Thomas Gonzalez

Antoine Blais

Nicolas Couëllan

Christian Ruiz

Multipath parameters estimation in physically based synthetic environment using robust deep neural regression

Keywords: Convolutional Neural Networks, correlator output images, distributional loss, GNSS, Kullback-Leibler divergence, multipath, multi-channel regression, synthetic environment, truncated Gaussian, Von Mises distribution

Engineer specialized in aerospace telecommunications and optimization for air transport, his current research focuses on the application of Deep Learning techniques to Global Navigation Satellite System (GNSS

I. INTRODUCTION

Multipath phenomenon has been and is still a matter of interest for the GNSS community. Its impact on the receiver can lead to serious deterioration on the positioning accuracy [START_REF] Quan | Convolutional neural network based multipath detection method for static and kinematic GPS high precision positioning[END_REF][START_REF] Vigneau | Neural networks algorithms prototyping to mitigate GNSS multipath for LEO positioning applications[END_REF]. Numerous investigations were conducted on multipath detection [START_REF] Blais | A novel image representation of GNSS correlation for deep learning multipath detection[END_REF][START_REF] Quan | Convolutional neural network based multipath detection method for static and kinematic GPS high precision positioning[END_REF][START_REF] Savas | Multipath detection based on k-means clustering[END_REF][START_REF] Suzuki | NLOS multipath detection using convolutional neural network[END_REF], multipath 1 mitigation [START_REF] Bhuiyan | Advanced multipath mitigation techniques for satellite-based positioning applications[END_REF][START_REF] Tao | Real-time multipath mitigation in multi-GNSS short baseline positioning via CNN-LSTM method[END_REF][START_REF] Phan | A unified framework for gps code and carrier-phase multipath mitigation using support vector regression[END_REF] and multipath estimation [START_REF] Fuchs | Multipath time-delay detection and estimation[END_REF][START_REF] Phan | A unified framework for gps code and carrier-phase multipath mitigation using support vector regression[END_REF]. The main motivation of this work is to estimate multipath parameters using DL algorithms. Few research works have already being conducted on applying Machine Learning (ML) techniques to multipath estimates. For example Support Vector Machines (SVM) have been employed to estimate the multipath code and carrier phase in [START_REF] Phan | A unified framework for gps code and carrier-phase multipath mitigation using support vector regression[END_REF]. Another ML application can be found in the work of [START_REF] Suzuki | NLOS multipath classification of GNSS signal correlation output using machine learning[END_REF] where a Non Line-Of-Sight (NLOS) multipath detection is based on Neural Networks (NN).

The vast majority of ML studies on multipaths focuses on constructing active features from the correlator output. Alternatively, we propose to apply DL directly on signal images. The images are constructed using correlator outputs. The idea is to compute more correlator outputs than usual to expand the information carried by the received signal after the correlation operation [START_REF] Munin | Convolutional neural network for multipath detection in GNSS receivers[END_REF]. The ultimate goal of this strategy is to accurately estimate the multipath components. In this work, two main assumptions are made. First, we assume that the multipath estimation takes place during the tracking step. In other words the direct signal, or Line-Of-Sight (LOS) signal, components are accurately estimated. This also means that the images built are not acquisition grids. Secondly, we only consider a LOS signal and a single multipath in a first step.

Object classification on images by ML techniques has been done successfully in many application domains such as multi-scale detection in [START_REF] Lin | Feature pyramid networks for object detection[END_REF], large scale recognition in [START_REF] Russakovsky | Imagenet large scale visual recognition challenge[END_REF] and even for audio application with a CNN structure in [START_REF] Hershey | CNN architectures for large-scale audio classification[END_REF]. CNN have proven to be very effective on images-like data due to their sparse connectivity and automatic feature extraction layers. They are able to extract high-level features from input images. The highly informative data are then processed into the fully-connected layers. Neurons in fully connected layers bring together the extracted information to form the final output.

CNN regression has been less studied than CNN classification. Among success of CNN regression applications, one can refer to the following works. [START_REF] Yi | Age estimation by multi-scale convolutional network[END_REF] have proposed based human age estimation on images analysis by CNNs, Human poses assessment by CNN was studied by [START_REF] Li | 3D human pose estimation from monocular images with deep convolutional neural network[END_REF] and another example of a CNN research in regression domain was conducted for real time 2D/3D registration in [START_REF] Miao | A CNN regression approach for real-time 2D/3D registration[END_REF].

The CNN usually requires a large amount of labelled data to be trained. Multipath labelled data are scarce and difficult to obtain. Therefore one solution is to randomly generate multipath components as in [START_REF] Munin | Convolutional neural network for multipath detection in GNSS receivers[END_REF]. It provides an estimate of the CNN estimation performance. To improve the model performance assessment, we propose to run experiments in 3D realistic environments. The multipath components generation-estimation chain will follow the diagram illustrated in Figure 1. Multipath signals will be generated from a specialized software in GNSS signal propagation in constrained environments. Implemented realistic urban environments give frame propitious to multipath phenomenon and all of them are then collected. The recovered multipath parameters are used as a reference base for multipath creation. Their reconstruction in the synthetic correlator outputs is made from the multipath parameters collected during the simulation. The CNN algorithm is fed with the generated dataset to carry out the multi-channel regression task.

The article organization is as follows. Section II explains the distinct elements contributing in the dataset generation chain. Section III deals with the DL used for the experiments. Section IV details the conducted experiments with numerical results. Section V concludes the article. In order to have enough labelled data to train the CNN, a solution is to generate synthetic multipaths via a generator. In this work, we propose to use the SE-Nav software (Oktal-SE, 2022). It simulates the GNSS signals propagation in constrained environments. The simulator generates realistic scenes like urban environments, suitable for multipath phenomenon. It computes the signal and multipaths propagation paths based on ray-tracing modeling. Multipaths parameters are available in the output files at the end of each simulation. The parameters are used in the synthetic correlator output generator to produce the I & Q image dataset. An alternative dataset construction, sharing the same correlator outputs generator, is illustrated in Figure 11 in Appendix A. Its process is detailed in subsection IV.1 a).

