Fabien Gogé
email: fabien.goge@inrae.fr

Laurent Thuriès

Youssef Fouad
email: youssef.fouad@agrocampus-ouest.fr

Nathalie Damay

Fabrice Davrieux

Géraud Moussard

Caroline Leroux

Séverine Trupin-Maudemain

Matthieu Valé

Thierry Morvan

Caroline Le Roux

A dataset of the chemical composition and near-infrared spectroscopy measurements of raw cattle, poultry and pig manure

Keywords:

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

a b s t r a c t

Combined with multivariate calibration methods, nearinfrared (NIR) spectroscopy is a non-destructive, rapid, precise and inexpensive analytical method to predict chemical contents of organic products. Nevertheless, one practical limitation of this approach is that performance of the calibration model may decrease when the data are acquired with different spectrometers. To overcome this limitation, standardization methods exist, such as the piecewise direct standardization (PDS) algorithm. The dataset presented in this article consists of 332 manure samples from poultry and cattle, sampled from farms located in major regions of livestock production in mainland France and Reunion Island. The samples were analysed for seven chemical properties following conventional laboratory methods. NIR spectra were acquired with three spec-trometers from fresh homogenized and dried ground samples and then standardized using the PDS algorithm. This important dataset can be used to train and test chemometric models and is of particular interest to NIR spectroscopists and agronomists who assess the agronomic value of animal waste.

© 2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Specifications

Value of the Data

• This dataset provides results of analyses of NIR spectroscopy and chemical composition of a large sample of cattle and poultry manures (collected from different livestock production regions). This dataset thus integrates the high compositional variability of these materials, which can be valuable in studies interested in assessing the agronomic value of these types of manure. • The data are of particular utility to NIR spectroscopists who test chemometric methods (e.g. regression, standardization) or agronomists who assess the agronomic value of animal waste. • The data are also particularly useful for studying effects of manure sample preparation (moisture, particle size) on standardized NIR spectra. • This article presents a case study of application of the piecewise direct standardization (PDS) algorithm, which can standardize data acquired with different spectrometers.

Data Description

The contents of dry matter (DM), total ammonium nitrogen (NH 4 ), total nitrogen (N), phosphorus pentoxide (P2O5), calcium oxide (CaO), magnesium oxide (MgO) and potassium oxide 1 , Fig. 1 ). The samples were also analysed by NIR spectroscopy using three instruments ( Table 2 ): two XDS Foss (belonging to CIRAD and ARVALIS, respectively) and one NIRFlex Buchi (belonging to LDAR), on fresh homogenized and dried ground samples. The XDS spectrometer of CIRAD was the reference instrument to which the other two were related. The absorbance spectra (1100-2498 nm) were standardized by the mean of three standard sealed cells (commercial compost, green waste compost and poultry litter) ( Fig. 2 ) and the PDS algorithm in order to use them in calibration models for future processing ( Fig. 3 ). The dataset is composed of .CSV and 3 Excel files ( Table 3 ). One .CSV file contains raw and standardized data of the three standard sealed cells acquired with the three spectrometers. Two Excel files contain standardized data of the 332 solid manures acquired from fresh homogenized and dried ground samples. One Excel file contains the chemical data. The dataset is available via the Data INRAE portal.

Experimental Design, Materials and Methods

Sample collection

A total of 196 cattle manure and 136 poultry manure samples were collected from major regions of livestock production in mainland France and Reunion Island. Broiler chicken were fed a mixture of cereals (wheat, maize, barley), oilseeds and their meals, vitamins and minerals in mainland France and on Reunion Island (the same feed). Dairy and feedlot cattle were fed hay or silage plus concentrated feed in mainland France and sugarcane residues plus concentrated feed on Reunion Island. Broiler chicken litter consisted of wheat straw in mainland France and Cryptomeria japonica woodchips on Reunion Island. The samples were sampled, collected along routes, prepared, preserved and analysed according to the following procedure adopted by the laboratories involved in this project. The samples were promptly frozen after collection to stop microbiological activity, which induces chemical transformations, and mineralization, which process that leads to causes NH4 losses during storage. In the laboratory, they were homogenized by crushing them in frozen form in a blender-cutter (Blixer Dito K45, Electrolux, Senlis, France). A subsample of each sample was dried in a forced-convection oven (at 40 °C for 4 days) and then ground (to 1 mm) in a cyclone mill (Cyclotec 1093, Foss, Silver Spring, MD, USA).

Chemical analytical methods

The samples collected in mainland France were analysed by LDAR, AUREA and ARVALIS laboratories, while those collected on Reunion Island were analysed by CIRAD's laboratory. In each laboratory, manure samples were analysed according to international (ISO) or French (NF) standards [START_REF]Amendement Organique -Dénominations[END_REF] for seven chemical components: DM, NH 4 , N, P 2 O 5 , CaO, MgO and K 2 O ( Table 1 ). DM was obtained by oven drying the partly dried (40 °C) samples at 103 ± 2 °C until they reached a constant weight. NH 4 and total N contents were measured by the Kjeldahl method (NF T90-015-AFNOR 1975). P 2 O 5 , CaO, MgO and K 2 O contents of only 158 of the 332 samples were measured by Inductively Coupled Plasma (Element XR Thermo Scientific). To address agronomic and environmental issues, we analysed N, NH4 and DM contents in all samples. To decrease analysis costs, the mineral compounds were analysed only in a subset of 158 samples that covered the variety manure types, farm types and sample-collection areas. All analyses were performed in duplicate, and values of the duplicates were averaged.

Spectroscopic analysis

Samples were scanned in triplicate using one of three spectrometers, depending on the laboratory: 80 samples scanned by an XDS device of CIRAD (Foss, Silver Spring, MD, USA), 96 samples scanned by the other XDS device of ARVALIS and 156 samples scanned by a NIRFlex device of LDAR (Büchi, Flawil, Switzerland). Each replicated measurement was the mean of 32 scans and was recorded in absorbance (Abs) using the following equation: Abs = log(1/Reflectance). For each sample, the final spectrum used in data processing and analysis was the mean of the triplicate.

NIR spectra were recorded for 1) fresh homogenized samples in a rectangular quartz cell (250 ml) and 2) dried ground samples in a Petri dish (10 cm diameter) for those scanned by ARVALIS and a circular quartz cell (5 cm diameter) for those scanned by CIRAD.

Standardization procedure

The spectral response (e.g. amplitude, wavelength shift) of a given sample analysed by NIR spectroscopy can differ among the types of instrument used (e.g. different technologies) and even between two spectrometers of the same brand (e.g. different production batches). Standardization methods ( [2-6] ) allow spectra measured by one instrument to be transformed as if they had been measured by another instrument.

The NIRFlex, based on Fourier-transform technology, measures absorbance as a function of wavenumber (cm -1 ), while the XDS Foss, based on dispersive technology, measures absorbance as a function of wavelength (nm). The wavenumber (40 0 0-12,50 0 cm -1 ) was converted to wavelength (80 0-250 0 nm). As the resolution step of wavelength of the NIRFlex was not constant after this conversion, the absorbance was recalculated at the wavelength of the XDS Foss using a shape-preserving piecewise cubic interpolation that preserves the convexity of scattered convex data.

To standardize the spectra measured by the spectrometers, three standard sealed cells (commercial compost, green waste compost and poultry litter) were measured with the three instruments. The common spectral range from 1100-2498 nm was maintained. The PDS algorithm [START_REF] Wang | Multivariate instrument standardization[END_REF] was used to standardize the spectra from the three instruments. The algorithm transforms data measured by a "slave" instrument to data that appears to have been measured by a "master" instrument. The algorithm has two steps: (1) wavelengths of the "slave" instrument are matched to those of the "master" instrument to correct for a possible wavelength shift, and (2) a transformation matrix F is created from the three spectra of the standard sealed cells measured by both the "slave" and "master" instruments. The transformation matrix F is composed of regression coefficients of multivariate linear (partial least squares) regression between the absorbance values at each λ i of the "master" spectra and those at ( λ i -k , λ i + k ) of the "slave" spectra, for the three spectra of the standard sealed cells ( k = 3 is the size of the wavelength window). Finally, the transformation is applied to the spectra of the samples measured by the "slave" instrument. In this study, the Foss XDS of CIRAD was the "master" instrument, while the two others were the "slave" instruments. Fig. 2 shows effects of the standardization based on the standard sealed cell spectra. Fig. 3 shows the standardized spectra for the fresh homogenized and dried ground samples.

Fig. 1 .

 1 Fig. 1. Boxplots of contents (% of wet weight) of chemical properties of manure samples.

Fig. 2 .

 2 Fig. 2. Near-infrared spectra of the three standard sealed cells measured with the three spectrometers before (commercial compost (1a), green waste compost (2a) and poultry litter (3a)) and after piecewise direct standardization (commercial compost (1b), green waste compost (2b) and poultry litter (3b)).

  Sample name (sample_name) Dry matter (% of wet weight).DM Total ammonium nitrogen (% of wet weight).NH4 Total nitrogen (% of wet weight).N Calcium oxide (% of wet weight).CaO Potassium oxide (% of wet weight).K2O Magnesium oxide (% of wet weight).MgO Phosphorus pentoxide (% of wet weight).P2O5 Type of manure (type_manure) Spectrometer Township Country spectra_standard_cells.csv The first column contains the sample identification code. The next two contain the spectrometer used for acquisition and whether the data are standardized or not, respectively. The other columns contain the absorbance data recorded by the spectrometers from 1100-2498 nm for the three standard sealed cells. spectra_FH_Abs_STD_1100_2498 nm_STD.xlsx The first column contains the sample identification code, and the others contain the standardized absorbance data recorded from 1100-2498 nm for the 332 fresh homogenized samples. spectra_DG_Abs_STD_1100_2498 nm_STD.xlsx The first column contains the sample identification code, and the others contain the standardized absorbance data recorded from 1100-2498 nm for the 332 dried ground samples.

Fig. 3 .

