Matthieu Guillaumin

Thomas Mensink

Jakob Verbeek

Cordelia Schmid

TagProp: Discriminative metric learning in nearest neighbor models for image auto-annotation

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Introduction

Image auto-annotation is an active subject of research [START_REF] Grangier | A discriminative kernel-based model to rank images from text queries[END_REF][START_REF] Li | Real-time computerized annotation of pictures[END_REF][START_REF] Liu | Image annotation via graph learning[END_REF][START_REF] Mei | Coherent image annotation by learning semantic distance[END_REF]. The goal is to develop methods that can predict for a new image the relevant keywords from an annotation vocabulary. These keyword predictions can be used either to propose tags for an image, or to propose images for a tag or a combination of tags. Such methods are becoming more and more important given the growing collections of user-provided visual content, e.g. on photo or video sharing sites, and desktop photo management applications. These large-scale collections feed the demand for automatic retrieval and annotation methods. Since the amount of images with more or less structured annotations is also increasing, this allows the deployment of machine learning techniques to leverage this potential by estimating accurate tag prediction models.

Although the general problem is a difficult one, progress has been made in the research community by evaluations on standardized annotated data sets. In the next section we will detail the related work that is most closely linked to ours. The main shortcomings of existing work are twofold. First, models are often estimated to maximize generative likelihood of image features and annotations, which might not be optimal for tag prediction. Second, many parametric models are not rich enough to accurately capture the intricate dependencies between image content and annotations. Non-parametric nearest neighbor like methods have been found to be quite successful for tag prediction [START_REF] Feng | Multiple Bernoulli relevance models for image and video annotation[END_REF][START_REF] Jeon | Automatic image annotation and retrieval using cross-media relevance models[END_REF][START_REF] Lavrenko | A model for learning the semantics of pictures[END_REF][START_REF] Makadia | A new baseline for image annotation[END_REF][START_REF] Pan | Automatic multimedia cross-modal correlation discovery[END_REF][START_REF] Zhang | SVM-KNN: Discriminative nearest neighbor classification for visual category recognition[END_REF]. This is mainly due to the high 'capacity' of such models: they can adapt flexibly to the patterns in the data as more data is available. However, existing nearest neighbor type methods do not allow for integrated learning of the metric that defines the nearest neighbors in order to maximize the predictive performance of the model. Either a fixed metric [START_REF] Feng | Multiple Bernoulli relevance models for image and video annotation[END_REF][START_REF] Zhang | SVM-KNN: Discriminative nearest neighbor classification for visual category recognition[END_REF] or adhoc combinations of several metrics [START_REF] Makadia | A new baseline for image annotation[END_REF] are used, despite many recent work showing the benefits of metric learning for many computer vision tasks such as image classification [START_REF] Jin | Learning a distance metric from multi-instance multi-label data[END_REF], image retrieval [START_REF] Hertz | Learning distance functions for image retrieval[END_REF], or visual identification [START_REF] Guillaumin | Is that you? Metric learning approaches for face identification[END_REF].

In this paper we present TagProp, short for Tag Propagation, a new nearest neighbor type model that predicts tags by taking a weighted combination of the tag absence/presence among neighbors. Our contributions are the following. First, the weights for neighbors are either determined based on the neighbor rank or its distance, and set automatically by maximizing the likelihood of annotations in a set of training images. With rank based weights the k-th neighbor always receives a fixed weight, whereas distance based weights decay exponentially with the distance. Our tag prediction model is conceptually simple, yet outperforms the current state-of-the-art methods using the same feature set. Second, contrary to earlier work, our model allows the integration of metric learning. This enables us to optimize e.g. a Mahalanobis metric between image features -or, less costly, a combination of several distance measuresto define the neighbor weights for the tag prediction task. Third, TagProp includes word-specific logistic discriminant models. These models use the tag predictions of the wordinvariant models as inputs and are able, using just two parameters per word, to boost or suppress the tag presence probabilities for very frequent or rare words. This results in a significant increase in the number of words that are recalled, i.e. assigned to at least one test image.

To evaluate our models and to compare to previous work, we use three data sets -Corel 5k, IAPR TC12 and ESP Game -and standard measures including precision, recall, mean average precision and break-even point. In Figure 1 we show several examples of images with their annotations, and predictions from our model. On all data sets and measures we show significantly improved accuracy of our method as compared to earlier work.

The rest of this paper is organized as follows. In the next section we give an overview of the related work. Then, in Section 3, we present our tag prediction models, and how we estimate their parameters. In Section 4 we present the three data sets, evaluation criteria as well as the image features we use in our experiments. The experimental results are presented in Section 5. In Section 6 we present our conclusions and directions for further research.

Related Work

In this section we discuss models for image annotation and keyword based retrieval most relevant for our work. We identify four main groups of methods: those based on topic models or mixture models, discriminatively trained ones, and nearest neighbor type models.

The first group of methods are based on topic models such as latent Dirichlet allocation, probabilistic latent semantic analysis, and hierarchical Dirichlet processes, see e.g. [START_REF] Barnard | Matching words and pictures[END_REF][START_REF] Monay | PLSA-based image autoannotation: Constraining the latent space[END_REF][START_REF] Yakhnenko | Annotating images and image objects using a hierarchical Dirichlet process model[END_REF]. These methods model annotated images as samples from a specific mix of topics, where each topic is a distribution over image features and annotation words. Parameter estimation involves estimating the topic mix for each image, and estimating the data distributions of the topics. Most often, a multinomial distribution over words is used, and a Gaussian over visual features from different regions of the image. Methods inspired by machine translation [START_REF] Duygulu | Object recognition as machine translation: Learning a lexicon for a fixed image vocabulary[END_REF], in this case translating from discrete visual features to the annotation vocabulary, can also be understood as topic models, using one topic per visual descriptor type.

A second family of methods uses mixture models to define a joint distribution over image features and annotation tags. To annotate a new image, these models compute the conditional probability over tags given the visual features by normalising the joint likelihood. Sometimes a fixed number of mixture components over visual features per keyword is used [START_REF] Carneiro | Supervised learning of semantic classes for image annotation and retrieval[END_REF], while other models use the training images as components to define a mixture model over visual features and tags [START_REF] Feng | Multiple Bernoulli relevance models for image and video annotation[END_REF][START_REF] Jeon | Automatic image annotation and retrieval using cross-media relevance models[END_REF][START_REF] Lavrenko | A model for learning the semantics of pictures[END_REF]. Each training image defines a likelihood over visual features and tags by a smoothed distribution around the observed values. These models can be seen as non-parametric density estimators over the cooccurrence of images and annotations. For visual features Gaussians are used, while the distributions over annotations are multinomials, or separate Bernoullis for each word.

Both families of generative models discussed above may be critisized because they maximize the generative data likelihood, which is not necessarily optimal for predictive performance. Therefore, discriminative models for tag prediction have also been proposed [START_REF] Cusano | Image annotation using SVM[END_REF][START_REF] Grangier | A discriminative kernel-based model to rank images from text queries[END_REF][START_REF] Hertz | Learning distance functions for image retrieval[END_REF]. These methods learn a separate classifier for each tag, and use these to predict for each test image whether it belongs to the class of images that are annotated with each particular tag. Different learning methods have been used, including support vector machines, multiple-instance learning, and Bayes point machines. Notable is [START_REF] Grangier | A discriminative kernel-based model to rank images from text queries[END_REF] which also addresses the problem of retrieving images based on multi-word queries.

Given the increasing amount of training data that is currently available, local learning techniques are becoming more attractive as a simple yet powerful alternative to parametric models. Examples of such techniques include methods based on label diffusion over a similarity graph of labeled and unlabeled images [START_REF] Liu | Image annotation via graph learning[END_REF][START_REF] Pan | Automatic multimedia cross-modal correlation discovery[END_REF], or learning discriminative models in neighborhoods of test images [START_REF] Zhang | SVM-KNN: Discriminative nearest neighbor classification for visual category recognition[END_REF]. A simpler adhoc nearest-neighbor tag transfer mechanism was recently introduced [START_REF] Makadia | A new baseline for image annotation[END_REF], showing state-of-the-art performance. There, nearest neighbors are determined by the average of several distances computed from different visual features. The authors also combine the base distances by learning a binary classifier separating image pairs that have several tags in common from images that do not share any tags. However, this linear distance combination did not give better results than an equally weighted combination.

