Luca Foppiano

Mato Tomoya

Terashima Kensei

Pedro Suarez

Tou Ortiz

Sakai Taku

Chikako

Wei-Sheng Wang

Amagasa Toshiyuki

Takano Yoshihiko

Ishii Masashi
email: ishii.masashi@nims.go.jp

Tomoya Mato

Kensei Terashima

Pedro Ortiz Suarez

Taku Tou

Chikako Sakai

Toshiyuki Amagasa

Yoshihiko Takano

Masashi Ishii

Semi-automatic staging area for high-quality structured data extraction from scientific literature

Keywords: materials informatics, superconductors, machine learning, database, tdm

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

Introduction

The emergence of new methodologies using machine learning for materials exploration has given rise to a growing research area called materials informatics (MI) [START_REF] Oliveira | Materials discovery with machine learning and knowledge discovery[END_REF]. This field leverages the knowledge of the materials data accumulated in the past to efficiently screen candidates of the materials with desired properties. As a matter of course, such an approach requires a larger amount of material-related data for training models. Researchers have been developing large aggregated databases of physical properties generated by first-principles calculations based on Density Functional Theory (DFT), such as Materials Project [START_REF] Jain | Commentary: The materials project: A materials genome approach to accelerating materials innovation[END_REF], JARVIS (Joint Automated Repository for Various Integrated Simulations) [START_REF] Curtarolo | Aflow: An automatic framework for high-throughput materials discovery[END_REF], NOMAD (Novel Materials Discovery) [START_REF] Draxl | The nomad laboratory: From data sharing to artificial intelligence[END_REF], that played a role of a strong driving force for the development of materials informatics. Using DFT data for machine learning (ML) in materials science has become popular since, in principle, it allows researchers to simulate and obtain various types of physical properties of the target materials only by knowing the crystal structures of the subjects. Those DFT codes are designed to reproduce/simulate the physical properties that should be observed by experiments in reality. Nonetheless, caution must be exercised while utilising these computed figures for constructing ML models aimed at steering experiments. This caution arises due to the potential lack of validity in their predictions when dealing with specific simplifications of the interactions between atoms and electrons in solids, such as electron-electron Coulomb correlation, spin-orbit coupling, and similar factors.

On the contrary, accumulated datasets of experimental data from scientific publications are still scarce, despite abundant publication availability, and exponential growth in materials science [START_REF] Pratheepan | Global publication productivity in materials science research: A scientometric analysis[END_REF]. Currently, only a few limited resources exist, such as the Pauling File [START_REF] Blokhin | The pauling file project and materials platform for data science: From big data toward materials genome[END_REF] and SuperCon [START_REF] Ishii | Structuring superconductor data with ontology: reproducing historical datasets as knowledge bases[END_REF], necessitating reliance on manual extraction methods. This scarcity can be attributed to inadequate infrastructure and a shortage of expertise in computer science within the materials science field.

The SuperCon database was built manually from 1987 [START_REF] Ishii | Structuring superconductor data with ontology: reproducing historical datasets as knowledge bases[END_REF] by the National Institute for Materials Science (NIMS) in Japan and it is considered a reliable source of experimental data on superconductors [START_REF] Roter | Predicting new superconductors and their critical temperatures using machine learning[END_REF][START_REF] Stanev | Machine learning modeling of superconducting critical temperature[END_REF][START_REF] Tran | Machine-learning approach for discovery of conventional superconductors[END_REF][START_REF] Konno | Deep learning model for finding new superconductors[END_REF]. However, the updates of SuperCon have become increasingly challenging due to the high publication rate. In response to the need for a more efficient approach to sustain productivity, we embarked on the development of an automated system for extracting material and property information from the text contained in relevant scientific publications. This automated process enabled the rapid creation of "SuperCon 2 Database", a comprehensive database of superconductors containing around 40000 entries, within an operational duration of just a few days [START_REF] Foppiano | Automatic extraction of materials and properties from superconductors scientific literature[END_REF]. Matching the level of quality seen in SuperCon while simultaneously automating the extraction of organised data can be achieved with a properly designed curation process. We use the term curation to describe the overall process of reviewing and validating database records, while correction refers to the specific action of altering the values of one or more properties within an individual record. At the moment of writing this article, we are not aware of any other curation tool focusing on structured databases of extracted information. There are several tools for data annotation, such as Inception [START_REF] Klie | The INCEpTION platform: Machine-assisted and knowledge-oriented interactive annotation[END_REF], and Doccano [START_REF] Nakayama | doccano: Text annotation tool for human[END_REF] which concentrate on text labelling and classification.

In this work, we designed and developed a workflow with a user interface, "SuperCon 2 Interface", crafted to produce structured data of superior quality and efficiency to the one obtained by the "traditional" manual approach consisting of reading documents and noting records, usually on an Excel file. We developed this framework around the specific use case of SuperCon, however, our goal is to be adapted to alternative data frameworks.

Our contributions can be summarised as follows:

• We developed a workflow and a user interface that allow the curation of a machine-collected database. We demonstrate that using it for data correction resulted in higher quality than the "traditional" (manual) approach. • We devise an anomaly detection process for incoming data lower rejection rate (false positive rate) from domain experts.

• We propose a mechanism that selects training data based on corrected records, and we demonstrate that such selections are rapidly improving the ML models.

The subsequent sections, Section 2 describes the curation workflow and Section 3 the user interface on top of it. Finally, we discuss our evaluation experiments and results in Section 4.

Curation workflow

The curation of the SuperCon 2 Database acts as a workflow where user actions result in database records state transitions (Figure 1). Allowed manual actions include a) mark as valid (validation) when a record is considered correct or corrected by someone else. When a record is not valid, users can: b) mark as invalid when considered "potentially" invalid (or the curator is not confident), c) perform manual correction to update it according to the information from the original PDF document, and d) remove the record when it was not supposed to be extracted.

Besides manual operations from users, this workflow supports also automatic actions: "anomaly detection" for pre-screening records (Section 2.2) and the "training data collector" for accumulating training data for improving ML models (Section 2.3).

Although only the most recent version of a record can be viewed on this system, the correction history is recorded (Section 3.3).

Workflow control

The workflow state is determined by the "curation status" (Section 2.1.1), the user action, and the error type (Section 2.1.2).

Curation status

The curation status (Figure 1) is defined by type of action, manual or automatic, and status, which can assume the following values:

• new: default status when a new record is created.

• curated: the record has been amended manually.

• validated: the record was manually marked as valid.

• invalid: the record is wrong or inappropriate for the situation (e.g., T m or T curie extracted as superconducting critical temperature). • obsolete: the record has been updated and the updated values are stored in a new record (internal status1 ). • removed: the record has been removed by a curator (internal status).

Error types

We first introduced error type in [START_REF] Foppiano | Automatic extraction of materials and properties from superconductors scientific literature[END_REF] and extended their scope in this work to consider data curation and anomaly detection.

Users are required to select one Error Type at every record update or removal. This information is stored in the "original" record and can be different at every record modification. The error type values can be summarised as follows:

• From table: the entities Material → T c → Pressure are identified in a table.

At the moment, table extraction is not performed • Extraction: The material, temperature, and pressure are not extracted (no box) or extracted incorrectly. • Linking: The material is incorrectly linked to the T c given that the entities are correctly recognised. • T c classification: The temperature is not correctly classified as "superconductors critical temperature" (e.g., Curie temperature, Magnetic temperature...). • Composition resolution: The exact composition cannot be resolved (e.g., the stoichiometric values cannot be resolved). • Value resolution: The extracted formula contains variables that cannot be resolved, even after having read the paper. This includes when data is from tables • Anomaly detection: The data has been modified by anomaly detection, which facilitates their retrieval from the interface. • Curation amends: The curator is updating the data which does not present issues due to the automatic system.

Anomaly detection

Anomaly detection is the process of identifying unusual events or patterns in data. In our context, this means identifying data that are greatly different from the expected values. This post-process was introduced in a limited scope to draw attention to certain cases during the curation.

The anomaly detection uses a rule-based approach and marks any record that matches the following conditions

• the extracted T c is greater than room temperature (273 K), negative, or contains invalid characters and cannot be parsed (e.g. "41]") • the chemical formula cannot be processed by an ensemble composition parser that combines Pymatgen [START_REF] Ong | Python materials genomics pymatgen : A robust open-source python library for materials analysis[END_REF], and text2chem [START_REF] Kononova | Text-mined dataset of inorganic materials synthesis recipes[END_REF] • the extracted applied pressure cannot be parsed or falls outside the range 0 -250 GPa.

Records identified as anomalies have status "invalid" and error type "anomaly detection" for easy identification. Since this process may find false positives, its output requires validation from curators. For example, in certain contexts, T c values above room temperature or applied pressure up to 500 GPa may be valid in researchers' hypotheses, calculations, or simulated predictions.

We ran the anomaly detection on the full SuperCon 2 Database (40324 records [START_REF] Foppiano | Automatic extraction of materials and properties from superconductors scientific literature[END_REF]). The anomaly detection identified 1506 records with invalid T c , 5021 records with an incomplete chemical formula, 304 records with invalid applied pressure, and 1440 materials linked to multiple T c values. Further analysis and cross-references with contrasting information may be added in future.

Automatic training data collector

The curation process is a valuable endeavour demanding significant knowledge and human effort. To maximise the use of this time for collecting as much information as possible. We integrated an automatic procedure in the curation process that, for every correction, accumulates the related data examples that can be used to improve the underlying ML models.

Training data collection

In the event of a correction (update, removal) in a database record, this process retrieves the corresponding raw data: the text passage, the recognised entities (spans), and the layout tokens information. This information is sufficient to be exported as training examples, which can be examined and corrected, and feedback to the ML model.

Training data management

We designed a specific page of the interface (Section 3) to manage the collected data (Figure 2) in which each row corresponds to a training example composed by the decorated text showing the identified entities, the document identifier, and the status. The users can examine the data, delete it, send it to the annotation tool to be corrected, and then export them. We integrated our interface with Label-studio [START_REF] Tkachenko | Label Studio: Data labeling software[END_REF] for the correction of the collected training examples. Label-studio is an open-source, python-based, and modern interface supporting many different TDM tasks (NER, topic modelling, image recognition, etc.).

Curation interface

The workflow is operated through the user interface, which offers several key features to facilitate the data curation process (Figure 1). It provides a comprehensive view of materials and their related properties as a table which includes search, filtering, and sorting functionality (Figure 3). The detailed schema, including examples, is reported in our previous work [START_REF] Foppiano | Automatic extraction of materials and properties from superconductors scientific literature[END_REF].

During the curation process, it is often necessary to switch back and forth between the database record and the related context in the paper (the related paragraph or sentence). Our interface provides a viewer for individual documents, which visualises in the same window a table with the extracted records and the original PDF document decorated with annotations that identify the extracted materials and properties (Figure 4).

Manual curation approach

In this section, we discuss our strategy concerning manual curation, which is still indispensable for developing high-quality structures.

We selected curators from domain experts in the field, to certify sufficient data quality. Nevertheless, as confirmed from our experiment in Section 4.3, the experience of each individual may have an impact on the final result. We followed two principles to guarantee robustness in the curation process. First, we built solid curation documentation as a form of example-driven guidelines with an iterative approach we first introduced in [START_REF] Foppiano | Supermat: construction of a linked annotated dataset from superconductors-related publications[END_REF]. Then, we used a double-round validation approach, in which the data was initially corrected by one person, and validated in a second round, by a different individual.

Curation guidelines

The guidelines consist mainly of two parts: the general principles and the correction rules with examples of solutions. The guidelines are designed to provide general information applied to corrections and very basic explanations containing illustrations for a faster understanding (e.g. the meaning of the colours of the annotations). Differently from our previous work [START_REF] Foppiano | Supermat: construction of a linked annotated dataset from superconductors-related publications[END_REF], these guidelines are divided into examples for different scenarios based on the error types mentioned in Section 2.1.2. Each example described the initial record, its context, the expected corrected record and a brief explanation, as illustrated in Figure 5.

Curation and processing logs

The Supercon 2 interface gives access to information regarding the ingestion (processing log) and the curation process (curation log). The processing log is filled up when the new data is ingested, it was built to have minimal functions able to explain why certain documents haven't been processed (Figure 6 top). For example, sometimes documents fail because they don't contain any text (image PDF documents) or they are too big (more than 100 pages).

