An approach for valid covariance estimation via the Fourier series

Abstract

The use of kriging for construction of prediction or risk maps requires estimating the dependence structure of the random process, which can be addressed through the approximation of the covariance function. The nonparametric estimators used for the latter aim are not necessarily valid to solve the kriging system, since the positive-definiteness condition of the covariance estimator typically fails. The usage of a parametric covariance instead may be attractive at first because of its simplicity, although it may be affected by misspecification. An alternative is suggested in this paper to obtain a valid covariance from a nonparametric estimator through the Fourier series tool, which involves two issues: estimation of the Fourier coefficients and selection of the truncation point to determine the number of terms in the Fourier expansion. Numerical studies for simulated data have been conducted to illustrate the performance of this approach. In addition, an application to a real environmental data set is included, related to the presence of nitrate in groundwater in Beja District (Portugal), so that pollution maps of the region are generated by solving the kriging equations with the use of the Fourier series estimates of the covariance.

Appendices

Appendix 1: A Fourier series approach

Let $f: E \longrightarrow I\!R$ be a bounded function, with E = [0, e ₁] × [0, e ₂], for some e ₁, e ₂ > 0. Then, a complete and countable orthonormal basis $\{ \psi_{i}: E \longrightarrow I\!R : i \in I\!N \},$ can be constructed, satisfying that:

$$ f(t)=\sum_{i \in I\!N } \theta_{f,i} \psi_{i} (t),\quad \hbox{for all }t \in E $$

(9)

where $\theta_{f,i}=\int_{E} f (t) \psi_i (t) \,\hbox{d}(t)$ is the ith Fourier coefficient of f. The linear expansion above is called a Fourier expansion of f.

The existence of such a basis can be derived from the following properties:

• The unidimensional cosine system $\psi_{i,e}(x)= a_{i} \cos (i \pi x e^{-1})$ is a complete orthonormal basis on [0, e], where a _i equals e ^−1/2 or (0.5 e)^−1/2, for i = 0 or i > 0, respectively.

• The set $\{ \psi_{i_1,i_2}: E \longrightarrow I\!R : i_1,i_2 \in I\!N \},$ with $\psi_{i_1,i_2}(t)=\psi_{i_1,e_1}(t_1)\psi_{i_1,e_2}(t_2)$ and t = (t ₁, t ₂), is a complete orthonormal basis on E (Zygmund 2002).

• A bijection $g: I\!N^2 \rightarrow I\!N$ can be established, referred to as Cantor’s diagonal function, with g(i ₁, i ₂) = 0.5 (i ₁ + i ₂)(i ₁ + i ₂ + 1) + i ₁, whose values are displayed in Table 3.

Table 3 Values of Cantor’s diagonal function

Then, a complete orthonormal basis on E would be given by:

$$ \psi_{i} (t)=\psi_{i_1,e_1} (t_1)\psi_{i_2,e_2} (t_2),\quad \hbox{with } (i_1,i_2) = g^{-1} (i) $$

(10)

For example, the first values of function g ⁻¹ are $g^{-1}(0)=(0,0), g^{-1}(1)=(0,1), g^{-1}(2)=(1,0), \ldots$

Furthermore, for every $\varepsilon>0$ there exists a number $M \in I\!N$ such that $| f(t) - f_m(t)| < \varepsilon$ for all m > M, with:

$$ f_m(t)=\sum_{i \leq m } \theta_{f,i} \psi_{i} (t), \quad \hbox{for all }t \in E $$

For practical reasons, it is customary to use the truncated partial sum f _m (t), instead of expansion (9), for approximation of f.

Appendix 2: Consistency of $\theta_{{\hat{C}},i}$

From relation (2) and the fact that the basis is orthonormal, one has:

$$ \begin{aligned} | \hbox{Bias} [ \theta_{{\hat{C}},i} ] |& = \left| \int\limits_{B} \hbox{Bias} [ {\hat{C}} (t) ] \psi_{i}(t) \,\hbox{d}t\right | \leq \mathop{\hbox{sup}}\limits_{{\rm t} \in E } | \hbox{Bias} [ {\hat{C}} (t) ]| \int\limits_{E} | \psi_{i}(t) | \,\hbox{d}t \\ & \leq b_i \mathop{\hbox{sup}}\limits_{{\rm t} \in E } | \hbox{Bias} [ {\hat{C}} (t) ]| \mathop{\longrightarrow }\limits^{n\to \infty} 0 \end{aligned} $$

