Figure 1.
Multilayer perceptron feedforward neural network with error backpropagation.
The information (cross entropy values of immunological parameters for each patient) is inserted in the input neurons. At the hidden layer, here with 6 neurons, we sum the information and transfer it (through the sigmoid function) to the outcome layer, where the sigmoid function exits an AgP or CP verdict. Bias neurons have a constant value and help the network to learn patterns. They are independent from other neurons and can shift the curve of the sigmoid function to the left or to the right. The classification error found at the outcome layer backpropagates in the network and synaptic weights are adapted accordingly as the network learns from its error and tries to minimize it.
Figure 2.
Univariate kernel density estimation (KDE) graphs.
Graphs A to C. Univariate KDE for radiographic bone loss measurements: modes (single, bimodal or multimodal) are defined as the values that appear more frequent. Graphs A & B from sample-1. In graph C (sample-2) we log transformed the confined data to find support in the interval (−∞, +∞) (see text S1). Graphs D to X. Univariate KDE for immunologic data: possible evidence of multimodality for the CD4/CD8 ratio, CD3, lymphocytes, monocytes, eosinophils, basophils and neutrophils counts (sample-2), IgG levels (sample-3), IL-2, IL-4, IL-6, INF-γ, TNF-α, IgG A.a. titers and IgG C.o. titers (sample-4). Mini clusters close to each other are detected for IL-1 and IgG P.g titers (sample-4).
Figure 3.
Bivariate kernel density estimation (KDE) for some selected parameters.
(A) Contour plot for bivariate KDE of longitudinal radiographic bone loss level (sample-1) in relation to age: this topographical-like plot shows a main cluster with 0.2 mm longitudinal bone loss and a small cluster with almost five times greater bone loss. (B) Contour plot for bivariate KDE: By estimating probability density for CD4/CD8 ratio by age (sample-2), we see two clusters although not separated distinctly, at modes of 1.5 and 1.9. (C) Contour plot for bivariate KDE: By estimating probability density for CD4/CD8 ratio (sample-2) by disease severity (% of teeth with bone loss ≥ of 50% of their root length), we reveal two distinct clusters of patients, with modes at x values of 1.5 and 1.9.
Table 1.
Characteristics of three artificial neural networks (ANN) built on immunological parameters.