 3 Fig. 3. Standardized near-infrared spectra for fresh homogenized samples (red) and dried ground samples (blue).

Table

	Subject	Chemistry, Near-Infrared (NIR) Spectroscopy
	Specific subject area	Cattle and poultry manure composition
	Type of data	.CSV and Excel files, figures, tables
	How data were acquired	Absorbance spectra were acquired with 3 NIR spectrometers (2 XDS Foss, 1
		NIRFlex Buchi). Conventional laboratory analyses (dry matter (DM), total
		ammonium nitrogen (NH 4 ), total nitrogen (N), phosphorus pentoxide (P2O5),
		calcium oxide (CaO), magnesium oxide (MgO) and potassium oxide (K2O))
	Data format	Chemical analysis data, raw and standardized spectral data of standard cells,
		standardized spectral data of manure
	Parameters for data collection	332 manure samples were analysed by NIR spectroscopy using 3
		spectrometers. Both fresh homogenized and dried ground samples were
		scanned to acquire spectra.
	Description of data collection	The 7 chemical contents were determined using conventional laboratory
		methods. NIR spectroscopy was combined with chemometric tools to
		standardize the spectral data, using the XDS Foss spectrometer of CIRAD as the
		reference instrument.
	Data source location	Manure samples were collected from different farms in mainland France and
		Reunion Island (French overseas territory)
	Data accessibility	Data presented in .CSV and Excel file formats are available in this article. Data
		are deposited in a public repository.
		Repository name: Data INRAE
		Data identification number: 10.15454/JIGO8R
		Direct URL to data: https://doi.org/10.15454/JIGO8R

Table 1

 1 Statistics of contents of dry matter and chemical properties of manure samples (% of wet weight). SD: standard deviation

	Property	n	Mean	SD	Min	Max	Range
	DM	332	37.28	20.06	11.25	82.48	71.22
	NH 4	332	0.262	0.276	0.001	1.086	1.085
	N	332	1.36	1.09	0.25	4.15	3.90
	CaO	158	0.57	0.56	0.09	3.11	3.01
	K 2 O	158	1.02	0.65	0.19	3.84	3.66
	MgO	158	0.23	0.19	0.06	1.05	0.99
	P 2 O 5	158	0.48	0.58	0.09	3.02	2.93

Table 2

 2 Basic features of the near-infrared spectrometers.

	Instrument	Foss XDS (CIRAD and ARVALIS)	Buchi NIRFlex (LDAR)
	Technology	Dispersive	Fourier transform
	Spectral range	40 0-250 0 nm	40 0 0-12,50 0 cm -1
	Spectral resolution	2 nm	8 cm -1
	Scans	32	32
	(K 2 O) of 332 poultry and cattle manure samples are available in this article ( Table

Table 3

 3 Contents of the dataset

Acknowledgments

This study was partly funded by the Ministry of Food, Agriculture and Fisheries (CasDar project no. 9109 ) and ADEME (French Environment and Energy Management Agency). Collection of samples from Reunion Island was partly supported by CIRAD and the Réunion Region (convention no. 20090885 ) via the European Regional Development Fund. G. Moussard and L. Thuriès were partly supported on Reunion Island by the European Agricultural Fund for Rural Development (FEADER, Measure no. 111.34, Convention no. DEE/20141575); the Conseil Regional de La Réunion; the French Ministry of Agriculture, Food and Fisheries; and CIRAD within the framework of the project "Services et impacts des activités agricoles en milieu tropical" (Siaam). The authors thank Virginie Parnaudeau (INRAE, Rennes); Anne Mouteau and Pierre Dardenne (formerly from CRAW, Gembloux, Belgium); Denis Bastianelli, Laurent Bonnal and Jean-Luc Marger (from CIRAD, Montpellier) and Hélène Ducept (formerly from LDAR, Laon). Samples in mainland France were collected with the collaboration of ITAVI, IFIP, IDELE, BCEL Ouest and the Chambers of Agriculture of Brittany, Lorraine, Loiret and Vendée. On Reunion Island, Avipole (federation of poultry farmers), the Réunion Chamber of Agriculture, and the Fédération Réunionnaise des Coopératives Agricoles (FRCA) facilitated sample collection. Samples were collected and prepared for spectra standardization with the help of Master's engineering student interns (Hanane Aït Aissa, Antoine Bazot and Thomas Rottatinti).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that have, or could be perceived to have, influenced the work reported in this article.


Outline

Context: linked selection and demographic inference

The IICR

Results

Panmictic model with several classes of (stationary) N e Population structure

Conclusions and perspectives

Outline

Context: linked selection and demographic inference

The IICR

Results

Panmictic model with several classes of (stationary) N e Population structure

Conclusions and perspectives

Context

Genetic diversity shaped by Genetic [START_REF] Drift | Population demography and structure (Nei et al[END_REF][START_REF] Drift | Population demography and structure (Nei et al[END_REF]. Population demography and structure (Nei et al, 1975;[START_REF] Wakeley | [END_REF][START_REF] Wall | Linked selection (Hill and Robertson, 1966) Linked selection (BGS or SW) pervasive[END_REF].

Linked selection (Hill and Robertson, 1966)

Context

Genetic diversity shaped by Genetic [START_REF] Drift | Population demography and structure (Nei et al[END_REF][START_REF] Drift | Population demography and structure (Nei et al[END_REF]. Population demography and structure (Nei et al, 1975;[START_REF] Wakeley | [END_REF][START_REF] Wall | Linked selection (Hill and Robertson, 1966) Linked selection (BGS or SW) pervasive[END_REF].

Linked selection (Hill and Robertson, 1966)

Context

Genetic diversity shaped by Genetic [START_REF] Drift | Population demography and structure (Nei et al[END_REF][START_REF] Drift | Population demography and structure (Nei et al[END_REF]. Population demography and structure (Nei et al, 1974;[START_REF] Wakeley | [END_REF][START_REF] Wall | Linked selection (Hill and Robertson, 1966) Linked selection (BGS or SW) pervasive[END_REF].

Linked selection (Hill and Robertson, 1966)

The variable N e hypothesis

Linked selection modelled as first approximation by a change of N e (Hill and Robertson, 1966;Charlesworth, 2009):

decreasing for BGS and SW. increasing for BalS.

Variable levels of N e genome-wide (Gossmann et al, 2011; Jimenez-Mena et al, 2016) reflecting the variations of selection constraints (recombination rate, functional density) .

Our approach

Assume linked selection can be approximated by modelling several classes of N e along the genome.

Study the genome-wide distribution of coalescence times (T 2 ) in this framework.

Influence of linked selection on demographic inference approaches such as PSMC (Li and Durbin, 2011). Denoting λ(t) = N(t) N(0) ,

P PSC (T 2 ≥ t) = e -t 0 1 λ(s) ds
We obtain f PSC (t)

P PSC (T 2 ≥ t) = 1 λ(t) → λ(t) Inverse Instantaneous Coalescence Rate (IICR). (inferred by PSMC).
Let F (t) = P(T 2 ≤ t), and

F (t) = f (t).
Introducing the IICR

λ(t) = 1 -F (t) f (t)
we obtain (Mazet et al, 2016)

F (t) = 1 -e -t 0 1 λ(s) ds = F PSC ,λ (t)
⇒ There always exists a PSC model mimicking perfectly the distribution of T 2 observed from two sequences. Class i evolves under the WF model with λ i N diploids.

Implications: inference

T i 2 ∼ E(µ i ), µ i = 1/λ i Genome-wide distribution of T 2 f (t) = K i=1 a i µ i e -µ i t IICR IICR(t) = - 1 -F (t) f (t) = K i=1 a i e -µ i t K i=1 a i µ i e -µ i t
. IICR (t) > 0 ∀t → spurious signal of population size decline (forward in time).

Starting value:

IICR(0) = 1 K i=1 a i λ i
→ harmonic mean of the λ i 's weighted by the a i 's IICR(t) → λ max as t → +∞ → IICR plateaus at λ max (whatever a max ).

λ 1 = 0.1 (BGS/SW), λ 2 = 1 (neutral).

→ BGS/SW impact recent past IICR.

λ 1 = 1 (neutral), λ 2 = 10 (BalS).

→ BalS strongly impacts intermediate to ancient past IICR, even if in small proportion. Scaled migration rate M = 4Nm (i.e. not affected by selection)

Analytical expression for f i () and IICR i () (Mazet et al, 2016).

IICR i (0) = λ i IICR i (+∞) ≈ n(λ i + 1 M ) for large n
Genome-wide IICR() deduced from the f i ()'s. → Limited effect of selection except in ancient past.

Under panmixia, linked selection leaves a signature of population decline from the largest genomic N e to the harmonic mean of the N e distribution.

Expected magnitude of this decline is about 3-5 fold for humans and Drosophila melanogaster.

BGS/SW impact recent past IICR, BalS ancient past.

Larger effect of BalS, even if in much lower proportions.

Linked selection partly masked by population strucure.

Outline

  Context: linked selection and demographic inferenceThe IICRResultsPanmictic model with several classes of (stationary) N e Population structureConclusions and perspectivesEstimation of past variations of N e through time from a single diploid genomeObserved values of H e along the genome ⇒ distribution of T 2 .t = time before present (backward), in units of 2N generations.

Use

  PSMC as a summary statistics to accept / reject demographic scenarios (Chikhi et al, 2018; Rodriguez et al, 2018). Piecewise stationary n-island model (Arredondo et al, 2021) with relative proportion a i .

λ 1

 1 = 0.1 (BGS/SW), λ 2 = 1 (neutral), λ 3 = 3 (BalS). a 3 = 0.01 a 1 = 0.5 → stronger effect of BalS vs BGS/sweeps for the same proportion. → intermediate plateau depending on a 1 and a 3 Gossmann et al (2011) → Very similar to the results obtained for humans. Elyashiv et al (2016): direct modelling of linked selection and full DFE using recombination maps and detailed annotation. → Bi-modal λ distribution and slight inflexion of the IICR → Quite similar to other IICRs (long term 4-fold decline) n-island model Class i evolves under an island model with n demes of diploid size λ i N.

n→

  = 10, non stationary M (4 epochs). three classes: λ 1 = 0.1, λ 2 = 1, λ 3 = 3. a 3 = 0.01 a 1 = 0.5 High proportions of BGS/SW (e.g. a 1 = 0.8) impact the IICR. → Effect of BalS in ancient past more dramatic. λ distribution from Gossmann et al (2011).

→ long term 6-fold decline from t = 10.

→ Similar to panmixia for M >> 1.

Model generalization

Panmictic constant size model with

replaced by

for a T 2 distribution with pdf f i ().

.

Outline

Context: linked selection and demographic inference

The IICR

Results

Panmictic model with several classes of (stationary) N e Population structure

Conclusions and perspectives

Where is the bug?

The decline signature comes for the variability of N e along the genome, not considered by single locus models (Zeng and Charlesworth, 2011; Walczak et al, 2012). models with regular and short scale alternance of neutral and selected loci [START_REF] Johri | Expected signature of population expansion based on: Theoretical predictions at single loci with (Zeng and Charlesworth, 2011) or without recombination[END_REF].

Variable N e hypothesis less appropriate for samples properties like the Site Frequency Spectrum (Ewing and Jensen, 2016).

excess of singletons in BGS vs neutral models (Charlesworth et al, 1993).

Pros and cons of the IICR approach

Limited by the validity of the variable N e approximation: heterozygosity / pairwise coalescence rate or strong purifying selection. Perspective: joint inference of demography and linked selection parameters?

Advertizing 1

PhD funding starting in september! Genomic consequences of habitat fragmentation.

