Conditional probability function with uncertainty estimates

The conditional probability function (CPF) and the conditional bivariate probability function (CBPF) are widely used and useful tools that aid in identifying pollution sources in atmospheric research: they can tell us whether the wind from a particular direction increases or decreases pollutant concentrations. For this conclusion to be reliable, the observed decrease or increase must be established as statistically significant. Our literature search has shown that this is never done. Even more, we found that the majority of published analyses calculate CPF and CBPF from a relatively small number of measurements when the statistical fluctuations are large. The combination of these two facts is dangerous because it can drastically increase the likelihood of incorrect conclusion. To resolve this important issue, we have developed two independent methods for estimating the significance of the CPF and CBPF results. The methods, called binomial ratio and bootstrapping, are based on the construction of confidence intervals. We validated the methods on large and real data sets. We found them to be in a very good agreement and to have good coverage properties, which is the main criterion for the validity of confidence interval construction. The calculation and visualisation of CPF, CBPF and the associated confidence intervals was done in "CPFU", our freely available open-source software written in R. We also freely provide software that facilitates the (otherwise difficult) extraction of data files from the EPA website.