Correlator output images generator

In addition to the classical Early-Prompt-Late correlator points, other correlator output values are computed to form a delay x Doppler frequency shift 2D grid as illustrated in Figure 2. By adopting such an approach, additional information is aggregated to the CNN data on both I and Q channels. As said in section I, the signal is assumed to be tracked, meaning that only multipath signal parameters remain unknown. This correlator output image generation process is more precisely explained in [START_REF] Munin | Convolutional neural network for multipath detection in GNSS receivers[END_REF]. In this work the dataset generator takes as inputs the multipath parameters from the GNSS signal propagation SE-Nav software. In realistic constrained environments which will be detailed in subsections IV.1 b) and IV.1 c), a moving receiver will collect the incoming LOS and its corresponding multipath signals.

III. CNN REGRESSION AND DISTRIBUTION LOSS

The CNN automatic feature extraction mechanism is based on convolution processed by kernel filtering (see [START_REF] Goodfellow | Deep learning[END_REF] for detailed explanations on filtering mechanism). The CNN model structure used in this work has been inspired by Visual Geometry Group (VGG)-16 architecture. VGG-16 networks have experimentally demonstrated their performances in image-recognition applications [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF]. The particularity of this network structure is the image size reduction while applying a growing number of filters. Figure 3 presents the CNN architecture we have used. The network can be divided in several convolutional blocks followed by dense layers. Each convolutional block is composed of consecutive convolutional layers which extract information from the images. A max pooling layer completes the convolutional block. It synthesizes the extracted information and accelerates data processing in the network. The fully-connected layers carry out the regression task on the extracted features from previous block.

The usual concept of regression task consists in building a model able to compute real responses from input x. The regression model N θ , where θ represents the model parameter, has to be trained in order to predict values close to the known labels y. The model parameter is calibrated by minimizing for example a square loss function l(θ , x, y) = (N θ (x)y) 2 . The target value y, considered as the ground-truth label of a given input x, is experimentally obtained or measured. This presence of noise during the labelling of the data can lead to a regression on a label value that is false, and thus hinder the generalization process. The soft labelling technique starts from the premise that the measured value y is a random observation from a prior distribution Y |x. The main idea of this method is the estimation of the underlying distribution of the target Y |x instead of estimating the sharp value y. Hence, the soft label of a given input x will be Y |x with y = E[Y |x] where Y |x is a prior distribution made on the target y.

The Y |x distribution will undergo a discretization in order to be implemented in a ML algorithm. The resulting histogram formed on K bins is the discrete version of Y |x. As it can be observed in Figure 3, the CNN will be trained to reproduce the soft labels thanks to the multiple outputs in the last layer. In the sequel of the article, CNN-HL will refer to a CNN with a histogram distribution output. To minimize the difference between the target distribution Y |x and the CNN-HL prediction, the square loss function is substituted by a histogram loss. Based on KL divergence, it computes the dissimilarities between two probability distributions. For two continuous probability distributions P and Q, the KL divergence is

D KL (P||Q) = P log( P Q ) (1)
The histogram loss, noted HL, is the score of dissimilarities between the predicted an real distributions. Its expression is derived from equation 1 and can be simplified as:

HL(θ , x) = - K ∑ i=1 p i log(q i (θ , x)) (2) with
• p i the probability mass of the target y in bin i,

• q i (θ , x) the i th output of the network for an input x,

• K the number of outputs of the CNN-HL.

Next, we discuss two relevant choices for the prior distributions p.

Truncated Gaussian distribution

Distinct probability distributions have been used in experiments by [START_REF] Imani | Improving regression performance with distributional losses[END_REF]. However they have experimentally shown that the truncated Gaussian is a good default choice. This means that for an input x with its ground-truth label y t , the probability density function p can be expressed as:

∀y ∈ [a, b], p(y|x) = 1 Z √ 2πσ 2 e - (y-y t ) 2 2σ 2 with Z = 1 2 (er f ( b -y t √ 2σ 2 ) -er f ( a -y t √ 2σ 2
))

(3) where • x → er f (x) is the Gauss error function,

• [a, b] is the data targets support set,

• y t the ground truth target value of the input x,

• σ 2 the variance parameter.

Von Mises distribution

In prior research work [START_REF] Gonzalez | Distributional loss for convolutional neural network regression and application to GNSS multi-path estimation[END_REF], it has been observed that the truncated Gaussian distribution show difficulties concerning the multipath phase estimation. It is not adapted for angular targets. To take in account the cyclical behaviour of angles, we opted for a Von Mises probability distribution. Figure 5 shows the difference between truncated Gaussian and Von Mises distributions for a multipath phase ground-truth value equals to 0 • . The truncated Gaussian distribution has only one peak for 0 • whereas the Von Mises distribution shows two peaks around 0 • and 360 • . It better reflects proximity of angles when a modulo 2π is applied. For this probability distribution, the probability density function p is:

p(y|x) = e κcos(y-y t ) 2πI 0 (κ) (4)
where • I 0 is the modified Bessel function of order 0,

• 1 κ is analogous to σ 2 for a normal probability distribution. IV. EXPERIMENTS

Experimental setup

Experiments were made using three distinct datasets. Each dataset contains 10 000 samples of labelled I and Q images pairs. Whitin a dataset the image size is constant. However, to observe the evolution in performance of our CNN-HL model with less correlator outputs (meaning decreasing image resolution), several variations of these datasets were created for 80x80, 40x40 and 20x20 pixels images. The datasets are further detailed in the following sections.

a) Dataset with randomly generated multipaths

A dataset with randomly generated multipaths is constructed as a baseline for CNN performance assessment. The random multipath parameters are generated as follow:

• α MP ∼ U [0.1, 0.9] the attenuation coefficient of the multipath with respect to the LOS amplitude,

• ∆τ MP ∼ U [0, 1.5Tc1] the code delay in excess compared with the useful signal,

• ∆ f MP ∼ N (0, 125 3 ) the difference in Doppler shift with the useful signal, • ∆φ MP ∼ U [0, 2π] the phase difference with the useful signal.