Demographic inference using IICR and AFS.

Mathematical modelling / methodological development.

contact lounes.chikhi@univ-tlse3.fr, omazet@insa-toulouse.fr.

Project starting in CBGP with M. Gautier, N. Rode, A. Estoup.

Genomic prediction of the adaptive potential of a population in a new environment.

Build from comparing allele frequencies in diverse environments.

Vulnerability to climate change (Bay et al, 2018).

Biological invasions, data in Drosophila suzukii and Harmonia Axyridis.

Comparison / development of prediction methods.

PhD funding starting in september!


Simon Boitard

Armando Arredondo

Lounès Chikhi

Olivier Mazet

Olivier Mazet Heterogeneity

Heterogeneity in effective size across the genome: effects on the Inverse Instantaneous Coalescence Rate (IICR) and implications for demographic inference under linked selection

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

Heterogeneity in effective size across the genome: effects on the Inverse Instantaneous Coalescence Rate (IICR) and implications for demographic inference under linked selection

Context

Genetic diversity shaped by Genetic Drift (Kimura, 1968;1983). Population demography and structure (Nei et al, 1974;Wakeley, 1999;Wall et al, 2002).

Linked selection (Hill and Robertson, 1966)

Inverse Instantaneous Coalescence Rate (IICR)

Arbitrary evolution model.

T 2 coalescence time between two haploid sequences, in units of 2N generations.

Let F (t) = P(T 2 ≤ t), and F (t) = f (t).

IICR(t) = 1 -F (t) f (

Panmictic model with "selection"

K genomic classes with relative proportion a i .

Class i evolves under the WF model with λ i N diploids.

T i 2 ∼ E(µ i ), µ i = 1/λ i Genome-wide distribution of T 2 f (t) = K i=1 a i µ i e -µ i t IICR IICR(t) = - 1 -F (t) f (t) = K i=1 a i e -µ i t K i=1 a i µ i e -µ i t
.

Mathematical properties IICR(t) increasing (backward in time) → spurious signal of population size decline.

Starting value:

IICR(0) = 1 K i=1 a i λ i IICR(t) → λ max as t → +∞ λ 1 = 0.1 (BGS/SW), λ 2 = 1 (neutral).
→ BGS/SW impact recent past IICR.

λ 1 = 1 (neutral), λ 2 = 10 (BalS).

→ BalS impacts intermediate to ancient past IICR, even if in small proportion.

Example 1: stationary n island model n = 10 islands, two classes: λ 1 = 0.1, λ 2 = 1.

→ Selection effect weaker than under panmixia for M ≤ 1.

Main conclusions

Under panmixia, linked selection leaves a signature of population decline from the largest genomic N e (with PSMC or MSMC).

BGS/SW impact recent past IICR, BalS ancient past.

Larger effect of BalS, even if in much lower proportions.

Linked selection partly masked by population strucure.

Ref: Genetics, 220(3), iyac008.

Panmictic model with λ 1 = 0.1, λ 2 = 1.

PSMC applied to genomes, simulated with variable N e (class 1, class 2, class 1 . . . ).

→ consistent for large scale N e variartions.

λ 1

 1 = 0.1 (BGS/SW), λ 2 = 1 (neutral), λ 3 = 3 (BalS). a 3 = 0.01 a 1 = 0.5 → stronger effect of BalS vs BGS/sweeps for the same proportion. → intermediate plateau depending on a 1 and a 3

  , including ...

	Selective sweeps (SW; Maynard Smith and Haigh, 1974;
	Gillespie, 1991).
	Background selection (BGS; Charlesworth et al, 1993).
	Balancing selection (BalS; Ford 1975).

Respective role of these forces? (Kern and Hahn 2018; Jensen et al 2019). Linked selection pervasive (Elyashiv et al, 2016; Pouyet et al, 2018). Linked selection biases demographic inference (Ewing and Jensen, 2016; Schrider et al, 2016; Pouyet et al, 2018; Johri et al 2021) Linked selection modelled by a change of N e (Hill and Robertson, 1966; Charlesworth, 2009): decreasing for BGS and SW. increasing for BalS. Variable levels of N e genome-wide (Gossmann et al, 2011; Jimenez-Mena et al, 2016) reflecting variations of selection constraints.

Objectives

Characterize the effects of linked selection on genome-wide patterns of genetic diversity.

Model linked selection through several classes of N e . Study the genome-wide distribution of coalescence times (T 2 ). Consequences for demographic inference with PSMC (Li and Durbin, 2011) or MSMC (Schiffels and Durbin, 2014).

More complex models

Genuine population size changes.

Population structure (stationary or non stationary).

Transient (rather than reccurent) selection.

Advertizing

PhD positions in population genomics available in september 2022.

Inference in spatial models.

Inference from genomic time series.

Genomic offset.

The decline signature comes for the variability of N e along the genome, not considered by single locus models (Zeng and Charlesworth, 2011;Walczak et al, 2012). models with regular and short scale alternance of neutral and selected loci (Johri et al, 2021).

Variable N e hypothesis less appropriate for samples properties like the Site Frequency Spectrum (Ewing and Jensen, 2016). excess of singletons in BGS vs neutral models (Charlesworth et al, 1993).

Realistic N e distributions: humans Gossmann et al (2011): N e distribution assumed log-normal and estimated from polymorphism and divergence data.

→ long term 6-fold decline from t = 10.


Melani Barrios
email: melani@fceia.unr.edu.ar

Gabriela Reyero

Mabel Tidball

Mihály Pituk

HARVEST MANAGEMENT PROBLEM WITH A FRACTIONAL LOGISTIC EQUATION

Keywords: fractional logistic equations, Caputo derivative, fractional optimal control problem, natural resource management ; Secondary 34A08, 49-XX, 91B76

In this article, we study a fractional control problem that models the maximization of the pro t obtained by exploiting a certain resource whose dynamics are governed by the fractional logistic equation. Due to the singularity of this problem, we develop di erent resolution techniques, both for the classical case and for the fractional case. We perform several numerical simulations to make a comparison between both cases.

INTRODUCTION

It is very well known that the logistic equation describes the population growth. The continuous Logistic model is described by a rst order ordinary di erential equation. The model describes the population growth that may be limited by certain factors like population density, [START_REF]An introduction to structured population dynamics[END_REF]. The continuous form of the logistic equation is,

̇ ( ) = ( ) 1 -( ) .

In the above equation, ( ) indicates population at time , > 0 represents the Malthusian parameter expressing the growth rate of species and denotes the carrying capacity. Motivated by its applications in di erent scienti c areas (electricity, magnetism, mechanics, uid dynamics, medicine, etc., [START_REF]A fractional calculus of variations for multiple integrals with application to vibrating string[END_REF][START_REF]On the initial-boundary-value problem for the time-fractional di usion equation on the real positive semiaxis[END_REF][START_REF]Applications of fractional calculus in physics[END_REF][START_REF]Theory and applications of fractional di erential equations[END_REF]), fractional calculus is in development, which has led to large growth in its study in recent decades. The fractional derivative is a nonlocal operator, [START_REF]The analysis of fractional di erential equations: An application-oriented exposition using di erential operators of Caputo type[END_REF][START_REF]Fractional di erential equations: an introduction to fractional derivatives, fractional di erential equations, to methods of their solution and some of their applications[END_REF], making fractional di erential equations good candidates for modeling situations in which it is important to consider the history of the phenomenon studied [START_REF]Study of a fractional-order model for HIV infection of CD4+ T-cells with treatment[END_REF], unlike the models with a classical derivative where this is not taken into account. There are several de nitions of fractional derivatives. The most commonly used are the Riemann-Liouville fractional derivative and the Caputo fractional derivative. It is important to note that while the Riemann-Liouville fractional derivative [START_REF]The fractional calculus theory and applications of di erentiation and integration to arbitrary order[END_REF], is historically the most studied approach to fractional calculus, the Caputo fractional derivative is more popular among physicists and scientists due to the fact that the formulation of initial value problems with this type of derivative is more similar to the formulation with the classical derivative.

The fractional order logistic equation has been discussed in the literature, [START_REF]Solving fractional-order logistic equation using a new iterative method[END_REF][START_REF]Numerical approximations and Padé approximants for a fractional population growth model[END_REF]. A detailed study of existence, uniqueness, stability and approximate solutions of this equation can be found in [START_REF] Pituk | Dynamical analysis of fractional-order modi ed logistic model[END_REF][START_REF]Numerical solution of a logistic growth model for a population with Allee e ect considering fuzzy initial values and fuzzy parameters[END_REF][START_REF]Estabilidad de una generalización de ecuaciones logísticas fraccionarias[END_REF][START_REF]On the fractional-order logistic equation[END_REF][START_REF]Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel[END_REF][START_REF]Numerical studies for solving fractional-order Logistic equation[END_REF].

When the logistic equation is used to describe the natural evolution of a species, it is logical to think about the exploitation of this resource. For this reason consider a natural resource management problem using fractional derivatives. The fractional equation can better represent the evolution of the resource, in fact in this work we want to see how a fractional derivative a ects the optimal extraction decision.

The principal objective of this work is to solve an optimal control problem of extraction management. Due to the nature of the problem, it can be turned into a Fractional Variational problem. In the last years numerous works have been developed extending the theory of the variational calculus [START_REF]The Calculus of Variations[END_REF] to problems of fractional variational calculus [START_REF]Formulation of Euler-Lagrange equations for fractional variational problems[END_REF][START_REF]Método numérico de tipo L1 para problemas variacionales fraccionarios[END_REF][START_REF]Calculus of variations involving Caputo-Fabrizio fractional di erentiation[END_REF][START_REF]A simple accurate method for solving fractional variational and optimal control problems[END_REF][START_REF]A general framework for optimal control of fractional nonlinear delay systems by wavelets[END_REF][START_REF]Discrete chebyshev polynomials for solving fractional variational problems[END_REF][START_REF]New aspects on the fractional euler-lagrange equation with non-singular kernels[END_REF]. With the purpose of solving fractional variational problems, there exist two theorems of optimality conditions: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives [START_REF]Generalized Euler-Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative[END_REF][START_REF]Fractional Euler-Lagrange di erential equations via Caputo derivatives[END_REF][START_REF]Necessary and su cient conditions for the fractional calculus of variations with Caputo derivatives[END_REF][START_REF]On exact solutions of a class of fractional Euler-Lagrange equations[END_REF][START_REF]Advanced methods in the fractional calculus of variations[END_REF][START_REF]Introduction to the fractional calculus of variations[END_REF], and another Euler-Lagrange equation that involves only Caputo derivatives [START_REF]Analysis of a class of boundary value problems depending on left and right Caputo fractional derivatives[END_REF][START_REF]An Euler-Lagrange Equation only Depending on Derivatives of Caputo for Fractional Variational Problems with Classical Derivatives[END_REF][START_REF]Numerical solution of Euler-Lagrange equation with Caputo derivatives[END_REF][START_REF]The DuBois-Reymond fundamental lemma of the fractional calculus of variations and an Euler-Lagrange equation involving only derivatives of Caputo[END_REF]. As it is not yet known which of these theorems is the most suitable for solving this type of problem, for qualitative reasons the second method will be used to solve our problem. The structure of this article is as follows: in Section 2, the classical control problem and its solution, the fundamental concept of fractional derivatives and fractional control and variational problems, are presented. In Section 3, the solution of a fractional control problem, the comparison between both problems and numerical approximations of the solutions are discussed. Finally, the last section, is dedicated to the conclusions.

