Fuzzy surfaces are surface models that account for the uncertainty that originates either in data or from user's uncertainty about the interpolation settings. So far, fuzzy surfaces have been visualized as 3D projections, profiles or as separate visualizations of three surfaces that show minimal, modal and maximal estimated value. Unfortunately, neither of these approaches towards fuzzy surface visualization is quite suitable for practical application. 3D projections and profiles manage to capture only part of the surface and three separate visualization are extremely challenging for users. In ideal case the user needs to obtain complete information (including uncertainty) about whole surface in the simplest visualization possible. As a result of these issues, a new method for fuzzy surface visualization is proposed. The method utilizes hue saturation lightness (HSL) colour model to depict the most important values of the predicted fuzzy surface. The method combines properties of the set of fuzzy numbers, that form the fuzzy surface, with properties of colour defined in HSL colour model. The resulting visualization utilizes continuous two-dimensional gamut to visualize the fuzzy surface. In cartography, however, the continuous gamuts are often consider as needlessly challenging for users (to obtain the correct information); so as a simplification a discrete variant of the two dimensional gamut is provided as well. The variant of visualization utilizing the discrete gamut is designed to be as customizable as possible, so that it is possible to adapt the visualization (in terms of complexity) for various groups of users from experts to non-experts. The proposed approaches are defined in terms of equations that allow visualization of arbitrary fuzzy surface with high level of visualization customization. The visualization method is presented on practical case study. A simple software, that allows testing of the proposed visualization, is provided as an appendix of the paper.

Keywords: fuzzy surface, uncertainty, visualization, HSL, geovisualization

1. Introduction

One of the main issues associated with spatial interpolation is the fact that the resulting visualizations often provide users with a false impression that the visible results are entirely precise while the opposite is true (Skeels, Lee, Smith, Robertson 2010). The predicted data contain uncertainty that may originate from data imprecision, vagueness or randomness. Another source of uncertainty is the interpolation method itself. Some interpolation methods like kriging even provide the estimation of uncertainty that arises directly from the prediction process. For kriging, this uncertainty is presented in a form of the standard deviation of prediction error (Hengl 2003). The interpolation method can also be affected by uncertainty as a result of user’s uncertainty regarding parameters settings for the interpolation (Loquin, Dubois 2010). However, for the decision making that commonly follows the spatial prediction, it is not important where in the previous process uncertainty originates but how it affects the results. (Edwards, Nelson 2001) state that data validity is the key to making credible decisions. As a consequence, it is necessary from the visualization point of view to pass on to the user not only the information about predicted values but also the information about certainty (or uncertainty) of the prediction (Skeels, Lee, Smith, Robertson 2010).

The presented research focuses on visualization of fuzzy surfaces. Fuzzy surfaces are special type of surfaces that incorporate uncertainty of data (Diamond 1989; Santos, Lodwick, Neumaier 2002; Waelder 2007), epistemic uncertainty of user regarding the settings of interpolation (Bardossy, Bogardi, Kelly 1990; Loquin, Dubois 2010) or both (Loquin, Dubois 2010; Soltani-Mohammadi 2015). Unlike the statistical approaches of representing uncertainty of surface, such as the one presented by (Hengl 2003), the uncertainty of fuzzy surface does not have to be symmetrical. This fact renders visualization methods like so called “whitening” (Hengl 2003) unsuitable for visualization of fuzzy surfaces. The problem of visualization is often neglected when dealing with fuzzy surfaces, mainly because most of the research focuses on construction of fuzzy surfaces (Caha, Vondráková 2014).

There has been some research regarding visualization of fuzzy information (Jiang 1996; Bastin, Fisher, Wood 2002; Pham, Brown 2005; Park, Park 2010) with the focus mainly on fuzzy sets and fuzzy logic systems. Unfortunately, the results obtained from the mentioned studies cannot be applied to visualization of fuzzy surfaces as the type of uncertainty differs significantly amongst those topics.