All the multipaths in the 'Random multipaths' dataset were generated at a C/N0 level fixed to 40dB -Hz to simulate a typical urban environment receiving condition.

b) Dataset with 'Toulouse center, France' 3D environment

The Capitole square of Toulouse is located in the center of the French city Toulouse. Characterized by its small streets surrounded by 3-4 floor buildings, Toulouse city center is propitious to multipath phenomenon. As it can be seen in Figure 6, main materials are brick and concrete for this scene. Four distinct trajectories were constructed for a mobile receiver to have massive signal collection with many cases of multipath. To supplement the dataset, the receiver follows those four trajectories twice, at different times to have distinct satellite constellation configurations.

The observed multipaths components distributions are plotted in Figure 7. Observations on multipath phase shift support the assumption made in section IV.1 a) with a uniform distribution of this parameter. Doppler frequency shift also follows a normal distribution, but with a different variance. The observations are interesting also for the other parameters since we can note that the delay in excess and the multipath attenuation seem to follow an exponential law. To form the various datasets, 10 000 multipaths are randomly picked among all the observed interference to reproduce in the synthetic correlator outputs image generator. The Manhattan scene in SE-Nav software is an environment characterized by sky-scrapper buildings and straight-lined streets. Even if streets are not as narrow as in Toulouse city, multipath phenomenon is present due to the high reflection potential of the high buildings. The receiver environment is mainly composed of glass and concrete (as it is shown in Figure 8). The observed multipaths distributions are different compared with the Toulouse collect. In Manhattan, multipaths are stronger and with larger delays than the multipaths generated in the French city. For the same reason than for Toulouse city dataset, the signals collect is made over four different trajectories at two distinct times of day. Doppler frequency shifts in NYC environment have a similar distribution (Figure 9) compared with the multipaths generated in Toulouse environment. Nevertheless, two parameters raise questions on them: the multipath magnitude coefficient seems to be attracted around two extreme values, 10% and 100% of the LOS signal. The amount of multipaths with a 100% attenuation coefficient (meaning with the same magnitude as the LOS signal), could be explained by the receiver local environment, surrounded by a large amount of metal structure and allowing a near 'perfect' reflection on those surfaces. The second parameter catching attention is the phase shift, with a uniform distribution of the multipath phases except between 225 • and 275 • , where the multipaths with such phase shift form a larger group.

Experiment setup

The figures or values provided in the next section are results averaged over 10 runs. For each run, the CNN-HL dataset is formed as follow: 8000 samples for the training set and 1000 samples for the validation set (both are used during the CNN-HL training phase). Then the last 1000 samples are to assess the performances of the trained network. The dataset is shuffled between each run to give distinct training conditions to the CNN-HL. The network is trained on 100 epochs with a batch size of 1000 samples. The CNN-HL learning rate is empirically fixed to 10 -3 . Convolutional filters are fixed to a 3x3 size while max-pooling kernels are 2x2 size as it is shown in Figure 3. The number of outputs of the neural network is fixed to K = 100 bins. Preliminary experiments were made on 80x80 images of all dataset to determine the 'best' σ and κ hyper-parameters for the estimation task. The corresponding tables and figures are available in Appendix C. Then the experiments on various image sizes were made with those optimal hyper-parameters.

Results

Performance evaluation is based on Mean Absolute Error (MAE) results averaged over 10 runs and computed as follows:

1 L L ∑ l=1 Ẽ[N θ (x l )] -y l with Ẽ[N θ (x l )] = K ∑ i=1 q i (θ , x l ).c i and
• L is the number of samples in the test dataset,

• x l is the l-th test sample,

• y l is the target value for the l-th sample.

• q i (θ , x l ) is the probability of the i-th softmax output neuron associated to x l ,

• c i is the center of the i-th bin for i ∈ [1, K] (where the interval [a, b] has been partitioned into K equal subdivisions).

Tables 1,2,3 and 4 gather the average MAE performances of the CNN-HL on the four multipath components for the various datasets. For those experiments, σ and κ values where set following the Figure 10 results. The plots in Figure 10 illustrate the optimal σ and κ parameters for CNN-HL on each 80x80 dataset and for each multipath parameter. The x-axis is labelled according to the bin width w since σ = β w, β ∈ R + . The estimation performances are plotted according to increasing σ for delay, Doppler frequency and attenuation, and κ for the phase. Tables 5,6, 7 and 8 in Appendix C show the numeric values of the plots. The Figure 10 gives an idea of the prior distributions parameter impact on estimation performances.

