Nonparametric estimation of circular trend surfaces with application to wave directions

Abstract

In oceanography, modeling wave fields requires the use of statistical tools capable of handling the circular nature of the data measurements. An important issue in ocean wave analysis is the study of height and direction waves, being direction values recorded as angles or, equivalently, as points on a unit circle. Hence, reconstruction of a wave direction field on the sea surface can be approached by the use of a linear–circular regression model, viewing wave directions as a realization of a circular spatial process whose trend should be estimated. In this paper, we consider a spatial regression model with a circular response and several real-valued predictors. Nonparametric estimators of the circular trend surface are proposed, accounting for the (unknown) spatial correlation. Some asymptotic results about these estimators as well as some guidelines for their practical implementation are also given. The performance of the proposed estimators is investigated in a simulation study. An application to wave directions in the Adriatic Sea is provided for illustration.

Appendix: Proof of Theorem 1

Proof

Before deriving the asymptotic variance of the estimator ${\hat{m}}_{{\varvec{H}}}({\varvec{x}},p)$, for $p=0,1$, stated in Theorem 1, some preliminary approximations are needed.

Firstly, defining $f_1({\varvec{x}})=\sin [m({\varvec{x}})]$ and $f_2({\varvec{x}})=\cos [m({\varvec{x}})]$, using sine and cosine addition formulas, the following relation between the covariance function $C_{n,3}$, defined from models (6) and (7), and the correlations ${\rho _{k,n}}$, $k=1,2,3$, directly derived from model (1) and given in (2), (3) and (4), can be obtained:

$$\begin{aligned} {C}_{n,3}({\varvec{X}}_i,{\varvec{X}}_j)&= f_1({\varvec{X}}_i)f_2({\varvec{X}}_j)\sigma ^2_2{\rho _{2,n}}({\varvec{X}}_i-{\varvec{X}}_j)\\&\quad- f_1({\varvec{X}}_i)f_1({\varvec{X}}_j)\sigma _{12}{\rho _{3,n}}({\varvec{X}}_i-{\varvec{X}}_j)\\&\quad+ f_2({\varvec{X}}_i)f_2({\varvec{X}}_j)\sigma _{12}{\rho _{3,n}}({\varvec{X}}_i-{\varvec{X}}_j)\\&\quad- f_2({\varvec{X}}_i)f_1({\varvec{X}}_j)\sigma ^2_1{\rho _{1,n}}({\varvec{X}}_i-{\varvec{X}}_j). \end{aligned}$$

Moreover, denoting

$$\begin{aligned}k_{1,n}({\varvec{x}})&= \dfrac{1}{n}\displaystyle \sum _{i=1}^{n} K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}}),\\ k_{2,n}({\varvec{x}})&= \dfrac{1}{n}\displaystyle \sum _{i=1}^{n} K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})({\varvec{X}}_i-{\varvec{x}}) ,\\ k_{3,n}({\varvec{x}})&= \dfrac{1}{n}\displaystyle \sum _{i=1}^{n} K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})({\varvec{X}}_i-{\varvec{x}})({\varvec{X}}_i-{\varvec{x}}){^\top },\\ s_{1,n}({\varvec{x}})&= \dfrac{1}{n^2}\left[ \displaystyle \sum _{i=1}^n K^2_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})c({\varvec{X}}_i)+\displaystyle \sum _{i\ne j} K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})\right. \left. \cdot K_{{\varvec{H}}}({\varvec{X}}_j-{\varvec{x}}) {C}_{n,3}({\varvec{X}}_i,{\varvec{X}}_j)\right] ,\\ s_{2,n}({\varvec{x}})&= \dfrac{1}{n^2}\left[ \displaystyle \sum _{i=1}^n K^2_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})({\varvec{X}}_i-{\varvec{x}}) c({\varvec{X}}_i)\right. \\&\left.\quad +\displaystyle \sum _{i\ne j} K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})K_{{\varvec{H}}}({\varvec{X}}_j-{\varvec{x}})({\varvec{X}}_i-{\varvec{x}}) \right. \left. \cdot {C}_{n,3}({\varvec{X}}_i,{\varvec{X}}_j) \right] , \\ s_{3,n}({\varvec{x}})&= \dfrac{1}{n^2}\left[ \displaystyle \sum _{i=1}^n K^2_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})({\varvec{X}}_i-{\varvec{x}})({\varvec{X}}_j-{\varvec{x}}) {^\top } c({\varvec{X}}_i)\right. \\&\left.\quad +\displaystyle \sum _{i \ne j} K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})K_{{\varvec{H}}}({\varvec{X}}_j-{\varvec{x}})({\varvec{X}}_i-{\varvec{x}})\right. \left. \cdot ({\varvec{X}}_j-{\varvec{x}}) {^\top } {C}_{n,3}({\varvec{X}}_i,{\varvec{X}}_j)\right] , \end{aligned}$$

and, after some calculations, it can be obtained that

$$\begin{aligned} k_{1,n}({\varvec{x}})&= f({\varvec{x}})+o_{{\mathbb {P}}}(1) , \end{aligned}$$

