Prediction of air pollutants PM₁₀ by ARBX(1) processes

Abstract

This work adopts a Banach-valued time series framework for component-wise estimation and prediction, from temporal correlated functional data, in presence of exogenous variables. The strong-consistency of the proposed functional estimator and associated plug-in predictor is formulated. The simulation study undertaken illustrates their large-sample size properties. Air pollutants PM₁₀ curve forecasting, in the Haute-Normandie region (France), is addressed by implementation of the functional time series approach presented.

Appendix

It is well-known that Besov spaces, $\left\{ \left( {\mathcal{B}}_{p,q}^{r}, \left\| \cdot \right\| _{p,q}^{r} \right) , \ r \in {\mathbb{R}},~1 \le p,q \le \infty \right\}$, and their norms can be characterized in terms of the wavelet transform (see, e.g., Triebel 1983). Specifically, for every $f\in {\mathcal{B}}_{p,q}^{r}$,

$$\begin{aligned} \left\| f \right\| _{p,q}^{r} \equiv \left\| \varphi _{J} *f \right\| _p + \left[ \sum _{j=J}^{\infty } \left( 2^{j r} \left\| \psi _j*f \right\| _p \right) ^{q} \right] ^{1/q} < \infty , \end{aligned}$$

(31)

where $\varphi$ and $\psi$ denote the father and mother wavelets, whose translations and dilations provide a multiresolution analysis of a suitable space of square-integrable functions. Particularly, consider the space $L^{2}([0,1])$, and its orthogonal decomposition from an $\left( \lceil r \rceil + 1 \right)$-regular Multiresolution Analysis, induced by an orthogonal basis of wavelets, for certain $r>0$. Then, father and mother wavelets belong to ${\mathcal{C}}^{\left( \lceil r \rceil + 1 \right) } ([0,1])$. For every $f\in L^{2}([0,1])$,

$$\begin{aligned} f(t) = \sum _{k = 0}^{ 2^J - 1} \alpha _{J,k}^{f} \varphi _{J,k} (t) + \sum _{j=J}^{K} \sum _{k=0}^{2^j - 1} \beta _{j,k}^{f} \psi _{j,k} (t), \quad t \in [0,1], \end{aligned}$$

(32)

where J is such that $2^J \ge 2^{\left( \lceil r \rceil + 1 \right) }$, and for $k=0,\ldots , 2^{j-1}, j=J,\ldots ,K$,

$$\begin{aligned} \alpha _{J,k}^{f}& = \int _{{\mathbb{R}}} f(x) \overline{\varphi _{J,k} (x)}dx, \quad \beta _{j,k}^{f} = \int _{{\mathbb{R}}} f(x) \overline{\psi _{j,k} (x)}dx \end{aligned}$$

(see Daubechies 1992). Here, K is the truncation parameter defining the last (or highest) resolution level considered in the finite-dimensional wavelet approximation (32).

As commented before, the following function spaces have been considered:

$$\begin{aligned} {\overline{B}}& = \left[ {\mathcal{B}}_{\infty ,\infty }^{0} ([0,1]) \right] ^{b+1}; \\ \widetilde{{\overline{H}}}& = \left[ H_{2}^{-\beta } ([0,1]) \right] ^{b+1}=\left[ {\mathcal{B}}_{2,2}^{-\beta }([0,1])\right] ^{b+1} \\ {\overline{H}}& = \left[ L^2 ([0,1])\right] ^{b+1}; \\ {\mathcal{H}}({\overline{X}})& = \prod _{i=1}^{b+1}H^{\gamma _{i}}_{2}([0,1])=\prod _{i=1}^{b+1}{\mathcal{B}}_{2,2}^{\gamma _{i} }([0,1]) \\ {\overline{B}}^{\star }& = \left[ {\mathcal{B}}_{1,1}^{0} ([0,1]) \right] ^{b+1}; \\ \widetilde{{\overline{H}}}^{\star }& = \left[ H_{2}^{\beta } ([0,1]) \right] ^{b+1}=\left[ {\mathcal{B}}_{2,2}^{\beta }([0,1])\right] ^{b+1} \\ {\overline{H}}^{\star }& = \left[ L^2 ([0,1])\right] ^{b+1}; \\ {[}{\mathcal{H}}({\overline{X}})]^{\star }& = \prod _{i=1}^{b+1}H^{-\gamma _{i}}_{2}([0,1])=\prod _{i=1}^{b+1}{\mathcal{B}}_{2,2}^{-\gamma _{i} }([0,1]), \end{aligned}$$

