=Paper=
{{Paper
|id=Vol-3309/paper5
|storemode=property
|title=Linear Regression Approximate Models for Predicting Severe Course of Bronchial Asthma
|pdfUrl=https://ceur-ws.org/Vol-3309/paper5.pdf
|volume=Vol-3309
|authors=Oleh Pihnastyi,Olha Kozhyna,Tetiana Kulik
|dblpUrl=https://dblp.org/rec/conf/ittap/PihnastyiKK22
}}
==Linear Regression Approximate Models for Predicting Severe Course of Bronchial Asthma==
https://ceur-ws.org/Vol-3309/paper5.pdf
Linear Regression Approximate Models for Predicting Severe
Course of Bronchial Asthma
Oleh Pihnastyi 1, Olha Kozhyna 2 and Tetiana Kulik 2
1
National Technical University "Kharkiv Polytechnic Institute", 2 Kyrpychova, Kharkiv, 61002, Ukraine
2
Kharkiv National Medical University, 4 Nauky Avenue, Kharkiv, 61022, Ukraine
Abstract
This paper discusses using of a multifactor linear regression model to predict bronchial asthma
severity. The study is aimed to develop the method of some five-factor linear regression
approximate models building and to substantiate the areas of their use. 142 factors obtained
during workup of 90 children at the age from 6 to 18 were analyzed. 70 children with bronchial
asthma of various degrees of severity as well as 20 healthy school-aged children were included
into the main group. The degree of qualitative and quantitative factors association being studied
was determined to select the predictors having an effect on severe course of bronchial asthma.
The following factors were used to build an approximate linear regression model: bronchial
asthma in relatives of second generation, atopic dermatitis, allergic rhinitis, sheep wool, rabbit
hair, domestic dust, severe. A comprehensive study of the value under investigation as well as
its dependence on a big number of factors is the basis of the proposed method. The developed
technique of linear regression multi-parameter models building allows us to simplify the
process of linear regression models building which can be used both for preliminary asthma
severity prediction and for detailed study of its course.
Keywords 1
Bronchial asthma, child, regression model, severe asthma, prediction, allergic rhinitis, atopic
dermatitis
1. Introduction
Allergic diseases prevalence in both children and adults increases year by year. Bronchial asthma
holds a specific place among allergic diseases. Currently, due to the rapid increase of asthma incidence,
this pathology can be considered as a global medical problem. According to GINA, in 2019, asthma
attacked 262 million people and caused death of 461000 people [1]. Bronchial asthma in children is the
most common chronical respiratory pathology. It is known that an early disease manifestation and
intensity of clinical symptoms are defined by the combination of genetic and environmental factors.
Multiple asthma manifestations were studied and their connection with various pathogenic mechanisms
was revealed [2]. Depending on the disease severity at an early age, the risk of a bronchopulmonary
pathology onset in adult life increases. Bronchial asthma wields major influence on a life quality of
patients, it leads to significant economic losses [3, 4]. A comprehensive study of the disease factors will
allow not merely to define a risk group among children but also to influence the disease severity.
Research in genetics, etiology, pathogenesis of asthma has expanded knowledge in this area, but there
is still a significant number of patients with severe asthma characterized by poor control [5]. The
diagnosis of asthma in a midchildhood is complicated due to the similarity of clinical manifestations
with other diseases. In applied medicine, predictive models based on anamnestic data are used to
ITTAP’2022: 2nd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24, 2022,
Ternopil, Ukraine
EMAIL: pihnastyi@gmail.com (A. 1); olga.kozhyna.s@gmail.com (A. 2); tv.kulik@knmu.edu.ua (A. 3)
ORCID: 0000-0002-5424-9843 (A. 1); 0000-0002-4549-6105 (A. 2); 0000-0002-8842-892X (A. 3)
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
improve the diagnosis of asthma at an early age [6]. However, asthma heterogeneity and its multiple
clinical manifestations reduce the accuracy of prediction.
2. Formal Problem Statement
Bronchial asthma, regardless of severity, is a chronic non-infectious inflammatory respiratory
disease. The inflammatory process in airways causes airway hyperresponsiveness, bronchial
obstruction and respiratory symptoms. Symptomatic manifestations of bronchial asthma, occurring
more than twice a year, require a thorough examination of a child to exclude asthma. Due to under-
diagnosis of asthma, the incorrect treatment is provided. Long-term ineffective therapy affects the
formation of nonreversible blocking of bronchi as a result of bronchial wall remodeling. Asthma with
a severe course calls a special attention and requires high drug dosage. [7, 8]. The multifactorial nature
of asthma contributes to an uncontrolled course of the disease development [9, 10]. Predicting of severe
asthma in children at risk will prevent exacerbations and improve a patient's quality of life. The
identification of factors that make asthma more difficult to control will help to eliminate causative or
precipitating factors [12, 13]. Using of linear regression models is a common approach to calculate a
probability of a severe asthma onset or an uncontrolled course of disease development [14]. Regression
models are used for preliminary estimation of bronchial asthma severity. As a rule, models contain from
three to five regressors [15] where special mention should go to asthma in parents, atopic dermatitis,
wheezes without a cold, specific Ig E, allergic rhinitis and a child's gender. To make a preliminary
assessment of bronchial asthma severity, low-dimension models are considered because of the fact that
the disease symptoms in a patient being studied are clearly caused by 5-7 factors. In most cases, these
factors appear to be qualitative. At the same time, more than a hundred factors influence severity of
bronchial asthma [16]. In most cases, the factors, on the one hand, are weakly related to each other, and
on the other hand, they make approximately the same contribution to the formation of the explained
quantity value [17]. The number of models required to describe possible situations corresponds to the
number of ways where 5 factors can be selected out of a hundred of factors is determined by the value
100!
