Valerio Basile
email: valerio.basile@inria.fr

Soufian Jebbara
email: sjebbara@cit-ec.uni-bielefeld.de

Elena Cabrio
email: elena.cabrio@unice.fr

Philipp Cimiano
email: cimiano@cit-ec.uni-bielefeld.de

Populating a Knowledge Base with Object-Location Relations using Distributional Semantics

The paper presents an approach to extract knowledge from large text corpora, in particular knowledge that facilitates object manipulation by embodied intelligent systems that need to act in the world. As a first step, our goal is to extract the prototypical location of given objects from text corpora. We approach this task by calculating relatedness scores for objects and locations using techniques from distributional semantics. We empirically compare different methods for representing locations and objects as vectors in some geometric space, and we evaluate them with respect to a crowd-sourced gold standard in which human subjects had to rate the prototypicality of a location given an object. By applying the proposed framework on DBpedia, we are able to build a knowledge base of 931 high confidence object-locations relations in a fully automatic fashion. 3

Introduction

Embodied intelligent systems such as robots require world knowledge to be able to perceive the world appropriately and perform appropriate actions on the basis of their understanding of the world. Take the example of a domestic robot that has the task of tidying up an apartment. A robot needs, e.g., to categorize different objects in the apartment, know where to put or store them, know where and how to grasp them, and so on. Encoding such knowledge by hand is a tedious, time-consuming task and is inherently prone to yield incomplete knowledge. It would be desirable to develop approaches that can extract such knowledge automatically from data.

To this aim, in this paper we present an approach to extract object knowledge from large text corpora. Our work is related to the machine reading and open information extraction paradigms aiming at learning generic knowledge from text corpora. In contrast, in our research we are interested in particular in extracting knowledge that facilitates object manipulation by embodied intelligent systems that need to act in the world. Specifically, our work focuses on the problem of relation extraction between entities mentioned in the text 4 . A relation is defined in the form of a tuple t = (e 1 ; e 2 ; ...; e n ) where the e i are entities in a predefined relation r within document D [START_REF] Bach | A Review of Relation Extraction[END_REF]. We develop a framework with foundations in distributional semantics, the area of Natural Language Processing that deals with the representation of the meaning of words in terms of their distributional properties, i.e., the context in which they are observed. It has been shown in the literature that distributional semantic techniques give a good estimation of the relatedness of concepts expressed in natural language (see Section 3 for a brief overview of distributional semantics principles). Semantic relatedness is useful for a number of tasks, from query expansion to word association, but it is arguably too general to build a general knowledge base, i.e., a triple like <entity1, relatedTo, entity2> might not be informative enough for many purposes.

Distributional Relation Hypothesis

We postulate that the relatedness relation encoded in distributional vector representations can be made more precise based on the type of the entities involved in the relation, i.e., if two entities are distributionally related, the natural relation that comes from their respective types is highly likely to occur. For example, the location relation that holds between an object and a room is represented in a distributional space if the entities representing the object and the room are highly associated according to the distributional space's metric.

Based on this assumption, as a first step of our work, we extract the prototypical location of given objects from text corpora. We frame this problem as a ranking task in which, given an object, our method computes a ranking of locations according how protoypical a location they are for this object. We build on the principle of distributional similarity and map each location and object to a vector representation computed on the basis of words these objects or locations co-occur with in a corpus. For each object, the locations are then ranked by the cosine similarity of their vector representations.

The paper is structured as follows. Section 2 discusses relevant literature, while Section 3 provides a background on word and entity vector spaces. Section 4 describes the proposed framework to extract relations from text. Section 5 reports on the creation of the goldstandard, and on the experimental results. Section 6 describes the obtained knowledge base of object locations, while conclusions end the paper.

Related Work

Our work relates to the three research lines discussed below, i.e.: i) machine reading, ii) supervised relation extraction, and iii) encoding common sense knowledge in domainindependent ontologies and knowledge bases.

The machine reading paradigm. In the field of knowledge acquisition from the Web, there has been substantial work on extracting taxonomic (e.g. hypernym), part-of relations [START_REF] Girju | Learning semantic constraints for the automatic discovery of part-whole relations[END_REF] and complete qualia structures describing an object [START_REF] Cimiano | Automatically learning qualia structures from the web[END_REF]. Quite recently, there has been a focus on the development of systems that can extract knowledge from any text on any domain (the open information extraction paradigm [START_REF] Etzioni | Open information extraction: The second generation[END_REF]). The DARPA Machine Reading Program [START_REF] Barker | Learning by reading: A prototype system, performance baseline and lessons learned[END_REF] aims at endowing machines with capabilities for lifelong learning by automatically reading and understanding texts (e.g. [START_REF] Etzioni | Machine reading at web scale[END_REF]). While such approaches are able to quite robustly acquire knowledge from texts, these models are not sufficient to meet our objectives since: i) they lack visual and sensor-motor grounding, ii) they do not contain extensive object knowledge. Thus, we need to develop additional approaches that can harvest the Web to learn about usages, appearance and functionality of common objects. While there has been some work on grounding symbolic knowledge in language [START_REF] Mooney | Learning to connect language and perception[END_REF], so far there has been no serious effort to compile a large and grounded object knowledge base that can support cognitive systems in understanding objects.

Supervised Relation Extraction. While machine reading attempts to acquire general knowledge by reading texts, other works attempt to extract specific relations applying supervised techniques to train classifiers. A training corpus in which the relation of interest is annotated is typically assumed (e.g. [START_REF] Bunescu | A shortest path dependency kernel for relation extraction[END_REF]). Another possibility is to rely on the so called distant supervision assumption and use an existing knowledge base to bootstrap the process by relying on triples or facts in the knowledge base to label examples in a corpus (e.g. [START_REF] Xu | Filling knowledge base gaps for distant supervision of relation extraction[END_REF][START_REF] Hoffmann | Learning 5000 relational extractors[END_REF][START_REF] Hoffmann | Knowledge-based weak supervision for information extraction of overlapping relations[END_REF][START_REF] Surdeanu | Multi-instance multi-label learning for relation extraction[END_REF]). Other researchers have attempted to extract relations by reading the Web, e.g. [START_REF] Blohm | Harvesting relations from the web -quantifiying the impact of filtering functions[END_REF]. Our work differs from these approaches in that, while we are extracting a specific relation, we do not rely on supervised techniques to train a classification model, but rather rely on semantic relatedness and distributional similarity techniques to populate a knowledge base with the relation in question.

Ontologies and KB of common sense knowledge. DBpedia5 is a large-scale knowledge base automatically extracted from semi-structured parts of Wikipedia. Besides its sheer size, it is attractive for the purpose of collecting general knowledge given the one-to-one mapping with Wikipedia ( allowing us to exploit the textual and structural information contained in there) and its position as the central hub of the Linked Open Data cloud.

YAGO [START_REF] Suchanek | Yago: A core of semantic knowledge[END_REF] is an ontology automatically created by mapping relations between WordNet synsets such as hypernymy and relations between Wikipedia pages such as links and redirects to semantic relations between concepts. Despite its high coverage, for our goals YAGO suffers from the same drawbacks of DBpedia, i.e. a lack of general relations between entities that are not instance of the DBpedia ontology, such as common objects. While a great deal of relations and properties of named entities are present, knowledge about, e.g. the location or the functionality of entities is missing.