(19)

$$\begin{aligned} k_{2,n}({\varvec{x}})&= {\mu _2}{{\varvec{\nabla }}} f({\varvec{x}}){\varvec{H}}^2+o_{{\mathbb {P}}}({\varvec{H}}^2{\varvec{1}}_d), \end{aligned}$$

(20)

$$\begin{aligned} k_{3,n}({\varvec{x}})&= {\mu _2}f({\varvec{x}}){\varvec{H}}^2+o_{{\mathbb {P}}}({\varvec{H}}{\varvec{1}}_{d\times d}{\varvec{H}}), \end{aligned}$$

(21)

$$\begin{aligned} s_{1,n}({\varvec{x}})&= \dfrac{1}{n|{\varvec{H}}|}{\nu _0}f({\varvec{x}})[c({\varvec{x}})+f({\varvec{x}})C_3({\varvec{x}})]+o_{{\mathbb {P}}}\left( \dfrac{1}{n|{\varvec{H}}|}\right) , \end{aligned}$$

(22)

$$\begin{aligned} s_{2,n}({\varvec{x}})&= \dfrac{1}{n|{\varvec{H}}|}o_{{\mathbb {P}}}({\varvec{1}}_d) \end{aligned}$$

(23)

$$\begin{aligned} s_{3,n}({\varvec{x}})&= \dfrac{1}{n|{\varvec{H}}|} o_{{\mathbb {P}}}({\varvec{1}}_{d\times d}), \end{aligned}$$

(24)

where ${\varvec{1}}_d$ and ${\varvec{1}}_{d\times d}$ denote the $d\times 1$ vector and the $d\times d$ matrix with every entry equal to 1, and

$$\begin{aligned} c({\varvec{x}})&= f_1({\varvec{x}})f_2({\varvec{x}})\sigma ^2_2-f_1^2({\varvec{x}}) \sigma _{12}+f_2^2({\varvec{x}})\sigma _{12}\nonumber \\&\quad-f_1({\varvec{x}})f_2({\varvec{x}})\sigma ^2_1, \end{aligned}$$

(25)

$$\begin{aligned} C_3({\varvec{x}})&= f_1({\varvec{x}})f_2({\varvec{x}})\sigma ^2_2\rho _{{\text {c}}_2}-f_1^2({\varvec{x}})\sigma _{12}\rho _{{\text {c}}_3}+ f_2^2({\varvec{x}})\sigma _{12}\rho _{{\text {c}}_3}\nonumber \\&\quad-f_1({\varvec{x}})f_2({\varvec{x}})\sigma ^2_1\rho _{{\text {c}}_1}. \end{aligned}$$

(26)

To derive the variance of ${\hat{m}}_{{\varvec{H}}}({\varvec{x}};p)$, for $p=0,1$, denoting by ${\varvec{\mathcal {X}}}=({\varvec{X}}_1,\dots ,{\varvec{X}}_n)$, using Taylor expansions and following similar arguments to those used in Di Marzio et al. (2013) and Meilán-Vila et al. (2020b), it can be obtained that

$$\begin{aligned}&{\mathbb {V}}{\text {ar}}\left[ {\hat{m}}_{{\varvec{H}}}({\varvec{x}};p)\mid {\varvec{\mathcal {X}}}\right] \nonumber \\&\quad =\dfrac{m_1^2({\varvec{x}})}{\big [m_1^2({\varvec{x}})+m_2^2({\varvec{x}})\big ]^2}{\mathbb {V}} {\text {ar}}[{\hat{m}}_{2, {\varvec{H}}}({\varvec{x}};p) \mid {\varvec{\mathcal {X}}}]\nonumber \\&\qquad + \dfrac{m_2^2({\varvec{x}})}{\big [m_1^2({\varvec{x}})+m_2^2({\varvec{x}})\big ]^2}{\mathbb {V}}{\text {ar}}[{\hat{m}}_{1, {\varvec{H}}}({\varvec{x}};p)\mid {\varvec{\mathcal {X}}}]\nonumber \\&\qquad - \dfrac{2m_1({\varvec{x}})m_2({\varvec{x}})}{\big [m_1^2({\varvec{x}})+m_2^2({\varvec{x}})\big ]^2}{\mathbb {C}}{\text {ov}}[{\hat{m}}_{1, {\varvec{H}}}({\varvec{x}};p),{\hat{m}}_{2, {\varvec{H}}}({\varvec{x}};p)\mid {\varvec{\mathcal {X}}}]\nonumber \\&\qquad + O\big \{[{\hat{m}}_{1, {\varvec{H}}}({\varvec{x}};p)-{m}_1({\varvec{x}})]^3\big \}+ O\big \{[{\hat{m}}_{2, {\varvec{H}}}({\varvec{x}};p)-{m}_2({\varvec{x}})]^3\big \}.\end{aligned}$$

