Josua Krause

Aritra Dasgupta

Jean-Daniel Fekete

Enrico Bertini

SeekAView: An Intelligent Dimensionality Reduction Strategy for Navigating High-Dimensional Data Spaces

Keywords: High-Dimensional Data, Subspace Exploration, Guided Visualization

Dealing with the curse of dimensionality is a key challenge in high-dimensional data visualization. We present SeekAView to address three main gaps in the existing research literature. First, automated methods like dimensionality reduction or clustering suffer from a lack of transparency in letting analysts interact with their outputs in real-time to suit their exploration strategies. The results often suffer from a lack of interpretability, especially for domain experts not trained in statistics and machine learning. Second, exploratory visualization techniques like scatter plots or parallel coordinates suffer from a lack of visual scalability: it is difficult to present a coherent overview of interesting combinations of dimensions. Third, the existing techniques do not provide a flexible workflow that allows for multiple perspectives into the analysis process by automatically detecting and suggesting potentially interesting subspaces. In SeekAView we address these issues using suggestion based visual exploration of interesting patterns for building and refining multidimensional subspaces. Compared to the state-of-the-art in subspace search and visualization methods, we achieve higher transparency in showing not only the results of the algorithms, but also interesting dimensions calibrated against different metrics. We integrate a visually scalable design space with an iterative workflow guiding the analysts by choosing the starting points and letting them slice and dice through the data to find interesting subspaces and detect correlations, clusters, and outliers. We present two usage scenarios for demonstrating how SeekAView can be applied in real-world data analysis scenarios.

Introduction

Let's start with an example scenario where a health-care analyst working with a care provider company is confronted with patient data, containing millions of hospital visit records, and hundreds of dimensions describing conditions and demographic information of patients. Her goal is to find patient groups with similar disease conditions or similar outcomes, to treat them in a similar way; either this group is "well managed", or that one "is suffering from lack of resources in the hospitals", or another one is "having an abnormal mortality rate". These outcomes are quite different in the dimensions and values that reveal them. Therefore, the analyst will need to gather different set of dimensions, filter over different values or categories, to find subspaces exhibiting interesting properties for her questions. Looking at all the dimensions for all the records at the same time is not only inefficient, but also unlikely to answer her questions.

For our work, we are interested in exploring datasets containing numerical or categorical dimensions, with n dimensions D = {d 1 , • • • , d n } (or features) and m items (also called records, or data points) R = {r 1 , • • • , r m }. The basic premise of our work is to let analysts build interesting k dimensional subspaces (kD) out of the n-dimensional data space (nD), where k n. For our purpose, k should be less than 20 to be visualized using either parallel coordinates or a scatter plot matrix; n is typically in the hundreds. Each subspace is usually only interesting for a filtered selection i over the m possible records, so analysts need to slice and dice through the dimensions and items to build interesting subspaces, where each subspace is a couple (K, I) where K j∈[1,...,n] = {d j } and I l∈[1,...,m] = {r l }.

While using automated approaches such as subspace clustering [START_REF] Kriegel | Clustering high-dimensional data: A survey on subspace clustering, pattern-based clustering, and correlation clustering[END_REF] is an option, these automatic methods give no control over the analytical process to the analyst but only provide a list of subspaces that may contain interesting combinations of dimensions and values. Besides,

• Josua Krause is with NYU. E-mail: josua.krause@nyu.edu.

• Aritra Dasgupta is with Pacific Northwest National Laboratory. E-mail: aritra.dasgupta@pnnl.gov • Jean-Daniel Fekete is with INRIA. E-mail: Jean-Daniel.Fekete@inria.fr.

• Enrico Bertini is with NYU. E-mail: enrico.bertini@nyu.edu.

Manuscript received 31 March 2007; accepted 1 August 2007; posted online 2 November 2007. For information on obtaining reprints of this article, please send e-mail to: tvcg@computer.org. subspaces clustering will only find one kind of pattern in multidimensional data: dense areas, while experts are interested in many more, such as correlated dimensions, or uncorrelated ones, dimensions following visible patterns, or exhibiting outliers. For addressing these problems, researchers have argued for a human-in-the-loop approach towards subspace exploration leading to more intelligent dimensionality reduction strategies [START_REF] Pang | Special issue on dimensionality reduction for visual big data[END_REF].

To this end, we propose SeekAView, a visual analytics system that leverages a transparent and flexible process-oriented approach for letting analysts build subspaces out of a high-dimensional data set. SeekAView is able to show hundreds of dimensions without reduction and provide suggestions to the analysts for letting them reconfigure different views, build interesting subspaces according to multiple kinds of patterns, and iteratively refine them based on their evolving analysis questions.

In this paper, we identify three main research gaps we aim to fill through SeekAView:

1. The lack of scalability of existing visualizations in the number of dimensions. By focusing on 1D plots as starting point and guiding the user in the construction and exploration of nD subspaces, we enable the analysis of large sets of combinations without overwhelming the user. Even simple visual representations like scatter plot matrices grow quadratically with the number of dimensions and can quickly become unmanageable. We support the analysts in tasks such as: "I want to look at an overview of all dimensions and find combinations revealing interesting patterns to explore in multidimensional views".

2. The lack of transparency [6] in methods that rely exclusively on dimensionally reduced dimensions/axes, such as PCA, t-SNE [START_REF] Van Der Maaten | Visualizing high-dimensional data using t-sne[END_REF], or MDS [18]. By making 1D plots of the original dimensions the main driver behind all visualizations and interactions and by letting the analyst interact with the output of the automated methods we increase transparency throughout the analysis process. This means the analyst can always ask questions like: "Which of the original dimensions are involved in this pattern?" and "What is it going to happen if I remove/add this group of dimensions from the visualization?"

3. Provide a flexible analytical workflow, that allows to enter the visual analytics loop through multiple starting points of analysis. By getting suggestions on the potential next steps of their analysis at every stage of their interaction analysts can have have different starting points: "I want to focus on this category and reconfigure different views for finding clusters" or "Suggest interesting subspaces to me and I refine them by using my domain knowledge".

Specifically, our contributions in this paper are threefold: i) we propose a framework for allowing analysts to build subspaces in a humanguided fashion with machine-generated suggestions-faithful to the visual analytics process; the analysis can start either from a set of suggestions from algorithms, or from a view selected by the analyst, and enter into the visual analysis loop then; ii) we focus on enabling key analysis tasks through transparent operations that analysts can understand and steer at any time; and iii) we provide a system that can effectively deal with high-dimensional spaces around hundred dimensions without information loss or dimensionality reduction.

Related Work

High-dimensional data analysis has a long history where different approaches have been pursued to handle the curse of dimensionality problem [START_REF] Buja | Interactive high-dimensional data visualization[END_REF][START_REF] Hoffman | A survey of visualizations for highdimensional data mining[END_REF][START_REF] Wong | 30 years of multidimensional multivariate visualization[END_REF]. In our work, we focus exclusively on the premise of generating interesting multidimensional views out of highdimensional data, commonly known as subspaces and letting analysts refine and reconfigure those subspaces. There are three main threads of research relevant to the work reported here, which are as follows:

Subspace Clustering and Search

Automated subspace search methods being heavily parameterized [START_REF] Kriegel | Clustering high-dimensional data: A survey on subspace clustering, pattern-based clustering, and correlation clustering[END_REF], visual analytics techniques have been proposed for handling this problem. Some of these methods focused on helping analysts select interesting feature subsets [START_REF] Guo | Coordinating computational and visual approaches for interactive feature selection and multivariate clustering[END_REF][START_REF] May | Guiding feature subset selection with an interactive visualization[END_REF] by interacting with the algorithm outputs. In these cases, the user is in control of the modification of the generated feature subset by being able to reconfigure them through addition or removal of features. These techniques are focused on displaying the outputs of the algorithms and not on the process. Users can modify the output, but there is no guidance from the system in being able to select more features for building different views. Moreover, they also do not allow users to manipulate the item space that would trigger a recomputation of the algorithms in the dimensions space. Another approach to this problem has been to devise intelligent subspace search methods [START_REF] Tatu | Combining automated analysis and visualization techniques for effective exploration of high-dimensional data[END_REF], where subspace clusters are shown to the user. This approach has high visual scalability in being able to provide an overview of the subspaces to the user. However, this approach also suffers from a lack of flexibility in letting the user modify the subspaces by either getting help from the system or by incorporating her domain knowledge. Our framework also displays outputs of subspace clustering algorithms, but lets the user reconfigure the output views through different suggestions. An exploratory approach towards generating interesting subspaces have been proposed by Wang et al. by clustering dimensions [START_REF] Wang | Interactive hierarchical dimension ordering, spacing and filtering for exploration of high dimensional datasets[END_REF] or showing values and relations at once in projected data spaces [START_REF] Yang | Value and relation display for interactive exploration of high dimensional datasets[END_REF] have also been proposed. However, the method is limited to a small number of dimensions.

Quality Metric Aided Visual Exploration

The problem with traditional techniques, such as scatter plot matrices [START_REF] Elmqvist | Rolling the dice: Multidimensional visual exploration using scatterplot matrix navigation[END_REF], parallel coordinates [START_REF] Inselberg | Parallel coordinates[END_REF][START_REF] Wegman | Hyperdimensional data analysis using parallel coordinates[END_REF], or radviz [27], is that they mostly do not scale beyond 10 to 20 dimensions. While there has been recent approaches to make scatter plot matrices more scalable [START_REF] Lehmann | Selecting coherent and relevant plots in large scatterplot matrices[END_REF], they can be visually complex for analysts to understand and navigate without any guidance. Moreover, most visual representations for highdimensional data exclusively focus on numerical attributes. For alleviating the scalability problem, researchers have proposed pixel-based displays [START_REF] Keim | Designing pixel-oriented visualization techniques: Theory and applications[END_REF], which are able to preserve the fidelity of the data. However in such a high-density display, showing relationships and guiding user interactions to selectively build views can be challenging.