ConceptNet6 [START_REF] Liu | Conceptnet — a practical commonsense reasoning tool-kit[END_REF] is a semantic network containing lots of things computers should know about the world. While it shares the same goals of the knowledge base we aim at building, ConceptNet is not a Linked Open Data resource. In fairness, the resource is in a graph-like structure, thus RDF triples could be extracted from it, and the building process provides a way of linking the nodes to DBpedia entities, among other LOD resources. However, we cannot integrate ConceptNet directly in our pipeline because of the low coverage of the mapping with DBpedia-of the 120 DBpedia entities in our gold standard (see Section 5) only 23 have a correspondent node in ConceptNet.

OpenCyC 7 attempts to assemble a comprehensive ontology and knowledge base of everyday common sense knowledge, with the goal of enabling AI applications to perform human-like reasoning. While for the moment in our work we focus on specific concepts and relations relevant to our scenario, we will consider linking them to realworld concepts in OpenCyc.

3 Background: Word and Entity Vector Spaces Word space models (or distributional space models, or word vector spaces) are abstract representations of the meaning of words, encoded as vectors in a high-dimensional space. A word vector space is constructed by counting cooccurrences of pairs of words in a text corpus, building a large square n-by-n matrix where n is the size of the vocabulary and the cell i, j contains the number of times the word i has been observed in cooccurrence with the word j. The i-th row in a cooccurrence matrix is a n-dimensional vector that acts as a distributional representation of the i-th word in the vocabulary. Words that appear in similar contexts often have similar representations in the vector space; this similarity is geometrically measurable with a distance metric such as cosine similarity, defined as the cosine of the angle between two vectors. This is the key point to linking the vector representation to the idea of semantic relatedness, as the distributional hypothesis states that "words that occur in the same contexts tend to have similar meaning" [START_REF] Harris | Distributional structure[END_REF]. Several techniques can be applied to reduce the dimensionality of the cooccurrence matrix. Latent Semantic Analysis [START_REF] Landauer | A solution to platos problem: The latent semantic analysis theory of acquisition, induction, and representation of knowledge[END_REF], for instance, uses Singular Value Decomposition to prune the less informative elements while preserving most of the topology of the vector space, and reducing the number of dimensions to 100-500.

In parallel, neural network-based models have recently began to rise to prominence. To compute word embeddings, several models rely on huge amounts of natural language texts from which a vector representation for each word is learned by a neural network. Their representations of the words are based on prediction as opposed to counting [START_REF] Baroni | Don't count, predict! a systematic comparison of context-counting vs. context-predicting semantic vectors[END_REF].

Vector spaces created on word distributional representations have been successfully proven to encode word similarity and relatedness relations [START_REF] Radinsky | A word at a time: Computing word relatedness using temporal semantic analysis[END_REF][START_REF] Reisinger | Multi-prototype vector-space models of word meaning[END_REF][START_REF] Ciobanu | Alternative measures of word relatedness in distributional semantics[END_REF], while word embeddings have proven to be a useful feature in many natural language processing tasks [START_REF] Collobert | Nlp (almost) from scratch[END_REF][START_REF] Le | Distributed Representations of Sentences and Documents[END_REF][START_REF] Santos | Learning character-level representations for part-of-speech tagging[END_REF] in that they often encode semantically meaningful information of a word.

Word Embeddings for Relation Extraction

This section presents our framework to extract relations from natural language text. The methods are based on distributional semantics, but present different approaches to compute vector representations of entities: one is based on a word embedding approach (Section 4.1), the other on a LSA-based representation of DBpedia entities (Section 4.2). We present one framework for which we test different ways of calculating the vector embeddings, each one having its own specificities and strengths.

A Word Space Model of Entity Lexicalizations

In this section, we propose a neural network-based word embedding method for the automatic population of a knowledge base of object-location relations. As outlined in Section 1, we frame this task as a ranking problem and score the vector representation for object-location pairs with respect to how prototypical the location is for the given object. Many word embedding methods encode useful semantic and syntactic properties [START_REF] Köhn | What's in an Embedding? Analyzing Word Embeddings through Multilingual Evaluation[END_REF][START_REF] Mitchell | Vector-based Models of Semantic Composition[END_REF][START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF] that we leverage for the extraction of object-location relations. In this work, we restrict our experiments to the skip-gram method [START_REF] Mikolov | Efficient Estimation of Word Representations in Vector Space[END_REF]. The objective of the skip-gram method is to learn word representations that are useful for predicting context words. As a result, the learned embeddings often display a desirable linear structure [START_REF] Mitchell | Vector-based Models of Semantic Composition[END_REF][START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF]. In particular, word representations of the skip-gram model often produce meaningful results using simple vector addition [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF]. For this work, we trained the skip-gram model on a corpus of roughly 83 million Amazon reviews [START_REF] Mcauley | Inferring networks of substitutable and complementary products[END_REF].

Motivated by the compositionality of word vectors, we derive vector representations for the entities as follows: considering a DBpedia entity such as Public_Toilet (we call this label the lexicalization), we clean it by removing parts in parenthesis, convert it to lower case, and split it into its individual words. We retrieve the respective word vectors from our pretrained word embeddings and sum them to obtain a single vector, namely, the vector representation of the entity: vector(public toilet) = vector(public) + vector(toilet). The generation of entity vectors is trivial for "singleword" entities, such as Cutlery or Kitchen, that are already contained in our word vector vocabulary. In this case, the entity vector is simply the corresponding word vector. With this derived set of entity vector representations, we compute cosine vector similarity score for object-location pairs. This score is an indicator of how typical the location for the object is. Given an object, we can create a ranking of locations with the most likely location candidates at the top of the list (see Table 1).

Distributional Representations of Entities

Vector representations of words (Section 4.1) are attractive since they only require a sufficiently large text corpus with no manual annotation. However, the drawback of focusing on words is that a series of linguistic phenomena may affect the vector representation. For instance, a polysemous word as rock (stone, musical genre, metaphorically strong person, etc.) is represented by a single vector where all the senses are conflated. NASARI [START_REF] Camacho-Collados | HLT-NAACL[END_REF], a resource containing vector representations of most of DBpedia entities, solves this problem by building a vector space of concepts. The NASARI vectors are actually distributional representations of the entities in BabelNet [START_REF] Navigli | Babelnet: The automatic construction, evaluation and application of a wide-coverage multilingual semantic network[END_REF], a large multilingual lexical resource linked to Wordnet, DBpedia, Wiktionary and other resources. The NASARI approach collects cooccurrence information of concepts from Wikipedia and then applies a LSA-like procedure for dimensionality reduction. The context of a concept is based on the set of Wikipedia pages where a mention of it is found. As shown in [START_REF] Camacho-Collados | HLT-NAACL[END_REF], the vector representations of entities encode some form of semantic relatedness, with tests on a sense clustering task showing positive results. Table 2 shows a sample of pairs of NASARI vectors together with their pairwise cosine similarity ranging from -1 (totally unrelated) to 1 (identical vectors). Following the hypothesis put forward in the introduction, we focus on the extraction of object-location relations by computing the cosine similarities of object and location entities. We exploit the alignment of BabelNet with DBpedia, thus generating a similarity score for pairs of DBpedia entities. For example, the DBpedia entity Dishwasher has a cosine similarity of .803 to the entity Kitchen, but only .279 with Classroom, suggesting that the appropriate location for a generic dishwasher is the kitchen rather than a classroom. Since cosine similarity is a graded value on a scale from -1 to 1, we can generate, for a given object, a ranking of candidate locations, e.g., the rooms of a house. Table 3 shows a sample of object-location pairs of DBpedia labels, ordered by the cosine similarity of their respective vectors in NASARI. Prototypical locations for the objects show up at the top of the list as expected, indicating a relationship between the semantic relatedness expressed by the cosine similarity of vector representations and the actual locative relation of entities.

