Ruiwen Liao
email: ruiwen.liao@lisn.fr

Franco Quezada
email: franco.quezada@usach.cl

Céline Gicquel
email: celine.gicquel@lisn.fr

Safia Kedad-Sidhoum
email: safia.kedadsidhoum@cnam.fr

A two-stage stochastic programming model for lot-sizing with onsite generation of renewable energy

One way to achieve energy efficiency in manufacturing consists of equipping plants with on-site renewable energy generation systems to partially power industrial processes. However, renewable energy sources are highly intermittent and their availability is difficult to predict accurately. We thus investigate an integrated industrial production and energy supply planning problem under uncertain renewable energy availability. We propose a two-stage stochastic programming model for this problem. The first decision stage consists of building a production plan for a proportional lot-sizing and scheduling problem in a singlemachine multi-item setting. The second decision stage considers a discrete set of scenarios representing potential realizations of the stochastic generation of renewable energy over the planning horizon and aims at building an energy supply plan for each scenario. Computational experiments will be presented to show the practical efficiency of the proposed approach.

Introduction

Industrial companies are increasingly under pressure to mitigate the CO2 and pollution emissions linked to the manufacturing of industrial products. They are also confronted with a sharp rise in the price of conventional energy sources (gas, grid electricity...) so that the availability and affordability of energy is becoming a critical parameter in manufacturing. One way to deal with these two challenges consists in powering industrial processes with electricity generated on-site from renewable sources.

However, the intermittence of renewable energy sources (wind, sun...) makes it impossible to accurately predict the amount of energy they will provide. Thus, an integrated energy supply and industrial production planning problem under uncertain renewable energy availability needs to be solved. This work is an attempt at modeling and solving such a problem by using a two-stage stochastic programming approach.

Problem description and modeling

Problem description We seek to plan production in an industrial plant producing several types of item to satisfy a time-varying external market demand. Backlogging is not allowed and demand must be met on time. The plant comprises a single production resource. This resource may produce only one type of item at a time. Changing the type of item in production requires to carry out startup operations. These operations incur a fixed startup cost but the startup time is assumed to be negligible.

Energy is consumed during both startup operations and manufacturing. The amount of energy consumed during a startup depends on the product that the machine will be setup for. The energy consumed during manufacturing is proportional to the number of items produced. No energy is consumed when the machine is idle or when it is setup for a given product but not producing.

The energy supply system comprises three main elements: the onsite power generation devices (photovoltaic panels) converting a renewable energy source into electricity, the on-site energy storage system (a set of batteries) and the main electricity grid from which electricity may be bought at a given time-varying buying price. The amount of electricity produced by the onsite power generation devices depends on the availability of the corresponding energy source and is thus subject to many uncertainties.

Problem modeling We propose to model the resulting stochastic lot-sizing problem via a two-stage stochastic programming approach. This approach relies on the assumption that the energy supply planning decisions do not have to be made 'hereand-now' but may rather be postponed to a later point in time at which more detailed information about the actual amount of the generated green energy over the whole planning horizon becomes available.

Thus, the first stage decisions consist of building a production plan for a singlemachine multi-product system. To this aim, the planning horizon is divided into a set of planning periods sufficiently short to track the variations in the green energy availability and grid electricity buying price with the necessary accuracy. The resulting small-bucket lot-sizing problem is modeled as a Proportional Lot-sizing and Scheduling Problem (PLSP). The startup and lot-sizing decisions embedded in the resulting production plan translate in an energy demand for each period of the planning horizon.

In order to represent the uncertainties on the available renewable energy, we use a finite set of randomly generated discrete scenarios. A scenario gives a potential realization of the quantity of energy generated by the PV (photovoltaic) panels for each period of the planning horizon. The second-stage decisions thus consist of building, for each considered scenario, an energy supply plan capable of meeting the energy demand of the system in each period. For each scenario, this energy supply plan determines the amount of energy to be bought from the grid, stored in or retrieved from the battery at each time period.

This modeling approach results in the formulation of a large-size mixed-integer linear program (MILP).

Solution approach and preliminary computational results

This MILP is solved by a Benders decomposition approach. This approach decomposes the problem into a master problem and a set of linear sub-problems. The master problem comprises the first-stage production planning decisions and is thus an MILP solving a deterministic PLSP. Each sub-problem corresponds to a second-stage scenario and is a small linear program taking as input data the energy demand coming from the first-stage decisions and building a minimal-cost feasible energy supply plan. More precisely, we develop a branch-and-Benders-cut algorithm. A primal acceleration technique called the in-out stabilization method [START_REF] Ben | Acceleration of cutting-plane and column generation algorithms: Applications to network design[END_REF] is used to accelerate the Benders procedure by generating more efficient Benders cuts. Extended valid inequalities, derived from the classical (l, S) inequalities in the lot-sizing problems, are also added to the MILP formulation at the root node of the branch-and-bound algorithm.

We carried out some numerical experiments to evaluate the performance of the proposed branch-and-Benders-cut algorithm. Numerical instances were randomly generated using data available in [START_REF] Gerhard Wichmann | Energy-oriented lot-sizing and scheduling considering energy storages[END_REF][START_REF] Zhang | Operational optimization of a grid-connected factory with onsite photovoltaic and battery storage systems[END_REF][START_REF] Golpîra | Robust smart energy efficient production planning for a general job-shop manufacturing system under combined demand and supply uncertainty in the presence of grid-connected microgrid[END_REF]. Regarding the scenarios, we generated them with the following procedure. First, for each time period, we randomly generated a value for the predicted PV generation quantity, by using a Gaussian process with means equal to the deterministic PV generation quantities given in [START_REF] Gerhard Wichmann | Energy-oriented lot-sizing and scheduling considering energy storages[END_REF]. We then used an Auto-Regression and Moving Average (ARMA) model to generate auto-correlated prediction errors.

The algorithm was implemented in Python and each instance was solved with the mathematical solver CPLEX 20.10. We compare the computational performance of our algorithm with the one of the Benders algorithm embedded in CPLEX as well as the one of a classical Benders approach. Our preliminary computational results first show that for small-size instances involving 3 products, 32 micro-periods and 1000 scenarios, the proposed Branch-and-Benders-Cut algorithm can find a relative good feasible solution with smaller MIP gap compared to the other Benders approaches. For the medium-size instances involving up to 5 products, 64 microperiods and 1000 scenarios, the other two Benders algorithms can hardly find a feasible solution within the time limit while the proposed method finds a feasible solution of acceptable quality.

