Hierarchical Data Visualization Based on Rectangular Cartograms

Abstract

As the diversity and complexity of geographic statistical data continue to increase, it becomes increasingly important to present multi-level information in order to meet a broader range of needs. In response to the limitations of existing visualization methods in representing the geographic distribution of statistical data, this paper proposes a geographical hierarchical data visualization method based on rectangular cartograms. First, a new rectangular cartograms construction algorithm is adopted in this paper, which can effectively preserve relatively accurate orientation and adjacency relationships between geographic regions, while also effectively preserving the statistical data features. Then, a treemap layout algorithm is applied within the rectangular cartogram to further partition the geographic regions, thereby visualizing the hierarchical structure of the data. Through experimental validation using real datasets and usability testing, the results demonstrate that the method presented in this paper excels in geographic distribution representation, hierarchical relationship visualization, and information readability. Compared to traditional thematic map methods, this approach demonstrates significant advantages in terms of information transmission efficiency and shows promising performance in expressive effectiveness, providing strong support for the analysis and decision making of geographical hierarchical data.

1. Introduction

1.1. Background

With the diversification of data sources and advancements in data collection technologies, various types of data often contain organizational structures and hierarchical relationships, such as corporate structures and administrative divisions. Hierarchical data visualization helps to clearly present the relationships between different levels, aiding users in understanding complex data structures, improving analysis efficiency, and supporting decision making [1,2].

Geographical hierarchical data exhibits a significant hierarchical structure, which is reflected both in the division of geographic regions and in the multi-level statistical variables [3]; see Figure 1 for an example [4]. These variables may involve spatial correlation, spatiotemporal trends, and other factors. Therefore, the visualization of geographical hierarchical data helps users understand the spatial location, scale, and distribution characteristics of geographic objects by displaying spatial hierarchical relationships while visually presenting the correlations, differences, and trends between levels, thus providing important references for regional analysis, decision-making support, and resource allocation [5,6].

1.2. Related Work

The traditional thematic map representation method conveys geographical hierarchical data through maps combined with graded colors, symbol sizes, and charts [7,8]. However, this representation method has certain limitations. On the one hand, the geographic area does not correspond to the statistical data, which leads to a lack of sufficient map space for effectively displaying areas with small geographic size but large statistical values [9]; on the other hand, the irregular shape of the geographical area itself may cause ‘visual confusion’ in spatial segmentation.

An area cartogram uses geometric transformations to make the area of regions proportional to the attribute data [10], while maintaining the relative positional relationships between regions, thus avoiding the issue of mismatched geographic area and statistical data. This visualization method not only enhances visual appeal but also improves information transmission efficiency [11]. Moreover, because the area itself represents a variable, it can better support multivariate expression and the comparison of geographic distribution patterns for different variables [12]. Area cartograms can be classified into four types based on their graphical features: simple contiguous [13,14], simple non-contiguous [15,16], complex contiguous [17], and complex non-contiguous cartograms [18].

A rectangular cartogram is a typical example of a simple contiguous area cartogram [19,20]. It uses regular rectangles to represent geographic regions, making them easy to segment and well suited for expressing the spatial distribution of geographical hierarchical data. Additionally, the rectangular shape facilitates the estimation and comparison of area sizes, making the rectangular cartogram particularly effective in data comparison and value estimation [21]. Treemap is a spatially efficient method for hierarchical data visualization. It divides the space into several rectangular blocks of varying sizes according to the object weights and displays the hierarchical relationships between objects through the nested relationships between these rectangles, thereby achieving efficient use of space while presenting hierarchical information [22,23]. In addition, applying treemaps to other cartographic methods can further improve spatial efficiency and enhance the practicality of mapping [24].

The rectangular cartogram naturally aligns with the treemap layout, and the combination of the two can fully utilize the map space, efficiently expressing geographic multi-level data. This method has been widely applied in the visualization of geographic multi-level data. Wood et al. [25] proposed the Spatially Ordered algorithm, which sorts the nodes based on the geographic distance between the data nodes before combining it with the Squarified algorithm [26] for the node layout. Although the Spatially Ordered algorithm can present the geographical distribution of the data nodes, its readability is poor, and the spatial positioning is rough. Tong et al. [27] combined a treemap with a cartogram to express the geographical distribution and hierarchical structure of UK healthcare data, but this method did not strictly maintain the adjacency relationship between data nodes. The Weighted Maps algorithm proposed by Ghoniem et al. [4] creates a rectangular layout based on the nodes’ latitude, longitude, and weights. Although it is better at maintaining the adjacency relationship, the expression of the orientation relationship is suboptimal. Buchin et al. [28] proposed an algorithm to convert rectangular layouts without hierarchical structure into spatial treemaps while maintaining adjacency relationships, which improves the recognizability of geographic data in a hierarchical state. However, Buchin et al. did not demonstrate and evaluate the algorithm on a complete geospatial map, which limits its applicability and practical use for geographical hierarchical data visualization discussed in this paper. Zhou Mengjie et al. [29] proposed a multi-level visualization method for thematic map attribute information based on the Level of Detail (LOD) technique, which adaptively adjusts the semantic and visual granularity of attribute data according to map scale. The method combines the RecMap algorithm [30] and treemaps to achieve a multi-level visualization of geographic data. However, this approach is not effective in representing the orientation and adjacency relationships between data nodes, which may lead to incorrect user perceptions.

1.3. Motivation and Objectives

In summary, existing methods that combine rectangular cartograms and treemaps still have certain limitations in the visualization of geographical hierarchical data, primarily reflected in two aspects: First, these approaches have not sufficiently addressed the accurate representation of spatial relative positions, leading to difficulties in accurately presenting spatial adjacency relationships, which in turn causes misunderstandings and cognitive biases regarding the locations of geographic areas. Second, the effectiveness of information transmission has not been fully validated, and the advantages and limitations of this visualization method have not been comprehensively compared and evaluated against traditional thematic map representation methods.