Evaluation

This section presents the evaluation of the proposed framework for relation extraction (Section 4). We collected a set of relations rated by human subjects to provide a common benchmark, and we test several methods with varying values for their parameters.

We then adopt the best performing method to automatically build a knowledge base and test its quality against the manually created gold standard dataset.

Gold Standard

To test our hypothesis, we collected a set of human judgments about the likelihood of objects to be found in certain locations. To select the objects and locations for this experiment, every DBpedia entity that falls under the category Domestic implements, or under one of the narrower categories than Domestic implements according to SKOS 8 , is considered an object; every DBpedia entity that falls under the category Rooms is considered a location. This step results in 336 objects and 199 locations.

To select suitable object-location pairs for the creation of the gold standard, we need to filter out odd or uncommon examples of objects or locations like Ghodiyu or Fainting room. For example, the rankings produced by the cosine similarity of NASARI vectors (Table 3) are cluttered with results that are less prototypical because of their uncommonness. An empirical measure of commonness of entities could be used to rerank or filter the result to improve its generality. To this extent, we use the URI counts extracted from the parsing of Wikipedia with the DBpedia Spotlight tool for entity linking [START_REF] Daiber | Improving efficiency and accuracy in multilingual entity extraction[END_REF]. These counts are derived, for each DBpedia entity, from the number of incoming links to its correspondent Wikipedia page. We use it as an approximation of the notion of commonness of locations, e.g., a Kitchen (URI count: 742) is a more common location than a Billiard room (URI count: 82). Table 4 shows an example of using such counts to filter out irrelevant entries from the ranked list of candidate locations for the entity Paper towel according to NASARI-based similarity. We rank the 66,864 pairs of Domestic implements and Rooms using the aforementioned entity frequency measure and select the 100 most frequent objects and the 20 most frequent locations (2,000 object-location pairs in total). Examples of pairs: (Toothbrush,Hall), (Wallet, Ballroom) and (Nail file, Kitchen).

In order to collect the judgments, we set up a crowdsourcing experiment on the Crowdflower platform 9 . For each of the 2,000 object-location pairs, contributors were asked to rate the likelihood of the object to be in the location out of four possible values:

--2 (unexpected): finding the object in the room would cause surprise, e.g., it is unexpected to find a bathtub in a cafeteria. --1 (unusual): finding the object in the room would be odd, the object feels out of place, e.g., it is unusual to find a mug in a garage.

-1 (plausible): finding the object in the room would not cause any surprise, it is seen as a normal occurrence, e.g., it is plausible to find a funnel in a dining room. -2 (usual): the room is the place where the object is typically found, e.g, the kitchen is the usual place to find a spoon.

Contributors are shown ten examples per page, instructions, a short description of the entities (the first sentence from the Wikipedia abstract), a picture (from Wikimedia Commons, when available), and the list of possible answers as labeled radio buttons.

After running the crowdsourcing experiment for a few hours, we collected 12,767 valid judgments (455 were deemed "untrusted" by Crowdflower's quality filtering system based on a number of test questions we provided). Most of the pairs have received at least 5 separate judgments, with some outliers collecting more than one hundred judgments each. The average agreement, i.e. percentage of contributors that answered the most common answer for a given question, is 64.74%. The judgments are skewed towards the negative end of the spectrum, as expected, with 37% pairs rated unexpected, 30% unusual, 24% plausible and 9% usual. The cost of the experiment was 86 USD.

Ranking Evaluation

The proposed methods produce a ranking on top of a list of locations, given an input object. To test the validity of our methods we need to compare their output against a gold standard ranking. The latter is extracted from the dataset described in Section 5.1 by assigning to each object-location pair the average of the numeric values of the judgments received. For instance, if the pair (Wallet, Ballroom) has been rated -2 (unexpected) six times, -1 (unusual) three times, and never 1 (plausible) or 2 (usual), its score will be about -1.6, indicating that a Wallet is not very likely to be found in a Ballroom. The pairs are then ranked by this averaged score on a per-object basis.

As a baseline, we apply two simple methods based on entity frequency. In the location frequency baseline, the object-location pairs are ranked according to the frequency of the location. The ranking is thus the same for each object, since the score of a pair is only computed based on the location. This method makes sense in absence of any further information on the object: e,g, a robot tasked to find an unknown object should inspect "common" rooms such as a kitchen or a studio first, rather than "uncommon" rooms such as a pantry. The second baseline (link frequency) is based on counting how often every object is mentioned on the Wikipedia page of every location and vice versa. A ranking is produced based on these counts. An issue is that they could be sparse, i.e., most object-location pairs have a count of 0, thus sometimes producing no value for the ranking for an object. This is the case for rather "unusual" objects and locations.

For each object in the dataset, we compare the location ranking produced by our algorithms to the gold standard ranking and compute the Normalized Discounted Cumulative Gain (NDCG), a measure of rank correlation used in information retrieval that gives more weight to the results at the top of the list than at its bottom. This choice of evaluation metric follows from the idea that it is more important to guess the position in the ranking of most likely locations for a given object than to the least likely locations. Table 5 shows the average NDCG across all objects: methods NASARI-sim (Section 4.2) and SkipGram-sim (Section 4.1), plus the two baselines introduced above. Both our methods outperform the baselines with respect to the gold standard rankings.

Precision Evaluation

The NDCG measure gives a complete account of the quality of the produced rankings, but it is not easy to interpret apart from comparisons of different outputs. To gain a better insight into our results, we provide an alternative evaluation based on the "precision at k" measure. This Information Retrieval measure is the number of retrieved items that are ranked in the top-k part of the retrieved list and of the relevance ranking. In our experiments, for a given object, precision at k is the number of locations among the first k of the produced rankings that are also among the top-k locations in the gold standard ranking. It follows that, with k = 1, precision at 1 is 1 if the top returned location is the top location in the gold standard, and 0 otherwise. We compute the average of precision at k for k = 1 and k = 3 across all the objects. The results are shown in Table 6. As for the rank correlation evaluation, our methods outperform the baselines. The location frequency baseline performs very poorly, due to an idiosyncrasy in the frequency data, that is, the most "frequent" location in the dataset is Aisle. This behavior reflects the difficulty in evaluating this task using only automatic metrics, since automatically extracted scores and rankings may not correspond to common sense judgment.

