Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Shattered set
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Shatter coefficient== {{Main|growth function}} To quantify the richness of a collection ''C'' of sets, we use the concept of ''shattering coefficients'' (also known as the ''growth function''). For a collection ''C'' of sets <math>s \subset \Omega</math>, <math>\Omega</math> being any space, often a [[sample space]], we define the ''n''<sup>th</sup> ''shattering coefficient'' of ''C'' as :<math> S_C(n) = \max_{\forall x_1,x_2,\dots,x_n \in \Omega } \operatorname{card} \{\,\{\,x_1,x_2,\dots,x_n\}\cap s, s\in C \}</math> where <math>\operatorname{card}</math> denotes the [[cardinality]] of the set and <math>x_1,x_2,\dots,x_n \in \Omega </math> is any set of ''n'' points,. <math> S_C(n) </math> is 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''. For example, if set ''A'' contains 3 points, its power set, <math>P(A)</math>, contains <math>2^3=8</math> elements. If ''C'' shatters ''A'', its shattering coefficient(3) would be 8 and S_C(2) would be <math>2^2=4</math>. However, if one of those sets in <math>P(A)</math> cannot be obtained through intersections in ''c'', then S_C(3) would only be 7. If none of those sets can be obtained, S_C(3) would be 0. Additionally, if S_C(2)=3, for example, then there is an element in the set of all 2-point sets from ''A'' that cannot be obtained from intersections with ''C''. It follows from this that S_C(3) would also be less than 8 (i.e. ''C'' would not shatter ''A'') because we have already located a "missing" set in the smaller power set of 2-point sets. This example illustrates some properties of <math>S_C(n)</math>: # <math>S_C(n)\leq 2^n</math> for all ''n'' because <math>\{s\cap A|s\in C\}\subseteq P(A)</math> for any <math>A\subseteq \Omega</math>. # If <math>S_C(n)=2^n</math>, that means there is a set of cardinality ''n'', which can be shattered by ''C''. # If <math>S_C(N)<2^N</math> for some <math>N>1</math> then <math>S_C(n)<2^n</math> for all <math>n\geq N</math>. The third property means that if ''C'' cannot shatter any set of cardinality ''N'' then it can not shatter sets of larger cardinalities.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)