The curation log provides a view of what, when and how a record has been corrected (Figure 6 bottom).

Results and evaluation

In this section, we illustrate the experiments we have run to evaluate our work. The evaluation is composed of three sets of results. The anomaly detection rejection rate (Section 4.1) indicates how many anomalies were rejected by curators after validation. Then, we demonstrate that the training data automatically selected contributed to improving the ML model with a small set of examples (Section 4.2) Finally, we evaluated the quality of the data extraction using the interface (and the semi-automatic TDM process) against the classical method of reading the PDF articles and noting the experimental information in an Excel file. In Section 4.3 we find out that using the interface improves the quality of the curated data by reducing missing experimental data.

Anomaly detection rejection rate

We evaluated the anomaly detection by observing the "rejection rate" which consists of the number of detected anomalies that were rejected by human validation. Running the anomaly detection on a database subset with 667 records, it found 17 anomalies in T c , 1 anomaly in applied pressure, and 16 anomalies in the chemical formulas. Curators examined each reported record and rejected 4 (23%) anomalies in T c , 6 anomalies (37%) in chemical formulas and 0 anomalies in applied pressure. This indicates an appropriate low rate of false positives although a study with a larger dataset might be necessary.

Training data generation

We selected around 400 records in the Supercon2 Database that were marked as invalid by the anomaly detection process and we corrected them following the curation guidelines (Section 3.2). Then, we examined the corresponding training data corrected by the interface (Section 2.3) and obtained a set of 352 training data examples for our ML models. We call the obtained dataset curation to be distinguished from the original SuperMat dataset which is referred to as base.

We prepared our experiment using SciBERT [START_REF] Beltagy | SciBERT: A pretrained language model for scientific text[END_REF] that we fine-tuned for our downstream task as in [START_REF] Foppiano | Automatic extraction of materials and properties from superconductors scientific literature[END_REF]. We trained five models that we evaluated using a fixed holdout dataset from SuperMat averaging the results to smooth out the fluctuations. We use the DeLFT (Deep Learning For Text) [START_REF]DeLFT contributors[END_REF] library for training, evaluating, and managing the models for prediction. A model can be trained with two different strategies:

(1) "from scratch": when the model is initialised randomly. We denote this strategy with an (s). (2) "incremental": when the initial model weights are taken from an already existing model. We denote this strategy with an (i).

The latter can be seen as a way to "continue" the training from a specific checkpoint. We thus define three different training protocols:

(1) base(s): using the base dataset and training from scratch (s).

(2) (base+curation)(s): using both the base and curation datasets and training from scratch (s). (3) base(s)+(base+curation)(i): Using the base dataset to train from scratch (s), and then continuing the training with the curation dataset (i).

We merge "curation" with the base dataset because the curation dataset is very small compared to "base", and we want to avoid catastrophic forgetting [START_REF] Kirkpatrick | Overcoming catastrophic forgetting in neural networks[END_REF] or overfitting. The trained models are then tested using a fixed holdout dataset that we designed in our previous work [START_REF] Foppiano | Automatic extraction of materials and properties from superconductors scientific literature[END_REF] and the evaluation scores are shown in Table 1.

This experiment demonstrates that with only 352 examples (2% of the Super-Mat dataset) comprising 1846 additional entities (11% of the entities from the Su-perMat dataset) (Table 2), we obtain an improvement of F1-score from 76.67% 2 to values between 77.44% (+0.77) and 77.48% (+0.81) for (base+curation)(s) and base(s)+(base+curation)(i), respectively. This experiment gives interesting insight relative to the positive impact on the way we select the training data. However, there are some limitations: the curation dataset is small compared to the base dataset. This issue could be verified by correcting all the available training data, repeating this experiment, and studying the interpolation between the size of the two datasets and the obtained evaluation scores. A second limitation is that the hyperparameters we chose for our model, in particular, the learning rate and batch size could be still better tuned to obtain better results with the second and third training protocols.

Data quality

We conducted an experiment to evaluate the effectiveness and accuracy of data curation using two methods: a) the user interface (interface), and b) the "traditional" manual approach consisting of reading PDF documents and populating an Excel file (PDF documents).

We selected a dataset of 15 papers, which we assigned to three curators -a senior researcher (SD), a PhD student (PS), and a master's student (MS). Each curator received 10 papers: half to be corrected with the interface and half with the PDF Document method. Overall, each pair of curators had 5 papers in common which they had to process using opposite methods. For instance, if curator A receives paper 1 to be corrected with the interface, curator B, who receives the same paper 1, will correct it with the PDF document method. After curation, a fourth individual manually reviewed the curated content. The raw data is available in the Appendix A.

We evaluated the curation considering a double perspective: time and correctness. Time was calculated as the accumulated minutes required using each method. Correctness was assessed using standard measures such as precision, recall, and the F1-score. Precision measures the accuracy of the extracted information, while recall assesses the ability to capture all expected information. F1-Score is a harmonic means of precision and recall.

Discussion

Overall, both methods required the same accumulated time: 185 minutes using the interface and 184 minutes using the PDF Document method. When the experiment was carried out, not all the curators were familiar with the interface method. Although they had access to the user documentation, they had to get acquainted with the user interface, thus the accumulated 185 minutes included such activities.

We examined the quality of the extracted data and we observed an improvement of +5.55% in precision and a substantial +46.69% in recall when using the interface as compared with the PDF Document method (Table 3). The F1-score improved by 39.35%.

The disparity in experience significantly influenced the accuracy of curation, particularly in terms of high-level skills. Senior researchers consistently achieved an average F1-Score approximately 13% higher than other curators (see Table 4). Furthermore, we observed a modest improvement between master's students and PhD students. These findings indicate also that for large-scale projects, employing master students instead of PhD students may be a more cost-effective choice. Thus, using only a few senior researchers for the second round of validation (Section 3.1).

Finally, the collected data suggest that all three curators had overall more corrected results by using the interface as illustrated in Table 5.

The results of this experiment confirmed that our curation interface and workflow significantly improved the quality of the extracted data, with an astonishing improvement in recall, thus preventing curators from overlooking important information.

Code availability

This work is available at https://github.com/lfoppiano/supercon2. The repository contains the code of the SuperCon 2 interface, the curation workflow, and the ingestion processes for harvesting the SuperCon 2 Database of materials and proper-ties. The guidelines are accessible at https://supercon2.readthedocs.io.

Conclusions

We built a semi-automatic staging area, called SuperCon 2 , to validate efficiently new experimental records automatically collected from superconductor research articles (SuperCon 2 Database [START_REF] Foppiano | Automatic extraction of materials and properties from superconductors scientific literature[END_REF]) before they are ingested into the existing, manually-build database of superconductors, SuperCon [START_REF] Ishii | Structuring superconductor data with ontology: reproducing historical datasets as knowledge bases[END_REF]. The system provides a curation workflow and a user interface (SuperCon 2 Interface) tailored to efficiently support domain experts in data correction and validation with fast context switching and an enhanced PDF viewer. Under the hood, the workflow ran "anomaly detection" to automatically identify outliers and a "training data collector" based on human corrections, to efficiently accumulate training data to be feedback to the ML model.

Compared with the traditional manual approach of reading PDF documents and extracting information in an Excel file, SuperCon 2 significantly improves the curation quality by approximately 6% and +47% for precision and recall, respectively. In future, this work can be expanded to support other materials science domains such as magnetic materials, spintronic and thermoelectric research and expanding the evaluation to a larger dataset. Table 1. F1-score from the evaluation of the fine-tuned SciBERT models. The training is performed with three different approaches. The base dataset is the original dataset described in [START_REF] Foppiano | Supermat: construction of a linked annotated dataset from superconductors-related publications[END_REF], and the curation dataset is automatically collected based on the database corrections by the interface and manually corrected. s indicate "training from scratch", while i indicate "incremental training". The evaluation is performed using the same holdout dataset from SuperMat [START_REF] Foppiano | Supermat: construction of a linked annotated dataset from superconductors-related publications[END_REF]. The results are averaged over 5 runs or train and evaluation.

Figures & Tables

Figure

  Figure 1. Schema of the curation workflow. Each node has two properties: type and status (Section 2.1.1). Each edge indicates one action. The workflow starts on the left side of the figure. The new records begin with "Automatic, New". Changes of state are triggered by automatic (Section 2.2) or manual operations (update, mark as valid, etc.. Section 3.1) and results in changes of the properties in the node. Each combination of property values identifies each state. "(*)" indicates a transition for which the training data are collected (Section 2.3)

1 .Figure 2 .

 12 Figure 1. Schema of the curation workflow. Each node has two properties: type and status (Section 2.1.1). Each edge indicates one action. The workflow starts on the left side of the figure. The new records begin with "Automatic, New". Changes of state are triggered by automatic (Section 2.2) or manual operations (update, mark as valid, etc.. Section 3.1) and results in changes of the properties in the node. Each combination of property values identifies each state. "(*)" indicates a transition for which the training data are collected (Section 2.3)

Figure 3 .

 3 Figure 3. Screenshot of SuperCon 2 interface showing the database. Each row corresponds to one material-Tc pair. On top, there are searches by attribute, sorting and other filtering operations. On the right (last column) there are curation controls (mark as valid, update, etc.). Records are grouped by document with alternating light yellow and white.

Figure 4 .

 4 Figure 4. PDF document viewer showing an annotated document. The table on top is linked through the annotated entities. The user can navigate from the record to the exact point in the PDF, with a pointer (the red bulb light) identifying the context of the entities being examined.

Figure 5 .

 5 Figure 5. Sample curation sheet from the curation guidelines. The sheet is composed of the following information: a) Sample input data: a screenshot of the record from the "SuperCon 2 Interface", b) Context represented by the related part of the annotated document referring to the record in exams. c) The Motivation, describing the issue, d) the Action to be taken, and the Expected output.

Figure 6 .

 6 Figure 6. Top: Processing log, showing the output of each ingestion operation and the outcome with the detailed error that may have occurred. Bottom: Correction log, indicating each record, the number of updates, and the date/time of the last updates. By clicking on the "Record id", is possible to visualise the latest record values.

Table 2 .

 2 Data support, the number of entities for each label in each of the datasets used for evaluating the ML models. The base dataset is the original dataset described in[START_REF] Foppiano | Supermat: construction of a linked annotated dataset from superconductors-related publications[END_REF], and the curation dataset is automatically collected based on the database corrections by the interface and manually corrected.

	base(s) (base+curation)(s) base(s)+(base+curation)(i)
	Nb total examples	16902		17254	16902(s), 17254 (i)
	<class>	70.41		73.02	71.86
	<material>	79.37		80.09	80.37
	<me method>	66.72		66.57	66.95
	<pressure>	46.43		48.42	47.23
	<tc>	80.13		80.92	80.34
	<tcValue>	78.29		78.41	79.73
	All (micro avg.)	76.67		77.44	77.48
	∆ avg. w/ baseline	-		+0.77	+0.81
			base base+curation	∆
	<class>	1646	1732	86
	<material>	6943	7580	637
	<me method> 1883	1934	51
	<pressure>	274	361	87
	<tc>		3741	4269	528
	<tcValue>	1099	1556	457
	Total	15586	17432	1846

Table 3 .

 3 Evaluation scores (P: precision, R: recall, F1: F1-score) between the curation using the SuperCon 2 interface (Interface) and the traditional method of reading the PDF document (PDF document).

	Method	P (%) R (%) F1 (%) # docs
	PDF document	87.83	45.61	52.67	15
	Interface	93.38	92.51	92.02	15

Table 4 .

 4 Evaluation scores (P: precision, R: recall, F1: F1-score) aggregated by experience (MS: master student, PD: PhD student, SR: senior researcher). Each person corrected 10 documents.

	Experience P (%) R (%) F1 (%) # docs # pages
	MS	90.03	60.26	64.50	10	96
	PD	83.33	65.69	69.45	10	100
	SR	98.45	81.22	83.08	10	96

Table 5 .

 5 Evaluation

	Experience	Method	P (%) R (%) F1 (%) # docs # pages
	MS	PDF Document Interface	94.58 83.19	36.55 95.83	48.67 88.25	6 4	46 50
	PD	PDF Document Interface	70.00 96.67	48.51 82.86	50.78 88.11	5 5	49 51
	SR	PDF Document 100.00 Interface 97.42	55.56 98.33	61.03 97.78	4 6	51 45

scores (P: precision, R: recall, F1: F1-score) listed by experience (MS: master student, PD: PhD student, SR: senior researcher), and method (PDF document, Interface).

