Abstract
Visualising data as diagrams using visual attributes such as colour, shape, size, and orientation is challenging. In particular, large data sets demand graphical display as an essential step in the analysis. In order to achieve comprehension often different attributes need to be displayed simultaneously. In this work a comprehensible bivariate, perceptually optimised visualisation scheme for highdimensional data is proposed and evaluated. It can be used to show fold changes together with confidence values within a single diagram. The visualisation scheme consists of two parts: a uniform, symmetric, twosided colour scale and a patch grid representation. Evaluation of uniformity and symmetry of the twosided colour scale was performed in comparison to a standard RGB scale by twentyfive observers. Furthermore, the readability of the generated map was validated and compared to a bivariate heat map scheme.
Introduction
Bioinformatics is a field with a broad range of applications and challenges (Augen 2005; Baldi and Brunak 2001). Based on the analysis of highdimensional data (e.g. gene expression profiles etc.) it can lead to a deeper understanding of fundamental mechanisms of oncogenesis, tumour progression, and metastasis with the potential to generate new hypotheses for diagnosis and therapeutics (Kim et al. 2012). Besides a plethora of different statistical and heuristic approaches, the inspection of visual data maps is often conducted as an important and typically first step in the analysis of data of this kind.
In these investigations a common scenario is the comparison of gene expression profiles from two different populations such as carcinoma versus inflammation (Buchholz et al. 2005). In such an instance the two populations are characterised by the difference in their gene expression levels. A basic method of describing the change of expression is using their logarithmized ratio, commonly called fold change in this context. Bilban et al. (2002) indicated that gene expression ratios are no reliable markers alone. Fold change scores can be enriched with a measure of confidence (Fensterer et al. 2004). Hence, the ultimate goal of this paper is to present a bivariate visualisation tool that encompasses both the fold changes and their standard errors.
Traditionally gene expression fold changes are visualised as red and green patches for positive and negative fold changes (see Fig. 1a). The commonly used redgreen colour palette is usually generated by uniformly sampling two channels of the standard RGB space (Eisen et al. 1998). The logarithmized fold changes are then linearly mapped onto the chosen colour palette.
The standard RGB space was designed as a colour mapping to control signals for technical devices such as computer monitors (Foley et al. 1997). It is rooted in the trichromatic colour theory (by Young and von Helmholtz (Young 1802)) which was originally based on the human ability to perceive any colour due to our photoreceptor sensitivity. The horseshoe like CIE xy chromaticity diagram, representing perceivable colours, does not show isotropic colour difference sensitivities, but depends on colour and location is this space (Lee 2005). Uniformly sampling such a colour space does therefore not result in perceptually equidistant stimuli in terms of just noticeable differences (JNDs) (Newman 1933), but rather a nonlinear mapping is necessary to achieve this. Certain values may become over or underemphasised alone by choosing the wrong colour scale. As an example Fig. 2 shows two equally sampled redgreen palettes with different perceptual properties: a standard RGB and a perceptually optimised scale (OPTIM). The commonly used bright tones of the standard RGB palette show a lower contrast to adjacent colours when compared to dark tones. This suggests the nonuniformness of the scale. Furthermore, the green halfscale appears much brighter than the red halfscale of the standard RGB palette indicating the nonsymmetry of the scale.
The colour scale in Fig. 2b was computed by the OPTSCALE algorithm (Kestler et al. 2006). Here, the corresponding colours of the red and the green halfscale reflect equal quantitative stimuli and neither of them shows a dominant effect. In each halfscale, the different colour tones are perceived as almost perfectly equidistant.
It was shown in Ware (Ware 2004, p. 136) that bivariate colour scales for encoding two dimensions are difficult to read. Based on these observations we propose the use of perceptually separable visual dimensions (Carswell and Wickens 1990): colour and size for encoding both values to achieve improved readability and the reduction of mapping artefacts in accordance with (Carswell and Wickens 1990).