Compared to the deterministic approach where we assume to be able to find the optimal solutions for each scenario, i.e., wait-and-see solutions, which is defined as the expected value of optimal solutions with perfect foresight of PV electricity obtained, we proved also the need to implement the aforementioned two-stage stochastic approach by calculating the Value of the Stochastic Solution (VSS). It was shown that the more the product-specific energy-consumption varies for different products and the PV-availability varies among scenarios, the more the necessity of implementing the stochastic approach is apparent.


Fluid Formalism

Lionel DUFAYE -Université Gustave Eiffel -LISAA EA4120

In his article on Mental Gesture, Antoine Culioli, quoting René Thom, points out that "any scientific endeavor is based on an aporetic situation which the scientist tries to find a way out of" (2011: 64)1 . The aporetic problem at stake in the present case could be formulated as follows: How can metalinguistic representations that have to be fixed on paper, at some point, be it by a diagram, a formula or a definition, account for the ever "overflowing"2 symbolic activity at work through linguistic markers? Indeed, since the metalinguistic level aims mostly at representing markers and operations as part of a given language-state, to what extent can it allow for the interpretation of diachronic evolution and semantic plasticity and impermanence? To quote Culioli:

Symbolic means that there is always something that turns into something else; but that transformation, that metamorphosis has some form of coherence. The problem is how to account for it. (2011: 68) Symbolic activity, which we understand is not to be confused with linguistic activity per se, is at the core of the epilinguistic process that "formalizes" thought into language:

Formalization is a tool of the mind, like algebra [ … ]; it "corporifies" ideas to borrow Lordat's words once again. (Laplane, 2000: 115 3 ) But there appears another paradox, in the sense that language, being made up of discreet markers, de facto fragments the diffuse process of thinking into discontinuous units. A consequence is that words should not be equated with the notions they point to:

The words of a text do not carry meaning in themselves, they are triggers of mental representations. (Weinberg (in Dortier 2012): 335) Or, as Culioli puts it:

Words are a form of condensed version of notional representations. They capture notions: with a word one can access a notion. The word refers to the notion as a whole but that relation is not symmetrical: a notion is only partially captured by a word. So again, this is not a one-to-one relationship. (Culioli 1985: 16) Following that line of reasoning, according to which "markers represent representations" (Culioli ibid) 4 , one can surmise that the metalinguistic representation of markers should also not be equated with the notions that they refer to. Actually, even beyond the issue of lexical and morphematic discreetness, it is not unreasonable to assume that pre-assembled phrases and structural specificities contribute to shape meaning into imposed patterns. For the sake of demonstration, let's imagine that someone falls asleep while reading a book or while watching a movie. They can describe what happened by saying I read myself to sleep, whereas *I watched myself to sleep is impossible. As we know, resultative structures accept only one NP in object position; however, contrary to an ambitransitive verb like read, with watch the object NP is mandatory, but of course *I watched a movie myself to sleep is ill-formed, since there would be not one but two NPs in object position. For the same reason, even though one can say that they read themselves to sleep, resultative structures do not make it possible for them to specify what they were actually reading, unless restructuring the syntax completely: *I read War And Peace myself to sleep. As Laplane observes:

Language is likely to freeze the fluidity of thought in its gel, precisely because it fixes it through its constraints, which are also habits. Language should no longer be seen as a means to express thought, but as a frame to define and organize it.

[ … ] Good writers are those who can develop their own use of language. Great thinkers are those who can detach themselves from its formalism. (2000: 118) On the face of it, the problem of metalinguistic modelling would seem to rest on a two-fold paradox: it aims at representing non-discreet mental activity (Level 1 in Culioli's theory) through the use of discreet concepts (level 3) based on the observation of discrete markers (level 2). In agreement with Culioli's approach, we will posit that, for it to account for semantic plasticity, metalinguistics has to rest on discreet dimensions from which dynamic relations can be drawn:

Fixist or physical approaches would map on langage descriptions such as those we find in physics. To say that we are dealing with a network means that we have a network system, whatever its meaning may be and it can be represented as something like this: Each of these elements stand for something; then we establish relations and ponderations among them. The simplest examplewhich is not even a network! consists in defining an occurrence with Qnt-Qlt5 dimensions, with either equiponderate value, or Qnt/Qlt (Qnt preponderant over Qlt), or inversely Qlt preponderant over Qnt. (64)(65) Although componential in its essence, this conception has little to do with a Katz and Fodor type of semantics (1963); the components that it rests on are not predefined semes (eg. animate, female, adult,…) but conceptual primitives drawn out from trans-linguistic and trans-categorial analyses. Although connexionist in appearance, it also has little to do with Collins and Quillian's 1969 semantic organization; as Culioli insists, it "is not even a network!". Contrary to a static meshwork of interwoven notions, Culioli's model is conceived as a modular representation of primitives allowing for dynamic operations between them. Mental gestures are to be construed as active processes whereby meaning is not so much "retrieved" as it is "built". 6In our view, for it to be in keeping with this approach, the concept of schematic form should be thought of as a modular configuration implying a set of abstract, fundamental features, rather than a synthetic "invariant" formula (precisely because "variance" needs to be accounted for). It is trite to mention that etymology and language change teach us that one same word base in two different languages will take different paths 7 , but the interesting conjecture that can be drawn from that is that although one cannot prospectively assess how a word will evolve, one may account retrospectively for why it evolved the way it did. In other words, language change may be unpredictable and random a priori, but it is nonetheless analysable and consistent a posteriori. It is through the "traces" (Culioli 2011: 68) left by markers that the linguist sets out to investigate the semantic consistency in actual use. For instance, drawing attention to the Catalan word gairebé, meaning almost, Culioli explains:

Gairebé is made of gaire, which is the same word (except for the meaning, of course) as guère ; and bé which means bien. [ … ] Gaire comes from a Germanic word (Frankish *waigaro) which meant "resist, refuse, not give in"; so a possible gloss, widely accepted actually, would be: to have the strength. In Old French the word became gaire "strong, much", then in the negative form ne…guère: "little, fairly little". All of this is the result of symbolic activity. Words are not set in stone once and for all ; they can be deformed, their initial value stretching into a different shape. (2011: 65-66) One among many, this example highlights the fact that there is no anticipating that the cooccurrence of two words meaning much and well can come to mean "almost", which shows that, on the epilinguistic level, symbolic activity comes up with a language-specific means of formalizing infralinguistic activity, exploitating semantic ductility to stretch meaning into new uses. Thus, metalinguistics requires a type of representation that makes it possible to specify from which semantic dimension any such derivation occurs. To do so, we posit that meaning has to be configurable in such a way that semantic variables can be singled out so as to make it possible for us to identify which specific dimension was foregrounded to extend the use of the marker. A further reason for elaborating schematic forms that can be broken down into discreet features pertains to the question of local synonymy. If one given context allows for two markers to alternate relatively freely (e.g. Elephants MAY be dangerous and Elephants CAN be dangerous) whereas another context prevents the same two markers from alternating or else brings forth radical differences (e.g. MAY the force be with you and *CAN the force be with you), one can hypothesize that one facet of their schematic form is shared while others are not. [START_REF] Deschamps | For A Topological Representation of the Modal System[END_REF]

apply this principle to propose an articulation of modal auxiliaries in English based on modular operations:

The assumed representation must first account for all the possible values generally attributed to each modal in all kinds of different contexts. It must also allow us to deal with the similarities, quasi-equivalences, compatibilities and incompatibilities of the modals and non-existing patterns, that is it must explain why commutations between modals are permitted (for instance SHALL and WILL, CAN and MAY, WILL and CAN, SHALL and MUST, WILL and MUST) as predicted by the are representations pertaining to corporal activity. (Culioli 1985: 23) ( les notions qui sont des représentations de l'ordre de l'activité du corps.) 7 Sometimes radically so, for instance, although complete opposite, black and blank, from French blanc meaning white derive from the same Indo-European root *bhel-. theoretical model itself, or prohibited (SHALL and CAN, SHALL and MAY, CAN and MUST, MAY and MUST…). (Deschamps et al. , 2009: 130) Based on this assumption, the authors propose to define each modal by a set of two operations a Quantitative (i.e. existential) one and a Qualitative (i.e. notional or subjective) oneeither one of which is shared by at least another modal8 .

When two connected modals appear in a context where the shared operation is compatible, an alternation is conceivable with related modal valuesrelated though never quite synonymous in the sense that the second operation preserves a kind of semantic ponderation9 . As such, this schematic type of representation can be seen as a diagram. However, Culioli insists that it is only under the form of a dynamic graph that this sort of representation can possibly modelize markers as the trace of mental gestures:

We are dealing with something that is not only deformable, but also malleable, etc. The first problem we face is that of the graph. It is oriented, therefore it is not a diagram. Our graph became more and more complex as it evolved. Its purpose was to resolve the problem of fixism that we exposed in our 1968 article10 . I wanted to avoid cases where we consider a word and say: "That word means what it means". (70) It might not be an unfitting comparison to say that, in the same way Chomsky reintroduced mentalism by positing syntactic movement between a deep and a surface structure, Culioli's theory on mental gesture posits cognitive movement among primitive components. With regard to Deschamps et al.'s representation, it is through a graph based on such diagrams that the logic behind modal operations and values can be "visualized". If we come back to the previous example where MAY and CAN alternate relatively freely: Elephants MAY / CAN be dangerous, the reason they appear to be interchangeable is because both operations are centered on the same quantitative scenario, namely the expression of both positive and negative validations in a generic context (or so called "sporadic modality"). Even in that context the two markers are never entirely synonymous, of course, in the sense that they correspond to different "gestures":

While MAY rests on a notional representation of positive and negative scenarios as both conceivable to envisage occurrential situations, CAN rests on situational experience or knowledge to predicate a potential notional property. Accordingly, An elephant can be dangerous (QLT oriented) is fine since an elephant is read as a notional occurrence, whereas ?An elephant may be dangerous (QNT oriented) would call for a form of situational anchoring: An elephant may be dangerous if you creep up on it.

To bring theory and practice together, we shall take this discussion on mental gestures further with an analysis of the prepositional/adverbial particle OVER. Cases of semantic proximity such as Fido slept with his paw over my leg / Fido slept with his paw on my leg or The night is over/ The night is through will serve to illustrate how meaning is built by gestures linking one dimension to another within a network of componential semantic features.

For the purpose of brevity, we shall not go into the heuristic intricacies or past experiences of other markers that led us to hypothesize that the various uses of OVER might be apprehended through a minimal set of three semantic components. Namely a topological component, a Qualitative component (henceforward QLT), Quantitative component (henceforward QNT), whose operations can be summed up as follows:

α. Topological: localization relative to the locator's surface (or "closure"). β. QNT : extensional path γ. QLT : asymmetrical valuation

The three components will be addressed alternatively, for each implies specific synonymic 11 areas, and for the purpose of argumentation we shall start with the QLT dimension (3d component in our list).

Component γ: asymmetrical valuation

As we know, in spatial contexts, OVER mostly localizes the locatum in an upper position relative to the locator 12 :

1. The sign over the door said "Exit". Yet the up-down axis is not semantically primitive but contextually derived from a more fundamental QLT alterity: X over Y implies an asymmetrical valuation in which the locatum X is associated to a superior value. This asymmetry may apply to any form of relation implying a hierarchy of values: salience, intensity, preference, ponderation, gradience, or event spatial configuration 13 . For instance: To go over the speed limit, to produce over a certain amount, to 11 It goes without saying that 'synonymic' is not to be understood as semantic 'sameness' (which is why we preferred it to the more colloquial adjective 'synonymous') ; in our approach, 'synonymy' is used for want of a better term to describe cases where possible substitutions have a similar enough meaning. 12 "Covering" values such as She placed the plywood over the hole in the ceiling imply that we take component 3 into account 13 Metaphorical uses like I'm down, I feel low, I hit the bottom This whole operation is going south, I'm under the weather… … as contrasted with I have the upper hand, I'm at the height of my fame, I'm at the top of my game… reveal explicitly how this axis is valued.

choose one thing over another… In spatial contexts, the asymmetry, which consistently gives prevalent value to the locatum, corresponds to a high (locatum) / low (locator) positioning.

