The Cumulative Synoptic Probability Density Function (CSPDF)

The probability density function (PDF) is intimately linked to the cumulative density function (CDF). The relationship between PDF and CDF is:

$$cdf(t) = \int_{-\infty}^t pdf(x) dx$$ (5)

$$pdf(t) = \frac{d(cdf(t))}{dt}$$ (6)

PDF and CDF are standard statistical terms. In the nomenclature of Ruhl and Hodges [2005], the specific case of a Gaussian kernel density estimator with $\alpha$ = 1 is named SPDF, and the corresponding CDF is named the Cumulative Synoptic Probability Density Function (CSPDF). Thus, the CSPDF is defined by using SPDF instead of PDF in Equation 5. Although SPDF and CSPDF are interchangeable from a statistical point of view, the CSPDF has recently gained considerable popularity for two reasons. First, the cumulative distribution has intuitive significance, as its shape mimics the shape of the the cumulative hypsometry (modulated by the age-elevation curve). A second advantage of cumulative plots is the ease of comparing different datasets by using the Kolmogorov-Smirnov (K-S) goodness-of-fit test. The K-S test determines if the maximum vertical distance between two cumulative distributions can be explained by random sampling effects alone. As illustrated by Amidon et al. [2005], using the CSPDF in combination with the K-S test is a useful tool for comparing two detrital datasets. However, we will next see that the CSPDF should not be used for the purpose of comparing a detrital age distribution with hypsometric predictions.

Revisiting the toy example of a ``diving board'' hypsometry, the CDF of the true ages is a step-function at t$_{true}$ = 10Ma (Figure 4). The theoretical CDF of the measured ages ($\approx$ CSPDF$_{t*m}$ curve of Ruhl and Hodges, 2005) is the cumulative normal distribution with mean 10Ma and standard deviation 1Ma (red curve on Figure 4). In contrast, the cumulative kernel density estimator CSPDF$_t$ (gray curve on Figure 4) is the cumulative normal distribution with mean 10Ma and standard deviation $\sqrt{2}$Ma. Thus, CSPDF$_t$ is not a good estimator of CSPDF$_{t*m}$, for the same reason why SPDF$_t$ is not a good estimator of SPDF$_{t*m}$ (Figure 3). In other words, the CSPDF is not a good tool for comparing detrital datasets with hypsometric predictions, which is exactly the goal of this paper. Fortunately, the misfit caused by the ``double smoothing'' of CSPDF$_t$ does not greatly affect the conclusions of Ruhl and Hodges [2005] and Stock et al. [2006], because the analytical uncertainties of their $^{40}$Ar/$^{39}$Ar and (U-Th)/He data are relatively small. The situation would be worse for the less precise AFT data presented here.

Figure 4: The cumulative equivalent of Figure 3.c. The dashed step-function is the cumulative hypsometry (CSPDF$_{z}$), corresponding to the ``true age distribution'', which could only be measured if we had access to errorless measurements. Gray circles are measurements, black circles are the Cumulative Age Distribution (CAD). The black curve is CSPDF$_{t*m}$, which coincides with theoretical ``predicted CAD'' obtained from an infinite number of synthetic measurements including measurement uncertainties. The observed CAD is an unbiased estimator of the predicted CAD. The gray curve is CSPDF$_{t}$ calculated from the 50 measurements and their respective uncertainties. CSPDF$_{t}$ is not a good estimator of CSPDF$_{t*m}$.

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The CSPDF-method can be ``fixed'' by smoothing the CSPDF$_{t*m}$ a second time. In practice, CSPDF$_{t*m}$ can be constructed by collecting a large number of synthetic ``measurements'' from the hypsometry and adding a synthetic measurement error to them. The second smoothing step would then involve stacking a bell curve on top of each of the sythetic measurements, as in Equation 2. Instead of this cumbersome ``fix'', the Cumulative Age Distribution (CAD) is introduced as a simpler alternative to the CSPDF method.