Cross-validation results for the craniometric dataset.

<p>For each of the 66 model categories, the best fitting model within the category is shown. The solid bar in color on the right represents the increase in relative to the poorest fitting category, and the solid white bar above it represents the estimated bias in the optimized log-likelihood of the model as an estimate of T_<i>n</i>. The bar on the left shows the AIC bias estimate based on the number of freely adjustable parameters in the model. The best fitting model is enclosed by a solid box. The dashed box encloses the model category that included the second and third best fitting models.</p

Estimation of bias in simulated data.

<p>Each plotted data point represents one million simulations and five such points are plotted for each model. In each simulation, <i>m</i> was two and the value of <i>n</i> is shown on the x-axis. Sampling along the x-axis is logarithmic but the scaling is proportional to . The biases were calculated by first solving for <b>W</b> using a set of <i>n</i> random deviates from the designated true source distribution and comparing the resulting log likelihood with the log likelihood estimated from a new independent set of <i>n</i> random deviates from the same distribution using the value of <b>W</b> derived from the first set. <b>A.</b> Results with both sources modeled as Gaussian. The solid line shows predicted results based on Fujikoshi and Satoh (1997). <b>B.</b> Results with both sources modeled as sub-Gaussian. <b>C.</b> Results with both sources modeled as super-Gaussian. The solid lines in B and C reflect the asymptotic Akaike prediction of a bias of 6.</p

Distributional results for the five best model categories for the iris dataset.

<p>Histograms of sources for the best fitting model in each of the five best model categories are shown in columns in order of with the corresponding model source distribution superimposed. Each species is shown in a different color, <i>I setosa</i> in red, <i>I versicolor</i> in green and <i>I virginica</i> in blue. Histograms have been displaced vertically when necessary to prevent members of one species from obscuring those of another species. The non-Gaussian sources from each model have been sorted such that the most strongly correlated sources from the different models appear in the same row. The percentage values reflect the contribution of each source to the total variance.</p

Heat map relating z-scores to sources and components for the craniometric data.

<p>Heat map intensities are derived from the best fitting model. Bright green values indicate that a higher (more positive) z-score contributes to a positive source or component score. Bright red values indicate that a higher (more positive) z-score contributes to a negative source or component score. The first four columns correspond sequentially to the sources shown in Figs. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118877#pone.0118877.g006" target="_blank">6</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118877#pone.0118877.g009" target="_blank">9</a>. The fifth column corresponds to the first principal component shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118877#pone.0118877.g010" target="_blank">Fig. 10</a>. The last four columns correspond to components accounting for increasingly smaller amounts of total variance (not shown). Raw data measures marked with asterisks are prominent contributors to the identified non-Gaussian sources and are discussed in the text.</p

Sub-Gaussian source separating the Far Eastern group and the European group from one another.

<p>Source scores are derived from the best fitting model. This source accounts for 10.0% of the variance of the original data set. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118877#pone.0118877.g006" target="_blank">Fig. 6</a> legend for additional details.</p

Mixtures of non-Gaussian and of Gaussian sources.

<p>Points with x- and y- coordinates drawn from two unmixed source distributions are plotted in the left panel. At the top, both sources were randomly drawn from a sub-Gaussian distribution; in the middle, both from a super-Gaussian distribution; and at the bottom, both from a Gaussian distribution. Corresponding images on the right show the distributions after the sources on the left were remixed with an orthonormal matrix representing a thirty degree rotation. For the non-Gaussian sources, the lack of statistical independence resulting from the rotation can be identified visually even without seeing the unrotated images as a result of the higher order distributional statistics. Statistical independence in the unmixed images is evident from the fact that the two-dimension distribution can be predicted as the product of the illustrated marginal distributions along each individual axis. When both sources are Gaussian, the two-dimensional distribution can be predicted from the marginal distributions of any arbitrary orthogonal pair of coordinate axes, so statistical independence cannot be used as a criterion for identifying the original Gaussian sources.</p

Super-Gaussian source separating the Eskimo population from other populations.

<p>Source scores are derived from the best fitting model. This source accounts for 6.4% of the variance of the original data set. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118877#pone.0118877.g006" target="_blank">Fig. 6</a> legend for additional details.</p

First principal component of the craniometric data six dimensional Gaussian subspace.

<p>Component scores are derived from the best fitting model. This component accounts for 20.0% of the variance of the original data set. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118877#pone.0118877.g006" target="_blank">Fig. 6</a> legend for additional details.</p

Super-Gaussian source separating the Bushman and Teita populations from other populations.

<p>Source scores are derived from the best fitting model. This source accounts for 18.0% of the variance of the original data set. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118877#pone.0118877.g006" target="_blank">Fig. 6</a> legend for additional details.</p

Super-Gaussian source separating the Buriat population from other populations.

<p>Source scores are derived from the best fitting model. This source accounts for 7.4% of the variance of the original data set. All members of each of the thirty populations are plotted on the same subpanel with males at the top of the subpanel and females at the bottom. The rectangular boxes enclose individuals with source scores within two standard deviations of the population mean. The vertical light gray lines are based on the observed mean and standard deviation for the entire population of 2524 subjects and positioned such that just one subject would be expected to fall outside those lines if the distribution were Gaussian. Populations are grouped according to Howells’ six main groups (Far Eastern, Polynesian, European, American, Austro-Melanesian and African), with Howells’ seven ungrouped populations together at the bottom.</p