Generalized Whittle–Matérn and polyharmonic kernels
Generalized Whittle–Matérn and polyharmonic kernels
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DOI：
10.1007/s10444-012-9277-9
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发表时间：
2013-07
影响因子：
1.7
通讯作者：
M. Bozzini;M. Rossini;R. Schaback
中科院分区：
文献类型：
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作者：
M. Bozzini;M. Rossini;R. Schaback
本文同时推广了两类标准的径向核，即与微分算子（- Δ）有关的多谐核和与微分算子（- Δ +I）m有关的whittle - matsamrn核。这是通过允许具有非零κj形式的一般微分算子并计算其相关核来实现的。结果表明，它们可以通过从缩放的whittle - mat<s:1>核开始并对其缩放取除差来显式给出。它们是正定的径向核它们在希尔伯特空间中复制核，范数等价。另一方面，我们证明了这种形式的广义逆多重二次核是正定的，并给出了它们的傅里叶变换。令人惊讶的是，这些傅里叶变换导致了惠特尔-马塔姆形式的核，其变量尺度κ(r)在κ1，…，κm之间。我们还考虑了一些基因消失的情况。这就产生了条件正定核，它是上述变尺度whittle - mat<s:1>核和多谐核的线性组合。文中还加了一些数值例子作说明。
This paper simultaneously generalizes two standard classes of radial kernels, the polyharmonic kernels related to the differential operator ( − Δ)mand the Whittle–Matérn kernels related to the differential operator ( − Δ +I)m. This is done by allowing general differential operators of the formwith nonzeroκjand calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle–Matérn kernels and taking divided differences with respect to their scale. They are positive definite radial kernels which are reproducing kernels in Hilbert spaces norm-equivalent to. On the side, we prove that generalized inverse multiquadric kernels of the formare positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle–Matérn form with a variable scaleκ(r) betweenκ1,...,κm. We also consider the case where some of theκjvanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle–Matérn kernels and polyharmonic kernels. Some numerical examples are added for illustration.