The NASARI-based similarities outperform the SkipGram-based method when it comes to guessing the most likely location for an object, as opposed to the better performance of SkipGram-sim in terms of precision at 3 and rank correlation (Section 5.2).

We explored the results and found that for 19 objects out of 100, NASARI-sim correctly guesses the top ranking location but SkipGram-sim fails, while the opposite happens 15 out of 100 times. We also found that the NASARI-based method has a lower coverage than the other method., due to the coverage of the original resource (NASARI), where not every entity in DBpedia is assigned a vector (objects like Backpack and Comb, and locations like Loft are all missing). The SkipGram-based method also suffer from this problem, however, only for very rare or uncommon objects and locations (as Triclinium or Jamonera). These findings suggest that the two methods could have different strengths and weaknesses. In the following section we show two strategies to combine them.

Hybrid Methods: Fallback Pipeline and Linear Combination

The results from the previous sections highlight that the performance of our two main methods may differ qualitatively. In an effort to overcome the coverage issue of NASARIsim, and at the same time experiment with hybrid methods to extract location relations, we devised two simple ways of combining the SkipGram-sim and NASARI-sim methods. The first method is based on a fallback strategy: given an object, we consider the pair similarity of the object to the top ranking location according to NASARI-sim as a measure of confidence. If the top ranked location among the NASARI-sim ranking is exceeding a certain threshold, we consider the ranking returned by NASARI-sim as reliable. Otherwise, if the similarity is below the threshold, we deem the result unreliable and we adopt the ranking returned by SkipGram-sim instead. The second method produces an object-location similarity scores by linear combination of the NASARI and SkipGram similarities. The similarity score for the generic pair o, l is thus given by sim(o, l) = αsim N ASARI (o, l) + (1 -α)sim SkipGram (o, l), where parameter α controls the weight of one method w.r.t. the other. Table 7 shows the obtained results, with varying values of the parameters threshold and α. The line labeled Max shows the result obtained by choosing the highest similarity between NASARI-sim and SkipGram-sim, for comparison. While the NDCG is basically not affected, both precision at 1 and precision at 3 show an increase in performance with respect to any of the previous methods.

Building a Knowledge Base of Object Locations

In the the previous section, we tested how the proposed methods succeed in determining the relation between given objects and locations on a closed set of entities (for the purpose of evaluation). In this section we return to the original motivation of this work, that is, to collect location information about objects in an automatic fashion. All the methods introduced in this work are based on some measure of relatedness between entities, expressed as a real number in the range [-1,1] interpretable as a sort of confidence score relative to the target relation. Therefore, by imposing a threshold on the similarity scores and selecting only the object-location pairs that score above said threshold, we can extract a high-confidence set of object-location relations to build a new knowledge base from scratch. Moreover, by using different values for the threshold, we are able to control the quality and the coverage of the produced relations.

We test this approach on the gold standard dataset introduced in Section 5, using the version with data aggregated by Crowdflower: the constributors' answers are aggregated using relative majority, that is, each object-location pair has exactly one judgment assigned to it, corresponding to the most popular judgment among all the contributors that answered that question.We extract two lists of relations from this dataset to be used as a gold standard for experimental tests: one list of the 156 pairs rated 2 (usual) by the majority of contributors, and a larger list of the 496 pairs rated either 1 (plausible) or 2 (usual). The aggregated judgments in the gold standard have a confidence score assigned to them by Crowdflower, based on a measure of inter-rater agreement. Pairs that score low on this confidence measure ( 0.5) were filtered out, leaving respectively 118 pairs in the "usual" set 496 pairs in the "plausible or usual" set.

We order the object-location pairs produced by our two main methods by similarity score, and select the first n from the list, with n being a parameter. We also add to the comparison the results of the two hybrid methods from Section 5.4, with the best performing parameters in terms of precision at 1, namely the fallback strategy with threshold on similarity equal to 0.6 and the linear combination with α = 0.4. For the location relations extracted with these methods, we compute the precision and recall against the gold standard sets, with varying values of n. Here, the precision is the percentage of correctly predicted pairs in the set of all predicted pairs, while the recall is the percentage of predicted pairs that also occur in the gold standard. Figures 1 and2 show the evaluation of the four methods evaluated against the two aggregated gold standard datasets described above. Figures 1c and2c, in particular, show F-score plots for a direct comparison of the performance. The precision and recall figures show similar performances for all the methods, with the SkipGram-sim method obtaining a generally higher recall. The SkipGram-sim method produces generally better-quality sets of relations. However, if the goal is high precision, the other methods may be preferable.

Given these results, we can aim for a high-confidence knowledge base by selecting the threshold on object-location similarity scores that produces a reasonably high precision knowledge base in the evaluation. For instance, the knowledge base made by the top 50 object-location pairs extracted with the linear combination method (α = 0.4) has 0.52 precision and 0.22 recall on the "usual" gold standard (0.70 and 0.07 respectively on the "usual" or "plausible" set, see Figures 1a and2a). The similarity scores in this knowledge base range from 0.570 to 0.866. Following the same methodology that we used to construct the gold standard set of objects and locations (Section 5.1), we extract all the 336 Domestic implements and 199 Rooms from DBpedia, for a total of 66,864 object-location pairs. Selecting only the pairs whose similarity score is higher than 0.570, according to the linear combination method, yields 931 high confidence location relations. Of these, only 52 were in the gold standard set of pairs (45 were rated "usual" or "plausible" locations), while the remaining 879 are new, such as (Trivet, Kitchen), (Flight bag, Airport lounge) or (Soap dispenser, Unisex public toilet). The distribution of objects across locations has an arithmetic mean of 8.9 objects per location and standard deviation 11.0.Kitchen is the most represented location with 89 relations, while 15 out of 107 locations are associated with one single object. 10This paper presents novel methods to extract object relations, focusing on the typical locations of common objects. The proposed approaches are based on distributional semantics, where vector spaces are built that represent words or concepts in a highdimensional space. We then map vector distance to semantic relatedness, and instantiate a specific relation that depends on the type of the entities involved (e.g., an object highly related to a room indicates that the room is a typical location for the object) 11 .

The NASARI-based scoring method is a concept-level vectors space model derived from BabelNet. The skip-gram model (Section 4.1) is trained on Amazon review data and offers a word-level vector space which we exploit for scoring object-location pairs. Experiments on a dataset of human judgments show that they offer different advantages. To combine their strengths, we test two combination strategies, and show an improvement on their performances. Finally, we select the best parameters to extract a new, high-precision knowledge base of object locations.