PRELIMINARIES

Some known facts are summarised for the reader's convenience.

A simple optimal control problem

We present a classical optimal control model, which consists of maximizing the total extraction (or harvest) of a certain renewable resource in the interval [0, ], where a rst order ordinary di erential equation describes the evolution of the dynamics,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max 0 -ℎ( ( )) d ̇ ( ) = ( ) 1 -( ) -ℎ( ( )) (0) = 0 ( ) = ℎ min ≤ ℎ( ( )) ≤ ℎ max .
(2.1) represents the nal time, 0 the initial condition, the nal condition, ℎ min and ℎ max minimum and maximum harvest and -represents a discount factor with ≥ 0 the instantaneous annual discount rate which can be zero.

The natural resource follows a logistic growth function, where > 0 is the intrinsic growth rate, > 0 the carrying capacity of the resource and ℎ( ) is the harvest. Now we will proceed to solve the problem (2.1).

Mathematica Pannonica New Series 27 /NS 1/ (2021) 2, 152-163

Clearing ℎ( ( )) from the dynamics and replacing it in the functional to maximize results,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max 0 - ( ) 1 -( ) -̇ ( ) d (0) = 0 ( ) = ℎ min ≤ ℎ( ( )) ≤ ℎ max , which is a variational problem with Lagrangian ( , , ̇ ) = - ( ) 1 - ( ) -̇ ( ) .
Then, its Euler-Lagrange equation results

- ̇ = 0, from which is obtained 1 - 2 ( ) -= 0.
That is, the optimal population is constant

* = 2 1 - , (2.2)
while the optimal harvest is

ℎ * = * 1 - * , (2.3)
then,

ℎ * = 4 2 -2 .
(2.4) REMARK 2.1. It can be seen that since the solution of the Euler-Lagrange equation * ( ), is a constant function, it will not be able to verify the imposed boundary conditions. For this reason, it can be said that the problem is singular.

For its resolution, the Nearest Feasible Paths theorem will be used, and its proof can be seen in [START_REF]Nearest feasible paths in optimal control problems: Theory, examples, and counterexamples[END_REF]. This theorem, as its name implies, shows that the optimal solution is to go as fast as possible to * ( ). With this result, the method consists in nding ( ) the solution to the problem that begins in (0) = 0 and uses the minimum harvest ℎ min if 0 < * ( ) or uses the maximum harvest ℎ max if 0 > * ( ), until it reaches the value * ( ) in the time min to be determined. Then it must be found ( ), a solution to the problem that uses the minimum harvest ℎ min if > * ( ) or uses the maximum harvest ℎ max if < * ( ), until it reaches the value ( ) = , from the time max to be determined so that it is the rst time when ( ) gets the value * ( ).

In summary, the following optimal solution is obtained

* ( ) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ( ) if 0 ≤ ≤ min 2 1 - if min ≤ ≤ max ( ) if max ≤ ≤ (2.5)
and the optimal harvest obtained with this procedure is

ℎ * ( ) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ℎ min if ( ) > * ( ) 4 2 -2 if ( ) = * ( ) ℎ max if ( ) < * ( ), (2.6)
taking into account that for it must be satis ed ℎ min ≤ 4 REMARK 2.2. The problem without discount is obtained as a particular case, writing = 0 in the equations (2.5) and (2.6).

In the next section, this problem will be solved but in its fractional version. To see more about fractional calculus refer to appendix at the end of the paper.

Fractional control and variational problems

To solve a fractional control problem, tools of fractional variational analysis will be used. For this, a brief introduction to them is presented.

Consider the following problem of the fractional calculus of variations: nd a function ∈ that optimizes (minimizes or maximizes) the functional

( ) = ( , , ) d , with a Lagrangian ∈ 1 ([ , ] × ℝ 2 ) and = { ∶ [ , ] → ℝ ∶ ∈ 1 ([ , ]), ∈ ([ , ])},
subject to the boundary conditions ( ) = , ( ) = . Now the Euler-Lagrange equation for this problem will be stated, its proof is in [START_REF]The DuBois-Reymond fundamental lemma of the fractional calculus of variations and an Euler-Lagrange equation involving only derivatives of Caputo[END_REF]. THEOREM 2.3. Let be an optimizer of in with ∈ 2 [ , ] × ℝ 2 subject to boundary conditions ( ) = , ( ) = , then satis es the fractional Euler-Lagrange di erential equation

+ = 0.
(2.7)

MAIN RESULT

Now we are ready to prove the following main result.

Fractional model

The fractional version of the problem proposed in the rst section is,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max 0 -ℎ( ( )) d 0 [ ] ( ) = ( ) 1 -( ) -ℎ( ( )) (0) = 0 ( ) = ℎ min ≤ ℎ( ( )) ≤ ℎ max . (3.1)
The only di erence with the problem exposed in Section 2.1, is that in the dynamic equation

0 [ ] ( ) = ( ) 1 - ( ) -ℎ( ( )), (3.2)
the rst-order derivative no longer appears, but now intervenes 0 [ ] ( ), the left Caputo fractional derivative of order 0 < ≤ 1.

The derivatives with fractional order contains partially or totally the history, temporary future or the spatial behavior of the function, averaged in some way. This transforms the fractional di erential equations on suitable candidates for the modeling of memory phenomenon or subsequent e ects, those in which what happens at a point on the space or at an instant of time depends on an interval (spatial or temporal) that contains the point or the instant.

The Riemann-Liouville fractional derivative had an important role in the development of the fractional calculus theory, and was used successfully in strictly mathematical applications. But when it was using for mathematical modeling of real physical phenomena with fractional di erential Unauthenticated | Downloaded 04/22/22 02:58 PM UTC equations, the problem of the initial conditions also of fractional order emerged. These types of conditions are not physically interpretable and present a considerable obstacle when making practical use of fractional calculus. The Caputo di erential operator, in contrast to the Riemann-Liouville operator, uses derivatives of integer order as initial conditions, that is, initial values that are physically interpretable as in the models with integer derivatives. The de nition that follows represented a notable practical advance in the study of physical phenomena such as those of the viscoelastic type and others.

Finally, the fractional derivative at of a function is a non-local operator, depending on past values of (left derivatives) or future values of (right derivatives). In physics, the right fractional derivative of ( ) is interpreted as a future state of the process ( ). For this reason, the right derivative is usually neglected in applications, when the present state of the process does not depend on the results of the future development. However, the left fractional derivative of ( ) is interpreted as a past state of the process ( ), in which memory e ects intervene.

Since the evolution of a certain resource depends on its past, we have decided to choose the left Caputo fractional derivative for modeling its evolution, see [START_REF]Duality for the left and right fractional derivatives[END_REF]. Now the problem (3.1) will be solved. Clearing ℎ( ( )) from the dynamic equation (3.2) and replacing it in the functional to maximize, it results,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max 0 - ( ) 1 - ( ) -0 [ ] ( ) d (0) = 0 ( ) = ℎ min ≤ ℎ( ( )) ≤ ℎ max .
This is a variational problem, now fractional, with fractional Lagrangian

( , , 0 [ ]) = - ( ) 1 - ( ) -0 [ ] ( ) ,
belonging to 2 [0, ] × ℝ 2 . Using Theorem 2.3, its fractional Euler-Lagrange equation (2.7) results,

+ 0 [ ] = 0, from which we have - 1 - 2 ( ) -( -) 1-- 1,2-( ( -)) = 0,
where 1,2-( ( -)) is the Mittag-Le er function of two parameters, de ned in equation (3.11). We can conclude that the optimal solution is *

( ) = 2 1 -( -) 1--( -) 1,2- ( -) . (3.3)
And the optimal harvest is obtained from this expression

ℎ * ( ) = * ( ) 1 - * ( ) -0 [ * ] ( ). (3.4)
Note that in the classical case the optimal population is constant, which is not happening in this case, therefore in the optimal harvest a term appears with the Caputo derivative of * ( ), that due to its di culty, we must calculate it numerically. REMARK 3.1. We can see that, as in the classical case, the optimal solution * ( ), although it is not a constant, will also not verify the established boundary conditions. This means that we are once again faced with a singular problem.

For its resolution, it would be possible to resort again to a Nearest Feasible Paths theorem, but in a fractional version. However, it has not been carried out yet because the proof of the theorem requires the use of a fractional Green theorem, which at the moment is only available for rectangular regions, [START_REF]Fractional variational calculus with classical and combined Caputo derivatives[END_REF], and its version is necessary for all types of regions. REMARK 3.2. The solution of the problem without discount ( = 0) is a particular case of the solution of the problem (3.1).

The optimal population is * ( ) = 2 and the optimal harvest is obtained from this expression ℎ * ( ) = 4 .

It can be observed that the solution is the same for the classical case. In the case with = 0, the solution and the optimal harvest is the same for all 0 < ≤ 1. The reason of this is that there is no time-dependent factor that multiplies 0

[ * ] ( ) in the fractional variational problem resulting from taking = 0.

Its fractional Euler-Lagrange equation is

1 - 2 ( ) + [(-1)] = 0, then, 1 - 2 ( ) = 0.
Since the Caputo derivative of a constant is also zero, as in the case of the classical derivative, then the same Euler-Lagrange equation is obtained for all 0 < ≤ 1 and its solution is independent of that value.

To perform a graphic analysis of the solutions to the discounted problem in both the fractional and classical cases, we must see a speci c example.

Example and comparison

Consider , , , 0 , , , ℎ min and ℎ max as in the example The Paci c Halibut Fishery, in [START_REF]Mathematical bioeconomics: the optimal management of renewable resources[END_REF], and 0 < ≤ 1.

The problem results, In Figure 1, the solutions corresponding to = 1, = 0.9, = 0.8, = 0.6 and = 0.4 were shown. It can be noticed that when tends to 1, the solutions converge to the classical solution. Also, it can be observed that the solutions not verify the established boundary conditions, their initial and nal values change.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
From now on, a xed value of is assumed to evade the problem of needing a fractional Nearest Feasible Paths theorem, being able to obtain the boundary conditions (0) and (10) of this fractional solution, in order to make a comparison with the classical problem (with = 1). The following problem will be considered, A comparison of the results will be made between = 1 (classical version) and = 0.6 (fractional version).

