论文标题
在紧凑型公制空间上渐近地对随机场的最佳最佳线性预测的必要条件
Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces
论文作者
论文摘要
随机字段$ \ {z（x）\} _ {x \ in \ Mathcal {x}} $由紧凑的度量码$（\ Mathcal {x}，d _ {\ nathcal {x {x {x {x} $ come oper -nare come oble of the nife factor， $ M \ COLON \ MATHCAL {X} \ to \ MATHBB {R} $和协方差函数$ \ Varrho \ Varrho \ colon \ Mathcal \ Mathcal {X} \ Times \ Times \ Times \ Mathcal {X}} \ to \ Mathbb {r Mathbb {r} $ of $ z $已知。我们考虑在某个位置预测$ z（x^*）$的问题$ x^*\ in \ int \ mathcal {x} $，基于在位置的观察值$ \ {x_j \} _ {x_j \} _ {j = 1}^n $，这些{ $ \ {φ_j（z）\} _ {j = 1}^n $用于线性函数$φ，φ_1，\ ldots，φ_n$）。我们的主要结果是基于不正确的二阶结构$（\ tilde {m}，\ tilde {\ varrho}）$的线性预测变量（$ n $增加）的渐近性能，而没有对$ \ varrho，\ varrho，\ tilde {\ tilde {\ varrho} $ sistarity的任何限制性假设。我们首次在$（\ tilde {m}，\ tilde {\ varrho}）$上提供必要和充分的条件，以均与相应的线性预测器相对于$φ$均匀地保持的相应线性预测变量的渐近最优性。这些一般结果通过$ \ Mathcal {x} \ subset \ Mathbb {r}^d $带有matérn或定期协方差函数的弱固定随机字段进行说明，以及在Sphere $ \ Mathcal {x} = \ Mathbb {s} s}^2 $ for Case case of Case case case case case cosotropic covariance covariance case case case。
Optimal linear prediction (aka. kriging) of a random field $\{Z(x)\}_{x\in\mathcal{X}}$ indexed by a compact metric space $(\mathcal{X},d_{\mathcal{X}})$ can be obtained if the mean value function $m\colon\mathcal{X}\to\mathbb{R}$ and the covariance function $\varrho\colon\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ of $Z$ are known. We consider the problem of predicting the value of $Z(x^*)$ at some location $x^*\in\mathcal{X}$ based on observations at locations $\{x_j\}_{j=1}^n$ which accumulate at $x^*$ as $n\to\infty$ (or, more generally, predicting $φ(Z)$ based on $\{φ_j(Z)\}_{j=1}^n$ for linear functionals $φ,φ_1,\ldots,φ_n$). Our main result characterizes the asymptotic performance of linear predictors (as $n$ increases) based on an incorrect second order structure $(\tilde{m},\tilde{\varrho})$, without any restrictive assumptions on $\varrho,\tilde{\varrho}$ such as stationarity. We, for the first time, provide necessary and sufficient conditions on $(\tilde{m},\tilde{\varrho})$ for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to $φ$. These general results are illustrated by weakly stationary random fields on $\mathcal{X}\subset\mathbb{R}^d$ with Matérn or periodic covariance functions, and on the sphere $\mathcal{X}=\mathbb{S}^2$ for the case of two isotropic covariance functions.
