Linear embeddings support interactive visual exploration by mapping high-dimensional (nD) data into a two-dimensional space. Despite their popularity, selecting meaningful projection parameters remains a key challenge due to the infinite 2$n$-dimensional parameter space. Once an informative projection is found, users often seek similar ones that emphasize specific items differently while preserving global structure. For instance: Do clusters become outliers under slight changes? Can grouped items separate-or merge-through parameter adjustments? Which changes to the embedding parameters lead to such projections - and do they exist at all? Answering these questions efficiently is critical for effective visual search. Yet, current methods-such as projection tours or manual parameter tuning-are time-consuming and risk overlooking important views, including those of specific interest. We propose Composition Operators, a mathematical foundation for a novel set-of-point manipulation concept for linear embeddings-such as Star Coordinates-as an alternative approach to selecting informative embedding parameters in a more controllable manner with respect to the desired outcome. Users specify item-based constraints on the projection result; the corresponding 2n parameters are then derived automatically, eliminating the need to exhaustively search the entire parameter space to get a similar outcome. Neither the embedding space nor the set of parameters is altered - only the mechanism for navigating and selecting parameters is redefined. We provide closed-form solutions for this and demonstrate our interactive prototype on nD datasets from the UCI repository.

This research was funded by the Spanish Research Agency, grant numbers PID2021-122392OB-I00, PID2019-106623RB-C41/AEI/10.13039/501100011033 and PID2019-107768RA-I00; and by Universidad Rey Juan Carlos (URJC) and Community of Madrid, Spain , grant number 2020-66.

Abstract Radial axes plots are projection methods that represent high‐dimensional data samples as points on a two‐dimensional plane. These techniques define mappings through a set of axis vectors, each associated with a data variable, which users can manipulate interactively to create different plots and analyze data from multiple points of view. However, updating the direction and length of an axis vector is far from trivial. Users must consider the data analysis task, domain knowledge, the directions in which values should increase, the relative importance of each variable, or the correlations between variables, among other factors. Another issue is the difficulty to approximate high‐dimensional data values in the two‐dimensional visualizations, which can hamper searching for data with particular characteristics, analyzing the most common data values in clusters, inspecting outliers, etc. In this paper we present and analyze several optimization approaches for enhancing radial axes plots regarding their ability to represent high‐dimensional data values. The techniques can be used not only to approximate data values with greater accuracy, but also to guide users when updating axis vectors or extending visualizations with new variables, since they can reveal poor choices of axis vectors. The optimal axes can also be included in nonlinear plots. In particular, we show how they can be used within RadViz to assess the quality of a variable ordering. The in‐depth analysis carried out is useful for visualization designers developing radial axes techniques, or planning to incorporate axes into other visualization methods.

Contemporary societies are suffering from negative population growth, with the consequent population aging. The prevalence of some chronic diseases, of slow progress and long duration, have become one of the main problems for healthcare systems. In particular, high blood pressure, diabetes mellitus, chronic obstructive pulmonary disease, and depression are health-status with a high economic and social burden. In collaboration with University Hospital of Fuenlabrada (Spain), we analyze in this work data (mainly diagnoses and drugs, both coded) from patients suffering from these chronic conditions. Given the high dimensionality of the data, we performed a hypothesis test with bootstrapping in order to select discriminative features that we subsequently analyzed using Multiple Correspondence Analysis (MCA). MCA allowed us to find associations among features and health-statuses, which may reveal not evident relationships. From the analysis carried out, on the one hand, some evidences are concluded, which can be used to validate the methodology followed in this work. On the other hand, we have drawn some conclusions that could assist in clinical decision-making, such as for example, offering more specialized care to patients stratified in the same health-status.

BACKGROUND: Chronic diseases are becoming more widespread each year in developed countries, mainly due to increasing life expectancy. Among them, diabetes mellitus (DM) and essential hypertension (EH) are two of the most prevalent ones. Furthermore, they can be the onset of other chronic conditions such as kidney or obstructive pulmonary diseases. The need to comprehend the factors related to such complex diseases motivates the development of interpretative and visual analysis methods, such as classification trees, which not only provide predictive models for diagnosing patients, but can also help to discover new clinical insights. RESULTS: In this paper, we analyzed healthy and chronic (diabetic, hypertensive) patients associated with the University Hospital of Fuenlabrada in Spain. Each patient was classified into a single health status according to clinical risk groups (CRGs). The CRGs characterize a patient through features such as age, gender, diagnosis codes, and drug codes. Based on these features and the CRGs, we have designed classification trees to determine the most discriminative decision features among different health statuses. In particular, we propose to make use of statistical data visualizations to guide the selection of features in each node when constructing a tree. We created several classification trees to distinguish among patients with different health statuses. We analyzed their performance in terms of classification accuracy, and drew clinical conclusions regarding the decision features considered in each tree. As expected, healthy patients and patients with a single chronic condition were better classified than patients with comorbidities. The constructed classification trees also show that the use of antipsychotics and the diagnosis of chronic airway obstruction are relevant for classifying patients with more than one chronic condition, in conjunction with the usual DM and/or EH diagnoses. CONCLUSIONS: We propose a methodology for constructing classification trees in a visually guided manner. The approach allows clinicians to progressively select the decision features at each of the tree nodes. The process is guided by exploratory data analysis visualizations, which may provide new insights and unexpected clinical information.

