Visualisation with treemaps and sunbursts in manyobjective optimisation
Abstract
Visualisation is an important aspect of evolutionary computation, enabling practitioners to explore the operation of their algorithms in an intuitive way and providing a better means for displaying their results to problem owners. The presentation of the complex data arising in manyobjective evolutionary algorithms remains a challenge, and this work examines the use of treemaps and sunbursts for visualising such data. We present a novel algorithm for arranging a treemap so that it explicitly displays the dominance relations that characterise manyobjective populations, as well as considering approaches for creating trees with which to represent multi and manyobjective solutions. We show that treemaps and sunbursts can be used to display important aspects of evolutionary computation, such as the diversity and convergence of a search population, and demonstrate the approaches on a range of test problems and a realworld problem from the literature.
Keywords
Manyobjective optimisation Visualisation Evolutionary computation1 Introduction
In the past decade, much work has been focussed on the development of methods that can visualise manyobjectives. The information that can be extracted from such methods varies depending on the type of visualisation. For example, in some methods the number of dimensions to be visualised is reduced so that a conventional visualisation can be employed (e.g., [7, 46]). Other methods avoid this loss of information by presenting the objective vectors in terms of the full set of objectives (e.g., [11, 12, 15, 20, 25, 36, 36, 46]) or visualising relationships between solutions (for example, conveying which solutions are superior to others) and are constructed in terms of the full objective set (e.g., [48]).

A new treemap layout algorithm is presented, specifically designed to visualise manyobjective populations with dominated solutions, and compared to an existing approach proposed by [26].

A quad tree from the literature [41] is used as the basis for a manyobjective visualisation.

The wellknown sunburst visualisation [40] is used to visualise manyobjective populations; demonstrations show that they can be used to convey information about the optimisation characteristics (e.g., convergence and diversity) as well as the solution quality of a mutually nondominating set.
Throughout the paper we present results for a selection of optimisation problems, including well known benchmark problems from the DTLZ problem suite [14], benchmark approximation sets proposed in [43] and solutions to a realworld test problem [24]. The remainder of this paper is organised as follows: Sect. 2 presents some relevant background material, describing existing approaches to manyobjective visualisation as well as introducing treemaps. Section 4 presents manyobjective sunbursts used for visualising mutually nondominating sets. Section 5 presents a short user experiment of the methods, and Sect. 6 provides an analysis of the properties of the introduced visualisation methods before concluding remarks are made in Sect. 7.
2 Background
2.1 Manyobjective visualisation
A challenge with visualising data in terms of the full set of objectives is that the visualisations are often too cluttered to observe useful information. Two examples of this are parallel coordinate plots [15, 20, 25] and pairwise coordinate plots [12]. Parallel coordinate plots represent a solution as a line, with the ordinal axis representing the problem objectives, and the abscissa conveying the objective value; an example is shown in Fig. 1. While this is scalable to any number of objectives and solutions the result is often too cluttered to allow a decision maker to extract useful information from it. Pairwise coordinate plots present a population of solutions according to each pair of objectives. This too is scalable, but relationships involving more than two objectives cannot be represented. Heatmaps are also a scalable approach to visualising all of the objectives within a population, and they can be enhanced to better convey the information within the data; for example, the rows and columns (representing solutions and objectives, respectively) can be rearranged to highlight the tradeoff between objectives [36, 46]. That said, one of the useful features of a manyobjective visualisation is the ability to observe dominance relations between pairs of individuals. That is not easily done using a heatmap, and such information is typically even more difficult to see using feature extraction techniques that have been used to visualise many objective solutions (e.g., selforganising maps [28, 35], generative topographic mappings [6, 18] and neuroscale [18, 32]). Presenting dominance relationships is one of the key aspects of this work.
As a close relative of the tree, methods presenting populations in terms of a graph are relevant to this work. One example of such a method is the Pareto shell visualisation demonstrated in [48]. Therein, a dominance graph is inferred on the population by ranking the individuals with Pareto sorting. Edges are then placed such that if a solution dominates an individual in the immediately inferior shell then \(W_{ij}\) indicates the probability of dominance, the number of objectives m for which \(y_{im} < y_{jm}\). That visualisation was enhanced with the use of colour, which is used to convey additional rank information. Various ranking schemes are discussed, and it is shown that the method can reveal useful structural information about the population, for example highlighting poor solutions that are extremely good on one objective and thus difficult to dominate. An extension of this method projected such a graph into the plane for visualisation as a 2dimensional scatter plot [17].
2.2 Treemaps and sunbursts
A treemap is a 2dimensional visualisation of hierarchical data constructed using a space filling algorithm. They are particularly effective for displaying clusters within data, and have been used in a variety of applications, such as visualising stock market information for identifying fraudulent transactions [23], visualising gene expression data [5], and representing file system hierarchies graphically [44]. Though we are aware of no cases in which treemaps have been used to visualise data arising from evolutionary computation (aside from [47]), they have been used to visualise the evolution of biological organisms [2] and biodiversity [22]. Hierarchies are common within evolutionary computation, examples being solutions to genetic programming problems and hierarchies of solutions defined in terms of the solutions’ relative quality, and for these reasons they are a natural choice for visualisation within evolutionary computation. We note that in their conventional square form treemaps are visually similar to mosaic plots [27]. These are not designed to convey hierarchical information, so are not considered herein.
