Approximation error due to the truncation

It is desired to estimate the errors of the approximations (X_ψ^A(x)−X_ψ(x) and X_ψ^A,M(x)−

X_ψ^A(x)) for a given x ∈ R^d, i.e. to calculate an upper bound for the absolute values of these differences. Because X_ψ(x), X_ψ^A(x) and X_ψ^A,M(x) are symmetric α-stable (SαS) random variables, their differences are SαS random variables, too. Therefore, as a mea-sure of the approximation errors, the scale parameters of their distributions (||X_ψ^A(x) − X_ψ(x)||_α and ||X_ψ^A,M(x) − X_ψ^A(x)||_α) have to be estimated. According to equation (3.4.4) in [10] (p. 122), such a norm for an α-stable integral is defined by ||R

R^df (ξ)W_α(dξ)||_α = R

R^d|f (ξ)|^αdξ
1/α

(this means that ||R

R^df (ξ)Wα(dξ)||^α_α =R

R^d|f (ξ)|^αdξ). Therefore, the scale parameter can be estimated by an estimation of this non-random integral.

Lemma 3.1. The scale parameter of the approximation error due to the truncation can be bounded by the following integral:

 

X_ψ(x) − X_ψ^A(x)   

α α ≤

Z

R^d\[−A,A]^d

ei<x,ξ>− 1 

αψ(ξ)^−αH−qdξ

Proof.

The error of approximation by truncation of the domain of integration can be estimated with the help of Lemma 2.1, Proposition 2.2 and Definition 2.3 as follows:

Theorem 3.2. There are positive real numbers C and δ⁰, which depend on the approx-imated random field X_ψ, (i.e. on the function ψ and on the parameters α and H), but not on A, so that

This estimation is valid for all dimensions d ≥ 2 and for all E^T-homogeneous functions ψ, even if ψ can’t be represented in the special form of (2.2) (in the more general case -if ψ is not in this special form - the value of λ_d in this estimation is the largest real part of eigenvalues of E).

Proof. According to Lemma 3.1, 

Using the fact that 

ei<x,ξ>− 1

In the polar representation depending on E^T (see chapter 2), ξ can be represented with g(A) := eC₄·A^1/λd−δ. According to Proposition 2.2, there exists a unique finite Radon measure σ on S₀ such that

This estimation has the drawback that it contains unknown constants (C and δ⁰), so that it can’t be used to calculate a concrete value for the error. However, it shows that

the error of estimation decreases with growing values of the parameter A, and converges to zero if A → ∞, and it gives an approximate estimate for the rate of convergence.

In a special case, if ψ is a ρ-norm, then an estimation can be found, which only contains calculable numbers instead of the unknown constants of the previous estimation, and thereby allows to easily calculate an upper bound for ||X_ψ(x)−X_ψ^A(x)||^α_α. This calculation uses the (d − 1-dimensional) Lebesgue measure of the || · ||∞-unit-sphere, i.e. of {x ∈ R^d : ||x||∞ = 1}. Therefore this integral is first considered in a lemma:

Lemma 3.3. The d − 1-dimensional Lebesgue measure of S_∞ := {x ∈ R^d: ||x||_∞ = 1}

is Z

S∞

1 dx = d · 2^d. (3.5)

Proof. The set {x ∈ R^d : ||x||∞ ≤ 1} = [−1, 1]^d is a d-dimensional hypercube whose edges have the length 2. Its surface {x ∈ R^d: ||x||∞= 1} consists of 2d d − 1-dimensional hypercubes whose edges have length 2, too. Each of this d − 1-dimensional hypercubes has a Lebesgue-measure of 2^d−1. Therefore the whole surface {x ∈ R^d : ||x||∞ = 1}

(1-sphere according to the || · ||∞ - norm) has the measure 2d · 2^d−1 = d · 2^d.