Researchers have also looked into augmenting high-dimensional data exploration with visual feedback [START_REF] Tatu | Combining automated analysis and visualization techniques for effective exploration of high-dimensional data[END_REF]. Approaches such as using feature statistics for producing ranked views have been proposed [START_REF] Seo | A rank-by-feature framework for interactive exploration of multidimensional data[END_REF], however they do not focus on guiding the user to build multivariate views. Analysts often need guidance on finding the salient hidden relationships in large high-dimensional data spaces. Quality metrics [7, [START_REF] Schneidewind | Pixnostics: Towards measuring the value of visualization[END_REF][START_REF] Tatu | Combining automated analysis and visualization techniques for effective exploration of high-dimensional data[END_REF][START_REF] Wilkinson | Graph-theoretic scagnostics[END_REF] have been proposed to deal with this issue by quantifying various data-space and screen-space relationships. Most of them however focus on two-dimensional relationships and do not let refinement of the metrics with respect to conditioning of the data.

Dimensionality Reduction Strategies

For handling the curse of dimensionality problem, one solution is to use dimensionality reduction methods like PCA, t-SNE [START_REF] Van Der Maaten | Visualizing high-dimensional data using t-sne[END_REF], or MDS [18]. However, with domain experts, reduction is often not an option as analysts need to use their domain knowledge to understand which combinations of dimensions make sense. They need a faithful representation of the data space with minimal abstraction and information loss.

Moreover, dimensionality reduction supports mainly one visualization task, that is, finding uniform groups of clustered objects, which covers only one of the possible visualization tasks with highdimensional data visualization. Furthermore, plots obtained through linear or nonlinear transformations are hard to interpret as dimensionality reduction works in a black box fashion. In the not too rare case in which dimensionality reduction returns results that are hard to interpret (e.g., a uniform cloud of points with no clear groupings), it is not clear whether the results reflect a real characteristic of the data or rather the method itself fails to represent the underlying trends correctly (that is, it's a false negative) [4]. Even more problematic, there is a lack of techniques able to guide the user through steps that can help disambiguate this situation with only a few exceptions [11].

For a visual approach to dimensionality reduction, researchers have proposed techniques like hierarchical clustering and interactive reduction of the search space by using information loss metrics [START_REF] Johansson | Interactive dimensionality reduction through user-defined combinations of quality metrics[END_REF]. Similar to the drawbacks of subspace search approaches, these exploratory methods are also focused on the results of the metrics and less focused on the process. A notable exception is the dual analysis model proposed by Turkay et al. [START_REF] Turkay | Brushing dimensions-a dual visual analysis model for high-dimensional data[END_REF], who allow a tight interaction between the item space and the dimension space. They are however, not focused on producing subspaces of dimensionality more than 2 and also do not follow a transparent approach of showing the properties of the dimensions themselves.

The focus of our framework is not only on the output of kD subspaces, but also on the transparent analytical process that analysts can follow to produce that output. As shown in Figure 1, we adapted the visual analytics schema proposed by Keim et al. [START_REF] Keim | Visual analytics: Definition, process, and challenges[END_REF] in the special case of subspace visualization. The framework aims to synergize user interactions with system generated suggestions for discovering and reconfiguring interesting multidimensional views. The reconfiguration process involves manipulating the dimension space and the item space for building subspaces with guidance from the system, using different strategies and flexible starting points. Additionally, we incorporate the dual analysis model proposed by Turkay et al. [START_REF] Turkay | Brushing dimensions-a dual visual analysis model for high-dimensional data[END_REF] with the suggestion system and allow for multiple paths through the schema.

Formally, our framework generates subspaces (K, I,C, T ) that extend the standard subspaces (K, I) described in Section 1. C is a partition over the filtered selection I, and corresponds to color associated with bins of values. T is a textual description provided by a user to document what is interesting in the subspace or any kind of annotation. Our framework is meant to produce colored bins of items with a set of dimensions and a textual representation. Dimensions are typically numerical or categorical and then usually mapped to a numerical value. We call conditioning as the transformations of the values of a dimension; it can be a normalization that will map a given range of values in the [0, 1] range, or a scaling that will transform values by applying e.g. a log or exponential function to them, or any combination thereof. In the rest of this paper, we assume that, when mentioned, conditioning is applied both for visualization and computation. For example, refer to the view in Figure 2. It contains of a set of nine dimensions, a coloring of four colors representing clusters in the item space, and an annotation describing the view.

As seen in Figure 1, there can be different starting points of analyses (A and B). Consider for instance a nutritionist wants to study the correlation patterns of nutrients among food items such as dairy products, meat products, etc.. She would like to know: which food products are similar in containing certain types of nutrients? (Q1), which dimensions are redundant? (Q2), which combination of nutrients are highly related to high Vitamin D content? (Q3), which combination of nutrients separate food groups the most? (Q4) etc.. Using our framework, analysts can either start from system generated suggestions (scenario A) or start by interacting through the visualization and then get help from the system on further analysis and exploration steps (scenario B). For supporting such flexible analysis, the suggestion component adapts to the different starting points and help analysts refine the subspaces by navigating and filtering both the dimension and item spaces. Suggestions: The suggestions encapsulate a set of metrics for guiding analysts towards finding interesting groups of dimensions. They also adapt to different starting points (subspaces and selections) an analyst can pursue. In case of the questions Q1 and Q2, the nutritionist does not have a starting point of the analysis and is just interested in finding interesting combinations of dimensions. In such cases, the nutritionist needs the system to guide her to potentially interesting patterns. This refers to the starting point A in Figure 1. In that case the system helps the analysts detect patterns by suggesting different subspaces which can be further iterated upon by the analysts using conditioning of the data and refinement of the generated subspaces (Figure 3). In our current implementation, the subspaces are generated by using either subspace clustering (Q1), generating groups of correlated dimensions (Q2), or manually selecting dimensions of interest.

In case of the questions Q3 and Q4, the nutritionist knows her starting point of analysis and she would focus on a given dimension in food products for visually inspecting relevant nutrients. Pure visual inspection is difficult though, as the larger the dimensionality (nutrients) and large data size (the number of food items), the larger the search space becomes. This refers to the starting point B in the figure. The analyst need guidance in reducing the search space for selecting interesting dimensions (Figure 5). This is supported by computing metrics for quantifying relationships between distributions i.e. distribution anomalies (Q3) and computing metrics for finding subspaces which separate classes well (Q4). We refer to this method as dimension filters. Refinement: Once subspaces are generated by automated methods, or interesting views are suggested to the analyst, our framework allows refinement of the produced output in different ways. For example in the scenarios mentioned above, the nutritionist can either try to incorporate her domain knowledge for refining the subspaces, e.g. removing uninteresting dimensions, or get help from the system through suggestions on how to refine the subspaces. The motivation of this task is similar to that in the dual analysis model proposed by Turkay et al. [START_REF] Turkay | Brushing dimensions-a dual visual analysis model for high-dimensional data[END_REF]. But in our case we provide higher transparency in letting users manipulate the data space and directly observe how the operations affect the relationship among dimensions and items. To ease visual inspection and steer analyses, analysts can adapt the scales of the different dimensions based on visual inspection of the distributions to either logarithmic scale or linear scale. One of the main refinement strategies is adding or removing dimensions to the subspace of interest based on the system suggestions. For example, by finding dimensions with our dimension filters, analysts can choose to add those dimensions to a subspace to investigate further their influence. Similarly they can remove dimensions which are of little or no interest to their analysis. Another refinement strategy is to condition the item space based on automatic or visual detection and removal of outliers. After modifying the item space, recomputations of suggestions can be triggered and different subspaces or dimensions are again suggested to the analyst, steered from to their interactive selections.

SeekAView: Instantiating the Subspace Exploration Framework

In this section we describe SeekAView, our implementation of the framework we described above. We first describe the visualization design and user interface and then ways of building views and the mechanisms used for suggesting and refining them. In this we cover four main data analysis tasks, namely: subspace clustering, correlation, anomaly detection, and class separation.

Visual Representations and User Interface

SeekAView is split into different panels (see Figure 2). On the left is a simple list showing all dimensions by name while the center shows all dimensions as small multiple frequency plots. On the right a magnified frequency plot shows the currently selected dimension. This target plot can be used to define targets by brushing. Targets are couples (I,C) of an item space I with a partition C of its items. This partition is called coloring of the items and is limited to up to 6 buckets i.e. colors (5 brush colors and the remaining non-brushed items). At the bottom a view generated via suggestions and refinements, as explained in Section 4.3, is shown in a number of multidimensional visualizations, including a PCA scatter plot, parallel coordinates, and a scatter plot matrix. Those views can be saved and annotated in the view manager at the bottom right.