The underlying task in visualising data with a treemap is to convey a sense of the scale or importance of a node by dividing the space within the treemap so that those nodes with high importance are represented by large regions of space, and those smaller or less important nodes receive less space. Various algorithms have been proposed to partition a space in the construction of treemaps. One of the most frequently used is the squarify algorithm, which divides a rectangular space into elements as closely as possible have an aspect ratio of 1 [8]. That work suggests that using square elements makes the comparison of pairwise elements’ size simpler, as well as providing a more efficient use of space. Treemaps do not have to be square or rectangular [45]; an alternative algorithm used in [3, 34] uses a Voronoi tessellation to divide the space. This is done by placing seed points that control the placement of irregular regions within the treemap. The regions are arranged so that they correspond to the data being visualised. It is argued by [22] that this type of treemap is more intuitive to a user, in that it is not constrained to fill a rectangular shape and can take more memorable and representative geometries. In this work, our main goal is to represent a multi or manyobjective population with a treemap so that dominance relationships are directly visible in the visualisation. We present two approaches for conveying this information, which are discussed in Sect. 3. An alternative to using the treemaps proposed herein is to exploit the sunburst visualisation [40], which divides space according to some quantified values in the way a treemap does, but nodes emanate from the centre of the visualisation. For a comprehensive review of hierarchical visualisation methods see [38].
In addition to the partitioning of the space, additional degrees of freedom can be employed to convey further information about the hierarchy. The obvious candidate is the colour of the node; in their use as a method for visualising clusters of data, treemaps are often coloured according to the cluster membership of a node [9]. The treemap can be further enhanced by using alternative rendering techniques to clarify aspects of the visualisation. An example is the use of “cushioned” nodes [44], which are intended to better highlight the hierarchical aspects of the data being represented. In this work we make use of node colour to represent additional aspects, such as solution quality, along with population convergence and diversity measures.
3 Multiobjective populations: treemaps
Before demonstrating how a spacedividing visualisations might be used to convey multi and manyobjective populations we first consider the intended workflow in which they will be incorporated. Two use cases are envisaged—one in which an evolutionary computation practitioner wishes to inspect the solutions within their algorithm’s search population, and the other in which the decision maker selects a solution generated by a MaOEA for implementation. The first use case is an important consideration, as visualising EA operation can enable better selection of algorithm parameters; the state of the search population is an important aspect of the algorithm’s operation. Additionally, interactive EAs are becoming more prevalent, and simple lightweight visualisations of the search population are an important inclusion into the user interface of such algorithms. This section considers the case of search populations, which can (and do) contain dominated solutions, and the following section describes the second use case—those examples deal exclusively with mutually nondominating solutions.
Having defined the tree’s root node, we order the population with Pareto sorting [39]. Pareto sorting begins by identifying the nondominated solutions, which become the first shell. They are then temporarily discarded, leaving a new nondominated set. This becomes the second shell, and these solutions are also discarded. Over time, the entire population is assigned to a shell. In Fig. 2, the first shell is comprised of A and B; the second shell has three members (C, D and E); and the final shell contains individual F. This produces a partial ordering of individuals, and as was done in [48] we infer a graph on the population by placing edges between the dominating and dominated individuals in adjacent shells. Additional edges are placed to connect each of the nondominated individuals to the root. The resulting network is not yet a tree; as shown in Fig. 2, it is possible for an individual to be dominated by two individuals in the superior shell. Using the nomenclature of trees, this means that a node can have two parents. In order to convert the network into a tree, we use the dominance distance [46] to identify which edges should be pruned. The dominance distance is a proper metric, and computes a distance in terms of the number of dominance relations two individuals share with the rest of the population. If the two individuals share most dominance relations then the individuals are said to be close; if they differ on a majority of the relations then they are distant. In order to prune an individual’s excess parents, we compute the dominance distance between the individual and all candidate parents in the superior shell, and retain the edge between it and the parent with which it is closest. The resulting tree structure contains, according to the dominance relation, the “best” solutions (those that are mutually nondominating) at the highest levels, and the “worst” solutions are the leaf nodes. In terms of the task of a decision maker, this is the most important structural characteristic of the tree as the solutions they are most likely to prefer are those that are mutually nondominating. The solutions at the lowest levels of the tree are unlikely to be of interest for a decision maker selecting an operating solution. We note that all nodes in the trees in this paper are unweighted.
Figure 3 illustrates a treemap which visualises an example population of 2objective individuals. Following the scheme outlined above, a tree is defined over the individuals; nondominated individuals are child nodes of the artificial root node. A region \(r_c\) is defined to specify the extent of the treemap occupied by the current node. The procedure by which the space is partitioned is based on the wellknown slice and dice method [26]. At the beginning of the partitioning procedure, the current node is the root and \(r_c=(0,1,0,1)\), defining the x origin, y origin, width and height. We initialise the partitioning direction to be vertical, however this is an arbitrary choice.
In order to make the treemap clearer, the order in which nodes are added to the visualisation can be controlled. Figure 5 shows three examples of ordered treemaps, in which the individuals have been ordered according to their value on the first objective. The lefthand panel shows the ordered version of the heatmap shown in Fig. 4. The other two treemaps show comparable populations for 3objective (centre) and 5objective (right) instances of DTLZ2. These visualisations present the individuals in a single visualisation, whereas multiple views are often required for a manyobjective visualisation using a conventional approach such as a scatter plot. That said, particularly in the 5objective case, there is an obvious lack of structure in the visualisation. Many of the individuals are mutually nondominating and do not dominate other members of the population. This means that the majority of the population is a direct child of the root node; this is because of the aforementioned lack of discrimination provided by dominance for manyobjective individuals. Given a case in which the entire population was mutually nondominating the treemap would consist entirely of Paretooptimal columns, and would impart very little information. We consider an approach to ameliorate this later in this paper.