Component α: localization relative the locator's topological surface (or "closure")

Just as the primitive QLT component γ can apply to either spatial and non-spatial values, topological configurations, although somehow at the core of spatial representations, can also further apply to non-spatial uses. Let's take a spatial context to begin with: A swarm of flies gathered over the pond for the sake of illustrationhere, the locatum (i.e. a swarm of flies) is in the neighborhood of the closure (i.e. the surface) of the locator (i.e. the pond). Put differently, we can say that the locatum is "in the Exterior of" the topological space derived from the locator, and yet it is still in its reference field. To better understand this compound localization, one may contrast OVER with THROUGH et ACROSS, for which Lindstrombreg provides the following comparison:

"If I hear that you walked through a stream, I assume you got your feet wet. If I hear that you walked across a stream, I do not know whether you did or did not [. . .]. If I hear that you walked (or rather jumped) over a stream, I can safely assume that your feet remain dry. (Lindstromberg, 1997: 127)" To rephrase this in more theoretical terms:

-Unlike THROUGH, with OVER the locatum is in the Exterior of the domain defined by the locator: the existence of the compound THROUGHOUT is evidence that THROUGH refers to the interior of the domain, whereas *OVEROUT does not exist ; -Unlike ACROSS, with OVER the locatum is localized relative to reference field of the locator: with ACROSS neither the Interior nor the Exterior is relevant, simply the dissociation of the locatum and the locator: She sat across from me (=There was a gap between her and me). In terms of localizing operations, THROUGH can be described as an identification, OVER as a differentiation, ACROSS as a disconnection. In the form of a topological space we get: Let's compare OVER to another marker which shares component γ, namely ABOVE. Their shared component (asymmetrical valuation) may make their meaning in spatial contexts hardly discernible: The sign over/above the door said "Exit". ABOVE has often been analyzed as the expression of a gap or an absence of contact (eg. [START_REF] Lindstromberg | English Prepositions Explained[END_REF][START_REF] Gilbert | Remarques autour de quelques prépositions[END_REF][START_REF] Stunell | The Construction Of Sense In English Particle And Prepositional Verbs: Focus On OVER[END_REF], contrary to OVER which is not incompatible with a representation where there is contact between the locatum and its locator. The difference here again is that OVER expresses differentiation (i.e. outside but relative to) while ABOVE expresses disconnection. Whereas

The sign above the door is a mere localization of the sign in a higher position relative to the door (shared component α), The sign over the door refers to a more holistic relationship between the sign and the door as forming a cohesive, or expected, representation. Accordingly, in the phrase over and above the relative alterity expressed by OVER is subsequently amplified by ABOVE; the same gradation effect applies to over and out, or out and away, sequences which cannot be reversed: *above and over, *out and over, *away and out. Lindstromberg provides another comment which reinforces this view: "(2) Dale's flat is somewhere above Jill's, in the same building.

That is, if Jill's flat is on the 3rd floor, then Dale's must be on some higher floor. It may be directly up from Jill's or on the opposite side of the building. If, on the other hand, one says, (2.1) Dale's flat is somewhere over Jill's... then that would suggest that Dale's flat was directly up from Jill's, on some higher floor." (Lindstromberg 1997: 112-113) With ABOVE the locatum (Dale's flat) is localized relative to a scale independent from the locator (Jill's flat); conversely, with OVER the locatum is localized within the extension of the locator, and thus remains within its reference space. These two modes of localization (i.e. OVER = differenciation vs. ABOVE = disconnection) provide a key to interpret the non-spatial forms of alternations between the two markers. Eric [START_REF] Gilbert | Across, by et through, Considérations sur les conditions de représentation métalinguistique des prépositions[END_REF] quotes Michael Swan14 : ABOVE is used in measurements of temperature and height, and in other cases where we think of a vertical scale.

The temperature is well above zero.

We usually use OVER, not ABOVE, to talk about ages and speeds, and to mean 'more than'.

You have to be over 18 to see this film.

In keeping with Gilbert's analysis, who also describes ABOVE in terms of disconnection and OVER in terms of differenciation, one can surmise that temperatures are calibrated relative to external scales (Celsius, Fahrenheit), whereas age corresponds to an intrinsic property, possibly associated with qualitative predicates such as being allowed to, being of age, having the duty to, etc. :

2. Children over the age of 12 must have full-price tickets.

Indeed, differentiation allows for the expression of alterity within cohesive representations; for that reason, OVER (unlike ABOVE) is prolifically used to derive notions: overhead, overspeed, overreact, overdose, overtime, overwork, overrated; or set phrases expressing prototypical situations: To go arse over tit / head over heels / hand over fist; or else compound adverbials: over the counter, over the top … The following example illustrates the difference between differentiation/relative alterity (OVER) and disconnection/absolute alterity (OUT):

3. "MILO, Air One, Air One, follow the Senator, roger? I'll take the sport boat, you take Baynard, over." "AIR ONE, That's a roger. Over and out."

OVER expresses discontinuity (end of the message) within a homogeneous domain (a communication). In contrast, OUT expresses the exit from that domain and consequently radical disconnection (end of the communication itself). Thus, OVER is suitable to express notional, cohesive linking between the locatum and its locator, and not just a positional relation between notionally distinct elements.

A structured network of semantic values is emerging from those alternations. Both OUT, ABOVE and ACROSS have been analyzed as markers of disconnection relative to the Interior, but only ABOVE shares the QLT component γ (asymmetrical valuation) with OVER, and only ACROSS shares component β (extensional path), as we shall see it in the next section. If we note component types α, β, γ … , and component subtypes 1, 2, 3…, we can put forward a heuristic diagram where each marker is defined by a conjunction of discrete component subtypes:

In the last section of this article, we shall try to make the case that mental gesture operates by either pondering certain subtypes and / or by establishing new links. But first, a few words should be said about the third component type β.

Component γ: extensional path

In an example like The dog ran over the lawn,, two semantic components are at stake: a localization relative to the locator's closure (i.e. the locatum (car) is on « the outside ») ; the locatum moves along the locator's closure (its surface). This latter scanning value is manifested in a variety of uses of OVER:

4. He would wander over the moors, losing all track of time. 5. Mind you don't spill coffee over my best tablecloth. 6. Come over here.

The extensional component accounts for the frequentand sometimes mandatorycooccurrence of OVER with the totalization quantifier ALL, which somehow reinforces the path by scanning the entirety of the locator's surface: He had blood all over his face. As it appears if we substitute ON for OVER, without the scanning operation conveyed by OVER, the totalization quantifier is incompatible: *He had blood all on his face. It is also noteworthy that ALL is likely to introduce an appreciative modality:

7. Dogs were barking all over the place. 8. If I can order something with garlic on the first date and she's all over it, she's a keeper.

The hyperbolic value that that modality may take on overshadows the asymmetrical valuation described in spatial (and other) uses:

She jumped over the table:

The qualitative asymmetry is preponderant (with an "on top" value). 10. She jumped all over the table: The extensional path is foregrounded with a hyperbolic value. Based on the same model, OVER and ON would appear to share two semantic components, but differ as far as the extensional dimension is concerned.