The design space of our framework facilitates progression from the selection of dimensions to defining targets and building multivariate views of the data. Three high-level goals motivate our key design choices: i) scalability with respect to a high number of dimensions (in the order of tens and hundreds), ii) the flexibility to represent mixed attribute types, like numerical, ordinal, categorical, or binary data, and iii) transparency with respect to user interaction for showing interesting subspaces and targets and letting analysts build multivariate views. In this section, we describe how our design choices satisfy these three criteria. Small Multiple Frequency Plots: Unlike conventional highdimensional data representations like scatter plot matrices and parallel coordinates which focus on showing pairwise relationships between features, our design is motivated by the fact that the starting points of an analysis is often exploring the 1D distributions. To this effect, we display small multiples [START_REF] Tufte | The Visual Display of Quantitative Information[END_REF] of each dimension. Within each small multiple, as shown in Figure 4, we use a beanplot-like layout [START_REF] Kampstra | Beanplot: A boxplot alternative for visual comparison of distributions[END_REF], which is a frequency-based representation of the attributes. Therefore, we name the plots as small multiple frequency plots. For numerical variables, the y-axis of each plot represents the frequency of the data 5), a PCA scatter plot (6), and a scatter plot matrix (7). Views can be selected, created, removed, and annotated in the lower right [START_REF] Elmqvist | Rolling the dice: Multidimensional visual exploration using scatterplot matrix navigation[END_REF]. Manual brushing can be configured at the top [START_REF] Guo | Coordinating computational and visual approaches for interactive feature selection and multivariate clustering[END_REF]. Subspace suggestions can be created with a one-to-many filter [START_REF] Hoffman | A survey of visualizations for highdimensional data mining[END_REF] or by computing subspace ("cluster") or dimension similarity ("stack") clusters (11). The current state is the end of the Communities and Crime case study (see Section 5.1) finding dimensions related to "Violent Crimes per Pop." and further clustering the item space.

values. A frequency-based representation provides the flexibility to meaningfully represent multiple attribute types. For ordinal or binary variables, we associate numerical values to each distinct value and for categorical variables, we divide the space into rectangles. The number of rectangles is equal to the number of categories of the feature and the area of each rectangle is scaled by the proportion of data points belonging to that category. Since there is no inherent order in the categories we order the categories by their frequency so that the most common category, i.e. with the most data points, is on top of the plot. The main motivation for showing the frequency was two-fold: i) each small multiple would have a very small resolution, therefore showing data items would only produce visual clutter, and ii) from a visual saliency pointof-view, detecting and comparing continuous shapes is relatively easier as deduced from the Gestalt principle of continuity [START_REF] Bileschi | Image representations beyond histograms of gradients: The role of gestalt descriptors[END_REF]. This and the fact that we cannot assume that dimensions are normal distributed is the reason why our representation is superior to e.g. box-plots. In a small multiple setting they would not produce salient distinguishable features and would not be visually scalable. Compared to, for example, scatter plot matrices, which grow quadraticly, 1D representations grow linearly with the number of features.

Target Plot: The small multiple plots are adaptive to user selections and let analysts select target dimensions for further exploration. These target dimensions then provide a focal point of analysis, while the small multiples serve as the context. Interaction between the two provides a focus-and-context analysis paradigm to the analysts in the high-dimensional data space, where they can explore multivariate relationships in a highly scalable manner. The target plot is a magnified frequency plot of the dimension. As small multiple frequency plots are displayed too small to add axes descriptions the user can hover over data points to see which value it corresponds to. Multivariate Views Panel: In the multivariate views panel a number of multidimensional visualizations show the current view created via suggestions and refinements (Section 4.3). The visualizations include a scatter plot showing the first two principal components of the dimension subspace, parallel coordinates, and a scatter plot matrix showing all pair-wise relations between dimensions of the subspace. The usage of those visualizations is feasible since the number of dimensions here is small compared to the overall number of dimensions. The scatter plot matrix helps users detect correlations and redundant dimensions which then can be removed if desired. As the screen space allotted for the scatter plot matrix is comparatively small the user can double click on the matrix to get a full page view of it. Analysts can brush on the parallel coordinates and scatter plots much like in the target plot and select dimensions like on the small multiples panel.

View Manager: The multivariate views can be saved and annotated in the view manager on the bottom right of the user interface. A text box allows to write comments about a view for future reference. The analyst can switch between created views also to go back steps or revisit findings that were made earlier.

The views can be created by the analyst using two strategies. Manual curation of subspaces by adding or removing single dimensions or a suggestion & refinement based approach.

Manual Curation of Subspace Views

Analysts can manually curate views with user interaction. We provide some statistical measures to ease this task and also the means of defining targets and filtering the dimension space as well as the item space. Sorting Small Multiples: Uniform and non-uniform distributions are of key interest to analysts. Uniform distributions usually denote high uncertainty or high entropy while non-uniform distributions usually denote low entropy. We compute the entropy of the dimensions using a pixel binning strategy. The number of bins is equal to the number of pixels in the magnified small multiple frequency plot, where the frequency variation can be seen at a higher resolution. In case of nonuniform distributions, it is useful for analysts to know whether high or low values dominate, i.e., whether there is a positive or negative skew. Skewed distributions are also easy to spot even in very low resolution. Therefore we use skewness as an interestingness measure for quantifying distributions. We provide both entropy and skewedness as a means to sort small multiple frequency plots. This points users to interesting dimensions for further inspection. Scaling Dimensions: A further form of dimension statistic mechanism is automatically deciding which scale (i.e. linear or logarithmic) to use for the representation of the data values. This conditioning reveals different insights about a dimension. The system provides the functionality to automatically use the representation with the higher entropy, however, analysts can manually change the scale if it is desired. After interacting with the data by e.g. removing outliers from the data set this operation can be performed again to update the optimal mapping for each dimension. Changing the scaling alters the outcome of subspace generating algorithms, the sorting of small multiples, and also helps an analyst get a clearer picture of the content of a dimension.

Brushing: Linking among the different panels and visualizations is done via brushes using differently colored data items (similar to the high-dimensional brushing [START_REF] Martin | High dimensional brushing for interactive exploration of multivariate data[END_REF]). Users have the flexibility to brush in the target frequency plot, the parallel coordinates, and scatter plots. Complex brushes are enabled by providing different operations, like replacing, joining, intersecting, or subtracting of brush operations. We provide the functionality to brush five quartiles of a dimension as automatic way to segment the data. For categorical attributes, we impose an order for the purpose of computing quartiles by sorting the categories by their frequency. This leads to the most common categories being in the highest "quartile", the second most common in the second "quartile", etc. (Figure 4). We also provide item space clustering (using k-means) w.r.t. to the current multivariate view which also provides an automatic brushing. This can be useful to look how item space clusters show up in other dimensions that are not part of the multivariate view.

Fig. 4. Small multiple frequency plots. On the left is the numerical dimension "Water" showing the distribution of percentages of water in various food products. On the right is the categorical dimension "FdGrp Desc" describing the food group of food products. The width of the rectangles representing distinct values is dependent on the number of data items having the respective value. The plots on the bottom show the quantiles of their respective data item distribution. For categorical values the most common categories are grouped together.

On a technical note, drawing parallel coordinates and scatter plots with multiple colors due to the brushes introduces a bias towards the last drawn color. In order to avoid this problem we randomize the data points before drawing scatter plots and parallel coordinates leading to a color distribution that reflects the actual distribution of the different brushes.

Brushes have multiple uses in our tool. They can be used to specify targets that are then used to suggest multivariate views. For class definitions different brush colors represent different classes in the item space. Brushing can also be used to define which items to retain when removing outliers or focus one's attention only on the defined classes. Lastly, brushes can also be used to inspect how a set of data items behaves in different dimensions.

Given brushes have multiple unrelated applications, users need to frequently reset and reassign the brushes for different tasks. For example, the user wants to remove some outliers so she brushes the data points that shall remain and removes the rest. After that, in order to define a target for further analysis the brushing has to be reset and newly assigned.

Suggestions & Refinements

Suggestions allow analysts to automatically create views which then can be refined either via manual curation or further automated suggestions. They guide analysts in performing four main data analysis tasks. Subspace clustering for finding dimensions with dense items, correlation to find redundant dimensions, anomaly detection to find outlier dimensions, and class separation for finding dimensions suitable for distinguishing labeled data. Subspace Clusters: The goal of subspace clustering is to find dimensions and items that are compact. Under the assumption that this typically occurs as hypercubes (i.e. ranges of items in dimensions) algorithms like CLIQUE (implemented in ELKI [START_REF] Achtert | ELKI: A software system for evaluation of subspace clustering algorithms[END_REF]) provide good results (see Figure 6). The resulting subspaces are shown in the small multiples plot as groups. This inevitably leads to dimensions showing up at multiple locations. Due to the use of the density plots same dimensions can easily be detected due to similar shapes of the distribution. Furthermore, next to the name of a dimension the number of times this dimension is shown in the small multiple plot is indicated. Also, selecting a dimension with multiple occurrences shows all occurrences of this dimension (e.g. "robberies" in Figure 6). The resulting subspace clusters also include associated brushes which are the actual dense items in the cluster. For this reason we create a view for every subspace cluster in the view management list to include the associated brushes. Correlated Dimensions: Finding highly correlated dimensions can be of interest when exploring relationships between dimensions. We use hierarchical clustering (DB-SCAN with single linkage) on the correlation between dimensions in order to find groups of similar dimensions. The correlation is computed using the Pearson coefficient converted into a metric by transforming the coefficient in a way such that highly (positive and negative) correlated features result in values close to 0 while features that are not correlated result in values close to 1.

The user can set the threshold for the clusters using a slider which are then used to group dimensions in the small multiples plot (see Figure 7). The resulting groups of dimensions are shown in the small multiples plot. DB-SCAN produces a partition of the available dimensions so no special care for dealing with dimensions showing up at multiple positions has to be done.

Anomaly detection: A way to detect outlier dimensions, i.e. dimensions with interesting properties according to a specified target, is by filtering for distributions of the specified items that are anomalous compared to the overall distribution of items in dimensions. To answer this we compute metrics comparing the distributions of colored items in a dimension to the overall distribution of items in the dimension. This provides an insight on how anomalous a given set of data items is w.r.t. a dimension and compares how the shapes of both data point distributions correlate. We compute those metrics by comparing the histogram of the selected data items to the histogram of all items in a given dimension. For the comparison we provide two alternatives: the Pearson coefficient applied to the distributions and the KL-divergence applied to the distributions. Both techniques give a slightly different result but both of them promote dimensions where the coloring has a different distribution than the rest of the items. Thus, the distribution of the colored items can be considered anomal for a dimension. Class Separating Dimensions: Finding what dimensions separate and thus predict classes defined by analysts is an important task. Suggestions to answer this question are based on both selections of a dimension as well as a target. We compute how well a given dimension is able to predict certain characteristics of a target with selected dimension. Those characteristics are formulated by defining targets that reflect for example different categories of a selected dimension. A metric that describes this "predictiveness" very well is Information Gain. Information Gain is the change of entropy if a dimension is included as source of information. The larger the change the more information is gained by including this dimension. We compute the Information Gain for all dimensions with respect to the defined target and the selected dimension. This dimension filter is useful for finding subspaces that separate classes well. The unique feature of SeekAView is that our system allows analysts to manipulate the results of view construction and influence their computation. This leads to a high-degree of transparency in the interpretation of the metrics. This can be done by changing the set of dimensions as well as the set of data items.

