Lovegrove Mathematicals
"Dedicated to making Likelinesses the entity of prime interest"
Lovegrove Mathematicals
"Dedicated to making Likelinesses the entity of prime interest"
There are several meanings for the word 'mode', not all of which are mathematical.
Two meanings are in common use in mathematics: local mode and global mode. Although there are exceptions, global modes tend to be used when working with data, local modes when working with functions. This can give rise to confusion, for example with authors using properties of local modes even though they have explicitly stated they are using global modes.
On finite sets the basic ideas are very simple.
It's all about which soldiers are so tall that no-one they check is taller.
Local modes on XN come in two flavours: strong and weak.
If f has domain XN (in particular, if f ∈ S(N) ) then m∈XN is:-
a strong local mode So called because the inequalities < and > are called the strong inequalities of f if
a weak local mode The inequalities ≤ and ≥ are called the weak inequalities of f if
If f has domain XN (in particular, if f ∈ S(N) ) then m∈XN is a Global mode of f if f(m)= max{f(i)|i∈XN}
These animations show random selections of unimodal distributions with mode 7 and degree 29, using, respectively, local and global modes.
The difference between strong and weak local modes is not important for our purposes since we shall be considering only injective unimodal distributions, so the question of equality will not arise.
In the wider context, however, which is used could be important, especially if local modes are found by counting how many times the different values occur in a data-set.