Tag Relevance Prediction Models

Our goal is to predict the relevance of annotation tags for images. Given these relevance predictions we can annotate images by ranking the tags for a given image, or do keyword based retrieval by ranking images for a given tag. Our proposed method is based on a weighted nearest neighbor approach, inspired by recent successful methods [START_REF] Feng | Multiple Bernoulli relevance models for image and video annotation[END_REF][START_REF] Jeon | Automatic image annotation and retrieval using cross-media relevance models[END_REF][START_REF] Lavrenko | A model for learning the semantics of pictures[END_REF][START_REF] Makadia | A new baseline for image annotation[END_REF], that propagate the annotations of training images to new images. Our models are learnt in a discriminative manner, rather than using held-out data [START_REF] Feng | Multiple Bernoulli relevance models for image and video annotation[END_REF], or using neighbors in an adhoc manner [START_REF] Makadia | A new baseline for image annotation[END_REF]. We assume that some visual similarity or distance measures between images are given, abstracting away from their precise definition.

Weighted Nearest Neighbor Tag Prediction

To model image annotations, we use Bernoulli models for each keyword. This choice is natural because keywords, unlike natural text where word frequency is meaningful, are either present or absent. The dependencies between keywords in the training data are not explicitly modeled, but are implicitly exploited in our model.

We use y iw ∈ {-1, +1} to denote the absence/presence of keyword w for image i, hence encoding the image annotations. The tag presence prediction p(y iw = +1) for image i is a weighted sum over the training images, indexed by j:

p(y iw = +1) = j π ij p(y iw = +1|j), (1)
p(y iw = +1|j) = 1 -ǫ for y jw = +1, ǫ otherwise, (2)
where π ij denotes the weight of image j for predicting the tags of image i. We require that π ij ≥ 0, and j π ij = 1.

We use ǫ to avoid zero prediction probabilities, and in practice we set ǫ = 10 -5 . To estimate the parameters that control the weights π ij we maximize the log-likelihood of the predictions of training annotations. Taking care to set the weight of training images to themselves to zero, i.e.

π ii = 0, our objective is to maximize

L = i,w c iw ln p(y iw ), (3)
where c iw is a cost that takes into account the imbalance between keyword presence and absence. Indeed, in practice, there are many more tag absences than presences, and absences are much noisier than presences. This is because most tags in annotations are relevant, but often the annotation does not include all relevant tags. We set c iw = 1/n + if y iw = +1, where n + is the total number of positive labels, and likewise c iw = 1/n -when y iw = -1.

Rank-based weights. In the case of rank-based weights over K neighbors we set π ij = γ k if j is the k-th nearest neighbor of i. The data log-likelihood (3) is concave in the parameters γ k and can be estimated using an EM-algorithm, or a projected-gradient algorithm. The derivative of Eq. ( 3) with respect to γ k equals

∂L ∂γ k = i,w c iw p(y iw |n ik ) p(y iw ) , (4)
where n ik denotes the index of the k-th neighbor of image i. The number of parameters equals the neighborhood size K. We refer to this variant as RK, for "rank-based".

Distance-based weights. The other possibility is to define the weights directly as a function of the distance, rather than the rank. This has the advantage that weights will depend smoothly on the distance, which is crucial if the distance is to be adjusted during training. The weights of training images j for an image i are redefined as

π ij = exp(-d θ (i, j)) j ′ exp(-d θ (i, j ′ )) , (5)
where d θ is a distance metric with parameters θ that we want to optimize. Note that the weights π ij decay exponentially with distance d θ to image i. Choices for d θ include Mahalanobis distances d M parametrized by a semi-definite matrix M, and d w (i, j) = w ⊤ d ij where d ij is a vector of base distances between image i and j, and w contains the positive coefficients of the linear distance combination. The number of parameters equals the number of base distances that are combined. In the rest of the paper we focus on this particular case. When we use a single distance, referred to as the SD variant, w is a scalar that controls the decay of the weights with distance, and it is the only parameter of the model. When multiple distances are used, the variant is referred to as ML, for "metric learning". Again, rather than using an EM-algorithm we directly maximize the log-likelihood using a projected gradient algorithm under positivity constraints on the elements of w.

Using the new definition of the weights, the gradient of the log-likelihood Eq. ( 3) with respect to w equals

∂L ∂w = i,j W i (π ij -ρ ij )d ij , (6)
where W i = w c iw and ρ ij denotes the weighted average over all words w of the posterior probability of neighbor j for image i given the annotation:

ρ ij = w c iw W i p(j|y iw ). (7)
To reduce the computational cost of training the model, we do not compute all pairwise π ij and ρ ij . Rather, for each i we compute them only over a large set, and assume the remaining π ij and ρ ij to be zero. For each i, we select K neighbors such that we maximise k * = min{k d }, where k d is the largest neighbor rank for which neighbors 1 to k of base distance d are included among the selected neighbors. In this way we are likely to include all images with large π ij regardless of the distance combination w that is learnt. Therefore, after determining these neighborhoods, our algorithm scales linearly with the number of training images.

Note the relation of our model to the multi-class metric learning approach of [START_REF] Globerson | Metric learning by collapsing classes[END_REF]. In that work, a metric is learnt such that weights π ij as defined by Eq. ( 5) are as close as possible in the sense of Kullback-Leibler (KL) divergence to fixed set of target weights ρ ij . The target weights were defined to be zero for pairs from different classes, and set to a constant for all pairs from the same class. In fact, when deriving an EM-algorithm for our model, we find the objective of the M-step to be of the form of a KL divergence between the ρ ij (fixed to values computed in the E-step) and the π ij . For fixed ρ ij this KL divergence is convex in w.

Word-specific Logistic Discriminant Models

Weighted nearest neighbor approaches tend to have relatively low recall scores, which is easily understood as follows. In order to receive a high probability for the presence of a tag, it needs to be present among most neighbors with a significant weight. This, however, is unlikely to be the case for rare tags. So, even if we are lucky enough to have a few neighbors annotated with the tag, we will predict the presence with a low probability.

To overcome this, we introduce word-specific logistic discriminant models that can boost the probability for rare tags and decrease it for very frequent ones. The logistic model uses weighted neighbor predictions by defining

p(y iw = +1) = σ(α w x iw + β w ), (8)
x iw = j π ij y jw , (9)
where σ(z) = (1 + exp(-z)) -1 and x iw is the weighted average of annotations for tag w among the neighbors of i, which is equivalent to Eq. ( 1) up to an affine transformation.

The word-specific models add 2 parameters to estimate for each word. The resulting modulated variants are referred to as σRK, σSD and σML, respectively. For fixed π ij the model is a logistic discriminant model, and the log-likelihood is concave in {α w , β w }, and can be trained per keyword. Using the new model, the gradient of the log-likelihood of the training annotations with respect to the parameters θ that control the weights equals

∂L ∂θ = i,w c iw α w p(-y iw )y iw ∂x iw ∂θ , (10)
and for the model based on rank or distance respectively

∂x iw ∂γ k = y n ik w , (11)
∂x iw ∂w = j π ij (x iw -y jw ) d ij . (12)
In practice we estimate the parameters θ and {α w , β w } in an alternating fashion. We observe rapid convergence, typically after alternating the maximization three times.

Data Sets and Experimental Setup

In this section we first present the data sets used in our experiments, then in Section 4.2 we describe the different features that we extract from images to compute distance measures between images, and in Section 4.3 we discuss the evaluation measures for image annotation and retrieval.

Data Sets

We consider three publicly available data sets that have been used in previous work, and allow for direct comparison. Table 1 summarizes some statistics of these data sets, example images are shown in Figure 1.

Corel 5k. This data set was first used in [START_REF] Duygulu | Object recognition as machine translation: Learning a lexicon for a fixed image vocabulary[END_REF]. Since then, it has become an important benchmark for keyword based image retrieval and image annotation. It contains around 5000 images manually annotated with 1 to 5 keywords. The vocabulary contains 260 words. A fixed set of 499 images are used as test, and the rest is used for training.

ESP Game. This data set is obtained from an online game where two players, that can not communicate outside the game, gain points by agreeing on words describing the image [START_REF] Von Ahn | Labeling images with a computer game[END_REF]. This way the players are encouraged to provide important and meaningful tags to images. We use the subset of 20.000, out of the 60.000 images publicly available, that was also used in [START_REF] Makadia | A new baseline for image annotation[END_REF]. This data set is very challenging, as it contains a wide variety of images including: logos, drawings, and personal photos. IAPR TC12. This set of 20.000 images accompanied with descriptions in several languages was initially published for cross-lingual retrieval [START_REF] Grubinger | Analysis and Evaluation of Visual Information Systems Performance[END_REF]. It can be transformed into a format comparable to the other sets by extracting common nouns using natural language processing techniques. We use the same resulting annotation as in [START_REF] Makadia | A new baseline for image annotation[END_REF].