Methods
Fold changes and their confidence measures
The definition of fold changes and confidence values depend on the specific application area. A fold change r is usually defined as a factor of change of a measurement value between two conditions (e.g., values at different times of a process or measurements among different environmental conditions). It could be defined by gains or losses in stock exchange relative to a basic level A representing such as
An associated confidence measure a may originate from a statistical model or be the total magnitude of the measurements, e.g., the basic level \(a=A\) or the mean measurements \(a=(A+B)/2\). In the analysis of gene expression data fold changes are commonly defined as log ratios \(M=\log _2 R/G\) and confidences as average log intensities \(A=\frac{1}{2} log_2 (R\cdot G)\) often displayed as MA plot (Dudoit et al. 2002) to shown intensity dependencies of singular fold changes. Here R and G can be intensity measurements within a microarray of a sample (e.g., tumour material) versus a control condition (healthy tissue) among multiple genes. Before rendering the dimensionality of data sets must be reduced. Feature selection methods try to omit attributes carrying no interesting information content such as those having no variation among all samples (Lausser et al. 2017; Müssel et al. 2016; Schirra et al. 2016; Völkel et al. 2015; Kraus et al. 2015; Lausser et al. 2013) or having low confidences over all samples (noise would dominate the signal in this case).
In the following we assume the existence of fold changes \({\hat{r}}_{ij} \in \mathbb {R}\) and associated confidences \({\hat{a}}_{ij} \in \mathbb {R}_0^+\) with features (such as, e.g., different genes) in the rows \(i=1\ldots m\) and samples (e.g., tissue probes of different subjects or probes at different times) in the columns \(j=1\ldots n\).
Dynamics reduction
In practice the dynamics of the raw fold changes \({\hat{r}}_{ij}\) and confidences \({\hat{a}}_{ij}\) needs to be limited and scaled to ranges \([1,1]\) and [0, 1], respectively, before further rendering. Otherwise outliers could dominate the map. Hence the following thresholded scaling functions are proposed:
with cutoff values \(\theta _a > 0\) and \(\theta _r > 0\) and hard thresholding function
Soft thresholding functions (e.g., sigmoids) can be used instead of Eq (4) as well.
Optimal colour scales
We previously described the OPTSCALE algorithm (Kestler et al. 2006) (an extension of the linear optimal scale algorithm of Levkowitz (Levkowitz 1997, pp.141)) which we briefly outline here. The procedure (see Alg 1) creates a bicoloured palette
comprising n colours \(\mathbf{c} ^{}\) representing negative log ratios (green tones), n colours \(\mathbf{c} ^{+}\) representing positive log ratios (red tones), and one central colour \(c_0\) (usually neutral dark grey or black) representing a log ratio of 0 in order to create a symmetric palette.
Hence a much larger perceptually ordered input palette (i.e. a rank can be assigned by the values)
is subsampled, thereby conserving the ordering. The input palette \(\hat{\mathbf{c }} \) is usually perceptually nonuniform. In the sample coding scheme shown here we used an 128times oversampling i.e. \(m=128\cdot n\) (we used \(n=64\)). The colours for the subscale are chosen according to a perceptual distance measure d) such that all adjacent colours have approximately constant distance \(\varDelta \):
Perceptual distance measures
Distance measures (e.g. \(\ell _2\)) between RGB coordinates do not correspond to perceptual differences between two stimuli. The Commission Internationale de l’Eclairage (CIE) proposed psychophysical derived colour spaces such as the CIE LUV and CIE LAB space in order to approximate a perceptually uniform colour space (CIE 2004). The benefit of these colour spaces is, that the \(\ell _2\) distance between two colour coordinates is nearly proportional to the perceptual stimulus difference i.e. for a difference within the \((L^*, a^*, b^*)\) coordinate space: \(\varDelta E^*_{ab} = \sqrt{(\varDelta L^*)^2+(\varDelta a^*)^2+(\varDelta b^*)^2}\). Here \(L^{*}\) is the lightness value of a colour. The pairs of values \((a^{*}, b^{*})\) (\((u^{*}, v^{*})\) for the LUV space) are the colour’s coordinates of the corresponding colour space. If a colour scale \(\hat{\mathbf{c }}\) is taken from one of these colour spaces, a subsampled scale \(\mathbf{c} \) built by OPTSCALE has the following properties:

Uniformity: Two adjacent colours \(c^{h}_{i1}, c^{h}_{i}\) within same halfscale \(h \in \{+,\}\) should always have similar perceptual distances: i.e. \(d(c^{h}_{i1}, c^{h}_{i}) \approx \varDelta \) \(\forall {i} \in \left\{ 2,\ldots ,n\right\} \) with a constant \(\varDelta > 0\). Uniformity is necessary in order to map fold changes as linearly as possible to visual stimuli.