As future work, we would like to employ retrofitting [START_REF] Faruqui | Retrofitting word vectors to semantic lexicons[END_REF] to enrich our pretrained word embeddings with concept knowledge from a semantic network such as Concept-Net or WordNet [START_REF] Miller | Wordnet: A lexical database for english[END_REF] in a post-processing step. With this technique, we might be able to combine the benefits of the concept-level and word-level semantics in a more sophisticated way to bootstrap the creation of an object-location knowledge base. We believe that this method is a more appropriate tool than the simple linear combination of scores. By specializing our skip-gram embeddings for relatedness instead of similarity [START_REF] Kiela | Specializing Word Embeddings for Similarity or Relatedness[END_REF] even better results could be achieved. Apart from that, we would like to investigate knowledge base embeddings and graph embeddings [START_REF] Weston | Connecting Language and Knowledge Bases with Embedding Models for Relation Extraction[END_REF][START_REF] Bordes | Learning Structured Embeddings of Knowledge Bases[END_REF][START_REF] Sun | Modeling mention, context and entity with neural networks for entity disambiguation[END_REF] that model entities and relations in a vector space in more detail. By defining an appropriate training objective, we might be able to compute embeddings that encode directly object-location relations and thus are tailored more precisely to our task at hand. Finally, we used the frequency of entity mentions in Wikipedia as a measure of commonality to drive the creation of a gold standard set for evaluation. This information, or equivalent measures, could be integrated directly into our relation extraction framework, for example in the form of a weighting scheme, to improve its predictions accuracy.

As main limitation of our current work, it needs to be stressed that the relation in question (here isLocatedAt) is predicted in all cases where the semantic relatedness is over a certain threshold. Thus, the method described is not specific for the particular relation given. In fact, the relation we predict is a relation of general (semantic) association. In our particular case, the method works due to the specifiy of the types invovled (room and object), which seem to be specific enough to restict the space of possible relations. It is not clear, however, to which other relations our method would generalize. This is left for future investigation. In particular, we intend to extend our method so that a model can be trained to predict a particular relation rather than a generic associative relationship.

Fig. 1 :

 1 Fig. 1: Evaluation on automatically created knowledge bases ("usual" locations).

Fig. 2 :

 2 Fig.2: Evaluation on automatically created knowledge bases ("plausible" and "usual" locations).

Table 1 :

 1 Locations for a sample object, extracted by computing cosine similarity on skip-gram-based vectors.

	Object	Location	Cosine Similarity
	Dishwasher Kitchen	.636
		Laundry room	.531
		Pantry	.525
		Wine cellar	.519

Table 2 :

 2 Examples of cosine similarity computed on NASARI vectors.

		Cherry Microsoft
	Apple	.917	.325
	Apple Inc. .475	.778

Table 3 :

 3 Locations for a sample object, extracted by computing cosine similarity on NASARI vectors.

	Object	Location	Cos. Similarity
	Dishwasher Kitchen	.803
		Air shower (room)	.788
		Utility room	.763
		Bathroom	.758

Table 4 :

 4 Locations for Paper towel, extracted by computing cosine similarity on NASARI vectors with URI count. Locations with frequency <100 are in gray.

	Location	URI count Cosine Similarity
	Air shower (room)	0	.671
	Public toilet	373	.634
	Mizuya	11	.597
	Kitchen	742	.589

Table 5 :

 5 Average NDCG of the produced rankings against the gold standard rankings.

	Method	NDCG
	Location frequency baseline	.851
	Link frequency baseline	.875
	NASARI-sim	.903
	SkipGram-sim	.912

Table 6 :

 6 Average precision at k for k = 1 and k = 3.

	Method	precision at 1 precision at 3
	Location frequency baseline	.000	.008
	Link frequency baseline	.280	.260
	NASARI-sim	.390	.380
	SkipGram-sim	.350	.400

Table 7 :

 7 Rank correlation and precision at k for the method based on fallback strategy.

	Method	NDCG precision at 1 precision at 3
	Fallback strategy (threshold=.4)	.907	.410	.393
	Fallback strategy (threshold=.5)	.906	.400	.393
	Fallback strategy (threshold=.6)	.908	.410	.406
	Fallback strategy (threshold=.7)	.909	.370	.396
	Fallback strategy (threshold=.8)	.911	.360	.403
	Linear combination (α=.0)	.912	.350	.400
	Linear combination (α=.2)	.911	.380	.407
	Linear combination (α=.4)	.913	.400	.423
	Linear combination (α=.6)	.911	.390	.417
	Linear combination (α=.8)	.910	.390	.410
	Linear combination (α=1.0)	.903	.390	.380
	Max	.911	.410	.413

The work in this paper is partially funded by the ALOOF project (CHIST-ERA program).

In the rest of the paper, the labels of the entities are identifiers from DBpedia URIs, stripped of the namespace http://dbpedia.org/resource/ for readability.

http://dbpedia.org

http://conceptnet5.media.mit.edu/

http://www.opencyc.org/; as RDF representations: http://sw.opencyc.org/

Simple Knowledge Organization System: https://www.w3.org/2004/02/skos/

http://www.crowdflower.com/

The full automatically created knowledge base is available at http://project.inria. fr/aloof/files/2016/04/objectlocations.nt_.gz

All the datasets resulting from this work are available at https://project.inria.fr/ aloof/data/


Vincent Magnin
email: vincent.magnin@iemn.univ-lille1.fr

Joseph Harari

Mathieu Halbwax

S Bastide

D Cherfi

J.-P Vilcot

Angle-dependent ray tracing simulations of reflections on

Keywords: texture, ray tracing, reflectance, solar cell

Introduction

Surface texturing is commonly used to reduce reflections from silicon solar cells and improve light absorption by light trapping [START_REF] Razykov | Solar photovoltaic electricity: Current status and future prospects[END_REF]. In particular, the record monocrystalline silicon cell (25% efficiency) has an inverted pyramid surface texture [START_REF] Zhao | 19.8% efficient "honeycomb" textured multicrystalline and 24.4% monocrystalline silicon solar cells[END_REF][START_REF] Green | Solar cell efficiency tables (version 42)[END_REF]. Pyramidal textures are typically produced by etching (100)-oriented monocrystalline wafers in an alkaline solution such as KOH or NaOH [START_REF] Moreno | A comparative study of wet and dry texturing processes of c-si wafers for the fabrication of solar cells[END_REF]. The etching being anisotropic gives theoretically [START_REF] Baker-Finch | Reflection distributions of textured monocrystalline silicon: Implications for silicon solar cells[END_REF] square-based pyramids with (111) facets inclined by 54.74 • from the horizontal plane.

The efficiency of surface textures is generally evaluated by characterizing or modelling under normally incident light, and it is well known that in those conditions inverted pyramids have a lower reflectance than upright ones [START_REF] Baker-Finch | Reflection of normally incident light from silicon solar cells with pyramidal texture[END_REF]. Sometimes the zenith incidence angle is considered [START_REF] Rodríguez | Random pyramidal texture modelling[END_REF][START_REF] Yu | Fabrication of large area subwavelength antireflection structures on si using trilayer resist nanoimprint lithography and liftoff[END_REF][START_REF] Byun | High-efficiency grid-type si solar cell structure[END_REF][START_REF] Mcintosh | Opal 2: Rapid optical simulation of silicon solar cells[END_REF]), but we found very few studies considering both zenith and azimuth incidence angles [START_REF] Parretta | Angle-dependent reflectance measurements on photovoltaic materials and solar cells[END_REF][START_REF] Gjessing | Comparison of periodic lighttrapping structures in thin crystalline silicon solar cells[END_REF]. In this paper, we study the directional-hemispherical reflectance of regular upright and inverted pyramids, taking into account both angles, and their bihemispherical reflectance (i.e. hemispherical-hemispherical reflectance, the first adjective relating to the collimation of the source, the second of the detector [START_REF] Nicodemus | Geometrical considerations and nomenclature for reflectance[END_REF][START_REF] Hapke | Theory of Reflectance and Emittance Spectroscopy[END_REF]).

