## 3 Algorithm  
The method of monotonic functions finds two solutions from the available data: the rule of nonparametric classification by analogy with the precedent and the rule of logical decision, which is formulated in the language of the subject domain. The method uses the following data model. There are two forms of the disease (classes) \(\omega_{1}\) and \(\omega_{2}\). We know of a limited sample of patients. Patients \(q=1,2,\ldots,Q_{1}\) belong to the class \(\omega_{1}\), and patients \(\overline{q}=1,2,\ldots,Q_{2}\) belong to the class \(\omega_{2}\). Each patient is described by a set of indicators (a set of features). Patients correspond to points of the \(I\)-dimensional feature space \(\mathbf{x}^{(q)}=\{x_{i}^{(q)}\}\) and \(\mathbf{x}^{(\overline{q})}=\{x_{i}^{(\overline{q})}\}\), \(i=1,\ldots,I\). Let the severity of the disease in patients of class \(\omega_{1}\) be lower than that in patients of class \(\omega_{2}\). For each feature, the interval of the indicator norm is known. Patients with indices \(\mathbf{x}=\{x_{i}\}\) whose deviations from the norm are component-wise less than or equal to deviations from the norm in indices of any point \(\mathbf{x}^{(q)}\in\omega_{1}\) have a disease severity no more than in a patient with indicators \(\mathbf{x}^{(q)}\) (monotonicity condition). For simplicity of explanation without loss of generality, we further assume that the monotonicity condition consists of the fact that all points \(\mathbf{x}\) of the feature space take values either from the intervals of the norms of the indicators, \(x_{i}\in[a_{i},b_{i}]\) or the larger right boundaries of these intervals, \(x_{i}>b_{i}\). Then, according to the condition of monotonicity, the patient, to whom the point \(\mathbf{x}=\{x_{i}\}\), \(i=1,2,\ldots,I\) corresponds, has a severity of the disease no higher than the severity of the disease in a patient of class \(\omega_{1}\) with indicators \(\mathbf{x}^{(q)}=\{x_{i}^{(q)}\}\) if \(x_{i}\leq x_{i}^{(q)}\) for all \(i=1,2,\ldots,I\). If at least one indicator \(i=1,2,\ldots,I\) occurs, \(x_{i}>x_{i}^{(q)}\), then the severity of the patient's disease is greater than that of a patient with indicators \(\mathbf{x}^{(q)}\). The monotonicity condition corresponds to an ideal situation in which all the indicators of objects necessary for classification are present and the values of indicators are measured accurately. In real situations, due to the incomplete description of the differential properties of objects and the inaccuracy in the measurements of indicators, inequalities \(x_{i}\leq x_{i}^{(q)}\), \(i=1,2,\ldots,I\) are not fulfilled for some pairs of objects. This is one of the reasons for misclassification. Consider a classification algorithm. With each point of the training sample \(\mathbf{x}^{(q)}\in\omega_{1}\), we connect the area of the feature space  
\[W^{(q)}=\{\mathbf{x}:x_{i}\leq x_{i}^{(q)},i=1,2,\ldots,I\}. \tag{1}\]  
We call this set the orthant \(W^{(q)}\) with apex at point \(\mathbf{x}^{(q)}\). Since the monotonicity condition is not satisfied for all objects, the orthant with apex at point \(\mathbf{x}^{(q)}\) can contain \(Q_{1}^{(q)}\) points of the class \(\omega_{1}\) and \(Q_{2}^{(q)}\) points of the class \(\omega_{2}\). Each orthant \(W^{(q)}\) with vertex \(\mathbf{x}^{(q)}\) corresponds to three parameters:  
\[P_{1}^{(q)}=Q_{1}^{(q)}/Q_{1}, \tag{2}\]  
\[P_{2}^{(q)}=Q_{2}^{(q)}/Q_{2}, \tag{3}\]  
\[L^{(q)}=P_{1}^{(q)}/(P_{2}^{(q)}+1/Q_{1}Q_{2}), \tag{4}\]  
where \(P_{1}^{(q)}\) is the probability that the points of the orthant belong to class \(\omega_{1}\), \(P_{2}^{(q)}\) is the probability that the points of the orthant belong to class \(\omega_{2}\) in the orthant, and \(L^{(q)}\) is the likelihood ratio, where the term \(1/Q_{1}Q_{2}\) corrects the value of \(L^{(q)}\) when \(P_{2}^{(q)}=0\). The machine-learning algorithm consists of the following steps:  
1.