Therefore, this paper proposes a method that strikes a balance between the accuracy of geographic distribution and the effective representation of multi-level structural features. Firstly, a new rectangular cartogram construction algorithm [31] is employed to effectively preserve the relatively accurate orientation and adjacency relationships between geographic regions, thereby enabling a more precise display of the geographic distribution of the data. Then, the space is divided by combining it with a treemap layout algorithm to present the multi-level structure of the data. Finally, this paper demonstrates the potential effectiveness of the proposed method through a case study based on a real dataset and provides a comparative analysis with traditional thematic mapping methods, highlighting its advantages in representing geographical hierarchical data.

2. Rectangular Cartogram Construction Method

In order to effectively display the geographical distribution of geographical hierarchical data, this paper adopts a new rectangular cartogram construction method. The core of the method lies in the integrated consideration of the spatial structure of geographic regions and the characteristics of statistical data, ensuring that the area of each region accurately reflects the relative size of its corresponding statistical data, and maximizing the presentation of the characteristics of the distribution of statistical data in geospatial space through the rectangular layout.

2.1. Problem Description

The construction of a rectangular cartogram is a complex combinatorial optimization process involving the following key constraints:

(1) Shape: The shape of each geographic region must be simplified into a rectangle, and no overlap is allowed between the rectangles.

(2) Area: The area of each rectangle should be proportional to the attribute value of the corresponding region, ensuring an accurate reflection of the statistical data for that region.

(3) Adjacency relationships: In the rectangular cartogram, the original adjacency relationships between regions must be preserved.

(4) Relative orientation: Ensure that the relative orientation between regions in the rectangular cartogram is consistent with that in the original map.

Let the set of regions in the original map be $R=\left\{r_1,r_2,\dots ,r_n\right\}$, and the set of regions in the rectangular cartogram be $R^{\mathrm{\prime }}=\left\{r_1^{\mathrm{\prime }},r_2^{\mathrm{\prime }},\dots ,r_n^{,}\right\}$. $r_i$ and $r_i^{\mathrm{\prime }}$ represent the same region, but their difference lies in the different representations in various cartographic methods. To construct the rectangular cartogram efficiently and address the above constraints, the following optimization objectives are adopted in the construction method:

(1) Minimizing the area error: Since the area of each rectangle must be proportional to the attribute value of the corresponding region, the difference between the actual area and the expected area needs to be minimized to ensure that the region’s statistics are accurately mapped. The area error is denoted as A(R′), and its calculation formula is shown in Equation (1) [32].

$$A\left(R^{\mathrm{\prime }}\right)={\displaystyle \frac{1}{n}}{\sum }_{r_i^{\mathrm{\prime }}\in R^{\mathrm{\prime }}}{\displaystyle \frac{\left|O\left(r_i^{\mathrm{\prime }}\right)-D\left(r_i^{\mathrm{\prime }}\right)\right|}{max\left\{O\left(r_i^{\mathrm{\prime }}\right),D\left(r_i^{\mathrm{\prime }}\right)\right\}}}$$

(1)

where $O\left(r_i^{\mathrm{\prime }}\right)$ is the actual area of region $r_i^{\mathrm{\prime }}$ in the rectangular cartogram, and $D\left(r_i^{\mathrm{\prime }}\right)$ is the expected area of region $r_i^{\mathrm{\prime }}$. The expected area refers to the area that each region should occupy in the rectangular cartogram, and its size is proportional to the statistics of the region to reflect the relative magnitude of the statistics.

(2) Minimizing the adjacency error: Since the adjacency relationships in the original map must be preserved, the difference between the adjacency relationships of regions in the rectangular cartogram and those in the original map must be minimized to ensure that the adjacency relationships remain intact. The adjacency error is denoted as E(R′), and its calculation is given by Equation (2) [30].

$$E\left(R^{\mathrm{\prime }}\right)={\displaystyle \frac{\left|a_{R^{\mathrm{\prime }}} \cup a_R\right| - \left|a_{R^{\mathrm{\prime }}} \cap a_R\right|}{\left|a_{R^{\mathrm{\prime }}} \cup a_R\right|}}$$

(2)

$a_R$ represents the set of adjacency relationships between regions in the original map, which is determined by the adjacency matrix of the original map. $a_{R^{\mathrm{\prime }}}$ represents the set of adjacency relationships between regions in the rectangular cartogram, which is defined by the adjacency matrix of the rectangular cartogram.

(3) Minimizing the relative orientation error: To ensure the accuracy of the relative positioning between regions, it is necessary to minimize the difference between the relative orientation of regions in the rectangular cartogram and the orientation in the original map. The relative orientation error is denoted as P(R′), and its calculation is given by Equation (3) [33].

$$P\left(R^{\mathrm{\prime }}\right)={\displaystyle \frac{2}{n\left(n-1\right)}}{\sum }_{i=1}^{n-1}{\sum }_{j=i+1}^n\left|\mathrm{\angle }(r_i,r_j)-\mathrm{\angle }(r_i^{\mathrm{\prime }},r_j^{\mathrm{\prime }})\right|$$

(3)

$\angle (r_i,r_j)$ denotes the angle between the centroids of regions $r_i$ and $r_j$ in the original map, and $\angle (r_i^{\prime },r_j^{\prime })$ denotes the angle between the centroids of regions $r_i^{\mathrm{\prime }}$ and $r_j^{\mathrm{\prime }}$ in the rectangular cartogram. This centroid-based method offers a concise and consistent approach for determining the relative orientation between regions. This is because the centroid can be regarded as the geometric center and serves as a stable reference point that reflects the overall spatial position of a region. Compared with using bounding boxes or boundary points, centroid-based orientation is less affected by irregular shapes, making it suitable for processing geographically irregular regions. Although, in some extreme cases—such as C-shaped or L-shaped regions—this method may deviate from human spatial perception, it still provides a sufficiently accurate and robust approximation of spatial orientation for most common administrative and geographic units.