Symmetry: The absolute intensity of a stimulus originating from the negative halfscale \(c^{}_{i}\) shall be similar to that of the corresponding positive stimulus \(c^{+}_{i}\). According to the central colour \(c_{0}\) in palette \(\mathbf{c} \) the stimulus difference \(d\left( c_{0}, c^{+}_{i}\right) \) and \(d\left( c_{0}, c^{}_{i}\right) \) should correspond in their magnitude, so that \(\left d\left( c_{0}, c^{+}_{i}\right) \right \approx \left d\left( c_{0}, c^{}_{i}\right) \right \) \(\forall {i} \in {1\ldots n}\).
Symmetry is necessary in order to let negative log ratios appear with similar intensity as positive log ratios of the same magnitude.
Bivariate heat maps
One approach for visualising fold changes together with confidence values is the use of a bivariate colour scale mapping of both attributes onto a single colour tone (cf. Figs. 1b and 5). As baseline and as a means of comparison for the proposed visualisation scheme a bivariate heat map based on the huesaturationvalue (HSV) colour space (Gonzalez and Woods 2002) is used. Here the hue value H is modulated by the foldchanges (ranging from green to red  a traditional colour scale for gene expressions). The confidence is encoded as brightness V. Saturation S was kept constant at its maximum. More reliable entities are represented by brighter colours than those of lower confidence.
The patch grid visualisation approach
A further possibility for showing log ratios together with confidences is the patch grid visualisation approach which we describe here and which is experimentally compared to the bivariate heat map described in the last section (see Figs. 1c and 4). The main idea behind this approach is the use of two (nearly) perceptually independent visual channels (size and colour) in order to optimise the perceived information content.
In the patch grid visualisation approach each entity (fold change \(r_{ij}\) together with confidence \(a_{ij}\)) is represented by a square patch of size \(s_{ij} \ge 0\) (edge length) representing the confidence filled by a colour \(c_{ij}\) encoding for the fold change. The squares (patches) are arranged as regular grid thereby showing the features (e.g., genes) in the rows and the samples in the columns. The background colour was chosen to be neutral dark grey. Spatial effects  i.e. the influence of adjacent stimuli  are not considered in this work as then the comparability to a common legend would have been lost (Robertson and O’Callaghan 1986, p. 30).
The patch colour \(c_{ij}\) is taken from the previously described OPTSCALE colour palette by a lookup of \(r_{ij}\) and a linear interpolation of the colour coordinates. The patch sizes \(s_{ij}\) were chosen according to Weber’s law (Weber 1905; Stuart et al. 1993) describing the threshold of perception of two physical stimuli I and \(I+\varDelta I\)
For a range of \(\left[ s_{min}, s_{max} \right] \subset \mathbb {R}^{+}\) a sequence of edge lengths \(s_{l}\), \(l \in \{1,\ldots , n\}\) is constructed that fulfils \((s_{l}s_{l1})/s_{l} = k\) for all \(l > 1\)
where \(s_{1}=s_{min}\) and \(s_{n}=s_{max}\).
Ordering
For enhancing the perceptual continuity of the generated visual maps the order of rows and columns can be rearranged. For this a plethora of different methods is available such as the leaf ordering of hierarchical clusterings (e.g., single, average, complete linkage clusterings) (Jain and Dubes 1988) as well as the application of traveling salesman problem (TSP) rearrangements (Climer and Zhang 2006). In contrast to clustering algorithms the TSP optimises the sum of all adjacent distances thereby creating a slightly different arrangement.
