Louis Martin
email: louismartin@fb.com

Samuel Humeau
email: samuelhumeau@fb.com

Pierre-Emmanuel Mazaré

Antoine Bordes
email: abordes@fb.com

Éric

Benoît Sagot
email: benoit.sagot@inria.fr

Reference-less Quality Estimation of Text Simplification Systems

The evaluation of text simplification (TS) systems remains an open challenge. As the task has common points with machine translation (MT), TS is often evaluated using MT metrics such as BLEU. However, such metrics require high quality reference data, which is rarely available for TS. TS has the advantage over MT of being a monolingual task, which allows for direct comparisons to be made between the simplified text and its original version. In this paper, we compare multiple approaches to reference-less quality estimation of sentence-level text simplification systems, based on the dataset used for the QATS 2016 shared task. We distinguish three different dimensions: grammaticality, meaning preservation and simplicity. We show that n-gram-based MT metrics such as BLEU and METEOR correlate the most with human judgment of grammaticality and meaning preservation, whereas simplicity is best evaluated by basic length-based metrics.

Introduction

Text simplification (hereafter TS) has received increasing interest by the scientific community in recent years. It aims at producing a simpler version of a source text that is both easier to read and to understand, thus improving the accessibility of text for people suffering from a range of disabilities such as aphasia [START_REF] Carroll | Practical simplification of english newspaper text to assist aphasic readers[END_REF] or dyslexia [START_REF] Rello | Simplify or help?: text simplification strategies for people with dyslexia[END_REF], as well as for second language learners [START_REF] Xia | Text readability assessment for second language learners[END_REF] and people with low literacy [START_REF] Massami Watanabe | Facilita: reading assistance for low-literacy readers[END_REF]. This topic has been researched for a variety of languages such as English [START_REF] Zhu | A monolingual tree-based translation model for sentence simplification[END_REF][START_REF] Sander Wubben | Sentence simplification by monolingual machine translation[END_REF][START_REF] Narayan | Hybrid simplification using deep semantics and machine translation[END_REF][START_REF] Xu | Problems in current text simplification research: New data can help[END_REF], French [START_REF] Brouwers | Syntactic sentence simplification for french[END_REF], Spanish [START_REF] Saggion | Text simplification in simplext. making text more accessible[END_REF], Portuguese [START_REF] Specia | Translating from complex to simplified sentences[END_REF], Italian [START_REF] Brunato | Design and annotation of the first italian corpus for text simplification[END_REF] and Japanese [START_REF] Goto | Japanese news simplification: Task design, data set construction, and analysis of simplified text[END_REF]. 1One of the main challenges in TS is finding an adequate automatic evaluation metric, which is necessary to avoid the time-consuming human evaluation. Any TS evaluation metric should take into account three properties expected from the output of a TS system, namely:

• Grammaticality: how grammatically correct is the TS system output?

• Meaning preservation: how well is the meaning of the source sentence preserved in the TS system output?

• Simplicity: how simple is the TS system output? 2

TS is often reduced to a sentence-level problem, whereby one sentence is transformed into a simpler version containing one or more sentences. In this paper, we shall make use of the terms source (sentence) and (TS system) output to respectively denote a sentence given as an input to a TS system and the simplified, single or multi-sentence output produced by the system.

TS, seen as a sentence-level problem, is often viewed as a monolingual variant of (sentencelevel) MT. The standard approach to automatic TS evaluation is therefore to view the task as a translation problem and to use machine translation (MT) evaluation metrics such as BLEU [START_REF] Papineni | Bleu: a method for automatic evaluation of machine translation[END_REF]. However, MT evaluation metrics rely on the existence of parallel corpora of source sentences and manually produced reference translations, which are available on a large scale for many language pairs [START_REF] Tiedemann | Parallel Data, Tools and Interfaces in OPUS[END_REF]. TS datasets are less numerous and smaller. Moreover, they are often automatically extracted from comparable corpora rather than strictly parallel corpora, which results in noisier reference data. For example, the PWKP dataset [START_REF] Zhu | A monolingual tree-based translation model for sentence simplification[END_REF] consists of 100,000 sentences from the English Wikipedia automatically aligned with sentences from the Simple English Wikipedia based on term-based similarity metrics. It has been shown by [START_REF] Xu | Problems in current text simplification research: New data can help[END_REF] that many of PWKP's "simplified" sentences are in fact not simpler or even not related to their corresponding source sentence. Even if better quality corpora such as Newsela do exist [START_REF] Xu | Problems in current text simplification research: New data can help[END_REF], they are costly to create, often of limited size, and not necessarily open-access. This creates a challenge for the use of referencebased MT metrics for TS evaluation. However, TS has the advantage of being a monolingual translation-like task, the source being in the same language as the output. This allows for new, nonconventional ways to use MT evaluation metrics, namely by using them to compare the output of a TS system with the source sentence, thus avoiding the need for reference data. However, such an evaluation method can only capture at most two of the three above-mentioned dimensions, namely meaning preservation and, to a lesser extent, grammaticality.

Previous works on reference-less TS evaluation include Štajner et al. (2014), who compare the behaviour of six different MT metrics when used between the source sentence and the corresponding simplified output. They evaluate these metrics with respect to meaning preservation and grammaticality. We extend their work in two directions. Firstly, we extend the comparison to include the degree of simplicity achieved by the system. Secondly, we compare additional features, including those used by Štajner et al. (2016a), both individually, as elementary metrics, and within multi-feature metrics. To our knowledge, no previous work has provided as thorough a comparison across such a wide range and combination of features for the reference-less evaluation of TS.

First we review available text simplification evaluation methods and traditional quality estimation features. We then present the QATS shared task and the associated dataset, which we use for our experiments. Finally we compare all methods in a reference-less setting and analyze the results.

2 Existing evaluation methods 2.1 Using MT metrics to compare the output and a reference TS can be considered as a monolingual translation task. As a result, MT metrics such as BLEU [START_REF] Papineni | Bleu: a method for automatic evaluation of machine translation[END_REF], which compare the output of an MT system to a reference translation, have been extensively used for TS [START_REF] Narayan | Hybrid simplification using deep semantics and machine translation[END_REF][START_REF] Štajner | A deeper exploration of the standard pb-smt approach to text simplification and its evaluation[END_REF][START_REF] Xu | Optimizing statistical machine translation for text simplification[END_REF].

Other successful MT metrics include TER [START_REF] Matthew G Snover | Ter-plus: paraphrase, semantic, and alignment enhancements to translation edit rate[END_REF], ROUGE [START_REF] Lin | Rouge: A package for automatic evaluation of summaries[END_REF] and METEOR [START_REF] Banerjee | Meteor: An automatic metric for mt evaluation with improved correlation with human judgments[END_REF], but they have not gained much traction in the TS literature. These metrics rely on good quality references, something which is often not available in TS, as discussed by [START_REF] Xu | Problems in current text simplification research: New data can help[END_REF]. Moreover, Štajner et al. (2015) and Sulem et al. (2018a) showed that using BLEU to compare the system output with a reference is not a good way to perform TS evaluation, even when good quality references are available. This is especially true when the TS system produces more than one sentence for a single source sentence.

Using MT metrics to compare the output and the source sentence

As mentioned in the Introduction, the fact that TS is a monolingual task means that MT metrics can also be used to compare a system output with its corresponding source sentence, thus avoiding the need for reference data. Following this idea, Štajner et al. (2014) found encouraging correlations between 6 widely used MT metrics and human assessments of grammaticality and meaning preservation. However MT metrics are not relevant for the evaluation of simplicity, which is why they did not take this dimension into account. [START_REF] Xu | Optimizing statistical machine translation for text simplification[END_REF] also explored the idea of comparing the TS system output with its corresponding source sentence, but their metric, SARI, also requires to compare the output with a reference. In fact, this metric is designed to take advantage of more than one reference. It can be applied when only one reference is available for each source sentence, but its results are better when multiple ref-erences are available.

Attempts to perform Quality Estimation on the output of TS systems, without using references, include the 2016 Quality Assessment for Text Simplification (QATS) shared task ( Štajner et al., 2016b), to which we shall come back in section 3. Sulem et al. (2018b) introduce another approach, named SAMSA. The idea is to evaluate the structural simplicity of a TS system output given the corresponding source sentence. SAMSA is maximized when the simplified text is a sequence of short and simple sentences, each accounting for one semantic event in the original sentence. It relies on an in-depth analysis of the source sentence and the corresponding output, based on a semantic parser and a word aligner. A drawback of this approach is that good quality semantic parsers are only available for a handful of languages. The intuition that sentence splitting is an important sub-task for producing simplified text motivated [START_REF] Narayan | Split and rephrase[END_REF] to organize the Split and Rephrase shared task, which was dedicated to this problem.

Other metrics

One can also estimate the quality of a TS system output based on simple features extracted from it.

For instance, the QUEST framework for quality estimation in MT gives a number of useful baseline features for evaluating an output sentence [START_REF] Specia | Quest-a translation quality estimation framework[END_REF]. These features range from simple statistics, such as the number of words in the sentence, to more sophisticated features, such as the probability of the sentence according to a language model. Several teams who participated in the QATS shared task used metrics based on this framework, namely SMH ( Štajner et al., 2016a), UoLGP [START_REF] Rios | Large scale translation quality estimation[END_REF] and UoW [START_REF] Béchara | Miniexperts: An svm approach for measuring semantic textual similarity[END_REF].

Readability metrics such as Flesch-Kincaid Grade Level (FKGL) and Flesch Reading Ease (FRE) [START_REF] Kincaid | Derivation of new readability formulas (automated readability index, fog count and flesch reading ease formula) for navy enlisted personnel[END_REF] have been extensively used for evaluating simplicity. These two metrics, which were shown experimentally to give good results, are linear combinations of the number of words per sentence and the number of syllables per word, using carefully adjusted weights.

Methodology

Our goal is to compare a large number of ways to perform TS evaluation without a reference. To this end, we use the dataset provided in the QATS shared task. We first compare the behaviour of elementary metrics, which range from commonly used metrics such as BLEU to basic metrics based on a single low-level feature such as sentence length. We then compare the effect of aggregating these elementary metrics into more complex ones and compare our results with the state of the art, based on the QATS shared task data and results.

The QATS shared task

The data from the QATS shared task ( Štajner et al., 2016b) consists of a collection of 631 pairs of english sentences composed of a source sentence extracted from an online corpus and a simplified version thereof, which can contain one or more sentences. This collection is split into a training set (505 sentence pairs) and a test set (126 sentence pairs). Simplified versions were produced automatically using one of several TS systems trained by the shared task organizers. Human annotators labelled each sentence pair using one of the three labels Good, OK and Bad on each of the three dimensions: grammaticality, meaning preservation and simplicity3 . An overall quality label was then automatically assigned to each sentence pair based on its three manually assigned labels using a method detailed in ( Štajner et al., 2016b). Distribution of the labels and examples are presented in FIGURE 1 andTABLE 1.