Table A1 .

 A1 Timetable recording the time spent for each of the 15 articles. Each row indicates the time and the event (Start, Finish) from each of the curators: Master Student (MD), PhD Student (PD), and Senior Researcher (SR). Duration is expressed in minutes.

	Time Event Document ID Curator Duration (mins)
	14:40 Start	02bf1b3db9	PS	0
	14:49 Finish	02bf1b3db9	PS	9
	14:53 Start	00b50fc0a8	PS	0
	14:58 Finish	00b50fc0a8	PS	5
	14:37 Start	0aa1b3161f	MS	0
	14:50 Start	0454e07f64	SR	0
	14:58 Finish	0454e07f64	SR	8
	15:01 Start	02cbc58819	PS	0
	15:06 Start	00c32076f4	SR	0
	15:07 Finish	0aa1b3161f	MS	30
	15:08 Finish	02cbc58819	PS	7
	15:08 Start	044939701d	PS	0
	15:12 Start	0021fd339f	MS	0
	15:15 Finish	00c32076f4	SR	9
	15:17 Finish	044939701d	PS	9
	15:17 Start	08e1cb8f4f	PS	0
	15:20 Start	0c7d3163ea	SR	0
	15:31 Finish	08e1cb8f4f	PS	14
	15:32 Finish	0021fd339f	MS	20
	15:32 Start	039105663f	MS	0
	15:37 Finish	0c7d3163ea	SR	17
	15:53 Finish	039105663f	MS	21
	15:55 Start	02c4f00127	MS	0
	15:58 Start	0454e07f64	PS	0
	16:02 Start	0da5febabf	SR	0
	16:08 Finish	0454e07f64	PS	10
	16:09 Finish	02c4f00127	MS	14
	16:11 Finish	0da5febabf	SR	9
	16:11 Start	0012333581	SR	0
	16:12 Start	00c32076f4	PS	0
	16:18 Start	021c413172	MS	0
	16:22 Finish	00c32076f4	PS	10
	16:23 Start	0c7d3163ea	PS	0
	16:30 Finish	0012333581	SR	19
	16:32 Finish	021c413172	MS	14
	16:37 Start	02bf1b3db9	MS	0
	16:38 Finish	0c7d3163ea	PS	15
	17:32 Finish	0021fd339f	SR	12
	17:34 Start	039105663f	SR	0
	17:55 Finish	039105663f	SR	21
	17:56 Start	02c4f00127	SR	0
	18:00 Finish	02c4f00127	SR	4
	18:00 Start	021c413172	SR	0
	18:09 Finish	021c413172	SR	9

Table A2 .

 A2 Evaluation scores obtained for each document and method (I: Interface, P: PDF) combination. TP: True positive, FP: False positive, FN: False negative. P: Precision, R: Recall, F1: F1-score

	Document ID # pages Method # TP # FP # FN	P	R	F1
			Senior Researcher (SR)
	0454e07f64	4	I	6	0	0	100.00 100.00 100.00
	00c32076f4	13	P	8	0	0	100.00 100.00 100.00
	0c7d3163ea	9	I	13	1	0	92.86 100.00 96.30
	0da5febabf	11	P	8	0	1	100.00 88.89	94.12
	0012333581	13	I	11	0	0	100.00 100.00 100.00
	0aa1b3161f	5	I	9	0	1	100.00 90.00	94.74
	0021fd339f	14	P	4	0	8	100.00 33.33	50.00
	039105663f	9	I	11	1	0	91.67 100.00 95.65
	02c4f00127	13	P	0	0	3	100.00	0.00	0.00
	021c413172	5	I	15	0	0	100.00 100.00 100.00
				PhD Student (PS)
	02bf1b3db9	7	I	5	0	2	100.00 71.43	83.33
	00b50fc0a8	11	P	2	0	7	100.00 22.22	36.36
	02cbc58819	4	I	4	0	3	100.00 57.14	72.73
	044939701d	12	P	4	0	2	100.00 66.67	80.00
	08e1cb8f4f	16	I	5	1	1	83.33	85.71	84.51
	0454e07f64	4	P	0	1	5	0.00	16.67	0.00
	00c32076f4	13	I	8	0	0	100.00 100.00 100.00
	0c7d3163ea	9	P	9	0	5	100.00 64.29	78.26
	0da5febabf	11	I	9	0	0	100.00 100.00 100.00
	0012333581	13	P	4	4	3	50.00	72.73	59.26
				Master Student (MS)
	0aa1b3161f	5	P	1	0	9	100.00 10.00	18.18
	0021fd339f	14	I	12	3	3	80.00 100.00 88.89
	039105663f	9	P	4	1	7	80.00	41.67	54.79
	02c4f00127	13	I	3	1	1	75.00 100.00 85.71
	021c413172	5	P	7	1	7	87.50	53.33	66.27
	02bf1b3db9	7	P	2	0	5	100.00 28.57	44.44
	00b50fc0a8	11	I	7	2	0	77.78 100.00 87.50
	02cbc58819	4	P	5	0	2	100.00 71.43	83.33
	044939701d	12	I	5	0	1	100.00 83.33	90.91
	08e1cb8f4f	16	P	1	0	6	100.00 14.29	25.00

"internal status" indicates that their records should be hidden in the interface

In our previous work[START_REF] Foppiano | Automatic extraction of materials and properties from superconductors scientific literature[END_REF] we reported 77.03% F1-score. There is a slight decrease in absolute scores between DeLFT 0.2.8 and DeLFT 0.3.0. One cause may be the use of different hyperparameters in version 0.3.0 such as batch size and learning rate. However, the most probable cause could be the impact of using the Huggingface tokenizers library which is suffering from quality issues https://github.com/kermitt2/delft/issues/150.

Acknowledgements

Our warmest thanks to Patrice Lopez, the author of Grobid [22], DeLFT [20], and other open-source projects for his continuous support and inspiration with ideas, suggestions, and fruitful discussions. We thank Pedro Baptista de Castro for his support during this work. Special thanks to Erina Fujita for useful tips on the manuscript.

Funding This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project (Digital Transformation Initiative Center for Magnetic Materials) Grant Number JPMXP1122715503.

Notes on Contributors

LF wrote the manuscript and KT helped with the editing. LF and POS discussed the ML results and experiments. LF implemented the workflow as a standalone service, and TM wrote the front end of the user interface. LF designed the user interface experiment with KT, TT and WS as curators. KT led the materials-science work on the data with CS, TT and WS. KT, TA, YT and MI revised the paper. YT and MI supervised the work of the respective teams.


Emilio Casanova Biscarri

Sophie Mercier
email: sophie.mercier@univ-pau.fr

Carmen Sang Üesa

A model for stochastic dependence implied by failures among deteriorating components

Keywords: reliability, Lévy process, inverse Gaussian process, functional dependence, stochastic comparison, aging property

A system of n components is here considered, with component deterioration modeled by non decreasing time-scaled Lévy processes. When a component fails, a sudden change in the time-scaling functions of the surviving components is induced, which makes the components stochastically dependent. We compute the reliability function of coherent systems under this new dependence model. We next study the distribution of the ordered failure times, and establish some positive dependence properties. We also provide stochastic comparison results in the usual multivariate stochastic order between failure times of two dependence models with different parameters. Finally, some numerical experiments illustrate the theoretical results.

Introduction

Safety and dependability is a crucial issue in many industries, which has lead to the development of a huge literature devoted to the so-called reliability theory. Traditionally, only lifetime data were available, from where appropriate lifetime models were inferred. See, e.g., [START_REF] Barlow | Mathematical theory of reliability[END_REF] for a pioneer reference work on the subject. Based on the on-going development of on-line monitoring which allows the effective measurement of a system deterioration, numerous papers nowadays model the degradation in itself, which allows for a better assessment of the system health and a better prediction of its future failure time. The deterioration level of a system is usually considered to be accumulating over time and it is often modeled by non decreasing (or with increasing trend) continuous-time stochastic processes. Most common models are Lévy processes such as gamma processes [START_REF] Abdel-Hameed | A gamma wear process[END_REF][START_REF] Bažant | Stochastic process for extrapolating concrete creep[END_REF][START_REF] Van Noortwijk | A survey of the application of Gamma processes in maintenance[END_REF], Wiener processes with trend [START_REF] Doksum | Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse gaussian distribution[END_REF][START_REF] Whitmore | Estimating degradation by a Wiener diffusion process subject to measurement error[END_REF][START_REF] Zhang | Degradation data analysis and remaining useful life estimation: A review on Wiener-process-based methods[END_REF] and inverse Gaussian processes [START_REF] Wang | An inverse gaussian process model for degradation data[END_REF][START_REF] Ye | The inverse gaussian process as a degradation model[END_REF] (see also [START_REF] Kahle | Degradation Processes in Reliability[END_REF][START_REF] Wang | Modeling multivariate degradation processes with time-variant covariates and imperfect maintenance effects[END_REF] for more references).

In this paper, we consider a system composed of n components, with component deterioration modeled by non decreasing time-scaled Lévy processes and component failures triggered by the reaching of a given known threshold. In the oldest literature, components were most of the time assumed to be independent, which highly facilitated the modeling and assessment of reliability indicators, using for instance structure functions (see, e.g. [START_REF] Barlow | Mathematical theory of reliability[END_REF]). Since then, many different kinds of dependence have been considered, see, e.g. [START_REF] Keizer | Condition-based maintenance policies for systems with multiple dependent components: A review[END_REF], where the authors review condition-based maintenance policies for systems with components subject to economic, structural, stochastic and resource dependence. Following their vocabulary, we here envision some "stochastic dependence" between components, which "means that the deterioration or failure processes of components are (partially) dependent" [START_REF] Keizer | Condition-based maintenance policies for systems with multiple dependent components: A review[END_REF]. To be more specific, we here assume that a failure among components has an impact on the future deterioration of the surviving components and we call it stochastic failure dependence. This kind of dependence has been envisioned in different papers such as [START_REF] Burkschat | Systems with failure-dependent lifetimes of components[END_REF][START_REF] Lee | On a new stochastic model for cascading failures[END_REF][START_REF] Lee | On construction of general classes of bivariate distributions[END_REF][START_REF] Lee | Point process approach to modeling and analysis of general cascading failure models[END_REF], where the authors model the impact of failures on the lifetimes of the surviving components (including possible cascading failures), see also [START_REF] Keizer | Condition-based maintenance policies for systems with multiple dependent components: A review[END_REF] for many other models and references. However, the papers involving Lévy deteriorating components together with stochastic failure dependence are fewer. One can quote [START_REF] Zhang | Condition-based maintenance for a k-out-of-n deteriorating system under periodic inspection with failure dependence[END_REF], where the authors consider that a component failure "acts as a shock to each surviving component by causing a random magnitude of damage which is additive to the degradation level of the survivor". See also [START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF] for a similar dependence model. One can also quote [START_REF] Dong | Reliability and availability analysis of stochastic degradation systems based on bivariate wiener processes[END_REF], where the system health is described by two correlated performance characteristics (or component deterioration levels) modeled by a bivariate Wiener process at a first stage. Once one of the two characteristics has reached its threshold (or once one of the two components has failed), the system operates defectively and enters its second stage, where the surviving characteristic evolves according to a new univariate Wiener process. Within this setting, the authors compute the system reliability and availability based on different inspection and maintenance policies.

In a similar spirit as [START_REF] Dong | Reliability and availability analysis of stochastic degradation systems based on bivariate wiener processes[END_REF], we here suggest that each failure among the n components entails an increase in the deterioration rate of the surviving components. As a typical example of such stochastic dependence, one can think about the case where the "multiple components share the total system load. If a component fails, the system keeps operating but the remaining components structurally need to work harder to realize the same output level. The failure of a component thus increases the load on the working components, which will hence deteriorate faster. In practice, this applies for example to a set of pumps that are used to distribute a certain amount of gas." [START_REF] Keizer | Condition-based maintenance policies for systems with multiple dependent components: A review[END_REF]. As another example, one can also think of a parallel-connected power supply system for a server farm, where each server is powered by its own power supply unit (PSU). When one PSU fails, the remaining PSUs need to compensate for the lost power output to meet the power demands of the servers, resulting in faster degradation of those surviving PSUs. One can also think of an electric vehicle powered by a lithium-ion battery pack composed of n identical battery cells. Each cell contributes to storing energy for the vehicle's operation. When a battery cell fails, the overall performance of the pack is disrupted and the operation of the remaining cells must be adjusted, putting additional strain on them and leading to an increased degradation rate. In the same way, the failure of one wind turbine within a wind farm can lead to adjustments in the operation of the remaining turbines, potentially accelerating the overall deterioration of the wind farm's mechanical and electrical components.

