Constrained Smoothing of Histograms and Scatterplots With Simulated Annealing

Histogram and scatterplot models are often required for statistical inference. In the field of petroleum engineering, stochastic simulation algorithms require, among other statistics, a model for the histogram of the petrophysical attribute (porosity/permeability) being simulated. Often this model is taken to be the deciustered distribution of the sample data. When there are many data (say, greater than l,000), this histogram may be reasonably informed. Most often, however, the sample histogram shows multiple sawtoothlike spikes that are not representative of the entire population; the sample histogram must be smoothed. A simulated annealing-based procedure is proposed for smoothing one-variable (univariate) histograms and two-variable scatterplots (bivariate histograms). The smoothed histograms are constrained to the sample mean, variance, specified quantiles, and a measure of smoothness. In the bivariate case, the distribution must be consistent with both marginal histograms and can be additionally constrained to a linear correlation coefficient. Several examples with real reservoir data are presented.