The goal of the shared task is, for each sentence in the test set, to either produce a label (Good, OK, Bad) or a raw score estimating the overall quality of the simplification for each of the three dimensions. Raw score predictions are evaluated using the Pearson correlation with the ground truth labels, while actual label prediction are evaluated using the weighted F1-score. The shared task is described in further details on the QATS website4 .

Features

In our experiments, we compared about 60 elementary metrics, which can be organised as follows:

• MT metrics

-BLEU, ROUGE, METEOR, TERp -Variants of BLEU: BLEU 1gram, BLEU 2gram, BLEU 3gram
, BLEU 4gram and seven smoothing methods5 from NLTK [START_REF] Bird | Nltk: the natural language toolkit[END_REF].

-Intermediate components of TERp inspired by ( Štajner et al., 2016a): e.g. number of insertions, deletions, shifts...

• Readability metrics and other sentence-level features: FKGL and FRE, numbers of words, characters, syllables...

• Metrics based on the baseline QUEST features (17 features) [START_REF] Specia | Quest-a translation quality estimation framework[END_REF], such as statistics on the number of words, word lengths, language model probability and ngram frequency.

• Metrics based on other features: frequency table position, concreteness as extracted from Brysbaert et al.'s 2014 list, language model probability of words using a convolutional sequence to sequence model from [START_REF] Gehring | Convolutional sequence to sequence learning[END_REF], comparison methods using pretrained fastText word embeddings [START_REF] Mikolov | Advances in pre-training distributed word representations[END_REF] or Skip-thought sentence embeddings [START_REF] Kiros | Skip-thought vectors[END_REF].

TABLE 2 lists 30 of the elementary metrics that we compared, which are those that we found to correlate the most with human judgments on one or more of the three dimensions (grammaticality, meaning preservation, simplicity).

Experimental setup

Evaluation of elementary metrics We rank all features by comparing their behaviour with human judgments on the training set. We first compute for each elementary metric the Pearson correlation between its results and the manually assigned labels for each of the three dimensions. We then rank our elementary metrics according to the absolute value of the Pearson correlation. 6 Training and evaluation of a combined metric We use our elementary metrics as features to train classifiers on the training set, and evaluate their performance on the test set. We therefore scale them and reduce the dimensionality with a 25-component PCA 7 , then train several regression algorithms 8 and classification algorithms 9 using scikit-learn [START_REF] Pedregosa | Scikit-learn: Machine Learning in Python[END_REF]. For each dimension, we keep the two models performing best on the test set and add them in the leaderboard of the QATS shared task (TABLE 4), naming them with the name of the regression algorithm they were built with.

Results

Comparing elementary metrics

FIGURE 3 ranks all elementary metrics given their absolute Pearson correlation on each of the three dimensions.

Grammaticality N -gram based MT metrics have the highest correlation with human grammaticality judgments. METEOR seems to be the best, probably because of its robustness to synonymy, followed by smoothed BLEU (BLEUSmoothed in 2). This indicates that relevant grammaticality information can be derived from the source sentence. We were expecting that information contained in a language model would help achieving better results (AvgLMProbsOutput), but MT metrics correlate better with human judgments. We deduce that the grammaticality information contained in the source is more specific and more helpful for evaluation than what is learned by the language model. 6 The code is available on Github at https: //github.com/facebookresearch/ text-simplification-evaluation 7 We used PCA instead of feature selection because it performed better on the validation set. The number of component was tuned on the validation set as well.

8 Regressors: Linear regression, Lasso, Ridge, Linear SVR (SVM regressor), Adaboost regressor, Gradient boosting regressor and Random forest regressor.

9 Classifiers: Logistic regression, MLP classifier (with L2 penalty, alpha=1), SVC (linear SVM classifier), Knearsest neighbors classifier (k=3), Adaboost classifier, Gradient boosting classifier and Random forest classifier.

Meaning preservation It is not surprising that meaning preservation is best evaluated using MT metrics that compare the source sentence to the output sentence, with in particular smoothed BLEU, BLEU 3gram and METEOR. Very simple features such as the percentage of words in common between source and output also rank high. Surprisingly, word embedding comparison methods do not perform as well for meaning preservation, even when using word alignment.

Simplicity Methods that give the best results are the most straightforward for assessing simplicity, namely word, character and syllable counts in the output, averaged over the number of output sentences. These simple features even outperform the traditional, more complex metrics FKGL and FRE. As could be expected, we find that metrics with the highest correlation to human simplicity judgments only take the output into account. Exceptions are the NBSourceWords and NBSour-cePunct features. Indeed, if the source sentence has a lot of words and punctuation, and is therefore likely to be particularly complex, then the output will most likely be less simple as well. We also expected word concreteness ratings and position in the frequency table to be good indicators of simplicity, but it does not seem to be the case here. Structural simplicity might simply be more important than such more sophisticated components of the human intuition of simple text.

Discussion Even if counting the number of words or comparing n-grams are good proxies for the simplification quality, they are still very superficial features and might miss some deeper and more complex information. Moreover the fact that grammaticality and meaning preservation are best evaluated using n-gram-based comparison metrics might bias the TS models towards copying the source sentence and applying fewer modifications.

Syntactic parsing or language modelling might capture more insightful grammatical information and allow for more flexibility in the simplification model. Regarding meaning preservation, semantic analysis or paraphrase detection models would also be good candidates for a deeper analysis.

Warning note We should be careful when interpreting these results as the QATS dataset is relatively small. We compute confidence intervals on our results, and find them to be non-negligible, yet without putting our general observations into question. For instance, METEOR, which performs best on grammaticality, has a 95% confidence interval of 0.36 ± 0.08 on the training set. These results are therefore preliminary and should be validated on other datasets.

Combination of all features with trained models

We also combine all elementary metrics and train an evaluation models for each of the three dimensions. It is surprising to us that the aggregation of multiple elementary features would score worse than the features themselves. However, we observe a strong discrepancy between the scores obtained on the train and test set, as illustrated by TABLE 3. We also observed very large confidence intervals in terms of Pearson correlation. For instance our lasso model scores 0.33 ± 0.17 on the test set for grammaticality. This should observe caution when interpreting Pearson scores on QATS.

F1-score for classifiers (assigning labels) On the classification task, our models seem to score best for meaning preservation, simplicity and overall, and third for grammaticality. This seems to confirm the importance of considering a large ensemble of elementary features including lengthbased metrics to evaluate simplicity.

Conclusion

Finding accurate ways to evaluate text simplification (TS) without the need for reference data is a key challenge for TS, both for exploring new approaches and for optimizing current models, in particular those relying on unsupervised, often MT-inspired models.

We explore multiple reference-less quality evaluation methods for automatic TS systems, based on data from the 2016 QATS shared task. We rely on the three key dimensions of the quality of a TS system: grammaticality, meaning preservation and simplicity.

Our results show that grammaticality and meaning preservation are best assessed using n-grambased MT metrics evaluated between the output and the source sentence. In particular, METEOR and smoothed BLEU achieve the highest correlation with human judgments. These approaches even outperform metrics that make an extensive use of external data, such as language models. This shows that a lot of useful information can be obtained from the source sentence itself.

Regarding simplicity, we observe that counting the number of characters, syllables and words provides the best results. In other words, given the currently available metrics, the length of a sentence seems to remain the best available proxy for its simplicity.

However, given the small size of the QATS dataset and the high variance observed in our experiments, these results must be taken with a pinch of salt and will need to be confirmed on a larger dataset. Creating a larger annotated dataset as well as averaging multiple human annotations for each pair of sentences would help reducing the variance of the experiments and confirming our findings.

In future work, we shall explore richer and more complex features extracted using syntactic and semantic analyzers, such as those used by the SAMSA metric, and paraphrase detection models.

Finally, it remains to be understood how we can optimize the trade-off between grammaticality, meaning preservation and simplicity, in order to build the best possible comprehensive TS metric in terms of correlation with human judgments. Unsurprisingly, optimizing one of these dimensions often leads to lower results on other dimensions [START_REF] Schwarzer | Human evaluation for text simplification: The simplicityadequacy tradeoff[END_REF]. For instance, the best way to guarantee grammaticality and meaning preservation is to leave the source sentence unchanged, thus resulting in no simplification at all. Improving TS systems will require better global TS evaluation metrics. This is especially true when considering that TS is in fact a multiply defined task, as there are many different ways of simplifying a text, depending on the different categories of people and applications at whom TS is aimed.

Figure 1 :

 1 Figure 1: Label repartition on the QATS Shared task

Table 1 :

 1 OriginalAll three were arrested in the Toome area and have been taken to the Serious Crime Suite at Antrim police station. good good good good syntactic Simple All three were arrested in the Toome area. All three have been taken to the Serious Crime Suite at Antrim police station. Examples from the training dataset of QATS. Differences between the original and the simplified version are presented in bold. This table is adapted fromŠtajner et al. (2016b).