Feature Extraction

We extract different types of features commonly used for image search and categorisation. We use two types of global image descriptors: Gist features [START_REF] Oliva | Modeling the shape of the scene: a holistic representation of the spatial envelope[END_REF], and color histograms with 16 bins in each color channel for RGB, LAB, HSV representations. Local features include SIFT as well as a robust hue descriptor [START_REF] Van De Weijer | Coloring local feature extraction[END_REF], both extracted densely on a multiscale grid or for Harris-Laplacian interest points. Each local feature descriptor is quantized using k-means on samples from the training set. Images are then represented as a 'bag-of-words' histogram. All descriptors but Gist are L1normalised and also computed in a spatial arrangement [START_REF] Lazebnik | Beyond bags of features: spatial pyramid matching for recognizing natural scene categories[END_REF]. We compute the histograms over three horizontal regions of the image, and concatenate them to form a new global descriptor, albeit one that encodes some information of the spatial layout of the image. To limit color histogram sizes, here, we reduced the quantization to 12 bins in each channel. Note that this spatial binning differs from segmented image regions, as used in some previous work.

This results in 15 distinct descriptors, namely one Gist descriptor, 6 color histograms and 8 bag-of-features (2 features types x 2 descriptors x 2 layouts). To compute the distances from the descriptors we follow previous work and use L2 as the base metric for Gist, L1 for global color histograms, and χ 2 for the others.

Evaluation Measures

We evaluate our models with standard performance measures, used in previous work, that evaluate retrieval performance per keyword, and then average over keywords.

Precision and recall for fixed annotation length. Following [START_REF] Duygulu | Object recognition as machine translation: Learning a lexicon for a fixed image vocabulary[END_REF], each image is annotated with the 5 most relevant keywords. Then, the mean precision P and recall R over keywords are computed. N+ is used to denote the number of keywords with non-zero recall value. Note that each image is forced to be annotated with 5 keywords, even if the image has fewer or more keywords in the ground truth. Therefore, even if a model predicts all ground-truth keywords with a significantly higher probability than other keywords, we will not measure perfect precision and recall.

Precision at different levels of recall. We also evaluate precision at different levels of recall as in [START_REF] Grangier | A discriminative kernel-based model to rank images from text queries[END_REF]. The break-even point (BEP), or R-precision, measures for each keyword w the precision among the top n w relevant images, where n w is the number of images annotated with this keyword in the ground truth. The mean average precision (mAP) over keywords is found by computing for each keyword the average of the precisions measured after each relevant image is retrieved.

Experimental Results

In this section we present a quantitative evaluation of TagProp and compare to previous work, qualitative results can be found in Figure 1. We first give a detailed presentation of results obtained on the Corel 5k data set, and compare them to previous work. In Section 5.2 we present our results for the IAPR TC12 and ESP Game data sets. Results for multi-word image retrieval are presented in Section 5.3.

Results for the Corel 5k data set

In a first set of experiments we compare the different variants of TagProp and compare them to the original results of [START_REF] Makadia | A new baseline for image annotation[END_REF], referred to as JEC, and also using our own features (JEC-15). That is, we take an equally weighted combination of our 15 normalized base distances to define image similarity.

From the results in Table 2 we can make several observations. First, using the tag transfer method proposed in [START_REF] Makadia | A new baseline for image annotation[END_REF] with our own features we obtain results very similar to the original work. Thus, other performance differences obtained using our methods must be due to the tag prediction methods. Our models that use this fixed distance combination to define weights (either directly in SD or using ranks in RK) perform comparably. Among these results, the ones of the sigmoidal model using distance-based weights (σSD) are the best, and they show a modest improvement over the results obtained with the more adhoc JEC-15.

More importantly, using our models that integrate metric learning (ML and σML), much larger improvements are obtained, in particular using the σML variant. Compared to the current state-of-the-art method using the same features, we obtain marked improvements of 5% in precision, 9% in recall, and count 20 more words with positive recall. This result shows clearly that nearest neighbor type tag prediction can benefit from metric learning. Interestingly, earlier efforts to exploit metric learning did not succeed [START_REF] Makadia | A new baseline for image annotation[END_REF], c.f . Section 2. The key to our successful use of metric learning is its integration in the prediction model.

Previsously reported results

TagProp CRM [START_REF] Lavrenko | A model for learning the semantics of pictures[END_REF] InfNet [START_REF] Metzler | An inference network approach to image retrieval[END_REF] NPDE [START_REF] Yavlinsky | Automated image annotation using global features and robust nonparametric density estimation[END_REF] SML [START_REF] Carneiro | Supervised learning of semantic classes for image annotation and retrieval[END_REF] MBRM [START_REF] Feng | Multiple Bernoulli relevance models for image and video annotation[END_REF] TGLM [START_REF] Liu | Image annotation via graph learning[END_REF] JEC [ 2. Overview of performance in terms of P , R, and N+ of our models (using K = 200), and those reported in a selection of earlier work. JEC-15 refers to our implementation of [START_REF] Makadia | A new baseline for image annotation[END_REF] using our 15 distances. We show results for our variants: RK and SD using the equal distance combination, ML which integrates metric learning, as well as their modulated extensions (σRK, σSD and σML, respectively).

In Figure 2 we show performance in terms of P , R, BEP, and mAP, for the distance based models as a function of the number of neighbors over which the weights are evaluated. From these results we conclude that consistently over all number of neighbors, with or without σ, the metric learning finds distance combinations that outperform the equal weighted combination. Furthermore, we see that σML has a major impact on the R measure, and also leads to improvements in the other measures. We observe that using large numbers of neighbors improves performance, at least up to 100. This is true in particular for the ML variants. This can be explained by the fact that, in ML, the ranking of neighbor images change with the learnt metric. Thus, in order to ensure that all useful training images are included in the initial neighborhoods (computed from the base distances, c.f . Section 3), these sets should be large enough.

In Figure 3 we further analyze which words benefit most from the improved recall in σ variants. As expected, the improvement is higher for rare words and only the few most frequent words are penalized in terms of recall, but even in this situation the performance remains high and many relevant images are still retrieved.

From these first experiments we identify our distance based variants as the most promising, and use them below for the other two data sets. We use K = 200 as a default choice for the number of neighbors as it shows good performance for all measures. Figure 3. Mean recall of words in the three data sets (from left to right: Corel 5k, ESP Game, and IAPR TC12) for ML (blue) and σML (yellow), grouped with respect to their frequency in the data set. Keywords are binned based on how many images they occur in: the first bin groups words with less then 5 relevant images, the second bin words with between 6 and 10 images, and so on. The lower bars show the fraction of keywords in each bin, the upper bars show the average recall for the words in a bin.

Results for ESP Game and IAPR TC12

Next we present experimental results obtained on the ESP Game and IAPR TC12 data sets. Table 3 compares the current state-of-the-art for these data sets to our distance based variants, using equal and learned distance combinations. For reference we also include our implementation of [START_REF] Makadia | A new baseline for image annotation[END_REF] using our feature set. Again we find that our models significantly improve over the current state-of-the-art per- formance. In contrast to the Corel data set, here we find most significant improvement for precision rather than recall: 18% (resp. 17%) on P , 6% (resp. 2%) on R, and 16 (resp. 15) on N+ on IAPR (resp. ESP).

As for Corel, we show in Figure 4 the influence of the neighborhood size on our different measures for both datasets for our σSD and σML variants. We can again observe the benefit of integrated metric learning.

Image Retrieval from Multi-word Queries

Up to this point, we looked at image retrieval performance for single word queries. Most existing work concentrates on this problem, as it is difficult as such. However, any realistic image retrieval system should support multiword queries as well. Here we present performance in terms of BEP and mAP on the Corel data set that include multiword queries. To allow for direct comparison, we follow the setup of [START_REF] Grangier | A discriminative kernel-based model to rank images from text queries[END_REF], which uses a subset of 179 words of the 260 annotation words of Corel 5k that appear at least twice in the test set. Images are considered relevant for a query when they are annotated with all words, and we consider all 2241 queries composed of one or more words such that the test set includes at least one image that is relevant. Further, the queries are divided into 'difficult' ones (1820) for which there are only one or two relevant images, and easy ones (421) with three or more relevant images.

To predict relevance of images for a multi-word query we compute the probability according to our model to observe all keywords in the query. Due to the probabilistic output of TagProp, this is easlily done by taking the product over the single keyword probabilities, as our model does not explicitly account for dependencies between words. In Table 4 we summarize our results, and compare to those of PAMIR [START_REF] Grangier | A discriminative kernel-based model to rank images from text queries[END_REF]. Our results improve by about 10% the mAP performance over all query types. Also in terms of BEP we gain 10% compared to PAMIR, which was found in [START_REF] Grangier | A discriminative kernel-based model to rank images from text queries[END_REF] to outperform a number of alternative approaches. 5. Overview of performance for the three data sets obtained with TagProp (σML variant with K = 200).

