Fig 1.
A tree-structured hybrid network.
The network maps binary input variables x ∈ {0, 1}9 to binary outputs y ∈ {0, 1}. Three first-layer black-box modules each have separate input variables, and a single black-box module processes the partial outputs of the first layer to compute the overall output of the network, which can then be interpreted as a decision for one of the two considered classes.
Fig 2.
The orthotope for the network structure of Fig 1.
The dimension of the orthotope is equal to the number of first-layer black-box modules of the tree-structured hybrid network of Fig 1. Each cell of the orthotope, characterized by three coordinates, represents an input data point holding the corresponding output label. Each intersection of a hyperplane with the orthotope holds input data with a constant input for a specific first-layer black-box module. For example, the uppermost horizontal blue slice of the orthotope illustrates all input vectors whose last three entries are 1, 1, 1.
Fig 3.
A 2-colorable graph representing the i-o function of a black-box module.
Vertices of the graph denote all inputs of the module. Different colors of the vertices represent different outputs of the module. The graph has 8 = 23 vertices because the module takes three binary variables as inputs.
Fig 4.
Schematic representation of the Label determination procedure.
The left figure shows an intersection of a hyperplane of constant inputs, say 0, 0, 0, for Module 2 with the orthotope of the network of Fig 1 (i.e., a “red slice”). The right figure represents the related conflict graph for Module 1. In accordance with Kőnig’s theorem, adding the dashed line to the edge set breaks the bipartiteness of the graph. Since assigning label 0 to the input vector (0, 0, 0, 0, 0, 0, 0, 0, 1) would imply the existence of an edge between (0, 0, 0) and (1, 0, 1) in the conflict graph, the label for the ‘?’ cell must be 1.
Table 1.
Biometric and physiological parameters of the studied COVID-19 patients on the ICU admission.
Values are represented as n (%) or median (interquartile-range).
Table 2.
Critical values for binarization of the most important COVID-19 patient features.
Table 3.
Classification results of the hybrid model on the synthetic data.
Fig 5.
The distribution of classification accuracies for binary classification on the synthetic data.
For each model and each size Ntr of the training data, we sampled 150 training data sets with input dimension d ∈ {8, 9, 10, 11, 12} and visualized the measured performance as box plots.
Table 4.
The comparison of the median of the binary classification measurement results on the synthetic data.
Table 5.
The Friedman test with significance level of 0.001.
Table 6.
The algorithms ranking.
Table 7.
Post-hoc test using the hybrid model as the control method.
Fig 6.
The influence of the dimensionality and the noise intensity of the synthetic data in the classification efficiency.
The effect of data dimensionality (left) and noise intensity (right) in the average of the classification accuracy for 150 experiments executed for each model and for N = 200 data points.
Fig 7.
The comparison of the average running time of the examined methods for 150 experiments executed for each model and for N = 200 data points.
Fig 8.
The SHM and the associated orthotope for the COVID-19 data.
The upper figure shows the hybrid network mapping five binarized patient features to their vital status. The lower figure depicts the associated orthotope consisting of 20 cells with labels and 12 cells without a label.
Fig 9.
The out-of-sample forecast performance for the vital status of COVID-19 patients.
The x-axis shows the percentage of the full input space that was used as the training data. For each training-data set size, we randomly sampled 1000 training data sets and measured the forecast performance. For randomly chosen training-data sets with a size of at least 56% of the entire valid input space, the median of the out-of-sample forecast performance equals 1.