(27)

The conditional variance of ${\hat{m}}_{j, {\varvec{H}}}({\varvec{x}};p)$, for $j=1,2$, and $p=0,1,$ for spatially correlated data, can be derived using similar arguments to those given in Liu (2001), which yield

$$\begin{aligned} {\mathbb {V}}{\text {ar}}[{{\hat{m}}}_{j, {\varvec{H}}}({\varvec{x}};p)\mid {\varvec{\mathcal {X}}}]&= \frac{{\nu _0}[s_j^2({\varvec{x}})+f({\varvec{x}})C_j({\varvec{x}})]}{n |{\varvec{H}}|f({\varvec{x}})}\nonumber \\&\quad+{o}_{{\mathbb {P}}}\left( \frac{1}{n |{\varvec{H}}|}\right) , \end{aligned}$$

(28)

where

$$\begin{aligned} s_1^2({\varvec{x}})&= f_1^2({\varvec{x}})\sigma ^2_2+2f_1({\varvec{x}})f_2({\varvec{x}})\sigma _{12}+f_2^2({\varvec{x}})\sigma ^2_1, \end{aligned}$$

(29)

$$\begin{aligned} s_2^2({\varvec{x}})&= f_2^2({\varvec{x}})\sigma ^2_2-2f_2({\varvec{x}})f_1({\varvec{x}})\sigma _{12}+f_1^2({\varvec{x}})\sigma ^2_1,\end{aligned}$$

(30)

$$\begin{aligned} C_1({\varvec{x}})&= f_1^2({\varvec{x}})\sigma ^2_2\rho _{{\text {c}}_2}+2f_1({\varvec{x}})f_2({\varvec{x}})\sigma _{12}\rho _{{\text {c}}_3}\nonumber \\&\quad+ f_2^2({\varvec{x}})\sigma ^2_1\rho _{{\text {c}}_1},\end{aligned}$$

(31)

$$\begin{aligned} C_2({\varvec{x}})&= f_2^2({\varvec{x}})\sigma ^2_2\rho _{{\text {c}}_2}-2f_1({\varvec{x}})f_2({\varvec{x}})\sigma _{12}\rho _{{\text {c}}_3}\nonumber \\&\quad+ f_1^2({\varvec{x}})\sigma ^2_1\rho _{{\text {c}}_1}. \end{aligned}$$

(32)

Moreover, using (19) and (22), it is easy to obtain that the conditional covariance between ${\hat{m}}_{1, {\varvec{H}}}({\varvec{x}};0)$ and ${\hat{m}}_{2, {\varvec{H}}}({\varvec{x}};0)$ is

$$\begin{aligned}&{\mathbb {C}}{\text {ov}}[{\hat{m}}_{1, {\varvec{H}}}({\varvec{x}};0),{\hat{m}}_{2, {\varvec{H}}}({\varvec{x}};0)\mid {\varvec{\mathcal {X}}}]\nonumber \\&\quad =\dfrac{\sum _{i=1}^{n}\sum _{j=1}^{n}K_{{\varvec{H}}}({\varvec{X}}_i- {\varvec{x}})K_{{\varvec{H}}}({\varvec{X}}_j-{\varvec{x}})}{\sum _{i=1}^{n}K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})\sum _{j=1}^{n}K_{{\varvec{H}}}({\varvec{X}}_j-{\varvec{x}})}\nonumber \\&\qquad \cdot {\mathbb {C}}{\text {ov}}[\sin (\varTheta _i),\cos (\varTheta _j)\mid {\varvec{\mathcal {X}}}]\nonumber \\&\quad = \dfrac{\sum _{i=1}^nK^2_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})c({\varvec{X}}_i)}{\left[ \sum _{i=1}^{n}K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})\right] ^2}\nonumber \\&\qquad + \dfrac{\sum _{i\ne j}K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})K_{{\varvec{H}}}({\varvec{X}}_j-{\varvec{x}}) C_{n,3}({\varvec{X}}_i,{\varvec{X}}_j)}{\left[ \sum _{i=1}^{n}K_{{\varvec{H}}}({\varvec{X}}_i-{\varvec{x}})\right] ^2} \nonumber \\&\quad =\dfrac{1}{n|{\varvec{H}}|f({\varvec{x}})}{\nu _0}[c({\varvec{x}})+f({\varvec{x}})C_3({\varvec{x}})] + o_{{\mathbb {P}}}\left( \dfrac{1}{n|{\varvec{H}}|}\right) . \end{aligned}$$

