of 95%level confidence intervals. Simulations are conducted using the statistical software R (R Core Team 2018), and all estimates are obtained by using the maxLik function via the Newton-Raphson algorithm (see [START_REF] Pho | Improvements of the Newton-Raphson method[END_REF].

Results. Tables 1 and2 present the simulation results for the two simulated cases. In general, the biases and RMSE are closer to zero and CPs are closer to 0.95 as the sample size increases. In most cases, the biases of β 0,n , β 1,n , β 2,n are closer to zero even for small sample sizes. Similarly, the CP of β0,n and β 1,n in case (i) needs a bigger sample size (n=500) to achieve the nominal coverage. Overall, the bias is generally slightly larger for the τ than for the β . The empirical coverage probabilities are close to the nominal confidence level. In addition, from these tables we can see that, the bias, SE SD and RMSE of all estimators decrease as sample size increases.

We assess empirically the Gaussian approximation stated in Theorem (1) by plotting normal Q-Q plots of the estimates. Figures (1) and ( 2) provide plots for case (ii) with n = 2000 and an average sample proportion of the ZI equal to 63% (plots for the other simulation scenarios are similar and are thus omitted). From these plots, the distribution of the MLE in the ZIPBell model is reasonable approximated by the Gaussian distribution.

3 -2 -1 0 1 2 3 1.2 1.6 2.0 Normal Q-Q plot for τ 1,n Theoretical Quantiles Sample Quantiles -3 -2 -1 0 1 2 3 -1.2 -1.0 -0.8 Normal Q-Q plot for τ 2,n Theoretical Quantiles Sample Quantiles -3 -2 -1 0 1 2 3 -0.4 -0.2 0.0 Normal Q-Q plot for τ 3,

Real Data Analysis

In this section, we describe an application of the ZIPBell model to the analysis of fish species occurrence records (FSOR) for surveys that were collected at different times in most of the aquatic ecosystems of Uganda. The dataset used here comes from the "Fish Species Occurrence Records for Uganda Mobilized Observation Archives", from the unpublished Mobilized Uganda Archives, available at https://www.gbif.org. The experiment is conducted on a single species, Lates niloticus. Lates niloticus, commonly known as "freshwater captain" or "Nile perch", is a fish whose flesh quality and rapid growth make it a highly prized species.

Therefore, this species is very unique and has many practical meanings. Indeed in recent years, fish farmers have turned their attention to this species, which also allows natural control of fish density. The fish is also world-famous, having been introduced to the Ugandan part of Lake Victoria in the 1950 to stimulate a declining fishery and thus contribute to the region economic development Lowe (1997). Stocks of Lates niloticus increased rapidly during the 1970, leading to the decline and even, in some areas, the complete disappearance of certain native species, including many endemic cichlids [START_REF] Witte | Species extinction and concomitant ecological changes in Lake Victoria[END_REF].

Hence it is interesting to study or learn about this issue in this study.

Next we discuss a little about how to collect the data under investigation. Note that, data were collected in the main river basins, in particular lakes Kwania, Kyoga, Nabugabo, Nakuwa, Nawampasa, Victoria and the Nile. This type of species is found in many types of freshwater, preferring warm tropical waters ( 27• N -7 • S) where they reach large sizes and are present in high densities. They are silvery in color with a blue tinge. They have a distinctive dark black eye with a bright yellow outer ring and generally weigh between 2 and 4 kg. Lates niloticus average 85 to 100 cm in length, but can reach 193 cm. Furthermore, females are generally larger than males.

Several measures were reported in this study, such as the number of Lates niloticus as the target variable. Informations useful for developing occurrences is also reported such as GPS coordinates (latitude and longitude) on the actual locations or sites where the fish were captured. The fish were captured using fishing gears such as gillnets, hooks and beach seines at specific sites of different waterbodies. A detailed description of the data can be found in [START_REF] Sidumo | Count Regression and Machine Learning Techniques for Zero-Inflated Overdispersed Count Data: Application to Ecological Data[END_REF]. Here we address the question of the influence of environmental factors on fish reproduction. Let Y i be the number of Lates niloticus species caught and we consider this as the response variable in this regression analysis. However, we can observe that most of the count data lies between 0 and 50 suggests that that a ZIPBell regression model should be used instead.