3.1 Circular treemaps
Though the procedure outlined above produced treemaps with which it was possible to view the relative quality of solutions, the arrangement of nodes made it difficult to observe the dominance relationships between dominated and dominating nodes. In this work, we propose a new treemap layout algorithm that addresses this issue. As noted in [45], there is no requirement for a treemap to follow the rectangular layout that is often used. We therefore consider a layout in which Pareto shells are arranged as layers within a circle. The outermost layer comprises the nondominated individuals, the next layer comprises the second shell, and so on. As before, the space allocated to a node reflects that node’s importance. In the case of the nondominated layer, this defines the proportion of the total layer that the solutions occupy. For child nodes, it defines the amount of its parent’s extent that the child occupies. By constraining child nodes to lie within their parent’s extent their dominance relationships are much clearer. As in the case of square treemaps, information is conveyed by the size of a rectangle representing a node; an individual with a large number of child nodes is represented by a larger node than one with a small number of dominating individuals, and the thickness of each layer decreases to show the diminishing importance of each subsequent Pareto shell. We note that the construction of these visualisations is similar to the icicle plot [29], which arranges clusters of nodes together so that they descend, in a similar way to how nodes here are arranged inwards. Both methods provide a similar view on the data; the circular design used herein is preferred as it keeps the extent of the visualisation constrained to a smaller space.
Figure 6 shows circular treemaps for the 2, 3 and 5objective DTLZ2 populations. Again, these visualisations clearly display the relationships between individuals and those that they dominate, however the effect of increasing numbers of objectives can be seen in the 5objective case. The number of dominated individuals within the population is reducing, and as such the number of nondominated individuals residing in the outer ring is increasing. As was the case with the square treemaps, this greatly reduces the usefulness of the treemap as a visualisation method. Given the prevalence of manyobjective optimisation problems and the continually increasing interest in manyobjective optimisation algorithms, it is important to consider methods by which treemaps can be used to visualise the data arising from such problems and algorithms.
4 Manyobjective mutually nondominating sets: sunbursts
The treemaps demonstrated in the section above are suitable for representing populations of solutions in which some of the solutions are dominated, such as the search population of a MaOEA. Another common use case is to visualise the solutions resulting from the execution of such an algorithm. Generally, these solutions represent the best approximation to a given problem’s Pareto front, and as such are mutually nondominated. Two problems occur when trying to visualise mutually nondominating objective vectors with the scheme outlined above. First, the tree construction procedure begins by performing nondominated sorting on the population. If all of the solutions are mutually nondominating, then they will all belong to the first Pareto shell, and the resulting treemap will comprise a single ring. This is not very informative to a decision maker, as it does not assist them with differentiating between the solutions in their approximated Pareto front. The second issue is that some of the solutions are given substantially smaller regions within the treemap, to indicate that they are less significant and concentrate the decision maker’s attention on the solutions of higher quality. This is an advantage of the proposed treemaps when dealing with dominated solutions, as typically a decision maker will be primarily interested in those residing in the superior Pareto shells, whose individuals are represented more prominently in the treemap. In the case of a mutually nondominated set, these “inferior” solutions do not exist. Such containment methods are not suitable in situations such as this where the deeper nodes are important to the visualisation (as opposed to the case described earlier, in which the outer rings were the most important).
4.1 Producing sunbursts from quad trees
The first of these issues, representing a mutually nondominating population with a tree, has been tackled within the evolutionary optimisation literature—though not from the standpoint of visualisation. A potentially computationally expensive task within a MaOEA is identifying whether a newly evolved solution is dominated by, or dominates members of the current Pareto front approximation. A naive approach is to compare each solution of the archive and check the dominance relationships between them and the new solution. Various attempts have been made to leverage the lower complexity of lookup within a tree. Two examples of nondominated trees are [19, 33].
The second issue, relating to the difficulty in seeing the innermost nodes of a circular treemap, is addressed by using a slightly different visualisation. The sunburst [40] operates in the same way as a circular treemap, representing a node’s children within the extent of the parent, however it places the root node at the centre and child nodes emanate from it. This allows all of the nodes to be visible, and as such no part of the the tree is implicitly less important than any other. The size of the node is again determined by the number of child nodes beneath it, as was done in the treemaps earlier. This work introduces the use of sunbursts for visualising manyobjective populations.
4.2 Sunburst examples
Several demonstrations of sunburst plots representing mutually nondominating sets follow. The examples are drawn from a number of sources, including 3 and 10objective benchmark approximation sets [43], samples from the discontinuous Pareto front of a test problem from the literature [14] and the solution set generated by using a MaOEA to optimise a realworld benchmark problem comprising 9 objectives, identifying good designs of radar waveforms [24].
