Hilbertian spatial periodically correlated first order autoregressive models

Abstract

In this article, we consider Hilbertian spatial periodically correlated autoregressive models. Such a spatial model assumes periodicity in its autocorrelation function. Plausibly, it explains spatial functional data resulted from phenomena with periodic structures, as geological, atmospheric, meteorological and oceanographic data. Our studies on these models include model building, existence, time domain moving average representation, least square parameter estimation and prediction based on the autoregressive structured past data. We also fit a model of this type to a real data of invisible infrared satellite images.

Appendix: Proofs

Proof of Theorem 3.1

First, we show that the random field ${{\mathbf {X}}}=\{{{\mathbf {X}}}_{\mathbf {t}},\ {\mathbf {t}}\in {{\mathbb {Z}}}^{2}\}$ defined by

$$\begin{aligned} {\mathbf {X}}_{i,j}\equiv {\left( X_{i,jT_{2}}, X_{i,jT_{2}+1}, \ldots , X_{i,jT_{2}+T_{2}-1}\right) }^{\prime } , \end{aligned}$$

is a ${\mathbf {T}}_1$-HSPC process. Let ${\mathbf {t}},{{\mathbf {s}}}$ and ${\mathbf {n}}\in {{\mathbb {Z}}}^{2}$. Then the cross-covariance matrix of $\mathbf X$ is $C_{{{{ \mathbf {X}}}}_{{\mathbf {t}}},{{{\mathbf {X}}}}_{\mathbf {s}}}=\left[ a_{i,j}(\mathbf{t,s})\right] _{ i,j=0}^{ T_{2}-1}$ where $a_{i,j}(\mathbf{t,s})=\mathbb {E}\left( X_{{\mathbf {t}}\odot {{\mathbf {T}}}_{2}+(0,i)}\otimes X_{{{\mathbf {s}}}\odot {{\mathbf {T}}}_{2}+(0,j)}\right)$. Since ${\mathbf {T}}={ \mathbf {T}}_{1}\odot {\mathbf {T}}_{2}$,

$$\begin{aligned} a_{i,j}(\mathbf{t,s})&= \mathbb {E}\left( X_{t_1,t_2T_2+i}\otimes X_{s_1,s_2T_2+j}\right) \\&= \mathbb {E}\left( X_{t_1+n_1T1,t_2T_2+i+n_2T_2}\otimes X_{s_1+n_1T1,s_2T_2+j+n_2T_2}\right) \\&= \mathbb {E}\left( X_{t_1+n_1T1, (t_2+n_2 )T_2+i}\otimes X_{s_1+n_1T1,(s_2+n_2)T_2+j}\right) \\&= \mathbb {E}\left( X_{\left( {\mathbf {t}}{\mathbf {+}}{\mathbf {n}}\odot {{\mathbf {T}}} _{1}\right) \odot {{\mathbf {T}}}_{2}+(0,i)}\otimes X_{\left( {{\mathbf {s}} \mathbf {+}}{\mathbf {n}}\odot {{\mathbf {T}}}_{1}\right) \odot {{\mathbf {T}}} _{2}+(0,j)}\right) \\&= a_{i,j}(\mathbf{t+n\odot T_1 ,s+n\odot T_1}). \end{aligned}$$

So $C_{{{{\mathbf {X}}}}_{{\mathbf {t}}},{{{\mathbf {X}}}}_{ \mathbf {s}}}=C_{{{{\mathbf {X}}}}_{{\mathbf {t}}{\mathbf {+}}{\mathbf { n}}\odot {{\mathbf {T}}}_{1}},{{{\mathbf {X}}}}_{{{\mathbf {s}} \mathbf {+}}{\mathbf {n}}\odot {{\mathbf {T}}}_{1}}}$. It means that ${{\mathbf {X}}}$ is a ${{\mathbf {T}}}_{1}$-HSPC process. By definition

$$\begin{aligned} {\varvec{\epsilon }}_{i,j}\equiv {\left( {\epsilon }_{i,jT_{2}}, {\epsilon }_{i,jT_{2}+1}, \dots , {\epsilon } _{i,jT_{2}+T_{2}-1}\right) }^{\prime }. \end{aligned}$$