The results are fully presented in Table 3. For this example, two models: ZIPBell and ZIBell, were used to study the influence of environmental variables on the abundance and distribution of the species Lates niloticus and to perform the MLE approach to estimate the parameters of these models. Besides, AIC and BIC are introduced to compare the fitting effects of these two models. In general, as we know the lower the AIC and BIC values, the better the fitting effect of the model. It is also important to note that the estimation results obtained for the ZIBell and ZIPBell models are not very different. To be more specific, then they are only slightly superior to each other. If we focus our comments on the models under consideration, based on the AIC and BIC results provided in the last two rows of Table 3. (Indeed, the lower the AIC and BIC values, the better the corresponding model). The effects of spatial factors (latitude, longitude) are used as a measure of spatial location. The environmental factors favoring fish reproduction in Uganda are waterBody, Latitude, and Longitude. Besides, in rivers and shallow lakes, water temperature is generally little different from air temperature. In deep lakes, its distribution in the water mass is less homogeneous.

In addition, Latitude and longitude are the two main factors determining water temperature, along with the local influence of atmospheric circulation. The latter explains the abundant production of the species in the lakes of Uganda. The ZIPBell model identifies almost the same set of significant covariates (waterBody and Longitude are significant, while Latitude is not). The magnitude and sign of significant parameters are similar in both models, leading to similar interpretations. Among the environmental parameters studied, temperature is one of the main parameters explaining spatial variations in Lates niloticus.

It should be noted that the ZIBell model and the proposed ZIPbell model are based on the same specifications for the susceptibility probability π i , when it comes to interpreting the τ parameter values in these models.

Discussions and Conclusions

This study is significant in many ways, first in terms of contributing to the literary treasure, we have proposed and introduced an additional tool for data analysis. Specifically, we first proposed the ZIPBell model. This can be seen as a new, meaningful and interesting regression model, which can work well for simulation and regression analysis for ZI count data sets. In this day and age, almost any profession or field has data, and therefore the addition of a new regression model is proposed, which means more tools to analyze the data in the most scientific way. From there we will provide the most accurate conclusions. In terms of practical applications, the ZIPBell model has shown great potential to be applied to real data sets. In this study, we have also demonstrated that the proposed model: ZIPBell is more suitable for the FSOR data set than the traditional ZIBell model.

In general, this work proposed a novel model for count data. This new model is briefly called the ZIPBell model. The ZIPBell model can be used to simulate and analyze traditional count data and can be used well in the simulation and analysis of count data with the phenomenon that the frequency for zero occupies a very high percentage. We provided the general formula, functions or related equations for the ZIPBell model. The results of asymptotic inferences for the ZIPBell model were also given in this work. MLE was used to check its performance of the proposed model. According to the numerical illustrations via several simulation researches and the analysis of a practical data set, it had been seen that the MLE method has offered correct and reliable inferences.

ℓ(θ) = n i=1 J i log F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) +(1 -J i ) Y i log[W (e β ⊤ x i )] + log[1 -F (τ ⊤ s i )] -e W (e β ⊤ x i ) .
The score function of the ZIPB regression model

Note that I(Y i > 0) = 1 -I(Y i = 0) and Y i I(Y i = 0) = 0, i = 1, . . . , n.
We have

S i1 (θ) = ∂ℓ i (θ) ∂τ = ∂ ∂τ I(Y i = 0) log F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) + ∂ ∂τ I(Y i > 0) Y i log[W (e β ⊤ x i )] + log[1 -F (τ ⊤ s i )] -e W (e β ⊤ x i ) = I(Y i = 0)    f (τ T X i ) -f (τ T X i ) exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )    + (1 -I(Y i = 0)) -f (τ T X i ) 1 -F (τ ⊤ s i ) = I(Y i = 0)    1 -exp 1 -e W (e β ⊤ x i ) f (τ T X i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )    - f (τ T X i ) 1 -F (τ ⊤ s i ) + I(Y i = 0) f (τ T X i ) 1 -F (τ ⊤ s i ) = I(Y i = 0)    1 -exp 1 -e W (e β ⊤ x i ) f (τ T X i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) + f (τ T X i ) 1 -F (τ ⊤ s i )    - f (τ T X i ) 1 -F (τ ⊤ s i ) = I(Y i = 0)      1 -exp 1 -e W (e β ⊤ x i ) f (τ T X i ) 1 -F (τ ⊤ s i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) (1 -F (τ ⊤ s i ))      + I(Y i = 0)      f (τ T X i ) F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) (1 -F (τ ⊤ s i ))      - f (τ T X i ) 1 -F (τ ⊤ s i ) = I(Y i = 0)    f (τ T X i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) (1 -F (τ ⊤ s i ))    - f (τ T X i ) 1 -F (τ ⊤ s i ) = I(Y i = 0). 1 F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) . f (τ T X i ) 1 -F (τ ⊤ s i ) - f (τ T X i ) 1 -F (τ ⊤ s i ) = f (τ T X i ) 1 -F (τ ⊤ s i ) . I(Y i = 0) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) - f (τ T X i ) 1 -F (τ ⊤ s i ) = f (τ T X i ) 1 -F (τ ⊤ s i ) . I(Y i = 0) -F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )
and

