Radial basis functions with compact support for multivariate geostatistics

Abstract

Matrix-valued radially symmetric covariance functions (also called radial basis functions in the numerical analysis literature) are crucial for the analysis, inference and prediction of Gaussian vector-valued random fields. This paper provides different methodologies for the construction of matrix-valued mappings that are positive definite and compactly supported over the sphere of a d-dimensional space, of a given radius. In particular, we offer a representation based on scaled mixtures of Askey functions; we also suggest a method of construction based on B-splines. Finally, we show that the very appealing convolution arguments are indeed effective when working in one dimension, prohibitive in two and feasible, but substantially useless, when working in three dimensions. We exhibit the statistical performance of the proposed models through simulation study and then discuss the computational gains that come from our constructions when the parameters are estimated via maximum likelihood. We finally apply our constructions to a North American Pacific Northwest temperatures dataset.

Appendix

Some proofs

Proof of Theorem 1.

Direct inspection shows that

$$ \psi_{\nu+\mu}(t) := \frac{\Upgamma(1+\nu+\mu)} {\Upgamma(\nu) \Upgamma(1+\mu)} \int\limits_0^1 \psi_{\nu-1,0,\beta}(t) g(\beta;\mu) \mathrm{d}\beta, \quad t \in [0,\infty), $$

(15)

for $\psi_{\nu-1,0,\beta}(\cdot)\in\Upphi(\mathbb{R}^d),\; \nu\ge\frac{1}{2} d+2$ and $\beta\mapsto g(\beta;\mu) := \beta^\nu(1-\beta)_+^\mu,\quad \mu\ge 0.$ This is evidently of the form in Eq. (8), being a special case of Theorem A, so that we only need to show that the matrix-valued mapping

$$ g(\beta; {\mu}):= \big[g(\beta;\mu_{ij})\big]_{i,j=1}^m $$

belongs to the class $\Upphi^m$ for all $0\le \beta \le 1$ and ${\mu} \in \mathbb{R}_+^{{\text {m}}({\text {m}}+1)/2}.$ To do this we appeal to the stronger statement of diagonal dominance and nonnegativity of the elements on the diagonal, which follows from noting that

$$ c_{ii} g(\beta;\mu_{ii}) \ge \sum_{j\ne i}|c_{ij}| g(\beta;\mu_{ii}) \ge \sum_{j\ne i} |c_{ij}| g(\beta; \mu_{ij}). $$

Because positive definiteness is preserved under scale mixtures and under Schur products with the positive-definite matrix $\mathbf{c}$ of coefficients $c_{ij}$ used in Eq. (9), the proof is complete.

Proof of Proposition 3.

The proof is again constructive. The structure in Eq. (11) is obtained through the Schur product of the matrix based on (9), namely

$$ \left( {\begin{array}{ll} {\sigma _{1}^{2} \frac{{\Gamma (2 + \nu + \mu _{{11}} )}}{{\Gamma (1 + \nu ){\text{ }}\Gamma (1 + \mu _{{11}} )}}} & {\rho _{{12}} \sigma _{1} \sigma _{2} \frac{{\Gamma (2 + \nu + \mu _{{12}} )}}{{\Gamma (1 + \nu ){\text{ }}\Gamma (1 + \mu _{{12}} )}}} \\ {\rho _{{12}} \sigma _{1} \sigma _{2} \frac{{\Gamma (2 + \nu + \mu _{{12}} )}}{{\Gamma (1 + \nu ){\text{ }}\Gamma (1 + \mu _{{12}} )}}} & {\sigma _{2}^{2} \frac{{\Gamma (2 + \nu + \mu _{{22}} )}}{{\Gamma (1 + \nu ){\text{ }}\Gamma (1 + \mu _{{22}} )}}} \\ \end{array} } \right), $$

with the matrix $\mathbf{C}(\cdot)$ defined in Eq. (9) for m = 2, so that we only need to verify that the matrix above is positive definite; this is indeed the case when the condition in Eq. (12) holds. The condition $\mu_{12} \le {\textstyle\frac{1}{2}}(\mu_{11}+\mu_{22})$ comes from the matrix $\mathbf{C} (\cdot)$ in Eq. (11) for the case m = 2 via a determinantal inequality.

Proof of Proposition 5.

We detail some algebra involving the convolution $(\psi_{{\nu}}* \psi_{{\mu}})(t),\quad t\in{\mathbb{R}},$ of the basic Askey functions $\psi_{{\nu}}(t):= (1 - |t|)_+^\nu$ under the restriction $\nu, \mu \in \mathbb N,$ namely

$$ (\psi_{{\nu}}* \psi_{{\mu}})(t):= \int\limits_{-1}^1 (1-|t-u|)_+^\nu (1-|u|)_+^\mu \mathrm{d} u. $$

(16)

As written in (16) the functions are defined on the interval [−1, 1] in ${\mathbb{R}}^1;$ the essential features are that they are nonnegative, are positive definite for $\nu,\mu$ greater or equal than two, and have compact support. The convolution at (16) also has compact support, albeit on [−2, 2].