	Version	Sentence

Table 2 :

 2 Brief description of 30 of our most relevant elementary metrics

	Short name		Description
	NBSourcePunct		Number of punctuation tokens in source (QUEST)
	NBSourceWords		Number of source words (QUEST)
	NBOutputPunct		Number of punctuation tokens in output (QUEST)
	TypeTokenRatio		Type token ratio (QUEST)
	TERp Del		Number of deletions (TERp component)
	TERp NumEr		Number of total errors (TERp component)
	TERp Sub		Number of substitutions (TERp component)
	TERp		TERp MT metric
	BLEU 1gram		BLEU MT metric with unigrams only
	BLEU 2gram		BLEU MT metric up to bigrams
	BLEU 3gram		BLEU MT metric up to trigrams
	BLEU 4gram		BLEU MT metric up to 4-grams
	METEOR		METEOR MT metric
	ROUGE		ROUGE summarization metric
	BLEUSmoothed		BLEU MT metric with smoothing (method 7 from nltk)
	AvgCosineSim		Cosine similarity between source and output pre-trained word embeddings
	NBOutputChars		Number of characters in the output
	NBOutputCharsPerSent	Average number of characters per sentence in the output
	NBOutputSyllables	Number of syllables in the output
	NBOutputSyllablesPerSent Average number of syllables per sentence in the output
	NBOutputWords		Number of words in the output
	NBOutputWordsPerSent	Average number of words per sentence in the output
	AvgLMProbsOutput	Average log-probabilities of output words (Language Model)
	MinLMProbsOutput	Minimum log-probability of output words (Language Model)
	MaxPosInFreqTable	Maximum position of output words in the frequency table
	AvgConcreteness		Average word concreteness Brysbaert et al.'s 2014 concreteness list
	OutputFKGL		Flesch-Kincaid Grade Level
	OutputFRE		Flesch Reading Ease
	WordsInCommon		Percentage of words in common between source and Output
	Grammaticality		Meaning Preservation		Simplicity
	Short name	Train ↓	Test Short name	Train ↓	Test Short name	Train ↓	Test
	Best QATS team		0.48 Best QATS team		0.59 Best QATS team		0.38
	METEOR	0.36	0.39 BLEUSmoothed	0.59	0.52 NBOutputCharsPerSent	-0.52 -0.45
	BLEUSmoothed	0.33	0.34 BLEU 3gram	0.57	0.52 NBOutputSyllablesPerSent	-0.52 -0.49
	BLEU 4gram	0.32	0.34 METEOR	0.57	0.58 NBOutputWordsPerSent	-0.51 -0.39
	BLEU 3gram	0.31	0.34 BLEU 2gram	0.57	0.52 NBOutputChars	-0.48 -0.37
	TERp NumEr	-0.30 -0.31 BLEU 4gram	0.57	0.51 NBOutputWords	-0.47 -0.29
	BLEU 2gram	0.30	0.34 WordsInCommon	0.55	0.50 NBOutputSyllables	-0.46 -0.42
	TERp	-0.30 -0.32 BLEU 1gram	0.55	0.52 NBOutputPunt	-0.42 -0.31
	ROUGE	0.29	0.29 ROUGE	0.55	0.47 NBSourceWords	-0.38 -0.21
	AvgLMProbsOutput	0.28	0.34 TERp	-0.54 -0.48 outputFKGL	-0.36 -0.37
	BLEU 1gram	0.27	0.33 TERp NumEr	-0.53 -0.49 NBSourcePunct	-0.34 -0.18
	WordsInCommon	0.27	0.30 TERp Del	-0.50 -0.52 TypeTokenRatio	-0.22 -0.04
	TERp Del	-0.27 -0.35 AvgCosineSim	0.44	0.34 AvgConcreteness	0.21	0.32
	NBSourceWords	-0.25 -0.07 AvgLMProbsOutput	0.39	0.36 MaxPosInFreqTable	-0.18	0.03
	AvgCosineSim	0.23	0.25 AvgConcreteness	-0.28 -0.06 MinLMProbsOutput	0.17	0.15
	MinLMProbsOutput	0.11 -0.07 NBSourceWords	-0.28 -0.13 OutputFRE	0.16	0.27

Table 3 :

 3 Pearson correlation with human judgments of elementary metrics ranked by absolute value on training set (15 best metrics for each dimension).

  TABLE 4a presents our two best regressors in validation for each of the dimensions and TA-BLE 4b for classifiers.

	Pearson correlation for regressors (raw scor-
	ing) Combining the features does not bring a
	clear advantage over the elementary metrics ME-
	TEOR and NBOutputSyllablesPerSent. Indeed
	our best models score respectively on grammati-
	cality, meaning preservation and simplicity: 0.33
	(Lasso), 0.58 (Ridge) and 0.49 (Ridge) versus 0.39
	(METEOR), 0.58 (METEOR) and 0.49 (NBOut-
	putSyllablesPerSent).

Table 4 :

 4 QATS leaderboard. Results in bold are our additions to the original leaderboard. We only select the two models that rank highest during cross-validation.

Note that text simplification has also been used as a preprocessing step for other natural language processing tasks such as machine translation[START_REF] Chandrasekar | Motivations and methods for text simplification[END_REF] and semantic role labelling(Vickrey and Koller

, 2008).2 There is no unique way to define the notion of simplicity in this context. Previous works often rely on the intuition of human annotators to evaluate the level of simplicity of a TS system output.

We were not able to find detailed information about the annotation process. In particular, we do not know whether each sentence was annotated only once or whether multiple annotations were produced, followed by an adjudication step.

http://qats2016.github.io/shared.html

https://www.nltk.org/api/nltk. translate.html#nltk.translate.bleu_ score.SmoothingFunction

Acknowledgments

We would like to thank our anonymous reviewers for their insightful comments.


Hugo Gruson

Christine Andraud

Willy Daney De Marcillac

Serge Berthier

Marianne Elias

Doris Gomez

Quantitative characterisation of iridescent colours in biological studies: a novel method using optical theory

Keywords: iridescence, structural colours, thin-film interferences, hummingbirds, Lepidoptera, visual signals

Iridescent colours are colours that change with viewing or illumination geometry. While they are widespread in many living organisms, most evolutionary studies on iridescence do not take into account their full complexity. Few studies try to precisely characterise what makes iridescent colours special: their angular dependency. Yet, it is likely that this angular dependency has biological functions and is therefore submitted to evolutionary pressures. For this reason, evolutionary biologists need a repeatable method to measure iridescent colours as well as variables to precisely quantify the angular dependency. In this study, we use a theoretical approach to propose five variables that allow to fully describe iridescent colours at every angle combination. Based on the results, we propose a new measurement protocol and statistical method to reliably characterise iridescence while minimising the required number of time-consuming measurements. We use hummingbird iridescent feathers and butterfly iridescent wings as test cases to demonstrate the strengths of this new method. We show that our method is precise enough to be potentially used at intraspecific level while being also timeefficient enough to encompass large taxonomic scales.

Most interactions between organisms, whether between different species (interspecific) or different individuals of the same species (intraspecific), involve communication. Communication can have different purposes (e.g. warning, camouflage, display) and use different channels (e.g. olfactory, acoustic, visual) [START_REF] Smith | Animal Signals. Reprint. Oxford series in ecology and evolution[END_REF]. In particular, colour is a specific kind of communication channel that can be produced through two non-mutually exclusive mechanisms: pigmentary colours are generated by the selective absorption of some wavelengths by special molecules called pigments while structural colours are generated by the physical interaction of light with matter, causing dispersion, diffraction or interferences [START_REF] Parker | '515 Million Years of Structural Colour[END_REF].

Among structural colours, iridescent colours change depending on the illumination or observation angle. They can be produced by interferences of light after reflection by a thin-film or multilayer structure, or diffraction on a grating. Iridescent colours are present in many taxa, and particularly widespread among bony fishes (Actinopterygii), insects, as well as some birds (see detailed review in table 3 for studies on each one of these taxa). Iridescent colours seem to be involved in many important biological processes [START_REF] Doucet | Iridescence: A Functional Perspective[END_REF] and their angular dependency is likely under selection to produce complex visual signals [START_REF] Stavenga | Dramatic Colour Changes in a Bird of Paradise Caused by Uniquely Structured Breast Feather Barbules[END_REF][START_REF] Wilts | Sparkling Feather Reflections of a Birdof-Paradise Explained by Finite-Difference Time-Domain Modeling[END_REF][START_REF] Simpson | Two Ways to Display: Male Hummingbirds Show Different Color-Display Tactics Based on Sun Orientation[END_REF][START_REF] Simpson | It's Not Just What You Have, but How You Use It: Solar-Positional and Behavioural Effects on Hummingbird Colour Appearance during Courtship[END_REF]. In some cases, however, angular dependency may be selected against [START_REF] Moyroud | Disorder in Convergent Floral Nanostructures Enhances Signalling to Bees[END_REF]. In all those cases, the study of the evolution of iridescent colours requires a pre-cise quantification of the angular dependency. However, the inherent physical complexity of iridescent colours has hampered the development of quantitative methods to fully describe them in the angle space.

We reviewed all studies that performed reflectance measurements of biological samples with iridescent colours produced by a multilayer or a thin-film structure in table 3. We notice two main trends: (i) many studies measure iridescence at a single fixed angle (first row in table 3). In these studies, authors generally remain cautious and warn they are not attempting to measure angle dependency. However, the multilayer or thin film producing iridescent colours may not be parallel to the sample surface [START_REF] Berthier | Morphological Structure and Optical Properties of the Wings of Morphidae[END_REF][START_REF] Edo | Unique Reflection Property Due to Bumpy Multilayer Structure in Elytra of Rhomborrhina Unicolor[END_REF][START_REF] Pirih | Spatial Reflection Patterns of Iridescent Wings of Male Pierid Butterflies: Curved Scales Reflect at a Wider Angle than Flat Scales[END_REF][START_REF] Wickham | Exaggeration and Suppression of Iridescence: The Evolution of Two-Dimensional Butterfly Structural Colours[END_REF][START_REF] Parnell | Wing Scale Ultrastructure Underlying Convergent and Divergent Iridescent Colours in Mimetic Heliconius Butterflies[END_REF], and the angle between them and the sample surface may vary between species or even between individuals of the same species [START_REF] Madsen | Morphology and Ornamentation in Male Magnificent Frigatebirds: Variation with Age Class and Mating Status[END_REF]. Hence, even though the angle of the measuring optical fibres relative to the macroscopic is constant, the angle relative to the structures is not. This jeopardises any biological interpretation of differences between samples because the effects of many different parameters are intertwined. (ii) Other studies take measurements at multiple angles but few attempt to precisely quantify angle dependency ('Literature review' folder in ESM). Even when angle dependency is quantified, variables never stem from a theoretical approach, which leads to a large diversity of custom variables for each author. This heterogeneity in the methhugo.gruson@normalesup.org ods, variable naming, and sign conventions have likely hindered the spread of new concepts and results among researchers working on iridescence in living organisms.

Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF] and Meadows et al. [START_REF] Meadows | Quantifying Iridescent Coloration in Animals: A Method for Improving Repeatability[END_REF] started to address this heterogeneity in measurement methods and advocated for the use of a goniometer to reliably measure colour in a controlled angle configuration. However, they did not propose a detailed protocol or statistical tools to study angular dependency. Here, we use the optical laws that govern iridescence to propose a set of parameters to characterise angle dependency of brightness, hue and saturation of iridescent colours. Next, we confirm the validity of these equations for complex biological structures using two highly different groups of organisms well-known for their iridescent colours: Trochilidae (hummingbirds) and Lepidoptera (i. e., butterflies and moths), including the iconic Morpho butterflies that harbour large wings with bright iridescent blue colours. The standard framework we propose here makes iridescent colours comparable across taxa and across studies, opening up new perspectives in the study of their biological functions.