Since the structure of definition ${\varvec{\epsilon }}=\left\{ \varvec{\mathbf {\epsilon }}_{\mathbf {t}},\ {\mathbf {t}}\in {{\mathbb {Z}}}^{2}\right\}$ is same as $\mathbf X$, one can conclude ${\varvec{\epsilon }}$ is also a ${{\mathbf {T}}}_{1}$-HSPC process. Moreover, the cross-covariance matrix of ${\varvec{\epsilon }}$ is $C_{{\varvec{\epsilon }}_{{\mathbf {t}} },{\varvec{\epsilon }}_{\mathbf {s}}}=\left[ b_{i,j}(\mathbf{s,t})\right] _{i,j=0}^{T_2-1}$ where $b_{i,j}(\mathbf{s,t})=\mathbb {E}\left( {\epsilon }_{{\mathbf {t}}\odot {{\mathbf {T}}}_{2}+(0,i)}\otimes {\epsilon }_{{\mathbf {s}}\odot {{\mathbf {T}}}_{2}+(0,j)}\right)$. For ${\mathbf {s}}\ne {{\mathbf { t}}}$ we have ${\mathbf {t }}\odot {{\mathbf {T}}}_{2}+(0,i)\ne \mathbf {s}\odot {{\mathbf {T}}} _{2}+(0,j)$ for all $i,j=0,1, \cdots , T_2-1$ and hence $b_{i,j}(\mathbf{s,t})=0$. Therefore, ${\varvec{\epsilon }}$ is a ${{\mathbf {T}}}_{1}$-HSPC-WN process. This also results that ${\varvec{\varepsilon }}_{i,j}={{{\mathcal {D}}}}_{i}{{\varvec{\epsilon }}}_{i,j}$, is a ${{\mathbf {T}}}_{1}$-HSPC-WN process with covariance operator ${C_{{{{ \varvec{\varepsilon }}}}_{i,j}}={{{\mathcal {D}}}} _{i}C_{{{\varvec{\epsilon }}}_{i,j}}{{{ \mathcal {D}}}}_{i}^{{{*}}}}$. To see $\mathbf X$ satisfies (3.1), by using (2.1) recursively, we have

$$\begin{aligned} X_{i,jT_{2}}&= {\alpha }_{i,jT_{2}}X_{i-1,jT_{2}}+{\beta }_{i,jT_{2}}X_{i,jT_{2}-1}+{\ \gamma }_{i,jT_{2}}X_{i-1,jT_{2}-1}+{\epsilon }_{i,jT_{2}}\\&= {\alpha }_{i,0}X_{i-1,jT_{2}}+{\beta }_{i,0}X_{i,jT_{2}-1}+{\ \gamma }_{i,0}X_{i-1,jT_{2}-1}+{\epsilon }_{i,jT_{2}},\\ X_{i,jT_{2}+1}&= {\alpha }_{i,jT_{2}+1}X_{i-1,jT_{2}+1}+{\beta }_{i,jT_{2}+1}X_{i,jT_{2}}+{ \gamma }_{i,jT_{2}+1}X_{i-1,jT_{2}}+{\epsilon }_{i,jT_{2}+1}\\&= {\alpha }_{i,1}X_{i-1,jT_{2}+1}+{\beta }_{i,1}X_{i,jT_{2}}+{ \gamma }_{i,1}X_{i-1,jT_{2}}+{\epsilon }_{i,jT_{2}+1}\\&= {\alpha }_{i,1}X_{i-1,jT_{2}+1}+{\beta }_{i,1}\left( {\alpha }_{i,0}X_{i-1,jT_{2}}+{\beta }_{i,0}X_{i,jT_{2}-1}+{\ \gamma }_{i,0}X_{i-1,jT_{2}-1}\right. \\&\left. +\,\,{\epsilon }_{i,jT_{2}}\right) +{ \gamma }_{i,1}X_{i-1,jT_{2}}+{\epsilon }_{i,jT_{2}+1}\\&= {\alpha }_{i,1}X_{i-1,jT_{2}+1}+\left( {\beta }_{i,1}{ \alpha }_{i,0}+{\gamma }_{i,1}\right) X_{i-1,jT_{2}}+{\beta } _{i,1}^{(2)}X_{i,jT_{2}-1} \\&+{\ \beta }_{i,1}{\gamma }_{i,0}X_{i-1,jT_{2}-1}+{\beta }_{i,1}{\epsilon } _{i,jT_{2}}+{\epsilon }_{i,jT_{2}+1}, \end{aligned}$$