First, the development and the validation of a 3D ray tracing model taking into account polarisation are presented. It is then used to compute and study the directional-hemispherical reflectance of regular upright and inverted pyramids versus zenith and azimuth angles. Then we compare these two textures under diffuse light by computing their bihemispherical reflectance.

Although chemical etching is imposing the base angle of pyramids, it could be interesting to obtain pyramids with a higher base angle in order to reduce reflections and to increase light trapping [START_REF] Hua | The effect of texture unit shape on silicon surface on the absorption properties[END_REF]. We therefore in a second part study the effect of the aspect ratio of both types of pyramids on their directional-hemispherical and bihemispherical reflectances.

Simulation

Model description

A ray tracing program was developed in Fortran using the test driven development methodology [START_REF] Gagliardi | Epistemological justification of test driven development in agile processes[END_REF], which is well adapted to scientific computing programs: for each needed function, automated tests are first written. Then the function is written, with the simplest possible code that can pass the tests. Then the code is refactored or optimised if needed, and the tests launched. The development process continues with another function. The automated tests allow the developer to code with more confidence. System tests can be added when the model is functional, in order to avoid any regression in the code. Following this methodology, we gradually developed a 2D model, then a 3D one, and finally we introduced a rigorous approach of the polarisation of light.

Indeed, a 3D model is necessary to study a texture illuminated under any incidence angle and azimuth angle, and the polarisation must be taken into account to study inverted pyramids because reflections on orthogonal facets produce polarisation rotations [START_REF] Trupke | Pyramidal micromirrors for microsystems and atom chips[END_REF][START_REF] Baker-Finch | Reflection of normally incident light from silicon solar cells with pyramidal texture[END_REF]. Our model can simulate any surface that can be mathematically described as a z = f (x, y) continuous function. This surface must be meshed with triangles: in the case of a square-based pyramids texture, each pyramid is simply meshed with four triangles. In this paper, unless otherwise specified, the simulated area is 100 × 100 µm, covered by one hundred 10 × 10 µm pyramids (Fig. 1), when computing the directional-hemispherical reflectance, and 400 × 400 µm for bihemispherical reflectance. These pyramids have a height or depth of 7.07 µm, their facets being inclined by 54.74 • from the horizontal plane, although recent studies have confirmed that in practice that angle can be slightly lower [START_REF] Yang | Characterization of 2-d reflection pattern from textured front surfaces of silicon solar cells[END_REF][START_REF] Baker-Finch | Reflection distributions of textured monocrystalline silicon: Implications for silicon solar cells[END_REF], a point that will be discussed later. The dimensions of the pyramids being much greater than the wavelengths absorbed by silicon, the geometric optics approximation can be considered valid. Under this hypothesis, the optical behaviour does not depend on the width of the pyramids but only on the inclination angle of their facets [START_REF] Byun | Analysis of light trapping effects in si solar cells with a textured surface by ray tracing simulation[END_REF][START_REF] Lien | Optimization of textured structure on crystalline silicon wafer for heterojunction solar cell[END_REF].

Each ray direction is defined using spherical coordinates as shown in Fig. 1. Light is considered unpolarised: each launched ray has its own complex electric field vector, orthogonal to the propagation direction (the medium is considered linear and isotropic), oriented in space using a random angle in the 0 -2π range and defined as a unit vector. The reflected elec-tric field vector is computed using the three-by-three polarisation ray tracing matrix as extensively described by [START_REF] Yun | Three-dimensional polarization ray-tracing calculus i: Definition and diattenuation[END_REF] and resumed by [START_REF] Baker-Finch | Reflection of normally incident light from silicon solar cells with pyramidal texture[END_REF]. This matrix is the product of three matrices: a first orthogonal matrix to transform between the global coordinate basis and the local s-p coordinates, the reflection Jones matrix and a third orthogonal matrix to transform between the local s-p coordinates and the global coordinate basis. The Jones Matrix includes the complex amplitude Fresnel coefficients computed for s and p polarisations [START_REF] Hapke | Theory of Reflectance and Emittance Spectroscopy[END_REF]. The cos(θ i ) geometric factor of the solar energy actually captured under an incidence angle θ i is omitted to better compare the actual light-trapping efficiency of the textures at various azimuth and incidence angles [START_REF] Gjessing | Comparison of periodic lighttrapping structures in thin crystalline silicon solar cells[END_REF].

In each simulation, one million incident rays are generated at random (x, y) positions over the surface. For studies with high incidence angles, these positions are restricted to the central pyramid in order to limit the number of rays exiting the computing window. For each ray, the algorithm searches the nearest intersected triangle, computes geometrically the incidence and reflection angles, then the reflected energy, until the ray quits the computing window. The number of bounces is recorded for statistics.

Transmitted rays are considered totally absorbed by silicon. Results computed under this hypothesis are generally in good agreement with measurements until a wavelength of 1 µm. Beyond, the absorption depth in silicon becomes rapidly greater than the thickness of the substrate and light can be easily reflected by the back surface one or several times and finally escape by the front surface.

The main output of the program is the directional-hemispherical reflectance at the considered wavelength. To lighten the text of this paper we will use the term hemispherical reflectance in the text and reflectance in the figures. Other outputs consist of the percentage of single bounces, double bounces, etc., the average number of bounces, and if necessary the complete path followed by each incident ray (directions, coordinates of each reflection point, facets numbers, energy, polarisation). The generated surface is recorded in a file using the STL (STereoLithography) format that can be imported in a 3D visualisation software (Fig. 1) or even printed with a 3D printer to obtain a macroscale prototype.

The complex refractive indexes of intrinsic silicon in the 0.25 µm-1.45 µm wavelength range are taken from the tabulation published by [START_REF] Green | Self-consistent optical parameters of intrinsic silicon at 300 k including temperature coefficients[END_REF]. Unless otherwise specified, all results in this paper are given for the λ = 0.7 µm wavelength. That will be discussed in the results section. The stan-dard deviation on the computed hemispherical reflectance expressed as a percentage has been evaluated to 0.005 by launching the same computing one thousand times with one million rays.