Perceptual threshold detection
Techniques for detecting a perceptual threshold reach back to Fechner (1860), (Palmer 1999, Appendix A) and include methods of adjustment, limits, and constant stimuli. Here, the best PEST method  a constant stimulus method with adaptive step size  was used (Pentland 1980; Lieberman and Pentland 1982; Treutwein 1995). The algorithm tries to find an optimum perceptual threshold by maximum likelihood estimation of a sensitivity function \(\Psi \) giving a relationship between stimulus \(X\in {\mathbb {R}}\) and a subject response \(Z\in \{0,1\}\):
The logistic function
was chosen with a fixed \(\sigma \). The \(50\%\) decision point \(\theta ^*\) is sought where the probability of a positive or negative response is equal. The subject’s response defines the random variable \(Z \in \{0,1\}\) to find the perceptual threshold denoted by
The likelihood function is defined as
with stimulus level \(x_i\in {{\mathbb {R}}}\) and subject response \(z_i \in \{0,1\}\) at trial \(i=1\ldots n\). The likelihood of a single stimulus is \({\mathcal {L}}(\theta  x_i) = \Psi (x_i\theta  z_i)\) with
The best PEST estimator shows at step n the stimulus being most likely the sought threshold value within range [a, b]:
The first presented stimulus is \(x_1=(a+b)/2\) and should be far enough from the sought threshold \(\theta \). For numeric stabilisation Eq. (13) is logarithmized. As this transformation is strictly monotone the location of the maximum is not changed. For an implementation the interval [a, b] is equally sampled at N points \(a=u_1<\cdots <u_N=b\) and a vector containing the current log likelihood function is generated (Lieberman and Pentland 1982).
Results
The following properties of the visualisation method were evaluated:

E1
Uniformity of the colour scale: the perceptual error is independent of the position within the scale.

E2
Symmetry of the colour scale: both parts of the bicoloured scale are balanced.

E3
Readability of the visualisation scheme: visual entities can be matched with a legend at low error.
The experiments for properties E1 and E2 were performed for the standard RGB scale and the optimised redgreen scale (OPTIM scale). The optimised scale was calculated by OPTSCALE using CIE LAB coordinates. Property E3 is compared for the described bivariate HSV scale and the patch grid method.
Evaluation and viewing conditions
The visualisation scheme was evaluated by 25 knowledgeable potential users from Ulm University (students, (8), staff personnel (2), research assistants (15), female/male = 6/19), age: 22–39 years (female), 24–37 years (male)) with an educational background of mathematics/computer science (19) and medicine/biology (6). All had normal or corrected to normal visual abilities. None of the participants had a dyschromatopsia. A 20 inch monitor was used. The viewing distance was about 70 cm. Colours were equalised using a calibration device (Bits++, Cambridge Research Systems) in combination with a colorimeter (ColorCAL, Cambridge Research Systems).
Uniformity (E1) and symmetry (E2)
The described best PEST scheme was used. In these evaluations two square shaped stimuli are shown (see Fig. 3): one fixed reference (target) stimulus on the left side and one adaptable stimulus on the right side. The observers’ task is to adjust the brightness of the right stimulus until it matches the brightness of the left stimulus. For E1 both stimuli were selected from the same side of the scale (green or red) whereas for E2 the stimuli were selected from opposite sides in order to match the brightness of a green colour tone with those of a red colour tone.
Six colour stimuli were equidistantly chosen from the interval [0.1, 0.7]. All stimulus values were scaled to [0, 1]. Each stimulus was shown twice, once for the RGB scale and once for the OPTIM scale (LAB space). At most 40 PEST adjustment steps had to be performed for each experiment. The 56 experiments (i.e. 14 colour tones \(\times \) 2 colour scale \(\times \) 2 experimental types  uniformity/symmetry) were randomly permuted for each evaluator. The target stimuli were chosen from the range [0.1, 0.7] for 6 identically spaced steps (it is assumed that the maximum stimulus range is defined by the interval [0, 1]). In each experiment a stimulus was shown twice, once for the RGB scale and once for the OPTIM scale. An observer had to do 28 adjustments for each experiment. All 56 single evaluations were shown in sequence, which was randomly permuted for each observer.
Initialization
The best PEST procedure is initialised such that the first presented stimulus \(x_1\) is far enough from the target stimulus \(x^*\) to be always detected as different. As the initial stimulus for the best PEST procedure is in the center \(m=a+\varDelta /2\) of the defined range \([a,a+\varDelta ]\), a was adapted such that \(m+2\sigma \le x^* \le m + 4\sigma \) or \(m4\sigma \le x^* \le m2\sigma \) uniformly distributed with equal probability. The used parameters for E1 and E2 are summarised in Table 1.