The main aim of this research is to provide a method for visualization that will allow a representation of the predicted value along with its uncertainty. According to the categorization of spatial uncertainty visualizations provided by (Kinkeldey, MacEachren, Schiewe 2014), the presented method falls in into categories of explicit (directly displaying uncertainty), intrinsic (altering existing symbology), visually integral (uncertainty cannot be separated from the data), coincident (both data and uncertainty are represented in one view) and static (no dynamic elements) representations. The organization of the rest of the paper is following section Theoretical background provides necessary information about fuzzy surfaces and HSL (hue saturation lightness) colour model. Section Proposed approach describes the procedure for fuzzy surface visualization. A practical example of the proposed approach is shown in section Case study. In the last section, several conclusions are drawn.

2. Theoretical background

This chapter summarizes the necessary concepts of fuzzy surfaces and the HSL colour model. Fuzzy surfaces are a particular type of surface that represent not only the surface but also its uncertainty. The uncertainty of the surface is modelled with the utilization of fuzzy numbers which are also described in this section. A thorough explanation of these concepts is essential for the visualization approach that is explained in section Proposed approach.

2.1. Fuzzy numbers

A fuzzy number is a unique case of a fuzzy set that represents the uncertain, imprecise or ill-know variable (Dubois, Prade 1979). Fuzzy numbers are used to model values that cannot be specified precisely, but there are indications of the most likely correct value as well as extreme values that provide limits. For example, value ’approximately two’ can be modelled as a fuzzy number with the most likely value 2 and limits 1 and 3. Given the uncertain description the user can only conclude that the vague value is higher than 1, lower than 3 and the most likely correct representation is 2. Such vague description of number is an ideal starting point for modelling using fuzzy numbers.

In the same way as a fuzzy set a fuzzy number is described by a membership function which is defined by mapping:

that indicates that element of universe has the membership value from the interval (Zadeh 1965). Element with does not belong to the fuzzy set, element with belongs to the set completely. Other values from the interval denote partial membership.

In order to be a fuzzy number, the fuzzy set has to satisfy several conditions (Hanss 2005). The universe on which the set is defined should be real numbers . The height of the fuzzy set according to

must be equal to 1. So that there is one and exactly one with full membership to the set . This value is called peak or modal value of a fuzzy number. Sometimes fuzzy numbers that have more than one with are also considered as fuzzy numbers. However, for the purpose of this research only fuzzy numbers with one with are considered.

Next condition is that the fuzzy set also has to be convex, which it is, if the condition

is satisfied. The support of the fuzzy set has to bounded. The last condition is that the membership function must be at least piecewise continuous. If the fuzzy set satisfy all those conditions it can be treated as a fuzzy number, which means that it can be used for further calculations (Hanss 2005).

Every fuzzy number consist of three values that are of specific interest to the decision maker. These values are the minimal, modal (or peak) and maximal value of the fuzzy number. Using these three values a fuzzy number can be described as triplet (Fig. 1). This triplet can also be a definition of a triangular fuzzy numbers (Hanss 2005; Santos, Lodwick, Neumaier 2002). From the decision-making perspective, it is important to note that the fuzzy number does not have to be symmetrical, which means that (Fig. 1).

Fig. 1. Example of symmetrical fuzzy number and nonsymmetrical fuzzy number with their minimal, modal and maximal values.

2.2. Fuzzy surfaces

The fuzzy surface is a surface that has for every coordinate the height defined not as a precise number but has as a fuzzy number (Santos, Lodwick, Neumaier 2002). The fuzzy number located at describes all the possible values that the surface can have at this location. If the surface is described is modelled as a grid, then each its cell is a fuzzy number. Fuzzy model of surface naturally captures uncertainty associated with the spatial prediction.

Fuzzy surfaces were first introduced almost concurrently by (Diamond 1989) and (Bardossy, Bogardi, Kelly 1990). Each of those authors utilized different approach towards the construction of fuzzy surfaces. While (Diamond 1989) proposed interpolation of uncertain data (modelled as fuzzy numbers), (Bardossy, Bogardi, Kelly 1990) propose interpolation of precise data by krigging method with uncertain parameters (again modelled as fuzzy numbers). Although these approaches provide the same result (fuzzy surface), they are internally very distinct with each of them focused on a different type of uncertainty in the interpolation process. Later subsequent approaches for fuzzy surface interpolation were proposed (Santos, Lodwick, Neumaier 2002; Lodwick, Santos 2003; Waelder 2007) and the approach suggested by (Bardossy, Bogardi, Kelly 1990) was further developed by (Loquin, Dubois 2010). Until recently there were no software tools available that would be able to carry out the computation necessary to construct fuzzy surface. However, that changed with the introduction of FuzzyKrig (Soltani-Mohammadi 2015).