Model validation

The model was validated first by computing well-known results. For a perfectly plane surface the computed hemispherical reflectance is 33.7% under normal incidence, each ray being reflected only once. In the case of regular upright pyramids and normal incidence, 88.8% of normally incident rays suffer a double bounce and 11.1% a triple bounce [START_REF] Baker-Finch | Reflection of normally incident light from silicon solar cells with pyramidal texture[END_REF], resulting in a 2.11 mean number of bounces and a 11.45% hemispherical reflectance. In the case of inverted pyramids, 59.2% of normally incident rays suffer a double bounce and 40.8% a triple bounce as obtained with OPAL 2 free online software [START_REF] Mcintosh | Opal 2: Rapid optical simulation of silicon solar cells[END_REF], resulting in a 2.41 mean number of bounces and a 8.70% hemispherical reflectance. As is well known, inverted pyramids are therefore better than upright pyramids when illuminated under normal incidence (Fig. 2). The angles of reflection for each possible path were also verified [START_REF] Baker-Finch | Reflection of normally incident light from silicon solar cells with pyramidal texture[END_REF]. We also compared our results with simulation and experimental data published in the literature [START_REF] Byun | Analysis of light trapping effects in si solar cells with a textured surface by ray tracing simulation[END_REF][START_REF] Lien | Optimization of textured structure on crystalline silicon wafer for heterojunction solar cell[END_REF] for bare silicon and upright pyramids. It should be noted that it is current in the literature to compare the modelling of regular upright pyramids with measurements of random upright pyramids, because a regular texture is far easier to mesh.

Secondly, the simulation results were compared with the first experiments made in our laboratories on that subject. Inverted pyramids were etched on silicon (Fig. 3). The samples were p-type (boron doped) Czochralski silicon wafers with 5 -10 Ω.cm resistivity, (100) orientation and 300 µm thickness. Their front side was polished. Silicon nitride (SiN x H y ) was first deposited by low-pressure chemical vapor deposition (LPCVD) on both sides. In a second step, 9 µm square patterns with 1 µm resist gaps were realized on front side using electron-beam lithography. Reactive-ion etching has been used to transfer the resist pattern into the SiN x H y film before this latter was also removed using a RIE process. KOH wet etching was then carried out at 80 • C. A rotating (150 rpm) magnetic stirrer was used during this etching step in order to obtain a uniform etch. The inverted pyramids measured width is 9.8 µm. The surface topography was characterized by scanning electron microscopy (Fig. 3). The total reflectivity was measured using a Hitachi UVVISNIR 4001 spectrophotometer equipped with an integrating sphere in the 0.35 -1 µm wavelength range at room temperature.

Simulation and experimental results are compared on Fig. 2. Concerning polished silicon, measurements and simulation are in very good agreement. Concerning inverted pyramids, the measured hemispherical reflectance is 9.8% when the computed value is 8.7% at a wavelength of 0.7 µm. A great part of the difference can be explained by the presence of a 200 nm wide planar zone between adjacent pyramids (see Fig. 3), due to the technological process. If we compute the weighted sum of the hemispherical reflectances of the pyramidal part and of the planar fraction, we obtain a 9.68% hemispherical reflectance, which is very close to measurements. However, it can not explain the greater difference in the UV wavelengths. At a wavelength of 0.35 µm the computed hemispherical reflectance is 21.5% and the measured one is 26.2%. This gap may come from a strong increase of the imaginary part of the complex optical index due to the doping of the silicon substrate (2 × 10 15 cm -3 ), the model using the optical index of intrinsic silicon [START_REF] Green | Self-consistent optical parameters of intrinsic silicon at 300 k including temperature coefficients[END_REF]. This hypothesis is consistent with the fact that the gap is lower for the plane surface, where the rays are bouncing only once.

Results and discussion

Influence of the incidence and azimuth angles

The mean number of bounces and the hemispherical reflectance of surfaces textured with regular upright pyramids (Fig. 4 and5) and regular inverted pyramids (Fig. 6 and7) have been computed for incidence angles between 0 and 90 • and azimuth angles between 0 and 45 • because of the four-fold rotational symmetry. The computing time for each kind of pyramids was four days on a personal computer. Note that at incidence angles beyond 80 • , the simulation window have been enlarged to 1000 × 1000 µm to avoid rays being lost outside that window. This can be the case for upright pyramids, rays being injected just above the plan containing the top of the pyramids, those travelling between two rows of pyramids (near 0 • azimuth angle) at high incidence angle can cross the window before reflecting upon a pyramid facet (Fig. 1).

Figures 4 and5 show that the azimuth angle can have a much more significant effect on the hemispherical reflectance and the mean number of bounces in the case of upright pyramids, for incidence angles above 40 • . There is even a dramatic peak of hemispherical reflectance for the 0 • azimuth angle and very high incidence angles (Fig. 5), because an important proportion of rays travelling between two rows of pyramids are reflected upward only once on the lateral pyramid facets at grazing incidence, which implies that Fresnel coefficients are very high. Some computations were made at other wavelengths than λ = 0.7 µm and showed that the global shape of the 3D reflectance plots were similar but with a shift toward higher or lower reflectances. Indeed, the values of the Fresnel coefficients are wavelength dependent but not the geometrical paths followed by the reflected rays. Which also means that the plots concerning the mean number of bounces are not dependent on wavelength.

Another result is that for upright pyramids the hemispherical reflectance is not minimal at normal incidence but between 15 • and 20 • , depending on the azimuth angle. Its minimal value is 10.6% for a zero azimuth angle and a 19 • incidence angle. It corresponds to the maximum of the mean number of bounces, whose value is 2.22. For inverted pyramids, the effect is smaller, the minimal hemispherical reflectance being 8.58% and the mean number of bounces being 2.48 for a 45 • azimuth angle and a 22 • incidence angle.

It is well known that at normal incidence inverted pyramids are better than upright ones. But this is not the case in all lighting conditions. The difference of hemispherical reflectance between upright and inverted pyramids is plotted Fig. 8. The upright pyramids hemispherical reflectance is higher at low and high incidences angles but can be lower in the middle range. For example, our results show that for a zero azimuth angle, upright pyramids have a lower hemispherical reflectance for incidence angles between 26 • and 85 • . At a 72 • incidence angle, its value is 19.8% for upright pyramids and 30.4% for inverted pyramids. It corresponds to the trough that can be seen in Fig. 5. For a 45 • azimuth angle, upright pyramids are better between 37 • and 57 • incidence angles, but the difference is far lower. At a 53 • incidence angle, the hemispherical reflectance is 25.0% for upright pyramids and 25.6% for inverted pyramids.

To compare globally these two textures we have computed their bihemispherical reflectance by a Monte Carlo method. The azimuth angle of each ray is randomly drawn between 0 and 360 • and the incidence angle between 0 and 90 • . For upright pyramids, the bihemispherical reflectance is then 19.6% and the mean number of bounces is 1.70, but for inverted pyramids the results are respectively 20.7% and 1.74. For such an isotropic diffuse light, upright pyramids are therefore better than inverted ones. The bihemispherical reflectance can give clues on the performances of solar cells under diffuse light (cloud-covered sky for example) [START_REF] Parretta | Optical loss of photovoltaic modules under diffuse light[END_REF][START_REF] Kelly | Increasing the solar photovoltaic energy capture on sunny and cloudy days[END_REF].

But of course, in the case of a solar cell module the encapsulant and the glass are strongly limiting the maximum incidence angle on the silicon surface. If the encapsulant optical index is 1.5, the maximum refraction angle will be 41.8 • . Moreover at high incidences, most of the light incident on the module will be reflected at the air/encapsulant interface. For those reasons, a multilayer model would be necessary to take into account rigorously these effects.