S i2 (θ) = ∂ℓ i (θ) ∂β = ∂ ∂β I(Y i = 0) log F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) + ∂ ∂β I(Y i > 0) Y i log[W (e β ⊤ xi )] + log[1 -F (τ ⊤ s i )] -e W (e β ⊤ x i ) = I(Y i = 0)     ∂ ∂β (-e W (e β ⊤ x i ) ). 1 -F (τ ⊤ s i ) . exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )     + (1 -I(Y i = 0))     Y i ∂ ∂β [W (e β ⊤ xi )] W (e β ⊤ xi ) - ∂ ∂β [W (e β ⊤ xi )].e W (e β ⊤ x i )     = I(Y i = 0)     ∂ ∂β [W (e β ⊤ xi )].(-e W (e β ⊤ x i ) ). 1 -F (τ ⊤ s i ) . exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )     + (1 -I(Y i = 0))     Y i ∂ ∂β [W (e β ⊤ xi )] W (e β ⊤ xi ) - ∂ ∂β [W (e β ⊤ xi )].e W (e β ⊤ x i )     = I(Y i = 0)     ∂ ∂β [W (e β ⊤ xi )].(-e W (e β ⊤ x i ) ). 1 -F (τ ⊤ s i ) . exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )     + (1 -I(Y i = 0)) . ∂ ∂β [W (e β ⊤ xi )]. Y i W (e β ⊤ xi ) -e W (e β ⊤ x i ) = I(Y i = 0)     ∂ ∂β [W (e β ⊤ xi )].(-e W (e β ⊤ x i ) ). 1 -F (τ ⊤ s i ) . exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )     + ∂ ∂β [W (e β ⊤ xi )]. Y i W (e β ⊤ xi ) -e W (e β ⊤ x i ) -I(Y i = 0). ∂ ∂β [W (e β ⊤ xi )]. Y i W (e β ⊤ xi ) -e W (e β ⊤ x i ) = I(Y i = 0)     ∂ ∂β [W (e β ⊤ xi )].(-e W (e β ⊤ x i ) ). 1 -F (τ ⊤ s i ) . exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )     + ∂ ∂β [W (e β ⊤ xi )]. Y i W (e β ⊤ xi ) -e W (e β ⊤ x i ) -I(Y i = 0).Y i .     ∂ ∂β [W (e β ⊤ xi )] W (e β ⊤ xi )     + I(Y i = 0). ∂ ∂β [W (e β ⊤ xi )].e W (e β ⊤ x i ) = -I(Y i = 0) ∂ ∂β [W (e β ⊤ xi )].    e W (e β ⊤ x i ) . 1 -F (τ ⊤ s i ) . exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )    + ∂ ∂β [W (e β ⊤ xi )]. Y i W (e β ⊤ xi ) -e W (e β ⊤ x i ) + I(Y i = 0). ∂ ∂β [W (e β ⊤ xi )].e W (e β ⊤ x i ) = -I(Y i = 0) ∂ ∂β [W (e β ⊤ xi )].    e W (e β ⊤ x i ) . 1 -F (τ ⊤ s i ) . exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) -e W (e β ⊤ x i )    + ∂ ∂β [W (e β ⊤ xi )]. Y i W (e β ⊤ xi ) -e W (e β ⊤ x i ) Note that e W (e β ⊤ x i ) . 1 -F (τ ⊤ s i ) . exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) -e W (e β ⊤ x i ) = -e W (e β ⊤ x i ) .F (τ ⊤ s i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) Thus S i2 (θ) = ∂ℓ i (θ) ∂β = -I(Y i = 0) ∂ ∂β [W (e β ⊤ x i )].   -e W (e β ⊤ x i ) .F (τ ⊤ s i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )   + ∂ ∂β [W (e β ⊤ x i )]. Y i W (e β ⊤ x i ) -e W (e β ⊤ x i ) = I(Y i = 0) ∂ ∂β [W (e β ⊤ x i )].   e W (e β ⊤ x i ) .F (τ ⊤ s i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )   + ∂ ∂β [W (e β ⊤ x i )] W (e β ⊤ x i ) . Y i -W (e β ⊤ x i ).e W (e β ⊤ x i ) = I(Y i = 0) ∂ ∂β [W (e β ⊤ x i )].
  e W (e β ⊤ x i ) .F (τ ⊤ s i )