Conclusions

We have introduced new models for image annotation and keyword based image retrieval. These models combine a weighted nearest-neighbor approach with metric learning capabilities in a discriminative framework. We add wordspecific logistic discriminant modulation to deal with the varying word frequencies in a data-driven manner.

We reported extensive experimental results on three data sets, using several performance measures. From these results we conclude that our σML sigmoidal variant of TagProp that uses distance based weights and integrates metric learning performs best. It combines good recall with high precision over all data sets. We refer to Table 5 for an overview of the performance obtained with σML with all five evaluation measures. Consistently, it outperforms previously reported results for all three data sets. It gives significant improvements over the same model applied to uniformly combined distances, σSD. This contrasts with earlier attempts to use metric learning for tag prediction, see e.g. [START_REF] Makadia | A new baseline for image annotation[END_REF], that were unsuccessful because metric learning was not integrated in the prediction model. Our word specific modulation significantly improves recall for rare words as well as the overall performance. On all three data sets, and all five evaluation measures, our model achieves performance well above all previously published results.

In future work, we will consider extending the model to assign tags to image regions, in order to address tasks such as image region labelling and object detection from imagewide annotations.

Figure 1 .

 1 Figure 1. Example test images from the three data sets. Next to each image, we show the ground truth annotation (left), and the five tags with highest relevance predictions (correct ones are underlined) given by our TagProp model (σML variant with K = 200). Note the large diversity between the data sets, and that the ground truth annotations do not always contain all relevant tags (e.g. 'water' for the bottom left image), and sometimes contain tags for which one can argue whether they are relevant or not (e.g. 'lot' for the bottom right image).

Figure 2 .

 2 Figure 2. Performance on the Corel data set of our distance-based models in terms of P , R, BEP, and mAP with respect to the neighborhood size K. The dashed curves correspond to the ML (red) and SD (blue) variants. The solid curves are the sigmoidal modulated extensions: σML (red) and σSD (blue).

Figure 4 .

 4 Figure 4. Performance of our sigmoidal modulated distance-based variants of TagProp in terms of P , R, BEP, and mAP, on the ESP (solid) and IAPR (dashed) datasets with respect to the neighborhood size K. Red curves correspond to the learnt combination, σML, and blue ones to an equal combination, σSD.

Table 1 .

 1 Statistics of the training sets of the three data sets. Image and word counts are given in the format mean / maximum. Statistics for the test sets resemble closely those of the training sets.

		Corel 5k	ESP Game	IAPR TC12
	Vocabulary size	260	268	291
	Nr. of images	4,493	18,689	17,665
	Words per img.	3.4 / 5	4.7 / 15	5.7 / 23
	Img. per word 58.6 / 1004 362.7 / 4553 347.7 / 4999

Table 3 .

 3 Comparison of performance in terms of P , R, and N+ on ESP and IAPR of the state-of-the-art[START_REF] Makadia | A new baseline for image annotation[END_REF] to our TagProp variants with distance based weights, using K = 200 neighbors.

				IAPR		ESP Game
			P	R	N+ P	R	N+
	MBRM [17] 24 23 223 18 19 209
	JEC [17]	28 29 250 22 25 224
	JEC-15		29 19 211 24 19 222
	TagProp	SD 50 20 215 48 19 212 σSD 41 30 259 39 24 232 ML 48 25 227 49 20 213
	σML 46 35 266 39 27 239

Table 4 .

 4 Comparison of TagProp variants (using K = 200) and PAMIR in terms of mAP and BEP. The mAP performance is also broken down over single-word and multi-word queries, easy and difficult ones. Only the 179 words that appear at least twice in test images are used, as in[START_REF] Grangier | A discriminative kernel-based model to rank images from text queries[END_REF].

			All-mAP	Single	Multi	Easy	Difficult	All-BEP
	PAMIR [7] 26 34 26 43 22	17
	TagProp	SD 32 40 31 49 28 σSD 31 41 30 49 27 ML 36 43 35 53 32	24 23 27
	σML 36 46 35 55 32	27
	TagProp		P	R	N+ BEP mAP
	Corel 5K	32.7 42.3 160 36.3	41.8
	IAPR		46.0 35.2 266 40.9	39.9
	ESP-Game 39.2 27.4 239 31.3	28.1
	Table

Acknowledgements

This work was supported by the European funded research project CLASS.


A Bouquet
email: alban.bouquet@qgg.au.dk

M Slagboom

J R Thomasen

Nicolas N C Friggens

M Kargo

Laurence Puillet

N C Friggens

Charlotte Gaillard

Coupling genetic and mechanistic models to benchmark selection strategies for feed efficiency in dairy cows: sensitivity analysis validating this novel approach

Keywords:

come

Coupling genetic and mechanistic models to benchmark

selection strategies for feed efficiency in dairy cows: sensitivity analysis validating this novel approach

Specification table

Subject

Breeding and Genetics Type of data Compiled Python code to simulate phenotypic data of milk production and feed efficiency traits in dairy cows, R code to manipulate datasets and carry out statistical analysis, shell script to chain all analyses.

How data were acquired Datasets simulated with AQAL software [START_REF] Puillet | Disentangling the relative roles of resource acquisition and allocation on animal feed efficiency: insights from a dairy cow model[END_REF]2021)

Parameters for data collection

Phenotypic trajectories of dairy cows were simulated for a grass-based production system with seasonal calving. Three nutritional environments simulated by changing the quantity of feed offered to cows.

Description of data collection

Simulated datasets

Data source location

Fill in the information available, and delete from this list as appropriate:

Introduction

For decades, mechanistic models (MMs) have been used to assist decision-making in agriculture. In the livestock sector, such models were developed to help optimising the technical performance of herds and reducing both costs and the environmental footprint of production [START_REF] Ellis | Review: Synergy between mechanistic modelling and data-driven models for modern animal production systems in the era of big data[END_REF]. In animal nutrition, MMs has been a useful tool to predict animal responses to changes in environmental factors (e.g. feed composition, feed restriction, herd management, climate). So far, the use of MMs have been limited in animal breeding. However, there is increased interest in combining mechanistic and genetic modelling to investigate how energy trade-offs could hamper selection response achieved in breeding schemes, either by exacerbating genetic antagonisms between traits or by generating genotype-by-environment (GÂE) interactions [START_REF] Douhard | Synergy between selection for production and longevity and the use of extended lactation: Insights from a resource allocation model in a dairy goat herd[END_REF]. Such an approach could be beneficial for the genetic improvement of feed and lactation efficiency in dairy cattle. These complex traits are highly dynamic and are affected by trade-offs between different biological functions. Such trade-offs occur when energy intake is restricted and prevent the cows from expressing their full potential for the different biological functions, for example during lactation in low-input production systems. Increasing milk production and efficiency traits in dairy cows may thus impair fitness traits such as reproduction [START_REF] Pryce | Estimation of genetic parameters using health, fertility and production data from a management recording system for dairy cattle[END_REF] and health [START_REF] Becker | Residual energy intake, energy balance, and liability to diseases: Genetic parameters and relationships in German Holstein dairy cows[END_REF] and lead to a shorter productive life. The magnitude and nature of trade-offs between biological functions are expected to differ across production systems and change phenotypic and genetic covariances between traits, thus generating GÂE interactions [START_REF] Friggens | Advances in predicting nutrient partitioning in the dairy cow: recognizing the central role of genotype and its expression through time[END_REF]. Moreover, the collection of feed intake data is costly and difficult, e.g. in outdoor production systems, leading to a scarcity of data needed for estimating reliable genetic parameters under a broad range of environments.

To broaden the scope of nutrition MMs to animal breeding issues, [START_REF] Puillet | Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation[END_REF] developed a model based on the resource acquisition and allocation theory which includes a genetic module. Four genetically driven parameters describe the variability in the way cows acquire and allocate energy for different functions such as lactation, growth, and reproduction over their lifetime. This model was used to explore the relative contribution of energy acquisition and allocation parameters (AAPs) on lactation efficiency in dairy cows and to evaluate the magnitude of GÂE interactions for these traits under various environments [START_REF] Puillet | Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation[END_REF]. In this study, the genetic parameters of AAPs were defined empirically with typical values for production traits [START_REF] Hill | The quantitative genetics of phenotypic variation in animals[END_REF]. Therefore, more insight is needed on the relative influence of genetic parameters assumed for AAPs and the effect of the nutritional environment on simulated traits before the model can be used in breeding applications. The objective of this study was to perform a sensitivity analysis to identify genetic parameters to be used as input for the simulation of dairy cow performance with realistic mean, variance and genetic correlation patterns among traits.

