-The paper considers the search for logical patterns by descriptions of objects in rectifying space. The rules of hierarchical agglomerative grouping are applied for the synthesis of latent features of this space. A pair of characteristics for combining into a group is chosen according to the maximum of criterion for characteristic values decomposition into disjoint intervals. The analytical form of arithmetic expressions for calculating latent features used to detect hidden patterns in the data of patients with chronic lymphocytic leukemia (CLL) is given.
The choice of space to describe objects through nonlinear transformations of features serves as a tool to detect latent patterns in data. When using such transformations, the structure of relations of objects in a new (latent) feature space changes. Quantitative estimators of the structure can be expressed in terms of compactness of objects of a class and the sample as a whole.
Several methods are proposed for evaluating compactness [1, 2]. In [2], connection between dimensionality of the feature space and the generalizing ability of recognition algorithms according to the nearest neighbor rule is shown through the measures of compactness of the objects of a class and the sample as a whole. The estimator of class compactness by a quantitative feature in [3] was calculated as an extremum of the criterion for decomposing the feature values into disjoint intervals.
For the data analysis, the numerical axis is considered as a universal scale with relations. The universal scale was used to study the relations between the objects of classes according to the results of the non-linear representation of their descriptions by the defined sets of features on the numerical axis [4]. Since the composition of each set is initially unknown, it was proposed to use the criterion of decomposing the feature values into disjoint intervals for its search.
Feature sets in [4] for further synthesis of latent features on their basis were sequentially formed according to the rules of hierarchical agglomerative grouping. The number of latent (groups) of features was determined upon the grouping results. A quality of informative orderliness of latent features makes them desirable for the analysis. This orderliness quality provides certain advantages for finding latent patterns in data.
The implementation of nonlinear transformations of features is considered as one of the stages of reducing the dimension of space [2]. When deciding whether to remove latent signs from the set, the property of their ordering is used. One of the goals of creating space through the removal of features is to solve the problem of retraining recognition algorithms. As a criterion for detecting the beginning of retraining in [2], a measure of compactness was used, calculated by solving the problem of minimal coverage of the sample with reference objects.
To detect hidden patterns in the data, you can use a combination of linear and nonlinear methods to reduce the dimension of the attribute space. The software implementation of many linear methods is presented in the Python language library [5]. As a linear display to the numerical axis, we consider the calculation of generalized object estimates [6] based on sets of heterogeneous attributes. Presentation of the results of data analysis on the numerical axis is a means for searching and recording logical patterns in the form of half-planes.
The issues of building information models in medicine are most often considered from the point of large or poorly structured systems [7]. Such systems require to take into account how various factors are connected and how they influence the processes in the body. The study explores the search for patterns for patients with CLL [8]. Latent features, synthesized according to the rules of a hierarchical agglomerative grouping, are considered to be factors affecting the duration of the patients actual survival. Here the results of the search for latent patterns in the data of CLL patients by latent features and the analytical form of arithmetic expressions (formulas) to calculate their values are presented. There are practically no publications describing other methods of forming an analytical representation (formulas) for calculating the values of latent features based on nonlinear transformations.
II. PROBLEM STATEMENT AND SOLUTION METHOD Let a set of objects E0={S1, …, Sm} be given containing representatives of two disjoint classes K1 and K2. Objects are described using a set of n quantitative features X(n)=(x1,…,xn). It is considered that the operator A is given on E0 to transform the descriptions of objects from X(n) to Y(k), k<n.
the number of latent features in Y(k); analytical form (equations) of arithmetic expressions to calculate the latent feature yi∊Y(k), i=1, …, k.
The analytical form of arithmetic expressions for calculating latent features is formed on the basis of the algorithm from [2]. The set of feature numbers in the description of the objects E0 will be identified as I = {1,...,n}. The rules of hierarchical agglomerative grouping nominal feature gradation. are applied to calculate the values of latent features. Latent features obtained at the p-th step of the grouping are denoted as , j∊I, p≥0. For p=0, |I|=n. We will divide the ordered set of feature values of objects from E0 into two intervals , 2 ] and ( 2 , 3 ], each of which is considered as a
Lets 1, 2 be the number of values of the feature , j∊I number of the element in an ascending order of
class intervals [
Ki, i = 1,2, respectively in the , 2 ] and ( 2 , 3 ] , |Ki|>1, v is the serial
1, … , , … , of values
of objects from boundaries of the intervals as = E0, which
defines the , 1
(5) ≥ max( , ) and u ≠ v, η= η +1 while η≤n; { } , ∈
by (5); u, v∊I}. If Φ=⍉, then go 9;
Step 3: Select Φ = {
| by Γη, η>0 being the subset of feature numbers from X (n).
A step-by-step implementation of the hierarchical agglomerative grouping algorithm will be as follows: Step 1: p=0, λc=0, η =1. Execute Γη ={η}, Marginη=–2, Step 2: Calculate the values of the elements of the matrix Let { } , ∈
, p ≥ 0 denote the square matrix of size (np)(n-p), whose element value at p = 0 is defined as . (∑2=1 ∑2=1 (| 3− | − 3− )) → 2| 1|| 2| 1 max < 2 < 3 the intervals [ allows to calculate the optimal value of the boundary 2 for
The extremum of criterion (
. When = 1, values of the feature for objects from classes K1 and K2 are mutually disjoint.
