## 2 Predictive Methodology, Terminology, and Analytical Procedures
We will be using five different models in our applications but the favorability function modeling and the processing structure are independent of the mathematical models that will be indicated later on. In essence, the likelihood ratios enable us to separate the presence of occurrence from their supposed absence within the SA. This separation uses another assumption. While the normalized frequencies or the density functions in the TA (the occurrence subarea) are related to observed/mapped landslide occurrences, the ones in the nonoccurrence subarea relate to both the absence of occurrences and the unknown occurrences. For the latter, we assume that their setting, due to being relatively rare, is diluted within the setting of the absence of occurrences. This assumption allows considering the ratios as a form of contrast between areas with presence and areas with a presumed absence of occurrences. Commonly, the TA with respect to the SA is three or four orders of magnitude smaller. A favorability function at each point of a SA should have two properties for modeling, i.e. : (1) should be able to measure a relative level of likelihood that a point \(i\) in the SA contains part of a future landslide of the given dynamic type and (2) should be able to provide a measure of uncertainty associated with the function by using only the part of all possible landslide occurrences present in the training area, TA. Now suppose that we use a given prediction model, still unspecified at this point, that generates integrated scores as predicted values for every pixel of the SA. The scores are in some a-dimensional units or values between a minimum and a maximum. How do we establish the very high, high, intermediate, low, and very low likelihood of future landslide occurrence? This is not a simple question to answer, and it leads to further assumptions depending on the kind of data available for the SA, database of DSP and ISPs, and additional information. Suppose, for instance, that we have temporal characterization for the many occurrences of the given specific dynamic type in the TA. We could use the occurrences from an older time interval to model a prediction pattern and then overlay it with the locations of the occurrences from the younger time interval. In this way, we obtain some statistics about their proportions within higher likelihood scores. Should some of them fall within the lower likelihood score areas, we would consider them as poorly predicted. Vice versa, well-predicted occurrences should fall on the higher likelihood score areas. We have termed this exercise as cross validation. In practice, this is a natural way to establishing how \"good\" our prediction pattern is as a \"predictor of future occurrences.\"  
Now let us suppose that the information on time partitioning of the occurrences, the DSP, is not available, as it is in most cases. How can we proceed with some other forms of cross validation? We could empirically pretend, for instance, not to know the existence of some of the occurrences, e.g., 25% of them, and use the remaining 75% to obtain a prediction pattern and then cross-validate it with the 25% we pretended not to know and that was not used as DSP for modeling. In this case, the \"future\" landslide occurrences for cross validation are the \"next\" 25%. Furthermore, we could operationally devise iterative strategies, depending on the number of occurrences available as DSP, such as (1) sequential exclusion of an arbitrary number of occurrences to be used for cross validation of the pattern produced with the remaining ones, (2) sequential selection of a number of occurrences for modeling and using the remainder for cross validation, or (3) random selection of a number of occurrences repeated a convenient number of times. All these strategies will provide different ways to predicting the next arbitrarily selected number or proportion of occurrences. Recall that we have not yet selected any particular mathematical model for predicting.