Usage Scenarios

In order to show the usefulness of SeekAView we demonstrate two usage scenarios. One scenario is exploring the unnormalized Communities and Crime data set. The other is based on the US Department of Agriculture (USDA) National Nutrient Database.

Communities and Crime

The Communities and Crime data set constructed from the 1990 US Census and the 1995 US FBI Crime report [START_REF]Communities and Crime Unnormalized Data Set[END_REF] contains 147 different dimensions and contains over 2000 items describing various crime and non-crime related attributes of communities in the United States of America. The data is provided by the University of California, Irvine (UCI) machine learning archive. Our hypothetical analyst wants to explore the data in two different ways, finding interesting subspaces and finding dimensions related to "Violent Crimes per Pop.".

As first step, the analyst clicks on "auto scale" which selects a good conditioning for each dimension automatically. This makes the distribution of items within the dimensions much clearer for her. Then, she initiates subspace clustering which results in clusters mostly related to "robberies" (see Figure 6 and6a). Inspecting the largest cluster she sees that "NumStreet" (number of homeless people) is highly correlated with the number of "robberies". However, the smaller the number of homeless people gets the larger is the variance of "robberies". In order to improve on her results she uses the quantiles selection on "robberies", brushes the middle three quartiles (see Figure 6b) and clicks on "show selection" in order to remove the outer quartiles. After removing those outliers she then re-runs the subspace clustering algorithm. In the largest cluster she sees a thin line that occupies most of the items. She uses brushing to verify that this line is consistently compact throughout the cluster (see Figure 6c). By reintroducing the earlier removed items she sees that this pattern is still present in the original data.

Next, our analyst wants to explore which factors play into the number of "Violent Crimes per Pop." so she selects the dimension and brushes for high values. She then uses the KL-Divergence selection shape similarity metric to find dimensions where this brushing is also of interest. After deciding on a threshold for the filter using the slider she constructs a multivariate view of those dimensions. Since "community name" is one of the available dimensions in the data set but has distinct values for every item, the KL-Divergence filter will always include this dimension since every selection of items has an entirely different shape for this feature. The analyst manually removes this dimension. The remaining dimensions in the view, upon inspection, are dimensions related to violent crimes, like "Assaults", "Burglaries per Pop.", "Robberies per Pop.", etc., and, surprisingly, dimensions related to the family situation of children, like "Pct. Family with 2 Parents", "Pct. Kids with 2 Parents", "Pct. Kids Born Never Married" (i.e. children born to parents that never got married), etc.. K-means clustering further reveals three clusters that connect high violent crimes with bad family situations and vice versa. The analyst notices an odd shape in the PCA projection and decides to brush this occurrence as fourth cluster (green brush in Figure 2). This brush corresponds to a peak in "Violent Crimes per Pop." with a relatively low value. However, this peak doesn't show up in any other dimension of this subspace. The large amount of communities in this range suggest that there is no correlation between violent crimes and family situation for most communities. For those, the norm also is a moderate amount of violent crimes. However, for communities with an exceptional amount of violent crimes there is a strong correlation between the number of crimes and the family situation. The analyst can write down her findings in the textual description associated with the view and save the view.

USDA National Nutrient Data

The US Department of Agriculture (USDA) National Nutrient Database [START_REF]USDA National Nutrient Database for Standard Reference[END_REF] contains information of about 50 different nutrients of over 8000 food products as well as their name and food group categorization. For the scenario we removed all food products that had incomplete records resulting in over 1800 remaining food products.

Our hypothetical analyst is interested in correlations of nutrients. After clicking on "auto scale" to choose optimal mapping functions for the dimensions she moves the "Stack" slider to group dimensions that are highly correlated. She inspects the biggest cluster which contains Fig. 7. Grouping of similar dimensions. A subspace of similar dimensions from the USDA nutrition data case study (see Section 5.2). The largest group is shown in the multidimensional view panel. The current target is computed using k-means item space clustering. The selected dimension "Food Group Desc." shows that those item space clusters roughly correspond to some food categories. This leads to a further analysis trying to better separate those categories (see Figure 8). seven dimensions. One of the dimensions is "NDB Number" which identifies food items. She removes this dimension. Then, she clusters the remaining subspace with k-means clustering (see Figure 7). Noticing, that the computed clusters roughly correspond to different food groups, our analyst wants to find out if she can find dimensions that separate food groups very well. She first saves this view with a textual description of her finding and next goal.

In order to do find dimensions that better separate the clusters, our analyst uses different brushes to define some food groups, such as "meats" ("beef", "pork", and "poultry"), "plant products" ("vegetables" and "fruits"), and "fish and shellfish". She then uses the "Information Gain" filter to find a subspace that predicts those categories well. Initially, her subspace has too many dimensions to be easily interpretable (see Figure 8a). Therefore, she uses the correlation similarity grouping, as well as the scatter plot matrix, to identify highly correlated i.e. redundant dimensions and filter them out. Furthermore, as she sees in the PCA scatter plot there are many items that are not in categories of her interest scattered around a central cluster. She uses brushing on the PCA view to filter-out those outliers. Without those outliers she finds and removes more dimensions that are not beneficial for the separation of her food classes (see Figure 8b). Also, she adds another category "sweets" to her target. Finally, she wants to see how well her classes separate without interference of unclassified items. In order to do this she clicks "show selection" to remove non-selected items (see Figure 8c). Looking at her result she sees that she can identify "meats" for example by having a relatively high "Vitamin B12" and "Iron", and low "Retinol" content. As nutritionist, she knows that "Potassium" would also be a good indicator but she removed this dimension earlier since it only provided redundant information which was also provided by the "Iron" content. "plant products" can be separated by having low "Vitamin B12", more "Carbohydrats" than "meats", and less "Lipids" than "meats". "fish and shellfish" have high "Retinol" and "Vitamin D1" contents and "sweets" are mostly identifiable by their high "Carbohydrats" contents.

Discussion

Through the two usage scenarios, we demonstrated how SeekAView helps analysts in configuring different views and building subspaces with the help of suggestions from the system. In this section we discuss our three claimed contributions (i, ii, and iii), and the benefits and limitations of the system with regard to the state-of-the-art.

Regarding claim i), SeekAView can effectively support the analyst in a human-guided fashion with machine-generated suggestions. At any time, the analyst can trigger a recomputation of the suggestions from a selected subspace, or can manually fix an automatically computed subspace using her domain expertise. Being able to enter the visual analytics loop by an automated analysis or by a human focus is essential to support important analytical tasks. Other systems such as [26] start from a subspace clustering and facilitate its exploration but never interact with the algorithm or its results.

Compared to the existing techniques for quality metric aided visual exploration, our approach is more scalable with respect to very high dimensionality (> 100), and flexible in allowing one-to-many comparison across all dimensions. The output of the suggestions, which encapsulate different metrics, are always groups of dimensions, thus focusing exclusively on visualizing multivariate patterns instead of just 1D or 2D patterns. Real-time guidance is provided on construction of different selections, the goal being similar to recommendation-based visual analytics [START_REF] Blaha | More bang for your research buck: toward recommender systems for visual analytics[END_REF].

Regarding claim ii) regarding transparency, SeekAView relies on visualizations that show the data items with understandable transformations, except for the PCA scatter plot that is used as a guide in addition to the other multivariate views. All the item filtering, binning, and dimension selections are always visible and represented as in the data table to enable analysts to relate the visualizations to the raw data, even if conditioning is applied. We consider conditioning operations as standard enough to be understandable by any analysts, unlike MDS for example.

Regarding claim iii) on the flexibility of SeekAView with regards to the multiple starting points of analyses, we have shown in the usage scenarios that handling more than 100 dimensions was manageable using multiple methods, either through guidance by starting with 1D distributions, through subspace clustering, or through elimination of redundant dimensions by grouping and filtering correlated dimensions. Compared to clustering dimensions [START_REF] Wang | Interactive hierarchical dimension ordering, spacing and filtering for exploration of high dimensional datasets[END_REF][START_REF] Yang | Value and relation display for interactive exploration of high dimensional datasets[END_REF], we provide a more flexible visual interface with which analysts can proactively choose dimensions of interest by looking at 100 dimensions at once, and interactively build multivariate plots from target features of different attribute types. Our visual design adapts to these different attribute types and addresses the challenge of visual representation of multiple attribute types at once.

Still, SeekAView can be improved in two directions: scalability on the number of items, and providing more automated algorithms to suggest interesting subspaces.

In terms of scalability, SeekAView is implemented in JavaScript and runs in a browser; it is therefore limited in the computational power of the browser that will undoubtedly improve but remains limited compared to a standard desktop application. Providing SeekAView as a web-based application facilitates its deployment while limiting its scalability; this is our selected trade-off but we are confident that the same system programmed as an application could handle ten to hundred thousands of items.

Conclusion and Future Work

We proposed SeekAView, a visual analytics system for letting analysts build and refine subspaces interactively with the help of suggestions from the system. We proposed and implemented an iterative workflow catered towards reconfiguration of interesting views for navigating multiple multidimensional data spaces with respect to both the dimensions and the items. Compared to the state-of-the-art in subspace visualization, we are able to achieve a much higher degree of flexibility in adapting to the different starting points of analyses. By providing suggestions on not only the quality of the current selections but also on next steps of analysis, there is a seamless interaction between analysts' mental models and the functionalities of our system.