We evaluate (16) for positive integers $\nu$ and $\mu.$ Observe that the function on the left-hand side is symmetric about the origin so that it is a function of $|t|.$ Indeed, inspection of the right-hand side shows that the first factor of the integrand is exactly $(1-t+u)_+^\nu$for $1<t<2,$and in this range it is nonzero only for $t-1<u<1.$ So for such t (and, indeed, for $|t| = t$) the integral equals

$$ \begin{aligned} \int\limits_{{t - 1}}^{1} {(1 - t + u)^{\nu } } (1 - u)^{\mu } {\text{d}}u =& \int\limits_{0}^{{2 - t}} {v^{\nu } } (2 - t - v)^{\mu } {\text{d}}v \\ = &\int\limits_{0}^{{2 - t}} {\frac{\nu }{{\mu + 1}}} v^{{\nu - 1}} (2 - t - v)^{{\mu + 1}} {\text{d}}v \\ =& \int\limits_{0}^{{2 - t}} {\frac{{\nu !\mu !}}{{(\mu + \nu )!}}} (2 - t - v)^{{\mu + \nu }} {\text{d}}v \\ =& \frac{{\nu !\mu !}}{{(\nu + \mu + 1)!}}(2 - t)^{{\nu + \mu + 1}} . \\ \end{aligned} $$

(17)

For $0<t<1,$ the set of values of u making positive contributions to the convolution expands to $-(1-t) < u < 1,$ while

$$ (1 - |t - u|)_{ + } = \left\{{\begin{array}{ll}{1 - t + u} \hfill & {{\text{for}}\; - (1 - t) < u < 0,} \hfill \\ {1 - t + u} \hfill & {{\text{for}}\;0 < u < t,} \hfill \\ {1 + t - u} \hfill & {{\text{for}}\;t < u < 1.} \hfill \\ \end{array} } \right.$$

Writing the convolution integral at (16) for this range of u as $\int_{-(1-t)}^1 \cdots = \big(\int_{-(1-t)}^0 + \int_0^t + \int_t^1\big)\cdots,$ we evaluate each of these contributions as below:

$$ \begin{aligned} & \int\limits_{t}^{1} {(1 + t - u)^{\nu } } (1 - u)^{\mu } {\text{d}}u \\ & \quad = \left. {\frac{{ - (1 + t - u)^{\nu } (1 - u)^{{\mu + 1}} }}{{\mu + 1}}} \right|_{t}^{1} - \int\limits_{t}^{1} {\frac{{\nu (1 + t - u)^{{\nu - 1}} (1 - u)^{{\mu + 1}} }}{{\mu + 1}}} {\text{d}}u \\ & \quad = \frac{{(1 - t)^{{\mu + 1}} }}{{\mu + 1}} - \frac{{\nu (1 - t)^{{\mu + 2}} }}{{(\mu + 1)(\mu + 2)}} + \int\limits_{t}^{1} {\frac{{\nu (\nu - 1)(1 + t - u)^{{\nu - 2}} (1 - u)^{{\mu + 2}} }}{{(\mu + 1)(\mu + 2)}}} {\text{d}}v \\ & \quad = \sum\limits_{{j = 0}}^{{\nu - 1}} {\frac{{\nu !\mu !( - 1)^{j} (1 - t)^{{\mu + 1 + j}} }}{{(\nu - j)!(\mu + 1 + j)!}}} + ( - 1)^{\nu } \int\limits_{t}^{1} {\frac{{\nu !\mu !}}{{(\mu + \nu )!}}} (1 - u)^{{\mu + \nu }} {\text{d}}v \\ & \quad = \sum\limits_{{j = 0}}^{\nu } {\frac{{\nu !\mu !( - 1)^{j} (1 - t)^{{\mu + 1 + j}} }}{{(\nu - j)!(\mu + 1 + j)!}}} ; \\ \end{aligned} $$

(18)

$$ \begin{aligned} & \int\limits_0^t (1-t+u)^\nu (1-u)^\mu \mathrm{d} u \\ & \quad= \frac{-(1-t+u)^\nu (1-u)^{\mu+1}}{\mu+1} \biggm|_0^t + \int\limits_0^t \frac{\nu(1-t+u)^{\nu-1}(1-u)^{\mu+1}}{\mu+1} \mathrm{d} u \\ & \quad = \sum_{j=0}^\nu \frac{\nu! \mu! [(1-t)^{\nu-j} - (1-t)^{\mu+1+j}]} {(\nu-j)! (\mu+1+j)!} ; \end{aligned} $$