(33)

On the other hand, the conditional covariance between ${\hat{m}}_{1, {\varvec{H}}}({\varvec{x}};1)$ and ${\hat{m}}_{2, {\varvec{H}}}({\varvec{x}};1)$ is

$$\begin{aligned}&{\mathbb {C}}{\text {ov}}[{\hat{m}}_{1, {\varvec{H}}}({\varvec{x}};1),{\hat{m}}_{2, {\varvec{H}}}({\varvec{x}};1)\mid {\varvec{\mathcal {X}}}]\nonumber \\&\quad ={\varvec{e}}_1^\top \left( {\varvec{X}}_{\varvec{x}}^\top {\varvec{W}}_{\varvec{x}}{\varvec{X}}_{{\varvec{x}}}\right) ^{-1}{\varvec{X}}_{\varvec{x}}^\top {\varvec{W}}_{\varvec{x}}{\varvec{\varSigma }} {\varvec{W}}_{\varvec{x}}{\varvec{X}}_{\varvec{x}}\left( {\varvec{X}}_{\varvec{x}}^\top {\varvec{W}}_{\varvec{x}}{\varvec{X}}_{\varvec{x}}\right) ^{-1}{\varvec{e}}_1, \end{aligned}$$

where ${\varvec{\varSigma }}$ is the covariance matrix of $\sin (\varTheta )$ and $\cos (\varTheta )$, whose (i, j)-entry is ${\varvec{\varSigma }}(i,j)={\mathbb {C}}{\text {ov}}[\sin (\varTheta _i),\cos (\varTheta _j)],$ $i,j=1,\dots ,n.$ Using (19), (20), (21), (22), (23) and (24), it follows that

$$\begin{aligned}&{\left( n^{-1}{\varvec{X}}_{\varvec{x}}^\top {\varvec{W}}_{{\varvec{x}}}{\varvec{X}}_{{\varvec{x}}}\right) ^{-1}}\\&\quad =\left( \begin{array}{ll} k_{1,n}({\varvec{x}}) &\quad {k^\top _{2,n}({\varvec{x}})} \\ k_{2,n}({\varvec{x}}) &\quad k_{3,n}({\varvec{x}}) \end{array}\right) ^{-1}\\&\quad = \left( \begin{array}{ll} \frac{1}{f({\varvec{x}})}+o_{{\mathbb {P}}}(1) &\quad \frac{-{\varvec{\nabla }}{^\top } f({\varvec{x}})}{f^2({\varvec{x}})}+o_{{\mathbb {P}}}\left( {\varvec{1}}^\top _d\right) \\ \frac{-{\varvec{\nabla }} f({\varvec{x}})}{f^2({\varvec{x}})}+o_{{\mathbb {P}}}({\varvec{1}}_d) &\quad \frac{1}{{\mu _2}f({\varvec{x}}){\varvec{H}}^2}+o_{{\mathbb {P}}}({\varvec{H}}{\varvec{1}}_{d\times d}{\varvec{H}}) \end{array} \right) ,\end{aligned}$$

and that

$$\begin{aligned}&{\dfrac{1}{n^2}{\varvec{X}}_{\varvec{x}}^\top }{\varvec{W}}_{{\varvec{x}}}{\varvec{\varSigma }}{\varvec{W}}_{{\varvec{x}}} {\varvec{X}}_{{\varvec{x}}}\\&\quad =\left( \begin{array}{ll} s_{1,n}({\varvec{x}}) &\quad {s^\top _{2,n}({\varvec{x}})} \\ s_{2,n}({\varvec{x}}) &\quad s_{3,n}({\varvec{x}}) \end{array}\right) \\&\quad =\dfrac{1}{n|{\varvec{H}}|}\left( \begin{array}{ll} {\nu _0}f({\varvec{x}})[c({\varvec{x}})+f({\varvec{x}})C_3({\varvec{x}})]{+o_{{\mathbb {P}}}(1)} & \quad o_{{\mathbb {P}}}\left( {{\varvec{1}}^\top _d}\right) \\ o_{{\mathbb {P}}}({\varvec{1}}_d) &\quad o_{{\mathbb {P}}}({\varvec{1}}_{d\times d}) \end{array} \right) .\end{aligned}$$

