Stationarity or Homogeneity of Random Fields

Let (Ω, F, P) be a probability space on which all random objects will be defined. A filtration {F_t : t ≥ 0} of σ-algebras, is fixed and defines the information available at each time t.

Random field: A real-valued random field is a family of random variables Z(x) indexed by x ∈ R^d together with a collection of distribution functions of the form F_x1,…,_xn which satisfy

F_x1,…,_xn(b₁,…,b_n) = P[Z(x₁) ≤ b₁,…,Z(x_n) ≤ b_n], b₁,…,b_n ∈ R

The mean function of Z is m(x) = E[Z(x)] whereas the covariance function and the correlation function are respectively defined as

R(x, y) = E[Z(x)Z(y)] − m(x)m(y)

c(x, y) = R(x, x)/√(R(x, x)R(y, y))

Notice that the covariance function of a random field Z is a non-negative definite function on R^d × R^d, that is if x₁, . . . , x_k is any collection of points in R^d, and ξ₁, . . . , ξ_k are arbitrary real constants, then

∑_l=1^k∑_j=1^k ξ_lξ_j R(x_l, x_j) = ∑_l=1^k∑_j=1^k ξ_lξ_j E(Z(x_l) Z(x_j)) = E (∑_j=1^k ξ_j Z(x_j))² ≥ 0

Without loss of generality, we assumed m = 0. The property of non-negative definiteness characterizes covariance functions. Hence, given any function m : R^d → R and a non-negative definite function R : R^d × R^d → R, it is always possible to construct a random field for which m and R are the mean and covariance function, respectively.

Bochner’s Theorem: A continuous function R from R^d to the complex plane is non-negative definite if and only if it is the Fourier-Stieltjes transform of a measure F on R^d, that is the representation

R(x) = ∫_R^d e^ix.λ dF(λ)

holds for x ∈ R^d. Here, x.λ denotes the scalar product ∑_k=1^d x_kλ_k and F is a bounded,  real-valued function satisfying ∫_A dF(λ) ≥ 0 ∀ measurable A ⊂ R^d

The cross covariance function is defined as R₁₂(x, y) = E[Z₁(x)Z₂(y)] − m₁(x)m₂(y)

, where m₁ and m₂ are the respective mean functions. Obviously, R₁₂(x, y) = R₂₁(y, x). A family of processes Z_ι with ι belonging to some index set I can be considered as a process in the product space (R^d, I).

A central concept in the study of random fields is that of homogeneity or stationarity. A random field is homogeneous or (second-order) stationary if E[Z(x)²] is finite ∀ x and

• m(x) ≡ m is independent of x ∈ R^d

• R(x, y) solely depends on the difference x − y

Thus we may consider R(h) = Cov(Z(x), Z(x+h)) = E[Z(x) Z(x+h)] − m², h ∈ R^d,

and denote R the covariance function of Z. In this case, the following correspondence exists between the covariance and correlation function, respectively:

c(h) = R(h)/R(o)

i.e. c(h) ∝ R(h). For this reason, the attention is confined to either c or R. Two stationary random fields Z₁, Z₂ are stationarily correlated if their cross covariance function R₁₂(x, y) depends on the difference x − y only. The two random fields are uncorrelated if R₁₂ vanishes identically.

An interesting special class of homogeneous random fields that often arise in practice is the class of isotropic fields. These are characterized by the property that the covariance function R depends only on the length ∥h∥ of the vector h:

R(h) = R(∥h∥) .

In many applications, random fields are considered as functions of “time” and “space”. In this case, the parameter set is most conveniently written as (t,x) with t ∈ R₊ and x ∈ R^d. Such processes are often homogeneous in (t, x) and isotropic in x in the sense that

E[Z(t, x)Z(t + h, x + y)] = R(h, ∥y∥) ,

where R is a function from R² into R. In such a situation, the covariance function can be written as

R(t, ∥x∥) = ∫_R ∫_λ=0^∞ e^itu H_d (λ ∥x∥) dG(u, λ),

where

H_d(r) = (2/r)^{(d – 2)/2} Γ(d/2) J_{(d – 2)/2} (r)

and J_m is the Bessel function of the first kind of order m and G is a multiple of a distribution function on the half plane {(λ,u)|λ ≥ 0,u ∈ R}.

Philosophizing Forgetful Functors: This Functor Forgets only Properties: Namely, the Property of Being Abelian + This Functor Forgets Both Structure (the generating set) and Properties (the property of being a free group).

A forgetful functor is a functor which is defined by ‘forgetting’ something. For example, the forgetful functor from Grp to Set forgets the group structure of a group, remembering only the underlying set.

In common parlance, the term ‘forgetful functor’ has no precise definition, being simply used whenever a functor is obviously defined by forgetting something. Many forgetful functors of this sort have left or right adjoints (and many are actually monadic or comonadic), leading to the paradigmatic adjunction “free ⊣⊣ forgetful.”