Material and methods

The sensitivity analysis applied in this study made intensive use of stochastic simulations. The compiled executable of the simulation tool and scripts to reproduce simulations and analyses can be retrieved from: https://data.mendeley.com/datasets/txx9zkb-d2g/2. This code needs to be run under Linux with the SLURM workload manager. In addition, it requires R statistical software [START_REF] Core | R: A language and environment for statistical computing[END_REF] for data manipulations and statistical analyses as well as DMU version 5.2 [START_REF] Madsen | DMU: A package for analyzing multivariate mixed models[END_REF] for the estimation of variance components. In this code, the seeds used for sampling random numbers were fixed at the beginning of each replicate to ensure the reproducibility of data.

Datasets were generated for 12 different scenarios, considering different combinations of input parameters based on three different nutritional environments, two levels of heritability and three levels of CV for AAPs (Table 1).

Dairy cow mechanistic model

The AQAL bioenergetic dairy cow model simulates phenotypic trajectories over a cow's lifetime for milk production, DM intake, body mass components including body reserves, as well as the timing of estruses and success of reproduction events [START_REF] Puillet | Disentangling the relative roles of resource acquisition and allocation on animal feed efficiency: insights from a dairy cow model[END_REF]. The model can accommodate different nutritional environments by modulating the DM offer and metabolisable energy (ME) content of feed over time. The dynamics of energy acquisition and allocation for the main energy sinks (maintenance, lactation, gestation, body reserves) is described deterministically [START_REF] Puillet | Disentangling the relative roles of resource acquisition and allocation on animal feed efficiency: insights from a dairy cow model[END_REF]. Four input parameters can be tuned to describe different dynamics of energy acquisition and allocation as observed for instance in different breeds. The first two acquisition parameters relate to the maximal potential DM intake of a non-lactating cow (basal acquisition abbreviated as BasAcq, in kg of DM/d) and the potential increase in DM intake during lactation (lactation acquisition or LactAcq, in kg of DM/d) in a non-limiting environment. The two other input parameters relate to the dynamics of energy allocation. The first one drives the dynamic of allocation to growth and corresponds to the rate of transfer between allocation to growth and allocation to survival (f prio G2S, dimensionless). A high value implies a fast decrease in allocation to growth when the female ages. The second one drives the allocation to lactation and corresponds to the intensity of the charge of allocation to lactation at parturition (LactAll, dimensionless). The mean values of BasAcq, LactAcq, f prio G2S and LactAll were determined to fit real data using a calibration procedure as described in [START_REF] Puillet | Disentangling the relative roles of resource acquisition and allocation on animal feed efficiency: insights from a dairy cow model[END_REF]. They were set to 7.00 kg/d, 10.25 kg/d, 3.5 10 À3 and 0.56 as in [START_REF] Puillet | Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation[END_REF]. Based on the individual cow model, a genetic module was developed to simulate phenotypic and genetic variability around the mean AAPs in a population of related cows. Thus, the variance observed around mean phenotypic trajectories at the population level was induced by assuming genetic and phenotypic variability for the AAPs. Only those four AAPs were assumed to be genetically driven as they were linked to the main energy acquisition processes and energy sinks. Further details concerning the model and its implementation can be found in [START_REF] Puillet | Disentangling the relative roles of resource acquisition and allocation on animal feed efficiency: insights from a dairy cow model[END_REF]2021).

Simulated production systems

As a case study, a grass-based production system with seasonal calving was modelled following the same set-up as [START_REF] Puillet | Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation[END_REF]. The population was simulated as a single cohort of females from birth to death. It comprised 20 000 cows reared in the same environment. The mating season started on October 10th each year and lasted for 10 weeks. Gestation lasted 282 d, and cows were dried-off 90d before calving or at latest on May 15th. First mating of heifers occurred at 424 d of age. The probability of conception was derived based on deterministic equations accounting for milk production and energy balance [START_REF] Puillet | Disentangling the relative roles of resource acquisition and allocation on animal feed efficiency: insights from a dairy cow model[END_REF]. Females that were not pregnant at the end of the mating season were culled. Moreover, females were culled when their body reserves were completely depleted due to insufficient energy acquisition or too high allocation to growth and lactation. Seasonality of feed resources in terms of quality and quantity was accounted for in the simulations. Variation in feed quality was defined to be consistent with ME content of grass in New Zealand [START_REF] Roche | Weather, herbage quality and milk production in pastoral systems. 4. Effects on dairy cattle production[END_REF]. The ME content varied from 10.85 to 12.45 MJ/kg DM with a yearly average of 11.70 MJ/kg DM. Three nutritional environments were defined by changing the DM offer as simulated by [START_REF] Puillet | Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation[END_REF]. The high and non-limiting environment (HS) was defined to provide sufficient energy to cover the nutritional requirements of all cows during lactation and gestation. Secondly, a moderate scenario (MS) was defined so that DM offer could be limiting for high acquisition cows. The yearly average DM offer was 12.2 kg/ d per cow with values ranging from 10 to 16.8 kg/d depending on the seasonality of grass production. Finally, a low nutritional environment (LS) was also simulated with a high constraint on DM offer. The yearly average DM offer was 9.8 kg/d per cow with values ranging from 8 to 13.4 kg/d per cow. 1 Abbreviations of scenarios were obtained by concatenating abbreviations of the nutritional environment, heritabilities of the two acquisition and allocation parameters and CV of the two acquisition and allocation parameters.

2 Parameter value for both basal acquisition and lactation acquisition parameters.

3 Parameter value for both allocation priority from growth to survival and lactation allocation parameters.

Genetic determinism of acquisition-allocation parameters

In AQAL, the variance of simulated traits was induced by assuming phenotypic variance for AAP. These parameters were assumed to be genetically driven meaning that part of phenotypic variability was explained by additive genetic effects. A half-sib pedigree structure was simulated in this study. The true breeding values (TBVs) of AAPs were sampled for a set of 200 unrelated sires and 20 000 unrelated dams from a multivariate normal distribution with assumed genetic parameters. Sires were mated to dams to generate a single generation of 200 half-sib families comprising 100 daughters each. Procedures to sample TBV of cows and reconstruct acquisition and allocation phenotypes are described in [START_REF] Puillet | Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation[END_REF].

First, the same genetic parameters for AAPs as in [START_REF] Puillet | Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation[END_REF] were used, i.e. heritability of 0.35, a phenotypic CV of 10% and no genetic or phenotypic correlation between the input traits. These parameters were used to simulate milk production and FE data in all three nutritional environments and served as a baseline.

Then, different sets of genetic parameters were considered (Table 1) to evaluate how sensitive the genetic and phenotypic correlations between simulated traits were to changes in the genetic parameters of AAPs. In all three environments, the effect of heritability was evaluated by decreasing it to 0.15 for all four AAPs. Alternatively, the effect of CV was evaluated by increasing it to 20% for all four AAPs, whilst keeping heritability to 0.35. These lower heritabilities and higher CVs are typical of fitness traits [START_REF] Hill | The quantitative genetics of phenotypic variation in animals[END_REF]. To have a better understanding on how the heritabilities (and CVs) of acquisition and allocation respectively influenced the (co)variances of simulated traits, we considered additional scenarios in the high nutritional environment only. First, we maintained heritabilities of acquisition to 0.35 and reduced heritabilities of allocation traits to 0.15, keeping CVs to 10% for all four parameters. Then, the CV of allocation parameters was increased to 30% whilst the CV of acquisition parameters kept to lower values (10 and 20%). Abbreviation of scenarios were built by concatenating the abbreviation of the nutritional environment, the heritability assumed for both acquisition and both allocation traits, and the CV assumed for both acquisition and both allocation traits (for instance HS_3535_2030).

Definition of phenotypes

Phenotypic trajectories for milk production, feed efficiency and body reserves were generated using the mechanistic model described above and the sampled AAPs for each nutritional environment. Thus, no TBVs were sampled for milk production, feed efficiency and body reserve traits. Five simulated phenotypes were considered for further genetic analyses and were estimated for cows in first and third lactation (variable names were suffixed with L1 and L3). These traits comprised total lactation milk production (Milk), BW at calving (BWcalv), daily DM intake across lactation (DMI), lactation efficiency (Lact_Eff), expressed as the ratio of energy invested in milk production over total energy intake during the lactation, and the minimum weight of body reserves during lactation (BRmin), expressed as the proportion of labile mass over empty BW. These traits were measured in first and third lactations. Records in both lactations were not analysed as repeated measurements but as two distinct traits.

Estimation of variance components

All traits were analysed using the following animal linear mixed model:

Y = Xb + Zu + e
where X is the incidence matrix-relating performances to the fixed effects; b is the vector of fixed effects that comprised the calving group (four levels) that defined the nutritional environment at the time of parturition for all traits and the number of days in milk as a covariate for Milk and Lact_Eff to correct for shorter lactations; Z is the incidence matrix-relating performances to animals; u is the vector of additive genetic values following a normal distribution N (0, Ar g 2 ) where A is the additive relationship matrix and r g 2 is the genetic variance of the trait; and e is the residual term following a normal distribution N(0, Ir e 2 ) where r e 2 is the residual variance of the trait and I the identity matrix.