4.2.1 3objective BAS & DTLZ6 plots
4.2.2 10objective BAS plots
Having demonstrated the potential for using the sunburst visualisation of a quad tree to identify regions of interest within a mutually nondominated set, it is important to consider how well the method scales to larger number of objectives. The methods used to generate BASs are scalable to any number of objectives, and a 10objective linear BAS is constructed. The set consists of 500 solutions. Two instances of a sunburst representing the BAS are shown in Fig. 10. In each case, the solutions are coloured according to their score on two of the objectives (objective 1, left, and objective 3, right). In both cases, regions (marked A) have been highlighted that indicate a cluster of solutions with a lower score on that objective than is found elsewhere in that sunburst. Though the clustering is less clear than it was in the case of the 3objective populations, especially in the righthand case, it is important to note that these populations are more complex given the higher number of objectives. Despite this it is possible to observe this relationship in the highdimensional space, which indicates that the sunburst is a useful manyobjective visualisation tool. As noted above, it is a trivial matter to recolour the nodes of the sunburst, and thus the visualisation could easily be incorporated into an interactive tool where it would be used to explore a high dimensional population such as this.
4.2.3 9objective radar plots
4.3 Visualising diversity and convergence
Beyond the visualisation of solution quality, aspects of evolutionary optimisation such as population diversity and convergence are an important consideration. We consider an approach in which solution diversity is evaluated in objective space using the crowding distance measure employed within the selection operator of NSGAII [13]. Crowding distance identifies the distance between a solution and its next nearest neighbour on each objective. In Ref. [47] demonstrated the use of evaluating Euclidean distances between parameter values to consider diversity in solution space. We note that this is a sensible alternative to the approach taken herein, and that a range of measures might be appropriate in order to consider different solution representations (e.g., permutationbased approaches).
Figure 12 illustrates this approach on a sample population comprising two Pareto shells arranged on the plane. In both shells, the distribution of points is uneven, so that the points in the centre of the shells are spread out and those at the edges are closer together. The effect of this is to give those central individuals a larger crowding distance than those at the edge, and that can be clearly seen in the treemap shown in the lower panel of Fig. 12. The individuals in the centre of the shells are those on the lefthand side of the treemap, coloured dark blue in both shells. The edges of the population are located on the righthand side of the visualisation. A second sample population is shown in Fig. 13. This data was generated by sampling from the true Pareto front of DTLZ2. In a Pdimensional chromosome, the first \(M1\) parameters control the position of a solution on the true Pareto front and the rest control the distance of the solution from the front. The sampled solutions were moved away from the Pareto front by adding a small amount of random noise to these distance parameters, to create a population with dominated solutions. Two variants of this population are shown. In the first, shown on the lefthand side of Fig. 13’s top panel, a large discontinuity has been induced by placing the individuals at the extreme edges of the population, essentially forming two clusters of solutions. The individuals on the inner edges of these clusters have a large crowding distance, and these individuals are clearly visible in the corresponding treemap, shown in the centre panel; they are the dark blue individuals. Conversely, the second population contains no such discontinuity, and the solutions are all much closer together. That population’s treemap, shown in the bottom panel of Fig. 13, has a much more uniform distribution of colours, and the colours are much lighter.
Figures 14, 15, 16 illustrate sunbursts of the results of these optimisation experiments. Each figure refers to a different selection mechanisim—Fig. 14 shows the random selection case; Fig. 15 shows the Pareto sorting example, and Fig. 16 shows the average rank results. The topleft panel shows the objective space image of the objective vectors. The solutions in all three final archives were ordered using average rank, and this information was used to colour the individuals in the population view as well as the sunburst visualisation next to it. The bottom row shows the same sunburst representation of the archive; the lefthand plot is coloured according to crowding distance to show diversity, while the righthand shows the age of the solution to indicate convergence. In the case of the diversity, the best results were achieved by the Pareto sorting algorithm. That algorithm’s sunburst has the lightest colouring, indicating that the crowding distance values are much more uniform and the solutions are better spread across the Pareto front. In both of the other cases, the treemaps feature dark blue colouring more, meaning that those Pareto front estimations contain more distant solutions. This is particularly the case for the average rank optimiser, which has explored very little of the Pareto front. This is supported by the convergence sunbursts, which, again, display the (normalised) age of the solutions in the archive. The age is specified by the generation number in which the solution was archived, so a low number indicates an old solution that was generated early on in the optimisation process. The premature convergence exhibited by this algorithm is shown clearly by the large group of solutions, shown in dark brown at the top of the sunburst, which have been in the archive since the beginning of the optimisation procedure. These correspond to the random solutions scattered at the bottom of the top panel’s scatter plot and are shown in region A. They correspond to the solutions with the largest crowding distance in the diversity sunburst. Beyond examining the colouring of the visualisations, these sunbursts convey more information through their structure than has been seen before. From examining all three it can be seen that the number of layers provide an indication of the degree to which the population has converged—the random selection algorithm has 10 layers, compared to 12 in the Pareto sorting example and 20 in the prematurelyconverged average rank case. There are also fewer missing regions in the sunburst representing the “ideal” case, using Pareto sorting, with the visualisation tending more towards the full circle shape seen in earlier examples sampled from the Pareto front and in the BAS examples.