In this paper, remembering that the deterioration of a component is modeled by a non decreasing time-scaled Lévy process, we suggest to model the impact of a component failure on the surviving components by some increase in the time-scaling functions of the corresponding Lévy processes. Note that up to our knowledge, such a stochastic failure dependence model with impact on the time-scaling functions of the surviving non decreasing Lévy-deteriorating components has not yet been considered in the literature.

Considering this dependence model, we compute the reliability of a coherent system, as well as the marginal and joint distributions of the component lifetimes and corresponding ordered statistics. These results are used to derive some dependence and stochastic comparison properties, which allow to better understand the impact of the model parameters. Numerical experiments are next given considering inverse Gaussian processes as baseline Lévy processes. These experiments show good agreements between the theoretical results and Monte-Carlo simulations (which require much longer computing times however). These experiments also illustrate the dependence and stochastic comparison results of the paper and highlight the requirement of several assumptions given for their derivation.

The paper is organized as follows: Section 2 provides a quick summary about time-scaled Lévy processes. The model for dependence is next introduced in Section 3. The reliability function of a coherent system with dependent components is studied in Section 4. The marginal and joint distributions of the component lifetimes are computed in Section 5, as well as the joint distribution of the corresponding ordered statistics. Section 6 is dedicated to dependence and stochastic comparison properties. The numerical experiments are given in Section 7. Section 8 ends the paper with some conclusion and perspectives for future works.

Please note that in all the paper, the terms "increasing" and "decreasing" mean "non-decreasing" and "non-increasing", respectively. Also, a function ϕ : I ⊆ R n -→ J ⊆ R is said to be increasing if ϕ(x) ≤ ϕ(y) whenever x ≤ y component-wise, for all x, y ∈ I .

Time-scaled Lévy processes

We here recall some basic facts about time-scaled Lévy processes and start with some definitions. Definition 1 A function Λ : R + -→ R + is said to be a time-scaling function if Λ is differentiable and strictly increasing, such that Λ (0) = 0 and lim t→+∞ Λ (t) = ∞. Its derivative function is denoted by λ = Λ ′ . Definition 2 Let Y = (Y (t)) t≥0 be a càdlag process, that is a right-continuous process with left-side limits. Then Y is a Lévy process if:

Y (0) = 0 almost surely, Y has independent and homogeneous increments: for all n ∈ N * , t 0 = 0 < t 1 < • • • < t n , the increments ∆Y (t i-1 , t i ) = Y (t i ) -Y (t i-1 ), i = 1, • • • , n are independent and ∆Y (t i-1 , t i ) is identically distributed as Y (t i -t i-1 ) for all i = 1, • • • , n, Y is stochastically continuous.
Definition 3 Let X = (X (t)) t≥0 be a càdlag process. Then X is called a time-scaled Lévy process if there exists a time-scaling function Λ and a baseline Lévy process Y such that

X (t) = Y (Λ (t))
for all t ≥ 0. In that case, X is said to be a time-scaled Lévy process with parameters (Λ, Y). Now, let us consider a component with deterioration level modeled by a time-scaled non negative Lévy process X = (X (t)) t≥0 with parameters (Λ, Y) as in the previous definition. Recall that both Y and X are increasing (and Y is a subordinator). Assume the component to be failed as soon as X (t) ≥ L, where L is a given failure threshold. Then, the component failure time is

τ X = inf (t ≥ 0 : X (t) ≥ L) with survival function Fτ X (t) = P (τ X > t) = P (X (t) ≤ L) = F X(t) (L) ,
where F X(t) is the cumulative distribution function (cdf) of X (t). Using similar notations for Y, one can see that

Fτ X (t) = P (Y (Λ (t)) ≤ L) = P (τ Y > Λ (t)) = Fτ Y (Λ (t)) . (1)
Assuming Fτ Y to be differentiable, then Fτ X also is differentiable with probability density function (pdf)

f τ X (t) = λ (t) f τ Y (Λ (t)) (2)
for all t ≥ 0.

The model for dependence

Let us consider n identical non-repairable components gathered in a system. Between failures among the n components, each component deteriorates according to a time-scaled Lévy process. A component fails as soon as its deterioration level crosses a given failure threshold L. Each failure entails some modification in the future deterioration of the still operating components, as they become more sollicited. This modification is modelled by a change in the time-scaling functions for the surviving components. Between failures, the instrinsic deterioration of the surviving components are assumed to be independent.

In order to define the model more specifically, let us introduce independent time-scaled non negative Lévy processes X i,j , i, j = 1, • • • , n, with parameters (Λ i,j , Y i,j ), i, j = 1, • • • , n, where: index i refers to the i-th component, index j refers to the number of already failed components (among the n components), the Λ i,j are time-scaling functions which only depend on the number of already failed components (and not on the component), that is:

Λ i,j = Λ j for all i, j = 1, • • • , n, (3)
the baseline Lévy processes Y i,j , i, j = 1, • • • , n are independent and identically distributed as a given non negative Lévy process Y:

P (Y i,j (t) ≤ x) = P (Y (t) ≤ x) for all i, j = 1, • • • , n, all t ≥ 0, all x ≥ 0. ( 4
)
At the beginning, each component deteriorates according to X i,1 , i ∈ I 1 = {1, • • • , n}. Then, after the first failure, the surviving components deteriorate according to X i,2 , i ∈ I 2 , where I 2 stands for the set of the n -1 surviving components. More generally, after the j-th failure with 1 ≤ j ≤ n -1, the surviving components deteriorate according to X i,j+1 , i ∈ I j+1 , where I j+1 stands for the set of the n -j surviving components. The failure times of the components are denoted by τ i , i = 1, • • • , n, and the successive failures happen at time τ (1) ≤ τ (2) ≤ • • • ≤ τ (n) , where τ (1) , • • • , τ (n) is the vector of order statistics of (τ 1 , • • • , τ n ). For j = 1, • • • , n, the period between τ (j-1) and τ (j) is called the j-th phase (where the 0-th failure is considered to happen at time τ (0) = 0). This is illustrated in Figure 1 in the case of four components, with τ (1) = τ 1 < τ (2)

= τ 3 < τ (3) = τ 2 < τ (4) = τ 4 .
The overall deterioration level of the i-th component is denoted by Z i = (Z i (t)) t≥0 , i = 1, • • • , n and we set D to be the down state for a failed component. In the specific case where τ 1 < τ 2 < • • • < τ n (for sake of simplicity) and setting τ 0 = 0, we have:

Z i (t) =                        X i,1 (t) = ∆X i,1 (τ 0 , t) if t < τ 1 , ∆X i,1 (τ 0 , τ 1 ) + ∆X i,2 (τ 1 , t) if τ 1 ≤ t < τ 2 , . . . . . . k-1 j=1 ∆X i,j (τ j-1 , τ j ) + ∆X i,k (τ k-1 , t) if τ k-1 ≤ t < τ k , . . . . . . i-1 j=1 ∆X i,j (τ j-1 , τ j ) + ∆X i,i (τ i-1 , t) if τ i-1 ≤ t < τ i , D if τ i ≤ t (5
) for all i = 1, • • • , n and t ≥ 0, where ∆X i,j (u, v) = X i,j (v) -X i,j (u) for all 0 ≤ u ≤ v and all i, j = 1, Technical assumptions 5 In all the paper, we make the following technical assumptions:

For all t ≥ 0, the random variable Y (t) admits a pdf with respect to Lebesgue measure,

The function F Y (t) (L) is differentiable with respect to t.

We end this section with some notations and a few direct consequences of the model definition, that will be used repeatedly in the remaining of the paper.

We set

∆Λ j (s, t) = Λ j (t) -Λ j (s)
for all 0 ≤ s ≤ t and j = 1, • • • , n, with ∆Λ j (0, t) = Λ j (t) when s = 0.

Based on (3), ( 4) and on the homogeneity of Y, it is easy to check that

P (∆X i,j (s, t) ≤ x) = P (Y (Λ j (t)) -Y (Λ j (s)) ≤ x) = P (Y (Λ j (t) -Λ j (s)) ≤ x) = P (Y (∆Λ j (s, t)) ≤ x)
for all 0 ≤ s < t and all x ≥ 0. When x is equal to the failure threshold L, we also have

P (∆X i,j (s, t) ≤ L) = P (τ Y > ∆Λ j (s, t)) = Fτ Y (∆Λ j (s, t)) .
Finally, in a similar way as in ( 1) and ( 2), we have

Fτ X i,j (t) = Fτ Y (Λ j (t)) , (6)
f τ X i,j (t) = λ j (t) f τ Y (Λ j (t)) , (7)
for all t > 0, i, j = 1, • • • , n.

Reliability of a coherent system

Considering n identical components with dependence modelled as in the previous section, we now assume that they are gathered in a coherent system and the point is to compute its reliability. To begin with, let us consider a specific coherent system, that is a k-out-of-n system (with 1 ≤ k ≤ n). In that case, the system is up as long as at least k components are operating and the system fails at the time of the n -k + 1-th failure among components. The lifetime of the system hence is the order statistics τ (n-k+1) and the reliability function is

R k,n (t) = P τ (n-k+1) > t ,
for all t ≥ 0.

Theorem 6 The reliability function of the k-out-of-n system is given by

R k,n (t) = n-k m=0 R m (t) , for all t ≥ 0, with R 0 (t) = Fτ Y (Λ 1 (t)) n (8)
and

R m (t) = n! (n -m)! R m + 1 {t 1 <t 2 <•••<tm<t} a m (t 1 , . . . t m ) × Fτ Y Λ * m+1 (t 1 , . . . , t m , t) n-m dt 1 • • • dt m (9)
for all t ≥ 0 and all m = 1, • • • , n -1, where

Λ ⋆ m (t 1 , . . . , t m ) = m j=1 ∆Λ j (t j-1 , t j ), (10)
a m (t 1 , . . . t m ) = m i=1 λ i (t i ) f τ Y (Λ ⋆ i (t 1 , . . . , t i )) , (11)
for all m = 1, • • • , n and all t 0 = 0 < t 1 < t 2 < • • • < t m .
Proof. Let us first divide the reliability function R k,n (t), according to the number of already failed components at time t :

R k,n (t) = n-k m=0 R m (t) , (12)
where

R m (t) = P τ (m) < t < τ (m+1) (13)
stands for the probability that exactly m components are failed at time t, with 0 ≤ m ≤ n -k (and τ (0) = 0). Let us first consider the case m = 0, where no components have failed before t :

R 0 (t) = P τ (1) > t = P min 1≤i≤n (τ i ) > t = P n i=1 {τ i > t} .
When no components have failed at time t, all components are in their first phase (see Figure 1) and they all deteriorate according to independent processes

X i,1 , i = 1, • • • , n up to time t. Hence R 0 (t) = P n i=1 {X i,1 (t) < L} = n i=1 P (X i,1 (t) < L) = Fτ Y (Λ 1 (t)) n . Now let us consider the generic term R m (t), with 1 ≤ m ≤ n -k and let A n,m be the set of sequences (i 1 , • • • , i m ) of m elements of {1, • • • , n} without repetition, with card (A n,m ) = n! (n -m)! .
Then, the reliability can be written as

R m (t) = (i 1 ,••• ,im)∈An,m P τ i 1 < τ i 2 < • • • < τ im < t < min j / ∈{i 1 ,••• ,im} τ j (14)
Noting that the probability in the right side of ( 14) does not depend on

(i 1 , • • • , i m ) (because all components are identical), we obtain R m (t) = n! (n -m)! S m (t) (15)
with

S m (t) = P τ 1 < τ 2 < • • • < τ m < t < min m+1≤i≤n τ i . Conditioning by τ 1 , • • • , τ m , we get S m (t) = E 1 {τ 1 <τ 2 <•••<τm<t} Φ (τ 1 , • • • , τ m ) (16)
with