Discussion

a) Tests on various σ and κ values

As it can be seen in Figure 10, σ values have an impact on the averaged parameter estimation accuracy. For example in Figure 10.a, a minimum in the delay estimation error performance can be observed for both 'Random multipaths' and NYC datasets whereas there is no significant influence on the delay estimation on the Toulouse multipath set. Figure 10.b shows lower Doppler frequency estimation errors for small σ on 'Random' and NYC datasets. On the contrary, the Doppler frequency estimation performs better for high σ values when estimating multipaths collected in Toulouse dataset. In Figures 10.c and 10.d, small σ and κ values lead to a raise of the parameter estimation error. For κ adjustment, the estimation error variance is consequent and does not allow the finding of an optimal κ value for the phase estimation. It can also be observed that for a same σ or κ value, the estimation performances can significantly vary depending on the multipath generation. An example of this is illustrated in Figure 10.a where the delay estimation error on Toulouse multipaths is always lower than the delay estimation error on NYC multipaths.

b) Multipath parameters estimation

The results on the tables show the impact of the image resolution on the parameter estimation errors. The error raises as the resolution of the input images deteriorates, with the example of the delay estimation in Table 1 on 'Random multipaths' and NYC datasets. We notice a steadiness in delay estimation for Toulouse multipaths where the estimation performance has a lower error compared with the Random multipaths. This consistency in the estimation error is also observable for the Doppler frequency when estimating Toulouse multipaths. A similar behaviour is noticeable for the multipath magnitude error performance for NYC dataset. For the delay and Doppler frequency estimation, we believe that Toulouse estimation performances are better due to the dataset distribution (in Figure 7 in section IV.1 b), ∆τ MP ∈ [0, 0.3Tc]). Generally, the CNN-HL estimation performances are similar between randomly generated multipaths and interference collected on simulations.

V. CONCLUSION

In this work we have proposed to estimate the multipath components using a CNN model that embeds a soft-labelling mechanism. Synthetic correlator output images were generated based on GNSS signal propagation simulations. The constructed I & Q images pair gathers information about the tracked signal and one multipath. Regression task on images is a difficult task for a DL algorithm. Soft labelling technique was applied to improve the multi-channel regression performances of the network. This was possible with a softmax output layer estimating the distribution of the parameters to estimate. The histogram loss function used during the training was constructed from the Kullback-Leibler divergence. It allowed to compare the target histograms and the predicted distributions. The experiments in section IV were conducted on several datasets in order to measure the stability and robustness of the algorithm facing multipaths from various environments. 'Random multipath' data was generated according to assumed hypothesis while NYC and Toulouse city sets were constructed on signal propagation simulator results.

The achieved averaged MAE demonstrates the capacity of the CNN-HL to estimate the delay, Doppler frequency, attenuation and phase of the multipath. The CNN-HL shows similar performances when estimating randomly generated multipaths or as outputs of physically-based signal propagation software. The experiments also shown the importance of the hyper parameters values to reach the optimal performances of the CNN-HL estimation. In addition, we have proposed the use of the prior Von Mises distribution for angular labels. Comparing with the results on the same dataset in [START_REF] Gonzalez | Distributional loss for convolutional neural network regression and application to GNSS multi-path estimation[END_REF], the new prior distribution gives improved performances. Results on phase estimation by CNN-HL based on Von Mises distribution may encourage the use of this technique in angular regression problems. From a model perspective, future investigations should focus on more realistic dataset with the estimate of several multipaths at the correlator outputs.

Figure 1 :

 1 Figure 1: Multipath generation and estimation workflow

Figure 2 :

 2 Figure 2: Correlator output images elaboration process. Firstly the received signal is divided into two orthogonal components. Then a grid G is generated from τ and δ f subset values. τ and δ f refer to the parameters of the local signal replica. The resulting 2D-images are used as an input for the CNN regression algorithm.

Figure 3 :

 3 Figure 3: CNN architecture used for regression task with distributional outputs on 80x80 images. The top figure illustrates the first convolution blocks. Bottom figure shows the last convolution block followed by the dense layers with a multiple outputs layer. Images generated with NN-SVG LeNail (2019).

Figure 4

 4 Figure 4 is an example with a truncated Gaussian distribution for a Doppler frequency shift target equal to 45Hz.

Figure 4 :

 4 Figure 4: Truncated Gaussian probability distribution with ∆ f D = 45 Hz.

Figure 5 :

 5 Figure 5: Comparison between truncated Gaussian (left) and Von Mises (right) distributions for ∆φ MP = 0 rad.

Figure 6 :

 6 Figure 6: Capitole place (Toulouse center) multipath environment, SE-Nav software

Figure 7 :

 7 Figure 7: Multipaths components distribution in the Toulouse city receiving simulations (10 722 in total).

Figure 8 :Figure 9 :

 89 Figure 8: Manhattan multipath environment, SE-Nav software

( a )

 a Averaged ∆τ MP estimation performances for 80x80 pixel images with respect to the bin width w. (b) Averaged ∆ f D estimation performances for 80x80 pixel images with respect to the bin width w. (c) Averaged α MP estimation performances for 80x80 pixel images with respect to the bin width w. (d) Averaged φ MP estimation performances for 80x80 pixel images with respect to κ.

Figure 10 :

 10 Figure 10: Averaged estimation performances for 80x80 pixel images with respect to σ and κ parameters

Table 1 :

 1 Average MAE ∆τ MP in 10 -2 Tc for different simulated environments.

	Dataset

Table 2 :

 2 Average MAE ∆ f D in Hz for different simulated environments.

	Dataset

Table 3 :

 3 Average MAE α MP in % for different simulated environments.

	Dataset

Table 4 :

 4 Average MAE ∆φ MP in degrees for different simulated environments.

	Dataset

1Tc is the chip period, a basic defining parameter of this type of signal. Tc = 1/1023 ms for the Global Positioning System (GPS) L1 C/A signal.