In order to achieve the optimization objectives above, the rectangular cartogram construction method used in this paper first constructs a rectangular segmentation map. In the rectangular segmentation map, the shapes of the geographic regions are simplified into rectangles, and the original relative orientation and adjacency relationships between regions are approximated and preserved as much as possible through the relative orientation and adjacency relationships between the rectangles. Then, based on this, a rectangular cartogram is constructed. During the construction process, the edge lengths and positions of each region in the rectangular cartogram are sequentially calculated by combining the positional relationships of the regions in the rectangular segmentation map with the corresponding statistical data, resulting in the generation of a complete rectangular cartogram. This process aims to approximate the optimization objectives outlined above as closely as possible.

However, the optimization problem discussed in this section does not guarantee a unique or global optimum. The algorithm aims to find a solution that is close to optimal by iteratively adjusting the layout of the regions. Although no formal proof is provided that the algorithm always converges to the global optimum, empirical results show that it performs well on real-world datasets. Future work could explore the convergence properties of the algorithm and provide further theoretical justification.

2.2. Algorithm Workflow

As shown in Figure 2, the construction process of the method is as follows:

Step 1: Perform basic data preprocessing of the original map.

Step 2: Generate the rectangular segmentation map from the original map.

Step 3: Create the corresponding rectangular cartogram based on the rectangular segmentation map and statistical data.

2.3. Construction of Rectangular Segmentation Map

A rectangular segmentation map is a representation used to simplify geospatial information, in which each rectangle corresponds to a geographical region. The areas of these rectangles do not reflect the actual data or geographic area but only indicate the spatial orientation and adjacency relationships between geographic regions. The positional relationships of the regions in the rectangular segmentation map provide support for subsequent rectangular cartogram construction.

2.3.1. Calculation of Relative Orientation of Regions

To reduce P(R′) as much as possible, the rectangular cartogram construction algorithm used in this paper first determines the relative orientation relationship between regions by calculating the angle between the centroid line between the regions and the horizontal axis.

As shown in Figure 3,the relative orientation relationships between the regions follow the four cardinal directions, i.e., east, south, west, and north. Let there be two regions A and B whose centroids are (x_A, y_A) and (x_B, y_B), respectively, and let the angle between the line connecting the two centroids and the horizontal axis be α. The specific formula is as follows:

$$\alpha =\left(arctan\left({\displaystyle \frac{y_B-y_A}{x_B-x_A}}\right)\times 180/\pi \right)$$

(4)

Based on the value of α, the relative orientation of B to A is determined: If 45° ≤ α < 135°, then region B is north of region A. If 135° ≤ α < 225°, then region B is west of region A. If 225° ≤ α < 315°, then region B is south of region A. If 315° ≤ α < 360° or 0° ≤ α < 45°, then region B is east of region A.

After determining the relative orientation between regions, when constructing the rectangular segmentation map, the position of the rectangles is determined according to this relative orientation. Moreover, in the subsequent construction process, the relative orientation between the rectangles is kept as consistent as possible so as to minimize the relative orientation error.

2.3.2. Calculation of Edge Length of Regions

After determining the relative orientations of the regions in the rectangular segmentation map, the next step is to calculate the side lengths of the regions in the rectangular segmentation map. It should be noted that two regions are considered adjacent only when they share a common edge. Contact at only the vertices is not regarded as a valid adjacency. Therefore, during the calculation of side lengths, the rectangles are gradually adjusted to preserve the original adjacency relationships between the regions, thereby minimizing E(R′) as much as possible.

Let M be the set of regions in the rectangular segmentation map, where $M=\left\{m_1,m_2,\dots , m_n\right\}$. For region $m_i$, let its length be $l_i$, its width be $w_i$, the number of adjacent regions in the south be $n_i^S$, the number of adjacent regions in the north be $n_i^N$, the number of adjacent regions in the east be $n_i^E$, and the number of adjacent regions in the west be $n_i^W$.

First, the initial length and width of the region in the rectangular segmentation map are computed. The initial length of the region is determined by the maximum number of adjacent regions in the south or north, as shown in Equation (5); the initial width is determined by the maximum number of adjacent regions in the east or west, as shown in Equation (6).

$$l_i=\left\{\begin{array}{c}Max\left(n_i^S,n_i^N\right) Max\left(n_i^S,n_i^N\right)>0\\ 1 Max\left(n_i^S,n_i^N\right)=0\end{array}\right.$$

(5)

$$w_i=\left\{\begin{array}{c}Max\left(n_i^E,n_i^W\right) Max\left(n_i^E,n_i^W\right)>0\\ 1 Max\left(n_i^E,n_i^W\right)=0\end{array}\right.$$

(6)

The side lengths of the regions are then adjusted in turn. In the adjustment process, the length and width of each region are jointly determined by the side lengths and number of its neighboring regions. Let ${S(m}_i)$ be the set of south-adjacent regions of $m_i$, and ${W(m}_i)$ be the set of west-adjacent regions of $m_i$. The length adjustment formula is shown in Formula (7), and the width adjustment formula is shown in Formula (8).

$$l_i^{\mathrm{\prime }}=\left\{\begin{array}{c}{l}_i, S\left(m_i\right)=\varnothing \\ \sum \left(l_j/n_j^N\right), m_j\in S\left(m_i\right)\end{array}\right.$$

(7)

$$w_i^{\mathrm{\prime }}=\left\{\begin{array}{c}{w}_i, W\left(m_i\right)=\varnothing \\ \sum \left(w_j/n_j^E\right), m_j\in W\left(m_i\right)\end{array}\right.$$

(8)

Repeat this adjustment process until the side lengths of all regions no longer change. Finally, the rectangular segmentation map is constructed by combining the relative orientation relationships between the regions to calculate the position of the regions in the rectangular segmentation map.