(11)

for some positive constant b _i , together with:

$$ \begin{aligned} \hbox{Var} [ \theta_{{\hat{C}},i} ] & = \int\limits_{E} \int\limits_{E} \hbox{Cov} [ {\hat{C}}(t), {\hat{C}}(t')] \psi_{i}(t) \psi_{i}(t') \,\hbox{d}t \,\hbox{d}t' \leq \left ( \int\limits_{E} \hbox{Var} [ {\hat{C}}(t) ]^{1/2} | \psi_{i}(t) | \,\hbox{d}t \right)^2\\ & \leq \mathop{\hbox{sup}}\limits_{{\rm t} \in E } \hbox{Var} [ {\hat{C}} (t) ] \left( \int\limits_{E} | \psi_{i}(t) | \,\hbox{d}t \right)^2 \leq b_{i}^{2} \mathop{\hbox{sup}}\limits_{{\rm t} \in E } \hbox{Var} [ {\hat{C}} (t) ] \mathop{\longrightarrow }\limits^{n\to \infty} 0 \end{aligned} $$

(12)

The fact that the bias and variance of $\theta_{{\hat{C}},i}$ tend to zero implies that $\theta_{{\hat{C}},i} \mathop{\longrightarrow}\limits^{P} \theta_{C,i}$ and, therefore, the consistency of $\theta_{{\hat{C}},i}.$

Appendix 3: MISE of ${\hat{C}}_{1,m}$

Observe that:

$$ \begin{aligned} \hbox{MISE} [ {\hat{C}}_{1,m}, C ]&=\hbox{E} \left[ \int ( {\hat{C}}_{1,m} (t)- C (t) )^2\,\hbox{d}t \right] =\sum_{i \leq m} \hbox{E} [ ( \theta_{{\hat{C}},i} - \theta_{C,i} )^{2} ] + \sum_{i>m } \theta_{C,i}^{2}\\ &=\sum_{i \leq m} ( \hbox{Var} [ \theta_{{\hat{C}},i} ]+ \hbox{Bias} [ \theta_{{\hat{C}},i} ]^2 ) + \sum_{i>m } \theta_{C,i}^{2} \end{aligned} $$

By Parseval’s identity (Efromovich 1999), one has:

$$ \sum_{i>m } \theta_{C,i}^{2} = \int\limits_{E} C(t)^2 \,\hbox{d}t - \sum_{i \leq m} \theta_{C,i}^{2} $$

Then:

$$ \hbox{MISE} [ {\hat{C}}_{1,m}, C ] =\int\limits_{B} C(t)^2 \,\hbox{d}t + \sum_{i \leq m} ( \hbox{Var} [ \theta_{{\hat{C}},i} ]+ \hbox{Bias} [ \theta_{{\hat{C}},i} ]^2 - \theta_{C,i}^{2} ) $$

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Appendix 4: Properties of ${\hat{C}}_{2,m}$

Firstly, the positive-definiteness of ${\hat{C}}_{2,m}$ will be proved. For the latter aim, take into account that each function in the basis $\{ \psi_{i} : i \in I\!N \}$ is obtained from the unidimensional cosine system, as given in (10). Then, ψ _i is positive-definite on account of the fact that $\cos(x-x')= \cos(x) \cos(x')+\sin(x)\sin(x').$ This means that for each $i \in I\!N$:

$$ c_i= \sum_{j=1}^{n} \sum_{k=1}^{n} d_{j} d_{k} \psi_{i} ( s_{j} - s_{k} ) \geq 0 $$

(14)

for any set of locations { s _j } ⁿ _j=1 and real numbers { d _j } ⁿ _j=1 .

By (14) and the definition of the weights w _i , one has:

$$ \sum_{j=1}^{n} \sum_{k=1}^{n} d_{j} d_{k} {\hat{C}}_{2,m} ( s_{j} - s_{k} )= \sum_{i \leq m} c_i w_i \theta_{{\hat{C}},i} \geq 0 $$

to conclude the positive-definiteness of ${\hat{C}}_{2,m}.$

By proceeding similarly as in Appendix 3, the MISE of ${\hat{C}}_{2,m}$ could be developed to yield:

$$ \begin{aligned} \hbox{MISE} [ {\hat{C}}_{2,m} , C ] & = E \left[ \int ( {\hat{C}}_{2,m} (t)- C (t) )^2\,\hbox{d}t \right]=\sum_{i \leq m} {\rm E} [ ( w_i \theta_{{\hat{C}},i} - \theta_{C,i} )^{2} ] + \sum_{i > m } \theta_{C,i}^{2}\\ &=\sum_{i \leq m} w_i E [ ( \theta_{{\hat{C}},i} - \theta_{C,i} )^{2} ] + \sum_{i \leq m} ( 1- w_i ) \theta_{C,i}^{2} + \sum_{i > m } \theta_{C,i}^{2} \\ & = \sum_{i \leq m} w_i ( \hbox{Var} [ \theta_{{\hat{C}},i} ]+ \hbox{Bias} [ \theta_{{\hat{C}},i} ]^2 - \theta_{C,i}^{2} ) + \int\limits_{E} C(t)^2 \,\hbox{d}t \end{aligned} $$

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