APPENDIX A. MULTIPATH GENERATION PROCESS

Figure 11: Representation of the different multipath I and Q images generation processes. The Fully Synthetic (FS) is the method employed for 'Random multipaths' dataset. The second approach, the Scenario Based Synthetic (SBS) method, uses a distinct parameters generation scheme. Based on the multipath parameters collected on the SE-Nav simulations, this strategy was used for Toulouse and Manhattan datasets.

APPENDIX B. HISTOGRAMS DEPENDING ON SIGMA VALUE


Dr Arnaud Viarouge
email: arnaud.viarouge@u-paris.fr

Hoyeon Lee

Grégoire Borst

Attention to number requires magnitude-specific inhibition

Keywords: Numerical cognition, numerical estimation, Inhibitory control, conflict adaptation

Recent studies have shown that the ability to process number in the face of conflicting dimensions of magnitude is a crucial aspect of numerosity judgments, relying in part on the inhibition of the nonnumerical dimensions. Here we report, for the first time, that these inhibitory control processes are specific to the conflicting dimension of magnitude. Using a non-symbolic numerical comparison task adapted to a conflict adaptation paradigm on a group of 82 adults, we show that congruency effects between numerical and non-numerical information were reduced only when the conflicting dimension was the same in the preceding incongruent trial. Attention to number thus involves inhibitory control processes acting at a specific level of information. These results contribute to better characterize the domain general abilities involved in numerical cognition, and provide evidence for a specific interaction between numerosity perception and inhibitory control.

Introduction

One of the main objects of investigation in the field of numerical cognition in the past decades has been our number sense, that is our ability to estimate and compare the quantities of objects in a set ("numerosities", [START_REF] Izard | Newborn infants perceive abstract numbers[END_REF][START_REF] Feigenson | Core systems of number[END_REF]. Classically, the non-symbolic numerical comparison task is used to assess the precision of our number sense. In this task, the participants are asked to decide which of two rapidly presented sets of items (usually dots) is the more numerous.

Because of the perceptual features of the sets of dots presented in the non-symbolic numerical comparison task, many studies have called into question its underlying mechanisms. Recently indeed, beyond the precision of our number sense, studies have suggested that the non-symbolic numerical comparison task also relies on the ability to process numerosities independently of competing dimensions of magnitude, such as size, density, or occupied space (hereafter referred to as "attention to number", [START_REF] Wilkey | Attention to number : The convergence of numerical magnitude processing, attention, and mathematics in the inferior frontal gyrus[END_REF]; see also Piazza et al., 2019). This stems in part from the systematic observation that the non-numerical dimensions of magnitude present in the stimuli, impact our performance in the task [START_REF] Gebuis | The interplay between nonsymbolic number and its continuous visual properties[END_REF]2013).

One of the proposed mechanisms of attention to number is inhibitory control, which allows blocking the irrelevant, non-numerical dimensions of magnitude when processing numerosities [START_REF] Fuhs | ANS acuity and mathematics ability in preschoolers from low-income homes : Contributions of inhibitory control[END_REF][START_REF] Fuhs | The Role of Non-Numerical Stimulus Features in Approximate Number System Training in Preschoolers from Low-Income Homes[END_REF][START_REF] Rousselle | The development of automatic numerosity processing in preschoolers : Evidence for numerosity-perceptual interference[END_REF]Szücs et al., 2013). Recently, using a nonsymbolic numerical comparison task adapted to a negative priming paradigm, Viarouge and colleagues (2019) provided evidence for the role of inhibitory control during the numerical comparison of two arrays of dots whereby number was incongruent with dots' size.

While inhibition seems to play a role when performing numerosity judgment tasks, studies are needed to better characterize the inhibitory control processes at play in this context. In particular, one opened question is the degree of specificity of the inhibitory control processes involved in attention to number. On one hand, one could postulate the existence of a common mechanism aimed at blocking any conflicting information of magnitude when processing numerosities. On the other hand, the inhibitory control processes at play when attending to number could be specific to the irrelevant dimension of magnitude conflicting with number.

One way to disentangle these hypotheses is by studying the congruency sequence effect, or Gratton effect [START_REF] Gratton | Optimizing the use of information: Strategic control of activation of responses[END_REF], consisting of a reduced congruency effect following incongruent trials. This effect has previously been observed in several classic inhibitory control tasks, such as the Stroop or the Flanker task [START_REF] Hommel | A feature-integration account of sequential effects in the Simon task[END_REF][START_REF] Larson | Conflict adaptation and cognitive control adjustments following traumatic brain injury[END_REF]. It is most commonly explained by the conflict adaptation account (Botvinick et al., 2001), which states that the cognitive control processes -including the inhibition of irrelevant informationinvolved in solving an incongruent trial facilitate the resolution of the next conflict.

Using such conflict adaptation paradigm, several studies have investigated the specificity of cognitive control processes in the context of numerical tasks, with mixed results. Some studies have concluded to a general inhibitory control process [START_REF] Borst | Inhibitory control in number-conservation and class-inclusion tasks : A neo-Piagetian inter-task priming study[END_REF][START_REF] Linzarini | When Stroop helps Piaget : An inter-task positive priming paradigm in 9-year-old children[END_REF]. For instance, using an inter-task conflict adaptation paradigm, Linzarini and colleagues (2015) observed that 9-year-olds performed better in the incongruent trials of a Piaget-like number conservation task (two rows of tokens with either identical numbers and different lengths, or different numbers and equal lengths) when they were preceded by incongruent Stroop trials. The authors concluded that their results supported a general inhibitory control process across both tasks in 9-year-old children. In their study however, [START_REF] Notebaert | Cognitive control acts locally[END_REF] observed a congruency sequence effect in a group of adults across two tasks (a Simon task using on laterally presented X's and a SNARC task on centrally presented symbolic digits) only when the relevant information was the same in both tasks, leading the authors to conclude for more specific cognitive control abilities. Since the irrelevant information remained identical throughout their study, the authors argued that their results showed the specificity of the processing of the relevant information, rather than of the inhibition of the irrelevant information. To date, no study has investigated this question in the context of a non-symbolic numerical task, with a constant relevant information (i.e. numerosity) and a varying irrelevant information (i.e. conflicting dimensions of magnitude).

