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.
Homogeneous data {f_{ij}} that are defined as follows:
F: E × C → R^{+}: (e_{i}, c_{j}) → f_{ij} 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 f_{i}_{j} 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 {A_{k}}:
A_{k} : E → S_{k}: e_{i} → A_{k}(e_{i}) = a_{ik} and 1 ≤ k ≤ l
Examples for realworld data that can be modeled using this class of matrix data are:

ItemClass Probabilities: Fuzzy classifiers compute the probability f_{ij} ∈ [0, 1] that an item e_{i} ∈ E belongs to class c_{j} ∈ C. The probabilities computed for the same item e_{i} 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 f_{ij} indicates the probability computed by the classifier that image e_{i} represents the handwritten digit c_{j}. In addition, each image i has a set of attribute values {A_{k}} that represent classification features extracted from this image. 
PointSet Memberships: Matrix data of this kind record how a large set of items E belong to a small number of nondisjoint subsets C. The binary value f_{ij} ∈ {0,1} denotes whether e_{i} ∈ c_{j} 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 twoway contingency table records the frequency of observations f_{ij} ∈ ℕ for each combination of categories (e_{i}, c_{j}) ∈ 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 f_{ij} represents the purchases of book e_{i} ∈ E in country c_{j} ∈ C. In addition, these books can have a set of attributes {A_{k}} such as release date, author(s) and publisher(s).
The tasks addressed in this project revolve around pattern discovery in large matrix data of the class described above:
 T1: Analyze the relations r_{ij}_{}between the row entities E and the columns C, in the light of the attribute values a_{ik}.
 T2: Analyze the similarity rc_{j1j2}_{}between columns based on their relations with the row entities.
Domain Expert (the same domain the data and the tasks come from) with sufficient background in data analysis.
A multilevel overview+detail exploration environment provides access to the matrix data f_{ij} the attribute values a_{ik} and any raw data aggregated in the matrix.
Several selection mechanisms allow marking interesting parts of the data.
The data presented by the visualization methods are the homogeneous data f_{ij}_{}of the class described above (with focus on the associations r_{ij}_{}between the row entities E and the columns C of the data table).
When performing task T1, the visualization is augmented with one of the attributes A_{k} to analyze the rowcolumn associations in the light of the its values a_{ik}.
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.
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 rowcolumn associations r_{ij}_{ }and the column similarities rc_{j1j2}_{}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.