Φ (t 1 , • • • , t m ) = P t < min m+1≤i≤n τ i |τ 1 = t 1 , • • • , τ m = t m = P m+1≤i≤n {t < τ i } |τ 1 = t 1 , • • • , τ m = t m = P m+1≤i≤n {Z i (t) < L} |τ 1 = t 1 , • • • , τ m = t m for t 1 < t 2 < • • • < t m < t. Now, given that τ 1 = t 1 < • • • < τ m = t m ,
all n -m still operating components are in their m + 1-th phase, and remembering [START_REF] Bažant | Stochastic process for extrapolating concrete creep[END_REF], their deterioration level is given by

Z i (t) = m j=1 ∆X i,j (t j-1 , t j ) + ∆X i,m+1 (t m , t) = st Y m j=1 ∆Λ j (t j-1 , t j ) + ∆Λ m+1 (t m , t) = st Y Λ * m+1 (t 1 , • • • , t m , t) (17)
for all t m ≤ t < τ (m+1) = min m+1≤k≤n τ k and all i = m + 1, • • • , n, where = st means equality in distribution. The second line is a direct consequence of the model assumptions described in Subsection 3. Noting that, given

τ 1 = t 1 < • • • < τ m = t m , the processes (Z i (t)) t≥tm , i = m + 1, • • •
, n are conditionally independent up to the next failure at time τ (m+1) , we obtain

Φ (t 1 , • • • , t m ) = m+1≤i≤n P Y Λ * m+1 (t 1 , • • • , t m , t) < L = Fτ Y Λ * m+1 (t 1 , • • • , t m , t) n-m .
Starting again from ( 16), we now have

S m (t) = R m + 1 {t 1 <t 2 <•••<tm<t} Fτ Y Λ * m+1 (t 1 , • • • , t m , t) n-m × f (τ 1 ,••• ,τm) (t 1 , • • • , t m ) dt 1 • • • dt m = R m + 1 {t 1 <t 2 <•••<tm<t} Fτ Y Λ * m+1 (t 1 , • • • , t m , t) n-m f τ 1 (t 1 ) × m i=2 f τ i |τ 1 ,••• ,τ i-1 (t i |t 1 , • • • , t i-1 ) dt 1 • • • dt m (18)
with [START_REF] Dong | Reliability and availability analysis of stochastic degradation systems based on bivariate wiener processes[END_REF]) and where

f τ 1 (t 1 ) = λ 1 (t 1 ) f τ Y (Λ 1 (t)) = λ 1 (t 1 ) f τ Y (Λ * 1 (t)) (see
f τ i |τ 1 ,••• ,τ i-1 (t i |t 1 , • • • , t i-1 ) stands for the conditional pdf of τ i given τ 1 = t 1 , • • • , τ i-1 = t i-1 .
Using a similar argument as in [START_REF] Samaniego | System signatures and their applications in engineering reliability[END_REF], we have

Fτ i |τ 1 ,••• ,τ i-1 (t i |t 1 , • • • , t i-1 ) = P (Z i (t i ) < L |τ 1 = t 1 , • • • , τ i=1 = t i-1 ) = P (Y (Λ * i (t 1 , • • • , t i )) < L) = Fτ Y (Λ * i (t 1 , • • • , t i ))
for all 2 ≤ i ≤ m, from where we easily derive that

f τ i |τ 1 ,••• ,τ i-1 (t i |t 1 , • • • , t i-1 ) = - ∂ ∂t i Fτ Y (Λ * i (t 1 , • • • , t i )) = λ i (t i ) f τ Y (Λ * i (t 1 , • • • , t i )) . (19)
This allows to conclude, based on ( 12), ( 15) and [START_REF] Sangüesa | Preservation of ageing classes in deterioration models with independent increments[END_REF].

Remark 7 In the specific case where Λ i (t) = Λ (t) for all t ≥ 0 and all i = 1, • • • , n, all components are independent. In that case, Formula (9) provides

R m (t) = n! (n -m)! R m + 1 {t 1 <t 2 <•••<tm<t} Fτ Y (Λ (t)) n-m × m i=1 λ (t i ) f τ Y (Λ (t i )) dt 1 • • • dt m = n! (n -m)! Fτ Y (Λ (t)) n-m 1 m! [0,t] m m i=1 λ (t i ) f τ Y (Λ (t i )) dt 1 • • • dt m = n! (n -m)! Fτ Y (Λ (t)) n-m 1 m! t 0 λ (t 1 ) f τ Y (Λ (t 1 )) dt 1 m = n m Fτ Y (Λ (t)) n-m (F τ Y (Λ (t))) m ,
which is coherent, as the distribution for the number of failed components at time

t is binomial B (n, F τ Y (Λ (t))).
Remark 8 Remembering that series and parallel systems are specific k-outof-n systems with k = n and k = 1, respectively, we easily derive that their reliability functions are

R n,n (t) = R 0 (t)
for the series case and

R 1,n (t) = n-1 m=0 R m (t)
for the parallel case, using the notations of Theorem 6.

Let us now assume that the components are gathered in a general coherent system, that is in a system such that all components are relevant and with a non decreasing structure function, see e.g. [START_REF] Samaniego | System signatures and their applications in engineering reliability[END_REF] for more details. Noting that the n components are exchangeable, we can observe from [START_REF] Navarro | Reliability and expectation bounds for coherent systems with exchangeable components[END_REF] that the structure of the system is caracterized by its signature p = (p 1 , • • • , p n ), where p i is the probability that the failure of the system is induced by the i-th failure among components, that is

p i = P T = τ (i)
for all i = 1, • • • , n, where T stands for the failure time of the system. Also, the signature p is known to be independent on the joint distribution of (τ 1 , • • • , τ n ) (please see [START_REF] Navarro | Reliability and expectation bounds for coherent systems with exchangeable components[END_REF] for more details). Finally, as simultaneous failures are not possible in our context, p is a probability vector, that is

n i=1 p i = 1.
We can now derive the system reliability as a direct application of [16, Lemma 1].

Proposition 9

The reliability of the coherent system with signature p = (p 1 , • • • , p n ) is given by:

R (t) = n k=1 p k R k,n (t) ,
for all t ≥ 0.

Distributions

In this section we compute the marginal and joint distributions of the component lifetimes, as well as the joint distribution of the corresponding ordered statistics.

Marginal distributions

Proposition 10 For each i = 1, • • • , n, the survival function of τ i is given by

P (τ i > t) = n-1 m=0 n -m n R m (t)
for all t > 0 (independent on i), where R m (t) has been introduced in Theorem 6 (see also [START_REF] Lee | Point process approach to modeling and analysis of general cascading failure models[END_REF]).

Proof. We may write

P (τ i > t) = n-1 m=0 P τ i > t|τ (m) < t < τ (m+1) R m (t) = n-1 m=0 n -m n R m (t) ,
for all t > 0, where the last equality comes from the fact that the probability that component i is one of the n -m surviving components is (n -m)/n (as all components are identical).

Joint distributions

We first compute the joint probability density function of (τ 1 , . . . τ n ).

Theorem 11

The probability density function of (τ 1 , . . . τ n ) is given by

f (τ 1 ,...τn) (t 1 , . . . , t n ) = n i=1 λ i t (i) f τ Y Λ * i t (1) , • • • , t (i) = a n t (1) , • • • , t (n)
for all distinct t 1 , t 2 , . . . , t n ∈ R + (0 otherwise), where (t (1) , . . . , t (n) ) is the sequence (t 1 , . . . , t n ) arranged in increasing order and where a n is defined in [START_REF] Lee | On a new stochastic model for cascading failures[END_REF].

Proof. Let us first compute the multivariate survival function of (τ 1 , . . . τ n ). Using a similar method as for the computation of R m (t) in the proof of Theorem 6, we distinguish the order in which failures occur, that is we write

P(τ 1 > t 1 . . . τ n > t n ) = (k 1 ,••• ,kn)∈An P(τ 1 > t 1 . . . τ n > t n , τ k 1 < • • • < τ kn )
for all t 1 , t 2 , . . . , t n ∈ R + , where A n := A n,n is the set of all permutations of {1, • • • , n}.

Noting that the set {τ 1 > t 1 . . . τ n > t n } does not depend on the ordering of the τ i 's, we now have

P (τ 1 > t 1 . . . τ n > t n ) = (k 1 ,••• ,kn)∈An P(τ k 1 > t k 1 , τ k 2 > t k 2 , . . . , τ kn > t kn , τ k 1 < • • • < τ kn ) = (k 1 ,••• ,kn)∈An P(τ k 1 > t k (1) , τ k 2 > t k (2) , . . . , τ kn > t k (n) , τ k 1 < • • • < τ kn ) (20) where t k (i) := t k 1 ∨ • • • ∨ t k i , i = 1, . . . , n.
Next, we know that

f (τ 1 ,...τn) (t 1 , . . . , t n ) = (-1) n ∂ n ∂t 1 . . . ∂t n P (τ 1 > t 1 . . . τ n > t n ) .
To compute this derivative, note that the sequence (t k (1) , t k (2) , . . . , t k (n) ) will contain repeated terms unless t k 1 < • • • < t kn , that is unless t i 's are all distinct and t (i) = t k i , i = 1, 2, . . . , n. As a consequence, the only term in [START_REF] Wang | Modeling multivariate degradation processes with time-variant covariates and imperfect maintenance effects[END_REF] which will not vanish after differentiation is the one in which (t k 1 , . . . , t kn ) = (t (1) , . . . , t (n) ).

Hence we have

f (τ 1 ,...τn) (t 1 , . . . , t n ) = (-1) n ∂ n ∂t 1 . . . ∂t n P(τ k 1 > t (1) , τ k 2 > t (2) , . . . , τ kn > t k (n) , τ k 1 < • • • < τ kn ) = (-1) n ∂ n ∂t 1 . . . ∂t n P(τ 1 > t (1) , τ 2 > t (2) , . . . , τ n > t n , τ 1 < • • • < τ n )
by symmetry. We next compute this expression using similar arguments as in ( 18) and ( 19), which provides

f (τ 1 ,...τn) (t (1) , . . . , t (n) ) = n i=1 f τ k i |τ k 1 ,••• ,τ k i-1 t (i) |t (1) , • • • , t (i-1) = n i=1 λ i t (i) f τ Y Λ * i t (1) , • • • , t (i) , (21)
and gives the result. We next derive the pdf of (τ (1) , . . . τ (n) ) in the following corollary.

Corollary 12

The pdf of (τ (1) , . . . τ (n) ) is given by

f (τ (1) ,...τ (n) ) (t 1 , . . . , t n ) = n! n i=1 λ i (t i ) f τ Y (Λ * i (t 1 , • • • , t i )) = n! a n (t 1 , • • • , t n ) for t 1 < t 2 < • • • < t n (0 otherwise).
Proof. The result is a direct consequence of Theorem 11, considering the n! possible orderings among the failures. Finally, we derive the probability density function of (τ (1) , . . . τ (k) ) for 1 ≤ k ≤ n, which is next used to derive the conditional distributions of the ordered failure times.

Proposition 13

For each 1 ≤ k ≤ n, the pdf of (τ (1) , . . . τ (k) ) is given by

f (τ (1) ,...τ (k) ) (t 1 , . . . , t k ) = n! (n -k)! Fτ Y (Λ * k (t 1 , • • • , t k )) n-k a k (t 1 , • • • , t k ) (22) = n! (n -k)! Fτ Y (Λ * k (t 1 , • • • , t k )) n-k k i=1 λ i (t i ) f τ Y (Λ * i (t 1 , • • • , t i )) , for t 1 < t 2 < • • • < t k (0 otherwise).
Proof. Note that ( 22) can be obtained by noticing that the first part is k! times the probability that n -k components have survived to time t k and the second part is the joint pdf of the first k failure times, divided by k!. For a more specific proof, we start from

f (τ (1) ,...τ (k) ) (t 1 , . . . , t k ) = ∞ t k ∞ u k+1 • • • ∞ u n-2 ∞ u n-1 f (τ (1) ,...τ (n) ) (t 1 , . . . , t k , u k+1 , . . . u n )du k+1 . . . du n (23)
Taking into account Corollary 12, the last integral is equal to

∞ u n-1 f (τ (1) ,...τ (n) ) (t 1 , . . . , t k , u k+1 , . . . u n ) du n = n! a n-1 (t 1 , . . . , t k , u k+1 , . . . u n-1 ) ∞ u n-1 λ n (t n ) f τ Y (Λ * n (t 1 , • • • , t k , u k+1 , . . . u n )) du n = n! a n-1 (t 1 , . . . , t k , u k+1 , . . . u n-1 ) Fτ Y Λ ⋆ n-1 (t 1 , . . . , t k , u k+1 , . . . u n-1 ) .
Substituting this expression into (23), we get

f (τ (1) ,...τ (k) ) (t 1 , . . . , t k ) = n! ∞ t k ∞ u k+1 • • • ∞ u n-2
a n-1 (t 1 , . . . , t k , u k+1 , . . . u n-1 )

× Fτ Y Λ ⋆ n-1 (t 1 , . . . , t k , u k+1 , . . . u n-1 ) du k+1 . . . du n-1 = n! ∞ t k ∞ u k+1 • • • ∞ u n-3 a n-2 (t 1 , . . . , t k , u k+1 , . . . u n-2 ) × 1 2! Fτ Y Λ ⋆ n-1 (t 1 , . . . , t k , u k+1 , . . . u n-2 ) 2 du k+1 . . . du n-2
by integrating the last integral. The conclusion follows integrating repeatedly the last expression.

