Types of Measues

• Distributive
• 주어진 데이터를 나누어(partitioning하여) 데이터를 small subset으로 만드는 measures. 각 subset을 계산하고 결과를 합치면 original dataset에 대한 결과물과 동일한 결과물을 도출할 수 있음.
• ex. sum, count, min, max
• Algebraic
• 하나이상의 distributive measures에 algegraic function을 적용한 measure.
• ex. mean = sum / count
• Holistic
• 반드시 전체 데이터셋(모두 사용하여)에 대해서 측정(measure)해야 함.
•  ex. median

Categorization of Measures - distributive, algebraic, holistic

Measures can be organized into three categories based on the kind of aggregate functions
used:

• distributive,

• algebraic,

• holistic.

Distributive. 
An aggregate function is distributive if it can be computed in a distributed manner. Suppose the data are partitioned into n sets. We apply the function to each partition, resulting in n aggregate values. If the result derived by applying the function to the n aggregate values is the same as that derived by applying the function to the entire data set (without partitioning), the function can be computed in a distributed manner.

For example, count() can be computed for a data cube by first partitioning the cube into a set of subcubes, computing count() for each subcube, and then summing up the counts obtained for each subcube. Hence, count() is a distributive
aggregate function. For the same reason, sum(), min(), and max() are distributive aggregate functions.

A measure is distributive if it is obtained by applying a distributive aggregate function. Distributive measures can be computed efficiently because they can be computed in a distributive manner.

Algebraic.
An aggregate function is algebraic if it can be computed by an algebraic function with m arguments (where m is a bounded positive integer), each of which is obtained by applying a distributive aggregate function.

For example, avg() (average) can be computed by sum()/count(), where both sum() and count() are distributive
aggregate functions. Similarly, it can be shown that min N() and max N() (which find the N minimum and N maximum values, respectively, in a given set) and standard deviation() are algebraic aggregate functions.

A measure is algebraic if it is obtained by applying an algebraic aggregate function.

Holistic.
An aggregate function is holistic if there is no constant bound on the storage size needed to describe a subaggregate. That is, there does not exist an algebraic function with m arguments (where m is a constant) that characterizes the computation.

Common examples of holistic functions include median(), mode(), and rank().

A measure is holistic if it is obtained by applying a holistic aggregate function.

<Sources: https://andyblg.wordpress.com/2010/05/05/categorization-of-measures/>