Fig. 2. Example of 3D representation of small fuzzy surface ( points).

There is still relatively little attention directed towards visualization of fuzzy surfaces in the existing literature. Usually, profiles and 3D projections (Fig. 2) of fuzzy surface (Bardossy, Bogardi, Kelly 1990; Diamond 1989; Lodwick, Santos 2003; Loquin, Dubois 2010; Santos, Lodwick, Neumaier 2002) or three separate surfaces (Fig. 3), representing values of and (Caha, Marek, Dvorský 2015; Soltani-Mohammadi 2015), are visualized; however, neither of these are quite suitable for user perception. Profiles and 3D representations only capture part of the fuzzy surface and the separate visualization for each significant value makes it complicated for user to assess the differences. Classic example of three separate surfaces is in Fig. 3; such visualization makes it really hard for the user to assess the differences amongst the surfaces and to identify the amount of uncertainty in specific locations. Thus, a new approach is necessary that will allow more suitable visualization of fuzzy surface.

Fig. 3. Example of fuzzy surface visualization using 3 surfaces. The dataset shows concentration of in . Variant a) represents , b) and c) .

3. HSL Colour Model

The typical colour systems integrated into cartography and GIS software are RGB (red green blue), HSV (hue saturation value), HSB (hue saturation brightness), HSL (hue saturation lightness) and CMYK (cyan magenta yellow black) (Brewer, Hatchard, Harrower 2003). In general, colours represent a hugely important area of cartography. Bertin (Bertin 2010) classified colour value and colour hue as a part of seven initial visual variables of a cartographic design. (MacEachren 2004) later extended this systematisation, and the importance of colour schemes is still emphasized. Colour models used for spatial visualization and cartographic design preferences are in the vast majority based on the request for final processing (Dent, Torguson, Hodler 2009). For printed maps model CMYK is principally used, for digital maps mostly RGB or HSB models are used.

(Joblove, Greenberg 1978) firstly described the HSL (hue saturation lightness) colour model. As other relative colour models (e.g. HSV (hue saturation value), HSB (hue saturation brightness)), it is based on the tiling of RGB cube so that the black colour is at the origin with white directly above it. The colours red, yellow, green, magenta, blue and cyan are mapped on a circle using interval . Hue values starts with red starting at ¹. The saturation describes the distance of the colour from the main (black - white) axis, and lightness specifies the amount of white present in the resulting colour. Both saturation and lightness take values from the interval . The model can be visualized as a cylinder (Fig. 4).

Fig. 4. Cylindrical model of HSL (left) and bicone variant that shows chroma instead of saturation (right). (based on: (Joblove, Greenberg 1978))

Fig. 5. Surface of the cylindrical model of HSL with colours with chrome lower than 0.3 shown. Note colours between hue values 300-360, that cannot be used to avoid confusion with colours close to value 0.

Important characteristics of colour in HSL model is its chroma, which is colourfulness relative to the white colour with similar lightness. If chroma is used instead of saturation when plotting the model, the cylinder becomes bicone (Fig. 4). This variant of visualization better captures the amount of possible colours for different values of lightness. Because if lightness is equal to 1 or 0 the colour is always white or black without relation to hue. The chroma of colour in HSL is calculated as:

where is a value of chroma, is lightness and denotes saturation. Low values of suggest that the colour is on black - grey - white scale and its hue will be hard to determine. Fig. 5 shows the surface of the cylindrical model from Fig. 4 with parts of the surface with low values (smaller than 0.3) of chroma denoted. It is apparent from this visualization that it would be impossible for the user to determine correct hue of colour with small values of . Quantifying this limit is directly dependent on the ability of map user to distinguish colors. Based on Eq. (4) it can be determined that ranges of lightness and should be omitted from the set of colours to provide distinguishable colours. Fig. 5 also illustrates the problem with the circular nature of hue that was mentioned previously.

The transformation from HSL colour model to RGB colour model as well as conversions to other models are in detail described by (Joblove, Greenberg 1978).