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max 10 0 -0.01 ℎ( ( )) d 0 [ ] ( ) = 0. 71 ( ) 1 -( ) 80.5 -ℎ( ( )) (0)
Solving the problem in its classical version as in the previous section, using (2.2) and (2.4) we have * ( ) = 39.6831 and ℎ * ( ) = 14.2859.

It can be observed that * ( ) does not verify the required boundary conditions, therefore, the Nearest Feasible Paths theorem will be used to solve the problem.

As explained in the last section we obtain the optimal solution as a most rapid approach to * , * ( )

= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
and the optimal harvest obtained with this procedure is In Figure 2, the optimal populations corresponding to both cases can be observed. It can be noticed that the optimal solution given by the fractional problem, * 0.6 ( ), is lower than the optimal population of the classical problem, * ( ), most of the time. It means that the version with = 0.6 shows a deterioration of the state of the stock with respect to the case = 1 which is only recovered at the end by the fact that it has to verify the nal condition. In Figure 3, the optimal harvests are considered. Since 0 0.6 * 0.6 ( ) cannot be obtained exactly, we will proceed to use a fractional numerical method of L1 type, [START_REF]Fractional calculus: models and numerical methods[END_REF][START_REF]Numerical methods for fractional calculus[END_REF]. It can be noticed that although with the fractional problem the extraction of the resource is smaller than with the classical case for most of the points, which is logical because in the fractional model the resource grows more slowly, it only decreases to a greater extent at the end of the interval near and until the nal extraction in turns out being larger than in the classical case. This way, it is possible to make a comparison of the pro t obtained in each case, Classical case pro t: 134.411. Fractional case pro t: 133.828.

ℎ * ( ) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 10 if 0 ≤ ≤ 0.
To make an analysis of this, consider the following graphic of the resources evolution without harvest with the given initial condition.

It can be noted that the use of a fractional dynamic equation, which makes the resource grow more slowly as in Figure 4, does not vary considerably the pro t compared to the classical case. Since this equation has no known exact solution, it will be approximated using the Adams fractional method, which consists of using Euler's method to obtain +1 (predictor), and the trapezoidal fraction rule to get +1 (corrector),

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ +1 = -1 =0 +1 ! 0 + =0 , +1 ( , ), +1 = -1 =0 +1 ! 0 + =0 , +1 (
, ) + +1, +1 ( +1 , +1 ).

For more details refer to [START_REF]Fractional calculus: models and numerical methods[END_REF][START_REF]Numerical methods for fractional calculus[END_REF]. In Figure 5, the population obtained from the fractional dynamic equation, taking the optimal harvest of the classical control problem, is lower than the obtained by taking the optimal harvest of the fractional problem, as we expected.

Unauthenticated | Downloaded 04/22/22 02:58 PM UTC Furthermore, if we assume that the "true evolution" of the resource is considering = 0.6 in the dynamic equation and that the harvesting agency considers the dynamic equation of = 1 to be erroneous, the loss is only due to the di erence between harvest. Also note that the di erence between the pro t obtained could be more signi cant if the instantaneous pro t function, which in this case is only the harvest, was also depending on the stock as in Remark 1.

CONCLUSIONS

In this article, we have studied a fractional control problem that models the maximization of the pro t obtained by exploiting a certain resource. An explanation of the proposed model has been made. Due to the singularity of the problem, di erent resolution techniques have been developed: for the classical case an Euler-Lagrange equation and the Nearest Feasible Paths theorem; and for the fractional case a fractional Euler-Lagrange equation and numerical methods. Although we have seen the need of a non-existent fractional Nearest Feasible Paths theorem, we have been able to make a comparison between the classical and fractional results for a certain value of the fractional order. It is also observed that the order of time fractional derivative signi cantly a ects the population growth. Hence, we conclude that fractional derivatives may be more suitable for modeling the evolution of natural resources that naturally have a resilience problem. As a future investigation, it is proposed that the extension of the fractional Nearest Feasible Paths theorem should be explored and the optimal control problem should be extended for more complex instantaneous pro t functions.

  Solving this as in the previous section, from(3.3) and (3.4), one obtains * ( ) = 40.

4 F 1 .

 41 Solutions for = 1, = 0.9, = 0.8, = 0.6 and = 0.4.

  = 38.6896 (10) = 40.25 10 ≤ ℎ( ( )) ≤ 15.

(3. 8 ) 9 )F 2 .

 892 The solution of its fractional version as in equation (3While the optimal harvest as in equation (3.6) is ℎ * 0.6 ( ) = 0.71 * 0.6 ( ) 1 Optimal populations of classic and fractional problems.

F 3 .

 3 Optimal harvests of classical and fractional problems.

6 F 4 .

 64 Populations of the classical and fractional problems without harvest.Lastly, it can be stated what happens to the population ( ) if we take the optimal harvest of the classic problem ℎ * 1 and consider it in the fractional dynamic equation of the resource.

F 5 .

 5 Optimal populations of the classical and fractional with di erent harvest problems.

Unauthenticated | Downloaded 04/22/22 02:58 PM UTC

-2 ≤ ℎ max . Unauthenticated | Downloaded 04/22/22 02:58 PM UTC

ACKNOWLEDGEMENTS

This work was partially supported by Universidad Nacional de Rosario a través del proyecto ING568 "Problemas de Control Óptimo Fraccionario" 80020190100005UR "Ecuaciones diferenciales fraccionarias en modelos de evolución poblacional". The rst author was also supported by CONICET through a PhD fellowship.

APPENDIX: INTRODUCTION TO FRACTIONAL CALCULUS

In this section certain de nitions and properties of the fractional calculus will be presented. For more details refer to [START_REF]The analysis of fractional di erential equations: An application-oriented exposition using di erential operators of Caputo type[END_REF][START_REF]The fractional calculus theory and applications of di erentiation and integration to arbitrary order[END_REF][START_REF]Fractional di erential equations: an introduction to fractional derivatives, fractional di erential equations, to methods of their solution and some of their applications[END_REF].


Ahmed Samet
email: ahmed.samet@fst.rnu.tn

Sadok Eric Lefevre
email: eric.lefevre@univ-artois.fr

Ben Yahia

Eric Lefèvre

Sadok Ben Yahia
email: sadok.benyahia@fst.rnu.tn

Ben Sadok

Yahia

Evidential Database: a new generalization of databases?

published or not. The documents may come

Introduction

Data mining is a technique that uses a variety of data analysis tools to discover, hidden but interesting patterns and relationships in data that may be used to make valid predictions. Thanks to its simple formulas, it associates performance and quality in its retrieved results. For this reason, it is used in various fields and attracted interest in different applications [START_REF] Liao | Data mining techniques and applications a decade review from 2000 to 2011[END_REF]. The first studies on data mining relies on a data model under which transactions captured doubtless facts about the items that are contained in each transaction. These binary databases have only two scenarios : 1 if an element exists, 0 otherwise. However, in many applications, the existence of an item in a transaction is better captured by likelihood measures. The obvious limits of the binary databases in handling such types of data led the data mining community to adopt imprecise frameworks in order to mine more pertinent knowledge.

In this paper, we present a non exhaustive review of existing data mining databases. The characteristics of binary, probabilistic, fuzzy and evidential databases are detailed. The support measures in the databases are presented. The aim of this paper is to demonstrate the pivotal role of the evidential database, which relies on the evidence theory [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF], in representing imprecision and uncertainty. The importance of using an evidential database rather than the other ones is justified. Indeed, we prove that the precise support measure [START_REF] Samet | Mining frequent itemsets in evidential database[END_REF] in evidential databases is a generalization of that of the classical ones. The remainder of the paper is organized as follows: in section 2, the key basic settings of the evidential database are recalled. In section 3, the binary database is studied and its relationship with the evidential database is highlighted. In section 4, probabilistic databases are scrutinized and the correlation between the precise support and the probabilistic support is highlighted. Section 5 stresses on the snugness connection between fuzzy databases with the evidential ones. Finally, we conclude and we describe issues for future work.

Evidential database and precise support

In this section, we detail the main concepts of evidential databases as well as as the notion of precise support.

Evidential Database concept

Introduced by Lee [START_REF] Lee | Imprecise and uncertain information in databases: an evidential approach[END_REF], the evidential database was aimed at modelling imperfect information. This type of database is supposed to handle imprecise and uncertain data. An evidential database is a triplet EDB = (A EDB , O, R EDB ). A EDB is a set of attributes and O is a set of d transactions (i.e., lines). Each column A i (1 ≤ i ≤ n) has a domain θ Ai of discrete values. R EDB expresses the relation between the j th line (i.e., transaction T j ) and the i th column (i.e., attribute A i ) by a normalized BBA as follows:

m ij : 2 θ A i → [0, 1] with    m ij (∅) = 0 ω⊆θ A i m ij (ω) = 1.
(1)

Table 1: Evidential transaction database EDB

Transaction Attribute A Attribute B T1 m(A 1 ) = 0.7 m(B 1 ) = 0.4 m(θ A ) = 0.3 m(B 2 ) = 0.2 m(θ B ) = 0.4 T2 m(A 2 ) = 0.3 m(B 1 ) = 1 m(θ A ) = 0.7
Table 1 illustrates an example of an evidential database. An item corresponds to a focal element. An itemset corresponds to a conjunction of focal elements having different domains. The inclusion operator is defined in [START_REF] Bach Tobji | Incremental maintenance of frequent itemsets in evidential databases[END_REF] such that for two itemsets X and Y , we have:

X ⊆ Y ⇐⇒ ∀x i ∈ X, x i ⊆ y i .
where x i and y i are the i th element of X and Y . For the same evidential itemsets X and Y , the intersection operator is defined as follows:

X ∩ Y = Z ⇐⇒ ∀z i ∈ Z, z i ⊆ x i and z i ⊆ y i .
An evidential associative rule R is a causal relationship between two itemsets that can be written in the following form R : X → Y such that X ∩Y = ∅.

Example 1. In Table 1, A 1 is an item and θ

A × B 1 is an itemset such that A 1 ⊂ θ A × B 1 and A 1 ∩ θ A × B 1 = A 1 . A 1 → B 1 is an evidential associative rule.
In the following subsection, we consider the precise support and confidence measures.

