3
<em>r.regression.multi</em> calculates a multiple linear regression from
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raster maps, according to the formula
5
<div class="code"><pre>
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Y = b0 + sum(bi*Xi) + E
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<div class="code"><pre>
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m = number of explaining variables
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Xi = {xi1, xi2, ..., xin}
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n = number of observations (cases)
19
<div class="code"><pre>
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b0 is the intercept, X0 is set to 1
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<em>r.regression.multi</em> is designed for large datasets that can not
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be processed in R. A p value is therefore not provided, because even
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very small, meaningless effects will become significant with a large
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number of cells. Instead it is recommended to judge by the estimator b,
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the amount of variance explained (R squared for a given variable) and
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the gain in AIC (AIC without a given variable minus AIC global must be
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positive) whether the inclusion of a given explaining variable in the
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<h4>The global model</h4>
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The <em>b</em> coefficients (b0 is offset), R squared or coefficient of
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determination (Rsq) and F are identical to the ones obtained from
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R-stats's lm() function and R-stats's anova() function. The AIC value
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is identical to the one obtained from R-stats's stepAIC() function
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(in case of backwards stepping, identical to the Start value). The
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AIC value corrected for the number of explaining variables and the BIC
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(Bayesian Information Criterion) value follow the logic of AIC.
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<h4>The explaining variables</h4>
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R squared for each explaining variable represents the additional amount
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of explained variance when including this variable compared to when
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excluding this variable, that is, this amount of variance is explained
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by the current explaining variable after taking into consideration all
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the other explaining variables.
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The F score for each explaining variable allows to test if the inclusion
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of this variable significantly increases the explaining power of the
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model, relative to the global model excluding this explaining variable.
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That means that the F value for a given explaining variable is only
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identical to the F value of the R-function <em>summary.aov</em> if the
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given explaining variable is the last variable in the R-formula. While
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R successively includes one variable after another in the order
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specified by the formula and at each step calculates the F value
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expressing the gain by including the current variable in addition to the
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previous variables, <em>r.regression.multi</em> calculates the F-value
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expressing the gain by including the current variable in addition to all
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other variables, not only the previous variables.
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The AIC value is identical to the one obtained from the R-function
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stepAIC() when excluding this variable from the full model. The AIC
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value corrected for the number of explaining variables and the BIC value
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(Bayesian Information Criterion) value follow the logic of AIC. BIC is
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identical to the R-function stepAIC with k = log(n). AICc is not
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available through the R-function stepAIC.
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Multiple regression with soil K-factor and elevation, aspect, and slope
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(North Carolina dataset). Output maps are the residuals and estimates:
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<div class="code"><pre>
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g.region raster=soils_Kfactor -p
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r.regression.multi mapx=elevation,aspect,slope mapy=soils_Kfactor \
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residuals=soils_Kfactor.resid estimates=soils_Kfactor.estim
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<p><i>Last changed: $Date: 2014-12-19 22:55:37 +0100 (Fri, 19 Dec 2014) $</i>