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U-shaped distributions

Introduction to U-shaped distributions

There are two possible approaches to U-shaped distributions: which one you choose depends on the information available in your background subject:-
  1. U-shaped distributions can be considered as a special type of Multimodal distribution, arising from two populations. You will want each distribution to come from one of those populations, with you controlling the proportion coming from each. It is worth emphasising that with this approach each of the distributions will be unimodal (to be precise, either ranked or reverse-ranked): the U-shape comes from the combination of the two populations rather than the intrinsic shape of the individual distributions.
  2. Alternatively, U-shaped distributions can be considered as an analogue to unimodal distributions, but based on the concept of a trough rather than a mode. With this approach, control is on the position of the trough and each distribution is U-shaped.
Because "the set of U-shaped distributions" must necessarily have elements which are U-shaped distributions, the notation etc concerned with such a set will inevitably be concerned with the second of these. It is recognised, however, that in practice the choice will often be of the first.

The set of U-shaped distributions

If f ∈ S(N) then m∈XN is a strong local trough of f if

  1. m=1 and f(1)<f(2); or
  2. 1<m<N and f(m-1)>f(m)<f(m+1); or
  3. m=N and f(N-1)>f(N).

By analogy with modes, there are also weak local troughs and global troughs. Global troughs are never referred to as such (simply as 'smallest value'), and we will have no interest in weak local troughs. Unless specified otherwise, a 'trough' will be a strong local trough.

The analogue to an unimodal distribution is a distribution which has precisely one trough. We could, perhaps, by analogy call these "unitroughal distributions" but we don't -we call them "U-shaped distributions". A U-shaped distribution is a distribution which has precisely one trough.

The set of all U-shaped distributions of degree N and trough m is denoted by U(m,N).

If A and B are elements of XN with B≥A then the set U(A to B,N) is defined by U(A to B,N)=U(A,N)∪U(A+1,N)∪...∪U(B,N)

If A=1 and B=N then we write U(N) rather than U(1 to N,N).

Algorithm to find injective f∈U(A,N) 
and injective f∈U(A to B,N)

The algorithms for U-shaped distributions are identical to those for Unimodal distributions, except that they start with an element of RR(N) rather than R(N).

U-shaped distributions are identical to Unimodal distributions, except that the inequalities have been reversed.

As this animation of LU(m,29)(i) demonstrates, on average the distribution is taller at the mode which is further from the trough. (Note: in this animation, the values do not actually go to zero at the trough.)

Changes in trough

The underlying set, here, is U(29). This is the U-shaped analogue to the unimodal "bell-shaped" distribution M(29).

U-shaped equivalent of bell-shaped