~ 108 , (1)
5! 100 5 !
Thus, despite a large number of publications devoted to predicting bronchial asthma severity, the
proposed models can be used for a relatively small group of patients characterized by a certain set of
regressors. To increase the scope of the model, it is possible to form some generalized factors that
combine several regressors or to build models with a higher dimension. However, this approach has not
become widely used in studies dealing with predicting bronchial asthma severity due to a significant
computational complexity associated with a large number of multiple regression models building. As
an alternative to these approaches, this paper proposes a technique for approximate regression models
containing five factors building. The proposed methodology is based on the assumption that
a) the value of correlation coefficient rxm xv between the model's regressors X m , X v is small:
K xm xv
rxm xv , rxm xv 0 , (2)
Dxm Dxv
n n n
X m i m x X v i m x
m v X mi X m
mi xm
2
(3)
K xm xv i 1 , m xm i 1 , D xm i 1 ;
n n n
b) the value of correlation coefficient rxm xv is far smaller than the value of correlation coefficient
r y xm between the observed value Y and the model's regressor X m :
K y xm
ry xm rxm xv , ry xm , (4)
D y D xm
n
n n
X m i m x Yi m y
m
Yi Y m
i y
2
(5)
K y xm i 1 , m y i 1 , D y i 1 ;
n n n
These circumstances make it possible to build a multiple regression approximate model in the form:
Y 0 1 X1 2 X 2 3 X 3 4 X 4 5 X 5 , (6)
Dy
m ry x 0 rx x
m m v
; (7)
D xm
5
0 my m ,
m 1
m xm (8)
where coefficients m of the regression model are determined to within the summands of the order of
smallness 0 rxm xv . The assumptions taken as the foundation in the method of approximate regression
models building for bronchial asthma severity predicting are confirmed by the research results [18].
The simplicity of models building is the basis for widespread use of the proposed methodology in
clinical trials
3. Literature Review
Linear regression models are the most common in predicting the risk of bronchial asthma in children.
The paper [19] considers a linear regression model that analyzes the dependence of the observed value
on five factors: wheezing after exercises; wheezing causing shortbreathing; coughing on exertion;
atopic dermatitis and allergic sensitization. An eight-factor linear regression model (male gender,
postterm delivery, parental education, parental inhalation medication, wheezing/dyspnea apart from
colds, wheezing frequency, respiratory tract infections, and doctor's diagnosis of eczema (ever) and
eczematous rash present) was used to analyze a group of 2877 children to determine the level of risk of
asthma at a school age development [20]. A ten-factor model (gender, age, wheezes without a cold,
wheezes rate, disruption of activity, shortbreathing, wheezes and cough caused by exercises and
aeroallergens, atopic dermatitis, asthma or bronchitis in parents) was used in the two-staged study of
risk of asthma at a school age and is described in the paper [21]. The regression model is based on a
noninvasive predicting method and an increased number of factors. The absence of quantitative factors
characterizing results of laboratory examinations in the model is a characteristic feature of these models.
A linear regression model containing three quantitative (hospitalization, eczema and atopy in parents)
and one qualitative factor (the positive and negative predictive value of specific Ig E to inhalant
allergens) was studied in the paper [22]. This allowed us to increase a prognostic value of the model.
In [23], a simplified technique that makes it possible to build a dependence of the observed value on
quantitative factors was considered. A significant simplification was obtained as a result of using a
combination of one-dimensional and three-dimensional models instead of the four-factor model. The
uniqueness of the proposed approach is in reducing the computational complexity of the regression
models building process. Another way to reduce the computational complexity is connected with the
use of approximate methods of regression models [16, 17]. In these studies, a technique of one- and
two-parameter approximate models building (TSLP, atopic dermatitis, allergic rhinitis, bronchial
asthma in relatives of the second generation, pillow feather, domestic dust, severe) was proposed, as
well as the use of their combinations to analyze the bronchial asthma severity in children predicting as
a way to replace multifactor linear regression models. This paper expands the field of application of
approximate methods for regression models building. The purpose of this paper is to develop a
technique for the five-factor linear regression approximate models building and to substantiate the area
of their use.