For analysing the results it is of interest if a subject categorised a colour square x erroneously to a response \(z \gg x\) or \(z \ll x\). To be more rigorous we define a quantisation function
mapping response failures \(Q_{\varDelta x}(z  x)\) to an integer k. The quantisation resolution matches the stimuli distance \(\varDelta x=0.1\). So \(k=0\) means that the colour patch was correctly classified, \(k=+1\) or \(k=1\) means that the adjacent bin was erroneously found.
Tables 2 (RGB) and 3 (OPTIM) categorise the response errors into the five classes \(\le \)2, 1\(, 0, 1, \ge 2\) for each stimuli s. There are the correct responses \(Q=0\) and the responses of the two adjacent stimuli \(Q=1\) or \(Q=1\), and those which are more than two stimuli steps away: \(Q\le 2\) or \(Q\ge 2\). For both scales the median number of correct matches is very similar \(76\%\) (RGB) versus \(84\%\) (OPTIM). In stimuli with higher magnitude s, more confusions between s and its neighbours can be seen for the RGB scale than for the OPTIM scale. The improvement was not significant (pvalue 0.1722) using a onesided paired Wilcoxon rank sum test.
Tables 4 and 5 show the number of correct matches and the adjacent matches for the RGB and OPTIM scales. For the RGB scale the responses tend to be higher than the original stimulus. 26 examples (of 350) have been classified correctly. Only few responses are classified as neighbouring stimuli (63). For the optimised scale the responses are more centred. More stimuli have been categorised correctly (82). Of the misclassified examples 126 are categorised as adjacent stimuli. Comparing the accuracies with a onesided paired Wilcoxon rank sum test showed a significant better accuracy of the OPTIM scale with pvalue 0.001652.
Readability (E3)
The readability of the patch grid visualisation scheme was investigated by presenting a random patch grid with a marked patch. Colour and size of the patch had to be matched independently to a legend at the side (see Fig. 4).
In the HSV colouring scheme the user had to match the selected value with a bivariate colour scale (see Fig. 5).
A set of 12 colour tones and 8 different patch sizes was used. The colours were chosen from a redgreen palette by using the OPTSCALE algorithm within the LAB space. The resulting colour scale contained 6 red and 6 green colours. The colour tones were equidistantly chosen within the range [0.05, 1.0] on both sides of the bicoloured scale. For each visualisation schemes 30 combinations were randomly drawn without replacement from all possible 96 ratio/confidence combinations (independently for each subject). The edge lengths (always quadratic patches) were subsampled from the interval \(\left[ 0.2, 1.0\right] \). A sequence of \(n=8\) edge lengths was constructed (Eq. 9).
The objective of the readability experiment was the matching of colour and patch size on two legends (one for the colour tone and one for the patch size) for the patch grid visualisation scheme. The bivariate colour map (HSV) had a two dimensional legend showing ordered colour tones in the rows (ranging from green to red) and varying brightness values among the columns.
Tables 6, 7, 8, and 9 summarise the results from the readability experiment as cooccurrence matrices. The median accuracy over all shown fold changes was 0.91 for the patch grid and 0.45 for the bivariate HSV colour map. A onesided paired Wilcoxon rank sum test reveals a significant better accuracy for the patch grid system (pvalue 0.002669). The confidence values could be read with median accuracy 0.905 in the patch grid approach and 0.37 in the bivariate HSV colour map (pvalue 0.007813).
Discussion
Heatmaprepresentations are a standard visualisation concept for a large variety of high dimensional molecular highthroughput profiles. Their correct interpretation can improve the analysis of mRNA or miRNA profiles as well as the analysis of methylation or mutation patterns (Gress et al. 2017; Ogechukwu et al. 2017; Taudien et al. 2016; Gress et al. 2011). However, heatmaps also provide an overwhelming amount of information. Ambiguous representations can blur the perception of existing information and can therefore be misleading (Fig. 1). In this work we evaluate a novel bivariate visualisation scheme which is mainly applied to gene expression data, but not limited to this application, showing fold changes and confidence values within a single diagram.