In the future, we will pursue research directions on further augmenting the guidance mechanism with more metrics and algorithms such as SVMs for providing more informed choices of subspaces to the analyst. Also, we plan to add functionality to restrict the number of subspaces relative to the display size. Furthermore, there is currently no in-system way of dealing with missing labels and necessary steps have to be performed in preprocessing steps which might be slow. We are planning to add a way of dealing with missing values in an interactive fashion. While we only have demonstrated the application of our approach using some usage scenarios, we are also in the process of deploying our tool in the fields of health-care and finance. We believe our tool will not only help domain experts in flexible multivariate analyses but also getting feedback from them will help us improve our framework and integrate more intelligent guidance strategies.

Acknowledgment

The research described in this paper is part of the Analysis in Motion Initiative at Pacific Northwest National Laboratory (PNNL)

Fig. 1 .

 1 Fig.1. Framework for guided reconfiguration of interesting views out of high-dimensional data: A tight integration of system generated suggestions with analysts' actions leads to flexibility in having multiple starting points of analyses (A and B), and transparency in being able to refine the suggestions and adapt both dimension and item space interactively.

Fig. 2 .

 2 Fig.2. SeekAView The user interface of SeekAView. (1) is the list of all dimensions while (2) shows them as small multiple frequency plots. Selected dimensions are high-lighted (3) and shown in the target plot (4). The current multivariate view can be seen in the multidimensional view panel consisting of parallel coordinates (5), a PCA scatter plot (6), and a scatter plot matrix (7). Views can be selected, created, removed, and annotated in the lower right[START_REF] Elmqvist | Rolling the dice: Multidimensional visual exploration using scatterplot matrix navigation[END_REF]. Manual brushing can be configured at the top[START_REF] Guo | Coordinating computational and visual approaches for interactive feature selection and multivariate clustering[END_REF]. Subspace suggestions can be created with a one-to-many filter[START_REF] Hoffman | A survey of visualizations for highdimensional data mining[END_REF] or by computing subspace ("cluster") or dimension similarity ("stack") clusters (11). The current state is the end of the Communities and Crime case study (see Section 5.1) finding dimensions related to "Violent Crimes per Pop." and further clustering the item space.

Fig. 3 .

 3 Fig. 3. Workflow: Building through dimensions of interest. With the help of the small multiple frequency plots of all dimensions analysts can detect dimensions with interesting distributions (a). Those dimensions can then be used to define targets that can be used as input for automatic suggestion mechanisms (b). This results in an automatically generated view illustrating interesting patterns (c).

2 Fig. 5 .

 25 Fig. 5. Workflow: Building and refining system-suggested subspaces. Subspace or correlation clustering groups dimensions together in the small multiples panel (a). The analyst can then explore those suggested subspaces (b) and decide on which one to focus. In this example, the analyst chose subspace (2) as view for further exploration. Via target definition and detection of further interesting dimensions this suggested view is refined by the analyst (c).

Fig. 6 .

 6 Fig.6. Refining subspace clusters. Example of system generated subspace clusters as described in the Communities and Crimes case study in Section 5.1. On the left, the results of the clustering in the Small Multiple Frequency Plot Widget can be seen. The selected dimension, "robberies", shows up in multiple clusters. In (a) the largest subspace cluster can be seen. Following the case study, in (b) only data items from the inner quartiles of "robberies" are brushed which is used to remove the non-brushed outliers. After that the subspace clustering is rerun resulting in (c) as the largest subspace cluster.

Fig. 8 .

 8 Fig.8. Class separating subspace. The process of finding a subspace separating user defined food categories in the USDA nutrition data case study (see Section 5.2). Using the Information Gain based filter to create the initial view (a). The target is specified using manual brushing in the "Food Group Desc." dimension (top right). Removing outliers seen in the PCA scatter plot and removing redundant or noisy dimensions results in the multivariate view seen in (b). Showing only the manually defined categories (c) by removing non-brushed data items from the view highlights the achieved class separation.

  This work was funded by the Laboratory Directed Research and Development Program (LDRD) at PNNL. Battelle operates PNNL for the U.S. Department of Energy (DOE) under contract DE-AC05-76RLO01830.


Evanthia Dimara
email: evanthia.dimara@gmail.com

Pierre Dragicevic
email: pierre.dragicevic@inria.fr

Anastasia Bezerianos
email: anastasia.bezerianos@lri.fr

Accounting for Availability Biases in Information Visualization

Keywords: H.5.2 [Information Interfaces and Presentation]: User Interfaces -Graphical user interfaces Design, Human Factors Availability Heuristic, Cognitive Bias, Visualizations, Decision-Making

The availability heuristic is a strategy that people use to make quick decisions but often lead to systematic errors. We propose three ways that visualization could facilitate unbiased decisionmaking. First, visualizations can alter the way our memory stores the events for later recall, so as to improve users' long-term intuitions. Second, the known biases could lead to new visualization guidelines. Third, we suggest the design of decisionmaking tools that are inspired by heuristics, e.g. suggesting intuitive approximations, rather than target to present exhaustive comparisons of all possible outcomes, or automated solutions for choosing decisions.

INTRODUCTION

We all want to make good decisions. However, decisionmaking judgments often involve approximate estimations of probabilities and frequencies. In order to reduce the complexity of estimation people rely on a limited number of strategies. One of these strategies is called Availability Heuristic [START_REF] Plous | The psychology of judgment and decisionmaking[END_REF].

The availability heuristic is a rule of thumb in which decision makers "estimate the frequency or probability by the ease with which instances or associations could be brought to mind" [START_REF] Tversky | Availability: A heuristic for judging frequency and probability[END_REF]. For example, news about a terrible plane crash may temporarily alter our feelings on flight safety. This heuristic simplifies some otherwise very difficult judgments, and it is usually effective since in principle it is easier to recall or imagine common events than uncommon ones. However, apart from the actual frequency or probability, other factors affect the ease of recalling instances, and thus estimating frequencies for making decisions. Some of these factors affecting recall, illustrated by Tversky and Kahneman [START_REF] Tversky | Judgment under uncertainty: Heuristics and biases[END_REF], often lead to systematic errors:

-Bias due to retrievability of instances: People evaluate the probability of an event as higher, when they retrieve its instances effortlessly. Schwarz et al [START_REF] Schwarz | Ease of Retrieval as Information: Another Look at the Availability Heuristic[END_REF] asked one group of participants to recall 12 examples of their past assertive behavior, and another group to recall only 6. After that, they rated their assertiveness.

The 12 examples were harder to be recalled than the 6, so the first group rated themselves as less assertive than the second group. Retrievability is often related to a) familiarity [START_REF] Tversky | Availability: A heuristic for judging frequency and probability[END_REF] or what Whittlesea [START_REF] Whittlesea | Illusions of immediate memory: Evidence of an attributional basis for feelings of familiarity and perceptual quality[END_REF] refers to as the "illusion of pastness"; b) saliency where one instance elicits more attention than another [START_REF] Taylor | Salience, attention and attribution: Top of the head phenomena[END_REF]; and c) recency where for example the serial presentation of information may affect memorization [START_REF] Ward | The display of information and the judgment of contingency[END_REF].

-Bias due to the effectiveness of the search set: The generation of a search set depends also on the performed search task. When we ask to compare the instances of the word 'love' with the word 'door' the first seems more frequent. A main reason for this is that besides the comparison of words, there is a hidden task of recalling contexts in which these words appear. It is generally easier to recall abstract contexts than concrete ones [START_REF] Galbraith | Perceived Frequency of Concrete and Abstract Words[END_REF].

-Bias of imaginability: When the frequency of an instance is not stored in memory, we sometimes generate this frequency according to some rule. For example when we want to estimate which is more frequent, the existence of committees of 8 members or of 2, we will mentally construct committees and rate them by the ease of this construction. The mental construction of 2 member committees is easier, and thus may be considered as most frequent [START_REF] Tversky | Availability: A heuristic for judging frequency and probability[END_REF]. In real life imaginability biases can lead us to overestimate some risks with vivid scenarios and underestimate dangerous risks that are hard to conceive.

-Bias due to illusory correlation: When two events co-occur people tend to overestimate the frequency of natural association. For example it is common to patients with paranoia to have peculiar eyes. This association misled undergraduate clinicians to diagnose as paranoid patients with no other symptoms related to paranoia in their medical data, simply because they were guided by a given picture of the patient with peculiar eyes [START_REF] Chapman | Genesis of popular but erroneous psychodiagnostic observations[END_REF] [START_REF] Chapman | Illusory Correlation as an Obstacle to the Use of Valid Psychodiagnostic Signs[END_REF].

Availability bias affects the decision-making ability in an unconscious way, and can lead people to irrational decisions. We believe we can better support decision-making, through the design of visualizations that take into account these factors that influence decision-making. We illustrate this in a voting scenario that we imagine takes place with and without hypothetical visualizations designed to account for biases.

THE VOTING DECISION

Imagine that one needs to decide which political candidate to vote for. There are three steps. As a first step, she shapes an opinion on which are the important personal and society issues based on her past exposure to information (media, social environment, personal experiences etc.). Second, she investigates the candidates' former actions, background and current positions, and estimates their ability and willingness to solve the important society issues. Finally, she compares all the alternatives and decides on a candidate.

In an ideal world, voters are aware of their position in the complex political landscape, understand statistical analysis data and micro-macro economics, and have endless memory capacity and time to process all the relative candidates' history. In reality, voters usually simplify this decision using heuristics. However, as we discussed, the common heuristics like availability may lead one to pick a candidate according to, for example: meaningless actions that media over-cover, without important impact in the society [START_REF] Combs | Newspaper coverage of causes of death[END_REF]; the sequence of their presentation in the public debate [START_REF] Miller | Recency and primacy in persuasion as a function of the timing of speeches and measurements[END_REF]; the vividness of the way they talk [START_REF] Reyes | Judgmental biases resulting from differing availabilities of arguments[END_REF][13] [START_REF] Borgida | The differential impact of abstract vs. concrete information on decisions[END_REF] or even whether their victory is an event easy to envision [START_REF] Carroll | The effect of imagining an event on expectations for the event: An interpretation in terms of availability heuristic[END_REF].