Consequently, by straightforward calculations, one gets that

$$\begin{aligned}&{\mathbb {C}}{\text {ov}}[{\hat{m}}_{1, {\varvec{H}}}({\varvec{x}};1),{\hat{m}}_{2, {\varvec{H}}}({\varvec{x}};1)\mid {\varvec{\mathcal {X}}}]\nonumber \\&= \dfrac{1}{n|{\varvec{H}}|f({\varvec{x}})}{\nu _0}[c({\varvec{x}})+f({\varvec{x}})C_3({\varvec{x}})] + o_{{\mathbb {P}}}\left( \dfrac{1}{n|{\varvec{H}}|}\right) . \end{aligned}$$

(34)

Using (27), (28), (33) and (34), one gets that, for $p=0,1$,

$$\begin{aligned}{\mathbb {V}}{\text {ar}}[{\hat{m}}_{{\varvec{H}}}({\varvec{x}};p)\mid {\varvec{\mathcal {X}}}]&= \frac{1}{n |{\varvec{H}}|}\dfrac{{\nu _0}}{f({\varvec{x}})}\dfrac{m_1^2({\varvec{x}})}{\left[ m_1^2({\varvec{x}})+m_2^2({\varvec{x}})\right] ^2}\\&\quad \cdot \left[ s_2^2({\varvec{x}})+f({\varvec{x}})C_2({\varvec{x}})\right] \\&\quad+ \frac{1}{n |{\varvec{H}}|}\dfrac{{\nu _0}}{f({\varvec{x}})}\dfrac{m_2^2({\varvec{x}})}{\left[ m_1^2({\varvec{x}})+m_2^2({\varvec{x}})\right] ^2}\\&\quad\cdot \left[ s_1^2({\varvec{x}})+f({\varvec{x}})C_1({\varvec{x}})\right] \\&\quad- \dfrac{2}{n|{\varvec{H}}|}\dfrac{{\nu _0}}{f({\varvec{x}})}\dfrac{m_1({\varvec{x}})m_2({\varvec{x}})}{\left[ m_1^2({\varvec{x}})+m_2^2({\varvec{x}})\right] ^2}\\&\quad\cdot \left[ c({\varvec{x}})+f({\varvec{x}})C_3({\varvec{x}})\right] +o_{{\mathbb {P}}}\bigg (\frac{1}{n |{\varvec{H}}|}\bigg ).\end{aligned}$$

Notice that it holds that

$$\begin{aligned} m_1({\varvec{x}})=f_1({\varvec{x}})\ell ({\varvec{x}})\quad {\text{ and }}\quad m_2({\varvec{x}})=f_2({\varvec{x}})\ell ({\varvec{x}}). \end{aligned}$$

(35)

Taking into account that $f^2_1({\varvec{x}})+f_2^2({\varvec{x}})=1$, it can be easily deduced that $\ell ({\varvec{x}})=[m^2_1({\varvec{x}})+m_2^2({\varvec{x}})]^{1/2}$. Therefore, using (25), (26), (29), (30), (31), (32) and (35), it follows that

$$\begin{aligned}&{m_1^2({\varvec{x}})\left[ s_2^2({\varvec{x}})+f({\varvec{x}})C_2({\varvec{x}})\right] + m_2^2({\varvec{x}})\left[ s_1^2({\varvec{x}})+f({\varvec{x}})C_1({\varvec{x}})\right] }\\&\qquad - 2m_1({\varvec{x}})m_2({\varvec{x}})[c({\varvec{x}})+f({\varvec{x}})C_3({\varvec{x}})] \\&\quad =\ell ^2({\varvec{x}})\sigma ^2_1\left[ 1+f({\varvec{x}})\rho _{{\text {c}}_1}\right] . \end{aligned}$$

Consequently, it can be directly obtained that

$$\begin{aligned}{\mathbb {V}}{\text {ar}}[{\hat{m}}_{{\varvec{H}}}({\varvec{x}};p)\mid {\varvec{\mathcal {X}}}]&= \dfrac{{\nu _0}\sigma ^2_1[1+f({\varvec{x}})\rho _{{\text {c}}_1}]}{n|{\varvec{H}}|\ell ^2({\varvec{x}})f({\varvec{x}})} + o_{{\mathbb {P}}}\bigg (\frac{1}{n |{\varvec{H}}|}\bigg ).\end{aligned}$$

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