Topological Susceptibility under Gradient Flow

We study the impact of the Gradient Flow on the topology in various models of lattice field theory. The topological susceptibility $\chi_{\rm t}$ is measured directly, and by the slab method, which is based on the topological content of sub-volumes ("slabs") and estimates $\chi_{\rm t}$ even when the system remains trapped in a fixed topological sector. The results obtained by both methods are essentially consistent, but the impact of the Gradient Flow on the characteristic quantity of the slab method seems to be different in 2-flavour QCD and in the 2d O(3) model. In the latter model, we further address the question whether or not the Gradient Flow leads to a finite continuum limit of the topological susceptibility (rescaled by the correlation length squared, $\xi^{2}$). This ongoing study is based on direct measurements of $\chi_{\rm t}$ in $L \times L$ lattices, at $L/\xi \simeq 6$.Comment: 8 pages, LaTex, 5 figures, talk presented at the 35th International Symposium on Lattice Field Theory, June 18-24, 2017, Granada, Spai

<i>Discrimination</i> plots (ROC curves) and <i>Calibration</i> plots for simulated models.

<p>(a) Perfect <i>discrimination</i> (i.e., AUC = 1) requires a classifier with perfect dichotomous predictions, which in the calibration plot has only one point (0,0) for negative observations and one point (1,1) for positive observations. (b) Poor <i>discrimination</i> (i.e., AUC = 0.530.02) and poor <i>calibration</i> (i.e.,  = 251.2765.2, <1e−10). (c) Good <i>discrimination</i> (i.e., AUC = 0.830.03) and excellent <i>calibration</i> (i.e.,  = 10.024.42,  = 0.260.82). (d) Excellent <i>discrimination</i> (i.e., AUC = 0.960.01) and mediocre <i>calibration</i> (i.e.,  = 34.462.77,  = 00.95). Note that a HL statistic smaller than 13.36 indicates that the model fits well at the significance level of 0.1.</p

Reliability diagrams and two types of HL-test.

<p>In (a), (b), and (c), we visually illustrate the reliability diagram, and groupings used for the HL-H test and the HL-C test, respectively.</p

Performance comparisons between three different models using breast cancer datasets.

<p>Performance comparisons between three different models using breast cancer datasets.</p

Confusion matrix of a classifier based on the gold standard of class labels.

<p>Confusion matrix of a classifier based on the gold standard of class labels.</p

ROC, AUC and its calculation.

<p>The horizontal line shows sorted probabilistic estimates on “scores” s. In (a) and (b), we show the ROC and the AUC for a classifier built from an artificial dataset. In (c) and (d), we show concordant and discordant pairs, where concordant means that an estimate for a positive observation is higher than an estimate for a negative one. The AUC can be interpreted in the same way as the c-index: the proportion of concordant pairs. Note that corresponds to an observation, represents its predicted score, and represents its observed class label, i.e., the gold standard. AUC is calculated as the fraction of concordant pairs out of a total number of instance pairs where an element is positive and the other is negative. Note that is the indicator function.</p

Tradeoffs between <i>calibration</i> and <i>discrimination</i>.

<p>(a) Perfect <i>calibration</i> may harm <i>discrimination</i> under a three-group binning. The numbers above each bar indicate the percentage of negative observations (green) and positive observations (orange) in each prediction group (0–0.33, 0.33–0.67, and 0.67–1). Note the small red arrows in the left figure indicate discordant pairs, in which negative observations ranked higher than positive observations. (b) Enforcing discrimination may also hurt calibration. The blue curve and error bars correspond to the AUC while the green curve and error bars represent the p-values for the Hosmer-Lemeshow C test (). Initially, as <i>discrimination</i> increases, -value of (<i>calibration</i>) increases but it quickly drops after hitting the global maximum. We use red arrows in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048823#pone-0048823-g005" target="_blank">Figure 5(b)</a> to indicate the location of optimal calibration and discrimination for the simulated data.</p

An example of outputs for two probabilistic classifiers and their ROC curves, which do not evaluate calibration.

<p>In (a) and (b), # indicates the observations, corresponds to the class membership, and represents the probability estimate. In (c) and (d), each red circle corresponds to a threshold value. Note that probabilistic classifier B has the same ROC as probabilistic classifier A, but their calibration differs dramatically: estimates for B are ten times lower than estimates for A.</p

Details of the training and test datasets in our first experiment.

<p>Details of the training and test datasets in our first experiment.</p

Performance comparison between using GSE2034 and GSE2990.

<p>Performance comparison between using GSE2034 and GSE2990.</p