Variance components were estimated using the Average Information Restricted Maximum Likelihood algorithm. At first, univariate analyses were carried out to estimate heritabilities, followed by bivariate analyses to estimate genetic and phenotypic covariances between traits. All calculations were carried out using the DMU software [START_REF] Madsen | DMU: A package for analyzing multivariate mixed models[END_REF] that can be retrieved from https://dmu.ghpc.au.dk/dmu/DMU/Linux/Previous/.

Comparison across scenarios

Each scenario was replicated 20 times to reduce the uncertainty around genetic correlation estimates that could be large for low heritability trait. Performances, heritabilities, and genetic and phenotypic correlations were compared across scenarios using a 1way ANOVA to evaluate the impact of each scenario, i.e. each combination of genetic parameters assumed for AAPs and environment. The normality of the response variable was checked at first within each scenario using the Shapiro-Wilks test. Then, the homogeneity of variances across scenarios was evaluated using the Levene test. In most cases, variances were different across scenarios. Therefore, a Welsh 1-way ANOVA was carried out followed by Games-Howell posthoc tests to determine the significance of contrasts between all pairs of scenarios. Statistical analyses were performed with the rstatix package [START_REF] Kassambara | rstatix: Pipe-Friendly Framework for Basic Statistical Tests[END_REF], and matrices were represented with the corrplot package [START_REF] Wei | R package 'corrplot': Visualization of a Correlation Matrix[END_REF] in R (R Core Team, 2021).

Results

Comparison of performance mean and variability

The number of cows starting their first and third lactation is presented in Table 1 for all scenarios. The number of heifers starting first lactation was not affected by the environment or heritability of AAPs. However, it was highly affected by the CV of AAPs. When CV of AAPs was increased to 20%, the number of heifers failing due to acute energy imbalance (i.e. too few body reserves) before starting first lactation increased from 1.4 to 4.8%. In the most extreme scenario regarding CV (HS_3535_2030), the proportion of heifers failing prematurely increased to 10.6%. These animals with a combination of too high allocation to growth and low basal acquisition failed to acquire and maintain sufficient body reserves to allow them to transition to viable lactation. The number of cows reaching third lactation depended on the nutritional environment and the CV assumed for AAPs but not on their heritability. In scenarios with a heritability of 0.35 and a CV of 0.10 for all AAPs, the proportion of primiparous cows reaching third calving was 74.5, 68.4 and 49.2% in the high, moderate and low nutritional environments. When CV of AAPs was increased to 20%, the proportion of primiparous cows reaching third lactation decreased to 62.8, 57.6 and 43.3% in the high, moderate and low nutritional environments.

The high nutritional environment mimicked a situation of ad libitum feeding. In the moderate nutritional environment, realised DMI was reduced by 2.0% compared to target intake during the first lactation meaning that DM offer was only slightly limiting compared to cow potential intake. This constraint was more important for third lactation cows which had higher nutritional requirements for maintenance and production (À12.9%). The LS environment was rather extreme with a DM offer that was much lower than potential intake both in first lactation (À12.5%) and third lactation (À27.8%).

Mean performances of cows were affected by the environment as well as the CV of AAPs but not their heritability. This was observed for first-parity cows (Table 2) as well as third-parity cows (Supplementary Table S1). However, for a given set of AAPs genetic parameters, average milk production, BW at calving and lactation efficiency measured in first lactation were significantly reduced in the MS and LS environments compared to the HS environments.

When CV of AAPs was increased from 10 to 20%, no significant difference was observed in milk production in first lactation in the high nutritional environment although BW at calving slightly increased, and lactation efficiency slightly decreased (Table 2). In the MS and LS environments, mean performance significantly differed for all traits when CV was increased from 10 to 20%. For third lactation cows, increasing CV of AAPs also induced small yet significant changes in all traits (Supplementary Table S1).

The environment also had a significant effect through restrictions exerted on DM intake in the suboptimal environments. Consequently, the CV of Milk was reduced in the MS and LS environments whereas CV of Lact_Eff and BRmin was increased, regardless of cow parity. In scenarios where different values were allocated to AAPs, the CVs of DMI and BWcalv were generally close to the CV of acquisition parameters in the non-constraining nutritional environment, whereas the CV of Milk and Lact_Eff was comprised between values set for CV of AAPs.

Comparison of heritabilities estimated across scenarios

Heritabilities estimated for the different traits and scenarios are presented in Table 3 for first-parity cows and in Supplementary Table S2 for third-parity cows. Heritability values estimated for the simulated traits depended simultaneously on the heritability and CV assumed for AAPs and on the environment.

In the high nutritional environment, the heritability of BWcalv was close to the one assumed for acquisition parameters. The her-itability estimates for Milk and BRmin were close to the average of assumed heritabilities for acquisition and allocation parameters. Indeed, these estimates were close to 0.35 and 0.15 when the heritability of all four AAPs was set to either 0.35 or 0.15, and they took intermediate values when the heritability of acquisition parameters was set to 0.35 and the heritability of allocation parameters was set to 0.15. Heritabilities estimated for Milk and BRmin were slightly reduced when considering the MS and LS environments. Unlike milk production, heritability estimated for BRmin was sensitive to the CV of AAPs, with lower estimates when CV increased.

As expected, the heritability of DMI was highly dependent on both assumed heritability for acquisition traits and the environment. In the high and non-constraining nutritional environment, the estimated heritability was close to heritabilities assumed for acquisition parameters. Estimated heritabilities were markedly reduced in the limiting nutritional environments, especially in the LS environment.

Heritability of lactation efficiency was lower than the heritability values assumed for AAPs. Furthermore, heritability estimates were slightly increased when the CV of allocation parameters was increased. Heritabilities estimated in the HS and MS nutritional environments were not significantly different whereas these estimates were slightly lower in the LS environment with an increased CV.

Change in genetic correlation patterns among traits across scenarios

The average of genetic and phenotypic correlations estimated in the baseline scenario are presented in Fig. 1 for the high nutritional environment. High genetic correlations were estimated between Milk and Lact_Eff as well as Milk and DMI regardless of cow parity. The genetic correlation between Milk and BWcalv was moderate to high in first lactation and much lower, but still positive, in third lactation. The genetic correlation between lactation efficiency and DMI was close to zero for both lactations. Genetic correlations between Lact_Eff and BWcalv, as well as BRmin, were negative and low in first lactation although phenotypic correlations were of higher magnitude. Genetic correlations between these traits were higher in third than in first lactation. The genetic correlation between DMI and BWcalv was high in first and third lactation. The genetic correlations between BRmin and DMI as well as BWcalv were moderate in first and third lactation. All genetic correlations estimated between the same trait measured in first and third lactation were very high (>0.86).

The nutritional environment had a large impact on the shaping of genetic correlations among simulated traits, as already shown by [START_REF] Puillet | Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation[END_REF]. When comparing the high and moderate nutritional environments, most changes in genetic correlations between traits measured in first lactation were limited (Fig. 2). The contrasts were larger for third lactation traits because thirdparity cows experienced higher feed restriction and hence stronger trade-offs than first-parity cows in the MS environment. The genetic correlations most impacted by the change in nutritional environment were those estimated between milk production and other traits, especially DMI, BWcalv and BRmin. These traits were generally more unfavourably correlated in more constraining environments due to increased trade-offs. In third lactation, the drop in genetic correlation between DMI and BRmin was also notable.

When comparing HS and LS environments, large changes in genetic correlations were observed whatever the cow parity. The genetic correlation between Milk and Lact_Eff increased and the genetic correlations between Milk and DMI, as well as BWcalv, strongly decreased and became negative indicative of a trade-off between milk production and BW. The genetic correlation between BRmin and BWcalv in third lactation was also substantially reduced. 1 See Table 1 for scenario abbreviations; HS = high nutritional environment; MS = moderate nutritional environment; LS = low nutritional environment.

2 SEs of the mean ranged from 0.004 to 0.01 for all traits and scenarios. Independently from the environment, the heritability defined for AAPs did not impact genetic correlations among simulated traits neither in first nor in third lactation (Fig. 3). Finally, changes in genetic correlations among simulated traits were limited when increasing the CV from 10 to 20% in the HS environment (Fig. 4). The largest change was observed between BRmin and lactation efficiency in third lactation. In the LS environment, increasing the CV of AAPs from 10 to 20% had a larger effect on estimated genetic correlations. However, those changes remained limited and much smaller than the changes induced by the change in environment.