Proposition 14

The conditional survival function of τ (i) given (τ (1) , . . . , τ (i-1) ) is provided by

P(τ (i) > t|τ (1) = t 1 , . . . , τ (i-1) = t i-1 ) = Fτ Y (Λ * i (t 1 , . . . , t i-1 , t)) Fτ Y Λ * i-1 (t 1 , . . . , t i-1 ) n-i+1 for all 0 < t 1 < • • • < t i-1 , all t > t i-1 and all i = 2, • • • , n.
Proof. Taking into account Proposition 13, we have

f τ (i) |τ (1) ,...τ (i-1) (t i |t 1 , . . . , t i-1 ) = f (τ (1) ,...τ (i) ) (t 1 , . . . , t i ) f (τ (1) ,...τ (i-1) ) (t 1 , . . . , t i-1 ) = (n -i + 1) Fτ Y (Λ * i (t 1 , . . . , t i )) n-i λ i (t i ) f τ Y (Λ * i (t 1 , . . . , t i )) Fτ Y Λ * i-1 (t 1 , . . . , t i-1 ) n-i+1
for all t i > t i-1 . As a consequence of the previous equality, we easily get

P τ (i) > t|τ (1) = t 1 , . . . , τ (i-1) = t i-1 = ∞ t f τ (i) |τ (1) ,...τ (i-1) (t i |t 1 , . . . , t i-1 )dt i = Fτ Y (Λ ⋆ i (t 1 , . . . , t i-1 , t)) n-i+1 Fτ Y Λ ⋆ i-1 (t 1 , . . . , t i-1 ) n-i+1 ,
from which the conclusion follows.

6 Dependence and stochastic comparison properties

Stochastic orders and aging properties

We here recall the definitions of the stochastic orders and aging properties that are used further in this section. We begin with univariate stochastic orders, please see [15, Chapter 1] for a more detailed account on these notions.

Definition 15 Let X and Y be non-negative random variables, with cdfs F and G, and survival functions F and Ḡ, respectively. X is said to be smaller than Y in the

(a) Usual stochastic order (X ≤ st Y ) if Ḡ(x) ≥ F (x) for all x ∈ R. (b) Hazard rate order (X ≤ rh Y ) if Ḡ(x) F (x)
is increasing in x (or equivalently Ḡ(y) F (x) -Ḡ(x) F (y) ≥ 0 for all x ≤ y, to avoid divisions by 0).

(c) Reversed hazard rate order (X ≤ rh Y ) if G(x) F (x) is increasing in x (or G(y)F (x) -G(x)F (y) ≥ 0 for all x ≤ y).
(c) Likelihood ratio order (X ≤ lr Y ) if X and Y are absolutely continuous with respective pdfs f and g such that g(x) f (x) is increasing in x (or g(y)f (x) -g(x)f (y) ≥ 0 for all x ≤ y).

Remark 16

We recall that the likelihood ratio order implies both the rh and hr orders and that both rh and hr orders imply the usual stochastic order.

In Section 6.3, we shall also use the multivariate stochastic order as defined in the next definition. Please see [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF]Chapter 3] for more details.

Definition 17 Let X = (X 1 , . . . , X n ) and Y = (Y 1 , . . . , Y n ) be two random vectors. We say that X is less or equal that Y in the multivariate stochastic order (written

X ≤ st Y) if E[ϕ(X)] ≤ E[ϕ(Y)] for all bounded increasing function ϕ : R n -→ R.
Remark 18 It is well known that, for n = 1 (univariate case), Definitions 15 (a) and 17 are equivalent (see [15, Thm. 1.2.13, p. 6]). However, in the multivariate case, even though X ≤ st Y implies that the survival functions or cdfs of both vectors are ordered (which correspond to the upper and lower orthant orders, respectively), these orders are not equivalent any more.

In Section 6.3, we will study dependence properties between the failure times of a system. Many dependence concepts can be used to this aim (see [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF]Section 3.10]). Here, we will use the concept of Conditionally Increasing In Sequence property, which we define below and where we use the following notation: given two random vectors X and Y, we write X ↑ Y if X is stochastically increasing in Y, namely if the conditional distribution of X given Y = y is increasing for all y ∈ R n . Definition 19 Let X = (X 1 , . . . , X n ) be a random vector. It is said to be Conditionally Increasing in Sequence (CIS) if X i ↑ (X 1 , . . . X i-1 ), for all i = 1, . . . n.

Remark 20

The concept of Conditionally Increasing in Sequence describes a positive dependence property between the random variables under consideration. Among these concepts, it is not the strongest (for instance, the Multivariate Totally Positive of order 2 (MTP2) property is stronger), but it implies many other common dependence concepts such as positive upper/lower orthant dependence or association, for instance. See [15, p. 146] for a summary of the relationships between these concepts of positive dependence).

We finally come to the Increasing Failure Rate property, which is used in reliability to describe the aging of a system which is deteriorating over time (see e.g. [14, p.104] for more details).

Definition 21 Let X be a non negative random variable with F and F the corresponding cdf and survival functions. The random variable X (or the cdf F ) is said to be Increasing failure rate (IFR) if P(X > t + h|X > t) is decreasing in t for all h > 0 and all t such that the previous expression makes sense, that is F (t + h)/F (t) is decreasing in t for all h > 0.

Remark 22 A function ϕ : R -→ R is said to be log-concave if log ϕ is a concave function (where finite). It is well known that ϕ is log-concave if and only if (see e.g. [14, p.696])

ϕ(x)ϕ(y + h) ≤ ϕ(y)ϕ(x + h), for all y ≥ x ≥ 0, h > 0. ( 24
)
Thus, the IFR property for X is equivalent to the log-concavity of the survival function F [14, p.105].

Dependence properties

The aim of this section is to study dependence properties of the ordered failure times (τ (1) , . . . , τ (n) ). Based on the fact that the failure times are ordered, we specifically investigate positive dependence properties. With that aim, we will need the following assumption for the derivatives λ j , j = 1, • • • , n of the time-scaling functions Λ j , j = 1, • • • , n :

λ i ≤ λ j for all i < j, (25)
which mostly means that the deterioration rate of the surviving components increases with the number of already failed components, and implies that the larger the number of observed failures, the shorter the waiting time for the next failure. Under this condition, and assuming also the Increasing Failure Rate (IFR) property for τ Y , we show that (τ (1) , . . . τ (n) ) is conditionally increasing in sequence (CIS). More specifically, we have the following result.

Proposition 23 Assume τ Y to be IFR. Under Assumption (25), the ordered vector (τ (1) , . . . τ (n) ) is conditionally increasing in sequence, that is, for i = 2, . . . , n,

τ (i) |τ (1) = t 1 , . . . , τ (i-1) = t i-1 ≤ st τ (i) |τ (1) = z 1 , . . . , τ (i-1) = z i-1 ,
for all (t 1 , . . . , t i ) ≤ (z 1 , . . . , z i ), where both vectors are arranged in increasing order.

Proof. Let us assume that both sequences t = (t 1 , . . . , t i-1 ) and z = (z 1 , . . . , z i-1 ) are arranged in increasing order (in order that the conditional probability makes sense). Let

A i t = {τ (1) = t 1 , . . . , τ (i-1) = t i-1 }; A i z = {τ (1) = z 1 , . . . , τ (i-1) = z i-1 }. ( 26
)
We have to show that

P τ (i) > t|A i t ≤ P τ (i) > t|A i z ( 27
)
for all t ≥ 0.

Let us first note that if t ≤ z i-1 , then P τ (i) > t|A i z = 1 so that ( 27) is true. Hence we can restrict the study to t > z i-1 .

Let us set

Λ 1 (t) = i k=1 Λ * k (t 1 , • • • , t k-1 , t) 1 [t k-1 ,t k ) (t) , (28)
Λ 2 (t) = i k=1 Λ * k (z 1 , • • • , z k-1 , t) 1 [z k-1 ,z k ) (t) (29)
for all t ≥ 0 with

t i = z i = ∞. Noticing that Λ 1 (t) = Λ * i (t 1 , • • • , t i-1 , t) = Λ 1 (t i-1 ) + ∆Λ i (t i-1 , t) = Λ 1 (t i-1 ) + ∆Λ i (t i-1 , z i-1 ) + ∆Λ i (z i-1 , t) = Λ 1 (z i-1 ) + ∆Λ i (z i-1 , t) for all t > z i-1 ≥ t i-1 , we get from Proposition 17 that P τ (i) > t|A i z = Fτ Y (Λ 2 (z i-1 ) + ∆Λ i (z i-1 , t)) Fτ Y (Λ 2 (z i-1 )) n-i+1 , ( 30
) P τ (i) > t|A i t = Fτ Y (Λ 1 (z i-1 ) + ∆Λ i (z i-1 , t)) Fτ Y (Λ 1 (t i-1 )) n-i+1 ≤ Fτ Y (Λ 1 (z i-1 ) + ∆Λ i (z i-1 , t)) Fτ Y (Λ 1 (z i-1 )) n-i+1 (31)
due to Λ 1 (z i-1 ) ≥ Λ 1 (t i-1 ) for the last inequality. Now, let us show that Λ 1 (t) ≥ Λ 2 (t) (32) for all t > z i-1 . Let λ 1 and λ 2 be the derivative of Λ 1 and Λ 2 on R + , respectively. At first, note that for each u ≥ 0, there exists j 1 such that t j 1 -1 ≤ u < t j 1 and j 2 such that z j 2 -1 ≤ u < z j 2 . Also j 1 ≥ j 2 due to (t 1 , . . . , t i ) ≤ (z 1 , . . . , z i ). This implies

λ j 1 (u) = λ 1 (u) ≥ λ j 2 (u) = λ 2 (u)
based on Assumption [START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF]. Hence λ 1 (u) ≥ λ 2 (u) for all u ≥ 0, which implies (32) for all t > z i-1 .

Remembering that τ Y is increasing failure rate, we know that

Fτ Y (y + h) Fτ Y (y) ≤ Fτ Y (x + h) Fτ Y (x) , for all y ≥ x ≥ 0, h > 0. ( 33
)
Based on (32), we derive that

Fτ Y (Λ 1 (z i-1 ) + ∆Λ i (z i-1 , t)) Fτ Y (Λ 1 (z i-1 )) ≤ Fτ Y (Λ 2 (z i-1 ) + ∆Λ i (z i-1 , t)) Fτ Y (Λ 2 (z i-1 )) ,
which taking into account (30) and ( 31) shows ( 27), and therefore the result.

Remark 24 Note that it is not possible to use similar arguments as in the previous proof to derive a conditionally decreasing in sequence property, because the reversed inequality of (27) would not be valid, as for t i-1 < t < z i-1 the second term in ( 27) is 1, whereas the first term can be less than 1.

Finally, let us mention that the IFR property for τ Y is quite a natural condition fulfilled by most usual Lévy processes, which can easily derived from the following Lemma. As an example, it is checked for the inverse Gaussian process (as we will see in the next section). It is also valid for the Gamma process, based on similar arguments.

Lemma 25 If the baseline Lévy process Y is such that Y (t) increases with t in the reversed hazard rate ordering, the crossing time τ Y of level L is Increasing Failure Rate (IFR).

Proof. Our aim is to use Proposition 3.1 from [START_REF] Sangüesa | Preservation of ageing classes in deterioration models with independent increments[END_REF]. At first, notice that the constant threshold L can be seen as an IFR random variable and that Y is a non negative Lévy process, and hence in the class IPII (independent positive increasing increments) in the sense of [START_REF] Sangüesa | Preservation of ageing classes in deterioration models with independent increments[END_REF]. As P(Y t = L) = 0, we have now checked all conditions from [18, Prop. 3.1 (a)], which allows to derive that τ Y is IFR.