5 Validation
As well as discussing the topological features that the treemaps and sunbursts allow the user to observe, it is important to quantify how useful the proposed methods are. To do this, a small user experiment was conducted to assess the extent to which the user can identify solutions of interest, as well as examining the accuracy of their selection. Using the nomenclature from [30], a controlled experiment was carried out. Sunburst plots were pit against three other manyobjective visualisation methods drawn from the literature. These are seriated heatmaps [46], parallel coordinate plots [25] and multidimensional scaling (MDS) [37] constructed using the dominance distance [46]. These methods were chosen as a crosssection of existing methods from the evolutionary computation visualisation literature, and include a method that shows the actual objective values (parallel coordinate plots); a method based on all M objectives (heatmaps); and a dimension reduction method (MDS). Three sets of 50 mutually nondominating solutions were constructed from datasets seen previously herein—the linear BAS, spherical BAS and samples drawn uniformly at random from the radar waveform optimisation solutions (giving different types of Pareto front geometry and different numbers of objectives (10, 3 and 9 objectives, respectively)). The 50 samples were generated at the start of the experiment, meaning that each user saw different datasets. That set of 50 samples was then displayed with each of the four visualisation types.
6 Discussion
The sections above demonstrate that both treemaps and sunburst plots can be used to convey useful information within evolutionary computation. In order to contextualise the methods with others used in the field, they are evaluated using the framework proposed by Tusar and Filipić [43]. They characterise nine properties of a population visualisation: the preservation of (1) dominance relations, (2) front shape, (3) objective range, and (4) the distribution of vectors; (5) robustness; (6) the ability to handle large sets; (7) the simultaneous visualisation of two or more populations; (8) scalability; and (9) simplicity. The visualisations proposed in this work are formed of two components—the treemap or sunburst visualisation itself and the underlying tree structure. The proposed methods are discussed with respect to the approaches taken herein, and may not apply to other tree structures.
The treemap visualisations preserve some dominance relations by including the dominated child within the extent of its parent solution. Those that are discarded in the tree construction process are lost, however care is taken to preserve the relation between the child and the dominating solution with which it is closest using the dominance distance. The sunburst visualisations were used to show mutually nondominating sets, so in terms of dominance all of the solutions are incomparable. Both visualisations are capable of preserving dominance relations, depending on the type of tree structure used to store the individuals. It is also possible to illustrate distribution of objective ranges can be included with colour. In this work the objective ranges have been normalised, however as has been discussed it is computationally cheap to update the colour of nodes. The distribution of vectors is shown both through the arrangement of nodes and by applying a colour indicating distances between points. In the mutually nondominating sets generated by the three MaOEAs different distributions of solutions were obtained; the well converged diverse population had a rounded arrangement, while the other two (showing poor convergence and premature convergence) had gaps. Both visualisations are capable of supporting large populations, both in terms of the number of solutions and the number of objectives; hence, they meet both the criteria relating to handling of large sets and scalability. In terms of simplicity, the treemaps and sunbursts are both constructed with a recursive function that is called linearly with the number of individuals in the population. The complexity of tree construction is also a consideration; the original purpose of storing mutually nondominating sets in tree structures was to enable fast lookup for checking dominance relations, which means the computationally expensive operations are carried out during tree construction. That said, the algorithm used herein completes tree construction in polynomial time. Thus, both treemaps and sunbursts meet the simplicity criterion, with the proviso that additional complexity can be introduced depending on the desired colour scheme.
Three of the characteristics are not observed in treemaps or sunbursts. The shape of the Pareto front is not conveyed in either the treemap or sunburst visualisations. According to [43], robustness refers to the ability to add solutions without changing the existing population. The visualisations herein are not robust because an additional solution will change the structure of the population. Finally, both treemaps and sunbursts represent a single population, therefore they are not suitable for comparing between different solution sets.
Beyond contextualising the methods in terms of the properties of a population visualisation, it is important to compare them to other methods in terms of the tasks they will be used to perform. A taxonomy of visualisation tasks was proposed [1] and we evaluate treemaps and sunbursts according to those tasks and compare them to other visualisation methods. The taxonomy provides ten tasks, some of which are not relevant to the general goal of identifying good solutions using the visualisation, but most of which are. The tasks defined by the taxonomy are: (1) retrieve value; (2) filter; (3) compute derived value; (4) find extreme values; (5) sort; (6) determine range; (7) characterise distribution; (8) find anomalies; (9) cluster; and (10) correlate.
Probably the most relevant to the overall goal are identifying extreme values and sorting. In both cases, both treemaps and sunbursts facilitate this through the colouring of nodes. In the examples demonstrated, we have shown that the nodes can easily be coloured according to the overall quality of a solution, or by the solutions’ value on an individual objective. In each case, the decision maker can look for the extreme colour to identify the best solution (as well as the worst), and can use the colour gradient between the maximum and minimum to infer an ordering of solutions. In a similar fashion the range of each objective can be identified by colouring the solutions by the relevant objectives. We note that in this paper the visualisations have shown normalised objective values—to identify the range of the objectives it would be necessary to visualise the objective range. A similar task is evaluating the distribution of objective values. This can be done, again, by considering the solutions’ colour. The authors’ description of distribution analysis [1] discusses the comparison of different classes—this relates to the comparison of different objectives, which in turn leads to tradeoff analysis. While it is possible to observe the tradeoff between different objectives, it relies on changing the node colouring between the objectives being compared, and in the methods’ current form it is not possible to visualise this without interaction. We do not feel that this is too much of a deficiency, however it is worth considering in future work. The ability to identify clusters within the data has been demonstrated within the examples shown earlier. In those examples, the data were coloured using a priori information—which region of the Pareto front the solutions belong to in Fig. 9 and which class of objective the solutions achieved the best rank on in Fig. 11. Clusters were also visible in the visualisations coloured according to average rank, in which regions of good and poor solutions could be observed, and in those coloured according to a specific objective. Again, regions of good performance on the objective at hand were easily discerned.