4. Proposed approach

The proposed approach is based and highly influenced by the work (Zehner, Watanabe, Kolditz 2010) and (Hengl 2003), who presented visualization methods for data with uncertainty based on changes of saturation, brightness and/or intensity. Suppose that there is a set of fuzzy numbers (e.g. set of fuzzy numbers that forms fuzzy surface) each with its set of important values . For each fuzzy number values of uncertainty between modal and both limit values can be calculated as:

Where is the magnitude of the uncertainty between the modal and minimal value of a fuzzy number and is a magnitude of uncertainty between maximal and modal value. Along with the modal (or peak) value of fuzzy number these two numbers describe the magnitude of uncertainty related to a fuzzy number (Fig. 6).

Fig. 6. Fuzzy number with its important points and magnitudes of uncertainty and .

From the set of fuzzy numbers mentioned previously and uncertainty values and four values that will be important for our visualization technique can be calculated:

The values and ) describe the range of modal values of all the fuzzy numbers from the set. The values , show the maximal magnitudes of uncertainty from modal value towards minimal and maximal values respectively.

In the proposed approach two components of HSL model are used to visualize the modal value of fuzzy surface and magnitudes of uncertainty to produce two maps that allow the user to observe the modal values as well as deviations from this value towards both sides. The modal value of a fuzzy number is associated with hue; is related to the difference of lightness from the mean value towards zero and is connected with the difference of lightness towards one. The value of saturation is considered as fixed with value 1.

To make the proposed approach as flexible as possible, it is not expected that whole ranges of hues or lightness have to be used in the visualization. Rather a user can specify ranges of hue and lightness that he/she wishes to use. Based on his decision variables , , , that describe ranges of hue and lightness are introduced. These values have to fulfil simple rules in order to provide meaningful results: and , where denotes a middle value of .

For every fuzzy number from the set of fuzzy numbers we can calculate:

The values of and denotes lightness for the two images. The example of the two-dimensional legend and two fuzzy numbers is shown in Fig. 7.

Fig. 7. An example of two fuzzy numbers on two-dimensional legend with the critical colour values shown.

For the two resulting maps, the colour for the map showing the magnitude of uncertainty towards smaller values is defined in HSL model as . The map showing the magnitude of uncertainty towards high values the colour is defined as . The value of is the same for both models. Fig. 7 shows an example of two fuzzy numbers with the legend and colours of minimal and maximal uncertainty values presented by lightness change.

5. Discrete colour scheme

In the theory of cartographic visualisation, it is needed to deal with user aspects of the final colour scheme (MacEachren, Robinson, Hopper, Gardner, Murray, Gahegan, Hetzler 2005). For many users, it is difficult to perceive a variety of information in a smooth transition gamut (Vondráková 2013). Concomitant use of multi-parameter changes colours makes this situation even more difficult for a significant part of users. To ensure proper perception and subsequent interpretation of the data it is reasonable to discretize the legend into the interval scheme (Vondráková, Caha 2014). According to the scale, purpose and target user groups of visualization it is possible to select an appropriate number of intervals for the map. The interval colour scheme is often used in thematic cartography, for example in choropleth maps. Therefore, this approach is the application of methods of thematic cartography, which are commonly used for its utility and efficiency (Dent, Torguson, Hodler 2009).

The method for visualization of fuzzy numbers described above that utilizes continuous scales of hue and lightness is computationally simple. However, such scale is not very suitable for users. The two-dimensional legend as described above might be too complicated, and thus, a simplification is necessary to make more comprehensible for the user. Such simplification can be done by categorization of both hue and lightness into and categories, which limits the number of used colours on the map to . For simplification of the visualization, is required to be an even number.

The example of this discretization is shown in Fig. 8. In this figure, an interval is divided into categories. The colour for each category is determined from the continuous legend as midpoints of the intervals. The first and last intervals get values of hue , so that whole range of hue is used. The resulting legend includes relatively distinct colours, especially when compared to a continuous legend. As a result, the information from the map is much easier to read for the user.

The specific values of hue for each interval are obtained as:

In this equation denotes the index of interval with first interval having index and thus its hue is \(minH\). Every modal value of fuzzy numbers is then categorized into respective category and the relevant hue value is assigned.

Fig. 8. The process of determination of hues for interval legend.

