Optimal versus asymptotic prediction intervals.

<p>Optimal versus asymptotic prediction intervals.</p

Optimal selection of parameter in terms of parameter .

<p>Constants associated with the controlled upper- to lower-bound ratio prediction intervals for , when ; in particular, for each and , and contain with a probability. For each , the smallest value of for which the equation:</p><p></p>admits a solution, is reported. Numerical values where determined using Maple 13.02.<p></p

Constants associated with 95% prediction intervals.

<p>Constants associated with upper-bound, conservative-lower, conservative-upper and lower-bound prediction intervals for , when and . By definition, this means that and . Furthermore, the constants are solutions to the equation:</p><p></p>solved numerically with Newton's method using Maple 13.02. This equation may have at most two different solutions, and star () denotes that the equation has no solution.<p></p

Rank curves associated with the rare biosphere simulation in the human-gut and -hand urn.

<p>Rank curves associated with <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0021105#pone-0021105-g006" target="_blank">Fig. 6</a> (green) and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0021105#pone-0021105-g007" target="_blank">Fig. 7</a> (blue).</p

Point predictions in a human-gut and exponential urn.

<p>Plots associated with a human-gut (top-row) and exponential urn (bottom-row). Left-column, sequential predictions of the conditional uncovered probability (black), as a function of the number of observations, using Robbins' estimator in equation (1) (green), Starr's estimator in equation (2) (orange), and the Embedding algorithm (blue, red), over a same sample of size from each urn. Starr's estimator was implemented keeping . Blue predictions correspond to consecutive outputs of the Embedding algorithm in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0021105#pone-0021105-t001" target="_blank">Table 1</a>, which was reiterated until exhausting the sample using the parameter . Red predictions correspond to outputs of the algorithm each time a new species was discovered. Right-column, correlation plots associated with consecutive predictions of the conditional uncovered probability (normalized by its true value at the point of prediction), under the various methods. The green and orange clouds correspond to pairs of predictions, 100-observations apart, using Robbins' and Starr's estimators, respectively. Blue and red clouds correspond to pairs of consecutive outputs of the Embedding algorithm, following the same coloring scheme than on the left plots. Notice how the red and blue clouds are centered around , indicating the accuracy of our methodology in a log-scale. Furthermore, the green and orange clouds show a higher level of correlation than the blue and red clouds, indicating that our method recovers more easily from previously offset predictions. In each urn, our predictions used the observations and a HPP with intensity one–simulated independently from the urn–to predict sequentially the uncovered probability of the first part of the sample. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0021105#pone-0021105-g004" target="_blank">Fig. 4</a> for the associated rank curve in each urn.</p

Rank curves associated with the human-gut, human-hand and exponential urn.

<p>In a rank curve, the relative abundance of a species is plotted against its sorted rank amongst all species, allowing for a quick overview of the evenness of a community. On the left, rank curves associated with the human-gut (blue) and -hand data (green) show a relatively small number of species with an abundance greater than 1%, and a long tail of relatively rare species. The right rank curve of the exponential urn (red) simulates an extreme environment, where relatively excessive sampling is unlikely to exhaust the pool of rare species.</p

Predictions in the human-hand urn when simulating the rare biosphere.

<p>Prediction of the conditional uncovered probability (black) in nine urns associated with a human-hand urn. Point predictions produced by the Embedding algorithm (blue), point predictions produced by the algorithm each time a new species was discovered (red), upper-bound interval (orange), and conservative-upper interval (green). The algorithm used the parameters . The different urns were devised as follows. For each (indexing rows) and (indexing columns), a mixture of two urns was considered: an urn with the same distribution as the microbes found in a sample from a human-hand and weighted by the factor , and an urn consisting of colors (disjoint from the hand urn), with an exponentially decaying rank curve and weighted by the factor . See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0021105#pone-0021105-g005" target="_blank">Fig. 5</a> for the rank curve associated with each urn.</p

Schematic description of the Embedding algorithm.

<p>Suppose that in a first sample from an urn you only observe the colors red, white and blue; in particular, . Let be the unknown proportion in the urn of balls colored with any of these colors i.e. . To estimate , sample additional balls from the urn until observing balls with colors outside . Embed the colors of this second sample into a homogeneous Poisson point process with intensity one; in particular, the average separation of consecutive points with colors outside are independent exponential random variables with mean . The unknown quantity can be now estimated from the random variable . As a byproduct of our methodology, conditional on , if denotes the relative proportion of color in the first sample then predicts the true proportion of color in the urn.</p

Predictions in the human-gut urn when simulating the rare biosphere.

<p>In a sample of size from a human-gut, species were discovered. Based on our methods, we estimate that of these species represent of that gut environment; hence, the remaining is composed by at least species. To test our predictions of the conditional uncovered probability (black), we simulated the rare biosphere by adding additional species and hypothesized that our point prediction could be offset by up to one order of magnitude: point predictions produced by the Embedding Algorithm (blue), point predictions produced by the algorithm each time a new species was discovered (red), upper-bound (orange), and conservative-upper interval (green). The predictions used the parameters . The different urns were devised as follows. For each (indexing rows) and (indexing columns), a mixture of two urns was considered: an urn with the same distribution as the microbes found in the gut dataset, and weighted by the factor , and an urn consisting of colors (disjoint from the gut urn), with an exponentially decaying rank curve and weighted by the factor . See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0021105#pone-0021105-g005" target="_blank">Fig. 5</a> for the rank curve associated with each urn.</p

Additional file 3: of Identifying and mitigating batch effects in whole genome sequencing data

Sample level summary statistics and annotations calculated by genotypeeval. (CSV 102 kb