Model

Choice of colour variables

Hue H R max Brightness B Saturation S Rmax 2 Wavelength (λ) Reflectance (R) Figure 1.
Graphical representation of the variables we used for hue H (wavelength at peak reflectance R max ; called H 1 in Montgomerie [START_REF] Montgomerie | Analyzing Colors[END_REF]), brightness B (average of reflectance over the wavelength range of interest; B 2 in [START_REF] Montgomerie | Analyzing Colors[END_REF]) and saturation S (full width at half maximum: no equivalent in [START_REF] Montgomerie | Analyzing Colors[END_REF]).

Since we want to produce a general method that does not depend on any specific vision system, we use variables directly derived from spectra, without computing vision models. We define brightness B as the average reflectance over a range between the minimal (λ min ) and maximal (λ max ) wavelengths (B 2 in Montgomerie [START_REF] Montgomerie | Analyzing Colors[END_REF]), saturation S as the full width at half maximum reflectance and hue H as the wavelength at which reflectance is maximal (H 1 in Montgomerie [START_REF] Montgomerie | Analyzing Colors[END_REF]). These three variables are represented in fig. 1 and are the most common measures of brightness, hue and saturation in studies about iridescence (see literature review in ESM).

Assumptions and equations

Our method relies on three assumptions that greatly simplify the equations for brightness, hue and saturation in the angle space. See Appendix 1 for mathematical proofs of the equations and the role of each one of these assumptions:

1. Small angles (≤ 30°). Outside of this range, the signal due to iridescence is often very low and all that remains is the effect of the underlying pigments, which can be measured through traditional methods. For all thin films, and in some multilayers (depending on chemical composition), it is possible to consider angles up to 45°, as illustrated in ESM. This may help producing more repeatable parameter estimates. For instance, a 45°angle can correspond to a viewer standing next to the viewed iridescent patch illuminated from above. Many previous studies have in this way mimicked the position of the bird relative to the sun in their measurements [START_REF] Simpson | It's Not Just What You Have, but How You Use It: Solar-Positional and Behavioural Effects on Hummingbird Colour Appearance during Courtship[END_REF][START_REF] Madsen | Morphology and Ornamentation in Male Magnificent Frigatebirds: Variation with Age Class and Mating Status[END_REF][START_REF] Meadows | Quantifying Iridescent Coloration in Animals: A Method for Improving Repeatability[END_REF][START_REF] Dakin | Eye for an Eyespot: How Iridescent Plumage Ocelli Influence Peacock Mating Success[END_REF][START_REF] Pérez I De Lanuza | Now You See Me, Now You Don't: Iridescence Increases the Efficacy of Lizard Chromatic Signals[END_REF][START_REF] Waldron | Experimental Evidence Suggests That Specular Reflectance and Glossy Appearance Help Amplify Warning Signals[END_REF].

2. The orientation of the layers within the multilayer structure is affected by Gaussian noise. Many developmental processes are controlled by a large array of independent factors of small effect each, causing subsequent errors to often be Gaussian (due to the central limit theorem). This assumption is also empirically supported by the results of Gur et al. [START_REF] Gur | The Mechanism of Color Change in the Neon Tetra Fish: A Light-Induced Tunable Photonic Crystal Array[END_REF] who looked at the orientation of guanine crystals in neon tetra fishes (Paracheirodon innesi) using wide-angle Xray scattering (WAXS). Fitting a Cauchy distribution (fatter tail distribution) instead of a Gaussian distribution yields similar values of parameter estimates. For simplicity, we here only present the results with Gaussian noise.

3. Multilayers are ideal, i. e. the optical thickness (layer thickness times optical index) of each layer is constant:

n 1 e 1 = n 2 e 2 .
This is a common assumption [START_REF] Berthier | Morphological Structure and Optical Properties of the Wings of Morphidae[END_REF][START_REF] Land | A Multilayer Interference Reflector in the Eye of the Scallop, Pecten Maximus[END_REF][START_REF] Brink | Thin-Film Biological Reflectors: Optical Characterization of the Chrysiridia Croesus Moth[END_REF][START_REF] Chae | Spectral Patterns of the Iridescence in the Males of Sapphirina (Copepoda: Poecilostomatoida)[END_REF][START_REF] Mäthger | Rapid Colour Changes in Multilayer Reflecting Stripes in the Paradise Whiptail, Pentapodus Paradiseus[END_REF][START_REF] Kinoshita | Structural Colors in Nature: The Role of Regularity and Irregularity in the Structure[END_REF][START_REF] Stavenga | Polarized Iridescence of the Multilayered Elytra of the Japanese Jewel Beetle, Chrysochroa Fulgidissima[END_REF][START_REF] Chandler | Structural Colour in Chondrus Crispus[END_REF][START_REF] Brink | Iridescent Colors on Seashells: An Optical and Structural Investigation of Helcion Pruinosus[END_REF] which is thought to be valid for most animal reflectors [START_REF] Land | The Physics and Biology of Animal Reflectors[END_REF] because it produces the brightest and most saturated signals with a minimal number of layers (but see Schultz and Rankin [START_REF] Schultz | The Ultrastructure of the Epicuticular Interference Reflectors of Tiger Beetles (Cicindela)[END_REF] and Parker et al. [START_REF] Parker | Multilayer Reflectors in Animals Using Green and Gold Beetles as Contrasting Examples[END_REF] for beetles, Kinoshita et al. [START_REF] Kinoshita | Physics of Structural Colors[END_REF] for neon tetra).

This set of assumptions allows us to formally derive simple analytic expressions of brightness B, hue H and saturation S (fig. 1) in the angle space (Φ inc , Φ col ). All variables used in this study with their notations and their possible values are listed in table 1 and illustrated whenever possible on fig. 2.

B(Φ inc , Φ col ) = B max exp - ( Φ inc -Φ col 2 -t ) 2 2γ 2 B ( 8 bis
) Hereafter, we focus on brightness B and hue H because saturation S is constant no matter the angle configuration. The brightness B(Φ inc , Φ col ) in the angle space is entirely defined by three parameters: B max , t and γ B . The tilt t is the angle between the multilayer structure and the sample surface (as illustrated in fig. 2). B max is the maximum reflectance produced by the multilayer of thin-film structure, reached when the fibres are placed in a symmetrical configuration relative to the normal of the multilayer. γ B is the parameter quantifying the disorder in the alignment of the multilayer structure. This disorder in the structure results in a reflected signal that is not purely specular but instead contains a diffuse component, meaning it can be seen at multiple angle configurations. For this reason, from a macroscopic point of view, γ B is correlated with the angular dependency of brightness. Earlier studies used a binary classification of iridescent colours depending on the angle range at which the colour was visible ('diffuse/directional' in Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF], 'wide-angle/flashing' in Huxley [START_REF] Huxley | The Basis of Structural Colour Variation in Two Species of Papilio[END_REF], 'limited view' of Vukusic et al. [START_REF] Vukusic | Limited-View Iridescence in the Butterfly Ancyluris Meliboeus[END_REF]). This classification is positively correlated with 1 γ B . The hue H(Φ inc , Φ col ) in the angle space is defined by two parameters: H max which is the hue at coincident geometry (when using a bifurcated probe for example) and γ H is the angular dependency of hue.

H(Φ inc , Φ col ) = H max cos(γ H Φ inc + Φ col 2 ) (18 bis) S(Φ inc , Φ col ) = S max (1)
The variations of brightness and hue in the angle space, according to eq. ( 8) and eq. ( 18) respectively are represented in fig. 3.

Angle and notation conventions

In the rest of this study, we measure the incoming light ray angles (θ i and Φ inc ) counter-clockwise and the outgoing light ray angles (θ r and Φ col ) clockwise. For both incoming and outgoing angles, the origin is the normal to the structures (θ i and θ r ) or the normal to the sample (Φ inc and Φ col ). These conventions are represented on fig. 2 where the direction of the arrows on angles represent the positive direction. The tilt t corresponds to the angle between the multilayer and the surface of the sample and is defined as t = Φ inc -θ i = θ r -Φ col (see Appendix 1 for more details about t). In other words, t is positive when the multilayer is tilted towards the illumination and negative otherwise (i.e. t is measured clockwise).

Methods

Study system: hummingbirds and butterflies

We used hummingbirds and butterflies (more precisely some Morpho and Papilio species) as study systems. Hummingbirds make an ideal example to test our framework for numerous reasons. First, they belong to a speciose family where all species are iridescent [START_REF] Hoyo | Handbook of the Birds of the World Alive[END_REF], which allows us to work on a large number of species that diverged fairly recently [START_REF] Mcguire | Molecular Phylogenetics and the Diversification of Hummingbirds[END_REF]. Upon visual examination, they display highly different types of iridescent colours, with either 'diffuse' (usually on dorsal patches) 'directional' (usually on facial or ventral patches) iridescence (sensu Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF]). In addition, many species have highly tilted multilayers, providing a good test case to estimate the tilt t [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF][START_REF] Meadows | Quantifying Iridescent Coloration in Animals: A Method for Improving Repeatability[END_REF]. Finally, most species are available in large numbers in museum collections. We obtained the authorisation from the Muséum National d'Histoire Naturelle to carefully cut feathers using surgical scissors. We selected one male from 36 species, evenly distributed across the phylogeny, from which we took feathers on two patches, one diffuse and one directional (sensu Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF]).

Because the exclusive use of hummingbirds as a test taxon for a new method has been criticised in previous studies [START_REF] Van Wijk | A Reliable Technique to Quantify the Individual Variability of Iridescent Coloration in Birds[END_REF], we also test our method on a very different group: butterflies. Butterflies are phylogenetically distant from birds and have different structures producing iridescence. For these reasons, the fact our method works in both taxa is a compelling argument for its universality. We used 17 butterfly species known to have multilayer structures [START_REF] Plattner | Optical Properties of the Scales of Morpho Rhetenor Butterflies: Theoretical and Experimental Investigation of the Back-Scattering of Light in the Visible Spectrum[END_REF][START_REF] Berthier | Iridescences: The Physical Colors of Insects[END_REF]. The full list of species we used for our measurements is available in ESM, for both hummingbirds and butterflies.

The method presented is also valid for whole specimens (whole birds instead of plucked feathers for example). We nonetheless opted for the use of single feathers to maximise repeatability. Indeed, the precision of the goniometer measurements relies on the fact that the sample is precisely located at the centre of rotation of both fibres, which is more difficult to ensure for whole specimens.