Thus, in the current study, we adapted a non-symbolic comparison task to a conflict adaptation paradigm, using different non-numerical dimensions of magnitude, in order to investigate whether the inhibitory control processes at play were specific to the irrelevant information, or general across the non-numerical dimensions of magnitude. We reasoned that, if attention to number requires dimension-general inhibition, then we should observe cross-dimension Gratton effects, i.e. reduced congruency effects in trials preceded by incongruent trials, regardless of the irrelevant dimension of magnitude conflicting with number. On the other hand, if the inhibitory control processes involved in attention to number are dimension-specific, we should only observe within-dimension Gratton effects, i.e. reduced congruency effects following incongruent trials involving the same irrelevant non-numerical dimension of magnitude.

Method

Participants

Participants were 82 undergraduate students at Université de Paris (4 males, mean age = 21 years, SD = 4.8), participating in the study for course credit.

The ethical board of the Faculty of Psychology ruled that in light of the potential risks for the participants of the present study, no formal ethical approval by one of the national ethical committee was needed in agreement with the Ethical law governing human research in France.

Participation was voluntary after obtaining signed informed consent. All participants were tested in accordance with national and international norms governing the use of human research participants.

Procedure

The task consisted in a non-symbolic numerical comparison task adapted to a conflict monitoring paradigm. Participants were asked to indicate as quickly and as accurately as possible which of two arrays of dots, presented on the left and right side of a computer screen, contained more dots, by pressing either the D (left array) or the K (right array) key on the keyboard. The task was programmed using the jsPsych JavaScript library (de [START_REF] De Leeuw | jsPsych: A JavaScript library for creating behavioral experiments in a web browser[END_REF], and the data collected online on the Cognition.run platform. Each presentation of a pair of dot arrays will thereafter be referred to as an item. A trial was defined as a sequence of two items, a prime item followed by a probe item (Figure 1).

In order to adapt the non-symbolic numerical comparison task to a negative priming paradigm, two non-numerical dimensions needed to be manipulated orthogonally, on a trial-by-trial basis. That is, the congruency of one dimension relative to number needed to be manipulated, at the trial level, independently of a third dimension. Such manipulation at the trial level would prove difficult using dimensions such as average item size and total surface area, which are, by definition, correlated. This is why the arrays of dots were generated with MATLAB using the Number, Size, and Spacing dimensions, as defined by DeWind and colleagues (2015), which form an orthogonal space whereby the three dimensions can be manipulated independently. That is, we manipulated a combination of aspects of average item size and total surface area on one hand (the Size dimension), and a combination of convex hull and density (the Spacing dimension) on the other. By doing so, for each item (prime or probe), one nonnumerical dimension of magnitude was either congruent or incongruent with Number (thereafter referred to as the "congruency-related dimension"), while the other was constant across the two arrays. Hence, the congruency-related dimension (e.g. Size) and its composing magnitudes (e.g. average item size and total surface area) are the only changing parameters between our conditions, at a trial-by-trial level (see Appendix A, Table A1, for a table showing the congruency and ratios of the four composing magnitudes in each condition, and Figures A1 andA2 for examples of stimuli in relation to our results). This led to four types of items depending on the congruency-related dimension (Size or Spacing), and its congruency with Number (congruent or incongruent). All possible combinations of the four types of items were presented, leading to 16 different trial conditions (i.e. prime-probe pairs). Consequently, for half of those 16 trial conditions, the congruency-related dimension was identical in the prime and in the probe items ("within-dimension trials"), while for the other half the congruency-related dimension was different between the prime and the probe ("between-dimension trials"). Each of the 16 trial conditions was presented 10 times in a random order, for a total of 160 trials.

Each trial started with a central fixation cross and two white circles (1000ms). The arrays of dots from the prime item were then presented in the white circles until the participants gave a response, for a maximum duration of 750ms, at which time they were replaced by a gray screen. As soon as the participants gave their response to the prime, a gray screen was presented for 1000ms, and then the fixation cross and the probe were presented. In order to separate the trials and minimize the potential effect of probes on the following primes [START_REF] Linzarini | When Stroop helps Piaget : An inter-task positive priming paradigm in 9-year-old children[END_REF], a visual mask was presented for 1500ms between each prime-probe pair.

Using the jsPsych calibration plugin, the stimuli were set to be presented at a fixed size of 14.7x11cm on the participants' screen. Using these settings, each array of black dots was presented on a white circle with 6.71cm diameter (visual angle 7.64° for a distance of 50cm from the screen). The center of the arrays of dots was at 4.14cm from the central fixation cross (visual angle 4.73°). In order to ensure a high enough level of performance for the analyses of the priming effects on the probe items' reaction times, all the items presented a numerical ratio of approximately 3:5. Different number pairs were presented (9-15, 10-17, 12-20, 14-23), so as to avoid recognition of the numerosities throughout the task. The ratio of the congruency-related dimension was always set at a third (with a 10% margin of error for the Spacing dimension) of the numerical ratio (i.e. approximately 1:5), and hence the underlying dimensions (item size and total surface area for Size, convex hull and density for Spacing) were also at a constant ratio with number In a Number-Size congruent item, both item size and total surface area were congruent with number, while they were both incongruent with number in a Number-Size incongruent item. The same was true for convex hull and density when Spacing was the congruency-related dimension. As mentioned above, the third dimension (not congruency-related) was constant between the two sets of dots, and across the items. For instance, Size was constant across the two sets of dots and took the same value in the Spacing-congruent and the Spacing-incongruent trials. The underlying dimensions (e.g. item size and total surface area for Size) were also at a constant ratio between the two sets of dots and across the items. The 16 combinations of number pairs were each presented ten times across the whole task.