and for each $k=0, 1, \dots , T_2-1,$ we can get

$$\begin{aligned} X_{i,jT_{2}+k}&= {\alpha }_{i,k}X_{i-1,jT_{2}+k}+\left( { \beta }_{i,k}{\ \alpha }_{i,k-1}+{\gamma }_{i,k}\right) X_{i-1,jT_{2}+k-1} \\&+{\beta }_{i,k}\left( {\beta }_{i,k-1}{\alpha }_{i,k-2}+{ \gamma }_{i,k-1}\right) X_{i-1,jT_{2}+k-2} \\&+\dots +{\beta }_{i,k}^{\left( k-1\right) }\left( {\beta }_{i,1}{ \alpha }_{i,0}+{\ \gamma }_{i,1}\right) X_{i-1,jT_{2}}+{\beta } _{i,k}^{\left( k+1\right) }X_{i,jT_{2}-1} \\&+{\beta }_{i,k}^{\left( k\right) }{\gamma } _{i,0}X_{i-1,jT_{2}-1}+\sum _{s=0}^{k}{{\ \beta }_{i,k}^{\left( k-s\right) }}{\epsilon }_{i,jT_{2}+s}. \end{aligned}$$

Writing this equations in matrix form, using definitions (2.6) and (2.7) gives

$$\begin{aligned} {{{{\mathbf {X}}}}}_{i,j}={{{\mathcal {A}}}} _{i}{{{\mathbf {X}}}}_{i-1,j}+{{{\mathcal { B}}}}_{i}{{{\mathbf {X}}}}_{i,j-1}+{{{ \mathcal {C}}}}_{i}{{{\mathbf {X}}}}_{i-1,j-1}+{{{\mathcal {D}}}}_{i}{{\varvec{\epsilon }}}_{i,j}, \end{aligned}$$

where ${{\mathcal {A}}}_{i}=\left[ {{a}}_{m,n}^{i}\right]$, ${{{ \mathcal {B}}}}_{i}=\left[ {{b}}_{m,n}^{i}\right]$, ${{ \mathcal {C}}}_{i}=\left[ {{c}}_{m,n}^{i}\right]$ and ${{{ \mathcal {D}}}}_{i}=\left[ {{d}}_{m,n}^{i}\right]$. $\square$

Proof of Theorem 3.2

By definition

$$\begin{aligned} \widetilde{\mathbf {X}}_\mathbf{t }&= {\left( {{\mathbf {X}}}_{t_1T_{1},t_2}, {{{\mathbf {X}}}}_{t_1T_{1}+1,t_2}, \dots , {{\mathbf {X}}}_{t_1T_{1}+T_{1}-1,t_2}\right) }^{\prime }\\&= {\left( {{\mathbf {X}}}_{{\mathbf {t}}\odot {{\mathbf {T}}} _{1}}, {{{\mathbf {X}}}}_{{\mathbf {t}}\odot {{\mathbf {T}}} _{1}+(1,0)}, \dots , {{\mathbf {X}}}_{{\mathbf {t}}\odot {{\mathbf {T}}} _{1}+(T_{1}-1,0)}\right) }^{\prime }. \end{aligned}$$

Therefore, for ${\mathbf {s,t}\in { \mathbb {Z}}}^{2}$, the cross-covariance matrix $C_{{{\widetilde{\mathbf {X}}}}_{{\mathbf {t }}},{{\widetilde{\mathbf {X}}}}_{{{\mathbf {s}}}}}$ is a $T_{1}\times T_{1}$ block matrix with $(i,j)$-th block is $C_{{{{\mathbf {X}}}}_{{\mathbf {t}}\odot {{\mathbf {T}}} _{1}+(i,0)},{{{\mathbf {X}}}}_{{{\mathbf {s}}}\odot {{\mathbf {T}}} _{1}+(j,0)}}$ for $i, j=0, 1, \dots , T_{1}-1$. Since by Theorem 3.1 ${{\mathbf { X}}}$ is a $\mathbf {T_1}$-HSPC process,

$$\begin{aligned} C_{{{{\mathbf {X}}}}_{{\mathbf {t}}\odot {{\mathbf {T}}}_{1}+(i,0)},{ {{\mathbf {X}}}}_{{{\mathbf {s}}}\odot {{\mathbf {T}}}_{1}+(j,0)}}=C_{ {{{\mathbf {X}}}}_{(i,0)},{{{\ \mathbf {X}}}}_{\left( {{ \mathbf {s}}\mathbf {-}}{\mathbf {t}}\right) \odot {{\mathbf {T}}}_{1}+(j,0)}}. \end{aligned}$$

