Bounding space

Post date: Mar 22, 2021 6:49:0 PM

Approximating is a means of estimating quantities in the face of uncertainties. It identifies or localizes values. Bounding can also be a means of estimating quantities under uncertainty, and thus it can be contrasted with approximation. Bounding can be considered to be a kind of outer approximating, in that it improves from the outside inward, whereas approximating doesn't control the direction of the approach. Rowe^[1] pointed out there are a variety of reasons that bounding may be more convenient than an approximation, including the fact that rigorous bounds are

• • usually relatively easy to compute compared to approximations which may involve integrals,

  • very simple to combine via intersection,

  • often sufficient to specify a decision,

  • obtainable even in situations where estimates are impossible, and

  • valuable because they are rigorous rather than approximative.

Exactly what is being bounded can make a substantial difference. For instance, the imprecise expression "France" can refer to somewhere in France as a fixed location whose actual position is not precisely known, or to anywhere in France as a dynamical range of a varying location that must at some point include every spot in France, or perhaps to the whole of France as the set of all locations in France at once and considered as one object. These interpretational issues are on top of the basal semantic imprecision in the word, as France may refer to Metropolitan France, Mainland France (the Hexagon), the modern French Republic, the country or its government at different points in history, or even the historical monarchs of France. Indeed, there are other possible meanings such as people, ships, or other entities with that name.

Mathematical representation

Let Ω be some universe, space or set of interest. Often, this is either the real numbers ℝ or the integers ℤ, but it could be any topological space. <<Does it even need a topology? It doesn’t need to be a metric space, does it? Doesn’t it at least need to be Hausdorff? Is separability an issue?>> Suppose A ⊂ Ω is the set of all values or solutions in Ω that we are trying to characterize via bounding.

A is a subset of Ω. Let B and C also denote subsets of Ω. Let ω, θ, and φ denote elements of Ω.

If ω ∈ A, we call ω a solution. If ω ∉ A, we say ω is not a solution. If θ ∈ B, we say θ is implied by B.

We say that B ⊂ Ω bounds A (and that A is bounded by B) if every element of A is also an element of B. If B bounds A then

A ⊂ B,

A ⊆ B, or

A = B.

We say that B is an inner bound for A if A bounds B. We say that C constrains A if either

C bounds A or

A bounds C,

and that we know which of these two statements is true.

Structured bounds

Structured bounds are less resolved than arbitrary sets but often far simpler to work with. Call D a structured bounding space for Ω if every subset of Ω is bounded by some element of D, i.e., if the following conditions hold:

D ⊆ 2^Ω, and

A ⊆ Ω implies A ⊆ B for some B ∈ D.

This definition implies that every element of Ω is covered by some element of D, and also that Ω itself is covered by some element of D which would be called a vacuous bound. Note that the empty set may or may not be an element of D.

The union of the elements in any subset of D is bounded by some element of D?

The bound B ∈ D is said to be a best-possible (relative to D) bound on A ⊆ Ω if both

A ⊆ B, and

there exists no C ∈ D such that A ⊆ C ⊂ B.

It may be further true that B is the unique best-possible bound in the sense that

A ⊆ C implies that B ⊆ C, for all C ∈ D.

Thus, B is the unique best-possible bound on A if B bounds A and it is the tightest bound in D that does so.

Some structured bounding spaces, such as intervals bounding sets of reals, have only unique best-possible bounds. Their elements form a tree-shaped upper lattice.

Examples

Intervals over the reals. If Ω is the real numbers, then 2^ℝ then is a trivial structured bounding space (i.e., one with no structure) because no subsets of Ω are missing. The set of traditional intervals^[2]

{ [a, b] : a ≤ b, a ∈ ℝ, b ∈ ℝ }

is not a structured bounding space for ℝ, because there is no interval that bounds the whole of ℝ itself. Likewise all half-bounded intervals of the form [a, ∞] and [−∞, b] are also not bounded by any element of the set of real intervals.

However, a structured bounding space for ℝ can be constructed as

{ [a, b] : a ≤ b, a ∈ ℝ*, b ∈ ℝ* }

where ℝ* = ℝ ∪ {−∞, ∞}.

Floating point ranges. A finite-precision number is <<?>>.

Significant-digit intervals are often suggested as a way to express uncertainty about quantities in science and engineering. Do such intervals represent a structured bounding space of possible measurements? Recall that significant digits imply an interval whose width is the magnitude of the last significant decimal place, so a significant-digit interval s(x) around an explicitly and finitely expressed numerical value x is the set of real numbers that are d-close to the value of x, where d = ½ <<>>.

x s(x)

0 [0, 0.5]

1 [ 0.5, 1.5]

9 [ 8.5, 9.5]

10 [ 5, 15]

23.7 [23.65, 23.75]

300 [ 250, 350]

8150 [ 8145, 8155]

23700 [23650, 23750]

1×10⁵ [ 5×10⁴, 1.5×10⁵]

2.3×10⁵ [ 2.25×10⁵, 2.35×10⁵]

Note that this convention breaks down if no digits of a measurement are significant. For example, there’s no scalar expression that encloses the range [2.34, 5.67]. In such cases, we may need to use two values to express the uncertainty such as

between 2.34 and 5.67

4.005 ± 1.665

4 ± 1.7

So are significant-digit intervals a structured bounding space for the reals? No, they are decidedly not. In fact, most simple intervals (which are subsets of the real line) cannot even be bounded by significant-digit intervals.

Probability boxes <<over the reals, or over the set of distribution functions>>

Notation

There are various notation systems in use for denoting open, closed and partially open intervals.

(a,b) = ]a,b[ = [a+, b-] = [a plus, b minus] (a,b] = ]a,b] = [a+, b] = [a plus, b] [a,b) = [a,b[ = [a, b-] = [a, b minus] [a,b] = [a,b] = [a, b] = [a, b]

The system using both parentheses and brackets is conventional, but is often confused with ordered sets. The backward-bracket notation ]a,b[ which was introduced by Bourbaki^[3] is efficient, but it is easy to misread. Notation should strive to be helpful in addition to being efficient. When something should be noticed, the notation should help readers notice it. Thus, we might prefer the plus-minus notation, which cannot be misread and, arguably, is the most intuitive of the three. The plus-minus notation is also more flexible, as it can designate a range which is a halo around a closed interval, such as [a minus, b plus].

<<Material on kinds of bounds: socks sleeves mittens gloves>>

<<Simultaneous (distributional) confidence bounds versus point-wise confidence bounds>>

<<Algebraic structure of bounding and uncertainty calculi See https://en.wikipedia.org/wiki/Magma_(algebra)#Types_of_magma>>

An algebra A over a set Ω is a non-empty set of sets of Ω that satisfies: (i) Ω∈A, (ii) if a∈A then a^c∈A, and (iii) if a,b∈A then a∪b∈A. Thus an algebra is a set containing Ω that is closed under complements and finite unions and intersections.

A σ-algebra is a non-empty set of sets that is closed under complements and countable unions and intersections.

A measurable space (Ω,F) consists of a set Ω and a σ-algebra of subsets of Ω.

References[edit]

• 1. ^ Rowe, N.C. (1988). Absolute bounds on the mean and standard deviation of transformed data for constant-sign-derivative transformations. SIAM Journal of Scientific Statistical Computing 9: 1098–1113.

1. ^

   1. ^ https://en.wikipedia.org/wiki/Interval_(mathematics)#Notations_for_intervals