Of the remaining five tasks that are not immediately facilitated by the treemaps or sunbursts, value retrieval, filtering, anomaly detection and correlation observation might easily be facilitated through interaction. The individual objective values are not present in the basic visualisation (though the visualisation is based on them), however they could be easily displayed for a selected solution alongside the visualisation. Likewise, additional work would be required to implement filtering of objective values, but this would be possible through the addition of a user interface. This feature could be extended to observe correlations by allowing composite filtering, however users are likely to be better served by providing a heatmap or parallel coordinate plot in such tasks. The same is true of anomaly detection. The computation of derived values is not something that a user engaged in tasks defined would use such a visualisation for, though, again, in theory it could be incorporated within the user interface.
7 Conclusion
Hierarchical information is common within evolutionary computation. This paper has presented treemaps and sunbursts for visualising data in evolutionary computation, focussing on populations of solutions to manyobjective problems. The visualisation of such data is an important task, as decision makers find comprehending solutions described by a large number of objectives difficult. Treemaps are a good choice of visualisation tool because of their flexibility. They have various degrees of freedom that can be exploited to convey the structure of a population. Though the standard form of a treemap is a square grid in which nodes are represented by rectangles, we have presented an alternative layout algorithm that is better suited to displaying the dominance relationships that characterise a manyobjective population. By using circular treemaps, the parent–child (and therefore dominance) relationships are much easier to observe. In addition to showing individual quality, we demonstrated that treemaps can be used to convey other information relevant to the operation of an MaOEA, such as the diversity within the search population and how well converged the solutions are.
We have presented a novel algorithm for building a tree of multiobjective solutions so that a treemap can be rendered. Based on dominance, the algorithm is shown to be suitable for multiobjective populations, but does not scale well to deal with manyobjective populations. This is because manyobjective individuals are generally incomparable under dominance, so instead a tree construction algorithm from the literature was employed. Whereas circular treemaps give most prominence to those individuals in a population that are nondominated, with less significance given to dominated solutions, a mutually nondominating set does not have such preferential individuals. As such, a sunburst visualisation is used instead of a treemap, which sees nodes radiating out from the centre rather than inwards as is the case with circular treemaps. Both of the methods result in graphs that convey useful information. We acknowledge that, unlike some of the other methods for constructing multiobjective trees in the literature, the purpose of these trees is to be used as a basis for visualisation. Were they to be used within an optimisation process, the computational complexity would likely be an issue. That said, for the construction of a oneoff visualisation this is not an issue, and we believe that they are fast enough to be used within an interactive visualisation too.
The main advantage offered by the approaches described herein are their flexibility. We have used three visualisation methods in combination with two tree representations of data arising within evolutionary computation to visualise aspects of that data, however the use of these methods is far from restricted to the approaches we have taken. As discussed in Sect. 2, other approaches to representing data as trees have been taken in evolutionary computation, and any of these tree representations could form the basis of a treemap visualisation. Likewise, the visualisation literature contains a plethora of approaches to arranging treemaps. Though the circular treemap proposed here is designed specifically for use in evolutionary computation, to illustrate dominance relationships between multi and manyobjective individuals, the selection of layout algorithm is largely problem specific. An aspect of future work is to consider other areas of evolutionary computation in which treemaps might be productively used, and design new ways of illustrating this hierarchical information.
In addition to considering other applications of treemaps and sunbursts, several aspects of future work are worthy of consideration. In terms of the mechanics of the visualisation, these fall into two groups. In the first, the tree used as the basis of the visualisation would be enhanced. As has been discussed, one of the benefits of the proposed method is that the visualisation is completely decoupled from the underlying tree, so it would be useful to consider whether there are characteristics of mutually nondominating sets that can be more effectively represented by using a strategy other than the successorbased quad tree demonstrated herein. Beyond this, there may be other data structures in use within MaOEAs that might inform or inspire a visualisation in the same way that this work was inspired by research into the use of trees to represent populations with a treemap. The second area of future work would consider alternative layout algorithms. This work has shown useful visualisations of solution sets, however some of the characteristics of the sets were lost, such as their shape. Mutually nondominating sets are characterised by their shape; this can be, for example, linear, convex or nonconvex, and is a piece of information that this work does not consider. Though it was possible to observe the tradeoff between objectives by updating node colourings to show different objective values, an ideal visualisation would incorporate this within a single visualisation without needing multiple views. We feel that the potential of presenting evolutionary computation data in this way is an exciting prospect, and likely to be extremely useful to evolutionary computation practitioners.
A final extension that we are currently exploring is how the visualisation can be more thoroughly evaluated. Elsewhere in evolutionary computation, such as algorithm development, rigorous benchmarking of methods is employed. In the realm of visualisation this has not historically been the case, and we are currently investigating how evaluation methods within the visualisation field might be applied within evolutionary computation to facilitate a more scientific investigation of methods such as those proposed herein, beyond examples such as the small user experiment used to gather quantifiable data used in this work. The study included herein would benefit from a random ordering of the visualisations, in order to eliminate the potential for visualisations presented later in the study to benefit from greater understanding on the part of the user. We feel that such investigation will lead to the development of much better visualisations for the wider evolutionary computing field.
Notes
Acknowledgements
The author would like to thank Prof. Ed Keedwell for his valuable comments on a draft of this paper, and was supported by EPSRC Grant EP/P009441/1 for some of this work.