Author's point of views

The main outcome of this study is a suite of scripts and a compiled version of the AQAL software to (1) simulate phenotypic trajectories of milk production and feed efficiency traits in a population of related cows and (2) estimate genetic parameters for traits derived from simulations. The sensitivity analysis performed in this study gives insight into the tuning of genetic parameters of AQAL input traits to simulate datasets of phenotypes with realistic means and genetic parameters. It confirmed that a heritability of 0.35 for all four AAPs and a phenotypic CV of 10% permitted to simulate datasets of phenotypes with realistic variance and covariance components, as suggested empirically by [START_REF] Puillet | Exploring underlying drivers of genotype by environment interactions in feed efficiency traits for dairy cattle with a mechanistic model involving energy acquisition and allocation[END_REF]. Such genetic parameters are typical for production traits in domesticated species in farm or natural conditions [START_REF] Hill | The quantitative genetics of phenotypic variation in animals[END_REF]. As expected, the heritability of AAPs did not influence the mean or variability of simulated performances but only the heritability of simulated underlying traits. The heritability of basal acquisition was the main driver of the heritability of BW at first calving. It had to be set to 0.35 to match heritability of BW at first calving reported in the literature [START_REF] Liinamo | Short communication: Genetic parameters for feed intake, production, and extent of negative energy balance in Nordic Red dairy cattle[END_REF]. The heritability of other traits depended jointly on the heritability of acquisition and allocation parameters. Defining a heritability of 0.35 for allocation parameters ensured reaching heritability in the range of 0.30-0.40 for milk production and 0.25-0.35 for body reserves as usually reported in the literature based on real data [START_REF] Vallimont | Genetic parameters of feed intake, production, body weight, body condition score, and selected type traits of Holstein cows in commercial tie-stall barns[END_REF][START_REF] Liinamo | Short communication: Genetic parameters for feed intake, production, and extent of negative energy balance in Nordic Red dairy cattle[END_REF][START_REF] Manzanilla-Pech | Genetic parameters across lactation for feed intake, fatand protein-corrected milk, and liveweight in first-parity Holstein cattle[END_REF][START_REF] Manzanilla-Pech | Genetic parameters between feed-intake-related traits and conformation in 2 separate dairy populations-the Netherlands and United States[END_REF][START_REF] Manafiazar | Genetic and phenotypic correlations among feed efficiency, production and selected conformation traits in dairy cows[END_REF][START_REF] Li | Genetic parameters for dry matter intake in primiparous Holstein, Nordic Red, and Jersey cows in the first half of lactation[END_REF]. Based on this parameterisation, the heritability obtained for lactation efficiency was equal to 0.22 in the high nutritional environment which is consistent with values reported in the literature for other measures of lactation and feed efficiency, such as feed conversion ratio or residual feed intake [START_REF] Manzanilla-Pech | Genetic parameters between feed-intake-related traits and conformation in 2 separate dairy populations-the Netherlands and United States[END_REF][START_REF] Manafiazar | Genetic and phenotypic correlations among feed efficiency, production and selected conformation traits in dairy cows[END_REF]. As expected, the CV of AAPs mainly influenced the variability of simulated traits. However, it had also an impact on the phenotypic mean and genetic parameters of simulated traits due to increased loss or early culling of females presenting extreme acquisition and allocation values. Even in the high nonlimiting nutritional environment, the proportion of females lost before first lactation was very high when CV of acquisition or allocation was set to 20% or higher. Those high culling rates are not realistic for heifers which suggested that, to be realistic, CV of acquisition and allocation input traits should be kept to values lower than 20%. This study confirmed that the nutritional environment has a much larger influence in the shaping of genetic and phenotypic covariances between traits than heritability and phenotypic CV of acquisition and allocation input parameters. Such information is critical for using the mechanistic AQAL software for breeding applications in the context of GÂE interactions. Indeed, the simulation of milk and feed efficiency performances should be relatively robust to deviations from true values of heritability and CV of AAP. The GÂE interactions emerging from the simulations are then mostly dependent on the nutritional environment specifications and resulting trade-offs in case of energy restriction. The significant changes in correlation among traits across environments suggest that the model adequately captures effects arising from GÂE interactions that are consistent with expectations, i.e. more unfavourable genetic correlations between traits conflicting for access to energy (for example Milk and BW) and a higher emphasis on allocation to fitness-related traits than milk production to achieve higher lifetime lactation efficiency in more unfavourable environments. The set of parameters used for the four AAPs enabled the simulation of cows with moderate BW and milk production in a typical grass-based production system. Hence, trade-offs for access to energy between production and reproduction traits were low in the non-limiting nutritional environment. This was consistent with the literature on trade-offs in ecology as does the increasing trade-offs in poorer environments leading to more unfavourable genetic correlations between traits conflicting for energy access. We expect that the script and software attached to this study will help breeders (1) explore the interest in novel phenotypes derived from phenotypic trajectories and (2) define new breeding strategies for feed efficiency in dairy cows, with a focus on the influence of GÂE interactions on the trait expression.

In conclusion, the code associated to this article enables the simulation of datasets of cow phenotypic trajectories for milk production and feed efficiency traits in different nutritional environments. The present sensitivity analysis confirmed the relevance of defining a heritability value of 0.35 and a CV of 10% for acquisition and allocation input parameters in the AQAL simulation tool to simulate traits with realistic heritability and genetic correlation structure. Furthermore, the heritability and CV of AAPs had a more limited impact than the nutritional environment on genetic correlations among simulated traits. Hence, the release of such a mechanistic simulation tool, coupled with genetic simulation, is an opportunity to explore the interest in new traits for genetic improvement and to define new breeding strategies under various environmental scenarios.

  production in first lactation; Lac_Eff_L1 = lactation efficiency in first lactation; DMI_L1 = DM intake in first lactation; BWcalv_L1 = BW at first calving; BRmin_L1 = minimal body reserves during first lactation. a-g Values within a row with different superscripts differ significantly at P < 0.05.

Fig. 1 .

 1 Fig. 1. Mean genetic and phenotypic correlations (upper and lower triangular matrices, respectively) estimated between simulated traits of dairy cows in the high nutritional environment. Traits: Milk_L1 = milk production in first lactation; Lac_Eff_L1 = lactation efficiency in first lactation; DMI_L1 = dry matter intake in first lactation; BWcalv_L1 = BW at first calving; BRmin_L1 = minimal body reserves during first lactation; Milk_L3 = milk production in third lactation; Lac_Eff_L3 = lactation efficiency in third lactation; DMI_L3 = DM intake in third lactation; BWcalv_L3 = BW at third calving; BRmin_L3 = minimal body reserves during third lactation. Correlations that were not significantly different from 0 at a 5% threshold based on the 20 replicates were crossed. SEs of mean genetic correlations ranged from 0.01 to 0.10 where SEs of genetic correlation estimates ranged from 0.05 to 0.58 for individual estimates within replicates.

Fig. 2 .

 2 Fig. 2. Contrasts between genetic correlations estimated for each trait measured on dairy cows in the moderate vs high nutritional environment (upper triangular) and in the low vs high nutritional environment (lower triangular), considering heritability of 0.35 and coefficient of variation of 10% for acquisition and allocation parameters. Traits: Milk_L1 = milk production in first lactation; Lac_Eff_L1 = lactation efficiency in first lactation; DMI_L1 = DM intake in first lactation; BWcalv_L1 = BW at first calving; BRmin_L1 = minimal body reserves during first lactation; Milk_L3 = milk production in third lactation; Lac_Eff_L3 = lactation efficiency in third lactation; DMI_L3 = DM intake in third lactation; BWcalv_L3 = BW at third calving; BRmin_L3 = minimal body reserves during third lactation. Contrasts that were not significantly different from 0 at a 5% threshold based on the 20 replicates were crossed.

Fig. 3 .

 3 Fig. 3. Contrasts between genetic correlations estimated for each trait measured on dairy cows in scenarios with heritability of acquisition and allocation parameters set to 0.35 vs 0.15 in the high nutritional environment (upper triangular) and in the low nutritional environment (lower triangular), considering a CV of 10% for acquisition and allocation parameters. Traits: Milk_L1 = milk production in first lactation; Lac_Eff_L1 = lactation efficiency in first lactation; DMI_L1 = DM intake in first lactation; BWcalv_L1 = BW at first calving; BRmin_L1 = minimal body reserves during first lactation; Milk_L3 = milk production in third lactation; Lac_Eff_L3 = lactation efficiency in third lactation; DMI_L3 = DM intake in third lactation; BWcalv_L3 = BW at third calving; BRmin_L3 = minimal body reserves during third lactation. Contrasts that were not significantly different from 0 at a 5% threshold based on the 20 replicates were crossed.

