NAME

       r.regression.multi  - Calculates multiple linear regression from raster maps.

KEYWORDS

       raster, statistics, regression

SYNOPSIS

       r.regression.multi
       r.regression.multi --help
       r.regression.multi      [-g]      mapx=name[,name,...]     mapy=name      [residuals=name]
       [estimates=name]   [output=name]   [--overwrite]  [--help]  [--verbose]  [--quiet]  [--ui]

   Flags:
       -g
           Print in shell script style

       --overwrite
           Allow output files to overwrite existing files

       --help
           Print usage summary

       --verbose
           Verbose module output

       --quiet
           Quiet module output

       --ui
           Force launching GUI dialog

   Parameters:
       mapx=name[,name,...] [required]
           Map for x coefficient

       mapy=name [required]
           Map for y coefficient

       residuals=name
           Map to store residuals

       estimates=name
           Map to store estimates

       output=name
           ASCII file  for  storing  regression  coefficients  (output  to  screen  if  file  not
           specified).

DESCRIPTION

       r.regression.multi  calculates a multiple linear regression from raster maps, according to
       the formula
       Y = b0 + sum(bi*Xi) + E
       where
       X = {X1, X2, ..., Xm}
       m = number of explaining variables
       Y = {y1, y2, ..., yn}
       Xi = {xi1, xi2, ..., xin}
       E = {e1, e2, ..., en}
       n = number of observations (cases)
       In R notation:
       Y ~ sum(bi*Xi)
       b0 is the intercept, X0 is set to 1

       r.regression.multi is designed for large datasets that can not be  processed  in  R.  A  p
       value  is therefore not provided, because even very small, meaningless effects will become
       significant with a large number of cells. Instead  it  is  recommended  to  judge  by  the
       estimator  b,  the  amount  of variance explained (R squared for a given variable) and the
       gain in AIC (AIC without a given variable minus AIC global must be positive)  whether  the
       inclusion of a given explaining variable in the model is justified.

   The global model
       The  b  coefficients (b0 is offset), R squared or coefficient of determination (Rsq) and F
       are identical to the ones obtained from R-stats’s  lm()  function  and  R-stats’s  anova()
       function. The AIC value is identical to the one obtained from R-stats’s stepAIC() function
       (in case of backwards stepping, identical to the Start value). The AIC value corrected for
       the  number  of  explaining  variables  and the BIC (Bayesian Information Criterion) value
       follow the logic of AIC.

   The explaining variables
       R squared for each explaining variable  represents  the  additional  amount  of  explained
       variance  when  including this variable compared to when excluding this variable, that is,
       this amount of variance is explained by the current explaining variable after taking  into
       consideration all the other explaining variables.

       The  F score for each explaining variable allows testing if the inclusion of this variable
       significantly increases the explaining power of the model, relative to  the  global  model
       excluding  this  explaining  variable.  That means that the F value for a given explaining
       variable is only identical to the F value of  the  R-function  summary.aov  if  the  given
       explaining  variable  is the last variable in the R-formula. While R successively includes
       one variable after another in the  order  specified  by  the  formula  and  at  each  step
       calculates  the  F value expressing the gain by including the current variable in addition
       to the previous variables, r.regression.multi calculates the F-value expressing  the  gain
       by  including  the  current  variable  in  addition  to  all other variables, not only the
       previous variables.

       The AIC value is identical  to  the  one  obtained  from  the  R-function  stepAIC()  when
       excluding  this  variable  from  the full model. The AIC value corrected for the number of
       explaining variables and the BIC value (Bayesian Information Criterion) value  follow  the
       logic  of  AIC.  BIC  is  identical to the R-function stepAIC with k = log(n). AICc is not
       available through the R-function stepAIC.

EXAMPLE

       Multiple regression with soil K-factor and elevation, aspect, and  slope  (North  Carolina
       dataset). Output maps are the residuals and estimates:
       g.region raster=soils_Kfactor -p
       r.regression.multi mapx=elevation,aspect,slope mapy=soils_Kfactor \
         residuals=soils_Kfactor.resid estimates=soils_Kfactor.estim

SEE ALSO

        d.correlate, r.regression.line, r.stats

AUTHOR

       Markus Metz

       Last changed: $Date: 2015-10-27 18:41:34 +0100 (Tue, 27 Oct 2015) $

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