, The value of the combination by a pair of features Sr∊E0 is calculated as ( ), 0≤ p <n, i, j∊I, i ≠ j of the object = { } ∈ , − 2 )/( 3 − 1 = ( (
− 2 )/( 3
− 1 )) + (1− ) − ( 1 )+ (
− 2 )/( 3 ), , ∈ , , , ∈ {−1,1}, ∈ [0; 1] −
−
(
, are the weights of the features determined
by (
max ∈ 0 − min ∈ 0
=
max , , ∈{−1,1}, ∈[0,1]
(
The extremum of functional (
In (
are scale independent. ,
,t ∊I | = λn and s<t}. Define a pair {i, j}, i<j as Step 4: Calculate = max ∈Φ . Select ∆ ={(s, t),s
{ , } = {{ , }, ( , ) ∈△
△, |△| = 1, ( , , ) >
max ( , )∈△∖( , ) ( , , ); go to 3; object = {
Step 5: If λn> λc or λn = λc and Margini < φ(p, i, j), then Γi =Γi ⋃ Γj, Γj =⍉, Margini = φ(p, i, j), go to 7; Step 6: Display feature numbers from Γi, Γi =⍉, I=I\{i}, Step 7: p = p+1,
I=I\max(i,j), k=min(i,j), λc = λn.
Replace the values of the features in the description of the −1}
∈ , r = 1, …, m with = { −1, ∈ ∖ { },
, = ;
Step 8: For each pair (u, v), u, v∊I determine the value
= {
(
If n-p> 1, then go to 3;
When implementing the algorithm described above, in order to form the analytical representation of arithmetic expressions the parameter values were used.
For example, here is the value η ∊ {0, 1} in (
If two or more nominal features are used to describe admissible objects, then latent quantitative features can be obtained based on combinations of their gradation [6]. The values of such latent features are calculated as generalized estimator of objects. The purpose of using generalized estimator is to abandon the point estimator of objects obtained on the basis of expert subjective criteria.
Let us consider one of the methods for calculating generalized estimates of objects using a set of latent attributes
Y(k)=(y1,…,yk), k <n.
Let
V={vi}i∊{1,…,k}
and
F={[ci1,ci2], (ci2,ci3]}i∊{1,…,k}, respectively, the set of weights
y1=1.2572*((0.1203*x0 – 0.2442*x1 – 0.5339*x2) –
The sequence of formation of the first latent feature from
and the set of boundaries of the intervals calculated from (
( ) = ∑ =1 ( ).
III. COMPUTATION EXPERIMENT
(
Additionally, the number of chemotherapy courses performed and the actual survival rate in months were recorded. Patients with HIV infection and other oncological conditions were excluded from the study.
By gender, two data samples were generated. The sample of these male patients consisted of 60 objects (64.6 ± 9.0 years old), female of 63 objects (67.0 ± 8.4 years old). The objects of each sample were divided into two disjoint classes K1 (actual survival rate is less than the prognosed overall survival) and K2 (actual survival rate is greater than or equal to the prognosed overall survival).
The strongest patterns were obtained on the male patients data. Classes K1 and K2 were represented by 36 and 24 objects respectively. Based on the results of computation experiments on two sets (identified as the first and second) of the initial features, nonlinear combinations were obtained to completely separate the objects of two classes. In the second set, there
was a feature of comorbidity index
missing. The complete separability of classes is confirmed
by the value of criterion (
Here below the sequence of formation of the first two latent features for each of the two sets of initial features is
The sequence of formation of the first latent feature from
given.
the first set:
x0=0.1428*(comorbidity index – 4.0);
x1=0.0175*(GFR – 76.0);
x2=1.9463*(x0*x1 + 0.01);
x0=0.0243*(age – 63.0);
x1=0.0175*(GFR – 76.0);
the result (
There is a functional dependence of GFR on age and creatinine value depending on gender. Determination of the points needed to calculate the comorbidity index demands additional effort from a user to collect data from patients. For practical reasons, the use of the second option (without the comorbidity index) for calculating the latent feature looks preferable for the prognosis. To prognoses the patient’s survival potential, it will be enough for the user to set the values of measurable indicators - age and creatinine.
The values of the first two latent features in [4] are
recommended to be used for visual representation of objects
on the plane. Recommendations are based on the analysis of
the results of breaking into disjoint groups of descriptions of
class objects in R2. Another argument when choosing the
first two latent features for visualization is their values
according to criterion (
Anomalous deviations in class objects can be visually
detected by their latent attribute values [11]. The probability
of such deviations is indicated by values (
The search for hidden patterns in the data of CLL patients is described. The algorithm to generate the analytical representation of arithmetic expressions for calculating the values of latent features is given. The prognosis of the deviation of the actual survival rate of male patients towards either decreasing or increasing from overall survival is determined by a logical regularity in the form of half-planes. The forecast of deviation of the actual terms of survival of patients in the direction of decreasing or increasing from the terms of overall survival is determined by logical patterns in the form of half-planes. The found patterns can be recommended for use in specialized medical institutions.