By foregrounding the extensional scanning, ALL both backgrounds locating and Qualitative components and explicitly blocks alternative extensional values. Beyond the specificity of micro-analyses, such examples serve to illustrate the more general principle that meaning is built through co-ocurrences and interactions that induce dynamic salience within isomeric relations. It is thanks to the contrastive study of markers within a paradigm of more or less synonymous contextual values that semantic structures emerge, thus making it possible for us to set a network under the guise of a diagram. Such diagrams, however, are at best a static cartography of the wefts and warps of the semantic fabric markers are made of. It is the representation of operations on such networks that account for semantic plasticity and ductability. Only graphs can indeed provide a representation of dynamic meta-operations (i.e. operations on components) such as the ponderation of certain components, the stretching of certain dimensions or else the local interaction with components induced by other markers. The distinction between static diagrams and dynamic graphs somehow echoes the concepts of khrema (what is available) and pragma (what we make of it) 15 .To reiterate our main point, we hypothesize, alongside Culioli's theory on mental gestures, that only a two-tier mode of representation is able to somehow modalize the shifting nature of language through fluid formalism.

Bibliographical references:

Our translation for this quote and the following ones by A. Culioli.

In French, Antoine Culioli uses the word débordant.

Our translation and the following ones by D. Laplane.

« Les marqueurs sont les représentants de représentations. »

Qnt stands for Quantity, i.e. the extensional or existential value of an occurrence, whereas Qlt stands for Quality, its notional or appreciative values.

Although the apprehension of semantic operations as mental gestures paralleling corporal activity was developed by Culioli in the course of the 2010s, its seed seemed to have been there as least since the seminal notes published in 1985: [ … ] notions

I and E stand for Interior (validation of the process) and Exterior (validation of the process). Each modal takes into account one or both scenarios. When a scenario is closed off (MUST and SHALL) it is conceivable but considered undesirable or incompatible.

See Deschamps et al. 2009 for detailed illustrations.

« La formalisation en linguistique », Cahiers pour l'analyse, n° 9, Paris, Éditions du Seuil, 1968, p. 106-117. Repris dans Culioli, Antoine, Pour une linguistique de l'énonciation [ci-après PLE], Formalisation et opérations de repérage, Paris, Éditions Ophrys, t. 2, Coll. « L'homme dans la langue animée par JanineBouscaren », 1999, p. 17-29.

Swan, M., (1995), Practical English Usage, Second Edition, Oxford University Press: 4.


domain: math stat

text:
Azzouz Dermoune
email: azzouz.dermoune@univ-lille.fr

Daoud Ounaissi
email: d.ounaissi@groupe-esa.com

Yousri Slaoui
email: yousri.slaoui@math.univ-poitiers.fr

Gaussian and Lerch Models for Unimodal Time Series Forcasting

Keywords: Gaussian model, Lerch model, least absolute deviation, daily infection, simplex algorithm, Nelder-Mead, optim function

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The least absolute deviations (LAD) method of curve-fitting proposed [START_REF] Bloomfield | Least Absolute Deviations Curve-Fitting[END_REF] consists of fitting the data (x i , y i ) to a function f (x i , θ), with i = 1, . . . , n. The parameter θ ∈ R p minimizes the sum of absolute deviations

n ∑ i=1 |y i -f (x i , θ)|.
According to [START_REF] Eisenhart | Boscovitch and the combination of observations[END_REF], in the linear regression case f (x i , θ) = ∑ p j=1 x ij θ j , the minimization of the quantity

n ∑ i=1 |y i - p ∑ j=1
x ij θ j | was suggested by Boscovitch (1757) (some asymptotic results are given in [START_REF] Koenker | On Boscovich's estimator[END_REF]), see also [START_REF] Rhodes | Reducing observations by the method of minima deviations[END_REF][START_REF] Karst | Linear curve fitting using least deviations[END_REF][START_REF] Sadovski | Algorithm As 74. L1-norm fit of straight line[END_REF]. The latter objective function is convex with respect to the parameter θ. Hence, it has only one minima but may have many minimizers.

The linear regression case of LAD optimization is inherently more complex than the minimization of the sum of squares. The interest in the LAD method is associated with the development of robust methods. The LAD method is more resistant to the outliers in the data (see [START_REF] Dielman | Least absolute value regression: recent contributions[END_REF][START_REF] Li | A maximum likelihood approach to least absolute deviation regression[END_REF]).

The aim of our work is to analyze LAD minimization using a nonlinear regression motivated by the daily infections of COVID-19 in China during the first wave. We denote I(t) the observed number of infected persons at time t ∈ [1, T] with T ≤ 60 (see Figure 1). The variable t = 1, . . . , T represents day 1, . . . , T.

Justified by the sigmoidal nature of a pandemic, we propose two models: the Gaussian model (see [START_REF] Barmparis | Estimating the infection horizon of COVID-19 in eight countries with a data-driven approach[END_REF])

I Gauss (t) = a exp(- (t -l) 2 s 2 )
and the Lerch model

I Lerch (t) = a z t (v + t) s ,
as a prediction of the observed number of infected persons I(t) at time t.

The Gaussian parameter θ = (a, l, s), with a, l, s denote, respectively, the peak, the location of the peak, and the width of the first wave of COVID-19.

The Lerch probability distribution on the non-negative integers is proportional to the function z t (v+t) s with t = 0, 1, . . ., the parameters z ∈ (0, 1) and v > 0. The Lerch probability distribution is strongly unimodal when s < -1 and v ≥ 1. In this case, its mode is at [•] signifies taking the integer part, see [START_REF] Aksenov | Some properties of the Lerch family of discrete distributions[END_REF].

[ 1 z 1 s -1 -v] + 1, where
To estimate the three parameters θ = (a, l, s) (respectively, θ = (a, z, v, s)) based on the T observations, we consider LAD nonlinear regression

f (T, θ) = ∑ T t=1 |I(t) -I m (t)| T ,
with the subscript m = Gauss (respectively, m = Lerch). The Gaussian model was studied in our previous work [START_REF] Dermoune | Confidence intervals from local minimas of objective function[END_REF].

As in our previous work [START_REF] Dermoune | Confidence intervals from local minimas of objective function[END_REF], we propose to solve our proposed LAD regression using the simplex Nelder-Mead algorithm implemented by the optim function in R software. The Nelder-Mead algorithm [START_REF] Dielman | Least absolute value regression: recent contributions[END_REF][START_REF] Gao | Implementing the Nelder-Mead simplex algorithm with adaptive parameters[END_REF][START_REF] Lagarias | Convergence properties of the Nelder-Mead simplex method in low dimensions[END_REF][START_REF] Nelder | A simplex method for function minimization[END_REF] is able to optimize functions without derivatives. It is a simplex method for finding a local minima of a function, the most widely used direct search method for solving optimization problems, and is considered one of the most popular derivative-free nonlinear optimization algorithms.