The 4 possible combinations of the side of the correct response were also counterbalanced across the total number of trials in the task.

Results

For each participant, we removed trials with reaction times faster than 150ms and slower than 3 standard deviations above the individual mean. This led to removing 2.54% of the full dataset. One participant was removed due to their average reaction time being slower than 3 standard deviations above the average reaction time of the sample. In order to ensure that participants were sufficiently attentive and engaged with the experiment to yield reliable/valid behavioral data, we restricted our analyses to participants who performed above 75% of accuracy rate in each of the four conditions (congruent or incongruent with Size or Spacing) to be able to analyze potential congruency sequence effect. Sixteen participants were removed based on this criterion. The analyses described below were thus run on a sample of 64 participants, meeting the requirements of our power analysis 1 .

Analyses of the prime items

A 2 (Dimension: Size, Spacing) x 2 (Congruency: congruent, incongruent) repeated measures ANOVA on the reactions times for the correctly responded prime items showed a main effect of Congruency, F(1, 63) = 117.12, p < .0001, η 2 p = .65, and a significant Congruency x Dimension interaction, F(1, 63) = 16.86, p < .0001, η 2 p = .21. Post-hoc one-way ANOVAs revealed that the congruency effects for each non-numerical dimension were both significant, with a stronger number/Spacing congruency effect

Analyses of the congruency sequence effects

The analyses of the congruency sequence effects were run on the reaction times for the correct responses on the probe items following correct responses on the prime items. We first ran a 2 (Prime dimension: Size or Spacing) x 2 (Probe dimension: Size or Spacing) x 2(Prime congruency: congruent or incongruent) x 2 (Probe congruency: congruent or incongruent) repeated measures ANOVA on the reaction times. As for the analyses of the prime items, we found a significant main effect of the probe's Congruency, F(1, 63) = 142.53, p < .0001, η 2 p = .69, as well as a significant interaction between the probe's Congruency and Dimension, F(1, 63) = 17.89, p < .001, η 2 p = .22, indicating again a stronger congruency effect for Spacing (incongruent: M = 653, 95% CI [636.7,669.3], congruent: M = 562, 95% CI [549.5, 574.5]), than for the Size dimension (incongruent: M = 627, 95% CI [612.9, 1 We ran an a priori power analysis using G*Power, Version 3.1; [START_REF] Faul | Statistical power analyses using G*Power 3.1 : Tests for correlation and regression analyses[END_REF] in order to estimate the sample size. Based on the effect size of the negative priming effect in the non-symbolic comparison task found in adults (ηp 2 = .29, Viarouge et al., 2019), we chose a conservative effect size of ηp 2 = .2. The analysis indicated that a sample size of 55 was necessary to detect such effect size in a 2 (irrelevant dimension in the prime) x 2 (irrelevant dimension in the probe) x 2 (prime congruency) x 2 (prime congruency) repeated measures ANOVA with a power of .95 (α = .05). These significant interactions reflected different patterns of results (see Table 1). When the irrelevant non-numerical dimension of the prime and the probe items were the same (withindimension trials), the congruency effects on the probe items were smaller for incongruent than for congruent prime items, reflecting a typical Gratton effect. However, this was not the case when the irrelevant non-numerical dimension of the prime and the probe items were different (betweendimension trials). In this condition, we observed a reversed Gratton effect, with a larger congruency effect on the probe items after an incongruent than after a congruent prime item.

Table 1. Mean reaction times (ms) and 95% confidence intervals for each of the 16 trial conditions.

Discussion

This study aimed at testing the specificity of the inhibitory control processes at play in the nonsymbolic numerical comparison task, using a conflict adaptation paradigm. First, we observed classic congruency effects between number and non-numerical dimensions of magnitude in the prime items. Interestingly, we found stronger congruency effects between number and Spacing than between number and Size, in line with studies indicating a stronger influence of extrinsic dimensions (such as convex hull or occupied space) than intrinsic dimensions (such as individual item size) in adults [START_REF] Gilmore | Congruency effects in dot comparison tasks : Convex hull is more important than dot area[END_REF]. Since the ratios between number and the non-numerical dimensions were identical for Size and Spacing, this result could indicate a higher salience of the Spacing dimension than the Size dimension.

Second, our analyses of the congruency sequence effects showed significantly different patterns between the within-and between-dimension pairs. When the irrelevant non-numerical dimension was identical between the prime and the probe items, the congruency effect was reduced after incongruent primes, showing a classic congruency sequence effect (i.e., Gratton effect). On the contrary when the irrelevant non-numerical dimensions in the primes and in the probes were different, we observed a reversed Gratton effect with larger congruency effects following the incongruent primes. Taken together, these results indicate that the inhibitory control processes involved in the non-symbolic comparison task are specific to the non-numerical dimension conflicting with number.

Previous studies have argued that conflict adaptation was mainly due to an improved processing of the relevant information [START_REF] Notebaert | Cognitive control acts locally[END_REF]. In our task, the relevant information remains constant throughout the task, thus an improved processing of the relevant information cannot in and by itself explain the different patterns observed between the within-and between-dimension trials, supporting the role of inhibitory control processes in the observed congruency sequence effect. It is important to note that other accounts based on binding effects have been proposed for the Gratton effect, which do not involve conflict adaptation [START_REF] Schmidt | Now you see it, now you don't: controlling for contingencies and stimulus repetitions eliminates the Gratton effect[END_REF]. While conflict adaptation is unlikely to be the sole underlying mechanism of the Gratton effect, several studies have shown that the latter remained even after controlling for possible binding confounds (Akcay & Hazeltine, 2007;[START_REF] Blais | The Gratton effect remains after controlling for contingencies and stimulus repetitions[END_REF].

