The data cube as a typed linear algebra operator

Abstract

There is a need for a typed notation for linear algebra applicable to the fields of econometrics and data mining. In this paper we show that such a notation exists and can be useful in formalizing and reasoning about data aggregation operations. One such operation-the construction of a data cube-is shown to be easily expressible as a linear algebra operator. The construction is shown to be type-generic and some of its properties are derived from its typed definition and proved using matrix algebra. Other forms of data aggregation such as eg. rollup and cross tabulation are shown to be algebraically derivable from data cubes.

Typed Linear Algebra for Efficient Analytical Querying

“…The typed LA approach to data representation [18] consists in representing projection functions by (Boolean) matrices, as follows: let f : A → B be any function, where A and B are finite. Function f can be represented by a matrix f with A-many columns and B-many rows such that, for any b ∈ B and a ∈ A, matrix cell 3 b f a = 1 if b = f a, otherwise b f a = 0.…”

“…As in [18] we shall abuse of notation and (very conveniently, as we shall see) drop the parentheses from f . This is consistent with writing f ▽ g to denote the operation above, which in fact corresponds to a well-known matrix operator.…”

“…More recently, it has been shown that data consolidation can be elegantly expressed in typed linear algebra [17,18], a categorial approach to linear algebra [16] in which matrices are represented by arrows m M G G n where M is a LA-expression denoting a matrix, m is the number of columns of M and n is the corresponding number of rows. The approach is strongly typed in the sense that it is free from matrix dimension errors, by construction [16].…”

“…Moreover, it is type-polymorphic, making room for proving properties of data constructions relying on types alone. For instance, the "free theorem" [27] of the data cube operator given in [18] is proved in that way.…”

“…The main aim of [17,18] was to show how analytical querying theory benefits from the typed LA approach, essentially from a foundation point of view. In the current paper, we propose, validate and evaluate such a typed LA approach as a means of data analysis programming.…”