Proof. The assumption Y (t) ≥ rh Z (t) for all t ≥ 0 means that

F Z(t) (x) F Z(t) (x + h) ≥ F Y (t) (x) F Y (t) (x + h) (37)
for all h, t ≥ 0 and x > -h, where F Y (t) and F Z(t) stand for the cumulative distribution functions of Y (t) and Z (t), respectively. The point is to show that τ Y ≤ hr τ Z , that is

P (τ Y > s + t) P (τ Y > t) ≤ P (τ Z > s + t) P (τ Z > t) (38)
for all s, t ≥ 0. Note that, based on the independent and homogenous increment property of Y, we have

Y (s + t) = st Y (t) + Y * (s) ,
where Y * is an independent copy of Y. Hence

P (τ Y > s + t) = P (Y (s + t) ≤ L) = P (Y (t) ≤ L -Y * (s)) = E F Y (t) (L -Y * (s)) = E F Y (t) (L -Y (s))
We derive

P (τ Y > s + t) P (τ Y > t) = E F Y (t) (L -Y (s)) F Y (t) (L) ≤ E F Z(t) (L -Y (s)) F Z(t) (L) , (39)
due to (37) for the last inequality, taking x = L -Y (s) and h = Y (s). Now, as Y (s) ≥ rh Z (s), we also have Y (s) ≥ st Z (s), which entails that

E F Z(t) (L -Y (s)) F Z(t) (L) ≤ E F Z(t) (L -Z (s)) F Z(t) (L) = P (τ Z > s + t) P (τ Z > t) (40)
and consequently (38) is true, gathering (39) and (40).

Lemma 29 Let Λ 1 and Λ 2 be two time-scaling functions such that Λ 1 ≥ Λ 2 . Let ϕ : R -→ R + be a log-concave decreasing function. Then, for all 0 ≤ s < t such that ∆Λ 1 (s, t) ≥ ∆Λ 2 (s, t) we have

ϕ Λ 1 (t) ϕ Λ 2 (s) ≤ ϕ Λ 2 (t) ϕ Λ 1 (s) . (41)
Proof. By assumption, we know that Λ 1 (s) ≥ Λ 2 (s). Then, using (24) for the second line and the decreasing property for the third one, we have

ϕ Λ 1 (t) ϕ Λ 2 (s) = ϕ Λ 1 (s) + ∆Λ 1 (s, t) ϕ Λ 2 (s) ≤ ϕ Λ 2 (s) + ∆Λ 1 (s, t) ϕ Λ 1 (s) ≤ ϕ Λ 2 (s) + ∆Λ 2 (s, t) ϕ Λ 1 (s) = ϕ Λ 2 (t) ϕ Λ 1 (s) ,
and the conclusion follows.

Proposition 30 Let us consider two deterioration models described by the pair of baseline Lévy processes Y and Z, with

Λ m j , j = 1, • • • , n and λ m j , j = 1, • • • , n , m = 1, 2
, as corresponding time-scaling functions and derivatives, respectively. Assume the following conditions to hold:

(a) Y (t) ≥ rh Z (t) for all t ≥ 0 and Λ 1 1 ≥ Λ 2 1 . (b) λ 1 j ≥ λ 2 j , for all j = 2, • • • , n.
(c) One of the deterioration models satisfies conditions in Proposition 23 (that is, Condition [START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF] for the time-scaling functions, as well as the IFR property for the failure time in the baseline Lévy process).

Then, setting (τ (1) , . . . τ (n) ) and (τ (1) , . . . τ (n) ) to be the ordered failure times in the first and second models, respectively, we have (τ (1) , . . . τ (n) ) ≤ st (τ (1) , . . . τ (n) ).

Proof. Taking into account condition (a) and that the fact that the hazard rate order implies the usual stochastic order, we know from Proposition 26 that τ (1) ≤ st τ [START_REF] Abdel-Hameed | A gamma wear process[END_REF] . As a second step, let us use the notation introduced in (26), and let Âi t and Âi z be the corresponding overcomes for the second model. With these notations, we want to show that Remark 32 Let us assume that the two deterioration models in Proposition 30 share the same baseline Lévy process Y and that Λ 2 j (t) = t for all t ≥ 0 and all j = 1, • • • , n. Then Condition ( 25) is true for the second deterioration process (which is the baseline Lévy process). Assume further that τ Y is IFR and that λ 1 j (t) ≥ (≤) 1, for all j = 1, • • • , n. Then, as Y (t) ≥ rh Y (t) for all t ≥ 0, we know from Proposition 30 that (τ (1) , . . . , τ (n) ) ≤ st (≥ st ) (τ (1) , . . . , τ (n) )

τ (i) |A i t ≤ st τ (i) | Âi z (
where τ (1) , . . . , τ (n) stand for the ordered failure times of the baseline Lévy process model. Proposition 30 hence provides us with conditions under which the ordered failure times of the deterioration model are stochastically smaller or larger than those of the baseline process. Based on Remark 31, the reliability functions of a specific coherent system for both deterioration and baseline models are also comparable under the same conditions.

Illustration in the case of inverse Gaussian processes

We here consider n identical and dependent components as in Section 3, with an inverse Gaussian process as underlying deterioration process Y. As a first step, we recall some well-known facts about inverse Gaussian processes.

The inverse Gaussian process

Following the parametrization of [START_REF] Wang | An inverse gaussian process model for degradation data[END_REF], we recall that an inverse Gaussian process Y is a Lévy process such that Y (t) ∼ IG (t, bt 2 ) with pdf and cdf

f t (y) = t b 2πy 3 exp - b (y -t) 2 2y , F t (y) = Φ (y -t) b y + e 2bt Φ -(y + t) b y
for all t, y > 0, respectively, where b > 0 and Φ is the cdf of the standard normal distribution.

The technical assumptions from Section 3 are clearly true and remembering that τ Y stands for the crossing time of level L, then Fτ Y (t) = F t (L) and

f τ Y (t) = - ∂F t (L) ∂t = b L ϕ (L -t) b L + e 2bt b L ϕ -(L + t) b L -2be 2bt Φ -(L + t) b L ,
where ϕ is the pdf of the standard normal distribution. Finally, based on [14, Section 13.A.e page 458], it is known that Y (t) increases with t in the likelihood hazard rate ordering, and consequently, it also increases in the reversed hazard rate ordering. Based on Lemma 25, we hence derive that the crossing time τ Y of level L is Increasing Failure Rate.

Finally

As a consequence, we now have all the material to illustrate the previous results considering an inverse Gaussian process as baseline Lévy process Y. Note however that IG (t, bt 2 ) is not monotonous in the reversed hazard rate ordering with respect to b, so that for Proposition 30, we have to consider the same inverse gaussian process as baseline Lévy process Y.

Numerical experiments

We assume in all this section that Y is an inverse Gaussian process such that Y (t) ∼ IG (t, t 2 ) (that is b = 1) and L = 1. The collection of time-scaling functions is denoted by {Λ j , j = 1, • • • , n}, as in Section 3.

As a first step, the results provided by Theorem 6 are compared to those obtained by Monte-Carlo simulations on an example.

Comparison with Monte-Carlo simulations

We here consider a 2-out-of-3 system. Then Theorem 6 provides

R 2,3 (t) = F Λ 1 (t) (L) 3 + 3 t 0 F Λ 1 (t 1 )+∆Λ 2 (t 1 ,t) (L) 2 λ 1 (t 1 ) f τ Y [Λ 1 (t 1 )] dt 1 (47)
for all t ≥ 0. We take

Λ 1 (t) = e t -1, Λ 2 (t) = e 2t -1, Λ 3 (t) = e 3t -1
for all t ≥ 0 and compute the reliability on [0, 1.75] through (47) and also through (MC) Monte-Carlo simulations, using Matlab software. As for the MC simulations, 10 4 trajectories of (Y (t)) 0≤t≤2.5 are simulated through random walk approximation with a small time-step (h = 0.001), in order to get mostly reliable crossing times for level L. This provides an empirical estimate of R 2,3 (t) denoted by R2,3 (t). The results are provided in Figure 2, where we can see that R2,3 (t) and R 2,3 (t) are mostly superimposed (left plot), with a difference less than 5 × 10 -3 (right plot). Hence, both methods provide similar results. However, the computation takes 242 cpu time on a standard laptop computer for the MC simulation and around 0.5 cpu time for the computation though (47). Even though the code might surely be bettered for the MC simulation, there clearly is a huge gap between the computing times through MC simulations and through (47). In the following, all computations are hence made through the theoretical results from the previous sections. Proposition 13, we obtain that F(τ (1) ,τ (2)

) (t 1 , t 2 ) = P τ (1) > t 1 , τ (2) > t 2 = n ∞ t 1 λ 1 (u 1 ) f τ Y (Λ * 1 (u 1 )) × ∞ t 2 1 {u 1 <u 2 } (n -1) λ 2 (u 2 ) f τ Y (Λ * 2 (u 1 , u 2 )) Fτ Y (Λ * 2 (u 1 , u 2 )) n-2 du 2 du 1 = n ∞ t 1 λ 1 (u 1 ) f τ Y (Λ 1 (u 1 )) Fτ Y (Λ * 2 (u 1 , max (t 2 , u 1 ))) n-1 du 1 = Fτ Y Λ 1 t (2) n + 1 {t 1 <t 2 } n t 2 t 1 λ 1 (u 1 ) f τ Y (Λ 1 (u 1 )) Fτ Y (Λ 1 (u 1 ) + ∆Λ 2 (u 1 , t 2 )) n-1 du 1 (48)
for all t 1 , t 2 ≥ 0, where we distinguish the two cases t 1 < t 2 and t 2 ≤ t 1 to derive the last line, remembering that t (2) = max (t 1 , t 2 ). Example 1 As a first example, 2-out-of-5 systems are considered, with

R 2,5 (t) = 3 m=0 R m (t) α 1 α 2 α 3 α 4 Set 1 (independent case) 1 1 1 1 Set 2 1 1.5 2 2.5 Set 3 1 2 3 4 Set 4
1 2.5 3.5 4.5

Table 1: The four parameter sets for Figure 5 and

R 0 (t) = F Λ 1 (t) (L) 5 (50) R 1 (t) = 5 t 0 Fτ Y (Λ 1 (t 1 ) + ∆Λ 2 (t 1 , t)) 4 λ 1 (t 1 ) f τ Y [Λ 1 (t 1 )] dt 1 , (51)
R 2 (t) = 20 [0,t] 2 1 {t 1 <t 2 } Fτ Y (Λ 1 (t 1 ) + ∆Λ 2 (t 1 , t 2 ) + ∆Λ 3 (t 2 , t)) 3 × λ 1 (t 1 ) f τ Y [Λ 1 (t 1 )] λ 2 (t 2 ) f τ Y [Λ 1 (t 1 ) + ∆Λ 2 (t 1 , t 2 )] dt 1 dt 2 , (52)
R 3 (t) = 60 [0,t] 3 1 {t 1 <t 2 <t 3 } Fτ Y (Λ 1 (t 1 ) + ∆Λ 2 (t 1 , t 2 ) + ∆Λ 3 (t 2 , t 3 ) + ∆Λ 4 (t 3 , t)) 2 × λ 1 (t 1 ) f τ Y [Λ 1 (t 1 )] λ 2 (t 2 ) f τ Y [Λ 1 (t 1 ) + ∆Λ 2 (t 1 , t 2 )] × λ 3 (t 3 ) f τ Y [Λ 1 (t 1 ) + ∆Λ 2 (t 1 , t 2 ) + ∆Λ 3 (t 2 , t 3 )] dt 1 dt 2 dt 3 (53)
(please see Theorem 6).

For illustration purpose, we take Λ j (t) = e α j t -1 for j = 1, 2, 3, 4 (note that Λ 5 is not involved in R 2,5 (t)) and consider four different sets of parameters α j 's, which are described in Table 1.

As can be seen, α j (and hence λ j ) increases with the set number, and for each parameter set, Assumption (25) is true. Based on the fact that R 2,5 (t) = P τ (4) > t , we hence expect R 2,5 (t) to decrease with the set number due to Proposition 30. This is illustrated in Figure 5, where R 2,5 (t) is plotted for the four parameter sets (including the independence case which corresponds to the first set of parameters), where we can see that the results are coherent with Proposition 30 and Remark 31.

