RESUMEN DEL CAPITULO

a sequence with uniform distribution on (0, 1) (cf. [59]) and the inverse transform method using F⁻¹_Y generates a time series {Yt, t = 1, 2, . . .} with the desired marginal distribution FY (cf. Section 2.2 and Figure 2.3).

In the following some properties of the extended ARTA process are established that relate statistical properties of the base process Zt to properties of the ARTA process Yt. For these observations we assume that the base process is constructed such that the {Zt; t= 1, 2, . . .} have standard normal distribution as mentioned above. After that it is explained how the base process can be constructed to fulfill these requirements.

4.1. Properties of Extended ARTA Processes

To use the extended ARTA processes from Definition 4.1 for fitting traffic data a re-lation between the autocorrere-lation structure of the extended ARTA process and the autocorrelation structure of the base process has to be established. For given autocorre-lations ˆρ= (ˆρ1,ˆρ2, . . . ,ˆρr) that have been estimated from a trace and a given marginal distribution FYone has to construct an ARMA(p, q) base process with autocorrelations ρ = (ρ1, ρ₂, . . . , ρr) such that the extended ARTA process has autocorrelations ˆρ.

[47] introduced such a relation for ARTA processes with an AR(p) base process (cf.

Equation 2.6) and proved that an autocorrelation ρh{<,=, >} 0 for the base process implies an autocorrelation ˆρh{<,=, >} 0 for the ARTA process. In the following it is shown that the same properties hold for extended ARTA models with an ARMA(p, q) base process as well.

The autocorrelation of an ARTA process and the base process are related by (cf.

Equation 2.6) Corr[Yt, Yt+h]= Corrh

F_Y⁻¹(Φ(Zt)), F_Y⁻¹(Φ(Zt+h))i, regardless of whether the base process is AR(p) or ARMA(p, q). Since (cf. [47])

Corr[Yt, Yt+h]= E[YtYt+h] − (E[Y])²

Var[Y] (4.2)

and E[Y] and Var[Y] are determined by the marginal distribution FY the important term for establishing the relation between base process and (extended) ARTA process is the joint moment E[YtY_t+h]. Recall the requirement for the {Zt; t= 1, 2, . . .} resulting from the base process to have standard normal distribution. Then any two elements of the time series (Zt, Zt+h) have a standard bivariate normal distribution with density function ϕρh and correlation ρh = Corr[Zt, Zt+h]. Now we have

E[YtYt+h] = E[F_Y⁻¹(Φ(Zt))F⁻¹_Y (Φ(Zt+h))] (4.3)

= Z ∞

−∞

Z ∞

−∞

F⁻¹_Y (Φ(zt))F_Y⁻¹(Φ(zt+h))ϕρh(zt, zt+h)dztdzt+h.

Observe from Equations 4.2 and 4.3 that the ARTA correlation ˆρh = Corr[Yt, Yt+h] is a function of the base process autocorrelation ρhthat appears in ϕρh(zt, zt+h). This function is denoted by ω(ρh) = ˆρh. Note, that it is possible to compute a ˆρh from a given base process autocorrelation ρh using Equation 4.3, but for actually fitting ARTA processes it is necessary to determine the base process autocorrelation ρh for a given (i.e. estimated from a trace) ˆρh. This is to be done numerically by a search algorithm and [47] established several properties of ω(ρh) that allow one to use such algorithms for ARTA processes with an AR(p) base process. The crucial requirement

CHAPTER 4. EXTENDED ARTA PROCESSES

for proving the properties of ω(ρh) for ARTA processes is the fact, that the {Zt; t = 1, 2, . . .} generated by the base process have standard normal distribution. Since this is required for the ARMA(p, q) base process of extended ARTA models as well, the proofs from [47] still work for extended ARTA processes. In the following the main results from [47] that are necessary for applying a search algorithm are summarized.

Proposition 4.1. For any extended ARTA process with marginal distribution F_Y and ARMA(p, q) base process we have that

• ω(0)= 0,

• ρh≤0 ⇒ ω(ρh) ≤ 0,

• ρh≥0 ⇒ ω(ρh) ≥ 0.

Proof. This follows from the proof of [47, Proposition 1].

• ρh= 0 implies that Zt and Zt+hare independent. Then E[YtYt+h] = Eh

F_Y⁻¹(Φ(Zt))F⁻¹Y (Φ(Zt+h))i

= Eh

F_Y⁻¹(Φ(Zt))i Eh

F⁻_Y¹(Φ(Zt+h))i = E[Yt]E[Yt+h]= E[Y]² and

Corr[Yt, Yt+h]= E[YtYt+h] − (E[Y])² Var[Y] = 0.

• According to [153] ρh≤(≥) 0 implies that Cov[g1(Zt, Zt+h), g2(Zt, Zt+h)] ≤ (≥) 0 for all nondecreasing functions g1 and g2. Since F_Y⁻¹(Φ(·)) is a nondecreasing function, the result immediately follows by setting g1(Zt, Zt+h) = F_Y⁻¹(Φ(Zt)) and g2(Zt, Zt+h)= F⁻¹_Y (Φ(Zt+h)).

 Theorem 4.1. For any extended ARTA process with marginal distribution F_Y and ARMA(p, q) base process the function ω(ρh) is nondecreasing for −1 ≤ ρh≤1.

