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Multivariate or multidimensional visualization plays an essential role in exploratory data analysis by allowing users to derive insights and formulate hypotheses. Despite their popularity, it is usually users' responsibility to (visually) discover the data patterns, which can be cumbersome and time-consuming. Visual Analytics (VA) and machine learning techniques can be instrumental in mitigating this problem by automatically discovering and representing such patterns. One example is the integration of classification models with (visual) interpretability strategies, where models are used as surrogates for data patterns so that understanding a model enables understanding the phenomenon represented by the data. Although useful and inspiring, the few proposed solutions are based on visual representations of so-called black-box models, so the interpretation of the patterns captured by the models is not straightforward, requiring mechanisms to transform them into human-understandable pieces of information. This paper presents multiVariate dAta eXplanation (VAX), a new VA method to support identifying and visual interpreting patterns in multivariate datasets. Unlike the existing similar approaches, VAX uses the concept of Jumping Emerging Patterns, inherent interpretable logic statements representing class-variable relationships (patterns) derived from random Decision Trees. The potential of VAX is shown through use cases employing two real-world datasets covering different scenarios where intricate patterns are discovered and represented, something challenging to be done using usual exploratory approaches.

Over the past decades, classification models have proven to be essential machine learning tools given their potential and applicability in various domains. In these years, the north of the majority of the researchers had been to improve quantitative metrics, notwithstanding the lack of information about models' decisions such metrics convey. This paradigm has recently shifted, and strategies beyond tables and numbers to assist in interpreting models' decisions are increasing in importance. Part of this trend, visualization techniques have been extensively used to support classification models' interpretability, with a significant focus on rule-based models. Despite the advances, the existing approaches present limitations in terms of visual scalability, and the visualization of large and complex models, such as the ones produced by the Random Forest (RF) technique, remains a challenge. In this paper, we propose Explainable Matrix (ExMatrix), a novel visualization method for RF interpretability that can handle models with massive quantities of rules. It employs a simple yet powerful matrix-like visual metaphor, where rows are rules, columns are features, and cells are rules predicates, enabling the analysis of entire models and auditing classification results. ExMatrix applicability is confirmed via different examples, showing how it can be used in practice to promote RF models interpretability., Comment: IEEE VIS VAST 2020

Kernel-based methods have experienced a substantial progress in the last years, tuning out an essential mechanism for data classification, clustering and pattern recognition. The effectiveness of kernel-based techniques, though, depends largely on the capability of the underlying kernel to properly embed data in the feature space associated to the kernel. However, visualizing how a kernel embeds the data in a feature space is not so straightforward, as the embedding map and the feature space are implicitly defined by the kernel. In this work, we present a novel technique to visualize the action of a kernel, that is, how the kernel embeds data into a high-dimensional feature space. The proposed methodology relies on a solid mathematical formulation to map kernelized data onto a visual space. Our approach is faster and more accurate than most existing methods while still allowing interactive manipulation of the projection layout, a game-changing trait that other kernel-based projection techniques do not have.

Multidimensional projection techniques have experienced many improvements lately, mainly regarding computational times and accuracy. However, existing methods do not yet provide flexible enough mechanisms for visualization-oriented fully interactive applications. This work presents a new multidimensional projection technique designed to be more flexible and versatile than other methods. This novel approach, called Local Affine Multidimensional Projection (LAMP), relies on orthogonal mapping theory to build accurate local transformations that can be dynamically modified according to user knowledge. The accuracy, flexibility and computational efficiency of LAMP is confirmed by a comprehensive set of comparisons. LAMP's versatility is exploited in an application which seeks to correlate data that, in principle, has no connection as well as in visual exploration of textual documents.

Most multidimensional projection techniques rely on distance (dissimilarity) information between data instances to embed high-dimensional data into a visual space. When data are endowed with Cartesian coordinates, an extra computational effort is necessary to compute the needed distances, making multidimensional projection prohibitive in applications dealing with interactivity and massive data. The novel multidimensional projection technique proposed in this work, called Part-Linear Multidimensional Projection (PLMP), has been tailored to handle multivariate data represented in Cartesian high-dimensional spaces, requiring only distance information between pairs of representative samples. This characteristic renders PLMP faster than previous methods when processing large data sets while still being competitive in terms of precision. Moreover, knowing the range of variation for data instances in the high-dimensional space, we can make PLMP a truly streaming data projection technique, a trait absent in previous methods.

Point placement strategies aim at mapping data points represented in higher dimensions to bi-dimensional spaces and are frequently used to visualize relationships amongst data instances. They have been valuable tools for analysis and exploration of data sets of various kinds. Many conventional techniques, however, do not behave well when the number of dimensions is high, such as in the case of documents collections. Later approaches handle that shortcoming, but may cause too much clutter to allow flexible exploration to take place. In this work we present a novel hierarchical point placement technique that is capable of dealing with these problems. While good grouping and separation of data with high similarity is maintained without increasing computation cost, its hierarchical structure lends itself both to exploration in various levels of detail and to handling data in subsets, improving analysis capability and also allowing manipulation of larger data sets.

The problem of projecting multidimensional data into lower dimensions has been pursued by many researchers due to its potential application to data analyses of various kinds. This paper presents a novel multidimensional projection technique based on least square approximations. The approximations compute the coordinates of a set of projected points based on the coordinates of a reduced number of control points with defined geometry. We name the technique least square projections (LSP). From an initial projection of the control points, LSP defines the positioning of their neighboring points through a numerical solution that aims at preserving a similarity relationship between the points given by a metric in mD. In order to perform the projection, a small number of distance calculations are necessary, and no repositioning of the points is required to obtain a final solution with satisfactory precision. The results show the capability of the technique to form groups of points by degree of similarity in 2D. We illustrate that capability through its application to mapping collections of textual documents from varied sources, a strategic yet difficult application. LSP is faster and more accurate than other existing high-quality methods, particularly where it was mostly tested, that is, for mapping text sets.