Remark 3.4. In the special cases d = 2 and d = 3, which are particularly relevant for real data and for simulation, the measure of S∞ is 2 · 2² = 8 if d = 2, and 3 · 2³ = 24 if d = 3. This can also be confirmed by a visualisation of this set in these cases: In the two-dimensional case, S∞ is the perimeter of the square [−1, 1]² and therefore consists of four edges of length 2 each, their sum being 8, and in the three-dimensional case, it is the surface of the cube [−1, 1]³, consisting of six squares with side length 2. Each of this squares has an area of 4, thus their total sum is 24.

This values of the Lebesgue measure of S∞ are used in the following estimation of the approximation error in the case of ψ being a norm:

Theorem 3.5. If ψ = || · ||_ρ with 1 ≤ ρ (i.e. if in the representation of equation (2.2) the parameters are choosen as λ₁ = . . . = λ_d = C₁ = . . . = C_d = 1, and the vectors θ₁, . . . , θ_d are the standard unit vectors of the R^d), then

 

X_ψ(x) − X_ψ^A(x)   

α

α≤ 2^α+d· d

αH · A^−αH which implies

 

X_ψ(x) − X_ψ^A(x)  

α ≤ 2 · 2^d· d αH

_α¹

· A^−H.

Proof. According to Lemma 3.1,

This integral is estimated as follows:

Z

Remark 3.7. As the exponent of A is negative (in Theorem 3.5), the estimated error of truncation converges to zero for A → ∞. For any ε > 0, a value for A can easily be

Not only in the special case of a norm, but also in the general case of an E^T-homogeneous function ψ in the representation of equation (2.2), an estimation of the error of approx-imation can be found without using the inequalities of Lemma 2.1 or Definition 2.5. Be the E^T-homogeneous function ψ defined as in equation (2.2), i.e.

ψ(x) =

d

X

k=1

C_k|< x, θ_k >|^ρ/λk

!^1/ρ

with linear independent vectors θ₁, . . . , θ_d ∈ R^d. Be assumed that 0 < λ₁ ≤ . . . ≤ λ_d (this is no limitation, as the λ_i have to be positive anyway, and the indices can be chosen according to the ordering of the λ_i), and that A > 1.

Without limitation of the possible functions ψ, we may assume that the vectors θk, 1 ≤ k ≤ d in this representation of ψ are of length 1 (i.e. ||θ_k||₂ = 1), as we show in the following lemma:

Lemma 3.8. If not ||θk||2 = 1 for all 1 ≤ k ≤ d, then ψ can be transformed in order to have this property, i.e. ψ can always be represented in such a form, by choosing C₁, . . . , C_d accordingly. Therefore it can be assumed that ||θ_k||₂ = 1 for all 1 ≤ k ≤ d.

Proof. Be ˜θ_k:= _||θ^θk

k||₂, so that θ_k = ||θ_k||₂· ˜θ_k and ||˜θ_k||₂ = 1. Then ψ(x) =

d

X

k=1

C_k|< x, θ_k >|^ρ/λk

!^1/ρ

=

d

X

k=1

C_k 

< x, ||θ_k||₂· ˜θ_k >

 

ρ/λk

!^1/ρ

=

d

X

k=1

C_k||θ_k||^ρ/λ₂ ^k· 

< x, ˜θ_k >

  

ρ/λ_k!1/ρ

=

d

X

k=1

C˜_k· 

< x, ˜θ_k >

  

ρ/λk

!^1/ρ

with ˜C_k := C_k||θ_k||^ρ/λ₂ ^k, and || ˜θ_k||₂ = 1 for all k ∈ {1, . . . , d}.

Before estimating an upper bound for the approximation error, we estimate a lower bound for the values of the function ψ (Since now, be C_min := min{C₁, . . . , C_d} and C_max := max{C₁, . . . , C_d}).

Lemma 3.9. Be C_min := min{C₁, . . . , C_d}, C_max := max{C₁, . . . , C_d} (Remark: The only difference between the two cases is in the exponent of ||ξ||₂.) Proof. From the assumption λ₁ ≤ . . . ≤ λ_d follows _λ^ρ If ||ξ||₂ ≤ 1, then ψ(ξ) can be estimated analogously:

ψ(ξ) ≥ C_min^1/ρ

If ||ξ||∞ ≥ A, then the first case is relevant, because ||ξ||₂ ≥ ||ξ||∞≥ A ≥ 1.