The output of the optim function depends on the initialization and is in general not a minimizer of the objective function. Restarting the Nelder-Mead algorithm from the last solution obtained (and continuing to restart it until there is no further improvement) can only improve the final solution and the latter is in general a local minimizer. Here is the iteration of the optim function until the convergence:

Probabilistic Interpretation of LAD Regression

Let us assume that

I(t) = I m (t) + e(t),
where the errors (e(t)) are i.i.d. with the common probability distribution

1 2λ exp(- |e| λ )
,

where e ∈ (-∞, ∞) and λ > 0 are location and scale parameters, respectively (see, e.g., [START_REF] Johnson | Continuous Univariate Distributions[END_REF][START_REF] Ord | Laplace Distribution[END_REF]). It was named after Pierre-Simon Laplace (1749-1827), as the distribution whose likelihood is maximized when the location parameter is set to the median. Based on the data (I(1), . . . , I(T)) the likelihood is equal to

T ∏ t=1 1 2λ exp(- |I(t) -I m (t)| λ ).
It comes that the maximum likelihood estimator of the parameters θ and λ are θ = arg min{ f m (T, θ) : θ} λ = f m (T, θ).

In practice, θ are given by an algorithm of optimization, and usually, they are only local minimizers. Having θ and the scale λ, we derive a confidence interval for I(t) with t > T as a solution of the equation

q -q 1 2 λ exp(- |e| λ )de = 0.95
given by q = -λ ln(0.05) = 2.995732 λ. We derive the confidence interval ) in the Gaussian case, and Îm (t) = â ẑt /( v + t) ŝ in the Lerch case.

IC 0.

LAD Regression Analysis Using Weighted Median

Before going forward, we recall the weighted median definition.

Weighted Median

We recall in the following proposition, the definition and the calculation of the weighted median. For more details, we advise the reader to see the work of [START_REF] Novoselac | The Component Weighted Median Absolute Deviations Problem[END_REF].

Proposition 1. Let us consider a sequence (x(t), w(t)) of real numbers with positive weighted w(t) > 0 and t = 1, . . . , T. The minimizer of the function a → ∑ T t=1 w(t)|ax(t)| (called the weighted median) is given as follows. We calculate the permutation p(1), . . . , p(T), which rearranges the sequence (x(t) : t = 1, . . . , T) into ascending order. We form the sequence (w(p(t)) : t = 1, . . . , T), then we find the largest integer k which satisfies

k ∑ t=1 w(p(t)) ≤ ∑ T t=1 w(t) 2 . If k ∑ t=1 w(p(t)) < ∑ T t=1 w(t) 2 ,
then the weighted median a = x(p(k + 1)).

If ∑ k t=1 w(p(t)) = ∑ n t=1 w(t) 2 , then the weighted median [x(p(k)), x(p(k + 1))] is equal to the interval [x(p(k)), x(p(k + 1))].

Back to Our Proposed LAD Regression

Both our LAD regressions have the form

f (T, a, b) = 1 T T ∑ t=1 |ag(t, b) -I(t)|,
with a > 0, and g : (0, +∞) × D → (0, +∞) is a continuous positive map with D is a Euclidean domain. Now, we can announce the following corollary. Proof. By definition of the local minimizer, there exists a neighborhood V of (a

: b}. If (a(b * ), b * ) is a local minimizer of the map (a, b) → f (T, a, b), then, in general, b * is not a local minimizer of the map b → f (T, a(b), b). However, if (a(b * ), b * ) is a global minimizer of (a, b) → f (T, a, b), then b * is also a global minimizer of b → f (T, a(b), b). Proof. For each fixed b, the minima of the map a → f (T, a, b) is attained at the point a(b), which implies that f (T, a, b) ≥ f (T, a(b), b) and achieves the proof of 1). Let (a * , b * ) be a local minimizer of the map (a, b) → f (T, a, b). It follows that f (T, a, b) ≥ f (T, a * , b * ) when a ∈ W, b ∈ V with W, V
(b * ), b * ) such that f (T, a, b) ≥ f (T, a(b * ), b * ) for each point (a, b) ∈ V. Necessarily, (a(b), b) is not in V for at least one point b near b * , if not f (T, a(b), b) ≥ f (T, a(b * ), b * ) for all

Numerical Results

In China, COVID-19 appeared on 23 December 2019 in the Wuhan region and after its fast-initial spreading, strict rules of social distancing were imposed almost one month later. Three months after the initially reported cases, the spreading in China subsided. The China data in Figure 1 were extracted from owid/COVID-19-data, available on the web. Figure 1 shows that the peak and location, which are equal, respectively, to a = 15136, and l = 22.

LAD Regression Using Gaussian Model with T = 10

By varying the initial condition and using the algorithm 1 with the objectif function equals (l, s) → f Gauss (T, a(T, l, s), l, s), we obtain only one minima: a(10, l * , s * ) = 2088.911, l * = 10.11930, s * = 5.712179 and f Gauss (T, a(10, l * , s * ), l * , s * ) = 319.5446. We recall that a(T, l, s) is the weighted median of the sequence (x(t) = I(t) exp( (t-l) 2 s 2 ) : t = 1, . . . , T) with the weights (w(t) = exp(-(t-l) 2 s 2 ) : t = 1, . . . , T). However, by varying the initial condition and using the Algorithm 1 with the objectif function equals (a, l, s) → f Gauss (T, a, l, s), we found several lists of minima. Table 1 shows some of them. The minima 331.487 corresponds to the minimizer a * = 5526.386, l * = 22.018, s * = 12.182. We recall that the observed location is l = 22. The minima 337.0095 corresponds to the minimizer a * = 15185.797, l * = 31.994, s = 15.614. We recall that the observed peak is a = 15136.

Algorithm 1

The output of the iteration of optim function until convergence Input: Fonction F 1: initialization θ 0 , 2: while F(θ 0 ) = F(T, optim(θ 0 , F, method = "Nelder -Mead") do 3:

θ(θ 0 ) =optim(θ 0 , F, method = "Nelder -Mead") // optim function applied to F with the initialization θ(θ 0 ). 4: end while output: θ(θ 0 ) and F(θ(θ 0 )).

