Visual Analytics

Visual Analytics of Large Homogeneous Data

Homogeneous multivariate data encompass multiple variables that have the same semantics. As example, these variables can represent the probabilities for a sample to belong to different classes, or item memberships of multiple sets.

With a large number of items, such homogeneous data tables become very rich of information that explains how the row entities are related to the different column variables, and how the columns are related to each other according to their relationships with the rows.

This project aims to develop visualization methods for analyzing homogeneous multivariate data. These methods should allow analyzing and selecting the row entities based on their relations with the different columns. Moreover, they emphasizes the column variables and the relations between them as the central part of the visualization, and allows analyzing these relations based on the row entities defining them.

Data

Homogeneous data {fij} that are defined as follows:

F: E × C → R+: (ei, cj) → fij and m = |C| ≪ n = |E|

where:

  • E is the relatively large set (thousands) of row items that typically represent single entities (individuals, samples, ..).
  • C is a relatively small set (tens) of column items that typically represent classes, labels, tags, or categories.
  • F is a bivariate function whose values fij define how the row items (entities) are related to the column items (classes).

In addition to the relationships with different classes, the entities E can also have a set of l numerical or categorical attributes {Ak}:

Ak : E → Sk: ei → Ak(ei) = aik and 1 ≤ k ≤ l

Examples for real-world data that can be modeled using this class of matrix data are:

  • Item-Class Probabilities: Fuzzy classifiers compute the probability fij ∈ [0, 1] that an item ei ∈ E belongs to class cj ∈ C. The probabilities computed for the same item ei with all different classes C sum up to 1.
    As an example, the items E can be a large set of sample images that represent handwritten digits. The classes C represent the digits. The value fij indicates the probability computed by the classifier that image ei represents the handwritten digit cj. In addition, each image i has a set of attribute values {Ak} that represent classification features extracted from this image.
  • Point-Set Memberships: Matrix data of this kind record how a large set of items E belong to a small number of non-disjoint subsets C. The binary value fij ∈ {0,1} denotes whether ei ∈ cj holds.
    As an example, the matrix data can denote how a large number of movies E belong to small number of genres C. A movie can belong to multiple genres and has attributes such as release date or director.
  • Large Contingency Tables: A two-way contingency table records the frequency of observations fij ∈ ℕ for each combination of categories (ei, cj) ∈ E×C of two categorical variables. The frequencies typically represent a statistic of each of the entities E computed for each of the columns C.
    As an example, E can be a large set of books, C a set of countries, and fij represents the purchases of book ei ∈ E in country cj ∈ C. In addition, these books can have a set of attributes {Ak} such as release date, author(s) and publisher(s).
Tasks

The tasks addressed in this project revolve around pattern discovery in large matrix data of the class described above:

  • T1: Analyze the relations rij between the row entities E and the columns C, in the light of the attribute values aik.
  • T2: Analyze the similarity rcj1j2 between columns based on their relations with the row entities.

 

Users

Domain Expert (the same domain the data and the tasks come from) with sufficient background in data analysis.

A multi-level overview+detail exploration environment provides access to the matrix data fij the attribute values aik and any raw data aggregated in the matrix.

Several selection mechanisms allow marking interesting parts of the data.

Data Level

The data presented by the visualization methods are the homogeneous data fij of the class described above (with focus on the associations rij between the row entities E and the columns C of the data table).

Task Level

When performing task T1, the visualization is augmented with one of the attributes Ak to analyze the row-column associations in the light of the its values aik.

When performing task T2, the visualization is augmented with with the column similarities rc to find out which columns exhibit similar associations with the rows.

Presentation Level

The visualization methods combine familiar visual representations to gain insights in the data, such as ring charts, histograms, stacked bar charts, star graphs, and arcs.

The row-column associations rij  and the column similarities rcj1j2 are computed using automated methods.

Depending on what the data represent, and on the tasks to be solved, these methods can employ different statistical or machine learning techniques.

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