Fig. 4 .

 4 Fig. 4. Contrasts between genetic correlations estimated for each trait measured on dairy cows in scenarios with CV for acquisition and allocation parameters set to 20 vs 10% in the high nutritional environment (upper triangular) and in the low nutritional environment (lower triangular), considering heritability of 0.35 for all acquisition and allocation parameters considering a CV of 10% for acquisition and allocation parameters. Traits: Milk_L1 = milk production in first lactation; Lac_Eff_L1 = lactation efficiency in first lactation; DMI_L1 = DM intake in first lactation; BWcalv_L1 = BW at first calving; BRmin_L1 = minimal body reserves during first lactation; Milk_L3 = milk production in third lactation; Lac_Eff_L3 = lactation efficiency in third lactation; DMI_L3 = DM intake in third lactation; BWcalv_L3 = BW at third calving; BRmin_L3 = minimal body reserves during third lactation. Contrasts that were not significantly different from 0 at a 5% threshold based on the 20 replicates were crossed.

Table 1

 1 Effect of the different scenarios on the average number of cows reaching first and third calving in the simulated dataset.

		Heritability		CV		Number of cows starting first	Number of cows starting third lactation
						lactation
	Scenario 1	Acquisition 2	Allocation 3	Acquisition 2	Allocation 3	Mean	SEM	% of born heifers	Mean	SEM	% of primiparous cows
	High nutritional environment (HS)
	HS_3535_1010	35	35	10	10	19 730	3.0	98.6%	14 771	20.0	74.9%
	HS_3515_1010	35	15	10	10	19 730	2.9	98.7%	14 773	16.1	74.9%
	HS_1515_1010	15	15	10	10	19 730	3.0	98.6%	14 774	17.0	74.9%
	HS_3535_2020	35	35	10	30	19 039	9.0	95.2%	11 953	31.1	62.8%
	HS_3535_1030	35	35	20	20	18 238	12.8	91.2%	10 768	37.9	59.0%
	HS_3535_2030	35	35	20	30	17 885	14.3	89.4%	10 457	31.6	58.5%
	Moderate nutritional environment (MS)
	MS_3535_1010	35	35	10	10	19 730	2.9	98.7%	13 486	22.3	68.4%
	MS_1515_1010	15	15	10	10	19 730	3.0	98.6%	13 848	18.6	70.2%
	MS_3535_2020	35	35	20	20	19 039	9.0	95.2%	10 973	30.5	57.6%
	Low nutritional environment (LS)
	LS_3535_1010	35	35	10	10	19 730	2.9	98.7%	9 707	33.9	49.2%
	LS_1515_1010	15	15	10	10	19 730	3.0	98.6%	9 686	25.6	49.1%
	LS_3535_2020	35	35	20	20	19 030	9.0	95.1%	8 267	33.7	43.4%

Table 2

 2 Mean and CV across replicates of simulated performances of cows in first lactation for the simulated traits and the different scenarios.

	DMI_L1 Mean High nutritional environment (HS) Scenario 1 SEM HS_3535_1010 14.1 a 0.01 HS_3515_1010 14.1 a 0.01 HS_1515_1010 14.1 a 0.00 HS_3535_1030 14.1 a 0.01 HS_3535_2020 14.3 b 0.01 HS_3535_2030 14.3 b 0.01 Moderate nutritional environment (MS) CV 2 9.37 d 9.37 d 9.36 d 9.52 c 15.20 a 15.15 a MS_3535_1010 13.8 c 0.01 8.57 e MS_1515_1010 13.8 c 0.00 8.57 e MS_3535_2020 13.7 d 0.01 12.45 b Low nutritional environment (LS) LS_3535_1010 12.3 e 0.00 8.03 f LS_1515_1010 12.3 e 0.00 8.02 f LS_3535_2020 12.2 f 0.00 9.54 c 60.38 A. Bouquet, M. Slagboom, J.R. Thomasen et al. BWcalv_L1 Milk_L1 Lac_Eff_L1 BRmin_L1 Mean SEM CV 2 Mean SEM CV 2 Mean SEM CV 2 Mean SEM CV 2 501.7 a 0.34 10.91 e 4 001.9 a 2.03 11.64 f 41.9 a 0.01 10.86 g 24.4 b 0.04 30.91 h 501.7 a 0.34 10.91 e 4 002.6 a 1.69 11.65 f 41.8 a 0.01 10.86 g 24.4 b 0.03 30.93 h 501.6 a 0.24 10.91 e 4 003.7 a 1.34 11.65 f 41.9 a 0.01 10.85 g 24.4 b 0.03 30.91 h 501.5 a 0.37 11.00 e 3 856.0 d 3.42 25.16 b 40.3 f 0.04 25.11 a 26.1 a 0.06 53.13 c 511.8 b 0.68 20.20 b 4 001.2 a 3.73 21.92 c 41.4 c 0.03 18.38 d 24.0 c 0.05 51.19 e 512.0 b 0.69 20.38 a 3 907.0 b 4.22 28.17 a 40.3 f 0.04 25.26 a 26.0 a 0.06 55.12 b 501.8 a 0.35 10.91 e 3 884.2 c 1.68 10.39 g 41.5 b 0.01 11.17 f 23.1 d 0.04 32.00 g 501.9 a 0.24 10.91 e 3 885.7 c 1.09 10.40 g 41.5 b 0.01 11.17 f 23.1 d 0.03 32.01 g 511.0 b 0.66 20.08 c 3 799.7 e 2.89 19.32 d 40.9 d 0.03 18.66 c 22.7 e 0.05 52.46 d 501.0 a 0.33 10.61 f 3 398.7 f 1.10 9.30 h 40.7 e 0.01 12.83 e 17.5 g 0.04 42.67 f 501.0 a 0.23 10.61 f 3 399.5 f 0.73 9.29 h 40.7 e 0.01 12.84 e 17.5 g 0.03 42.72 f 501.7 a 0.56 18.26 d 3 286.7 g 1.96 17.10 e 40.0 g 0.03 19.41 b 18.7 f 0.05 animal -open space 1 (2022) 100017

a Milk_L1 = milk production in first lactation; Lac_Eff_L1 = lactation efficiency in first lactation; DMI_L1 = DM intake in first lactation; BWcalv_L1 = BW at first calving; BRmin_L1 = minimal body reserves during first lactation. a-h Values within a column with different superscripts differ significantly at P < 0.05.

1 See Table

1

for scenario abbreviations.

2 SEM of CV ranged from 0.01 to 0.13% for the five analysed traits.

Table 3

 3 Heritabilities estimated for simulated traits measured on dairy cows in first lactation across scenarios.

	Scenario 1	Milk_L1 2	BWcalv_L1 2	Lac_Eff_L1 2	DMI_L1 2	BRmin_L1 2
	HS_3535_1010	0.34 a	0.35 a	0.22 b	0.24 b	0.34 a
	HS_3515_1010	0.23 d	0.35 a	0.10 d	0.24 b	0.19 c
	HS_1515_1010	0.15 e	0.15 c	0.10 d	0.15 d	0.14 d
	HS_3535_1030	0.29 bc	0.35 a	0.26 a	0.21 b	0.19 c
	HS_3535_2020	0.32 ab	0.32 ab	0.27 a	0.27 a	0.23 b
	HS_3535_2030	0.29 bc	0.33 ab	0.26 a	0.28 a	0.18 c
	MS_3535_1010	0.34 a	0.35 a	0.22 b	0.19 c	0.33 a
	MS_1515_1010	0.14 e	0.15 c	0.10 d	0.08 ef	0.14 d
	MS_3535_2020	0.31 ab	0.32 ab	0.26 a	0.22 b	0.23 b

Author ORCIDs AB: https://orcid.org/0000-0002-5567-7284 MS: https://orcid.org/0000-0002-0296-356X JRT: https://orcid.org/0000-0001-7068-0572 MK: https://orcid.org/0000-0003-1981-2722 LP: https://orcid.org/0000-0003-2358-7302

Acknowledgements

None.

Financial support statement.

This study was funded by the European Union's H2020 Research and Innovation program GenTORE, under grant agreement no 727213.

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Supplementary material

Supplementary material to this article can be found online at https://doi.org/10.1016/j.anopes.2022.100017.

Ethics approval

This study did not require ethics approval because it relied on simulated data.

Author contributions AB: Conceptualization, Formal analysis, Investigation, Writing original draft; MS: Conceptualization, Writing reviewing and editing; JRT: Conceptualization, Writing reviewing and editing; NCF: Conceptualization, Writing reviewing and editing, Funding acquisition; MK: Conceptualization, Writing reviewing and editing; LP: Conceptualization, Software, Investigation, Writing reviewing and editing.

Declaration of interest

None.


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