## 10 Spatial Predictions, Prediction Uncertainty, and Their Visualization
### Prediction Uncertainty  
Prediction uncertainty in environmental modeling may refer to various aspects of the modeling process and is used to encompass many concepts [117; 118; 119; 120]. It can result from various sources or factors as previously discussed [121; 17; 120]. In this study, the uncertainty which is produced by a predictive model is about spatial predictions. Prediction uncertainty is increasingly required for decision-makingand many methods are used to produce such uncertainty. In this study, I focus on the uncertainty produced by some commonly used methods: OK, LM, and RF.  
For OK, prediction variances can be produced [58]. However, it was shown that the variances produced are independent of the actual predicted values [122]. Thus, the resulting variances should not be used to measure uncertainty, although many studies used them for such purpose. Since they reflect the variations in spatial departures among samples, they can be used as good indicators where samples are sparse and, thus, may provide useful information for selecting future sampling locations.  
For LM, prediction uncertainty (i.e., prediction intervals) produced are much wider than confidence intervals of a model fitted [101]. Such uncertainty, however, has little to do with the predictive accuracy of the model. For instance, it was found that the models developed according to goodness of fit could be misleading when they were used as predictive models [6]; hence, its prediction uncertainty could also be misleading, and further studies are recommended.  
For RF, many types of uncertainty could be produced, which actually reflect the difference in sampling strategies [123; 124; 125; 126]. For example, prediction uncertainty produced for RF in a previous study based on Monte Carlo resampling [127] was, in fact, measuring the variation in predictions among individual trees rather than by RF. A further example for RF is that an ensemble of equally probable realizations was generated and the differences amongst the realizations were used as a measure of uncertainty [128]. This type of uncertainty only measures the differences among the results of various runs of RF, that is, measuring the difference resulted from the randomness associated with each run. Hence, these values do not relate to predictive accuracy and do not measure prediction uncertainty.  
In addition, for any of the spatial predictive methods above, predictive errors based on validation can be produced for a predictive model developed. However, this leads to only one error value, and all predictions would have the same uncertainty value if it is used as an uncertainty measurement [129].  
It is apparent that the uncertainty values produced above are either not measuring prediction uncertainty, or they depend on various factors as discussed above. This consequently results in the need to question the uncertainty of uncertainty. In short, how to assess prediction uncertainty needs further study. Any uncertainty measures that can incorporate the information of predictive accuracy are worth further investigation and recommended for future studies.