Support and confidence in evidential database

Several definitions for the support's estimation have been proposed for the evidential itemsets such as [START_REF] Bach Tobji | Incremental maintenance of frequent itemsets in evidential databases[END_REF][START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF]. Those definitions assess the support based on the belief function Bel(). The based belief support is constructed from the Cartesian product applied to the evidential database. Interested readers may refer to [START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF]. The support is computed as follows:

Support EDB (X) = Bel EDB (X) (2)
such that:

Bel : 2 θ → [0, 1] (3)
Bel(A) = ∅ =B⊆A m(B). (4)
In a previous work [START_REF] Samet | Mining frequent itemsets in evidential database[END_REF], we introduced a new metric for support estimation. The latter has been shown to provide more accuracy and to overcome several drawbacks of using the belief function. This measure is called Precise support P r and it is defined by: P r :

2 θi → [0, 1] (5)
P r(x i ) = x⊆θi |x i ∩ x| |x| × m ij (x) ∀x i ∈ 2 θi . (6)
The evidential support of an itemset

X = i∈[1...n]
x i in the transaction T j (i.e., P r Tj ) is then equal to:

P r Tj (X) = xi∈θi,i∈[1...n] P r(x i ). (7)
Thus, the evidential support Support EDB of the itemset X becomes:

Support EDB (X) = 1 d d j=1 P r Tj (X). (8)
Additionally, in [START_REF] Samet | Classification with evidential associative rules[END_REF], we introduced a new measure of confidence for evidential associative rules that we called the precise confidence measure. Let us assume an evidential association rule such as R : R a → R c , where R c and R a respectively, denote the conclusion and the antecedent (premise) part of the rule R. The precise confidence measure can be written as follows:

Conf idence(R : R a → R c ) = d j=1 P r Tj (R a ) × P r Tj (R c ) d j=1 P r Tj (R a ) . (9)
In the following sections, we highlight the relationships between evidential databases and the main other ones. The link between existing measures and the evidential precise one is also demonstrated.

Binary data mining

The first database variants studied from a data mining view are the binary ones. A binary database can be represented by a triplet BDB = (A, O, R BDB ).

A represents the set of n binary attributes (i.e., columns). R BDB is the relation that reflects the existence of an item in a transaction by only the values 0 and 1. R BDB (A i , T j ) = 1 means that the item A exists in the transaction T j and R BDB (A i , T j ) is set equal to 0 otherwise. Since the inception of the Apriori algorithm [START_REF] Agrawal | Fast algorithm for mining association rules[END_REF], several other approaches have been introduced to reduce the computational complexity. The support of an item A i in a transaction T j is defined as follows:

Support Tj (A i ) = R BDB (A i , T j ). ( 10
)
The support of an item A i in those binary databases is still computed with the same manner:

Support(A i ) = d j=1 R BDB (A i , T j ) = count(A i ). ( 11
)
The same goes for an itemset A∪B (or A×B if we keep the product notation):

Support(A × B) = count(A ∪ B). (12)
Thus, the support is computed by counting the number of transactions having both A and B. From the support, the confidence measure of a rule R : R a → R c is computed as follows:

conf idence(R : R a → R c ) = count(R a ∪ R c ) count(R a ) . (13)
A binary database can be constructed by redefining the R EDB as a precise BBA. Indeed, each item A i ∈ A can be redefined as an evidential item having the following frame of discernment θ Ai = {∃, ∃}. ∃ and ∃ denote respectively the existence and absence of the attribute A i in the considered transaction. Such a BBA can be written as follows:

m ij ({∃}) = R BDB (A i , T j ) m ij ({ ∃}) = 1 -R BDB (A i , T j ) ( 14
)
where m ij is equivalent to a certain BBA. In that case, the support measure proposed in [START_REF] Samet | Mining frequent itemsets in evidential database[END_REF] is equivalent to the binary support equation defined in Equation [START_REF] Samet | Mining frequent itemsets in evidential database[END_REF]. To demonstrate that equivalence, let us consider a binary database D and the evidential database EDB constructed as in the described procedure. Suppose that R BDB (A i , T j ) = 1 such that A i ∈ A, then the corresponding evidential attribute is an A i ∈ A EDB with θ Ai = {∃, ∃}:

P r Tj (∃) = |∃ ∩ ∃| |∃| m ij ({∃}) + | ∃ ∩ ∃| | ∃| m ij ({ ∃}) = m ij ({∃}) = R BDB (A i , T j ).
(15) From this point, we deduce that the evidential precise support is a generalization of the binary one. The same goes for the precise confidence given in Equation ( 9) that generalizes binary confidence since they both rely on the same support fraction.

Example 2. In this example, Table 2 shows how to create an evidential database from a binary one.

A B C T 1 X X T 2 X X T 3 X X (a) A B C T 1 m 11 ({∃}) = 0 m 21 ({∃}) = 1 m 31 ({∃}) = 1 m 11 ({ ∃}) = 1 m 21 ({ ∃}) = 0 m 31 ({ ∃}) = 0 T 2 m 12 ({∃}) = 1 m 22 ({∃}) = 1 m 32 ({∃}) = 0 m 12 ({ ∃}) = 0 m 22 ({ ∃}) = 0 m 32 ({ ∃}) = 1 T 3 m 13 ({∃}) = 0 m 23 ({∃}) = 1 m 33 ({∃}) = 1 m 13 ({ ∃}) = 1 m 23 ({ ∃}) = 0 m 33 ({ ∃}) = 0 (b)
The equivalency of the support measure is shown for the itemset B × C.

The support of the itemset B × C from the transactions of Table 2.a is Support(B × C) = 2 3 . In the evidential database, it is computed as follows: Thus, the support retrieved from the binary database is the same as the precise support computed from the evidential database.

Support EDB (B × C) = 1

Probabilistic data mining

Probabilistic data mining [START_REF] Aggarwal | Managing and Mining Uncertain Data[END_REF] was introduced to represent imperfect information thanks to the probability support. It can be represented by a triplet PDB = (A PDB , O, R PDB ). The degree of existence of the item A i in the transaction T j is measured through the probability function p(A i , T j ) ∈ [0, 1]. The support of an itemset X ∈ A PDB in such type of database is defined as follows [START_REF] Chui | Mining frequent itemsets from uncertain data[END_REF]:

p(X, T j ) = i∈X p(i, T j ). (16)
Thus, the support of an itemset X in a database is the sum of its expected probability in the transaction:

Support PDB (X) = d j=1 p(X, T j ). (17)
An equivalent evidential database can be constructed through using Bayesian BBA 1 . The BBA can be modeled on a two-member-based frame of discernment θ i = {∃, ∃} where ∃ indicates that A i belongs to the considered transaction, whereas ∃ performs the opposite. Such a BBA can be constructed as follows:

m ij ({∃}) = p(i, T j ) m ij ({ ∃}) = 1 -p(i, T j ). ( 18
)
With this construction, the probabilistic support defined in Equation ( 17) is equivalent to the proposed precise support. Indeed, the assertion can be verified i.e.:

P r Tj (∃) = |∃ ∩ ∃| |∃| m ij ({∃}) + | ∃ ∩ ∃| ∃ m ij ({ ∃}) = m ij ({∃}) = p(i, T j ). ( 19
)
As is the case for a binary database, the Evidential Data mining Algorithm (EDMA) generalizes the probabilistic version of Apriori: i.e., U-Apriori [START_REF] Chui | Mining frequent itemsets from uncertain data[END_REF].

Example 3. Table 3 shows how to create an evidential database from a probabilistic one. The equivalency of the support measure is shown for the itemset B × C. The support of the itemset B × C from the transactions of the Table 3.a is Support(B × C) = (0.7×0.8)+(0.7×0.1)+(0.8×0.7) 3 = 0.4. In the evidential 1 A BBA is called Bayesian only if all its focal sets are singletons.

   m ij (ω 1 ) = µ Tj (ω 1 ) m(∪ k ω k ) = 1 -µ Tj (ω 1 ). (23)
We can obviously remark that:

T (µ(A i ), µ(A j )) = min(Bel(A i ), Bel(A j )) ( 24
)
where T is a minimum t-norm. Thus, the fuzzy support can be retrieved in an evidential database as follows:

Support F DB (X) = Tj ∈O min{Bel(x i ), x i ∈ X} d . (25)
Interestingly enough, an equivalent to fuzzy database support in evidential database does exists.

Example 4. Table 4 shows how to create an evidential database from a fuzzy one. The equivalency of the support measure is shown for the itemset B × C. The support of the itemset A ω1 × B ω2 from the Table 4.a is Support(A ω1 × B ω2 ) = 0.3+0.5+0.2 3 = 1.0. In the evidential database, Table 4.b, it is computed as follows: Despite the fact that the precise support is not equivalent to the fuzzy support, it is still possible to recover the same value with the use of the Equation (25).

Support EDB (A ω1 × B ω2 ) = 1

Conclusion

In this paper, we detailed the data mining measures such as the support and the confidence on the several databases such as binary, probabilistic, fuzzy databases. We have proven the generalization relation between precise measures in evidential databases and measures used in other databases. In future works, we aim to study the evidential transformation of other imperfect databases such as fuzzy-possibilistic database [START_REF] Weng | Mining fuzzy association rules from uncertain data, springerverlag new york, inc. new york[END_REF].

3 3 j=1Pr

 33 Tj (A) × P r Tj (B) Support EDB (B × C) = 1 3 (m 21 ({∃}) × m 31 ({∃}) + m 22 ({∃}) × m 32 ({∃})+ m 23 ({∃}) × m 33 ({∃})) = 2 3

T 1 m

 1 11 ({∃}) = 0 m 21 ({∃}) = 0.7 m 31 ({∃}) = 0.8 m 11 ({ ∃}) = 1 m 21 ({ ∃}) = 0.3 m 31 ({ ∃}) = 0.2 T 2 m 12 ({∃}) = 0.9 m 22 ({∃}) = 0.7 m 32 ({∃}) = 0.1 m 12 ({ ∃}) = 0.1 m 22 ({ ∃}) = 0.3 m 32 ({ ∃}) = 0.9 T 3 m 13 ({∃}) = 0 m 23 ({∃}) = 0.8 m 33 ({∃}) = 0.7 m 13 ({ ∃}) = 1 m 23 ({ ∃}) = 0.2 m 33 ({ ∃}) = 0.3 (b)

T 1 m

 1 11 (ω 1 ) = 0.3 m 21 (ω 2 ) = 0.7 m 31 (ω 1 ) = 0.1 m 41 (ω 2 ) = 0.8 m 11 (Ω) = 0.7 m 21 (Ω) = 0.3 m 31 (Ω) = 0.9 m 41 (Ω) = 0.2 T 2 m 12 (ω 1 ) = 0.5 m 22 (ω 2 ) = 0.2 m 32 (ω 1 ) = 0.3 m 42 (ω 2 ) = 0.8 m 12 (Ω) = 0.5 m 22 (Ω) = 0.8 m 32 (Ω) = 0.7 m 42 (Ω) = 0.2 T 3 m 11 (ω 1 ) = 0.8 m 21 (ω 2 ) = 0.1 m 31 (ω 1 ) = 1.0 m 41 (ω 2 ) = 0.2 m 11 (Ω) = 0.2 m 21 (Ω) = 0.9 m 31 (Ω) = 0 m 41 (Ω) = 0.8 (b)

3 3 j=1

 33 min(Bel(A ω1 ), Bel(A ω2 )) Support EDB (A ω1 × B ω2 ) = 1 3 (Bel T1 (A ω1 ) + Bel T2 (A ω1 ) + Bel T2 (B ω2 )) = 1.0

Table 2 :

 2 The evidential transformation of BDB (Table (a)) to EDB (Table (b))

Table 3 :

 3 The evidential transformation of PDB (Table (a)) to EDB (Table (b))

	A B C
	T 1 0.0 0.7 0.8
	T 2 0.9 0.7 0.1
	T 3 0 0.8 0.7

Table 4 :

 4 The evidential transformation of F DB (Table (a)) to EDB (Table (b))

			Support	A		B
	Support	A	B	ω 1	ω 2	ω 1	ω 2
		ω 1 ω 2 ω 1 ω 2
	T 1	0.3 0.7 0.1 0.8
	T 2	0.5 0.2 0.3 0.8
	T 3	0.8 0.1 1.0 0.2
		(a)

database, it is computed as follows: Thus, the support retrieved from the probabilistic database is the same as the precise support computed from the evidential database.