Symbol

Range Meaning

θ i [ -π 2 ; π 2 ] Incident light angle relative to the multilayer θ r [ -π 2 ; π 2 ]
Reflected light angle relative to the multilayer

θ 1 [ 0; π 2 ]
Angle between the incident ray and the interface between layers 1 and 2

θ 2 [ 0; π 2 ]
Angle between the transmitted ray and the interface between layers 1 and 2 Angle between the incident ray and the interface between layers

Φ inc [ -π 2 ; π 2 ]
Angle between incidence fibre and sample surface (measured counterclockwise)

Φ col [ -π 2 ; π 2 ]
Angle between collection fibre and sample surface (measured clockwise) cst R Used to denote a constant whose value is not important for the calculations Table 1. List of parameters used in this study, with their domains of definition and their meanings.

Reflectance measurements

We measured reflectance at various angles using a purpose-built goniometer, following the recommendations of Meadows et al. [START_REF] Meadows | Quantifying Iridescent Coloration in Animals: A Method for Improving Repeatability[END_REF]. The light emitted by a Xenon lamp (300 W) over the 300 nm to 700 nm range of wavelengths to which birds are sensitive [START_REF] Chen | Four Spectral Classes of Cone in the Retinas of Birds[END_REF] was brought to the sample through an illuminating UV-VIS optical fibre collimated to get a 1 mm light spot at normal illumination. Light reflected by the sample was then collected by a second identical collimated optical fibre and conducted toward an Oceanoptics USB4000 spectrophotometer. This setup allows for a precise independent rotation of the illumination and the collection fibres, necessary for the measurements of iridescent colours. Our previous mathematical exploration revealed that hue is constant along the Φ inc + Φ col = cst line ('constant span') and brightness along the Φ inc -Φ col = cst line ('constant angle bisector'), as illustrated in fig. 3. We thus only need to take measurements in two orthogonal directions: in the direction Φ inc -Φ col = cst to quantify hue variation and in the direction Φ inc + Φ col = cst to quantify brightness variations. This will allow us to infer all parameters controlling hue and brightness, and therefore to potentially compute all values of hue and brightness in the entire angle space (Φ inc , Φ col ).

The shape and size of the light spot on the sample depends on the position of the illuminating fibre relatively to the sample. As the angle of illumination θ i increases, the light spot becomes more and more elongated, according to a θ i cosine function. This means the amount of light received by the spectrometer decreases when θ i increases, independently of sample characteristics. This can also be empirically observed by taking measurements of the white reference (which is a Lambertian surface, i. e., reflectance does not depend on the angle) at different angles. To control for this, we took white reference measurements at several angle configurations (detailed in the protocol below). The white stand-ard for this study was an Avantes reference tile WS-2. Because this is a diffuse (Lambertian) white reference and because some iridescent colours are very directional (i.e., all reflected light is focused in a single direction), it is expected to sometimes get values of brightness that can be over 100 %.

The detailed protocol we used for our measurements is similar to Waldron et al. [START_REF] Waldron | Experimental Evidence Suggests That Specular Reflectance and Glossy Appearance Help Amplify Warning Signals[END_REF] and inspired from Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF] and Meadows et al. [START_REF] Meadows | Quantifying Iridescent Coloration in Animals: A Method for Improving Repeatability[END_REF]. A detailed walkthrough of the measurement protocol is presented in box 1.

We repeated each measurement twice, on different days, by two different experimenters for hummingbirds and butterflies. We performed statistical analyses after the completion of the measurement session to prevent experimenter bias.

Statistical analyses

As explained in the previous section, the angle configuration changes the shape of the light spot and thus the total possible amount of light collected by the collection fibre. To address this issue, we first pre-processed spectra to normalise count data using the appropriate reference white spectrum (script available in ESM). Resulting csv files were then imported in pavo R package [START_REF] Maia | Pavo: An R Package for the Analysis, Visualization and Organization of Spectral Data[END_REF]. Hue values were discarded (i.e. converted to NA) when brightness was lower than 8.5 % because hue is not defined for black colours.

Iridescence parameters can be estimated using various methods, including least squares optimisation and Bayesian non linear regression. We used a least squares optimisation as it is more common in biological sciences. We tested the Bayesian approach as well but it returned similar results and it is therefore not presented here.

We used two indices to estimate the variability of the parameters resulting from our method: (i) Relative standard deviation (RSD, also called coefficient of variation or This position depends on the tilt t of the multilayer and is therefore different for every sample. Once this is done, this means the angle bisector of the two fibres is close to the normal to the multilayer structure (red line in fig. 2).

2. While keeping the same angle bisector, take measurements at different angular spans (orange line Φ inc -Φ col = t in fig. 3d). These measurements will be used to estimate hue parameters. To have a sample size large enough for reliable estimation and to stay at small angles, we recommend measurements at (Φ inc , Φ col ) ∈ {(t + 5°, t + 5°), (t + 10°, t + 10°), (t + 15°, t + 15°), (t + 20°, t + 20°), (t + 25°, t + 25°)}.

3. Take measurements while keeping the angular span between the two fibres constant (for example Φ col -Φ inc = 20°) and moving the angle bisector (if you cannot do this, because for example, one of your fibres is not mobile, see section 6.2). This will be used to estimate parameters related to brightness. We recommend 3 measurements on each side of the supposed normal to the multilayer structure (7 measurements in total) and a span of 20°: 4. Take white reference measurements with the same angular spans as before but using the normal to the goniometer as angle bisector (same measurements as in 2 but with t = 0°). If you have followed our advice for measurements, you should now take white measurements at (Φ inc , Φ col ) ∈ {(5°, 5°), (10°, 10°), (15°, 15°), (20°, 20°), (25°, 25°), (30°, 30°)}.

(Φ inc , Φ col ) ∈ {(t -
5. Take white reference measurements with a constant span but various angle bisectors (same measurements as in 3 but with t = 0°). If you have followed our advice of 3 measurements on each side to the supposed normal to the multilayer structure and a span of 20°, you should now take white measurement at

(Φ inc , Φ col ) ∈ {(-5°, 25°), (0°, 20°), (5°, 15°), (10°, 10°), (15°, 5°), (20°, 0°), (25°, -5°)}
CV) as the standard deviation divided by the absolute value of the mean. (Absolute) standard deviation SD is a common measure of the noise in a dataset. RSD is a way to quantify the signal-to-noise ratio. Because it is normalised by the mean value of the parameter, it is dimensionless and can be compared between parameters.

It represents the precision of the experimental and statistical framework and does not depend on the sample population.(ii) Repeatability as the intra-class coefficient (ICC) computed with the rptR package [START_REF] Nakagawa | Repeatability for Gaussian and Non-Gaussian Data: A Practical Guide for Biologists[END_REF]. ICC assesses whether the method allows to discriminate individual samples among the population by comparing intra-and inter-samples standard deviation. ICC is therefore highly dependent on the sample population and on the biological question. RSD and ICC complement each other. A very precise method can still lead to non-repeatable measurements is there is no variability in the population. Conversely, a coarse method can work well enough to discriminate between samples and be repeatable if the variability between samples is high.

Results and discussion

Spectra from measurement along the 'constant span' (Φ inc + Φ col = 20°) and 'constant angle bisector' (Φ inc -Φ col = cst) lines after correction by the appropriate white reference are displayed in fig. 4 for the iridescent blue of the breast of the hummingbird Heliomaster furcifer. We also show values of hue H and brightness B along these two measurement lines as well as the result from parameters estimation.

Relative error and repeatability

Variability and repeatability results are summarised in table 2. We find low values of RSD for hue-related variables for both hummingbirds and butterflies, indicating that our framework provides precise estimations of parameters. For brightness-related parameters, RSD is higher, as it is usually the case, even for non-iridescent colours [START_REF] Evans | Age, Sex and Beauty: Methodological Dependence of Age-and Sex-Dichromatism in the Great Tit Parus Major[END_REF][START_REF] Midamegbe | Female Blue Tits with Brighter Yellow Chests Transfer More Carotenoids to Their Eggs after an Immune Challenge[END_REF][START_REF] Charmantier | Colour Ornamentation in the Blue Tit: Quantitative Genetic (Co)Variances across Sexes[END_REF]. Despite relatively high RSD, all values for brightness remain repeatable, expected tilt t for butterflies because of a low inter-species variability, as demonstrated by the low value of standard deviation SD.

Correlation between parameters

Correlation between B max and γ B

Madsen et al. [START_REF] Madsen | Morphology and Ornamentation in Male Magnificent Frigatebirds: Variation with Age Class and Mating Status[END_REF] noticed a negative relationship between brightness angular dependency and maximum brightness. From an evolutionary point of view, this means there is a trade-off between the signal brightness at a given angle and the range of angle at which it is not black (i.e. directionality sensu Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF]).

This correlation can also be proved theoretically. Indeed, the total energy of light that is reflected by the sample cannot exceed the received light energy. In other words, if absorption is similar across samples, the total

(Φ inc + Φ col ) 2
Hue H (nm) (f) . Spectra (top row) and corresponding values of brightness (middle row) and hue (bottom row) at different angle configurations for the breast patch of the hummingbird Heliomaster furcifer along the Φ inc -Φ col = cst (left column; data points with round shape) and Φ inc + Φ col = cst (right column; data point with square shape) lines. Colours correspond to the conversion of the spectra in human vision using the CIE10 visual system. As expected, brightness is constant when Φ inc -Φ col = cst and has a Gaussian shape when Φ inc + Φ col = cst. Conversely, hue has a cosine shape wen Φ inc -Φ col = cst and is constant when brightness reflected in all directions is constant across samples:

Φ inc +Φ col = cst.
∫ ∫ B(Φ inc , Φ col ) dΦ inc dΦ col = cst (2)
The value of this double integral is known (B(Φ inc , Φ col ) is a bivariate Gaussian function) and when we compute it, we find:

B max √ 2πγ 2 B = cst (3)
B max ∝ 1 γ B ( 4
)
We indeed find a positive correlation between B max and 1 γ B in the empirical data (F = 147.0742, df = 1, p < 0.0001), illustrated in fig. 5. We also notice an effect of the taxon (butterflies or hummingbirds) on the slope of the correlation (F (1) = 8.3198, p = 0.0057). Because the link between B max and 1 γ B was proven when ignoring absorption (eq. ( 4)), this may suggest that absorption is higher in hummingbirds than in butterflies.

Correlation between angular dependency for hue γ H and and other parameters

Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF] found that γ H and γ B are negatively correlated among 15 bird species from different families. We do not find support for such correlation for either the hummingbirds or the butterflies (F (1) = 3.1994, p = 0.074, figure in ESM). Additionally, as discussed later in section 6.3.2, many studies use variables that are correlated to H max to quantify hue angular dependence. On the contrary, we find that the parameters used in our method, H max and γ H are not correlated (F (1) = 0.5167, p = 0.47, figure in ESM).

