An adaptive algorithm for clustering cumulative probability distribution functions using the Kolmogorov–Smirnov two-sample test

This paper proposes an adaptive algorithm for clustering cumulative probability distribution functions (c.p.d.f.) of a continuous random variable, observed in different populations, into the minimum homogeneous clusters, making no parametric assumptions about the c.p.d.f.’s. The distance function for clustering c.p.d.f.’s that is proposed is based on the Kolmogorov–Smirnov two sample statistic. This test is able to detect differences in position, dispersion or shape of the c.p.d.f.’s. In our context, this statistic allows us to cluster the recorded data with a homogeneity criterion based on the whole distribution of each data set, and to decide whether it is necessary to add more clusters or not. In this sense, the proposed algorithm is adaptive as it automatically increases the number of clusters only as necessary; therefore, there is no need to fix in advance the number of clusters. The output of the algorithm are the common c.p.d.f. of all observed data in the cluster (the centroid) and, for each cluster, the Kolmogorov–Smirnov statistic between the centroid and the most distant c.p.d.f. The proposed algorithm has been used for a large data set of solar global irradiation spectra distributions. The results obtained enable to reduce all the information of more than 270,000 c.p.d.f.’s in only 6 different clusters that correspond to 6 different c.p.d.f.’s.

Comparison Between Dcrit Considering the Abrupt Variation and Inflexion in the Concrete Mercury Intrusion Porosimetry Curve

Mercury intrusion porosimetry (MIP) has been widely used to evaluate the quality of concrete through the pore size distribution parameters. Two of these parameters are the critical pore diameter (Dcrit) and the percentage of the most interconnected net of pores compared to the total volume of pores. Some researchers consider Dcrit as the diameter obtained from the inflexion point of the cumulative mercury intrusion curve while others consider Dcrit as the diameter obtained from the point of abrupt variation in the same curve. This study aims to analyze two groups of concretes of varying w/c ratios, one cast with pozzolanic cement and another with high initial strength cement, in order to determine which of these diameters feature a better correlation with the quality parameters of the concretes. The concrete quality parameters used for the evaluations were (1) the w/c ratios and (2) chloride diffusion coefficients measured at approximately 90 days. MIP cumulative distributions of the same concretes were also measured at about 90 days, and Dcrit values were determined (1) from the point of abrupt variation and alternatively, (2) from the inflexion point of each of these plots. It was found that Dcrit values measured from the point of abrupt variation were useful indicators of the quality of the concrete, but the Dcrit values based on the inflexion points were not. Hence, it is recommended that Dcrit and the percentage of the most interconnected volume of pores should be obtained considering the point of abrupt variation of the cumulative curve of pore size distribution.