Fuzzy Data mining

Let us assume the triplet FDB = (A F DB , O, R F DB ) that denotes a fuzzy database. R F DB denotes the fuzzy relationship between an item and a transaction expressed through a membership function. The membership function µ Tj (i) = α (α ∈ [0, 1]) rates the degree of membership of the considered item to the transaction T j . The support computation in such databases is done by the use of the count() function in the following manner [START_REF] Hong | A fuzzy AprioriTid mining algorithm with reduced computational time[END_REF]:

The support of item i in the fuzzy database is found as follows:

Thus, for an itemset X of size q such that x i ∈ X and i ∈ [1, q], the support becomes:

The numerator of the support could be seen as the Gödel t-norm (minimum t-norm). Assuming a fuzzy database is available, it is possible to construct an evidential database. In addition, the precise support sustains fuzzy support in its formulation. Indeed, as can be seen in Equation ( 8), the precise support is also equal to the sum of the transactional support divided by the database size.

In the following, we show how to obtain analogous evidential support of the fuzzy support. Assuming an attribute A i ∈ A EDB having a frame of discernment θ Ai such that ω 1 ⊂ • • • ⊂ ω n ⊆ θ Ai , the corresponding BBA for a fuzzy relation R F DB (ω 1 , T j ) = µ Tj (ω 1 ) is constructed in this form:


Ahmed Samet
email: ahmed.samet@fst.rnu.tn

Eric Lefèvre
email: eric.lefevre@univ-artois.fr

Sadok Ben Yahia
email: sadok.benyahia@fst.rnu.tn

Classification with evidential associative rules

Keywords: Evidential database, Confidence, Associative classification, Evidential Apriori

Mining database provides valuable information such as frequent patterns and especially associative rules. The associative rules have various applications and assets mainly data classification. The appearance of new and complex data support such as evidential databases has led to redefine new methods to extract pertinent rules. In this paper, we intend to propose a new approach for pertinent rule's extraction on the basis of confidence measure redefinition. The confidence measure is based on conditional probability basis and sustains previous works. We also propose a classification approach that combines evidential associative rules within information fusion system. The proposed methods are thoroughly experimented on several constructed evidential databases and showed performance improvement.

Introduction

Data mining domain allows extracting pertinent information within databases [START_REF] Agrawal | Fast algorithm for mining association rules[END_REF]. The provided information are represented in a set of rules, where each one is associated with a pertinence measure denoted Confidence. Among their purposes, those associative rules are used for data classification [START_REF] Li | CMAR: Accurate and efficient classification based on multiple class-association rules[END_REF][START_REF] Bouzouita | GARC: A new associative classification approach[END_REF]. The classification process from those associative rules is denoted associative classification. Associative classification offers one of the best classification rate and measure membership [START_REF] Bouzouita | GARC: A new associative classification approach[END_REF]. Recently, new databases have appeared proposing data suffering from imperfection. Those types of data fit reality where opinions are no longer represented with Boolean values. In addition, it has added more complexity in their treatment. The imperfection is handled with several theories such as fuzzy [START_REF] Zadeh | Fuzzy sets[END_REF] and evidential theory [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. In [START_REF] Lee | Imprecise and uncertain information in databases: an evidential approach[END_REF], the author introduced a new type of databases that handle both imprecise and uncertain information thanks to the evidential theory. Those types of databases were denoted as the Evidential database. The evidential databases were shortly studied from a data mining view [START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF] and not so much attention was paid to that issue. In literature, two major works [START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF][START_REF] Bach Tobji | Incremental maintenance of frequent itemsets in evidential databases[END_REF] stand by proposing new measures for itemsets' support. Indeed, in [START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF], Hewawasam et al. proposed a methodology to estimate itemsets' support and modelize them in a tree representation: Belief Itemset Tree (BIT). The BIT representation brings easiness and rapidity for the estimation of the associative rule's confidence. In [START_REF] Bach Tobji | Incremental maintenance of frequent itemsets in evidential databases[END_REF], the authors introduced a new approach for itemset support computing and applied on a Frequent Itemset Maintenance (FIM) problem. Only [START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF] paid attention to associative classification where the authors introduced evidential associative rules. A new measure for rule's confidence was introduced based on conditional belief [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. In this work, evidential data mining problem is tackled by putting our focus on the associative classification. We highlight problems existing in current measure of evidential rule's confidence which are based on conditional belief. A new confidence measure is proposed based on Bayesian assumption. We also introduce a new associative classification method that reduces the overwhelming number of generated rules. The retained rules are then used for classification purposes and tested on several benchmarks. This paper is organized as follows: in section 2, the main principles of the evidential database are recalled. In section 3, several state of art works on confidence measure are scrutinized and we highlight their limits. In addition, we introduce an alternative confidence measure based on probabilistic definitions. In section 4, we introduce a new method for evidential rule generation. The provided rules are filtrated and combined through a fusion system. The performance of this algorithm is studied in section 5. Finally, we conclude and we sketch issues of future work.

Evidence database concept

An evidential database stores data that could be perfect or imperfect. Uncertainty in such database is expressed via the evidence theory [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. An evidential database, denoted by EDB, with n columns and d lines where each column i (1 ≤ i ≤ n) has a domain θ i of discrete values. Cell of line j and column i contains a normalized BBA as follows:

m ij : 2 θi → [0, 1] with    m ij (∅) = 0 A⊆θi m ij (A) = 1. (1)
Table 1. Evidential transaction database EDB

Transaction Attribute A Attribute B T1 m11(A1) = 0.7 m21(B1) = 0.4 m11(θA) = 0.3 m21(B2) = 0.2 m21(θB) = 0.4 T2 m12(A2) = 0.3 m22(B1) = 1 m12(θA) = 0.7
In an evidential database, as shown in Table 1, an item corresponds to a focal element. An itemset corresponds to a conjunction of focal elements having different domains. Two different itemsets can be related via the inclusion or the intersection operator. Indeed, the inclusion operator for evidential itemsets [START_REF] Bach Tobji | Incremental maintenance of frequent itemsets in evidential databases[END_REF] is defined as follows, let X and Y be two evidential itemsets:

X ⊆ Y ⇐⇒ ∀x i ∈ X, x i ⊆ y i .
where x i and y i are the i th element of X and Y . For the same evidential itemsets X and Y , the intersection operator is defined as follows:

X ∩ Y = Z ⇐⇒ ∀z i ∈ Z, z i ⊆ x i and z i ⊆ y i .
An Evidential associative rule R is a causal relationship between two itemsets that can be written in the following form R :

X → Y fulfilling X ∩ Y = ∅. In Table 1, A 1 is an item and {θ A B 1 } is an itemset such that A 1 ⊂ {θ A B 1 } and A 1 ∩ {θ A B 1 } = A 1 . A 1 → B 1 is an evidential associative rule.
Several definitions for the support estimation were defined for the evidential itemsets such as [START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF][START_REF] Bach Tobji | Incremental maintenance of frequent itemsets in evidential databases[END_REF]. Those methods assess the support based on the belief function applied on the evidential database BBA m EDB3 :

Support EDB (X) = Bel EDB (X) (2)
such that:

Bel : 2 θ → [0, 1] (3)
Bel(A) = ∅ =B⊆A m(B). (4)
In a previous work [START_REF] Samet | Mining frequent itemsets in evidential database[END_REF], we introduced a new metric for support estimation providing more accuracy and overcoming several limits of using the belief function. The Precise support P r is defined by:

P r : 2 θ i → [0, 1] (5)
P r(x i ) = x⊆θi |x i ∩ x| |x| × m ij (x) ∀x i ∈ 2 θi . (6)
The evidential support of an itemset

X = i∈[1...n]
x i in the transaction T j (i.e., P r Tj ) is then computed as follows:

P r Tj (X) = xi∈θi,i∈[1...n] P r(x i ) (7)
Thus, the evidential support Support EDB of the itemset X becomes:

Support EDB (X) = 1 d d j=1 P r Tj (X). ( 8
)
The confidence is the measure assigned to the associative rules and it represents its relevance [START_REF] Agrawal | Fast algorithm for mining association rules[END_REF]. As originally introduced in Boolean databases, the confidence measure was relying on conditional probability [START_REF] Agrawal | Fast algorithm for mining association rules[END_REF]. Indeed for a rule R : R a → R c , such that R c and R a are respectively the conclusion and the antecedent (premise) part of the rule R, the confidence is expressed as follows:

Conf idence(R) = P (R c |R a ) = d i=1 P (R a ∩ R c ) d i=1 P (R a ) (9)
In addition, even in fuzzy data mining, the associative rule's confidence is built with conditional fuzzy measures [START_REF] Hong | A fuzzy AprioriTid mining algorithm with reduced computational time[END_REF]. In this respect, evidential associative rules were initially introduced in [START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF]. The authors defined the structure of an evidential associative rule and estimated its relevance following a confidence metric. The confidence of a rule R in the set of all rules R, i.e., R ∈ R, is computed as follows:

Conf idence(R) = Bel(R c |R a ) (10)
where Bel(•|•) is the conditional Belief. The proposed confidence metric is hard to define where several works have tackled this issue and different interpretations and formulas were proposed such as those given respectively in [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF][START_REF] Fagin | A new approach to updating beliefs[END_REF]. In [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF], the conditional belief is defined as follows:

Bel(R c |R a ) = Bel(R c ∪ R a ) -Bel(R a ) 1 -Bel(R a ) (11)
In [START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF], the authors used Fagin et al.'s conditional belief such that:

Bel(R c |R a ) = Bel(R a ∩ R c ) Bel(R a ∩ R c ) + P l(R a ∩ Rc ) . ( 12
)
where P l() is the plausibility function and is defined as follows:

P l(A) = B∩A =∅ m(B). ( 13
)
Example 1. Through the following example, we highlight the inadequacy of the conditional belief use. We consider the Transaction 1 of Table 1 from which we try to compute the confidence of A 2 → B 1 (i.e., Bel(B 1 |A 2 )). The conditional belief introduced in [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF] gives the following results:

Bel(B 1 |A 2 ) = Bel(B 1 ∪ A 2 ) -Bel(A 2 ) 1 -Bel(A 2 ) = Bel(B 1 ) 1 = 0.4
The result of the belief of B 1 knowing A 2 is true is equal to that of Bel(B 1 ) due to the independence between A 2 and B 1 . On the other hand, both hypothesis might be correlated so that the event B 1 does not occur knowing already the happening of A 2 .