Example 2 We next consider a 5-component system with a bridge structure, as shown in Figure 6.

The corresponding signature is given by p = (0, 1/5, 3/5, 1/5, 0), please see Example 4.1 page 50 in [START_REF] Samaniego | System signatures and their applications in engineering reliability[END_REF]. We hence derive from Proposition 9 that 0.2 0.4 0.6 0.8 R m (t) + 1 5

1 m=0 R m (t) = R 0 (t) + R 1 (t) + 4 5 R 2 (t) + 1 5 R 3 (t)
We take again Λ j (t) = e α j t -1 for j = 1, 2, 3, 4 and consider α 1 = 1, α 2 = 2, α 3 = 3, α 4 = 4 as a first parameter set and α 1 = 1, α 2 = 1.75, α 3 = 5, α 4 = 5 as a second parameter set.

We can see that the corresponding time-scaling functions both fulfill Assumption (25). However we do not have λ 1 j ≤ λ 2 j for all j, nor λ 1 j ≥ λ 2 j for all j so that we cannot apply the results of Proposition 30 and Remark 31. The corresponding reliability functions are plotted in the left plot of Figure 7 together with their difference in the right plot. We can see that the two reliability functions are crossing, showing that they are not comparable.

Conclusion and perspectives

Mimicking some existing lifetime models where failures among components entail an increase in the failure rates of surviving components, we have here suggested to consider Lévy-deteriorating components, with an increase in the time-scaling functions of the Lévy processes for the surviving components. The reliability of a coherent system with components linked by this specific stochastic dependence has been computed, together with the joint distribution of the component lifetimes. Some dependence and comparison results have been provided and illustrated considering inverse Gaussian processes as baseline Lévy processes.

We remind that for the stochastic comparison result of Proposition 30, we have not been able to conclude about the necessity of Assumption [START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF] for the result to hold. Hence, the question remains open whether this assumption is necessary, and this requires further research. Also, this paper has focused on the most usual multivariate stochastic order for comparison purpose. It should be possible to consider other types of multivariate stochastic orders and try to derive other results. Beyond that, one could also think about other types of conditions to derive stochastic comparison results, such as conditions on the Lévy measures of the baseline Lévy processes, in a similar spirit as [START_REF] Bergenthum | Comparison of semimartingales and Lévy processes[END_REF].

Furthermore, beyond the few papers quoted in the introduction [START_REF] Dong | Reliability and availability analysis of stochastic degradation systems based on bivariate wiener processes[END_REF][START_REF] Zhang | Condition-based maintenance for a k-out-of-n deteriorating system under periodic inspection with failure dependence[END_REF][START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF]] and the present paper, there remains many work left for adapting previous dependence models (following the vocabulary of [START_REF] Keizer | Condition-based maintenance policies for systems with multiple dependent components: A review[END_REF]) to the case of Lévy-deteriorating components. At first, we have here chosen to model the impact of a failure as a change in the time-scaling function of the Lévy processes. Other possibilities include the multiplication by a given function of the Lévy process itself or maybe use similar ideas as those developed by [START_REF] Kamps | A concept of generalized order statistics[END_REF] for sequential order statistics of lifetimes. Also, we could consider both sudden increase in the component deterioration levels as in [START_REF] Zhang | Condition-based maintenance for a k-out-of-n deteriorating system under periodic inspection with failure dependence[END_REF][START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF] and some change in the Lévy process for the surviving components as in the present paper. Other ideas include trying to adapt existing models for cascading failures, load-sharing or spare-part-sharing systems, starting for instance from the literature provided in [START_REF] Keizer | Condition-based maintenance policies for systems with multiple dependent components: A review[END_REF].

Figure 1 : 4 . 4

 144 Figure 1: Example of trajectories for the component deterioration levels up to failure, case n = 4 and τ 1 < τ 3 < τ 2 < τ 4 .

Figure 2 :Figure 3 :

 23 Figure 2: R2,3 (t) and R 2,3 (t) (left and R2,3 (t) -R 2,3 (t) (right plot)

Figure 4 :

 4 Figure4: F2 -F1 (t 1 , t 2 ) (left plot) and (F 1 -F 2 ) (t 1 , t 2 ) (right plot)

Figure 5 :Figure 6 :

 56 Figure 5: The reliability function R 2,5 (t) for the different parameter sets of Example 1 (2-out-of-5 system)

Figure 7 :

 7 Figure 7: The reliability functions for the two parameter sets of Example 2 (left plot) together with their difference (right plot)

  • • • , n.

	L	𝑋 1,1 (𝑡)		𝑋 3,2 (𝑡)		𝑋 2,3 (𝑡)
								𝑋 4,4 (𝑡)
			𝑋 2,2 (𝑡)
		𝑋 2,1 (𝑡)			𝑋 4,3 (𝑡)
		𝑋 3,1 (𝑡)
				𝑋 4,2 (𝑡)
		𝑋 4,1 (𝑡)
	0	Phase 1	𝜏 1	Phase 2	𝜏 3	Phase 3	𝜏 2	𝜏 4 Phase 4	t

  42)for all (t 1 , . . . t n ) ≤ (z 1 , . . . z n ) and all 2 ≤ i ≤ n.

	marginals (see [15, Thm.3.3.10, p.94]), we immediately deduce from Propo-
	sition 30 that R k,n (t) ≤ Rk,n (t), k = 2, . . . , n. In particular, this includes
	stochastic comparisons for parallel systems (k = n). Moreover, if R(t) and
	R(t) are the reliability functions of a specific coherent system for both models,
	we have by Proposition 9 that R(t) ≤ R(t).
	Assume first that Condition (c) is satisfied for the first model. Based on
	Proposition 23, we then have τ (i) |A i t ≤ st τ (i) |A i z , so that (42) will follow
	if we show that
	τ (i) |A i z ≤ st	τ (i) | Âi z .	(43)

Acknowledgments

The authors would like to thank the anonymous referees for their constructive remarks on a previous version of the paper. The third author acknowledges support from the Spanish research projects PID2021-123737NB-100 and PGC2018-097621-B-100 (MINECO/FEDER) and for Gobierno de Aragón project E48 20R.

Stochastic comparisons

In this section, we provide conditions to compare the failure times in two different deterioration models, with, possibly, different baseline processes and time-scaling functions. We begin with a simple result comparing the times of the first failures among components, or equivalently, comparing the failure times of series systems.

Proposition 26 Let us consider two deterioration models described by the pair of baseline Lévy processes Y and Z, with Λ m j , j = 1, • • • , n the corresponding time-scaling functions. Assume that Y (t) ≥ st Z (t) for all t ≥ 0 and Λ 1 1 ≥ Λ 2 1 . Then, setting τ (1) and τ (1) to be the first failure times in the first and second models, respectively, we have

for all t > 0. Gathering (35) and (36) shows (34).

Remark 27 As a specific case, assume that the two deterioration models in Proposition 26 share the same baseline Lévy processes Y.

, we get that τ (1) ≤ st τ [START_REF] Abdel-Hameed | A gamma wear process[END_REF] . Considering Λ m 2 (t) = t for all t ≥ 0 and all m (which means that the second deterioration process is the baseline process), we can see that if Λ 1 1 (t) ≥ (≤)t for all t, then the time of the first failure in the deterioration model is stochastically smaller (larger) than the time of the first failure in the baseline model. Equivalently, this provides comparison results between the lifetimes τ X (deterioration model) and τ Y (baseline model) in case of a common series structure.

We next study the multivariate stochastic ordering property of the ordered failure times. As previous steps, we prove two technical lemmas, which have some interest in themselves.

Lemma 28 Let Y and Z be two baseline Lévy processes such that

Then, due to (30), we have

Condition (a), together with the condition λ

. As τ Y is increasing failure rate, conditions in Lemma 29 are satisfied and we have

Now based on Y (t) ≥ rh Z (t) for all t ≥ 0, we can derive from Lemma 28 that τ Y ≤ hr τ Z , and consequently (38) is true. This provides

for all t > z i-1 . Then, from (45) and (46), we conclude that

and this last inequality together with (44) show (43). Finally, the result follows by (34) and (42) (see [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF]Thm. 3.3.7,p. 93]). In the case where Condition (b) is satisfied for the second model, it is easy to check that similar arguments remain valid.

Remark 31 Let R k,n (t) and Rk,n (t) be the reliability functions of a k-out-ofn system for two deterioration models as defined in Proposition 30. Proposition 26 gives conditions for the case k = 1 (series system). Moreover, as the multivariate stochastic order implies the univariate stochastic order of the

Conditionally Increasing in Sequence property

The point here is to illustrate the results from Proposition 23. For each 2 ≤ i ≤ 9, t = (t j ) 1≤j≤i-1 and z = (z j ) 1≤j≤i-1 ≥ t, we set

for all u > t i-1 and a similar expression for G z (u), with t substituted by z.

Considering n = 10, k = 6 and t j = 0.1 j for 1 ≤ j ≤ 5, we envision two different sets of scaling functions and two different z. As a first case, we take Λ j (t) = 0.1 e tα j -1 for all t ≥ 0, with α j = 0.1 j for 1 ≤ j ≤ 10, so that Assumption (25) is fulfilled. We set z = 4t ≥ t. The functions G t (u) and G z (u) are plotted in Figure 3 (left plot). As expected, we can see that G t (u) ≤ G z (u) for all u > z i-1 , which is coherent with the fact that (τ (1) , . . . τ (n) ) is conditionally increasing in sequence (CIS property), as shown in Proposition 23.

As a second case, we take Λ j (t) = 0.1 t α j with α j = 0.2 (11 -j) for 1 ≤ j ≤ 10 and z = 10t. Note that Assumption [START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF] is not fulfilled any more. The functions G t (u) and G z (u) are plotted in Figure 3 (right plot). It can be seen that the functions G t (u) and G z (u) are crossing, which shows that the CIS property does not hold any more for (τ (1) , . . . τ (n) ).

Hence, Assumption [START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF] appears to be a necessary assumption, in order to guaranty that (τ (1) , . . . τ (n) ) is conditionally increasing in sequence.

Stochastic comparisons

The point here is to illustrate the results from Proposition 30. We here consider n components and look at the survival and cumulative distribution functions of (τ (1) , τ (2) ). Starting from the joint pdf of (τ (1) , τ (2) ) provided in Similar computations provide

for all t 1 , t 2 ≥ 0, where we remind that t (1) = min (t 1 , t 2 ). Using the notations of Proposition 30, we take Y (t) = Z(t) ∼ IG (t, t 2 ) (based on the remark at the end of Section 7.1), Λ m j (t) = β m j e tα m j -1 for all t ≥ 0 and j = 1, • • • , n, m = 1, 2, with

Then it is easy to check that Assumption (25) is true for {Λ 1 1 , Λ 1 2 }, but not for {Λ 2 1 , Λ 2 2 }. Also λ 1 j ≥ λ 2 j for j = 1, • • • , n. Let Fm (t 1 , t 2 ) and F m (t 1 , t 2 ), m = 1, 2 be the survival and cumulative distribution functions of (τ (1) , τ (2) ) and (τ (1) , τ (2) ), respectively, which are computed through (48) and (49). The differences F2 -F1 (t 1 , t 2 ) and (F 1 -F 2 ) (t 1 , t 2 ) are plotted in Figure 4. As expected from Proposition 30, we can see that F2 (t 1 , t 2 ) ≥ F1 (t 1 , t 2 ) and F 1 (t 1 , t 2 ) ≥ F 2 (t 1 , t 2 ) for all t 1 , t 2 ≥ 0, which is coherent with the fact that (τ (1) , τ (2) ) ≤ st (τ (1) , τ (2) ) (recall Remark 18).

Remark 33 Note that we have not been able to find any counter-example such that λ 1 j ≥ λ 2 j j = 1, 2, where neither Λ 1 j , j = 2, • • • , n nor Λ 2 j , j = 2, • • • , n fulfills Assumption [START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF], and such that (τ (1) , τ (2) ) is not stochastically smaller than (τ (1) , τ (2) ). Hence, Assumption (25) may be not required for the stochastic comparison result to hold, but we have not been able to prove it without this assumption. Then the question remains open whether Assumption [START_REF] Zhang | Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence[END_REF] is required to hold for one of the two models or not.