The rectangular segmentation map effectively maintains the relative orientation and adjacency relationships of the regions in the original map. Therefore, during the subsequent construction of the rectangular cartogram, the rectangular layout is determined based on the positional information in the rectangular segmentation map so as to minimize the adjacency error and relative orientation error.

2.4. Construction of Rectangular Cartogram

During the construction of a rectangular cartogram, the rectangular cartogram construction algorithm gradually calculates the edge length and placement of each region in the cartogram based on the expected area and its position within the rectangular segmentation map, and sequentially places them into the rectangular cartogram, thus achieving the construction of the rectangular cartogram. During construction, the method marks regions already placed into the rectangular cartogram as ‘placed regions’, and marks regions yet to be placed as ‘unplaced regions’.

In a rectangular segmentation map, if two regions are adjacent, then the following three positional relationships can exist: (1) two rectangles share two common vertices and have one overlapping edge; (2) two rectangles share one common vertex and have one overlapping edge; (3) two rectangles have only one overlapping edge. Figure 4 illustrates the positional relationship between adjacent regions. In this figure, the regions are arbitrarily labeled as ‘A’ and ‘B’, and this labeling is made without loss of generality. When calculating the position of an unplaced region, if the region has common vertices and overlapping edges with the placed region in the rectangular segmentation map, then this positional relationship is also preserved in the rectangular cartogram so that the rectangular layout can efficiently maintain relatively accurate adjacency relationships and relative orientation.

Therefore, based on the positional relationship between regions in the rectangular segmentation map, i.e., the common vertices or edges of the unplaced region and the placed region, the coordinates of some vertices of the unplaced region in the rectangular cartogram and the edge length of one of its sides can be deduced. Then, by combining the expected area and positional relationships, the remaining vertices’ coordinates and edge lengths of the unplaced region are further calculated to ultimately determine its location. Determining the region’s position based on the expected area allows the area of the region in the rectangular cartogram to reflect the region’s statistical data as accurately as possible, thereby minimizing area errors.

3. Algorithmic Design of Treemap Layout in Rectangular Cartograms

3.1. Treemap Layout Algorithm

In the rectangular cartogram, the hierarchical relationship of statistical data can be effectively visualized using a treemap layout algorithm within each rectangle. As the rectangular cartogram contains multiple rectangles and the structure of the statistical data is more complex. Therefore, two aspects should be focused on in terms of the visualization outcome:

(1) Data visualization effect inside the rectangles: The treemap layout needs to clearly express the hierarchical relationship inside each rectangle to avoid difficulties in reading the data due to a confusing layout.

(2) Overall coordination of the rectangular cartogram layout: The rectangular cartogram contains multiple rectangular regions, which require spatial division within each rectangle, and if the layout strategy is inefficient, it may lead to visual clutter in the overall graphic.

In order to meet the above requirements, the treemap layout must exhibit good continuity, stability, and readability. Good continuity ensures that neighboring sub-nodes in the layout remain adjacent, ensuring the unity and logic of the treemap structure. Good stability can provide a stable layout when data dynamically changes, avoiding significant layout shifts caused by data fluctuations, and making it easier for users to track specific nodes [34].

Among the various treemap layout algorithms, the slice and dice method [35], though suboptimal in terms of aspect ratio, outperforms many other algorithms in continuity, stability, and readability [36]. Therefore, this algorithm is chosen as the treemap layout scheme in the rectangular cartogram.

The core principle of the slice and dice layout algorithm is dividing the rectangular region of the parent node according to the weight ratio of the child nodes in order to visualize the hierarchical structure of the data. The algorithm fills the rectangles in a predetermined order of node placement, using either left-to-right or top-to-bottom layout directions. In this paper, the original arrangement order of the data is used as the placement order of the data nodes; by keeping the node order consistent with the original sequence, it helps users understand the distribution of the data more intuitively, thus avoiding difficulties in conveying information due to the confusion of the order.

To clarify the method, the definitions involved in the construction method of the treemap in this paper are presented first. In this construction method, we define a parent node in the treemap as V_P and the n children of V_P are defined as (v_c1, v_c2, …, v_cn). The weight of the parent node V_P is denoted as S_P and the weight of the child node v_ci as s_ci. Let the rectangular area that the parent node V_P should occupy in the treemap be denoted as A_P and the rectangular area occupied by the child node v_ci in the treemap be denoted as a_ci, after which a_ci is computed by the following formula:

$$a_{ci}=A_P\left({\displaystyle \frac{s_{ci}}{S_P}}\right)$$

(9)

Let the length of the rectangle of the parent node V_P be defined as L_P and the width as W_P. Let the length of the rectangle of the child node v_ci be defined as l_ci and the width as w_ci. Let the rectangle aspect ratio of the parent node V_P be denoted as r_P and the rectangle aspect ratio of the child node v_ci as r_ci, after which the aspect ratio is computed by the following equation:

$$\left\{\begin{array}{c}{r}_P={\displaystyle \frac{max(L_P,W_P)}{min(L_P,W_P)}}\\ r_{ci}={\displaystyle \frac{max(l_{ci},w_{ci})}{min(l_{ci},w_{ci})}}\end{array}\right.$$

(10)

To adapt to different data scenarios, this paper divides the slice and dice layout algorithm into two layout strategies:

(1) Short-side filling strategy: the child node v_ci is arranged along the short side of the parent node V_P rectangle, immediately to the left or top edge of the parent node’s rectangle, using a left-to-right or top-to-bottom order. The side length of the child node is calculated as

$$\left\{\begin{array}{c}\left(l_{ci}=L_P,w_{ci}={\displaystyle \frac{a_{ci}}{l_{ci}}}\right) if L_P<W_P\\ \left(w_{ci}=W_P,l_{ci}={\displaystyle \frac{a_{ci}}{w_{ci}}}\right) if W_P<L_P\end{array}\right.$$

(11)

(2) Long-side filling strategy: arrange the child node v_ci along the long side of the parent node V_P rectangle, immediately to the left or top edge of the parent node’s rectangle, using a left-to-right or top-to-bottom order. The side length of the child node is calculated as

$$\left\{\begin{array}{c}\left(l_{ci}=L_P,w_{ci}={\displaystyle \frac{a_{ci}}{l_{ci}}}\right) if L_P>W_P\\ \left(w_{ci}=W_P,l_{ci}={\displaystyle \frac{a_{ci}}{w_{ci}}}\right) if W_P>L_P\end{array}\right.$$

(12)

3.2. Implementation of Treemap Layout in Rectangular Cartograms

Each rectangle in the rectangular cartogram represents a geographical region, and the treemap layout algorithm divides and fills each rectangle based on the region’s data hierarchy, effectively presenting the data’s hierarchical structure within each region.

The statistical values of each geographic region serve as the root node in the treemap layout, and the child nodes are hierarchically arranged based on the region’s data hierarchy. Taking Figure 5 as an example, the root node rectangle is filled according to the hierarchical relationship of data nodes. Two methods can be used for this layout process:

Method 1: Sort the nodes according to the tree’s hierarchy and arrange them at each level according to the short-edge filling strategy, i.e., sequentially fill the child nodes along the short edges of the parent node’s rectangle. As shown in Figure 6.

Method 2: Sort the nodes according to the tree’s hierarchy and arrange them in the even-numbered levels using the long-edge filling strategy, i.e., fill the child nodes along the root node’s long edge. For the nodes in the odd-numbered levels, arrange them using the short-edge filling strategy. As shown in Figure 7.

Method 3: Sort the nodes according to the tree’s hierarchy, using the long-edge filling strategy for nodes at odd-numbered levels, and the short-edge filling strategy for nodes at even-numbered levels. As shown in Figure 8.

Fill the root node’s rectangle using each of the three methods mentioned above. Each method generates a set of rectangle layouts. For each layout method, calculate the average aspect ratio of its leaf node rectangles, and select the layout method with the average aspect ratio closest to 1 as the final layout for the root node’s rectangle.

To enhance the readability of the treemap and improve the visualization of the data, this paper further highlights the hierarchical relationships and category differences by using colors in the layout design. Node colors in the treemap are used to represent the hierarchical relationships and category differences of the data. To highlight category differences, distinct colors are used for the nodes of each category. Additionally, to effectively differentiate the levels of data, changes in saturation indicate the level of each node.

4. Visualization and Evaluation

4.1. Experiment Data

To comprehensively evaluate the visualization effectiveness of the proposed method from different perspectives, this study utilizes two distinct datasets. The first dataset is the 2022 tuberculosis-positive population dataset of Wuhan, characterized by small geographic areas in the central urban districts but relatively large statistical values, with limited variation among regions. The second dataset is the 2022 population dataset of Hubei Province, which features considerable differences in both geographic areas and statistical data values across regions.

As shown in Figure 9,the hierarchical structure of the 2022 tuberculosis-positive population dataset of Wuhan is as follows:

(1) First-level nodes: total affected population in each district.

(2) Second-level nodes: subdivision of the affected population by gender into males and females in each district.

(3) Third-level nodes: subdivision by age range, with 15–34 years as the youth group, 35–59 years as the middle-aged group, and 60 years and above as the elderly group.

As shown in Figure 10, the hierarchical structure of the 2022 population dataset of Hubei Province is as follows:

(1) First-level nodes: the total population of each city.

(2) Second-level nodes: the population of each city divided by gender into male and female groups.

(3) Third-level node: further division by age group within each gender—ages 0–14 as children and adolescents, 15–34 as youth, 35–59 as middle-aged, and 60 and above as elderly.

4.2. Visualization Results

4.2.1. Single-Level Visualization Results

A rectangular cartogram was constructed based on the number of tuberculosis cases in each district of Wuhan. The number of cases in each district is represented by the area of the corresponding rectangle, while the adjacency and relative orientation between rectangles reflect the geographical distribution of the districts. This forms a single-level visualization. The visualization result is shown in Figure 11.

A rectangular cartogram was constructed based on the population of each city in Hubei Province, where the population of each city is represented by the area of its corresponding rectangle. Due to the extremely low population of the Shennongjia Forestry District compared to other regions, its visual proportion in the cartogram would be minimal. Therefore, in this visualization, Shennongjia was merged with the geographically adjacent Shiyan City. As the population of Shennongjia accounts for only a very small proportion, this merging has a negligible impact on the overall population value of Shiyan. The visualization result is shown in Figure 12.

4.2.2. Multi-Level Visualization Results

Building upon the single-level visualization results, treemaps are further overlaid to reveal the more detailed statistical structure within each region. In the multi-level visualization design presented in this study, blue is used to represent males and pink to represent females, while color saturation is used to distinguish age groups; the higher the saturation, the older the age group. The visualization results are shown in Figure 13 and Figure 14.

4.3. Usability Evaluation

In order to test and evaluate the effectiveness of the proposed method, two sets of experiments were designed to compare it with traditional thematic statistical maps. The experiments provided a comprehensive assessment of the method’s effectiveness through task correctness, completion time, and subjective preference. The participants were 80 undergraduate students majoring in computing-related and geography-related fields. These students possessed a certain level of foundational knowledge in data visualization or spatial information. Among them, students from computing-related majors had completed coursework in Geographic Information Systems (GIS), giving them a basic understanding of geographic information visualization. Although this sample selection may limit the generalizability of the experimental results to a broader population, it helps ensure that participants can accurately understand the task requirements, thereby avoiding potential interference caused by a lack of domain-specific knowledge.