Conclusion

Using both a theoretical and an experimental approach we find that hue and brightness can be easily characterised for all angle configurations using a set of 5 parameters (H max and γ H for hue; B max , t and γ B for brightness). Additionally, we show that a relatively small number of measurements is sufficient to reliably estimate these parameter values. This is made possible by the fact that hue is constant when the angular span between the two fibres remains constant (Φ inc -Φ col = cst), and that brightness is constant for small angles as long as the angle bisector remains in the same position (Φ inc + Φ col = cst) (as illustrated in fig. 3 and fig. 4). These properties have been previously noticed empirically for hue H 1 by Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF] on 15 bird species sampled from different families and Meadows et al. [START_REF] Meadows | Quantifying Iridescent Coloration in Animals: A Method for Improving Repeatability[END_REF] on Calypte anna. Without being formalized, it had been illustrated for brightness in Eliason and Shawkey [START_REF] Eliason | A Photonic Heterostructure Produces Diverse Iridescent Colours in Duck Wing Patches[END_REF] and Stavenga et al. [START_REF] Stavenga | Magnificent Magpie Colours by Feathers with Layers of Hollow Melanosomes[END_REF] on B 3 , Stavenga et al. [START_REF] Stavenga | Structural Coloured Feathers of Mallards Act by Simple Multilayer Photonics[END_REF] for B 1 .

Our contribution unlocks new perspectives for studies on iridescent colours, such as the evolution of complex visual signals leveraging angular dependency properties of iridescent colours.

The proofs for the equation in this article are based on the multilayer theory. However, it is possible that parts of it may work for iridescence from diffraction gratings. Future studies should aim at integrating iridescence from diffraction into our framework. This would allow for a standard set of variables to describe iridescence, no matter its physical origin. Further investigation is also required to assess whether it is possible to relax some of the assumptions made in the manuscript under certain conditions. as well as V. Debat for lending us Morpho specimens to explore Morpho iridescence. Finally, we thank the Living Light conference organisers for putting together this special issue on structural colours in living organisms.

5 Appendix 1: mathematical proof of the equations 5.1 Brightness B in the angle space (Φ inc , Φ col )

For a perfectly regular multilayer, all the reflected signal is focused in the specular direction, at an angle θ r equal to the incident angle θ i . The brightness B is proportional to the reflected signal intensity, meaning:

B(θ i , θ r ) = { B(θ i ) if θ i = θ r 0 if θ i ̸ = θ r ( 5
)
where B(θ i ) is defined by the Fresnel factor in the case of a thin-film structure (equation and R code to compute the Fresnel factor available in ESM). However, because we are dealing with small angles (hypothesis 1), we can approximate B(θ i ) to a constant B max (as illustrated in ESM):

B(θ i , θ r ) ≈ { B max if θ i = θ r 0 if θ i ̸ = θ r (6)
But because biological structures are not entirely flat, and because the different layers of the multilayer structure are not perfectly aligned, there is also some amount of light reflected outside of the specular reflection (often referred as diffuse reflection). We thus assume a Gaussian decay of the brightness B around the specular position θ i = θ r (assumption 2), controlled by a parameter γ B related to the disorder of the multilayer:

B(θ i , θ r ) ≈ B max exp - ( θi-θr 2 ) 2 2γ 2 B (7)
In the case of an perfectly regular multilayer with no disorder, we have γ B = 0 and we find eq. ( 6). Conversely, if γ B = +∞, the value of brightness is the same for all angle configurations, which means we are dealing with a Lambertian surface.

Additionally, the multilayer structure is not always parallel to the sample surface. It is the case for example for hummingbirds included in this study, as well as for Morpho butterflies [START_REF] Berthier | Morphological Structure and Optical Properties of the Wings of Morphidae[END_REF],for the rainbow stag beetle, Phalacrognathus muelleri, structures described in Edo et al. [START_REF] Edo | Unique Reflection Property Due to Bumpy Multilayer Structure in Elytra of Rhomborrhina Unicolor[END_REF], 6 pierid butterflies in Pirih et al. [START_REF] Pirih | Spatial Reflection Patterns of Iridescent Wings of Male Pierid Butterflies: Curved Scales Reflect at a Wider Angle than Flat Scales[END_REF], 10 species of butterflies in Wickham et al. [START_REF] Wickham | Exaggeration and Suppression of Iridescence: The Evolution of Two-Dimensional Butterfly Structural Colours[END_REF], and for 6 species of Heliconius butterflies in Parnell et al. [START_REF] Parnell | Wing Scale Ultrastructure Underlying Convergent and Divergent Iridescent Colours in Mimetic Heliconius Butterflies[END_REF]). So the illuminating angle Φ inc and the collection Φ col at the macroscopic scale do not necessarily match θ i and θ r (as illustrated in fig. 2). If we denote t the angle between the multilayer surface and the macroscopic sample surface (called tilt hereafter, as in Madsen et al. [START_REF] Madsen | Morphology and Ornamentation in Male Magnificent Frigatebirds: Variation with Age Class and Mating Status[END_REF] and Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF]), we get:

B(Φ inc , Φ col ) ≈ B max exp - ( Φ inc -Φ col 2 -t ) 2 2γ 2 B ( 8
)
Using eq. ( 8), we only have 3 parameters (B max , t and γ B ) to estimate to be able to reconstruct all values of brightness B in the angle space defined by (Φ inc , Φ col ). The resulting brightness in this space in plotted in fig. 3.

Hue H in the angle space (Φ inc , Φ col )

We defined the hue H as the wavelength for which reflectance is maximal. In the context of interferences, it is therefore the wavelength for which reflected light interferes constructively. For a regular multilayer, this happens when:

mH(θ 1 , θ 2 ) = 2(n 1 e 1 cos θ 1 + n 2 e 2 cos θ 2 ) ( 9
)
where m is an integer (interference order), θ 1 is the angle between the incident light ray and the multilayer structure at the interface between layer 1 and 2, θ 2 is the angle between the transmitted ray after going through the first interface between layers 1 and 2 and the multilayer structure, n 1 and n 2 are the optical indices of the layers, and e 1 and e 2 the thickness of the layers. The product n 1 e 1 and n 2 e 2 is often called optical thickness of the layers 1 and 2 (respectively).

The relationship between θ 1 and θ 2 is given by Snell's law:

n 1 sin θ 1 = n 2 sin θ 2 (10)
Because

θ 1 ∈ [0; π 2 ]
, hue H increases when angle θ 1 decreases according to eq. ( 9). This means a maximum value for hue H max is achieved when θ 1 = 0 (in this case θ 2 = 0 as well because of Snell's law; eq. ( 10)):

mH max = 2(n 1 e 1 + n 2 e 2 ) (11)
We can replace n 1 e 1 and n 2 e 2 in eq. ( 9) using eq. ( 11):

mH(θ 1 , θ 2 ) = mH max (cos θ 1 + cos θ 2 ) -2(n 1 e 1 cos θ 2 + n 2 e 2 cos θ 1 ) (12)
By adding eq. ( 12) and eq. ( 9), we get:

2mH(θ 1 , θ 2 ) = mH max (cos θ 1 + cos θ 2 ) + 2(cos θ 1 -cos θ 2 )(n 1 e 1 -n 2 e 2 ) (13)
We consider here the case of an ideal multilayer, meaning that n 1 e 1 = n 2 e 2 (assumption 3). This allows us to simplify eq. ( 13) into:

H(θ 1 , θ 2 ) = H max cos θ 1 + cos θ 2 2 (14)
Because we are working with small angles (assumption 1), Snell's law (eq. ( 10)) can be approximated by:

θ 2 ≈ n 1 n 2 θ 1 (15)
H(θ 1 , θ 2 ) ≈ H max cos θ 1 + cos n1 n2 θ 1 2 (16)
For small angles (assumption 1), this sum of cosine functions can be approximated by a single cosine function with twice the amplitude (numerical proof in ESM):

H(θ 1 , θ 2 ) ≈ H max cos γ H θ 1 (17)
where

γ H ≈ √ 1+ ( n 1 n 2 ) 2 2
(after identification of the coefficients of the second-order Taylor series expansions in eq. ( 16) and eq. ( 17)).

This reasoning is valid for ideal thin film structures and multilayers and tells what happens at the specular position. But as explained in the previous section, biological structures are noisy and there is signal outside the specular position. As previously, if there is signal, this means that there is a multilayer for which the position of the fibres is specular. And in this case, we can apply eq. ( 17) as well.

H(Φ inc , Φ col ) = H max cos ( γ H Φ inc + Φ col 2 ) (18)
We only need two parameters (H max and γ H ) to plot all hue values in the angle space (Φ inc , Φ col ) as in fig. 3. In the case of non-iridescent colours, we have γ H = 0.

Saturation

S in the angle space (Φ inc , Φ col )

Along the 'constant span' direction (Φ inc

+ Φ col = cst)
We know that along the Φ inc + Φ col = cst direction (constant span), hue is constant (as shown in eq. ( 18) and fig. 3b). Using a similar reasoning as in section 5.1, we find that the reflectance R for a wavelength λ at a given angle configuration (Φ inc , Φ col ) is given by:

R(Φ inc , Φ col , λ) = R bisector (λ) exp - ( Φ inc -Φ col 2 -t ) 2 2γ 2 B ( 19
)
This means that reflectance spectra at all angle configurations along the 'constant span' axis (Φ inc + Φ col = cst) can be derived by scaling of the spectrum at another angle configuration.

The saturation S(Φ inc , Φ col ) is defined as the full width at half maximum of the reflectance spectrum R(Φ inc , Φ col , λ). Let us call R the reflectance spectrum at a given angle configuration (Φ pos1 inc , Φ pos1 col ). Then the saturation S at this configuration is:

     S = λ 1 -λ 2 R(λ 1 ) = R(λ 2 ) = Rmax 2 λ 1 > λ 2 (20)
If the reflectance spectrum R ′ at (Φ pos2 inc , Φ pos2 col ) is equal to R scaled by a factor s. Then the saturation S ′ is:

     S ′ = λ ′ 1 -λ ′ 2 R ′ (λ ′ 1 ) = R ′ (λ ′ 2 ) = R ′ max 2 λ ′ 1 > λ ′ 2 (21)
where

     R ′ (λ ′ 1 ) = R(λ ′ 1 ) s R ′ (λ ′ 2 ) = R(λ ′ 2 ) s R ′ max = Rmax s ( 22
)
From this, we find that:

R(λ ′ 2 ) s = R(λ ′ 1 ) s = R max 2s (23) R(λ ′ 2 ) = R(λ ′ 1 ) = R max 2 (24)
This means that λ ′ 1 = λ 1 and λ ′ 2 = λ 2 . In other words, the full width at half max is stable by scaling, which results in the saturation S remaining constant along the Φ inc + Φ col = cst axis (constant span).

