# Getting Negative values in Regression based Biomass model!

I am using Multi Linear Regression (stepwise method) to estimate above ground biomass. Different spectral indices and other variables mostly derived from Landsat 8 OLI are used to check potential relationship with the biomass samples. (The biomass samples range between 7 to 37 tons). The adjusted R Square hovers around 0.6 which is respectable.

The resultant equation contains coefficients which are statistically significant. However, the input equation produces raster with values ranged between -24 to 57.

I am not sure what might be the source of these negative values in the raster and how should I deal with these negative values?

• What GIS software are you using?
– PolyGeo
Jul 28 '17 at 5:30
• Are all calculations raster based? Or are you also using points/lines/polygons?
– LMB
Jul 28 '17 at 8:10

The crux is most likely in the fact that you are using a MLR, as this method can indeed result in values below 0, while a negative biomass obviously does not exist in real life.

As for the statistical background, values below zero are caused by the combination of your predictors (independent variables). Where the outputs are below 0, your predictors in combination with your betas (the values that determine how strong and what direction the relation between your preditors and dependent variable is) result in the output being below zero. You should therefore investigate where the negative values are, and what the predictor input values are at that location. If you have the regression equation you will get a mathematical explanation why the predicted value at that location is below 0. From that point you know what predictor is/ predictors are causing the negative value. You will have to apply your common sense/biological expertise to see what you should do next. Several options are:

1. Remove the predictor that causes the output to be negative
2. Investigate wheter the relationship between the predictor and the independent variable is indeed linear. You can check this with a error risidual plot. The risiduals should be evenly distributed along the predictors range (as is in the last example of the picture). If the relation is not linear (the first or second example), you should not use a linear relationship. If your software allows it you should change the relationship to a exponential or other type that better fits the data. Also, it is worth posing youself the question what the goal of your analysis is, as this will influence your statistical methods. See this or this article. In short:

• if you are predicting, a data driven approach can be better. Multicolinearity and significance are less of a problem, the only criterion is how well you predict (but do be carefull for overfitting)
• if you are explaining, strict assumtpions exist, and there is need for a theory driven model.

Also pose yourself the question whether or not the use of 1 model through the research area is valid. Perhaps you should use a geographically weighted model (either linear regression or otherwise).

• I am grateful for your time and contribution, please let me add that most of the input variables were showing linear trend so I used Multi Linear Regression as best option. Secondly, please also let me know how would I know which variable is producing negative values?
– Rex
Aug 1 '17 at 5:24
• What are fundamental difference between Geographically Weighted Model and MLR?
– Rex
Aug 1 '17 at 5:25
• Fundamental differences between the GWLR (geo-weighted) and the MLR is that the MLR is one single model. i.e. there is one single formula with values. A GWLR is acutally a unique equation on each geographical location. i.e. the betas change over geographic space. To know what variables are causing the negative values you should find the negative outputs, and determine the inputs at that location. Fill them in the formula for yourself and that you should see what variable(s) produce the negative output.
– LMB
Aug 27 '17 at 19:27