On the other hand, from the perspective of stuff, structure, property, every functor is regarded as a forgetful functor and classified by how much it forgets (namely, stuff, structure, or properties). From this perspective, the forgetful functor from GrpGrp to SetSet forgets the structure of a group and the property of admitting a group structure, as usual; but its left adjoint (the free group functor) is also forgetful: if you identify SetSet with the category of free groups with specified generators, then it forgets the structure of a set of free generators and the property of being free.

There are many cases in which we want to say that one kind of mathematical object has more structure than another kind of mathematical object. For instance, a topological space has more structure than a set. A Lie group has more structure than a smooth manifold. A ring has more structure than a group. And so on. In each of these cases, there is a sense in which the first sort of object – say, a topological space – results by taking an instance of the second sort – say, a set – and adding something more – in this case, a topology. In other cases, we want to say that two different kinds of mathematical objects have the same amount of structure. For instance, given a Boolean algebra, one can construct a special kind of topological space, known as a Stone space, from which one can uniquely reconstruct the original Boolean algebra; and vice-versa.

These sorts of relationships between mathematical objects are naturally captured in the language of category theory, via the notion of a forgetful functor. For instance, there is a functor F : Top → Set from the category Top, whose objects are topological spaces and whose arrows are continuous maps, to the category Set, whose objects are sets and whose arrows are functions. This functor takes every topological space to its underlying set, and it takes every continuous function to its underlying function. We say this functor is forgetful because, intuitively speaking, it forgets something: namely the choice of topology on a given set.

The idea of a forgetful functor is made precise by a classification of functors due to Baez et al. (2004). This requires some machinery. A functor F : C → D is said to be full if for every pair of objects A, B of C, the map F : hom(A, B) → hom(F (A), F (B)) induced by F is surjective, where hom(A, B) is the collection of arrows from A to B. Likewise, F is faithful if this induced map is injective for every such pair of objects. Finally, a functor is essentially surjective if for every object X of D, there exists some object A of C such that F(A) is isomorphic to X.

If a functor is full, faithful, and essentially surjective, we will say that it forgets nothing. A functor F : C → D is full, faithful, and essentially surjective if and only if it is essentially invertible, i.e., there exists a functor G : D → C such that G ◦ F : C → C is naturally isomorphic to 1_C, the identity functor on C, and F ◦ G : D → D is naturally isomorphic to 1_D. (Note, then, that G is also essentially invertible, and thus G also forgets nothing.) This means that for each object A of C, there is an isomorphism η_A : G ◦ F (A) → A such that for any arrow f : A → B in C, η_B ◦ G ◦ F(f) = f ◦ η_A, and similarly for every object of D. When two categories are related by a functor that forgets nothing, we say the categories are equivalent and that the pair F, G realizes an equivalence of categories.

Conversely, any functor that fails to be full, faithful, and essentially surjective forgets something. But functors can forget in different ways. A functor F : C → D forgets structure if it is not full; properties if it is not essentially surjective; and stuff if it is not faithful. Of course, “structure”, “property”, and “stuff” are technical terms in this context. But they are intended to capture our intuitive ideas about what it means for one kind of object to have more structure (resp., properties, stuff) than another. We can see this by considering some examples.

For instance, the functor F : Top → Set described above is faithful and essentially surjective, but not full, because not every function is continuous. So this functor forgets only structure – which is just the verdict we expected. Likewise, there is a functor G : AbGrp → Grp from the category AbGrp whose objects are Abelian groups and whose arrows are group homomorphisms to the category Grp whose objects are (arbitrary) groups and whose arrows are group homomorphisms. This functor acts as the identity on the objects and arrows of AbGrp. It is full and faithful, but not essentially surjective because not every group is Abelian. So this functor forgets only properties: namely, the property of being Abelian. Finally, consider the unique functor H : Set → 1, where 1 is the category with one object and one arrow. This functor is full and essentially surjective, but it is not faithful, so it forgets only stuff – namely all of the elements of the sets, since we may think of 1 as the category whose only object is the empty set, which has exactly one automorphism.

In what follows, we will say that one sort of object has more structure (resp. properties, stuff) than another if there is a functor from the first category to the second that forgets structure (resp. properties, stuff). It is important to note, however, that comparisons of this sort must be relativized to a choice of functor. In many cases, there is an obvious functor to choose – i.e., a functor that naturally captures the standard of comparison in question. But there may be other ways of comparing mathematical objects that yield different verdicts.

For instance, there is a natural sense in which groups have more structure than sets, since any group may be thought of as a set of elements with some additional structure. This relationship is captured by a forgetful functor F : Grp → Set that takes groups to their underlying sets and group homomorphisms to their underlying functions. But any set also uniquely determines a group, known as the free group generated by that set; likewise, functions generate group homomorphisms between free groups. This relationship is captured by a different functor, G : Set → Grp, that takes every set to the free group generated by it and every function to the corresponding group homomorphism. This functor forgets both structure (the generating set) and properties (the property of being a free group). So there is a sense in which sets may be construed to have more structure than groups.