Along the 'constant angle bisector' direction (Φ inc -Φ col = cst)

Additionally, along the Φ inc -Φ col = cst axis (constant angle bisector), brightness is constant and only hue changes. This means spectra are translations of one another. The full width at half max is also stable by translation so the saturation S remains constant along Φ inc -Φ col = cst axis (constant angle bisector).

In the general case

All points in the (Φ inc , Φ col ) space can be reached by a combination of moves along the orthogonal 'constant span' (Φ inc + Φ col = cst) and 'constant angle bisector' (Φ inc -Φ col = cst) axes. We just showed the saturation S is constant along these two axes so it is actually constant in the whole (Φ inc , Φ col ) space.

6 Appendix 2: Comparison with other methods

Measurements at fixed angle configuration

The angle t between the multilayer structure and the normal to the surface of the feather (tilt) is highly variable between species of the same family (sd = 19.36°in hummingbirds, as reported in table 2). This is in agreement with Osorio and Ham [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF] who found tilt values t ranging from -20°to 40°. Even if the angle configuration (Φ inc , Φ col ) is constant at the macroscopic scale, the configuration relative to the multilayer structure (θ i , θ r ) may not be constant because of the variation in the tilt t between samples. This means measurements at fixed geometry cannot be compared between samples. For this reason, we warn against measurements of iridescent colours at a fixed angle, even when angular dependency is not studied.

Parameters estimation using constant illumination

Some goniometers only allow for the rotation of the collection fibre while the illumination fibre stays at a fixed position. Measurements realised with a such protocol can still be used with our method but this leads to a loss of statistical power.

If illumination is provided at a fixed angle Φ inc = α:

B(Φ col ) = B max exp - ( α-Φ col 2 -t ) 2 2γ 2 B = B max exp - (Φ col + 2t -α) 2 8γ 2 B ( 25
)
So, B(Φ col ) is still a normal function of Φ col with the same maximum value B max but with parameters t * = 2t -α and γ * B = 2γ B for mean and standard deviation respectively.

Because the estimation of the parameters of a normal function through a regression is more reliable is the standard deviation is low, using anything else than a fixed normal as measurement line, such as a fixed illumination, to measure brightness parameters will result in a loss of statistical power.

Additionally, depending on the exact value of α, it may not be possible to have a fibre configuration where α+Φ col 2

= t but the span between the fibres is still less than 90°(small angles assumption). In this case, data points never reach the maximum B max , which makes parameters estimation very unreliable.

Finally, the new value of the mean t * does not have a direct biological and physical interpretation, as opposed to t which is the tilt of the multilayer of thin-film structure.

For hue, if illumination is at fixed angle α:

H(Φ col ) = H max cos(γ H α 2 + γ H 2 Φ col ) (26)
The equation for hue at fixed illumination has a shape different from its general form depending on the span between the fibres, Φ inc +Φ col

2

. There is a constant term in the cosine function and the new term for hue angular dependency is γ * H = γ H 2 . As we explain in the next section, the estimation of the parameters is more reliable for high values of γ H . For this reason, the parameters estimation at fixed illumination may not be as precise as along the Φ inc + Φ col = cst line.

Link with other variables of angular de-

pendency for hue

Linear regression

Linear regression instead of cosine regression to estimate H max and γ H is common [START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF][START_REF] Rutowski | Diversity in Structural Ultraviolet Coloration among Female Sulphur Butterflies (Coliadinae, Pieridae)[END_REF][START_REF] Eliason | How Hollow Melanosomes Affect Iridescent Colour Production in Birds[END_REF][START_REF] Igic | A Nanostructural Basis for Gloss of Avian Eggshells[END_REF]. Because the curvature of the cosine function in eq. ( 18), defining hue depending on the angular span, is often small, we obtain congruent results using either cosine or linear regression. However, this creates a systematic bias where H max is more overestimated for samples with larger angle dependency γ H . Indeed, a linear regression overestimates more the intercept value as the curvature of the function increases.

Difference between two angle configurations with the same angle bisector

The difference in hue between two angle configurations is sometimes used as a proxy for iridescence [START_REF] Maia | Iridescent Structural Colour Production in Male Blue-Black Grassquit Feather Barbules: The Role of Keratin and Melanin[END_REF]. However, it is problematic because it leads to a very high correlation between hue and iridescence, as reported in Dakin and Montgomerie [START_REF] Dakin | Eye for an Eyespot: How Iridescent Plumage Ocelli Influence Peacock Mating Success[END_REF] (R 2 > 0.95).

We can prove mathematically this linear correlation. Let us focus on the difference between hue H pos1 at a given angle configuration (Φ 1 inc , Φ 1 col ) and hue H max at coincident geometry (i.e. Φ inc +Φ col = θ i +θ r = 0). It follows from eq. ( 18) that defines the hue at any angle configuration that:

H pos1 -H max = H max [ cos ( γ H Φ 1 inc + Φ 1 col 2 ) -1 ] (27)
From this equation, we see that if γ H is constant or display low variability between samples, H pos1 -H max is proportional to H max :

H pos1 -H max ∝ H max ( 28
)
We can apply the same reasoning and prove the difference H pos2 -H max between hue H pos2 at (Φ 2 inc , Φ 2 col ) and H max is proportional to H max :

H pos2 -H max ∝ H max (29)
Thus (doing eq. ( 28) -eq. ( 29)) the difference in hue between any two angle configurations

(Φ 1 inc , Φ 1 col ) and (Φ 2 inc , Φ 2 col ) is proportional to H max H pos1 -H pos2 ∝ H max (30)
This correlation between the two variables characterising hue in the angle space can lead to errors in subsequent statistical inferences. On the opposite and we showed previously, the parameters proposed in this study (H max and γ H ) do not have the same issue.

Link with other variables of angular dependency for brightness

We are providing the following comparison with variables that have previously used in the literature to describe brightness angular dependency. This means that values from previous studies using these variables can still be used in a meta-analysis or a discussion using our new variables B max , t and γ B . We however explain why they are less precise, less versatile and/or more time consuming than those measured under our unified framework.

FWHM and angular breadth

We have shown brightness is a Gaussian function of standard deviation γ B along the line of 'constant span' (Φ inc + Φ col = cst direction). Many studies previously characterised angular dependency in this direction using the full width at half max (hereafter FWHM) [START_REF] Pirih | Spatial Reflection Patterns of Iridescent Wings of Male Pierid Butterflies: Curved Scales Reflect at a Wider Angle than Flat Scales[END_REF][START_REF] Wickham | Exaggeration and Suppression of Iridescence: The Evolution of Two-Dimensional Butterfly Structural Colours[END_REF][START_REF] Osorio | Spectral Reflectance and Directional Properties of Structural Coloration in Bird Plumage[END_REF][START_REF] Stavenga | Polarized Iridescence of the Multilayered Elytra of the Japanese Jewel Beetle, Chrysochroa Fulgidissima[END_REF][START_REF] Brink | Structural Colours from the Feathers of the Bird Bostrychia Hagedash[END_REF]. For a Gaussian function, there is an easy link between standard deviation and FWHM:

FWHM = 2γ * B √ 2 ln 2 = 4γ B √ 2 ln 2 ≈ 4.71γ B (31)
Similarly, some studies use what they call angular breadth [START_REF] Van Wijk | A Reliable Technique to Quantify the Individual Variability of Iridescent Coloration in Birds[END_REF][START_REF] Kemp | Stressmediated Covariance between Nanostructural Architecture and Ultraviolet Butterfly Coloration[END_REF][START_REF] Rutowski | Iridescent Ultraviolet Signal in the Orange Sulphur Butterfly (Colias Eurytheme): Spatial, Temporal and Spectral Properties[END_REF][START_REF] Kemp | Condition Dependence, Quantitative Genetics, and the Potential Signal Content of Iridescent Ultraviolet Butterfly Coloration[END_REF][START_REF] Kemp | Potential Direct Fitness Consequences of Ornament-Based Mate Choice in a Butterfly[END_REF][START_REF] Kemp | Female Mating Biases for Bright Ultraviolet Iridescence in the Butterfly Eurema Hecabe (Pieridae)[END_REF][START_REF] White | Signal Design and Courtship Presentation Coincide for Highly Biased Delivery of an Iridescent Butterfly Mating Signal[END_REF][START_REF] Van Wijk | The Influence of Iridescent Coloration Directionality on Male Tree Swallows' Reproductive Success at Different Breeding Densities[END_REF], which they define as the range of angle where brightness is higher than 3 % of its maximum (threshold at 10 % for White et al. [START_REF] Parker | Multilayer Reflectors in Animals Using Green and Gold Beetles as Contrasting Examples[END_REF] We see that these variables are proportional to γ B in theory. However because they are computed from raw data, without any pre-processing or curve fitting, they are more sensitive to noise.

Hunter's specular gloss and integrating sphere

Multiple studies [START_REF] Igic | A Nanostructural Basis for Gloss of Avian Eggshells[END_REF][START_REF] Maia | What Makes a Feather Shine? A Nanostructural Basis for Glossy Black Colours in Feathers[END_REF][START_REF] Iskandar | Morphological Basis of Glossy Red Plumage Colours[END_REF]] use Hunter's gloss [START_REF] Hunter | Methods of Determining Gloss[END_REF], defined by the ratio of specular to diffuse reflectance. This method is convenient because it can easily be achieved using an integrating sphere to capture the needed spectra in two measurements only (one at specular position without the sphere and one with the sphere to capture diffuse and specular reflectance). This is equivalent of keeping the illumination at a fixed angle and measuring reflectance at all collection angles. We already know the brightness at the specular position is B max . The diffuse reflection is the integral on all angle configurations of the brightness. Hence Hunter's specular gloss G using the notation defined in this study is :

G = B max ∫∫ B(Φ inc , Φ col ) dΦ inc dΦ col (33)
The integral of brightness for every angle configurations is B max γ * B √ 2π (integral of the normal function with maximum B max and standard deviation γ * B ), which gives:

G = 1 γ * B √ 2π = 1 2γ B √ 2π (34)
However, this is assuming the measurement of B max was actually done at the normal to the multilayer

Φ inc +Φ col 2 = t.
But there is no way to know whether it is the case without doing several goniometer measurements with different normal positions. Once this is done, γ B can be estimated without doing additional integrating sphere measurements.

