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  "body": "\n## Definition\n\nLet $D \\subseteq \\R^d$ and let $(Z(s))_{s \\in D}$ be a real random field of the form $Z(s) = \\mu + \\delta(s)$.\nHere $\\mu$ is an unknown constant and $\\delta$ is intrinsically stationary with [[variogram|ZF8AA2JVKV]] $2\\gamma$ and $E[\\delta(s)] = 0$.\nLet $s_1, \\dots, s_n, s_0 \\in D$, and write $\\gamma_{ij} = \\gamma(s_i - s_j)$ for $0 \\le i, j \\le n$.\n\nThe *ordinary kriging predictor* of $Z(s_0)$ is $\\hat Z(s_0) = \\sum_{i=1}^{n} \\lambda_i Z(s_i)$, where the weights minimise the [[mean squared error|05NZ1HHQMV]] $E[(Z(s_0) - \\hat Z(s_0))^2]$ subject to $\\sum_{i=1}^{n} \\lambda_i = 1$.\nEquivalently, $\\lambda_1, \\dots, \\lambda_n$ and a number $m$ solve the linear system\n$$\\sum_{j=1}^{n} \\lambda_j \\gamma_{ij} + m = \\gamma_{i0} \\quad (1 \\le i \\le n), \\qquad \\sum_{j=1}^{n} \\lambda_j = 1.$$\n\n## Remarks\n\nThe constraint $\\sum_i \\lambda_i = 1$ makes the predictor unbiased, $E[\\hat Z(s_0)] = \\mu = E[Z(s_0)]$, whatever the value of $\\mu$.\nThe minimal mean squared error, the kriging variance, is $\\sum_{i=1}^{n} \\lambda_i \\gamma_{i0} + m$.\nSimple kriging assumes $\\mu$ known and drops the constraint; universal kriging replaces $\\mu$ by a linear combination of known functions of $s$.\n\nExample: with one observation, $n = 1$, the constraint forces $\\lambda_1 = 1$, so $\\hat Z(s_0) = Z(s_1)$.\nThe first equation then gives $m = \\gamma_{10} - \\gamma_{11} = \\gamma(s_1 - s_0)$, since $\\gamma(0) = 0$, so the kriging variance is $2\\gamma(s_1 - s_0)$, which is indeed $E[(Z(s_0) - Z(s_1))^2]$.\n\nNon-example: inverse distance weighting also predicts by a weighted average with weights summing to $1$.\nIts weights are fixed by the distances alone and do not minimise the mean squared error for the given variogram, so it is not kriging.\n",
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