I have a question. As you know, we use Geographically Weighted Regression (GWR) to discover the association between exploratory and dependent variables. Now the question: Do the standardized residual values (shown in red and blue graduated map) indicate a positive or negative association and the direction of this association ? *Can it be concluded that where the standardized residual values are high and red, in these areas the correlation between the variables is positive and significant?? (I want only interpret the standardized residual values).
The high negative and positive StdResiduals are areas that the model poorly fits the data. They do not indicate positive or negative association. See the ArcGIS documentation here, where you will see the exact same map you included in your question.
The documentation states:
(C) Examine the output feature class residuals.
Over- and underpredictions for a well-specified regression model will be randomly distributed. Clustering of over- and/or underpredictions is evidence that you are missing at least one key explanatory variable. Examine the patterns in your OLS and GWR model residuals to see if they provide clues about what those missing variables might be. Run the Spatial Autocorrelation (Moran's I) tool on the regression residuals to ensure that they are spatially random. Statistically significant clustering of high and/or low residuals (model under- and overpredictions) indicates that the GWR model is misspecified.