The identical process can be done for the values of lightness, resulting in discretized legend for uncertainty of fuzzy numbers. As mentioned, the number of categories for lightness denoted as , is required to be an even number. That way it is easier to define the categories with below the and the same number of categories above . The values of for categories are calculated as:

In this equation, denotes the index of the interval with the first interval having index . In the same way as hue is categorized in the description above, the values of and are categorized to obtain the value of lightness.

Graphical representation of this process is in Fig. 9. The combination of discretized lightness and hue values results in the two-dimensional legend categorical legend.

Fig. 9. The process of determination of lightness values for interval legend.

An example of how lightness is determined for three fuzzy numbers with identical modal value is shown in Fig. 10. The figure shows lightness divided into six intervals. Intervals with negative values on axis show deviation towards smaller values while the positive values indicate uncertainty towards higher values. The final colours used in visualization for the particular fuzzy number is determined by the intervals in which the values of and fall in.

Fig. 10. Three fuzzy numbers with equal modal value and with varying amounts of uncertainty shown on a lightness scale.

6. Possible further extensions of the proposed approach

The utilization of proposed approach can be done in three main directions. These directions are a) visualization of another information from fuzzy number, b) visualization of vector data, c) visualization of another type of uncertainty.

6.1. Visualization of another information from fuzzy number

The presented approach is focused on visualization of triangular fuzzy numbers and uses three most important values of a triangular fuzzy number to do so. The reason for using this type of fuzzy number is that is the most commonly implemented in practical applications (Hanss 2005) and it is also the most illustrative example for users. However, there is an interesting potential to visualize either other types of fuzzy numbers (i.e. Gaussian or trapezoidal) as well as other parts of fuzzy numbers. In order to do so, we need to introduce so-called -cut which is an interval with specified membership value . For more details about -cuts see (Zadeh 1965) and (Hanss 2005). The proposed approach utilizes the whole range of fuzzy numbers by the use of its limit values. However, by substituting with in Eq. (5) and by with in Eq. (6) the differences can be calculated to another -cut. This allows the visualization of the only portion of uncertainty. Such approach can be favourable in certain decision-making situations. It also allows users to assess differences among various levels of uncertainty. In chapter Fuzzy numbers it is mentioned that only fuzzy numbers containing one with are considered. Fuzzy numbers with more than one that satisfy this condition also exist in the literature (see (Hanss 2005)). Such fuzzy numbers do not have the single value of . These can also be visualized using the proposed approach in case that their modal value is simplified into a single value. For example using this equation:

In this situation, the midpoint of \(\alpha\)-cut with value 1 (the core of fuzzy number) is used as the main reference value.

6.2. Visualization of vector data

The proposed approach focuses on visualization of surface data represented as raster. The nature of calculations necessary to obtain both colours for visualization is general enough so that it can be calculated from a vector dataset as well. In the beginning of section Proposed approach there is a mention about a set of fuzzy number. This set can be obtained from various types of data. This property allows simple extension of this visualization approach to other types of data. For example, polygons with values of attribute predicted in a form of fuzzy number can be visualized using the proposed approach.

6.3. Visualization of another type of uncertainty

The proposed approach focuses on visualization of uncertainty modelled by fuzzy numbers. In theory, another uncertainty theories can be used as well. The uncertain value has to have some kind of modal or most likely value as well as lower and upper limit. If those three values can be determined than the proposed method of visualization can be used. However, the proposed approach might not be the best case when all the uncertain values are symmetrical around the modal value. In such case, there is no need to visualized three parameters because two parameters (modal value and uncertainty) would be enough.

7. Case Study

The fuzzy surface used for the visualization in the case study is taken from (Caha, Marek, Dvorský 2015)². The fuzzy surface represents the monthly average of concentrations of air pollutant in Czech Republic for October 2013. The values of concentration are in in . The data from 182 locations are used to construct fuzzy surface by interpolation with fuzzy variogram (Fig. 11). The method used to calculate the fuzzy surface was proposed by (Loquin, Dubois 2010) and the detailed process of calculation is described by (Caha, Marek, Dvorský 2015). A FuzzyKrig tool (Soltani-Mohammadi 2015) can be used to calculate fuzzy surface based on the described data and fuzzy variogram.