In a first step the used OPTIM colour scale (Kestler et al. 2006) was investigated by comparing its uniformity (E1) and symmetry (E2) property to a standard RGB scale commonly used in the rendering of gene expression data. Here uniformity is the property of perceptual equidistance within a colour scale and symmetry is the property of balancedness between two halfscales. As a consequence reddish and green colour tones are matched by their brightness such that the encoded value is equally well perceived independent of its sign. These are necessary prerequisites for a bicolour visualisation that conveys meaning in the display of gene expression fold changes. The uniformity experiments (E1) showed a slightly higher accuracy among all stimuli for the OPTIM scale (median accuracy: 0.84) compared to the standard RGB scale (mean accuracy: 0.76). The symmetry experiments (E2)  the matching of the brightness of reddish and green colour tones  revealed a strong imbalance within the RGB scale; a red stimulus was matched with a considerable darker green (mean accuracy: 0.04). The OPTIM scale showed a better balance and much more stimuli were categorised correctly (mean accuracy: 0.24) and is therefore less likely to overemphasise positive or negative fold changes of expression levels.
For an increasing magnitude the confusion of neighbouring stimuli increased for the standard RGB scale. Here, the perceptual differences between single colours decrease, which makes it harder to distinguish single stimuli. This effect was not observable in the OPTIM scale experiments where colours were chosen perceptually equidistant. For small magnitudes the perceptual differences in the RGB scale are disproportionately high. Here, more adjustments were registered for the RGB scale than for the OPTIM scale. Considering the complete scale the confusion with adjacent stimuli is more uniform for the OPTIM scale compared to the standard RGB colour scale. Overall the OPTIM scale seems to be more appropriate for visualising quantitative information than the standard RGB scale. Especially in a symmetry task the OPTIM scale is much more useful than the RGB scale. The OPTIM scale therefore allows an improved discrimination of different fold changes.
In a second step the readability (E3) of the proposed patch visualisation scheme compared to a bivariate colour coding (using the HSV colour space) was evaluated. Here, the ability of each observer for directly matching both encoded attributes as a single visual entity with a legend was evaluated. In the patch grid approach the readability showed a much higher accuracies on fold changes (median accuracy: 0.91) and confidences (median accuracy 0.905) compared to the HSV visualisation: here the fold changes (coded as colour hue) were categorised with a median accuracy of 0.45 and the confidences (coded as brightness value) of only 0.37.
The improved interpretability of the proposed patch grid visualisation approach compared to a simple bivariate colour scale was demonstrated. The focus was here to add a confidence value to a standard visualisation approach without disturbing the readability of the so generated map by introducing a second attribute. Showing confidences together with fold changes is important in rendering gene expressions but may be of interest in other application areas as well, such application domains include analysing economic growth rates or assessment of investment strategies; here, the observed dimensions may include time and geographical region, business area, or business strategy. Another example could be the analysis of the agricultural output of different crops among different environmental conditions (soil, fertiliser, temperature, humidity etc.).
Availability
This visualisation scheme is implemented as the Rproject R Core Team (2016) package colorpatch available on CRAN (https://cran.rproject.org/package=colorpatch). In addition to the red/green colour palette an optimised blue/yellow palette is also included.
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Acknowledgements
The research leading to these results has received funding from the German Research Foundation (DFG, SFB 1074 project Z1) and the Federal Ministry of Education and Research (BMBF; e:Med, conFirm, id 01ZX1708C) all to HAK.
Funding
Open Access funding provided by Projekt DEAL.
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Contributions
AM, LL, HN and HAK conceived the concept and study design, AM designed and implemented the colorpatch package. LL and HAK performed data acquisition and analyses. AW, TR and MP performed data interpretation. AM, LL and HAK wrote the manuscript. All authors reviewed the manuscript.
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Müller, A., Lausser, L., Wilhelm, A. et al. A perceptually optimised bivariate visualisation scheme for highdimensional foldchange data. Adv Data Anal Classif 15, 463–480 (2021). https://doi.org/10.1007/s11634020004165
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Keywords
 Colour scales
 Bivariate
 Visualisation
Mathematics Subject Classification
 9204
 62H35
 62P10
 92C42