INFOVIS ON A COMPLEX DECISION

The actual challenge on the three decision-making steps in the voting problem is how to filter, understand, recall and compare information. In principle, this challenge is related to the infovis objectives. But how could visualizations actually assist a voter to reduce availability biases?

Visualizations to aid recall: Let's think of a scenario with simple hypothetical visualizations involved in the voting problem. Consider also two alternative policies: a consolidation of the national health system focusing on cancer cure, and a high-cost terrorism counteraction. People tend to make wrong estimations on most probable causes of death in their country [START_REF]Death Odds[END_REF], misled by media. In contrast, imagine the voter had access to a map with stacked (men/women) bar charts of death causes (society issues awareness) that she can filter. For the simplicity of the argument here we assume a voter who wants to maximize her own selfinterest. Thus, the displayed causes can be filtered according to her family's medical records and other individual characteristics (personal issues awareness). The user interacts with the visualization, composing a view that is focused on her interests. From the amount of information that she was able to process, she captures a snapshot of her self-constructed view of the visualization to save among her personal notes. In the last step of the decision-making, she evaluates the policies of each candidate. Based on her memorable interaction experience with the visualization, she intuitively evaluates which election promise has greater impact in her life.

Visualizations could thus educate decision makers to develop unbiased intuitions of their surroundings. However, as we saw in the previous example, this does not only imply visual comparisons of choices and consequences. The availability heuristic succeeds when memory stores the frequent events in an easy-to-recall way. In our example, the visualization facilitates the user's memory by capturing her self-constructed summary. Thus, visualizations could go beyond simply showing all possible alternatives. Visualizations should also help decision makers easily recall the important take-away information.

Visualizations to remove biases: So visualizations can help deal with biases. But visualization designers can also reuse the knowledge of robust biases already studied in psychology literature to make better visualizations. Studies on how the candidates' order in the set of ballot papers affects the vote rank, confirm that "recency effect" [START_REF] Ward | The display of information and the judgment of contingency[END_REF] occurs also in visualizations [START_REF] Wood | BallotMaps: Detecting name bias in alphabetically ordered ballot papers[END_REF]. Thus visualizations can reinforce the importance of some information due to some known biases (e.g., presenting them last).

Moreover, if the magnitude of a visual variable does not reflect its real impact, we may reinforce not only visual perception bias (bigger is more important), but also retrievability biases (bigger may be easier to remember). For example, consider the two political candidates suggest either the increase of unemployment allowance, or the tax exemption of families with a lot of children. Both are fair measures, but unemployment applies to a larger part of the population. However, families with many children may not be able to survive with the current tax policy. A visualization where the visual variable depends exclusively on the population size is legible, but may lead voters to evaluate the policies only according to the population criterion, even for voters who have 10 children themselves. Visualizations, in addition to what the media can offer, should be able to also display a customized perspective and alternative views of the data. These customized views of the data will be the ones most likely retrieved from their memory during the decision process.

The visualization design should also take into account the biases due to imaginability. When it comes to radical ideas, the mind's inability to construct the outcome of this idea can lead a person to consider it as impossible to happen. In the voting problem, if a candidate proposes "decentralization of state power to local communities", the voters may reject it not only because they disagree, but also because the outcome is an event hard to envision. A conceptualized, but vivid, map representation of the idea of decentralization could alter the voter's willingness to accept a change.

Visualizations inspired by heuristics:

We mostly discussed so far how to present the information to lead to effective and unbiased decisions. However, the decision-making process itself can be hard even when all the information we need is available. Automated decision-making tools that give explicit answers according to probability computations, may not always feel intuitive and understandable even by experts [START_REF] Gersh | Cognitive Engineering: Undertanding Human Interaction with Complex Systems[END_REF]. They often restrict users by expecting a very particular input, or ignoring other context-relevant information that the users may have [START_REF] Teevan | The Perfect Search Engine Is Not Enough: A Study of Orienteering Behavior in Directed Search Computer Human Interaction -CHI[END_REF]. On the other hand common visualization tools often hide uncertainty in the data [START_REF] Streit | A Spreadsheet Approach to Facilitate Visualization of Uncertainty in Information[END_REF] [1] and do not actually shield decision makers against perceptual and cognitive biases [START_REF] Hullman | Visualization Rhetoric: Framing Effects in Narrative Visualization[END_REF][14] [START_REF] Kahneman | Thinking, fast and slow[END_REF] Visualization tools are currently designed and evaluated based on data retrieval and insight tasks, rather than on the ultimate and crucial task of decision-making [START_REF] Amar | Knowledge Precepts for Design and Evaluation of Information Visualization[END_REF]. Thus we could drive some inspiration of how heuristic strategies like availability, simplify decision tasks, find the effective tradeoff among simplicity and accuracy, and apply this analogy on new visualization tools. That is, decision tools may need to allow some imperfection for the sake of understandability.

CONCLUSION

We discussed so far how information visualization could eliminate biases due to availability, when we visualize the actual probabilities, rearrange the sequence of information, increase their saliency or filter a subset of them for later recall. While designing such systems, our goal should not be to eliminate the use of heuristics that can cause these biases altogether, but rather to exploit them helping users make better decisions.

To sum up, we suggest that information visualization can reduce

  DECISIVe : Workshop on Dealing with Cognitive Biases in Visualisations, IEEE VIS2014, Nov 9th 2014, Paris Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-329436

availability biases and assist decision-making in three ways. First, we can take advantage of the good use of availability heuristics and improve users' long-term intuitions. Second, visualizations could provide design techniques that eliminate the known availability biases. Third, we can investigate new decision-making tools that target to inspired by rather than replace the way that availability heuristics work.


Richard Pasquetti
email: richard.pasquetti@unice.fr.

Viscous stabilizations for high order approximations of Saint-Venant and Boussinesq flows

. We conclude with a parallel between these two stabilization techniques.

Introduction

Simulations of flows that can develop stiff gradients generally suffer from numerical instabilities if nothing is done to stabilize the computation. Here we address shallow water flows and the turbulent wake of a sphere in a thermally stratified fluid. Shallow water flows are approximately governed by the Saint-Venant equations, i.e. by a non linear hyperbolic system that can yield shocks depending on the initial conditions. Moreover, one generally expects the numerical scheme to be well balanced and that it can support the presence of dry zones. A large literature is devoted to this questions, see e.g. [START_REF] Berthon | A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes[END_REF][START_REF] Xing | Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes[END_REF] and references herein. Our simulation of the stratified turbulent wake of a sphere relies on the incompressible Navier-Stokes equations coupled, within the Boussinesq approximation, to an advection-diffusion equation for the temperature. No shocks are in this case expected, but because the smallest scales of the flow cannot be captured by the mesh, here also a stabilization is required. Such a problem is generally addressed using the large-eddy simulation (LES) methodology, see e.g. [START_REF] Sagaut | Large Eddy Simulation for Incompressible Flows[END_REF]. In both cases, in the frame of high order methods, typically spectrally accurate methods, standard approaches are generally not useful, because implying an unacceptable loss of accuracy.

This paper describes two stabilization techniques that allow, at least formally, to preserve the accuracy of high order methods by introducing relevant viscous terms: The entropy viscosity method (EVM) and the spectral vanishing viscosity (SVV) technique. In Section 2 we develop a spectral element method (SEM) for the Saint-Venant system and we stabilize it using the EVM. In Section 3 we use the SVV technique to carry out a Fourier-Chebyshev large-eddy simulation of the turbulent stratified wake of a sphere, on the basis of stabilized Boussinesq equations. To conclude, we provide in Section 4 a parallel between these two stabilization techniques.

Entropy viscosity stabilized approximation of the Saint-Venant system

This part follows a previous paper [START_REF] Pasquetti | Stabilized spectral element approximation of the Saint-Venant system using the entropy viscosity technique[END_REF] where the one-dimensional (1D) Saint-Venant system was considered and where we especially focused on problems involving dry-wet transitions. Here this work is extended to the 2D case. Moreover, the treatment of dry zones is improved and the well balanced feature of the scheme is focused on. Comparisons are done with an analytical solution involving dry-wet transitions and the result of a problem combining shocks with dry zones is presented. The Saint-Venant system results from an approximation of the incompressible Euler equations which assumes that the pressure is hydrostatic and that the perturbations of the free surface are small compared to the water height. Then, from the mass and momentum conservation laws and with Ω ⊂ R 2 for the computational domain, one obtains equations that describe the evolution of the height h : Ω → R + and of the horizontal velocity u : Ω → R 2 : For (x,t) ∈ Ω × R + :

∂ t h + ∇ • (hu) = 0 (1) ∂ t (hu) + ∇ • (huu + gh 2 I/2) + gh∇z = 0 (2)
with I, identity tensor, uu ≡ u ⊗ u, g, gravity acceleration, and where z(x) describes the topography, assumed such that ∇z 1. Let us recall the following properties:

• The system is nonlinear and hyperbolic, which means that discontinuities may develop; • Assuming that the inlet flow-rate equals the outlet flow-rate, the total mass is preserved; • The height h is non-negative; • Rest solutions are stable; • There exists a convex entropy (actually the energy E) such that

∂ t E + ∇ • ((E + gh 2 /2)u) ≤ 0, E = hu 2 /2 + gh 2 /2 + ghz. (3)
Set q = hu and let h N (t) (resp. q N (t)) to be the piecewise polynomial continuous approximation of degree N of h(t) (resp. q(t)). The proposed stabilized SEM relies on the Galerkin approximation of the Saint-Venant system completed with mass and momentum viscous terms. For any v N , w N spanning the same approximation spaces, in semi-discrete form:

(∂ t h N + ∇ • q N , v N ) N = -(ν h ∇h N , ∇v N ) N (4) (∂ t q N + ∇ • I N (q N q N /h N ) + gh N ∇(h N + z N ), w N ) N = -(ν q ∇q N , ∇w N ) N (5)
where ν h ∝ ν q = ν, with ν : entropy viscosity (in the rest of the paper we simply use ν h = ν q ). The usual SEM approach is used here: I N is the piecewise polynomial interpolation operator, based for each element on the tensorial product of Gauss-Lobatto-Legendre (GLL) points, and (., .) N stands for the SEM approximation of the L 2 (Ω ) inner product, using for each element the GLL quadrature formula which is exact for polynomials of degree less than 2N -1 in each variable. The following remarks may be expressed:

• Mass conservation is ensured by the present SEM approximation: Set v N = 1 in the equation ( 4); If Γ q N • dΓ = 0, where Γ is the boundary of Ω , then the equalities

Ω (∂ t h N + ∇ • q N ) dΩ = Ω ∂ t h N dΩ + 0 = d t Ω h N dΩ = 0
still hold after the SEM discretization. Note however that it is assumed that the Jacobian determinants of the mappings from the reference element (-1, 1) 2 to the mesh elements are piecewise polynomials of degree less than N. • On the contrary, the expected conservation of energy for smooth problems is approximate; This results from the presence of nonlinear terms in (3). • A stabilization term appears in the mass equation (4). This is required when a high order approximation is involved, i.e. when the scheme numerical diffusion becomes negligible. • In the momentum equation [START_REF] Dommermuth | Numerical simulation of the wake of a towed sphere in a weakly stratified fluid[END_REF] we do not use the viscous term ∇ • (hν∇u), which turned out to be less efficient. Indeed, for stabilization purposes the physically relevant stabilization may not be the best suited, see e.g. [START_REF] Guermond | Viscous regularization of the Euler equations and entropy principles[END_REF] for the Euler equations. • Thanks to using ∇•I N (gh 2 N I/2) ≈ gh N ∇h N (while h 2 N is generally piecewise polynomial of degree greater than N), and thus grouping in [START_REF] Dommermuth | Numerical simulation of the wake of a towed sphere in a weakly stratified fluid[END_REF] the pressure and topography terms, a well balanced scheme is obtained by construction: If q N ≡ 0 and h N = 0, then h N + z N = Constant.

• Another difficulty comes from the positivity of h N . This point is addressed at the end of the present Section.

It remains to define the entropy viscosity ν. To this end we make use of an entropy that does not depend on z but on ∇z, which is of interest, at the discrete level, to get free of the choice of the coordinate system. Taking into account the mass conservation equation one obtains:

∂ t Ẽ + ∇ • (( Ẽ + gh 2 /2)u) + ghu • ∇z ≤ 0 , Ẽ = hu 2 /2 + gh 2 /2 . (6)
At each time-step, we then compute the entropy viscosity ν(x) at the GLL grid points, using the following three steps procedure:

• Assuming all variables given at time t n , compute the entropy residual, using a backward difference formula, e.g. the BDF2 scheme, to approximate

∂ t ẼN r E = ∂ t ẼN + ∇ • I N (( ẼN + gh 2 N /2)q N /h N ) + gq N • ∇z N where ẼN = q 2 N /(2h N ) + gh 2 N /2.
Then set up a viscosity ν E such that:

ν E = β |r E |δ x 2 /∆ E ,
where δ x stands for the local GLL grid-size, ∆ E for a reference entropy, and where β is a user defined control parameter. • Define a viscosity upper bound based on the wave speeds : λ ± = u ± √ gh :

ν max = α max Ω (|q N /h N | + gh N )δ x
where α is a O(1) user defined parameter (recall that for the advection equation α = 1/2 is well suited). • Compute the entropy viscosity:

ν = min(ν max , ν E )
and smooth: (i) locally (in each element), e.g. in 1D:

(ν i-1 + 2ν i + ν i+1 )/4 → ν i ; (ii)
globally, by projection onto the space of the C 0 piecewise polynomials of degree N. Note that operation (ii) is cheap because the SEM mass matrix is diagonal.

The positivity of h is difficult to enforce as soon as N > 1, so that for problems involving dry-wet transitions the present EVM methodology must be completed. The algorithm that we propose is the following: In dry zones, i.e. for any element Q dry such that at one GLL point min h N < h thresh , where h thresh is a user defined threshold value (typically a thousandth of the reference height):

• Modify the entropy viscosity technique, by using in Q dry the upper bound first order viscosity:

ν = ν max in Q dry
• In the momentum equation assume that:

h N g∇(h N + z N ) ≡ 0 in Q dry
• Moreover, notice that the upper bound viscosity ν max is not local but global, and that the entropy scaling ∆ E used in the definition of ν E is time independent. This has improved the robustness of the general approach described in [START_REF] Guermond | Entropy viscosity method for non-linear conservation laws[END_REF].

The numerical results presented hereafter have been obtained using the standard SEM for the discretization in space (GLL nodes for interpolations and quadratures) and the usual forth order Runge-Kutta (RK4) scheme for the discretization in time.

To outline the efficiency of the present EVM for stabilization of Saint-Venant flows involving dry-wet transitions, we first consider a problem for which an analytical solution is available: The planar fluid surface oscillations in a paraboloid [START_REF] Thacker | Some exact solutions to the nonlinear shallow-water wave equations[END_REF]. The topography is defined by z(x) = D 0 x 2 /L 2 and the exact solution is given by:

h = max(0, 2η D 0 L ( x 1 L cos(ωt) - x 2 L sin(ωt) - η 2L ) + D 0 -z) u = -ηω(sin(ωt) , cos(ωt))
where η determines the amplitude of the motion and with ω = √ 2gD 0 /L. Here we use L = 1 m, D 0 = 0.1 m and η = 0.5 m. The computational domain is a circle centered at the origin and of diameter 4. The discretization parameters are the following, number of elements: 2352, polynomial approximation degree: N = 5, resulting number of grid-points: 59081, time-step: 2.96 10 -4 , EVM control parameters: α = 1, β /∆ E = 10. Moreover, h thresh = max Ω h 0 /500, with h 0 ≡ h(t = 0), and at the initial time one sets h N = I N (max(h 0 , h thresh /2)). Since here the exact solution is only C 0 continuous, the computation cannot be spectrally accurate and so the convergence rate would be disappointing. This is why, in order to outline the interest of using a high order method like the present stabilized SEM, we prefer to compare the EVM results to results obtained with a first order viscosity. One looks in Fig. 1 at the error |h N -h|(x) at time t ≈ 6π/ω, i.e. after three loops of the fluid surface inside the paraboloid. Fig. 1 (left) gives this error field when using the first order viscosity, that is with ν h = ν q = ν max , whereas Fig. 1 (right) shows the result obtained when using the EVM. Clearly, the naive approach that would consist of simply using a first order viscosity for stabilization yields a result by far worse than the one obtained with the EVM. Animations of these error fields clearly confirm such a conclusion. Another test case provided in [START_REF] Thacker | Some exact solutions to the nonlinear shallow-water wave equations[END_REF] has also been investigated, namely the axisymmetric oscillations in a paraboloid, and the conclusion is quite similar. To conclude this Section we give the results of a simulation that shows axisymmetric oscillations in a paraboloid and that involves both dry-wet transitions and shocks. The initial condition is the following:

h = max(1. -x 2 , 0) , u = 0 ,
and at the boundary an impermeability condition together with an homogeneous Neumann condition for h are imposed. The geometry and the space discretization are those used in the first example. Calculations have been made till time t = 5 with time-step 10 -4 , and the EVM control parameters are: α = 1, β /∆ E = 20. As previously, h thresh = max Ω h 0 /500 and at the initial time h N = I N (max(h 0 , h thresh /2)). Such a flow is alternatively expanding and then retracting towards the paraboloid axis. Fig. 2 shows the flow at three different times, during the first retractionexpansion phase: At t ≈ 1.4 the velocity field is inwards, at t ≈ 1.65 it is close to reversal and at t ≈ 1.9 it is outwards. The height h N (at left) and the entropy viscosity ν (at right) are visualized. As desired, the entropy viscosity saturates in dry zones and also focuses at the shock.

3 Spectral vanishing viscosity for large-eddy simulation of the stratified wake of a sphere

To demonstrate the interest of using the SVV technique for the computation of turbulent flows, we consider the turbulent wake of a sphere in a thermally stratified fluid. Here we just focus on the main characteristics of the SVV technique and illustrate its capabilities for this particular geophysical flow. Details concerning the physical study may be found in [START_REF] Pasquetti | Temporal / spatial simulation of the stratified far wake of a sphere[END_REF].

The SVV technique was initially developed to solve with spectral methods (Fourier / Legendre expansions) hyperbolic problems (non-linear, scalar, 1D, typically the Burgers equation), while (i) preserving the spectral accuracy and (ii) providing a stable scheme [START_REF] Maday | Legendre pseudo-spectral viscosity method for nonlinear conservation laws[END_REF][START_REF] Tadmor | Convergence of spectral methods for nonlinear conservation laws[END_REF]. Later on, say in the 2000's, the SVV technique turned out to be of interest for stabilization of the Navier-Stokes equations and so for the large-eddy simulation (LES) of turbulent flows, see e.g. [START_REF] Karamanos | A spectral vanishing viscosity method for large-eddy simulation[END_REF][START_REF] Kirby | Stabilisation of spectral / hp element methods through spectral vanishing viscosity: Application to fluid mechanics[END_REF][START_REF] Koal | Adapting the spectral vanishing viscosity method for large-eddy simulations in cylindrical configurations[END_REF][START_REF] Pasquetti | Spectral vanishing viscosity method for large-eddy simulation of turbulent flows[END_REF][START_REF] Pasquetti | From stratified wakes to rotorstator flows by an SVVLES method[END_REF][START_REF] Xu | Stabilized spectral element computations of high Reynolds number incompressible flows[END_REF] and references herein.

The basic idea of the SVV stabilization technique is to add some artificial viscosity on the highest frequencies, i.e., to complete the conservation law of some given quantity u, assumed to be scalar for the sake of simplicity, with the SVV term :

V N ≡ ε N ∇ • (Q N (∇u N ))
where N is again the polynomial degree, u N the numerical approximation of u, ε N a O(1/N) coefficient and Q N the so-called spectral viscosity operator, defined to select the highest frequencies: If Q N is omitted in the definition of V N , then one recovers a first order viscosity term.