(19)

$$ \begin{aligned} \int\limits_{-(1-t)}^0 (1-t+u)^\nu (1+u)^\mu \mathrm{d} u =& \frac{(1-t+u)^\nu (1+u)^{\mu+1}}{\mu+1}\biggm|_{-(1-t)}^0 - \int\limits_{-(1-t)}^0 \frac{\nu(1-t+u)^{\nu-1} (1+u)^{\mu+1}}{\mu+1} \mathrm{d} u \\ =& \sum_{j=0}^{\nu-1} \frac{\nu! \mu! (-1)^j (1-t)^{\nu-j}}{(\nu-j)! (\mu+1+j)!} + (-1)^\nu \frac{\nu! \mu!}{(\mu+\nu)!} \int\limits_{-(1-t)}^0 (1+u)^{\mu+\nu} \mathrm{d} u \\ =& \sum_{j=0}^\nu \frac{\nu! \mu! (-1)^j (1-t)^{\nu-j}}{(\nu-j)! (\mu+1+j)!} + (-1)^{\nu+1} \frac{\nu! \mu! t^{\mu+\nu+1}}{(\mu+\nu+1)!} . \\ \end{aligned} $$

(20)

Putting together (18)–(20) gives for this integration, when $0<t<1,$

$$ 2\nu !\mu !\left( {\sum\limits_{{j = 0}}^{{\left[ {\frac{1}{2}\nu } \right]}} {\frac{{(1 - t)^{{\nu - 2j}} }}{{(\nu - 2j)!(\mu + 1 + 2j)!}}} - \sum\limits_{{j = 0}}^{{\left[ {\frac{1}{2}(\nu - 1)} \right]}} {\frac{{(1 - t)^{{\mu + 2 + 2j}} }}{{(\nu - 1 - 2j)!(\mu + 2 + 2j)!}}} + \frac{{\frac{1}{2}( - 1)^{{\nu + 1}} t^{{\mu + \nu + 1}} }}{{(\mu + \nu + 1)!}}} \right). $$

Finally then, $(\psi_{{\nu}}*\psi_{{\mu}})(t)$ is zero except when $|t|<2$ where it is as given at (13) and (14) of Proposition 5 for $\nu_i=\nu, \nu_j=\mu$ and $|t|=|x|.$

Derivation of convolution formulae in $\mathbb{R}^3.$

The argument sketched below follows the lines of Theorem 3.c.1 in Gaspari and Cohn (1999), and shows that the convolution $(C_i *C_j)(z)$ of two radial functions compactly supported in the unit ball of $\mathbb{R}^3,$ when their centres are distance z apart, is expressible

$$ z: = x \mapsto C_{{ij}} (z) = \left\{ {\begin{array}{ll} {\frac{{2\pi }}{z}\int_{0}^{1} r C_{i} (r)\int_{{|r - z|}}^{{r + z}} s C_{j} (s){\text{d}}s{\text{d}}r,} \hfill & {z > 0,} \hfill \\ {\int_{0}^{1} {C_{i} } (r)C_{j} (r)4\pi r^{2} {\text{d}}r,} \hfill & {z = 0.} \hfill \\ \end{array} } \right. $$

(21)

Appealing to the formula at (21), we calculate the convolution $(\psi_{{\nu}}* \psi_{{\mu}})(z)$ of two Wendland–Gneiting functions as

$$ \frac{2\pi}{z}\int\limits_0^{1} r (1-r)_+^{\nu} \int\limits_{|r-z|}^{r+z} s (1-s)_+^{\mu} \mathrm{d} s \mathrm{d} r. $$

(22)

The inner integrand at (22) equals

$$ \int\limits_{\max(r-z,z-r)}^{\min(r+z,1)} s(1-s)^\mu \mathrm{d} s, $$

where the upper limit replaces the truncation in the integrand. For the case $r>z$ this equals

$$ \int\limits_{{r - z}}^{{\min (1,r + z)}} s (1 - s)^{\mu } {\text{d}}s = \left. {\left( {\frac{{ - s(1 - s)^{{\mu + 1}} }}{{\mu + 1}} - \frac{{(1 - s)^{{\mu + 2}} }}{{(\mu + 1)(\mu + 2)}}} \right)} \right|_{{r - z}}^{{\min (1,r + z)}} $$

after integration by parts, and depending on $r+z>$ or $\le 1,$ this equals

$$ \begin{array}{ll} \begin{gathered} \frac{{(r - z)(1 - r + z)^{{\mu + 1}} }}{{\mu + 1}} + \frac{{(1 - r + z)^{{\mu + 2}} }}{{(\mu + 1)(\mu + 2)}}, \hfill \\ \frac{{ - (r + z)(1 - r - z)^{{\mu + 1}} + (r - z)(1 - r + z)^{{\mu + 1}} }}{{\mu + 1}} \hfill \\ \end{gathered} & {{\text{if }}r + z > 1,} \\ \quad { + \frac{{ - (1 - r - z)^{{\mu + 2}} + (1 - r + z)^{{\mu + 2}} }}{{(\mu + 1)(\mu + 2)}},} & {{\text{if }}r + z \le 1.} \\ \end{array} $$

