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. 2021 Mar 22;11:6482. doi: 10.1038/s41598-021-85758-6

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Characterizing the variation in the output of each stochastic NN (intra-NN variance or V_RA) and the differences in predicted tumor likelihood amongst the NN configurations (inter-NN variance, V_ER). The use of 3 configurations of stochastic NNs provides two useful variance estimates for assessing NN reliability and enhancing tumor detection likelihood estimates. The output variance of each stochastic NN when run through 10 complete train-test cycles with random weight resets for each cycle provides an intra-NN output variance estimate (V_RA) for each of the three NN configurations. The lower the V_RA, the more certain the NN is of its prediction. The inter-NN variance (V_ER) is then calculated as the variance in the average output probabilities (P_Tumor, N = 10) generated by the 3 NN configurations while determining the V_RA variance for each. V_ER provides an immediate detection of disagreements in diagnostic predictions for the three NN configurations. (A) Depicts V_RA for each of the three NN configurations as a function of P_Tumor between the NN training boundaries (P_Tumor ~ 0 = > healthy, P_Tumor ~ 1 = > tumor). P_Tumor estimates are divided into those with V_RA < 1σ and V_RA > 1σ. Vertical dotted lines indicate the point where a 5^th order polynomial fit to the data intersects at the 1σ level. Significant increases in V_RA (V_RA > 1 σ) appear as expected in the boundary zone between the two classes (i.e., ~ 0.2 < P_Tumor < ~ 0.8). (B) Depicts V_RA as a function of V_ER. Significant inter-NN variance (V_ER > 1σ) occurred in 7 targets (shaded red), all from tumor or boundary regions. In six of the seven high V_ER samples, the HW NN exhibits the least variance in its predictions (V_RA between 0.001 and 0.10). For the other target (s7) FP exhibits the minimal variance across ten trials (V_RA ~ 0.003). Spectra corresponding to these 7 targets appear in Fig. 6. See Table 6 for a comparison of the cancer probability predictions and the NN variances. Another 17 targets (yellow shading) exhibit high V_RA (> 1σ), but low V_ER (< 1σ). See Table S2 in Supplementary Material.