F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )   + ∂ ∂β [W (e β ⊤ x i )] W (e β ⊤ x i ) . Y i -F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) W (e β ⊤ x i ).e W (e β ⊤ x i ) = I(Y i = 0) ∂ ∂β [W (e β ⊤ x i )].
  e W (e β ⊤ x i ) .F (τ ⊤ s i )

F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i )   + ∂ ∂β [W (e β ⊤ x i )]
W (e β ⊤ x i ) . Y i -1 -F (τ ⊤ s i ) W (e β ⊤ x i ).e W (e β ⊤ x i )

- ∂ ∂β [W (e β ⊤ x i )]
W (e β ⊤ x i ) . F (τ ⊤ s i ).W (e β ⊤ x i ).e W (e β ⊤ x i ) = ∂ ∂β [W (e β ⊤ x i )].e W (e β ⊤ x i ) .F (τ ⊤ s i ).

  I(Y i = 0) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) -1   + ∂ ∂β [W (e β ⊤ x i )] W (e β ⊤ x i ) . Y i -1 -F (τ ⊤ s i ) W (e β ⊤ x i ).e W (e β ⊤ x i ) = ∂ ∂β [W (e β ⊤ x i )
].e W (e β ⊤ x i ) .F (τ ⊤ s i ).

I(Y i = 0) -F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) + ∂ ∂β [W (e β ⊤ x i )]
W (e β ⊤ x i ) . Y i -1 -F (τ ⊤ s i ) W (e β ⊤ x i ).e W (e β ⊤ x i )

Let S i (θ) = ∂ℓ i (θ)/∂θ = S T i1 (θ) , S T i2 (θ) T , (i = 1, . . . , n) be the score function of the ZIPBell model. Then S i1 (θ) = ∂ℓ i (θ)/∂τ and S i2 (θ) = ∂ℓ i (θ)/∂β are respectively given by

S i1 (θ) = f (τ T X i ) 1 -F (τ ⊤ s i )
.

I(Y i = 0) -F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) , S i2 (θ) = D.e W (e β ⊤ x i ) .F (τ ⊤ s i ).

I(Y i = 0) -F (τ ⊤ s i ) + 1 -F (τ ⊤ s i ) exp 1 -e W (e β ⊤ x i ) F (τ ⊤ s i ) + (1 -F (τ ⊤ s i )) exp 1 -e W (e β ⊤ x i ) + D W (e β ⊤ x i )

. Y i -1 -F (τ ⊤ s i ) W (e β ⊤ x i ).e W (e β ⊤ x i )

with D = ∂ ∂β [W (e β ⊤ x i )].

Proof of Theorem 1

To demonstrate that θ F is a consistent estimator of θ, note that

U F,n (θ) = 1 √ n n i=1 S i (θ), G F,n (θ) = - 1 √ n ∂U F,n (θ) ∂θ T = - 1 n n i=1 ∂S i (θ) ∂θ T .
Basing on the WLLN, we have:

G F,n (θ)
p → G F (θ) and (K2), the convergence of G F,n (θ) to G F (θ) is uniform in a neighborhood of the true θ.

Depending on the inverse function theorem of [START_REF] Foutz | On the unique consistent solution to the likelihood equations[END_REF], it can be inferred that a unique consistent solution of U F,n (θ) = 0 exists in a neighborhood of the true.

Consequently, θ F is a consistent estimator of θ.

The asymptotic distribution of √ n( θ F -θ) is inferred right here.

As we know θ F is the unique root of U F,n (θ) = 0, and perform the Taylor's expansion of U F,n ( θ F ) at θ to have the main concern:

0 = U F,n ( θ F ) = U F,n (θ) + 1 √ n ∂U F,n (θ) ∂θ T √ n( θ F -θ) + o p (1).
This can lead to an inference:

√ n( θ F -θ) = G -1 F,n (θ)U F,n (θ) + o p (1) = G -1 F (θ)U F,n (θ) + [G -1 F,n (θ) -G -1 F (θ)]U F,n (
θ) + o p (1).

For i = 1, 2, ..., n, V ar[S i (θ)] = E [[S i (θ)] ⊗2 ] and E[S i (θ)] = 0.

