# Reasons for GLM ('identity') performing better than GLM ('gamma') for predicting a gamma distributed variable?

I am investigating different methods for fitting my target variable (observed wind speed: positive, real, with small values being most probable) using generalized linear modeling (GLM) and - in a first instance - a single predictor (wind speed output from a numerical weather prediction model).

I found that a GLM based on the 'identity' link function (for normal distributions, equal to ordinary least squares regression) performs better than a GLM based on the 'reciprocal' link function (for gamma distributions): 1/mu=Xb. I also tried 'inverse Gaussian: 1/mu^2=Xb.

As a measure of model performance I use skill scores calculated as SS= 1-mse/mse_(ref); where mse is the cross-validation squared error of the GLM and mse_(ref) is the cross-validation squared error of the reference model, with the reference model being the mean value of the observed wind speed.

I tried calculating SS using the expectation value of the gamma distribution as the reference model (E(y)=omega*phi, with omega and phi being the estimated shape and scale parameters of a gamma distribution fitted to Y), but found out that this is exactly the mean of Y.

While all three gamma, inverse Gaussian and Gaussian - based GLMs show significant prediction skill, the Gaussian option shows definitely the highest skill. What could be the reasons why a gamma GLM does not improve, but even lower the model performance?

• What makes you think that the output variable should be gamma distributed? May 8, 2015 at 12:31
• Because wind speed values are positive and real (in general), and the distribution is skewed towards low values (close to zero - in my case study), while values equal to zero do not occur. In literature, gamma GLMs are often used in statistical postprocessing (downscaling) of numerical weather prediction output for wind speed. May 13, 2015 at 13:53

## 1 Answer

Your outcome variable is measured wind speed, while the predictor is wind speed calculated from a numerical weather forecast model. So, input and output has the same units of measurement, and, assuming the forecasting model is good, the result of the model should be some small adjustment (maybe also depending on some other predictors).

The identity link than says that expectation of wind speed is a linear function of forecasted wind speed, which seems natural. The log link would imply that the expectation of wind speed is the exponential of a linear function of forecasted wind speed, which seems unnatural.

Note that you can use identity link function also with a gamma model, there is no need to use the (canonical) inverse link. It is not clear from your post if you did try that!