Before the experiment began, participants were given detailed instructions to ensure that they fully understood the tasks and procedures. The experiment was conducted on a professional online survey platform, which provided an automatic timing function. The response time of each participant was accurately recorded and free from human interference, ensuring the reliability of the data. In each round of the experimental tasks, both the question order and the option order were randomized to ensure that each participant received a different sequence of questions and answer choices. This approach effectively reduced potential learning effects caused by fixed ordering.

4.3.1. Evaluation Task Design

In the evaluation of the expressive effectiveness of the visualization method, this study refers to the ten evaluation tasks proposed by Nusrat et al. [37] for assessing cartogram performance, including difference detection, location, recognition, identification, comparison, finding extremes, filtering, adjacency identification, classification, and summarization. Based on the characteristics of rectangular cartograms, the following four tasks were designed to evaluate the effectiveness of the proposed visualization method:

1. Comparison: This task evaluates the performance of the rectangular cartogram in conveying the relative differences in data magnitude between regions. By comparing values, it tests the accuracy of data variation representation through differences in the rectangle area.

2. Estimation: This task assesses how effectively the rectangular cartogram presents quantitative data. It examines users’ accuracy in estimating data values based on the area of the rectangles.

3. Extrema identification: This task aims to evaluate the ability of the rectangular cartogram to highlight extreme values. It tests whether extreme data values can be effectively displayed through the relative sizes of rectangles.

4. Spatial localization: This task assesses the spatial layout performance of the rectangular cartogram, examining its ability to preserve the original geographic positional relationships between regions.

4.3.2. Evaluation of Single-Level Visualization Performance

(1) Purpose

This experiment focuses on evaluating the expressive effectiveness of rectangular cartograms in single-level geographical data visualization. It aims to examine whether the distortion of the original region shapes in the rectangular cartogram affects users’ ability to interpret the data, thereby assessing its effectiveness in geographical information visualization.

(2) Comparative design of visualization methods

In this experiment, the proportional symbol map [38]—a common form of traditional thematic statistical mapping—is compared with the rectangular cartogram to evaluate the effectiveness of both visualization methods in representing geographical statistical information. To ensure a fair and comparable experimental design, the proportional symbol map was constructed using an administrative boundary map as the geographic base, with the area of circular symbols encoding the statistical values of each region. This design strategy minimizes variation in experimental variables, enabling participants to objectively assess the expressive performance of the proposed visualization method within a consistent geographic context. The specific mapping procedure for the proportional symbol map is described as follows:

As shown in Figure 15 and Figure 16, circular symbols of varying sizes are overlaid within each administrative region to represent statistical data values using proportional encoding. In cases where symbols in smaller or adjacent regions may overlap and obscure administrative boundaries or create visual clutter, some symbols are appropriately displaced outside the region boundaries and connected with dashed lines to ensure positional clarity and accurate data representation.

Finally, the expressive effectiveness of the rectangular cartogram in single-level geographical information visualization is evaluated by comparing Figure 11 with Figure 15, and Figure 12 with Figure 16.

(3) Question design

Based on the four evaluation tasks mentioned above, the following questions were designed to assess the effectiveness of the rectangular cartogram in visualizing single-level data. The questions based on the tuberculosis-positive population dataset of Wuhan are listed in

Table 1. Evaluation questions for single-level visualization of the tuberculosis-positive population dataset of Wuhan.

Tasks	Question
Comparison	WH-P1 Which district has a higher number of tuberculosis patients: Jiangan (JA) district or Hanyang (HY) district?
Spatial localization	WH-P2 Which direction outside of Hongshan (HS) district has the highest number of tuberculosis patients?
Estimation	WH-P3 The number of tuberculosis patients in Jiangxia (JX) district is approximately how many times that in Hannan (HN) district?
Extrema identification	WH-P4 Which district has the lowest number of tuberculosis patients?
Estimation	WH-P5 Please select the three districts with the highest number of tuberculosis patients.

, while those based on the population dataset of Hubei Province are shown in

Table 2. Evaluation questions for single-level visualization of the population dataset of Hubei province.

Tasks	Question
Comparison	HB-P1 Which region has a larger population: Yichang (YC) or Jingzhou (JZ)?
Comparison	HB-P2 Which region has a smaller population: Enshi (ES) or Huanggang (HG)?
Extrema identification	HB-P3 Which city has the largest population?
Extrema identification	HB-P4 Which city has the smallest population?
Spatial localization	HB-P5 Which city is both adjacent to Yichang (YC) and located to the east of Yichang (YC)?
Spatial localization	HB-P6 Which of the following cities does not share a direct border with Xiaogan (XG)?
Estimation	HB-P7 If Wuhan (WH) has approximately 12 million people, estimate the population of Xiaogan (XG).

.

(4) Comparison results and analysis: tuberculosis-positive population in Wuhan

During the experiment, participants answered the questions listed in Table 1 using Figure 11 and Figure 15, respectively. The time taken and accuracy of their responses were recorded to evaluate the expressive effectiveness of the different visualization methods.

The experimental results indicate that, in the visualization of single-level data, the rectangular cartogram did not interfere with users’ ability to interpret the data compared to traditional thematic statistical maps, demonstrating a clear advantage in terms of interpretation efficiency. This highlights the practical value and expressive effectiveness of rectangular cartograms in geographic information visualization. Specifically:

In terms of response time, the average time taken by participants to complete the tasks using the rectangular cartogram was significantly lower than that using the traditional thematic statistical map (as shown in Figure 17). An analysis of variance (ANOVA) yielded an F-value of 44.15 with a p-value significantly less than 0.05, indicating that the rectangular cartogram offers a clear advantage in task completion efficiency compared to the traditional method.