Confidence Intervals Using Six minimas with T = 10 in the Gaussian Model

We recall that the confidence interval of I(t) from the minimizer ( â, l, ŝ) is given by

IC 0.95 [I(t)] = â exp(- (t -l) 2 (ŝ) 2 ) -2.995732 λ; â exp(- (t -l) 2 (ŝ) 2 ) + 2.995732 λ with t > 10.
In Figure 2, we present the confidence intervals for the global minima and five local minimas among the list T = 10. The predictions using the minimas 335.047 and 336.92 are clearly bad. However, their predictions at the location l = 22 are close to the real peak among the six minimas. The percentage of predictions is given in Table 1. An R source code is given in Appendix A, which can be used to determine the confidence intervals for the other values of T once the list of minimas is determined by using one of the three considered methods.

In Figure 2, we present the confidence intervals for the global minima and six local minimas among the list T = 10 in the Gaussian model.

LAD Regression Using the Lerch Model with T = 10

By varying the initial condition and using the Algorithm 1 with the function (z, v, s) → f Lerch (T, a(T, z, v, z, v, s) we found a huge number of minimas. Table 2 shows some of them. We recall that a(T, z, v, s) is the weighted median of the sequence (x(t) = I(t)(v + t) s /z t :t = 1, . . . , T) with the weights w(t) = z t /(v + t) s : t = 1, . . . , T . We also recall that in the Gaussian case, the surface (l, s) → f Gauss (T, a(T, l, s), l, s) has only one minima equal to the global minima of the map (a, l, s) → f Gauss (T, a, l, s).

Figures 4 and5 show, respectively, the optimal time series for the Lerch and Gauss models for T = 10, using our list of local minimas. We can observe that the Lerch model fits better with the prediction of I(t) for t > 10. But the Gauss model fits better with the prediction of the location and the size of the peak. In Table 3, we report the columns of mode positions and the corresponding amplitudes for each minimizer. The percentage of predictions is given in Table 2. Observe that the best percentage of predictions occurs at the global minima of the Lerch model, but the best percentage of predictions occurs at the local minimizer 325.186 of the Gauss model.

In Figure 6, we present the confidence intervals for the global minima and six local minimas among the list T = 10 of the Lerch model.

The Case T > 10

In this section, we consider the cases T = 20 and T = 60. In the Lerch model case, LAD regression still has many minimizers for T = 20. See Table 4. But in the Gauss model, LAD regression has only one minimizer for T = 20. See Table 5. From Tables 4 and5, we can observe that the percentage of prediction of the Lerch model is better than the percentage of prediction of the Gauss model.

Finally, the minima for T = 60 for both models (Gauss and Lerch) are unique and respective to (3318.433, 16.94084, 10.19735, 617.2386) and (0.05691053, 0.7445283, 3.3413322, -5.2886448, 570.7861098).

Figures 7 shows the optimal time series for the Lerch and Gauss models for T = 60, using our local minima. We observe that the Lerch approximation has a heavier tail than the Gaussian approximation.

Conclusions

In this work, we considered LAD nonlinear regression f (T, θ) = s 2 ) that the map f (T, a, l, s) has a huge number of local minimas, but the surface S has only a global minima, which is also the global minima of the map f (T, •). However, in the Lerch case g(t, z, v, s) = z t (t+v) s , contrary to the Gaussian case, we showed that the maps (a, z, v, s) → f (T, a, z, v, s) and (z, v, s) → S(T, z, v, s) have each a huge number of local minimas. We derived confident intervals for the daily infections from each local minima. Our message is that each local minima contains a part of the information and can be used for the prediction of a part of the parameters.

In future work, we will deal with the log-transformation ln(I Gauss (t)) = β 0 + β 1 tβ 2 t 2 , where (β 0 , β 1 , β 2 ) are in bijection with the parameters a, l, s of the Gaussian model. This allows us to consider the linear model ln(I(t)) = β 0 + β 1 tβ 2 t 2 + ε(t), with the errors (ε(t)), which are i.i.d. We will deal with this problem using other robust regression estimators as the least trimmed median estimator (LTM) proposed in [START_REF] Croux | Positive-breakdown regression by minimizing nested scale estimators[END_REF].

Corollary 1 . 3 . 1 . 2 .

 1312 For each fixed b, the minima of the function a → f (T, a, b) is attained at the weighted median a(b) of the sequence (x(t) = I(t)/g(t, b) : t = 1, . . . , T) endowed with the weights (w(t) = g(t, b) : t = 1, . . . , T). Moreover, if (a * , b * ) is a local minimizer of the function (a, b) → f (T, a, b), then a * is the weighted median of (x * (t) = I(t)/g(t, b * ) : t = 1, . . . , T) endowed with the weights (w * (t) = g(t, b * ) : t = 1, . . . , T). Proof. We observe that for fixed b, the map a → f (T, a, b) is a convex function. Now, let us assume that (a * , b * ) is a local minimizer of the function (a, b) → f (T, a, b). Then, a * is the global minimizer of the convex function a → f (T, a, b * ). Hence, a * is the weighted median of (x * (t) = I(t)/g(t, b * ) : t = 1, . . . , T) endowed with the weights (w * (t) = g(t, b * ) : t = 1, . . . , T). Comparison of the Minimizers of the Map b → f (T, a(b), b) and the Minimizers of the Map (a, b) → f (T, a, b) The following proposition is obvious. Proposition 2. (1) For each fixed a, the map b → f (T, a, b) is above the map b → f (T, a(b), b) and they intersect at the curve a = a(b). (2) If b * is a local minimizer of the map b → f (T, a(b), b), then (a(b * ), b * ) is also a local minimizer of the map (a, b) → f (T, a, b). (3) The local minimizers of the map (a, b) → f (T, a, b) belong to the set {(a(b), b)

Proposition 3 .

 3 are some neighborhoods, respectively, of a * and b * . We derive that the minima of a ∈ W → f (T, a, b * ) is attained at b * . As a ∈ V → f (T, a, b * ) is convex, it has the unique minima b * = a(b * ). Which achieves the proof of 3). The proof of 2) works as follows. There exists a neighborhood V of b * such that f (T, a(b), b) ≥ f (T, a(b * ), b * ) for each b ∈ V. By definition of a(b), we have f (T, a, b) ≥ f (T, a(b), b) for all a. It follows that (a(b * ), b * ) is a local minimizer of (a, b) → f (T, a, b). Assume that b → f (T, a(b), b) has only a global minimizer. Then, the map (a, b) → f (T, a, b) may have many local minimizers, and b → a(b) is discontinuous at any point b * such that (a(b * ), b * ) is a local minimizer of the map (a, b) → f (T, a, b).

  point b near b * , and then b * is a local minimizer of the map b → f (T, a(b), b). This is absurd because b → f (T, a(b), b) has only a global minimizer.