So the entries of $C_{{\widetilde{{\mathbf {X}}}}_{{\mathbf {t}}},{\widetilde{{\mathbf {X}}}}_{{{ \mathbf {s}}}}}$ depend on ${\mathbf {t}},{{\mathbf {s}}}$ through ${{\mathbf {s} }}-{\mathbf {t}}$ and this means ${\widetilde{\mathbf {X}}}=\{{\widetilde{\mathbf {X}}}_{\mathbf {t}}, {\mathbf {t}}\in {{\mathbb {Z}}}^{2}\}$ is an HSS process. This result is also true for $\widetilde{\varvec{\epsilon }}=\{\widetilde{\varvec{\epsilon }}_{\mathbf {t}}, {\mathbf {t}}\in {{\mathbb {Z}}}^{2}\}$ which is defined as

$$\begin{aligned} \widetilde{\varvec{\epsilon }}_{i,j}={\left( {\varvec{\epsilon }}_{iT_{1},j}, {\varvec{\epsilon }} _{iT_{1}+1,j}, \dots , {\varvec{\epsilon }}_{iT_{1}+T_{1}-1,j}\right) }^{\prime }. \end{aligned}$$

Moreover, by the same technique as used in the proof of Theorem 3.1, it can be shown that $\widetilde{\varvec{\epsilon }}$ is an HSS-WN process. Therefore, $\widetilde{\varvec{\varepsilon }}_{{\mathbf {t}}}={\varvec{\widetilde{\mathcal {D}}}}\varvec{\mathcal {E}}\varvec{\widetilde{\epsilon } }_{{\mathbf {t}}}$ is an HSS-WN process with covariance operator $C_{{\widetilde{\varvec{\varepsilon }}}}=\varvec{\widetilde{\mathcal {D }}}\varvec{{\mathcal {E}}}C_{\widetilde{\varvec{\epsilon }}}\varvec{\mathcal {E} }^{*}\varvec{{\widetilde{\mathcal {D}}}}^{*}$. On the other hand, by (2.6) and (2.7),$\ {{ {\mathcal {D}}}}_{iT_{1}+k}={{{\mathcal {D}}}}_{k}$, therefore, ${ {{\varvec{\varepsilon }}}}_{iT_{1}+k,j}={{{\mathcal {D}} }}_{iT_{1}+k}{{\varvec{\epsilon }}}_{iT_{1}+k,j}={{{\mathcal {D}} }}_{k}{{\varvec{\epsilon }}}_{iT_{1}+k,j}$. Now using Eq. (3.1) recursively, gives

