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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Regression-kriging ： ウィキペディア英語版 Regression-kriging In applied statistics, regression-kriging (RK) is a spatial prediction technique that combines a regression of the dependent variable on auxiliary variables (such as parameters derived from digital elevation modelling, remote sensing/imagery, and thematic maps) with kriging of the regression residuals. It is mathematically equivalent to the interpolation method variously called ''universal kriging'' and ''kriging with external drift'', where auxiliary predictors are used directly to solve the kriging weights. == BLUP for spatial data == Regression-kriging is an implementation of the best unbiased linear predictor for spatial data, i.e. the best linear interpolator assuming the universal model of spatial variation. Matheron (1969) proposed that a value of a target variable at some location can be modeled as a sum of the deterministic and stochastic components: :$$ Z(\mathbf) = m(\mathbf) + \varepsilon '(\mathbf) + \varepsilon '' which he termed ''universal model of spatial variation''. Both deterministic and stochastic components of spatial variation can be modeled separately. By combining the two approaches, we obtain: :$$ \hat z(\mathbf_0 ) = \hat m(\mathbf_0 ) + \hat e(\mathbf_0 )= \sum\limits_^p + \sum\limits_^n \lambda_i \cdot e(\mathbf_i ) where $\backslash hat\; m(\backslash mathbf\_0)$ is the fitted deterministic part, $\backslash hat\; e(\backslash mathbf\_0)$ is the interpolated residual, $\backslash hat\; \backslash beta\; \_k$ are estimated deterministic model coefficients ($\backslash hat\; \backslash beta\; \_0$ is the estimated intercept), $\backslash lambda\_i$ are kriging weights determined by the spatial dependence structure of the residual and where $e(\backslash mathbf\_i)$ is the residual at location $\_\backslash mathtt\; =\; \backslash left(\; \backslash mathbf^\backslash mathbf\; \backslash cdot$ \mathbf^ \right)^^\mathbf \cdot \mathbf^ where $\backslash mathbf\_\backslash mathtt$ is the vector of estimated regression coefficients, $\backslash mathbf$ is the covariance matrix of the residuals, $$ is the vector of measured values of the target variable. The GLS estimation of regression coefficients is, in fact, a special case of the geographically weighted regression. In the case, the weights are determined objectively to account for the spatial auto-correlation between the residuals. Once the deterministic part of variation has been estimated (regression-part), the residual can be interpolated with kriging and added to the estimated trend. The estimation of the residuals is an iterative process: first the deterministic part of variation is estimated using OLS, then the covariance function of the residuals is used to obtain the GLS coefficients. Next, these are used to re-compute the residuals, from which an updated covariance function is computed, and so on. Although this is by many geostatisticians recommended as the proper procedure, Kitanidis (1994) showed that use of the covariance function derived from the OLS residuals (i.e. a single iteration) is often satisfactory, because it is not different enough from the function derived after several iterations; i.e. it does not affect much the final predictions. Minasny and McBratney (2007) report similar results—it seems that using more higher quality data is more important then to use more sophisticated statistical methods. In matrix notation, regression-kriging is commonly written as: :$$ \hat z_\mathtt(\mathbf_0 ) = \mathbf_\mathbf^\mathbf \cdot \mathbf_\mathtt + \mathbf_\mathbf^\mathbf \cdot (\mathbf - \mathbf \cdot \mathbf_\mathtt ) where $\backslash hat\; z(\}\_0$, $\}$ is the vector of $p+1$ predictors and $\backslash mathbf\_^2\; (\backslash mathbf\_0)$ = (C_0 + C_1 ) - \mathbf_\mathbf^\mathbf \cdot \mathbf^\mathbf \cdot \mathbf_\mathbf + \left( \mathbf_\mathbf - \mathbf^\mathbf \cdot \mathbf^_\mathbf \right)^\mathbf \cdot \left( \mathbf^\mathbf \cdot \mathbf^ \right)^\mathbf \cdot \left(\mathbf_\mathbf - \mathbf^\mathbf \cdot \mathbf^_\mathbf \right) where $C\_0\; +\; C\_1$ is the sill variation and $$) becomes an identity matrix. Likewise, if the target variable shows no correlation with the auxiliary predictors, the regression-kriging model reduces to ordinary kriging model because the deterministic part equals the (global) mean value. Hence, pure kriging and pure regression should be considered as only special cases of regression-kriging (see figure). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Regression-kriging」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.