Figure 1 .

 1 Figure 1. Daily infections of COVID-19 in China.

Figure 2 .Remark 1 .

 21 Figure 2. Confidence intervals using six minimas with T = 10 in the Gaussian model. Remark 1. Let us conclude this paragraph with a comparison between the output of the optim function and the output of the iteration of the optim function until the convergence using the same initialization with T = 10. As initialization, we use (a = 25, 325.01, l = 41.78141, s = 19.85630) for the two approaches; the use of the optim function leads to the following minima (23, 034.33, 36.03957, 16.80575, 338.2104), while the iteration of the optim function until convergence leads to (2137.277, 10.98621, 6.457341, 320.9898).Figure 3 illustrates the plots of daily infections t ∈ [1, 60] → I(t) of COVID-19 in China, the LAD regression

Figure 3 .

 3 Figure 3. Optimal time series for T = 10, for the Gaussian model with and without iterations.

Figure 4 .Figure 5 .

 45 Figure 4. Optimal time series for T = 10 for Gauss.

4. 4 .

 4 Confidence Intervals Using Six Minimas with T = 10 in the Lerch Model

Figure 6 .

 6 Figure 6. Confidence intervals using six minimas with T = 10 of the Lerch model.

Figure 7 .

 7 Figure 7. Optimal time series for T = 60 for Lerch and Gauss.

  ∑ T t=1 |I(t) -I m (t)| T , with the subscript m = Gauss (respectively, m = Lerch). The two models have the form (a, b) → f (T, a, b) = ∑ T t=1 |ag(t, b) -I(t)|. We have associated f (T, •) with the map S(T, •) : b → f (T, med(x(b), w(b)), b) with the sequence x(b) = (I(t)/ f (t, b), t = 1, . . . , T), and the weights w(b) = (g(t, b), t = 1, . . . , T), and med(x(b), w(b)) denotes the weighted median of x(b) endowed with the weights w(b). We showed that if b * is a local minimizer of S, then (med(x(b * ), w(b * )), b * ) is also a local minimizer of f (T, •). The converse is, in general, false, i.e., if (a * , b * ) is a local minimizer of f (T, •), then b * is not, in general, a local minimizer of S. However, if (a * , b * ) is the global minima of f (T, •), then b * is also the global minima of S. We showed that if S has only a global minima, then b → f (T, med(x(b), w(b)), b) is discontinuous at each local minimizer. Using the data of the daily infections of COVID-19 in China during the first wave, we showed numerically in the Gaussian case g(t, l, s) = exp(-(t-l) 2

Table 1 .

 1 A list of minimizers of (a, l, s) → f Gauss (T, a, l, s) with T = 10.

	a *	l *	s *	minima	% Predictions
	2088.911	10.119	5.712	319.545	0.66
	2137.277	10.98621	6.457	320.9898	0.66
	2301.818	12.186	7	322.358	0.68
	3229.843	16.346	9.609	325.995	0.88
	5526.387	22.018	12.183	331.487	0.60
	15,185.797	31.994	15.614	337.009	0.12

Table 2 .

 2 A list of minimizers of (a, z, v, s) → f Lerch (T, a, z, v, s) with T = 10.

	a *	z *	v *	s *	Minima % Predictions
	5.990886 ×10 -01	0.81023163	3.201870	-3.976447	311.9606	0.84
	1.354215 ×10 -08	0.59318684	8.174956	-10.683932	314.7595	0.70
	1.555525 ×10 -14	0.50711308	10.701817	-15.255833	315.4869	0.68
	8.235320 ×10 -34	0.35420137	16.512305	-28.741171	316.5241	0.66
	2.185815 ×10 -103	0.1647315	28.956740	-71.553876	317.5824	0.66
	7.23785 ×10 -266	0.05804243	45.948388	-160.673224	318.2142	0.66

Table 3 .

 3 Mode position and amplitude using Lerch.

	a *	z *	v *	s *	Mode Position	Amplitude
	5.990886 ×10 -01	0.81023163	3.201870	-3.976447	16.19885	2618.210
	1.354215 ×10 -08	0.59318684	8.174956	-10.683932	12.78678	2236.537
	1.555525 ×10 -14	0.50711308 10.701817	-15.255833	12.26928	2186.184
	8.235320 ×10 -34	0.35420137 16.512305	-28.741171	11.68264	2134.780
	2.185815 ×10 -103	0.1647315	28.956740	-71.553876	11.22172	2101.053
	7.23785 ×10 -266	0.05804243 45.948388 -160.673224	11.05461	2090.932

Table 4 .

 4 A list of minimizers of (a, z, v, s) → f Lerch (T, a, z, v, s) with T = 20. The last column contains the percentage of predictions.

	a *	z *	l *	s *	Minima % Predictions
	4.366541e-268	0.1073967	55.31169	-154.51447 392.5692	0.825
	3.422419e-262	0.1387492	47.81770	-121.91023 393.5286	0.825
	2.572282e-256	0.1708497	41.39999	-97.69044	394.7033	0.825
	1.155039e-253	0.1860852	38.76658	-88.52397	395.2740	0.825
	3.035334e-234	0.3069971	23.34634	-43.86187	400.4082	0.825
	4.970115e-230	0.3344249	20.71433	-37.77693	401.7952	0.825
	1.667726e-233	0.3634773	22.30914	-36.60867	403.6049	0.825

Table 5 .

 5 The unique minimizer of (a, l, s) → f Gauss (T, a, l, s) with T = 20.

	a *	l *	s *	Minima	% Predictions
	3712.1297	14.2556	7.60744	386.0318	0.80

Entropy 2023, 25, 1474. https://doi.org/10.3390/e25101474 https://www.mdpi.com/journal/entropy

Acknowledgments:

We would like to thank the Editor-in-Chief of Entropy, the associate Editor, and two anonymous reviewers for their helpful comments, which helped us to focus on improving the original version of the paper. The authors would like to thank the Editor, the Associate Editor, and the three referees for their very helpful comments, which led to a considerable improvement of the original version of the paper and a more sharply focused presentation.

Funding: This research received no external funding.

Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.

Data Availability Statement: Data are unavailable due to privacy.

Conflicts of Interest:

The authors declare no conflict of interest.

Appendix A. R Source Code

Here, we provide a source code for a better understanding of the proposed method.