Applying the central limit theorem and (K2), we can demonstrate that:

U F,n (θ) d → N (0, Q F (θ)),

  In this paper, we are interested in proposing a new model in the family of the ZI count models, which is shortly called the ZI Probit Bell (ZIPBell) model. This proposed model can be executed to simulate count data or ZI count data. The ZIPBell as well as the ZIBell model has numerous important meanings and applications in practice. Therefore, it is interesting and makes a lot of sense to propose this new regression model in the family of ZI count models. Everything from formulating the general formula for the proposed model, the necessary equations for the estimation, the asymptotic properties, numerical analyzes for the ZIPBell model are presented very fully in this work. The layout of this job is set up as follows. Section 2 briefly presents the general form the ZIBell model and its estimating equation (EE). The general formual and related function of the proposed model are introduced in Section 3. The results of asymptotic inferences for the ZIPBell model are also given in Section 4. Numerous simulation studies is executed in Section 5. Section 6 performs the IBC data set to the ZIPBell model. Some discussions and conclusions of this work are offerred in the last section.

  θ) is an unbiased ESF, and thus the MLE θ F of θ can be obtained by solving U F,n (θ) = 0.

Figure 1 :

 1 Figure 1: Normal Q-Q plots of β 1,n , • • • , β 6,n for case (ii) with n = 2000 and average sample proportion of the ZI equal to 0.63.

-

Figure 2 :

 2 Figure 2: Normal Q-Q plots of τ 1,n , • • • , τ 6,n for case (ii) with n = 2000 and average sample proportion of the ZI equal to 0.63.

Table 1 :

 1 Simulation results (case 1). Note that all results are based on N = 1000 simulated samples.

			βn			τn
		β0,n	β1,n	β2,n	τ0,n	τ1,n	τ2,n
	200
	bias	-0.0182 0.0009 0.0053	-0.0558 0.0633 -0.0017
	SD	0.3602 0.0938 0.1626	0.3080 0.2572 0.1994
	SE	0.3557 0.0946 0.1597	0.3143 0.2518 0.1885
	RMSE	0.5065 0.1332 0.2279	0.4435 0.3653 0.2743
	CP	0.9530 0.9480 0.9420	0.9580 0.9550 0.9660
	500
	bias	0.0140 0.0028 -0.0086	-0.0105 0.0230 -0.0118
	SD	0.2195 0.0576 0.0973	0.1674 0.1326 0.1099
	SE	0.2209 0.0580 0.0992	0.1678 0.1332 0.1063
	RMSE	0.3116 0.0818 0.1392	0.2372 0.1893 0.1533
	CP	0.9500 0.9500 0.9470	0.9600 0.9550 0.9640
	1000
	bias	0.0053 0.0007 -0.0034	-0.0136 0.0136 -0.0002
	SD	0.1485 0.0398 0.0668	0.1187 0.0941 0.0714
	SE	0.1551 0.0405 0.0695	0.1161 0.0921 0.0718
	RMSE	0.2148 0.0568 0.0964	0.1666 0.1324 0.1012
	CP	0.9570 0.9520 0.9590	0.9480 0.9500 0.9590
	2000
	bias	-0.0030 0.0001 0.0005	-0.0019 0.0051 -0.0025
	SD	0.1105 0.0292 0.0488	0.0781 0.0639 0.0483
	SE	0.1096 0.0285 0.0491	0.0806 0.0637 0.0499
	RMSE	0.1556 0.0408 0.0692	0.1122 0.0904 0.0695
	CP	0.9470 0.9390 0.9560	0.9560 0.9470 0.9570

Table 2 :

 2 Simulation results (case 2). Note that all results are based on N = 1000 simulated samples.

	βn

Table 3 :

 3 Ecological data analysis results.

		Proposed ZIPBell		ZIBell
		Estimates	S.E.	p-values Estimates	S.E.	p-values
	β
	intercept	2.1487	0.0741 < 0.0001	2.182	0.0744 < 0.0001
	waterBody	0.4444	0.0738 < 0.0001
	Latitude	-0.0087	0.0425	0.8370	0.1566	0.032 < 0.0001
	Longitude	0.0265	0.0025 < 0.0001	0.0239	0.0024 < 0.0001
	τ
	intercept	-1.4064	0.1284 < 0.0001	-2.7474	0.5583 < 0.0001
	waterBody	1.0779	0.2882	0.0002	2.4628	0.7335	0.0008
	Latitude	-0.6281	0.1924	0.0011	-1.4462	0.4904	0.0031
	Longitude				0.0096	0.0205	0.6401
	AIC		-5439.436			-5406.259
	BIC		-2485.436			-2452.259

Appendix

The log-likelihood

For the sake of simplicity and similarity in the study by [START_REF] Lemonte | Zero-inflated Bell regression models for count data[END_REF], the above can be rewritten as follows:

In the case of the probability of ZI is modeled by the probit model as follows:

where

p → 0 by the WLLN and G F (θ) is nonsingular by (K2), we execute the Slutsky's theorem to show that

Using the Slutsky's theorem we obtain √ n( θ F -θ)