In the following, we propose a new metric for the confidence estimation based on our Precise support measure [START_REF] Samet | Mining frequent itemsets in evidential database[END_REF] and probability assumption:

Conf idence(R) = d j=1 P r Tj (R a ) × P r Tj (R c ) d j=1 P r Tj (R a ) ( 14
)
where d is the number of transactions in the evidential database. Thanks to its probabilistic writing, the proposed metric sustains previous confidence measure such as that introduced in [START_REF] Agrawal | Fast algorithm for mining association rules[END_REF].

Example 2. Let us consider the example of the evidential database in Table 1. The confidence of the evidential associative rule R 1 : A 1 → B 1 is computed as follows:

Conf idence(R 1 ) = P r T1 (A 1 ) × P r T1 (B 1 ) + P r T2 (A 1 ) × P r T2 (B 1 ) P r T1 (A 1 ) + P r T2 (A 1 ) = 0.75

The generated rules with their confidence could find several applications. In the following, we tackle the classification problem case and a based evidential rule classifier is introduced.

Associative Rule Classifier

One of the main characteristics of the evidential database is the great number of items that it integrates. The number of items depends from the frame of discernment of each column. This asset makes from the evidential database more informative but more complex than the usual binary database. In [START_REF] Samet | Mining frequent itemsets in evidential database[END_REF], we have shown the significant number of generated frequent patterns that may be drawn even from small databases. Indeed, from a frequent itemset, of size k, 2 k -2 potential rules are generated. In order to use the generated evidential rules for a classification purposes, we first have to reduce their number for a more realistic one. In the following, we propose two processes for classification rule's reduction.

Classification rules

From the obtained rules, we retain only the classification ones. From a rule such that i∈I X i → j∈J Y j , we only keep those matching a class hypothesis at the conclusion part (i.e., Y j ∈ θ C and θ C is the frame of discernment).

Example 3. Let us consider the following set of the association rules

S = {A 1 → C 1 ; A 1 , B 2 → C 1 ; A 1 → B 1 } and the class frame of discernment θ C = {C 1 , C 2 }.
After classification rule reduction, the set

S becomes S = {A 1 → C 1 ; A 1 , B 2 → C 1 }.

Generic and Precise rules

Generic rules: the rule's reduction can assimilate the redundant rules. A rule R 1 is considered as a redundant rule if and only if it does not bring any new information having at hand a rule R 2 . R 2 is considered as more informative as far as its antecedent part is included in that of R 1 . The retained rules from the reduction process constitute the set of Generic rules R extracted from the set of frequent itemsets FI.

Example 4. Let us consider the previous set of the association rules

S = {A 1 → C 1 ; A 1 , B 2 → C 1 ; A 1 → B 1 }. After redundant rule reduction, the set S becomes S = {A 1 → C 1 ; A 1 → B 1 }.
Precise rules: A rule is considered as precise if the rule's premise is maximized. Thus, from the set of all possible rules, we retain only those having the size of their premise part equal to n (number of columns of EDB).

Algorithm 1 sketches the process of rule's generation as well as rule reduction. The algorithm relies on the function Construct Rule(x, θ C ) (Line 10) that generates associative rules and filtrates out them by retaining only the classification ones. The function F ind Conf idence(R, P r T able) (Line 22) computes the confidence of the rule R following the P r T able that contains all transactional support of each item (for more details see [START_REF] Samet | Mining frequent itemsets in evidential database[END_REF]). Finally, the function Redundancy(R, R) (Line 42) builds the set of all classification rules R which are not redundant and having the confidence value greater than or equal to the fixed threshold minconf .

Classification

Let us suppose the existence of an instance X to classify represented a set of BBA belonging to the evidential database EDB such that:

X = {m i |m i ∈ X, x j i ∈ θ i } (15)
where x j i is a focal element of the BBA m i . Each retained associative rule, in the set of rules R, is considered as a potential piece of information that could be of help for X class determination. In order to select rules that may contribute to classification, we look for rules having a non null intersection with X such that:

RI = {R ∈ R, ∃x j i ∈ θ i , x j i ∈ R a } (16)
Each rule found in the set RI constitutes a piece of information concerning the instance X membership. Since several rules can be found and fulfilling the return R 50: end function intersection condition, it is of importance to benefit from them all. In our work, we assume that all information is valuable and should be handled within the information fusion problem. From the set RI, we extract the set of generic or precise classification rules (see Subsection 4.2). Indeed, each rule from the computed set R l ⊂ RI, l ∈ [1 . . . L] and L < |RI|, that brings a new information (different R a ) is transformed into a BBA following the frame of discernment θ C (frame of discernment of R c ):

m θ C R l ({R c }) = α × conf idence(R l ) m θ C R l (θ C ) = 1 -(α × conf idence(R l )) (17)
where R c is the conclusion part of the rule R l and α ∈ [0, 1] is a discounting factor.

The L constructed BBA are then fused following the Dempster rule of combination [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF] as follows:

m ⊕ = ⊕ L l=1 m θ C R l . (18)
⊕ is the Dempster's aggregation function where for two source's BBA m 1 and m 2 :

m ⊕ (A) = 1 1-K B∩C=A m 1 (B) • m 2 (C) ∀A ⊆ Θ, A = ∅ m ⊕ (∅) = 0 ( 19
)
where K is defined as:

K = B∩C=∅ m 1 (B) • m 2 (C). ( 20
)

Experimentation and results

In this section, we present how we managed to conduct our experiments and we discuss comparative results.

Evidential database construction

In order to perform experimental tests, we construct our own evidential databases from UCI benchmarks [START_REF] Frank | UCI machine learning repository[END_REF] based upon ECM [START_REF] Masson | ECM: An evidential version of the fuzzy c-means algorithm[END_REF]. Interested reader may refer to [START_REF] Samet | Mining frequent itemsets in evidential database[END_REF] for more details on evidential database construction. The transformation was operated on Iris, Vertebral Column, Diabetes and Wine databases. The studied databases are summarized on Table 2 in terms of number of instances and attributes.

Comparative results

In the following, we compare the classification result performance between the Generic and Precise rules. Table 3 shows the difference in classification result between the generic and the precise associative rules. The precise rules highlight better results than do the generic ones. Indeed, the larger the rule's premise is, the more pertinent the rule is. On the other hand, the generic rule based approach fuse much more rules than do the precise one. In addition, all generic rules are considered with the same weight within the fusion process despite their pertinence difference. These characteristics with Dempster's combination behavior mislead the fusion process to errors. Indeed, as shown in Figure 1, the high number of fused rules depends highly from the minsup value. Unlike the generic approach, the number of precise rule is defined by number of larger premise's rule which is dependent from the treated evidential transaction.

Conclusion

In this paper, we tackled associative rule's extraction from evidential databases. We proposed a new confidence measure for associative rules in evidential databases.

The proposed measure is based on Precise support (i.e., probability measure) providing coherence and sustains previous work on fuzzy and binary databases. The rules are then filtrated to retain only classification and non redundant rules. A classification method based on evidential associative rules is introduced. The classification approach is based on a fusion system that represents interesting rules. As illustrated in the experimentation section, the proposed method provides an interesting performance rates. In future work, we plan to study the development of a new method to estimate the reliability of each combined associative rule. Indeed, each rule has a precision relatively to the instance to classify. The precision is measured by the intersection between the premise and the instance itself. A reliability measure for rule BBA is under study.

Fig. 1 .

 1 Fig. 1. Generic associative rule's number for different support values

  Algorithm 1 Evidential Associative Rule Generation algorithm

	Require: P r T able, minconf, FI, θC	27:	den ← 1
	Ensure: R		28:	for all i ∈ P r(j).f ocal element
	1: for all x ∈ FI do	do
	2:	R ← Construct Rule(x, θC )	29:	if P r(j).f ocal element ∈
	3:	if R = ∅ then	R.premise then
	4:	Conf	← F ind Conf iden-	30:	num	←	num ×
		ce(R, P r T able)	P r(j).val
	5:	if Conf > minconf then	31:	den ← den × P r(j).val
	6:	R ← Redundancy(R, R)	32:	else
	7:	end if		33:	if P r(j).f ocal element ∈
	8:	end if		R.conclusion then
	9: end for		34:	end if
	10: function Construct Rule(X , θC )	35:	end if
	11:	for all x ∈ X do	36:	end for
	12:	if x / ∈ θC then	37:	numer ← numer + num
	13:	prem ← prem + {x}	38:	denom ← denom + den
	14:	else		39:	end for
	15:	concl ← concl + {x}	40:	return numer denom
	16:	end if		41: end function
	17:	end for		42: function Redundancy(R,R)
	18:	R.premise ← prem	43:	for all rule ∈ R do
	19:	R.conclusion ← concl	44:	if R.premise ⊂ rule.premise &
	20:	return R		R.conclusion = rule.conclusion then
	21: end function		45:	R ← R\rule
	22: function Find Confidence(R, P r)	46:	R ← R ∪ R
	23:	numer ← 0		47:	end if
	24:	denom ← 0		48:	end for
	25:	for j=1 to d do	49:
	26:	num ← 1

Table 2 .

 2 Database characteristics

	Database	#Instances #Attributes #Focal elements
	Iris EDB	150	5	40
	Vertebral Column EDB	310	7	116
	Diabetes EDB	767	9	132
	Wine EDB	178	14	196

Table 3 .

 3 Comparative result between Generic and Precise classification rules

	Database Iris EDB Vertebral Column EDB Diabetes EDB Wine EDB
	Precise rules 80.67%	88.38%	83.20%	100%
	Generic rules 78.67%	67.74%	65.10%	51.68%

A BBA constructed from Cartesian product applied on the evidential database. Interested readers may refer to[START_REF] Hewawasam | Rule mining and classification in a situation assessment application: A belief-theoretic approach for handling data imperfections[END_REF].