Fig. 11. Fuzzy variogram used in interpolate fuzzy surface. (source: (Caha, Marek, Dvorský 2015))

The fuzzy surface is represent using proposed approach with continuous scale (Fig. 12) as well as using discrete colour scheme (Fig. 13). For this case study values of ), , and were selected for definition of colour ranges according to Eqs. (11,12, 13). From the set of fuzzy numbers the following statistics:

are obtained and used to construct the visualization. The resulting maps are visualized in Fig. 12 using continuous legend with hue limit values - , and lightness limit values - and .

Fig. 13 shows visualization using discrete colour scheme with the same limit values as continuous scale and 5 categories of hue and 10 categories of lightness. The number of categories in the case study was selected based on expert opinion as a compromise between number of categories and amount of information in the resulting visualization. The number of classes is selected with respect to the type of users and main aim of the map. For example if the users are not well acquainted to the uncertainty problematic the number of lightness categories can be reduced to four. For more experienced users the values around ten should be comprehensible. The continuous scale quite possible suitable only for experts and for interactive visualizations that would allow users to query the data directly through interaction. For the visual assessment of surface uncertainty the discreate colour scheme should be preferred.

Fig. 12. Visualization with applications of whitening and darkening approach — continuous colour scale.

Fig. 13. Visualization with applications of whitening and darkening approach — interval colour scale.

Both visualizations (Figs. 12, 13) show the nonsymmetrical character of uncertainty in the example. Clearly there are more significant deviations towards than . The differences almost make impression that there is no uncertainty towards higher values, especially on the continuous scale (Fig. 12).

8. Conclusion

The presented research focuses on the fuzzy surfaces visualization with emhpasis on the asymmetry of fuzzy numbers. Hue allows the map user to estimate the modal (most possibly true) value of the fuzzy surface, and whiteness and darkness provide him the estimation of minimal and maximal value (deviations from the modal value caused by uncertainty). Information perception of fuzzy surfaces visualizations made by internal graphic variables HSL model should be more suitable, than the most commonly used approaches used so far (profiles, 3D visualizations and separate representation of important values).

The general perception of geovisualization is significantly affected by a particular user, his/hers knowledge, skills and other characteristics, but the main aim of the researcher is to provide as accurate information as possible, which is enabled by the suitable methods of visualization.

Even though the presented visualization method is shown on the fuzzy surface, it is defined universally, so that it can be used for other types of uncertainty visualization. The proposed approach can also be applied for example to statistical data by replacing modal value of fuzzy number by mean value and using differences amongst mean and minimum and maximum as the two measures of uncertainty. Further research should focus on testing users attitude towards the proposed visualization method by objective means such as eye-tracking.

Acknowledgement

The authors gratefully acknowledge the support by the Operational Program Education for Competitiveness - European Social Fund (projects CZ.1.07/2.3.00/20.0170 and CZ.1.07/2.2.00/28.0078 of the Ministry of Education, Youth and Sports of the Czech Republic).

Appendix A: Software - Fuzzy Data Visualizer

For practical testing of the proposed approach of visualization of fuzzy data a software was programmed in C# language. The software is called Fuzzy Data Visualizer and allows construction of two dimensional legend using both continuous and discretized scale of the legend, as described in section Proposed approach. The graphical user interface of the legend design is shown in Fig. 14.

The legend can be created independently or it can be based on fuzzy dataset provided as a “.csv” file. An example file is provided with the software for testing purposes. The data after being loaded into the software are interactively visualized in the tab “Data Visualization” using the currently specified legend. The Data Visualization part of the graphical user interface is shown in Fig. 15.

The described software is freely available from the https://github.com/JanCaha/FuzzyDataVisualizer as an appendix to this paper.

Fig. 14. Graphical user interface of Fuzzy Data Visualized -- legend design.

Fig. 15. Graphical user interface of Fuzzy Data Visualized — data interactive visualization.

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Due to the circular nature of hue, it is worth to note both values and are red. That means that hues after cyan are gain by mixing the cyan with red (Fig. 5). As this may pose problems for cartographic visualization, we recommend using only the interval for practical applications. (Hengl 2003) suggests the similar solution of this problem.↩
The dataset used in this paper is available from https://github.com/JanCaha/FuzzySurface.↩