References
 1.R. Amar, J. Eagan, J. Stasko, Lowlevel components of analytic activity in information visualization, in IEEE Symposium on Information Visualization (INVOVIS’05) (2005)Google Scholar
 2.A. Arvelakis, M. Reczko, A. Stamatakis, A. Symeonidis, I.G. Tollis, in Proceedings of International Symposium on Biological and Medical Data Analysis (ISBMDA) (Springer, 2005), pp. 283–293Google Scholar
 3.M. Balzer, O. Deussen, Voronoi treemaps, in IEEE Symposium on Information Visualization (INVOVIS’05) (2005), pp. 49–56. https://doi.org/10.1109/INFVIS.2005.1532128
 4.P.J. Bentley, J.P. Wakefield, Finding acceptable solutions in the Paretooptimal range using multiobjective genetic algorithms, in Soft Computing in Engineering Design and Manufacturing (1998), pp. 231–240Google Scholar
 5.J. Bernhardt, S. Funke, M. Hecker, J. Siebourg, Visualizing gene expression data via voronoi treemaps, in Sixth International Symposium on Voronoi Diagrams, 2009. ISVD ’09 (2009), pp. 233–241Google Scholar
 6.C. Bishop, M. Svensén, C. Williams, GTM: the generative topographic mapping, Neural Comput. 10, 215–235 (1998)CrossRefMATHGoogle Scholar
 7.D. Brockhoff, D.K. Saxena, K. Deb, E. Zitzler, On handling a large number of objectives a posteriori and during optimization, in Multiobjective Problem Solving from Nature: From Concepts to Applications, ed. by J. Knowles, D. Corne, K. Deb (Springer, Berlin, 2007), pp. 377–403Google Scholar
 8.M. Bruls, K. Huizing, J. van Wijk, Squarified treemaps, in In Proceedings of the Joint Eurographics and IEEE TCVG Symposium on Visualization (Press, 1999), pp. 33–42Google Scholar
 9.Y. Chen, X. Zhang, Y. Feng, J. Liang, H. Chen, Sunburst with ordered nodes based on hierarchical clustering: a visual analyzing method for associated hierarchical pesticide residue data. J. Vis. 18(2), 237–254 (2015)CrossRefGoogle Scholar
 10.A.R.R. de Freitas, P.J. Fleming, F.G. Guimaraes, Aggregation trees for visualization and dimension reduction in manyobjective optimization. Inf. Sci. 298, 288–314 (2015)CrossRefGoogle Scholar
 11.A.R.R. de Freitas, P.J. Fleming, F.G. Guimaraes, Aggregation trees for visualization and dimension reduction in manyobjective optimization. Inf. Sci. 298, 288–314 (2015)CrossRefGoogle Scholar
 12.K. Deb, MultiObjective Optimization using Evolutionary Algorithms, WileyInterscience Series in Systems and Optimization (Wiley, Chichester, 2001)MATHGoogle Scholar
 13.K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGAII. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)CrossRefGoogle Scholar
 14.K. Deb, L. Thiele, M. Laumanns, E. Zitzler, Scalable multiobjective optimization test problems, in Proceedings of IEEE Congress on Evolutionary Computation, vol 1, (2002), pp. 825–830Google Scholar
 15.M. D’Ocagane, Coordonnées parallles et axiales: Méthode de transformation géométrique et procédé nouveau de calcul graphique déduits de la considération des coordonnées parallèlles (GauthierVillars, 1885) reprinted by Kessinger PublishingGoogle Scholar
 16.M. Farina, P. Amato, On the optimal solution definition for manycriteria optimization problems, in 2002 Annual Meeting of the North American Fuzzy Information Processing Society Proceedings (2002), pp. 233–238Google Scholar
 17.J. Fieldsend, R. Everson, Visualising highdimensional Pareto relationships in twodimensional scatterplots, in Evolutionary Multicriterion Optimization (EMO 2013) (2013), pp. 558–572Google Scholar
 18.J.E. Fieldsend, R.M. Everson, Visualisation of multiclass ROC surfaces, in Proceedings of the ICML 2005 Workshop on ROC Analysis in Machine Learning (2005), pp. 49–56Google Scholar
 19.J.E. Fieldsend, R.M. Everson, S. Singh, Using unconstrained elite archives for multiobjective optimization. IEEE Trans. Evol. Comput. 7(3), 305–323 (2003)CrossRefGoogle Scholar
 20.C.M. Fonseca, P.J. Fleming, Genetic algorithms for multiobjective optimization: formulation, discussion and generalization, in Proceedings of the Fifth International Conference on Genetic Algorithms (Morgan Kauffman, 1993), pp. 416–423Google Scholar
 21.M. GarzaFabre, G. ToscanoPulido, C.A.C. Coello, Two novel approaches for manyobjective optimization, in Proceedings of the IEEE Congress on Evolutionary Computation (2010), pp. 4480–4487Google Scholar
 22.M.S. Horn, M. Tobiasz, C. Shen, Visualizing biodiversity with voronoi treemaps, in Sixth International Symposium on Voronoi Diagrams, ISVD 2009, Copenhagen, Denmark, June 23–26, 2009 (2009), pp. 265–270Google Scholar
 23.M.L. Huang, J. Liang, Q.V. Nguyen, A visualization approach for frauds detection in financial market, in Proceedings of the 2009 13th International Conference Information Visualisation (2009), pp. 197–202Google Scholar
