General Formulation

Let j} j = 1,…,m be an order statistic of a variable with quasi punctual support taking values into the natural classes of precision d, m ≤ N(d), and let Θ0 < Θ1 be the minimum chosen value to take care of the tail effect. Let Θmin = Θ0 and Θmax = Θm. Then:

(1)

Θj – 1 < Θj, j = 1,…, m

Making the correspondence of {Θj} to the m classes:

(2)

(Θj – 1 < Θj ], j = 1,…, m

each class will contain one or more values.

Let’s define an application that makes the correspondence of each class (Θj – 1, Θj] to one and only one extension Tj ∈ IR. Defining:

(3)

T(Θi) = ∑ij=1Tj

T(Θi) is the summation of all field extensions whose punctual variable value is less or equal to Θi. Obviously T(Θi – 1) < T(Θi). Calling:

(4)

T = T(Θm)

a set of weights {ωj} can be defined in such a way that:

(5)

Tj = ωjT,   0 ≤ ωj ≤ 1

A standardized cumulative global variable can be defined as:

(6)

τi = T(Θi) / T

Obviously:

(7)

τi = (∑ij=1Tj) / T

Let g(Θ) be a discrete non decreasing function, defined over the whole interval [Θ0, Θm], so that:

(8)

g(Θi) = τi

The values τi can be understood as the values of g(Θ) in the points chosen to represent the interval (Armony, 2000). Then g(Θ0) = 0, g(Θm) = τm = 1, and we may define τ0 = 0. The function g will be called the extension function. The cumulative global variable τ will be used for modelling and estimation as the application {Θi} → {τi } is isomorphic.

The point variable Θ can also be rescaled to the interval [0,1]:

(9)

ζk = (Θk – Θmin) / (Θmax – Θmin),  k = 1, …, N(d)

and ζ can be the point variable to work with.

Application to Natural Conditional Classes

Let {ti(u)} be an order statistic with n samples taking values on Q, the set of rational numbers, measured in a region A ⊂ IRk, where u means the sample location, u ∈ A. Then:

ti(uα) ≤ ti + 1(uβ); ∀ i = 1,…, n – 1

To the set {ti(u)} corresponds the set {Θj},  j = 1,…, m; m ≤ n + 1. The set {Θj} is an order statistic of m different values of the set {ti(u)} plus Θmax = Θm, the maximum value chosen for tail effect correction. Then, Θj – 1 < Θj, ∀ j = 1 … m, where Θ0 = Θmin. Such a set containing all the measured values with a given precision d, and only measured values will be called the set of natural conditional classesassociated to a precision d. The natural conditional classes are given by (2): (Θj – 1, Θj],  j = 1,…, m. So, all the expressions from (3) to (9) apply to this case.
The weight ωj defined in (5) is composed of three factors:

  • the factor concerning the number of samples with value Θj; in other words, the frequency of Θj;
  • the factor referring to spatial arrangement that expresses the extension to be associated to each sample with a measured value Θj. For example, the factor related to the declustering process or the area of influence;
  • an additional factor for representativeness. For example, the specific gravity, when tonnage is taken as the global variable, or sample size or type.

The weight wjcan be written as:

ωj= (1/n)∑μj k=1αkβk

where μjis the number of samples with value Θj, αk and βk are respectively a spatial weight and an internal factor for representativeness, and n is the total number of samples.