# Dictionary Definition

shattering adj : seemingly loud enough to break
something; violently rattling or clattering; "shattering rain
striking the windowpanes"; "the shattering tones of the enormous
carillon"; "the shattering peal of artillery" n : the act of
breaking something into small pieces [syn: smashing]

# User Contributed Dictionary

## English

### Verb

shattering- present participle of shatter

# Extensive Definition

The concept of shattering of a set of points
plays an important role in Vapnik-Chervonenkis
theory, also known as VC-theory. Shattering and VC-theory are
used in the study of empirical
processes as well as in statistical
computational learning theory.

## Definition

Suppose we have a class C of sets and a given set A. C is said to shatter A if, for each subset T of A, there is some element U of C such that- U \cap A = T.\,

Equivalently, C shatters A when the power set P(A)
is the set .

For example, the class C of all discs in
the plane
(two-dimensional space) cannot shatter every set A of four points,
yet the class of all convex sets in
the plane shatters every finite set on the (unit) circle. (For the
collection of all convex sets, connect the dots!)

We employ the letter C to refer to a "class" or
"collection" of sets, as in a Vapnik-Chervonenkis class (VC-class).
The set A is often assumed to be finite
because, in empirical processes, we are interested in the
shattering of finite sets of data points.

## Shatter coefficient

To quantify the richness of a collection C of sets, we use the concept of shattering coefficients (also known as shatter coefficients or the growth function). For a collection C of sets s \subset Ω, Ω being any space, often a probability space, we define the nth shattering coefficient of C as- S _C(n) = \max_ \operatorname \

where "card" denotes the cardinality, that is the
number of elements of a set.

S _C(n) is equal to the largest number of subsets
of any set A of n points that can be formed by intersecting A with
the sets in collection C .

It is obvious that

- 1.S_C(n)\leq 2^n for all n.
- 2. If S_C(n)=2^n, that means there is a set of cardinality n, which can be shattered by C.
- 3. If S_C(N) for some N>1 then S_C(n) for all n\geq N