A×B ={a ,...,a }×{b ,...,b } 1n1m

={a b ,...,a b ,
11 1m
a b ,...,a b ,...,
21 2m
a b ,...,a b }
n1 nm
A×P ={a ,...,a }×P
1n

={a P,...,a P} 1n

 

with those domain values. If we define R to be the dataset being analyzed, r to be a record, and A(r) to be the value of the field A for the record r, then we can define the nest operator as follows:

 

A/B ={a b | ∃r ∈R st A(r)= a & B(r)= b } ij i j

 

The intuitive interpretation of the nest operator is “B within A”. For example, given the fields quarter and month, the expression quarter/month would be interpreted as those months within each quarter, resulting in three entries for each quarter. In contrast, quarter × month would result in 12 entries for each quarter. Data cubes represent hierarchies explicitly and there is no need to compute the nest relationship.

Using the above semantics for each operator, every expression in the algebra can be reduced to a single sequence, with each entry in the sequence being an ordered term consisting of zero or more ordinal values with zero or more quantitative field names. We call this set evaluation of an expression the

normalized form. The normalized form of an expression determines one axis of the table: the table axis is partitioned into columns (or rows or layers) so that there is a one-to-one correspondence between set entries in the normalized set and columns. Figure 2 illustrates the configurations resulting from a number of expressions.

Analysts can also combine multiple data sources in a single Polaris visualization. When multiple data sources are imported, each data source is mapped to a distinct layer (or set of layers). While all data sources and all layers share the same configuration for the x and y axes of the table, each data source can have a different expression for partitioning its data into layers.

In retrospect, the Polaris table algebra is very similar to the operations in the MDX query language for data cubes.

3. 2. types of graphics

Given a table configuration, the next step is to specify the type of graphic in each pane. We have developed a taxonomy of graphics that results in an intuitive and concise specification of graphic types. This taxonomy is based on both the axes within each pane (implicitly specified from the table configuration via the role and scale of the innermost field in the sequence) as well as the mark type representing a tuple (e.g., text, shape, bar, etc.). We group this taxonomy into three families (illustrated in Figure 3) based on the axes: ordinal–ordinal, ordinal–quantitative, and quantitative–quantitative.

Each family contains a number of variants depending

 

figure 2: the graphical interpretation of several expressions in the table algebra. each expression in the table algebra can be reduced to a single sequence of terms, and that sequence can then be directly mapped into a configuration for an axis of the table.