Using these lemmas, the error of truncation can be estimated as follows:

Theorem 3.10. Be the E^T-homogeneous function ψ defined as in equation (2.2), i.e.

ψ(x) = (this is no limitation, as the λ_i have to be positive anyway, and the indices can be chosen according to the ordering of the λ_i), that A > 1 (which we may assume as we are only interested in error estimates for large values of A), and that ||θ_k||₂ = 1, 1 ≤ k ≤ d (compare Lemma 3.8). Then



Proof. According to Lemma 3.1, 

which is not more than Z

=2^α· C

Remark 3.11. Because the exponent of A is assumed to be negative, the error estimate converges to zero for A → ∞.

dθ_ξ is dependent only on the param-eters of the approximated random field X (like H and the paramparam-eters of the function ψ), but independent of the approximation parameter A. As no estimation in a simpler form was found for if d ≥ 3, it has to be approximated numerically in each concrete case (which can be done in a sufficient precision on a modern computer within a fraction of a second). For the case d = 2, the integral can also be estimated as follows (however, the numerical approximation of the integral gives a much better result also in this case):

Corollary 3.13. In the two-dimensional case (if d = 2), the Integral ˜I can be estimated by ˜I ≤ 2π · 2^αH+qλ1 · (1 − | < θ₁, θ₂ > |²)

−αH−q

2λ1 , so that the inequality

||X_ψ(x) − X_ψ^A(x)||^α_α follows (provided that the exponent of A is negative).

Proof. Be α₁ the angle between the vectors θ_ξ and θ₁, α₂ the angle between the vectors θ_ξ and θ₂, and α₁₂ the angle between θ₁ and θ₂. Then (using the fact that ||θ₁||₂ =

||θ2||2 = ||θξ||2 = 1) it follows that

| < θ_ξ, θ₁ > | = | cos(α₁)|, | < θ_ξ, θ₂ > | = | cos(α₂)|, | < θ₁, θ₂ > | = | cos(α₁₂)|

Then

|< θ_ξ, θ₁ >|^ρ/λ1 + |< θ_ξ, θ₂ >|^ρ/λ11/ρ

≥

max{|< θ_ξ, θ₁ >|^ρ/λ1, |< θ_ξ, θ₂ >|^ρ/λ1}1/ρ

=

(max{|< θ_ξ, θ₁ >| , |< θ_ξ, θ₂ >|})^ρ/λ11/ρ

= (max{|cos(α1)| , |cos(α2)|})^1/λ1

= (max{|cos(α₁)| , |cos(α₁+ α₁₂)|})^1/λ1

≥ min

β∈R

n

(max{|cos(β)| , |cos(β + α₁₂)|})^1/λ1o

= min

β∈R

n

(max{|cos(β − π/2)| , |cos(β + α₁₂− π/2)|})^1/λ1o

= min

β∈R

n

(max{|sin(β)| , |sin(β + α₁₂)|})^1/λ1o

= (max{|sin(−α₁₂/2)| , |sin(α₁₂/2)|})^1/λ1

=  

sinα₁₂ 2

  