Abstract The UN Comtrade database is the world's largest repository of bilateral trade data. Their complexity poses a challenge to visualization systems, leading to issues such as scalability and visual clutter. Thus, we propose a radial layout‐based visual exploration system to enable the user to smoothly explore the change over time and to explore different commodity classes at once by using a novel edge bundling concept. We evaluated our system with the aid of a group of domain experts.

Abstract Radial axes plots are multivariate visualization techniques that extend scatterplots in order to represent high‐dimensional data as points on an observable display. Well‐known methods include star coordinates or principal component biplots, which represent data attributes as vectors that define axes, and produce linear dimensionality reduction mappings. In this paper we propose a hybrid approach that bridges the gap between star coordinates and principal component biplots, which we denominate “adaptable radial axes plots”. It is based on solving convex optimization problems where users can: (a) update the axis vectors interactively, as in star coordinates, while producing mappings that enable to estimate attribute values optimally through labeled axes, similarly to principal component biplots; (b) use different norms in order to explore additional nonlinear mappings of the data; and (c) include weights and constraints in the optimization problems for sorting the data along one axis. The result is a flexible technique that complements, extends, and enhances current radial methods for data analysis.

RadViz and star coordinates are two of the most popular projection-based multivariate visualization techniques that arrange variables in radial layouts. Formally, the main difference between them consists of a nonlinear normalization step inherent in RadViz. In this paper we show that, although RadViz can be useful when analyzing sparse data, in general this design choice limits its applicability and introduces several drawbacks for exploratory data analysis. In particular, we observe that the normalization step introduces nonlinear distortions, can encumber outlier detection, prevents associating the plots with useful linear mappings, and impedes estimating original data attributes accurately. In addition, users have greater flexibility when choosing different layouts and views of the data in star coordinates. Therefore, we suggest that analysts and researchers should carefully consider whether RadViz's normalization step is beneficial regarding the data sets' characteristics and analysis tasks.

Star coordinates is a well-known multivariate visualization method that produces linear dimensionality reduction mappings through a set of radial axes defined by vectors in an observable space. One of its main drawbacks concerns the difficulty to recover attributes of data samples accurately, which typically lie in the [0], [1] interval, given the locations of the low-dimensional embeddings and the vectors. In this paper we show that centering the data can considerably increase attribute estimation accuracy, where data values can be read off approximately by projecting embedded points onto calibrated (i.e., labeled) axes, similarly to classical statistical biplots. In addition, this idea can be coupled with a recently developed orthonormalization process on the axis vectors that prevents unnecessary distortions. We demonstrate that the combination of both approaches not only enhances the estimates, but also provides more faithful representations of the data.

This paper presents the evaluation of an educational tool focused on the visualization of the symbol table in the context of a compiler course. In a first evaluation we used simulation exercises and tested basic concepts of symbol tables. We detected efficiency improvements, students who used the tool completed the exercises with the same grading and significantly faster than the students who did not use the tool. In addition students' opinion was positive. In a second evaluation we used more active tasks, and tested students' skills on writing parser specifications regarding symbol table management. We have detected significant improvements. Students who used the tool outperformed those who did not use the tool in a 22%.

No abstract available.

The design of many tail recursive algorithms can involve thinking about the status of variables and parameters, and how these change with execution flow. In other words, tail recursion is closely related to iteration and imperative programming. However, it is possible to derive tail recursive functions by exclusively using concepts inherent in recursion, such as declarative programming, induction, or problem decomposition. This paper proposes a simple methodology for designing tail recursion functions by using a declarative approach and the concept of function generalization. We have carried out an evaluation of the technique with second and third-year computer science students. Results suggest that this new point of view improves students' ability to design tail recursive programs, helps them understand the distinction between the imperative and declarative paradigms, and may reinforce their programming skills in general. Furthermore, students found the methodology easy to learn and apply, simpler than more sophisticated formal methods, and described it as fast and methodic or mechanical, as it involves a sequence of well-defined steps.

The design of tail recursive algorithms may require thinking about iteration rather than recursion. This paper provides a methodology for deriving tail recursive functions that is based on declarative programming and the concept of function generalization, which allow to avoid iterative thinking.

This paper proposes the use of several classes of simple combinatorial problems that share the same solution for teaching problem equivalence and recursion. Our focus is on counting problems that involve Fibonacci numbers. While these problems have simple recursive solutions, we propose that - for teaching purposes - they can also be solved by finding other isomorphic problems for which the solution is known.

Recursion is an important topic in computer science curricula. It is related to the acquisition of competences regarding problem decomposition, functional abstraction and the concept of induction. In comparison with direct recursion, mutual recursion is considered to be more complex. Consequently, it is generally addressed superficially in CS1/2 programming courses and textbooks. We show that, when a problem is approached appropriately, not only can mutual recursion be a powerful tool, but it can also be easy to understand and fun. This paper provides several intuitive and attractive algorithms that rely on mutual recursion, and which have been designed to help strengthen students' ability to decompose problems and apply induction. Furthermore, we show that a solution based on mutual recursion may be easier to design, prove and comprehend than other solutions based on direct recursion. We have evaluated the use of these algorithms while teaching recursion concepts. Results suggest that mutual recursion, in comparison with other types of recursion, is not as hard as it seems when: (1) determining the result of a (mathematical) function call, and, most importantly, (2) designing algorithms for solving simple problems.

This paper addresses the relationship between recursion and combinatorial problems, which may benefit teaching recursion in CS1/2 courses. Problems whose solutions are related to Fibonacci numbers are particularly interesting, since some can be decomposed by using different strategies, which may be based on the numerous Fibonacci identities or even on the concept of mutual recursion.