Influence of the aspect ratio

In geometrical optics, the hemispherical reflectance of a texture depends on the complex optical index of the material, the reflection angles and the number of reflections. For a given material, the optimisation must therefore concern the geometry of the texture. For example, although alkaline etching of silicon imposes a pyramid slope near the theoretical 54.74 • angle, alternative methods could be used to obtain pyramids or other patterns with different slopes, for example femtosecond laser structuration [START_REF] Crouch | Comparison of structure and properties of femtosecond and nanosecond laser-structured silicon[END_REF], reactive ion etching [START_REF] Yoo | Reactive ion etching (rie) technique for application in crystalline silicon solar cells[END_REF] or nanoimprint lithography [START_REF] Yu | Fabrication of large area subwavelength antireflection structures on si using trilayer resist nanoimprint lithography and liftoff[END_REF]. Indeed, higher aspect ratios would have some advantages in the case of solar cells: there would be more light trapped by reflections on the encapsulant/semiconductor interface and the path length of rays would be extended, the absorption of infrared light would thus be improved [START_REF] Baker-Finch | Reflection distributions of textured monocrystalline silicon: Implications for silicon solar cells[END_REF]. In this section, we study the effect of the aspect ratio (height over width), i.e. the slope of upright and inverted pyramids on the hemispherical reflectance of the texture. Note that a 54.74 • slope corresponds to a 0.707 aspect ratio, the base angle α being related to the height h and the width w of the pyramids by

α = arctan(2 • h w ) (1)
The pyramids width is fixed to 10 µm and their height or depth varies in the 0 -25 µm range, corresponding to an aspect ratio in the 0 -2.5 range. They are illuminated at normal incidence at λ = 0.7 µm. The computed hemispherical reflectance and the mean number of bounces of regular upright and inverted pyramids versus their aspect ratio are shown Fig. 9 and 10. Inverted pyramids give a lower hemispherical reflectance for aspect ratios above 0.51, and upright pyramids are better in the 0.29 -0.51 range (Fig. 9).

Concerning upright pyramids, the mean number of bounces versus aspect ratio is a step curve, because new paths with more bounces appear at discrete values of the aspect ratio (Fig. 10). For inverted pyramids the curve is smoother, with a step ending at a 0.29 aspect ratio. And above a 0.51 aspect ratio, corresponding to a base angle of 45.6 • , the mean number of bounces on inverted pyramids is always higher than on upright ones.

It can be noted that the mean number of bounces is not sufficient to characterize the two types of surfaces, because the aspect ratio determines the possible trajectories of the rays, and therefore the Fresnel coefficients via the incidence angles. And it must be recalled that the reflectance results are given at λ = 0.7 µm. Some computations were made at other wavelengths but are not presented, the global shape of the curves being similar but with a shift toward higher or lower reflectances.

We have also computed for both upright and inverted pyramids the bihemispherical reflectance versus aspect ratio. The results show that inverted pyramids have a very slightly lower bihemispherical reflectance up to an aspect ratio of 0.23 and always a higher one above.

Some experimental studies have shown that the base angle of pyramids can be lower than the theoretical 54.74 • angle [START_REF] Baker-Finch | Reflection distributions of textured monocrystalline silicon: Implications for silicon solar cells[END_REF][START_REF] Yang | Characterization of 2-d reflection pattern from textured front surfaces of silicon solar cells[END_REF], being around 52 • for regular inverted pyramids and 50 -52 • for random upright pyramids. In our study, the computed hemispherical reflectance for a 0.6 aspect ratio (50 • base angle) is 12.2% for upright pyramids and 10.7% for inverted pyramids. But we have not taken into account the fact that upright pyramids can moreover not being really square-based [START_REF] Baker-Finch | Reflection distributions of textured monocrystalline silicon: Implications for silicon solar cells[END_REF].

A limitation of this study on the effect of the aspect ratio is that the transmitted rays are supposed to be totally absorbed by silicon. A more rigorous approach would require a multilayer ray tracing model because some of these rays can exit from silicon, especially for high aspect ratios, for rays intersecting the region around the top of a pyramid or for infrared wavelengths. But this problem is quite limited by the light trapping effect, due to the total internal reflections, except for infrared wavelengths.

Conclusions

We have developed, using a test driven development methodology, a 3D ray tracing program taking into account the polarisation of light that can compute the reflectance of a textured surface, in the limits of geometrical optics. It was validated using both literature and measurements on silicon with pyramidal textures fabricated in our laboratories.

Using this model, we have computed the directional-hemispherical reflectance versus the azimuth and incidence angles for both regular upright and inverted pyramids, with (111) facets, and we have presented the results at a wavelength of 0.7 µm. A first result is that the hemispherical reflectance is not minimal at normal incidence but for an incidence angle near 20 • for both textures. A second result is that upright pyramids can have a lower hemispherical reflectance than inverted ones for incidence angles in the middle range. We have also computed the bihemispherical reflectance, by simulating an hemispherical diffuse lighting, and obtained 19.6% for regular upright pyramids and 20.7% for inverted ones. These results show that in terms of reflectance of a bare textured silicon surface, inverted pyramids are not necessarily always more efficient in real world conditions.

In the second part of this work, we explored the effect of the pyramids aspect ratio on the hemispherical reflectance at normal incidence. The mean number of bounces increases and the hemispherical reflectance decreases with the aspect ratio of both upright and inverted pyramidal textures. At a wavelength of 0.7 µm, above an aspect ratio of 0.51, corresponding to a base angle of 45.6 • , inverted pyramids have a lower reflectance. But in the case of an hemispherical lighting, the bihemispherical reflectance of inverted pyramids is lower only for aspect ratios below 0.23.

Further work will involve the modelling and optimisation of other geometrical patterns that could be fabricated by new technological processes. We also plan to develop a multilayer ray tracing model to take account light trapping effects in the substrate, the effect of the encapsulant and glass of a module, and of an anti-reflection coating.

Figure 1 :Figure 2 :

 12 Figure1: Three-dimensional model of textured surfaces: regular inverted pyramids (top) and upright pyramids (bottom). In the model, the direction of a light ray is defined by the polar angle θ and the azimuth angle φ. The incidence angle is therefore π -θ.

Figure 3 :

 3 Figure 3: SEM micrographs of inverted pyramids etched on Si.

Figure 4 :

 4 Figure 4: Mean number of reflections on a surface textured with regular upright pyramids versus the azimuth angle and the incidence angle.

Figure 5 :

 5 Figure 5: Hemispherical reflectance of a surface textured with regular upright pyramids versus the azimuth angle and the incidence angle at λ = 0.7 µm.

Figure 6 :

 6 Figure 6: Mean number of reflections on a surface textured with regular inverted pyramids versus the azimuth angle and the incidence angle.

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: Hemispherical reflectance of a surface textured with regular inverted pyramids versus the azimuth angle and the incidence angle at λ = 0.7 µm.

Figure 10 :

 10 Figure 10: Number of reflections on a surface textured with regular upright or inverted pyramids versus the pyramid aspect ratio.

Acknowledgements

This work was supported by the DEFI ENRS CNRS contract PRONOS-TIQUES. The authors would also like to thank the Polytech Lille engineering school for the use of its computing power.