The Cumulative Age Distribution (CAD)

One of the most significant advantages of using cumulative distributions instead of probability density estimates is not exploited by the CSPDF. Whereas it is impossible to visualize probability densities without using at least some degree of smoothing (either by binning or by stacking Gaussian kernels), this is not the case for cumulative distributions. The Cumulative Age Distribution (CAD) is a discrete step-function with the sorted detrital age measurements on the x-axis and their respective ranks on the y-axis (Figure 4). The (measured or ``observed'') CAD is an unbiased estimator of Ruhl and Hodges' [2005] CPDF$_{t*m}$, which will hereafter be named the ``predicted CAD''.

At first glance, it seems like the CAD does not properly account for variable measurement uncertainties. The CSPDF explicitly takes into account analytical uncertainties ( $\hat{\sigma}(t_i)$ in Equation 2), by ``spreading out'' imprecise measurements with wider Gaussian kernels. In contrast with this, the observed CAD gives equal ``weight'' to all measurements. The observed CAD gets around the need for kernel smoothing by shifting the uncertainty weighting to the predicted CAD. The predicted CAD is computed from the hypsometry by simulating the detrital sampling process. First, a number of synthetic ages are generated from the hypsometry. Then, synthetic measurement errors are added to each of these values, comparable in size to the real analytical uncertainties. From these synthetic measurements, a predicted CAD is calculated in exactly the same way as the observed CAD. Thus, the measurement uncertainties are properly accounted for by the predicted CAD, and there is no need to add them to the observed CAD a second time. Note that the same procedure is also part of the CSPDF-method, because the ``predicted CAD'' is essentially the same as Ruhl and Hodges' [2005] CSPDF$_{t*m}$.