In spectral space (Fourier, Chebyshev, Legendre, or any other hierarchical basis) the operator Q N is defined by a set of coefficients Q k , 0 ≤ k ≤ N. Thus, in the 1D periodic case and if Fourier expansions are concerned (trigonometric polynomials are involved in this case):

( V N ) k = -ε N Q k k 2 ( u N ) k , ( V N ) -k = ( V N ) k
with ( .) k for the k-Fourier component, (.) for complex conjugate, and where the coefficients Q k are such that Q k = 0 for k ≤ m N , with m N a threshold value, and 0 < Q k ≤ 1 , Q k being monotonically increasing, for k > m N . In the seminal paper [START_REF] Tadmor | Convergence of spectral methods for nonlinear conservation laws[END_REF], Q k was simply chosen as a step function, i.e. with Q k = 1 if k > m N , but it quickly appeared of interest to rather use a smooth approximation. In Fig. 3 two SVV kernels are shown, using the smooth approximation proposed in [START_REF] Maday | Legendre pseudo-spectral viscosity method for nonlinear conservation laws[END_REF] and with m N = N/2 and m N = N/4. One may remark that if defining differently the Q k coefficients, one recovers alternative approaches. Thus, the hyperviscosity stabilization, i.e. such that V N ∝ -∆ 2 u N , so that ( V N ) k ∝ k 4 ( u N ) k , can be recasted in the SVV frame by stating that Q k = (k/N) 2 . In this case, the interpolating curve Q k (k) is simply a parabola, see Fig. 3. Also, if choosing ε N ∝ N -4/3 and with the curve Q k (k) shown in Fig. 3, one recovers the Chollet-Lesieur spectral viscosity [START_REF] Chollet | Parametrisation of small scales of three-dimensional isotropic turbulence utilizing spectral closures[END_REF], developed for LES in the early 80's and based on the Eddy Damped Quasi-Normal Markovian (EDQNM) theory. This latter approach strongly differs from SVV, since Q k = 0 even for k = 0, but both approaches also differ from SVV because the associated kernels nolonger constitute an approximation of a step function. As a result, it is no longer a Laplacean which is acting in the high frequency range. Indeed, if the SVV kernel is simply a step function, then the stabilization term also writes

V N = ε N ∆ u H N , where u H N is the high frequency part of u N .
Results obtained with a multi-domain spectral Chebyshev-Fourier solver with SVV stabilization are presented hereafter, see [START_REF] Pasquetti | Temporal / spatial simulation of the stratified far wake of a sphere[END_REF] for details. The turbulent wake is generated by a sphere moving horizontally and at constant velocity in a stably stratified fluid, with constant temperature gradient. The flow is assumed governed by the Boussinesq equations and the control parameters are Pr = 7, Re = 10000, F = 25, for the Prandtl, Reynolds and internal Froude number, respectively. The study is carried out in two steps: First we make a space development study, that is the Galilean Fig. 4 Temperature field in the median vertical plane at the end of the space development study (color scale: -4 to 4, from blue to red) [START_REF] Pasquetti | Temporal / spatial simulation of the stratified far wake of a sphere[END_REF].

frame is associated to the moving sphere; Second, we make a time development study, with a Galilean frame at rest. Such an approach is required to compute the far wake without needing a very elongated computational domain. With respect to anterior works, see e.g. [START_REF] Diamessis | A spectral multidomain penalty method model for the simulation of high Reynolds number localized incompressible stratified turbulence[END_REF][START_REF] Dommermuth | Numerical simulation of the wake of a towed sphere in a weakly stratified fluid[END_REF], the originality of the present study is to make use of the result of the space development study for setting up the initial condition of the temporal development study, thus avoiding the use of a synthetic initial condition.

For the space development study, the computational domain is (-4.5, 30.5) × (-4, 4) × (-4, 4). At the initial time, the fluid is stably stratified: T 0 = y. The sphere, of unit diameter and centered at (0, 0), is modeled by using a volume penalization technique. For the boundary conditions one has: Dirichlet conditions at the inlet, advection at the mean velocity at the outlet, free-slip / adiabaticity conditions on the upper and lower boundaries. The mesh makes use of 12. For the temporal development study the computational domain is (-18, 18) × (-4, 4) × (-12, 12). Note it has been enlarged, since we expect, from the confinement effect due to the stratification, a strong expansion of the wake in the horizontal plane. The initial conditions are set up from the spatial development study, by extraction of the fields obtained at the final time in (6.5, 24.5) × (-4, 4) × (-4, 4). The boundary conditions are: periodicity in streamwise direction, free-slip and adiabaticity elsewhere. The mesh makes use of 27.7 10 6 grid-points. The SVV parameters have not been changed. The flow, at the beginning and at the end of the temporal study, is visualized in Fig. 5. Fig. 6 Wake amplitude (left) and velocity deficit (right) vs time (N, buoyancy angular frequency) [START_REF] Pasquetti | Temporal / spatial simulation of the stratified far wake of a sphere[END_REF].

We conclude with some more quantitative results: In Fig. 6-left one has the evolution of the wake amplitude, both in the vertical and horizontal planes, which clearly points out the confinement effect of the stratification (here Nt is a dimensionless time, with N buoyancy angular frequency); Fig. 6-right shows the evolution of the velocity defect. This latter curve is in reasonable agreement with the "universal curve" of [START_REF] Spedding | The evolution of initially turbulent bluff-body wakes at high internal Froude number[END_REF], where three phases in the development of sphere stratified wakes are described: First the 3D phase, then the non equilibrium (NEQ) phase and finally the quasi two 2D (Q2D) phase. Tab. 7 compares the experimental results to the present numerical ones, in terms of characteristic quantities of the velocity defect evolution.

Nt I

Nt II 3D rate NEQ rate Q2D rate [START_REF] Spedding | The evolution of initially turbulent bluff-body wakes at high internal Froude number[END_REF] 1.7 ± 0.3 50 ± 15 -2/3 -0.25 ± 0.4 -0.76 ± 0.12 SVV-LES 2.4 30 ∼ -2/3 -0.2 ∼ -0.76 Fig. 7 Critical times Nt I and Nt II that correspond to the 3D-NEQ and NEQ-Q2D transitions, respectively, and rates of variation of the velocity defect in each of the three phases.

Concluding parallel between the EVM and SVV stabilizations

Two viscous stabilizations, namely the entropy viscosity method (EVM) and the spectral vanishing viscosity (SVV) technique, have been successfully implemented in high order approximations of geophysical flows: With EVM shallow water flows involving dry-wet transitions have been addressed and with SVV the turbulent wake of a sphere in a thermally stratified fluid has been investigated. We conclude with a parallel between these two approaches:

• Both SVV and EVM are viscosity methods first developed for hyperbolic problems, see [START_REF] Guermond | Entropy-based nonlinear viscosity for Fourier approximations of conservation laws[END_REF][START_REF] Tadmor | Convergence of spectral methods for nonlinear conservation laws[END_REF]. • EVM is non-linear while SVV is linear. SVV is thus not costly, since its implementation is done in preprocessing step. Moreover, because of this linear feature it is very robust and so well adapted to the LES of turbulent flows. • Both EVM and SVV preserve the accuracy of the numerical approximation. This is of course essential when high methods are concerned. • SVV is not Total Variation Diminishing (TVD) and EVM is not fully TVD, since this depends on the values of the EVM control parameters (α and β ). For SVV, a post processing stage for removing spurious oscillations has been suggested [START_REF] Maday | Legendre pseudo-spectral viscosity method for nonlinear conservation laws[END_REF]. • A theory exists for SVV [START_REF] Tadmor | Convergence of spectral methods for nonlinear conservation laws[END_REF], whereas no complete theory is available for EVM.

Some theoretical results, restricted to some specific time schemes and space approximations, are however available [START_REF] Bonito | Stability analysis of explicit entropy viscosity methods for non-linear scalar conservation equations[END_REF]. • EVM may be used with various numerical methods, since based on a physical argument, including the standard finite element method (FEM), finite volume methods etc.. SVV is restricted to spectral type methods, e.g. high order FEMs like the SEM. • SVV has proved to be of interest for LES (SVV-LES). Preliminary numerical experiments are now available for EVM [START_REF] Guermond | Validation of an entropy-viscosity model for large eddy simulation, Direct and Large Eddy-Eddy Simulation IX[END_REF], but additional tests and comparisons remain needed to check if EVM is not too diffusive and robust enough when turbulent flows are concerned.

Fig. 1

 1 Fig. 1 Planar oscillations in a paraboloid: Error on the height and at the final time for the first order viscosity (left) and EVM (right) solutions.

Fig. 2

 2 Fig. 2 Axisymmetric oscillations with shocks in a paraboloid: Height (left) and entropy viscosity (right), at t ≈ 1.4 (top), t ≈ 1.65 (middle) and t ≈ 1.9 (bottom).

Fig. 3 Q

 3 Fig.3Q k for two SVV kernels, and if using an hyperviscosity (HV) or the Chollet-Lesieur (C-L) subgrid scale model.

  4 10 6 grid-points. For the velocity components, the SVV parameters are: m N = N/2, ε N = 1/N, where N is here associated to each of the three axis, whereas for the temperature, m N = √ N , ε N = 1/N. The temperature field obtained at the end of the space development study is shown in Fig.4

Fig. 5

 5 Fig. 5 Temperature and vorticity fields at the beginning (left panel) and at the end (right panel) of the temporal development study. Up: Temperature in the median vertical plane; Middle: Transverse component of the vorticity in the same plane; Bottom: vertical component in the median horizontal plane. The vorticity being much weaker at the final time, the vorticity color scales differ [16].

Acknowledgements Part of this work was made at the Dpt of Mathematics of National Taiwan University in the frame of the Inria project AMOSS.