Difference/Quotient between max and another position with the same span

Some studies [START_REF] Van Wijk | A Reliable Technique to Quantify the Individual Variability of Iridescent Coloration in Birds[END_REF][START_REF] Meadows | Dietary Protein Level Affects Iridescent Coloration in Anna's Hummingbirds, Calypte Anna[END_REF][START_REF] Loyau | Iridescent Structurally Based Coloration of Eyespots Correlates with Mating Success in the Peacock[END_REF] use the difference or the quotient between the brightness at the fibre position where it is maximum and another position. With this approach, they find t and B max .

The difference or the quotient between these two positions can easily be linked to γ B because we know that B(Φ inc , Φ col ) is a normal function of parameters t and γ B .

However, this is very sensitive to noise and measurement error because B max and t are estimated with only one data point and γ B (or its equivalent variable) with only two data points.

Appendix 3: structural colours with pigmentary component

The framework we presented here focuses on purely structural iridescent colours. However many colours integrate both pigmentary and structural components [START_REF] Prum | Anatomy, Physics, and Evolution of Structural Colors[END_REF][START_REF] D'alba | Relative Contributions of Pigments and Biophotonic Nanostructures to Natural Color Production: A Case Study in Budgerigar (Melopsittacus Undulatus) Feathers[END_REF]. If there is a pigmentary component, it adds constant term B pigment to brightness B:

B(Φ inc , Φ col ) = B irid + B pigment (35) B(Φ inc , Φ col ) = B max exp - ( Φ inc -Φ col 2 -t ) 2 2γ 2 B + B pigment (36)
This can easily be investigated using our protocol and statistical framework. The only difference is that 4 parameters (B max , t, γ B and B pigment ) instead of 3 need to be estimated by running a non-linear regression on eq. ( 36) instead of eq. [START_REF] Moyroud | Disorder in Convergent Floral Nanostructures Enhances Signalling to Bees[END_REF].

There are cases where the structural and pigmentary components of colour act on very different regions of the light spectrum. This happens for example in Colias eurytheme [START_REF] Rutowski | Pterin Pigments Amplify Iridescent Ultraviolet Signal in Males of the Orange Sulphur Butterfly, Colias Eurytheme[END_REF], where iridescence is restricted to the UV region while the visible region colour is caused by pigments. In this case, our method can be applied directly by restricting the studied wavelength range to the region of interest (this option is available in the code provided in ESM).

  Angle of the illumination fibre inc Angle of the collection fibre col (a) Angle of the illumination fibre inc Angle of the collection fibre col (b) Angle of the illumination fibre inc Angle of the collection fibre col (c) Angle of the illumination fibre inc Angle of the collection fibre col s t a n t a n g l e b i s e c t o r

Figure 3 .

 3 Figure 3. Colour variables ((a): brightness; (b): hue; (c) and (d): hue & brightness) of an iridescent multilayer (with tilt t ̸ = 0) in the angle space relative to the sample surface (Φ inc , Φ col ). The colour lines in (d) indicate alternative bases: the angle space relative to the multilayer structure (θ i , θ r ) in blue and (Φ inc +Φ col = 0, Φ inc -Φ col = t) in orange and illustrates the terms 'constant illumination', 'constant collection', 'constant angle bisector' and 'constant span' used in table 3 and throughout this article.

Box 1 :

 1 Measurement protocol1. Move one of the two fibres of the goniometer to find the position where you get a signal of maximal intensity.

Figure 4

 4 Figure 4. Spectra (top row) and corresponding values of brightness (middle row) and hue (bottom row) at different angle configurations for the breast patch of the hummingbird Heliomaster furcifer along the Φ inc -Φ col = cst (left column; data points with round shape) and Φ inc + Φ col = cst (right column; data point with square shape) lines. Colours correspond to the conversion of the spectra in human vision using the CIE10 visual system. As expected, brightness is constant when Φ inc -Φ col = cst and has a Gaussian shape when Φ inc + Φ col = cst. Conversely, hue has a cosine shape wen Φ inc -Φ col = cst and is constant when Φ inc +Φ col = cst. The red lines correspond to the fit of the functions after parameters estimation, with the values of the parameters. The R script to produce this figure is available in ESM.

Figure 5 .

 5 Figure 5. Correlation between B max and directionality 1/γ B . The dots are the data points. The lines show the result of the linear model.

  [START_REF] White | Signal Design and Courtship Presentation Coincide for Highly Biased Delivery of an Iridescent Butterfly Mating Signal[END_REF]): ang. breadth = 2γ * B √ 4 ln 10 -2 ln 3 = 4γ B √ 4 ln 10 -2 ln 3 ≈ 10.59γ B

  40°) and incoming and reflected light rays relative to the multilayer structure (with angles θ i and θ r respectively) and relative to the sample surface (with angle Φ inc and Φ col respectively). There is a relationship involving the tilt t between angles relative to the multilayer structure (θ i and θ r ) and angles relative to the sample surface (Φ inc and Φ col ): θ i = Φ inc -t and θ r = Φ col +t. The positive direction for each angle is figured by an arrowhead. The multilayer is composed of an alternance of two layers characterised by the optical indices n 1 and n 2 and their thicknesses e 1 and e 2 .

	Collection
	Normal to light ray
	the multilayer		Normal to
	Illumination	Φ col	the sample surface
	light ray
	Φ inc			Macroscopic
				sample surface
		θ i θ r t	e 1 e 2
	n 1 n 2 n 1
	Figure 2. Schematic representation of a tilted multilayer
	(angle between the multilayer and the sample surface or tilt
	t = A schematic
	representation at a different scale, focusing on the goniometer
	is available in ESM.

  5°, t + 25°), (t°, t + 20°), (t + 5°, t + 15°), (t + 10°, t + 10°), (t + 15°, t + 5°), (t + 20°, t + 0°), (t + 25°, t -5°)}. Depending on how directional your sample is, it may be needed to increase the resolution of the measurement grid and only move the angle bisector of 2.5°or 5°at each step.

Table 2 .

 2 Repeatability (intra-class coefficient ICC with likelihood ratio and permutation p-values) and standard deviations (standard deviation SD and relative standard deviation RSD) of iridescence parameters for hummingbird and butterflies.

	Taxon	Variable	Param. mean	SD	RSD (%)	ICC	p (likel.)	p (perm.)
	Hummingbirds	Brightness	B max t γ B	36.60 14.61 13.67	47.54 18.21 7.85	14.79 7.428 11.19	0.947 0.968 0.875	<0.0001 <0.0001 0.0009	0.001 0.001 0.002
		Hue	H max γ H	556.80 65.66 0.64 0.18	0.3004 2.281	0.997 0.689	<0.0001 0.028	0.001 0.098
	Butterflies	Brightness	B max t γ B	148.80 99.78 2.94 4.83 5.35 5.12	6.91 32.96 4.76	0.936 0.268 0.769	<0.0001 0.18 <0.0001	0.001 0.098 0.004
		Hue	H max γ H	492.69 27.87 0.73 0.14	0.2484 2.993	0.993 0.853	<0.0001 <0.0001	0.001 0.001

The red lines correspond to the fit of the functions after parameters estimation, with the values of the parameters. The R script to produce this figure is available in ESM.

Acknowledgements

We are grateful to C. Doutrelant and two anonymous reviewers for their valuable comments on this manuscript. We would also like to thank the Muséum National d'Histoire Naturelle and in particular J. Fuchs, P. Boussès, and A. Previato for letting us sample feathers from museum hummingbird specimens,

Data accessibility Data used in this study as well as scripts to apply the described method are available at https://doi.org/10.5281/zenodo.1412735.

Competing interests

The authors declare no competing interests.

Author contributions HG conducted the study (model construction, data analysis) and wrote the first version of this manuscript. HG performed measurements on hummingbirds and DG on butterflies. WDdM designed and built the goniometer. CA, DG, ME, SB contributed to the design of the goniometer. CA, DG and WDdM helped with measurement protocol. DG and ME participated in the discussion for biological significance and pitfalls. All authors contributed to the final version of this article. Bony fishes [START_REF] Gur | The Structural Basis for Enhanced Silver Reflectance in Koi Fish Scale and Skin[END_REF]; Mammals [START_REF] Snyder | Iridescent Colour Production in Hairs of Blind Golden Moles (Chrysochloridae)[END_REF];

Number of measurements

Plants [START_REF] Chandler | Structural Colour in Chondrus Crispus[END_REF][START_REF] Lee | Ultrastructural Basis and Function of Iridescent Blue Colour of Fruits in Elaeocarpus[END_REF][START_REF] Lee | Structural Fruit Coloration in Delarbrea Michieana (Araliaceae)[END_REF][START_REF] Vignolini | Pointillist Structural Color in Pollia Fruit[END_REF] Single measurement relative to the structures orientation ( Bacteria [START_REF] Kientz | Glitter-like Iridescence within the Bacteroidetes Especially Cellulophaga Spp.: Optical Properties and Correlation with Gliding Motility[END_REF] Constant collection (2)

[18]

[9]

- Bony fishes [START_REF] Mäthger | Rapid Colour Changes in Multilayer Reflecting Stripes in the Paradise Whiptail, Pentapodus Paradiseus[END_REF];

Lizards [START_REF] Pérez I De Lanuza | Now You See Me, Now You Don't: Iridescence Increases the Efficacy of Lizard Chromatic Signals[END_REF][START_REF] Pérez I De Lanuza | Iridescent (Angle-Dependent Reflectance) Properties of Dorsal Coloration in Podarcis Muralis (Laurenti, 1768)[END_REF] Multiple measurement lines Multiple constant illuminations (4)

[141] [START_REF] Pirih | Spatial Reflection Patterns of Iridescent Wings of Male Pierid Butterflies: Curved Scales Reflect at a Wider Angle than Flat Scales[END_REF][START_REF] Plattner | Optical Properties of the Scales of Morpho Rhetenor Butterflies: Theoretical and Experimental Investigation of the Back-Scattering of Light in the Visible Spectrum[END_REF] Bacteria [START_REF] Kientz | A Unique Self-Organization of Bacterial Sub-Communities Creates Iridescence in Cellulophaga Lytica Colony Biofilms[END_REF] Multiple constant collections (1) Gastropods [START_REF] Brink | Iridescent Colors on Seashells: An Optical and Structural Investigation of Helcion Pruinosus[END_REF] Constant illumination, span & bisector (

-Table 3. Review of the methods used in the literature to study iridescent colours from multilayer or thin-films structure. The criteria we used for studies to be included in the table were the following (i) at least one quantitative reflectance measurement using a spectrometer (ii) functioning with white light (no monochromatic illumination) and (iii) the patch measured had to be described as iridescent in the article. A more detailed version of this table, with all angle configurations and colour variables used for each study is available in ESM. The terms 'constant illumination', 'constant collection', 'constant angle bisector' and 'constant span' are defined in fig. 3d.