In terms of response accuracy, as shown in Figure 18, the average accuracy of answers using the rectangular cartogram was slightly higher than that of the traditional thematic statistical map. An analysis of variance (ANOVA) produced an F-value of 0.198 and a p-value of 0.657, indicating that the difference in accuracy between the two cartographic methods is not statistically significant. Therefore, it can be inferred that the rectangular cartogram does not negatively impact the users’ ability to interpret the data.

(5) Comparison results and analysis: population data of Hubei province

During the experiment, participants answered the questions listed in Table 2 using Figure 12 and Figure 16, and their response time and accuracy were recorded.

The results from this dataset were largely consistent with those of the Wuhan tuberculosis-positive population dataset. The findings indicate that, compared with traditional thematic statistical maps, the rectangular cartogram does not introduce significant interference in data interpretation and demonstrates a clear advantage in interpretation efficiency. Specifically:

In terms of response time, the average time required by participants to complete the tasks using the rectangular cartogram was significantly lower than that using the traditional thematic statistical map (as shown in Figure 19). An analysis of variance (ANOVA) yielded an F-value of 50.83 with a p-value significantly less than 0.05, indicating that the rectangular cartogram offers a clear advantage in task completion efficiency compared to the traditional method.

In terms of response accuracy, as shown in Figure 20, the average accuracy using the rectangular cartogram was slightly higher than that of the traditional thematic statistical map. An analysis of variance (ANOVA) produced an F-value of 2.66 and a p-value of 0.105, indicating that the difference in accuracy between the two mapping methods is not statistically significant. Therefore, it can be inferred that the rectangular cartogram does not negatively impact users’ ability to interpret the data.

4.3.3. Evaluation of Multi-Level Visualization Performance

(1) Purpose

In multi-level data visualization, it is necessary not only to present the overall statistical value of a region but also to clearly convey the hierarchical data within the region. This experiment aims to evaluate the expressive effectiveness of the proposed method in visualizing multi-level data, with a focus on whether it outperforms traditional thematic statistical maps in terms of information extraction efficiency and accuracy. The goal is to verify the method’s effectiveness in the context of geographical hierarchical data visualization.

(2) Comparative design of visualization methods

This experiment compares the proposed visualization method with traditional thematic statistical maps. As shown in Figure 21 and Figure 22, circular symbols are first used to represent the statistical values of each region. Within each circle, sectors of different colors and sizes are used to indicate the distribution proportions according to gender and age group. This design clearly presents the distribution characteristics of different demographic groups. Finally, the expressive effectiveness of the proposed method in multi-level geographical information visualization is evaluated by comparing Figure 13 with Figure 21 and Figure 14 with Figure 22.

(3) Question design

Based on the four evaluation tasks mentioned above, the following questions were designed to assess the effectiveness of the proposed visualization method in representing multi-level data. The specific questions are listed in

Table 3. Evaluation questions for multi-level visualization of the tuberculosis-positive population dataset of Wuhan.

Tasks	Question
Extrema identification	WH-Q1 In the male tuberculosis-positive population, which age group has the highest number of patients?
Comparison	WH-Q2 Which district has a higher number of tuberculosis patients: Jiangan (JA) district or Xinzhou (XZ) district?
Comparison	WH-Q3 Which district has a higher number of elderly male tuberculosis patients: Hongshan (HS) district or Dongxihu (DXH) district?
Comparison	WH-Q4 Which district has a higher number of middle-aged female tuberculosis patients: Caidian (CD) district or Hannan (HN) district?
Comparison	WH-Q5 Which district has a higher number of young female tuberculosis patients: Qiaokou (QK) district or Hanyang (HY) district?
Estimation	WH-Q6 In which district does the proportion of elderly male tuberculosis patients account for the largest share of the total in that district?

and

Table 4. Evaluation questions for multi-level visualization of the population dataset of Hubei Province.

Tasks	Question
Comparison	HB-Q1 Which region has fewer elderly females: Tianmen (TM) or Qianjiang (QJ)?
Extrema identification	HB-Q2 Across the entire province, which age group accounts for the largest proportion of Hubei’s total population?
Extrema identification	HB-Q3 In the western region of Hubei Province, which area has the largest population of young and middle-aged males (youth + middle-aged)?
Estimation	HB-Q4 Based on the visualization, approximately what proportion of Wuhan’s (WH) total population is in the middle-aged group?
Estimation	HB-Q5 Based on the visualization, estimate the total male population of Xiaogan (XG).

.

(4) Comparison results and analysis: tuberculosis-positive population in Wuhan

During the experiment, participants answered the questions listed in Table 3 using Figure 13 and Figure 21, respectively, and their response time and accuracy were recorded.

The experimental results show that, in visualizing multi-level data, the proposed visualization method significantly outperforms traditional thematic statistical maps in both task completion time and accuracy. This indicates that the proposed method offers strong expressive effectiveness and high practical value in the visualization of geographical hierarchical data. Specifically:

In terms of response time, the average time required to complete the tasks using the proposed visualization method was significantly lower than that of the traditional thematic statistical map, as shown in Figure 23. An analysis of variance (ANOVA) yielded an F-value of 13.74 with a p-value significantly less than 0.05, indicating that the proposed method can significantly improve users’ information extraction efficiency compared to traditional thematic statistical maps.

In terms of response accuracy, the proposed visualization method also outperformed the traditional thematic statistical map, as shown in Figure 24. An analysis of variance (ANOVA) produced an F-value of 99.104 with a p-value significantly less than 0.05, indicating that the proposed method can significantly improve the accuracy of information extraction by users.