$$\begin{aligned} {\mathbf {X}}_{iT_{1},j}&= {\mathcal {A}}_{iT_{1}}{\mathbf {X}}_{iT_{1}-1,j}+{ {{\mathcal {B}}}}_{iT_{1}}{{{\mathbf {X}}}}_{iT_{1},j-1}\mathrm { +}{{{\mathcal {C}}}}_{iT_{1}}{{{\mathbf {X}}}}_{iT_{1}-1,j-1} \mathrm {+}{{{\mathcal {D}}}}_{iT_{1}}{{{\varvec{ \epsilon }}}}_{iT_{1},j}\\&= {\mathcal {A}}_{0}{\mathbf {X}}_{iT_{1}-1,j}+{ {{\mathcal {B}}}}_{0}{{{\mathbf {X}}}}_{iT_{1},j-1}\mathrm { +}{{{\mathcal {C}}}}_{0}{{{\mathbf {X}}}}_{iT_{1}-1,j-1} \mathrm {+}{{{\mathcal {D}}}}_{0}{{{\varvec{ \epsilon }}}}_{iT_{1},j},\\ {\mathbf {X}}_{iT_{1}+1,j}&= \mathcal {A}_{iT_{1}+1}{\mathbf {X}}_{iT_{1},j}+\mathcal {B}_{iT_{1}+1}{\mathbf {X}}_{iT_{1}+1,j-1}+\mathcal {C}_{iT_{1}+1}{\mathbf {X}}_{iT_{1},j-1}+{{{\mathcal {D}}}}_{iT_{1}+1}{{\varvec{\epsilon }}}_{iT_1+1,j}\\&= \mathcal {A}_{1}{\mathbf {X}}_{iT_{1},j}+\mathcal {B}_{1}{\mathbf {X}}_{iT_{1}+1,j-1}+\mathcal {C}_{1}{\mathbf {X}}_{iT_{1},j-1}+{{{\mathcal {D}}}}_{1}{{\varvec{\epsilon }}}_{iT_1+1,j}\\&= \mathcal {A}_{1}\left( {\mathcal {A}}_{0}{\mathbf {X}}_{iT_{1}-1,j}+{ {{\mathcal {B}}}}_{0}{{{\mathbf {X}}}}_{iT_{1},j-1}\mathrm { +}{{{\mathcal {C}}}}_{0}{{{\mathbf {X}}}}_{iT_{1}-1,j-1} \mathrm {+}{{{\mathcal {D}}}}_{0}{{{\varvec{ \epsilon }}}}_{iT_{1},j}\right) \\&+\mathcal {B}_{1}{\mathbf {X}}_{iT_{1}+1,j-1}+\mathcal {C}_{1}{\mathbf {X}}_{iT_{1},j-1}+{{{\mathcal {D}}}}_{1}{{\varvec{\epsilon }}}_{iT_1+1,j}\\&= {{{\mathcal {A}}}}_{1}^{{(2)}}{ {{\mathbf {X}}}}_{{{iT}}_{{1}}-{ 1,j}}+{{{\mathcal {A}}}}_{1}^{{(1)}}{ {{\mathcal {C}}}}_{0}{{{\mathbf {X}}}}_{{i}{ {T}}_{1}-{1,j-1}}+{{{ \mathcal {A}}}}_{1}^{(0)}\left( {{{\mathcal {A} }}}_{1}{{{\mathcal {B}}}}_{0}+{{{\mathcal {C}} }}_{1}\right) {{{\mathbf {X}}}}_{i{{T}}_{1} {,j-1}} \\&+{{{\mathcal {B}}}}_{1}{{{\mathbf { X}}}}_{{i}{{T}}_{1}+1,j-1}+{ {{\mathcal {D}}}}_{1}{{{\varvec{\epsilon }}}}_{iT_{1}+1,j} +{{{\mathcal {A}}}}_{1}{{{\mathcal {D }}}}_{0}{{{\varvec{\epsilon }}}}_{iT_{1},j}, \end{aligned}$$

and for each $k=0, 1, \dots , T_1-1,$ we can get

$$\begin{aligned} {{{\mathbf {X}}}}_{iT_{1}+k,j}&= {{{\mathcal {A}}}} _{k}^{(k+1)}{{{\mathbf {X}}}}_{{iT}_{1}-1,j}+{{{ \mathcal {A}}}}_{k}^{(k)}{{{\mathcal {C}}}}_{0}{ {{\mathbf {X}}}}_{iT_{1}-1,j-1} \\&+{{{\mathcal {A}}}}_{k}^{(k-1)}\left( {{{ \mathcal {A}}}}_{1}{{{\mathcal {B}}}}_{0}+{{{\mathcal {C} }}_{1}}\right) {{{\mathbf {X}}}}_{iT_{1},j-1}+{{{ \mathcal {A}}}}_{k}^{(k-2)}\left( {{{\mathcal {A}}}}_{2}{ {{\mathcal {B}}}}_{1}+{{{\mathcal {C}}}}_{2}\right) { {{\mathbf {X}}}}_{iT_{1}+1,j-1} \\&+\dots +{{{\mathcal {A}}}}_{k}^{\left( 0\right) }\left( { {{\mathcal {A}}}}_{k}{{{\mathcal {B}}}}_{k-1}+{ {{\mathcal {C}}}}_{k}\right) {{{\mathbf {X}}}} _{iT_{1}+k-1,j-1}\\&+{{{\mathcal {B}}}}_{k}{{{ \mathbf {X}}}}_{iT_{1}+k,j-1}+\sum _{s=0}^{k}{{{{\mathcal {A}}}}_{k}^{\left( k-s\right) }{{{\mathcal {D}}}}_{s}{{\varvec{ \epsilon }}}_{iT_{1}+s,j}}. \end{aligned}$$