 24.E.J. Hughes, Radar waveform optimisation as a manyobjective application benchmark, in Proceedings of the 4th International Conference on Evolutionary Multicriterion Optimization, EMO’07 (Springer, Berlin, 2007), pp. 700–714Google Scholar
 25.A. Inselberg, Parallel Coordinates: Visual Multidimensional Geometry and Its Applications (Springer, Berlin, 2009)CrossRefMATHGoogle Scholar
 26.B. Johnson, B. Shneiderman, Treemaps: a spacefilling approach to the visualization, in Proceedings of the 2nd International IEEE Visualization Conference (1991), pp. 284–291Google Scholar
 27.B. Kleiner, J.A. Hartigan, Representing points in many dimensions by trees and castles. J. Am. Stat. Assoc. 76(374), 260–269 (1981)CrossRefGoogle Scholar
 28.T. Kohonen, SelfOrganising Maps (Springer, Berlin, 1995)CrossRefGoogle Scholar
 29.J.B. Kruskal, J.M. Landwehr, Icicle plots: better displays for hierarchical clustering. Am. Stat. 37(2), 162–168 (1983)Google Scholar
 30.H. Lam, E. Bertini, P. Isenberg, C. Plaisant, S. Carpendale, Empirical studies in information visualization: seven scenarios. IEEE Trans. Vis. Comput. Graph. 18(9), 1520–1536 (2012)CrossRefGoogle Scholar
 31.B. Li, J. Li, K. Tang, X. Yao, Manyobjective evolutionary algorithms: a survey. ACM Comput. Surv. 48(1), 13 (2015). https://doi.org/10.1145/2792984 CrossRefGoogle Scholar
 32.D. Lowe, M.E. Tipping, NeuroScale: novel topographic feature extraction using RBF networks, in NIPS (1996), pp. 543–549Google Scholar
 33.J.B. Mendes, J.A. de Vasconcelos, Using an adaptation of a binary search tree to improve the NSGAII nondominated sorting procedure, in Simulated Evolution and Learning (Springer, 2010), pp. 558–562Google Scholar
 34.A. Nocaj, U. Brandes, Computing voronoi treemaps: faster, simpler, and resolutionindependent. Comput. Graph. Forum 31(3pt1), 855–864 (2012)CrossRefGoogle Scholar
 35.S. Obayashi, Pareto solutions of multipoint design of supersonic wings using evolutionary algorithms, in Adaptive Computing in Design and Manufacture V, ed. by I.C. Parmee (Springer, London, 2002), pp. 3–15CrossRefGoogle Scholar
 36.A. Pryke, S. Mostaghim, A. Nazemi, Heatmap visualization of population based multi objective algorithms, in Evolutionary Multicriterion Optimization (EMO 2006) (2006), pp. 361–375Google Scholar
 37.J.W. Sammon, A nonlinear mapping for data structure analysis. IEEE Trans. Comput. 18(5), 401–409 (1969)CrossRefGoogle Scholar
 38.H.J. Schulz, S. Hadlak, H. Schumann, The design space of implicit hierarchy visualization: a survey. IEEE Trans. Vis. Comput. Graph. 17(4), 393–411 (2011)CrossRefGoogle Scholar
 39.N. Srinivas, K. Deb, Multiobjective optimization using nondominated sorting in genetic algorithms. Evol. Comput. 2(3), 221–248 (1994)CrossRefGoogle Scholar
 40.J. Stasko, R. Catrambone, M. Guzdial, K. McDonald, An evaluation of spacefilling information visualizations for depicting hierarhcical structures. Int. J. Hum. Comput. Stud. 53, 663–694 (2000)CrossRefMATHGoogle Scholar
 41.M. Sun, R.E. Steuer, Interquad: an interactive quad tree based procedure for solving the discrete alternative multiple criteria problem. Eur. J. Oper. Res. 89, 462–472 (1996)CrossRefMATHGoogle Scholar
 42.S. Tiwari, N. Chakraborti, Multiobjective optimization of a twodimensional cutting problem using genetic algorithms. J. Mater. Process. Technol. 173, 384–393 (2006)CrossRefGoogle Scholar
 43.T. Tušar, B. Filipič, Visualization of Pareto front approximations in evolutionary multiobjective optimization: a critical review and the prosection method. IEEE Trans. Evol. Comput. 19(2), 225–245 (2015)CrossRefGoogle Scholar
 44.J.J. Van Wijk, H. van de Wetering, Cushion treemaps: visualization of hierarchical information, in Proceedings of the 1999 IEEE Symposium on Information Visualization, INFOVIS ’99 (1999), pp. 73–78Google Scholar
 45.R. Vliegen, J.J. van Wijk, E.J. van der Linden, Visualizing business data with generalized treemaps. IEEE Trans. Vis. Comput. Graph. 12(5), 789–796 (2006)CrossRefGoogle Scholar
 46.D.J. Walker, R.M. Everson, J.E. Fieldsend, Visualising mutually nondominating solution sets in manyobjective optimization. IEEE Trans. Evol. Comput. 17(2), 165–184 (2013)CrossRefGoogle Scholar
 47.D.J. Walker, Visualising manyobjective populations, in Proceedings of the 2015 Genetic and Evolutionary Computation Conference (2015), pp. 451–458Google Scholar
 48.D.J. Walker, R.M. Everson, J.E. Fieldsend, Visualising and ordering of manyobjective populations, in 2010 Congress on Evolutionary Computation (2010), pp. 3664–3671Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.