Proof. For positive correlations 0 ≤ ρh ≤ 1 this follows from [153, Theorem 5.3.10], which states that for two normal variables Z1 and Z2 with correlation ρh we have Corr[g(Z1), g(Z2)] is nondecreasing in ρh for all functions g(·). Setting Yt = g(Zt) = F_Y⁻¹(Φ(Zt)) and Yt+h = g(Zt+h) = F⁻¹_Y (Φ(Zt+h)) we obtain that ω(ρh) = Corr[Yt, Yt+h] is nondecreasing for 0 ≤ ρh ≤ 1. A similar result has been obtained for negative

correlations −1 ≤ ρh ≤0 in [47, Theorem 1]. 

Theorem 4.2. For any extended ARTA process with marginal distribution FY and ARMA(p, q) base process the function ω(ρh) is continuous, if there exists  > 0 such that E[|YtY_t+h|^1+] < ∞ for all −1 ≤ ρh ≤1.

Proof. In [47] Theorem 4.2 has been proven for ARTA processes with AR(p) base process. We will summarize the basic ideas and show that the proof also holds for an

4.1. PROPERTIES OF EXTENDED ARTA PROCESSES

ARMA(p, q) base process.

First observe, that requiring E[|YtYt+h|^1+] < ∞ for all −1 ≤ ρ ≤ 1 is equivalent to Z ∞

−∞

Z ∞

−∞

ρ∈[−1,1]sup

n|F⁻_Y¹[Φ(z1)]F_Y⁻¹[Φ(z2)]|^1+ ×ϕ_ρ(z1, z₂)o

dz₁dz₂< ∞. (4.4) Now, let Z1and Z3be iid standard normal random variables and assume that ρ ∈ [−1, 1]

is fixed. Furthermore, let {ρn}^∞_n=1 be a sequence with ρn ∈ [−1, 1], n = 1, 2, . . . and ρn→ρas n → ∞. Define

Z_1n ≡ Z₁, Z_2n≡ρnZ₁+ q 1 − ρ²n

!

Z₃, Z₂≡ρZ₁+ q 1 − ρ²

! Z₃. Observe, that Z1and Z2are standard normal random variables with correlation ρ and Z_1nand Z2nare standard normal random variables with correlation ρn[90]. Now, let

Y_1n≡ F_Y⁻¹[Φ(Z1n)] and Y_2n≡ F⁻¹_Y [Φ(Z2n)]

and define

h z₁ z₂

!

≡ F⁻¹_Y [Φ(z1)]F_Y⁻¹[Φ(z2)].

For fixed z2the function h is monotone in z1 and vice versa. Therefore h has only a countable number of discontinuities and from

Z_1n Z_2n

!

⇒ Z₁ Z₂

!

as n → ∞

we get by application of the mapping theorem (cf. [27, Theorem 29.2]) the following convergence in distribution

h Z_1n Z_2n

!

⇒ h Z₁ Z₂

!

as n → ∞.

For Y1≡ F⁻_Y¹[Φ(Z1)] and Y2 ≡ F_Y⁻¹[Φ(Z2)] this is equivalent to

Y_1nY_2n⇒ Y₁Y₂ as n → ∞. (4.5)

It then follows from Equations 4.4 and 4.5 and [27, Theorem 25.12] that E[Y1nY_2n] → E[Y1Y₂] as n → ∞, implying that ω(ρ) is a continuous function since ω(ρn) → ω(ρ) as n → ∞. Setting Z1= Zt, Z₂= Zt+h, Y₁ = Yt, Y₂= Yt+hand ρ= ρhproves Theorem 4.2.

Observe, that the proof holds for any Zithat have standard normal distribution. There-fore, the presented proof from [47] is valid for an ARTA process with ARMA base process as well, as long as the observations generated by the ARMA process have standard normal distribution, which we required in Definition 4.1.  Since the autocorrelation structure of the extended ARTA process is a nondecreasing and continuous function according to Theorems 4.1 and 4.2 any search procedure can be applied to find the base process autocorrelation that results in the desired extended ARTA autocorrelation. From Proposition 4.1 the initial bounds can be obtained, i.e.

for a positive ARTA autocorrelation only base process autocorrelations 0 ≤ ρh ≤ 1

CHAPTER 4. EXTENDED ARTA PROCESSES

have to be considered and for a negative ARTA autocorrelation only −1 ≤ ρh ≤ 0 has to be treated.

Aside from the properties above, which are all related to the autocorrelation of an extended ARTA model, several other characteristics are of interest to describe a stochastic process or to assess its fitting quality. The probability density function, the cumulative distribution function and moments, variance, etc. are all completely deter-mined by the marginal distribution of the extended ARTA model and thus, cannot be given in general. Appendix B contains a brief description of several distributions that can be used as marginal distribution for an extended ARTA model. Another measure of a stochastic process that is related to the dependence of consecutive arrivals are joint moments, which have for example been successfully used for MAP fitting (cf. Sec-tion 2.3.6). For extended ARTA models EquaSec-tion 4.3 can be generalized to compute arbitrary joint moments, i.e.

E[Yt^kY_t^l_+h] = E

F_Y⁻¹(Φ(Zt))k

F⁻¹_Y (Φ(Zt+h))l

(4.6)

= Z ∞

−∞

Z ∞

−∞

F_Y⁻¹(Φ(zt))k

F_Y⁻¹(Φ(zt+h))l

ϕ_ρh(zt, zt+h)dztdz_t+h. Recall from Definition 4.1 that we required the ARMA(p, q) base process to be sta-tionary, i.e. that Zt are invariant under time shifts. This of course implies, that the (extended) ARTA process is stationary as well, because the Yt are only a transforma-tion of the Zt.

The previous observations on the relation of the autocorrelation structure of the base process and the autocorrelation of the extended ARTA process can be employed to construct the base process, which will be described in the following section.