Although observed in previous studies [START_REF] Notebaert | Cognitive control acts locally[END_REF][START_REF] Brown | A computational model of fractionated conflict-control mechanisms in task-switching[END_REF], the increased congruency effects following incongruent prime items observed in the between-dimension conditions is can be surprising. One possible interpretation, further supporting the dimension-specific nature of the inhibitory control involved in our task, is in terms of switch cost between the prime and the probe items when the irrelevant non-numerical dimensions changes. This account predicts in particular that, in between-dimension trials, the performances are impaired in an incongruent probe immediately following an incongruent prime, due to the cost of switching from the specific inhibition of the previously conflicting information of magnitude to the currently conflicting new information of magnitude. While [START_REF] Notebaert | Cognitive control acts locally[END_REF] proposed a similar account in terms of switch cost at the level of responses (see also [START_REF] Brown | A computational model of fractionated conflict-control mechanisms in task-switching[END_REF] for a similar account at the task level), here our results suggest that this switch cost can not only impact the activation of the relevant information in the task, but also the processing of the irrelevant information. Following this account, the presence of a reverse Gratton effect further supports a dimension-specific inhibitory control in our task.

We would like to stress the fact that our study evidenced a specificity of inhibitory control solely in regards to the manipulated Size and Spacing dimensions, which are each defined as a combination of two covarying dimensions of magnitudes (average item size and total surface for Size, convex hull and density for Spacing), commonly used in the literature (Gebuis & Reynvoet, 2011). Our design does not allow to disentangle the role of these four dimensions in our study, and whether one or a combination of the two dimensions composing our manipulated dimensions drove the observed effects. Further studies would be very interesting to explore the specificity of inhibitory control relative to each of these dimensions.

Our study is the first to demonstrate that the mechanisms involved in the attention to number are specific to the irrelevant information conflicting with number, and contributes to better characterizing the executive mechanisms at play in numerical cognition. Using brain imaging methods, further research could allow to further identify the precise neural networks underlying this specificity. More generally, by using a task whereby the relevant information is maintained constant, our study demonstrates that the inhibition of the irrelevant information plays an important role in the Gratton effect, which appears to not be solely driven by the increased activation of the relevant information.

Table A1. Congruency (C: congruent, I: incongruent) and ratios for each magnitude composing the Size and Spacing dimensions, in each of the four experimental conditions. Ratios are calculated as the smaller value over the larger value for a given dimension. Values for the magnitudes composing the congruency-related dimension are in bold.

Size

Figure 1 :

 1 Figure 1: Examples of between-dimension trials. Both in the prime and the probe items of the trial, the congruency-related dimension can be congruent or incongruent with Number. In a betweendimension trial, the congruency-related dimensions in the prime and the probe items are different.

  (incongruent : M = 623, 95% CI [595.5, 650.5], congruent: M = 533, 95% CI [514.6, 551.4]), F(1,63) = 113, p < .0001, η 2 p = .64), than number/Size congruency effect (incongruent: M = 598 , 95% CI [574.4, 621.6], congruent: M = 559, 95% CI [533.3, 584.7]), F(1, 63) = 19, p < .0001, η 2 p = .23.

  Figure2).

Figure 2 :

 2 Figure 2: Congruency Sequence Effects. reaction times (ms) in the congruent and incongruent probe items depending on the congruency of the prime items, for the within-dimension (on the left) and between-dimension (on the right) trials. Error bars represent 95% confidence intervals.

Figure A1 .

 A1 Figure A1. Example of stimuli in the within-dimension conditions, with Spacing as the congruencyrelated dimension. Next to each stimulus is the congruency of item size (IS), total surface area (TSA), convex hull (CH) and sparsity (Spars). + : congruent with number ; -: incongruent with number. Dimensions composing the manipulated dimension are in bold.

Figure A2 .

 A2 Figure A2. Example of stimuli in the between-dimension conditions, with Size as the congruencyrelated dimension in the prime items. Next to each stimulus is the congruency of item size (IS), total surface area (TSA), convex hull (CH) and sparsity (Spars). + : congruent with number ; -: incongruent with number. Dimensions composing the manipulated dimension are in bold.

Acknowledgments

Declarations of interest: none.

Appendix A.

Below is a table showing the congruency and the ratios for each of the four magnitudes composing the Size and Spacing dimensions manipulated in our study (Table A1), as well as examples of the stimuli for a within-dimension and a between-dimension conditions (Figure A1 andA2).

The ratios in the table are always calculated from our stimuli, as the smaller value over the larger value of a given dimension (e.g. for item size, ratio = (item size in the set with smaller dots) / (item size in the set with larger dots)). Thus, when the item is congruent ("C" in the table), this ratio corresponds to the value of the dimension in the less numerous set over the value in the more numerous set, and vice versa for incongruent items ("I" in the table). In some cases, there is a small jitter between the items within a condition. It was easier to generate sets of dots with precise values for Size and its sub-dimensions than for Spacing. For instance, ratio of item size in number-Size congruent items and ratio of total surface area in number-Size incongruent items are exactly the same in all the items. Since our numerical ratio was approximately the same across all the trials (~3:5), but using different number pairs, the ratios differ slightly between the four pairs. The same applies for Spacing and its sub-dimensions (convex hull and sparsity), with a bit less precision due to our program generating sets of dots until the convex hull surface falls within a small range of values.