(33)

where the parameters $\{\gamma _{i}\}_{i=1,\ldots ,b+1}$ reflect the second-order local regularity of the functional random components of ${\overline{X}}=\{{\overline{X}}_{n},\ n\in {\mathbb{Z}}\}$ in Eq. (5). From embedding theorems between Besov spaces, the following continuous inclusions hold (see Triebel 1983):

$$\begin{aligned} {\mathcal{H}}({\overline{X}})\hookrightarrow \widetilde{{\overline{H}}}^{*} \hookrightarrow {\overline{B}}^{*} \hookrightarrow {\overline{H}} \hookrightarrow {\overline{B}} \hookrightarrow \widetilde{{\overline{H}}}\hookrightarrow [{\mathcal{H}}({\overline{X}})]^{\star }, \end{aligned}$$

(34)

for $\gamma _{i}>2\beta >1, i=1,\ldots ,b+1$. Thus, Assumptions A4–A5 are satisfied. The ${\overline{B}}$ and ${\overline{B}}^{\star }$ norms are then computed from the following identities: For every ${\overline{f}} = \left( f; f_{1},\ldots ,f_{b} \right) ,\ {\overline{g}} = \left( g; g_{1},\ldots , g_{b} \right) \in {\overline{B}}\subset \widetilde{{\overline{H}}}$,

$$\begin{aligned} \left\| {\overline{f}} \right\| _{{\overline{B}}}& = \sup _{j\ge J}\sup _{k =0,\ldots , 2^{j}-1} \sup \left( \left| \alpha _{J,k}^{f}\right| ,\left| \beta _{j,k}^{f}\right| , \sup _{i =1,\ldots , b}\left| \alpha _{J,k}^{f_{i}}\right| , \right. \\&\quad \left. \sup _{i =1,\ldots , b}\left| \beta _{j,k}^{f_{i}}\right| \right) \\ \left\| {\overline{g}} \right\| _{{\overline{B}}^{*}}& = \left[ \sum _{k=0}^{2^J-1} \left| \alpha _{J,k}^{g}\right| + \sum _{j=J}^{K} \sum _{k=0}^{2^j - 1} \left| \beta _{j,k}^{g}\right| \right] \\&\quad + \left[ \sum _{k=0}^{2^J-1} \sum _{i=1}^{b} \left| \alpha _{J,k}^{g_{i}}\right| + \sum _{j=J}^{K} \sum _{k=0}^{2^j - 1}\sum _{i=1}^{b} \left| \beta _{j,k}^{g_{i}}\right| \right] , \end{aligned}$$

(35)

where for $f\in B$, and $g\in B^{\star }$,

$$\begin{aligned} \left\| f \right\| _{B}& = \sup \left\{ \left| \alpha _{J,k}^{f}\right| ,~k =0,\ldots ,2^{J}-1; ~\left| \beta _{j,k}^{f}\right| ,\right. \\&\left. k=0,\ldots ,2^{j}-1,~j=J,\ldots ,K \right\} , \\ \left\| g \right\| _{B^{*}}& = \sum _{k =0}^{2^{J}-1} \left| \alpha _{J,k}^{g}\right| + \sum _{j=J}^{K} \sum _{k =0}^{2^{j}-1} \left| \beta _{j,k}^{g}\right| . \end{aligned}$$