1

λ1 ≥ | sin(α₁₂)|

2

_λ1¹

= 2^−λ11 | sin(α₁₂)|^λ11

=2^−λ11 ·p

1 − cos²(α₁₂)_λ1¹

= 2^−λ11 · 1 − | < θ₁, θ₂ > |²
_2λ1¹

which implies

I =˜ Z

S2

|< θ_ξ, θ₁ >|^λ1ρ + |< θ_ξ, θ₂ >|^λ1ρ ^−αH−q_ρ dθ_ξ

≤ Z

S2



2^−λ11 · 1 − | < θ₁, θ₂ > |²
_2λ1¹ ^−αH−q dθ_ξ

≤ 2^αH+qλ1 · 1 − | < θ₁, θ₂ > |^2
−αH−q_2λ1

· Z

S2

1 dθ_ξ

≤ 2^αH+qλ1 · 1 − | < θ₁, θ₂ > |^2
−αH−q_2λ1

· 2π

For a numerical approximation of the integral ˜I, be ν = _λ^ρ

1 and γ = ^−Hα−q_ρ . Then I =˜

Z

S2

(|< θξ, θ1 >|^ν + |< θξ, θ2 >|^ν)^γ dθξ

which can be approximated (with n support points) by the sum S˜n := 2π

n ·

n

X

k=1

< (cos(ζk), sin(ζk))^T, θ1 >



ν +

< (cos(ζk), sin(ζk))^T, θ2 >



ν
γ

with ζ_k := ^2π_n · k.

Table 3.1 shows approximated values of ˜I for the parameter values ν = 1.0 and γ = −3.0.

One of the vectors which are parameters to the integral is set to θ₁ = (1, 0)^T, and the other is θ₂ = (cos(v₂), sin(v₂))^T for different values of v₂ (v₂ ∈ {₂₀¹ π, . . . ,¹⁹₂₀π}). Thereby, I is the estimation of ˜ˆ I according to Corollary 3.13, while ˜S_4k and ˜S_60k are the numerical approximations of ˜I with 4000 and 60000 points of support.

From this table, several conclusions can be drawn (for the used sample of parameter values):

(a) The difference of the approximations ˜S_4k and ˜S_60k are relatively small. Thus it may be assumed that only a few thousand (e.g. 4000) points of support are necessary to obtain a sufficient approximation of the integral for most purposes.

(b) The estimation of ˜I in Corollary 3.13 overestimates the integral by a large factor.

(c) The values of ˜S_60k · sin(v₂)² didn’t show large differences for different values of v₂. They were all in the interval [3.0, 3.25]. Therefore, the term 3.25 · sin(v₂)⁻² seems to be a relatively good approximation for ˜I with the parameters ν = 1.0 and γ = −3.0. A comparison with similar tables of numerical approximations for other values of γ and ν suggests that, generally, ˜I may be estimated quite well by I ≤ C˜ _γ,ν · sin(v₂)^γν+1, where C_γ,ν is a number which depends on γ and ν (but not on v₂).

v₂ Iˆ S˜_4k S˜_60k S˜_60k· sin(v₂)² 0.05 · π 13130.2360 123.2474 123.2392 3.0159 0.10 · π 1703.4240 31.9048 31.9043 3.0466 0.15 · π 537.1920 14.9577 14.9576 3.0829 0.20 · π 247.5220 9.0309 9.0309 3.1201 0.25 · π 142.1723 6.3106 6.3106 3.1553 0.30 · π 94.9286 4.8684 4.8684 3.1864 0.35 · π 71.0603 4.0458 4.0458 3.2120 0.40 · π 58.4320 3.5720 3.5720 3.2309 0.45 · π 52.1687 3.3239 3.3239 3.2425 0.50 · π 50.2655 3.2465 3.2465 3.2465 0.55 · π 52.1687 3.3239 3.3239 3.2425 0.60 · π 58.4320 3.5720 3.5720 3.2309 0.65 · π 71.0603 4.0458 4.0458 3.2120 0.70 · π 94.9286 4.8684 4.8684 3.1864 0.75 · π 142.1723 6.3106 6.3106 3.1553 0.80 · π 247.5220 9.0309 9.0309 3.1201 0.85 · π 537.1920 14.9577 14.9576 3.0829 0.90 · π 1703.4240 31.9048 31.9043 3.0466 0.95 · π 13130.2360 123.2474 123.2392 3.0159

Table 3.1: Estimation and numerical approximation of ˜I with ν = 1.0 and γ = −3.0, where v₂ is the angle between the vectors θ₁ and θ₂. ˆI is the estimation of ˜I according to Corollary 3.13, while ˜S_4k and ˜S_60k are the numerical approxima-tions of ˜I with 4000 and 60000 points of support.

In document Simulation and estimation of operator scaling stable random fields (Page 22-34)