The case $z>r$ equals

$$ \begin{aligned} & \int\limits_{z-r}^{\min(1,r+z)} s(1-s)^\mu \mathrm{d} s\\ & \quad = \bigg(\frac{-s(1-s)^{\mu+1}}{\mu+1} - \frac{(1-s)^{\mu+2}}{(\mu+1)(\mu+2)} \bigg)\bigg|_{z-r}^{\min(1,r+z)} , \\ &\quad = \frac{(z-r)(1-z+r)^{\mu+1}} {\mu+1} + \frac{(1-z+r)^{\mu+2}} {(\mu+1)(\mu+2)}, \quad\hbox{if }r+z > 1, \\ & \quad= \frac{-(r+z)(1-r-z)^{\mu+1} + (z-r)(1-z+r)^{\mu+1} }{\mu+1} \\ & \quad\quad+ \frac{-(1-r-z)^{\mu+2} + (1-z+r)^{\mu+2} } {(\mu+1)(\mu+2)}, \quad\hbox{if }r+z\le 1. \end{aligned} $$

Here the case that $r+z>1$ can be given as the single expression

$$ \frac{(1 - |z-r|)^{\mu+2}}{(\mu+1)(\mu+2)} + \frac{|z-r|(1 - |z-r|)^{\mu+1}}{\mu+1}; $$

the case $r+z\le 1$ also gives a single expression in terms of $|z-r|.$

To evaluate the expression at (22) write $\int_0^1 (\cdots) \mathrm{d} r = \Big(\int_0^{1-z} + \int_{1-z}^1 \Big) (\cdots) \mathrm{d} r.$ This gives two integrals

$$ J_1:\,= \int\limits_{1-z}^1 r(1-r)^\nu \bigg[ \frac{(1 - |z-r|)^{\mu+2}} {(\mu+1)(\mu+2)} + \frac{|z-r|(1 - |z-r|)^{\mu+1}} {\mu+1}\bigg] \mathrm{d} r $$

and

$$ \begin{aligned} J_2 & := \int\limits_0^{1-z} r(1-r)^\nu \bigg[ \frac{|z-r|(1 - |z-r|)^{\mu+1} - (r+z)(1 - r - z)^{\mu+1}}{\mu+1} \\ & \quad +\frac{(1 - |z-r|)^{\mu+2} - (1-r-z)^{\mu+2}}{(\mu+1)(\mu+2)} \bigg] \mathrm{d} r. \\ \end{aligned} $$

To evaluate J ₁ we must distinguish the two cases according as $1-z>$or$< z,$ i.e. according as $z<\frac{1}{2}$ or $z>\frac{1}{2}.$ For the simpler case $z<\frac{1}{2},$

$$ J_1 = \int\limits_{1-z}^1 \frac{r(1-r)^\nu}{\mu+1} \bigg[\frac{(1-r+z)^{\mu+2}}{\mu+2} + (r-z)(1 - r + z)^{\mu+1}\bigg] \mathrm{d} r, $$

while for the case $z>\frac{1}{2}$ we use $\int_{1-z}^1 = \int_{1-z}^z + \int_z^1$ and find

$$ \begin{aligned} J_1 =& \int\limits_{1-z}^z \frac{r(1-r)^\nu}{\mu+1} \bigg[\frac{(1-z+r)^{\mu+2}}{\mu+2} + (z-r)(1-z+r)^{\mu+1}\bigg] \mathrm{d} r \cr & {}+ \int\limits_z^1 \frac{r(1-r)^\nu}{\mu+1} \bigg[\frac{(1-r+z)^{\mu+2}}{\mu+2} + (r-z)(1-r+z)^{\mu+1}\bigg] \mathrm{d} r \\ :=\, & J_{11} + J_{12} . \end{aligned} $$

The rest of the formula is then deduced through simple, albeit tedious, algebra. Table 2 shows some special cases of the formulae above, obtained for $\mu$ and $\nu$ positive integers.

Table 2 A detailed calculation of the convolution formulae in ${\mathbb{R}^3}$ for special values of the parameters $\mu$ and $\nu$