Writing the above equations in matrix form, using definitions (2.8) and (2.9) gives

$$\begin{aligned} {\widetilde{{\mathbf {X}}}}_{i,j}=\varvec{\widetilde{\mathcal {A}}}{{\widetilde{\mathbf {X}}}}_{i-1,j}+ \varvec{\widetilde{\mathcal {B}}}{\widetilde{\mathbf {X}}}_{i,j-1}+\varvec{\widetilde{\mathcal {C}}}{ \widetilde{\mathbf {X}}}_{i-1,j-1}+{{\widetilde{\varvec{\varepsilon }}}}_{i,j}, \end{aligned}$$

where $\varvec{\widetilde{\mathcal {A}}}=\left[ \widetilde{\mathbf {a}}_{m,n}\right] , \varvec{ \widetilde{\mathcal {B}}}=\left[ \widetilde{\mathbf {b}}_{m,n}\right] ,\ \varvec{\widetilde{\mathcal {C}}}=\left[ \widetilde{\mathbf {c }}_{m,n}\right]$ and $\varvec{\widetilde{\mathcal {D}}}=\left[ \widetilde{\mathbf {d}}_{m,n}\right]$. $\square$

Proof of Theorem 3.4

By Corollary 3.5, ${\mathbf {Y}}$ is an HSS-AR(1) process on $H^{T_{1}T_{2}}$ satisfies

$$\begin{aligned} {{\mathbf Y}}_{i,j}={\mathbb A}{{\mathbf Y}}_{i-1,j}+{\mathbb B}{{ \mathbf Y}}_{i,j-1}+{\mathbb C}{{\mathbf Y}}_{i-1,j-1}+{\varvec{\widetilde{\epsilon }}}_{i,j}. \end{aligned}$$

Now, by the results of Ruiz-Medina (2011a), under the Assumptions A, $\mathbf Y$ has a unique stationary solution given by

$$\begin{aligned} {{\mathbf {Y}}}_{i,j}=\sum _{k=0}^{\infty }{\sum _{l=0}^{\infty }{ \sum _{r=0}^{\infty }{\ \frac{\left( k+l+r\right) !}{k!l!r!}{{\mathbb {A}}}^{k} {{\mathbb {B}}}^{l}{{\mathbb {C}}}^{r}}}}{\varvec{\widetilde{\epsilon } }} _{i-k-r,j-l-r}. \end{aligned}$$

Since ${{\widetilde{\mathbf {X}}}}_{{i,j}}={\widetilde{\varvec{\mathcal {D}}}} {\varvec{\mathcal {E}}}{{\mathbf {Y}}}_{{i,j}}$,

$$\begin{aligned} {\widetilde{{\mathbf {X}}}}_{i,j}=\sum _{k=0}^{\infty }{\sum _{l=0}^{\infty }{ \sum _{r=0}^{\infty }{\frac{\left( k+l+r\right) !}{k!l!r!}{\widetilde{\varvec{\mathcal {D}}}}{ \varvec{\mathcal {E}}}{{\mathbb {A}}}^{k}{{\ \mathbb {B}}}^{l}{{\mathbb {C}}}^{r}}}}{ \varvec{\widetilde{\epsilon } }}_{i-k-r,j-l-r}. \end{aligned}$$

(8.1)

By definition (2.4)

$$\begin{aligned} \widetilde{\mathbf {X}}_{i,j}&= {\left( {{\mathbf {X}}}_{iT_{1},j}, {{{\mathbf {X}}}}_{iT_{1}+1,j}, \dots , {{\mathbf {X}}}_{iT_{1}+T_{1}-1,j}\right) }^{\prime } ,\\ {\mathbf {X}}_{i,j}&= {\left( X_{i,jT_{2}}, X_{i,jT_{2}+1}, \ldots , X_{i,jT_{2}+T_{2}-1}\right) }^{\prime }. \end{aligned}$$