(5) Comparison results and analysis: population data of Hubei province

During the experiment, participants answered the questions listed in Table 4 using Figure 14 and Figure 22, and their response time and accuracy were recorded.

The experimental results show that, in multi-level visualization tasks, the proposed method demonstrates a significant advantage in task completion time, while its advantage in response accuracy is not statistically significant. This suggests that the method performs consistently well in improving efficiency, but its impact on accuracy may be influenced by the characteristics and complexity of the dataset. Specifically:

In terms of response time, the average time required to complete the tasks using the proposed method was significantly lower than when using the traditional thematic statistical map, as shown in Figure 25. An analysis of variance (ANOVA) yielded an F-value of 30.86 with a p-value significantly less than 0.05, indicating that the proposed method can significantly improve users’ efficiency in information extraction compared to the traditional approach.

In terms of response accuracy, although the proposed method achieved a higher average accuracy than the traditional thematic statistical map, as shown in Figure 26, the difference was not statistically significant. The ANOVA results showed an F-value of 0.84 and a p-value of 0.36. This suggests that, while the proposed method demonstrated a slight advantage in accuracy in this experiment, the difference may be influenced by dataset characteristics or other factors and does not constitute a statistically significant improvement.

(6) Subjective satisfaction evaluation

In addition to quantitatively measuring response time and accuracy, this paper also collected satisfaction ratings from the subjects for the visualization method proposed in this paper and for the traditional thematic statistical map, as shown in Figure 27. This evaluation approach is inspired by the work of both Dent and Sui [39,40]. The subjects generally reported that obtaining information from traditional thematic statistical maps is difficult, especially in regions with small geographic areas, because there is insufficient space to place statistical charts, making it difficult to intuitively find the corresponding statistics, resulting in a cluttered map that is hard to interpret. In contrast, the visualization method proposed in this paper received higher ratings for readability. The method preserves relatively accurate geographic location information while directly mapping statistical data size through rectangular areas, making the data presentation more intuitive. Moreover, the method makes the map more esthetically pleasing and concise, enhancing the user’s visual experience.

In addition, the subjects generally found that the visualization method proposed in this paper performed better in data comparison. They reported that traditional thematic statistical maps are more difficult to use for data comparison, especially when dealing with multiple regions, which can easily trigger visual confusion, leading to impatience and erroneous conclusions. In contrast, the method proposed in this paper makes data comparison easier through a more intuitive graphical design, enabling them to be more patient and make more detailed comparisons.

4.4. Evaluation Results

This chapter evaluated the effectiveness of the proposed visualization method from both single-level and multi-level perspectives. By selecting two real-world datasets with different structural characteristics, the comprehensiveness and representativeness of the evaluation were ensured. The experimental results demonstrate that the method significantly improves the users’ task completion efficiency while maintaining data interpretability, particularly excelling in multi-level data visualization. Although improvements in accuracy did not reach statistical significance in some experiments, the overall performance was not inferior to that of traditional thematic statistical maps. Subjective satisfaction ratings also indicate that users generally found the proposed method to have clear advantages in readability and ease of data comparison.

Numerous previous studies have explored the readability and user perception of different cartographic representations. For example, Sui and Holt [40] proposed a perceptual evaluation framework for cartograms, emphasizing clarity and interpretability; Nusrat and Kobourov [41] compared various types of cartograms in terms of accuracy and user comprehension. The usability experiments conducted in this study are generally consistent with these findings, further confirming that well-structured cartogram designs can enhance users’ task performance. Moreover, this study expands on previous research by introducing the evaluation perspective into the context of geographical hierarchical data visualization, as well as systematically analyzing the combined performance of rectangular cartograms and treemaps in terms of readability and expressiveness.

5. Discussion and Conclusions

This paper proposes a geographical hierarchical data visualization method based on rectangular cartogram to address the limitations of the current methods. The method uses a rectangular cartogram to visually present the geographic information and statistical data of a region, generating a stable and orderly treemap layout within the rectangular cartogram using the slice and dice layout algorithm, thereby enabling multi-level geographic data visualization. This method improves the accuracy of the geographic distribution of statistical data in the visualization while also presenting the multi-level information of the statistical data, making the visualization more intuitive and easier to understand.

Through experimental validation using real datasets and a comparison with traditional thematic statistical maps. The results suggest that the visualization method proposed in this paper addresses some of the limitations of traditional methods and shows promising potential in representing both the geographic distribution and hierarchical structure of the data. The usability experiment results suggest that the proposed visualization method supports multiple levels of map reading tasks—including comparison, extrema identification, and spatial localization—with significantly improved information extraction efficiency compared to traditional thematic statistical maps and comparable accuracy in information interpretation.

However, the proposed visualization method in this paper performs poorly in terms of proportional perception. This is mainly because the method focuses primarily on the area accuracy and relative positioning of the rectangles during the layout process without considering the aspect ratio of the rectangles as an optimization objective. As a result, there is a significant difference in the aspect ratios of the rectangles, which affects users’ intuitive perception of the area proportions. Extremely elongated or narrow rectangles can negatively impact the users’ ability to perceive area differences and reduce the overall visual balance of the layout. This is particularly important in hierarchical visualization, where nested structures must remain visually interpretable. Future work could explore the incorporation of aspect ratio constraints during the cartogram construction process, such as limiting the maximum aspect ratio or optimizing for more uniform shapes. Such constraints could be integrated into the optimization framework to achieve a better balance between shape regularity, adjacency preservation, and data accuracy. Additionally, the proposed method uses four primary directions (north, south, east, and west) to determine spatial orientation, which simplifies the alignment process but may overlook finer directional distinctions. Introducing an eight-directional model could improve orientation accuracy, though it may increase complexity and ambiguity. Balancing directional granularity and algorithm simplicity is a potential direction for future research.