For $(i,j)\in {{\mathbb {Z}}}^{2}$, let $i=i^{*}+\left[ \frac{i}{T_{1}}\right] T_{1}$ and $j=j^{*}+\left[ \frac{j}{T_{2}}\right] T_{2}$ it is resulted that $X_{i,j}$ is $i^{*}T_{2}+j^{*}$-th member of the vector ${{\widetilde{\mathbf {X}}}}_{\left[ \frac{i}{T_{1}}\right] , \left[ \frac{j}{T_{2}}\right] }$. i.e., ${\pi }_{i^{*}T_{2}+j^{*}}{{\widetilde{\mathbf {X}}}}_{\left[ \frac{i}{T_{1}}\right] ,\left[ \frac{j}{T_{2}}\right] }=X_{i,j},$ and by $i^{*}T_{2}+j^{*}=\left( i-T_{1}\left[ \frac{i}{T_{1}}\right] -\left[ \frac{j}{T_{2}} \right] \right) T_{2}+j$, we have

$$\begin{aligned} {\pi }_{\left( i-T_{1}\left[ \frac{i}{T_{1}}\right] -\left[ \frac{j}{T_{2}} \right] \right) T_{2}+j}{{\widetilde{\mathbf {X}}}}_{\left[ \frac{i}{T_{1}}\right] ,\left[ \frac{j}{T_{2}}\right] }=X_{i,j}. \end{aligned}$$

(8.2)

Using Eqs. (8.1) and (8.2) gives

$$\begin{aligned} X_{i,j}{=}\sum _{k{=}0}^{\infty }{\sum _{l=0}^{\infty }{\sum _{r=0}^{\infty }{\frac{ \left( k+l+r\right) !}{k!l!r!}{\pi }_{\left( i-T_{1}\left[ \frac{i}{T_{1}} \right] -\left[ \frac{j}{T_{2}}\right] \right) T_{2}+j}{ \widetilde{\varvec{\mathcal {D}}}}{\varvec{\mathcal {E}}}{{\mathbb {A}}}^{k}{{\mathbb {B}}}^{l}{{\mathbb {C}}} ^{r}\widetilde{{\varvec{\epsilon }}}_{\left[ \frac{i}{T_{1}}\right] -k-r,\left[ \frac{j}{T_{2}}\right] -l-r} }}}, \end{aligned}$$

$\square$

Proof of Lemma 4.1

For $h\in H$ define ${\left\langle \varvec{\theta } ,h\right\rangle }_{H} =\left[ \left\langle {\theta }_1,h\right\rangle _{H} ,\dots ,\left\langle {\theta } _m,h\right\rangle _{H} \right] ^{\prime }$. If $\left\{ e_1,e_2,\dots \right\}$ be an orthogonal basis for ${H}$. Using Parseval’s identity for (4.1) gives

$$\begin{aligned} S(\varvec{\theta })=\sum ^{\infty }_{k=1}{\sum ^n_{i=1}{{ \left| \left\langle Y_i-{{\mathbf {x}}}_i\varvec{\theta },e_k \right\rangle _{H} \right| }^2}}. \end{aligned}$$

But the minimization of $\sum ^n_{i=1}{{\left| \left\langle Y_i- {{\ \mathbf {x}}}_i\varvec{\theta } ,e_k\right\rangle _{H} \right| }^2}$ is by $\widehat{\varvec{\theta }} \mathrm {=}{\left( {\mathbf {\mathcal {X}}}^{\prime }{\mathbf {\mathcal {X}}}\right) }^{{-1}} {\mathbf {\mathcal {X}}}^{\prime }\mathbf {Y}$. The reasoning is that the corresponding regression equation is

$$\begin{aligned} \left\langle Y_i,e_k\right\rangle _{H} =\left\langle {{\mathbf {x}}}_i\varvec{\theta } ,e_k\right\rangle _{H} +\left\langle {\varepsilon }_i,e_k\right\rangle _{H} ={{ \mathbf {x}}}_i\left\langle \varvec{\theta } ,e_k\right\rangle _{H} +\left\langle { \varepsilon }_i,e_k\right\rangle _{H} ,\ \ i=1,\dots ,n. \end{aligned}$$

The least square solution of this system is

$$\begin{aligned} {\left\langle \widehat{\varvec{\theta }},e_k\right\rangle }_{H}={\left( {\mathbf {\mathcal {X}}}^{\prime }{\mathbf {\mathcal {X}}}\right) }^{{-1}} {\mathbf {\mathcal {X}}}^{\prime }\left\langle \mathbf {Y},e_k\right\rangle _{H} =\left\langle {{\left( {\mathbf {\mathcal {X}}}^{\prime }{\mathbf {\mathcal {X}}}\right) }^{{-1}}{\mathbf {\mathcal {X}}}^{\prime }\mathbf {Y},e_k} \right\rangle _{H}. \end{aligned}$$

$\square$