4. Analysis Data Preparation
To demonstrate the technique for five parametric regression approximate models building, we will
use the results taken from a clinical study of the severity of bronchial asthma in children of Kharkov
region, 2017 [18]. The study was conducted with respect for human rights and in accordance with
international ethical requirements; it doesn't violate any scientific ethical standards and standards of
biomedical research [24]. The group for asthma severity analyzing included 90 children at the age from
6 to 18. The structure of the group was as follows: the main subgroup contained 70 children with
bronchial asthma and the control group with 20 healthy children. The data of the parents survey about
the symptoms of patients, characteristic to bronchial asthma, as well as the history of the patients'
diseases were analyzed. This information was the basis for the values of the qualitative factors used to
build the regression model formation. The clinical features of the disease course were studied. The
results of laboratory research were used to form the values of the model's quantitative factors.
During the analysis, 142 factors were under consideration. A detailed analysis of the factors is given
in [16]. Using a set of clinical research data, the most significant factors in the study were identified to
build a linear regression approximate model. For these factors, correlation coefficients rxm xv between
the model regressors and correlation coefficients ry xm between the regressor and the observed value are
given in Table 1
Table 1
Correlation Coefficients Values
Atopic Bronchial Allergic Sheep Domestic Rabbit Pillow Bronchial
dermatitis asthma in rhinitis wool dust hair feather asthma in
father
Atopic - -0.076 0.738 0.1346 0.158 0.1533 0.0181 0.1084
dermatitis
Bronchial -0.076 - 0.1025 0.1483 0.2571 0.0069 0.2842 -0.087
asthma in
Allergic 0.738 0.1025 - 0.0058 0.1732 0.1107 0.1032 0.2704
rhinitis
Sheep 0.1346 0.1483 0.0058 - 0.1507 0.2658 0.3004 0.104
wool
Domestic 0.158 0.2571 0.1732 0.1507 - 0.0424 0.1899 -0.040
dust
Rabbit
hair 0.1533 0.0069 0.1107 0.2658 0.0424 - 0.0727 0.3211
Pillow
feather 0.0181 0.2842 0.1032 0.3004 0.1899 0.0727 - -0.033
Bronchial
asthma in 0.1084 -0.087 0.2704 0.104 -0.040 0.3211 -0.033 -
father
Severe 0.3767 0.4157 0.3223 0.3373 0.3116 0.2236 0.3681 0.0309
Taking into account the selection criterion, we take six factors from Table 1 to build six proposed
models
ry x max , rx x min , m, v 1...M .
m m v
(9)
The numerical characteristics mxm , m y , Dxm , D y for the selected factors and the observed value (Severe)
determining the mathematical expectation and variance are given in Table 2.
Table 2
Regressor numerical characteristics and the explained value
№ Regressor mx , m y m
Dx m , D y
1 Sheep wool 0.5217 0.4234
2 Rabbit hair 0.5652 0.7965
3 Bronchial asthma in relatives of second
0.0658 0.0614
generation
4 Allergic rhinitis 0.4494 0.2474
5 Domestic dust 2.2319 1.280
6 Atopic dermatitis 0.0562 0.053
7 Severe 0.1124 0.0997
The numerical characteristics given in Table 1 and Table 2 were obtained as a result of clinical
studies analysis, we used them to build the linear regression approximate models (6).
5. Linear Regression Approximate Model Building
To build a linear regression equation, we introduce some dimensionless parameters:
Yi m y X m i m xm
, m , (10)
Dy D xm
corresponding to the dimensionless value of the observed value and dimensionless values of the model
regressors. Then the equation (6), taking into account the equality (7), can be written in its
dimensionless form:
11 2 2 3 3 4 4 5 5 , (11)
Dxm
m m . (12)
Dy
We use the least squares method [16] to determine the values of coefficients. From the condition for
minimum of a mean square root error in the observed value predicting
n
min
i 1
i 1 1i 2 2i 3 3i 4 4i 5 5i
2
(13)
the system of equation follows:
n n n n n n
i 1
1 1i 1i 2
i 1
1i 2i 3 1i 3i 4 1i 4i 5 1i 5i
i 1 i 1 i 1
i 1
1i i ,
n
n n n n n
1
i 1
2i 1i 2
i 1
2i 2i 3
i 1
2i 3i 4
i 1
2i 4i 5
i 1
2 i 5i
i 1
2i i ,
n n n n n n
i 1
1 3i 1i 2
i 1
3i 2i 3 3i 3i 4 3i 4i 5 3i 5i
i 1 i 1
i 1
i 1
3i i , (14)
n n n n n n
i 1
1 4i 1i 2
i 1
4i 2i 3 4i 3i 4 4i 4i 5 4i 5i
i 1 i 1 i 1
i 1
4i i ,
n n n n n n
1
i 1
5i 1i 2
i 1
5i 2 i 3
i 1
5i 3i 4
i 1
5i 4 i 5
i 1
5 i 5i
i 1
5i i ,
Its solution allows us to obtain the expressions for the values of coefficients m definition. Using
dimensionless designations (10), as well as the form of expressions K xm xv (3), K y xm (5), we obtain a
formula for correlation coefficients through the introduced dimensionless parameters:
n n
K xm xv 1 K y xm 1
rxm xv mi vi , ry xm mi i , (15)
Dx Dx n i 1 D y Dx n
m v m i 1
This allows us to represent the system of equations (14) in a simplified form:
1 rx x 2 rx x 3 rx x 4 rx x 5 ry x ,
1 2 1 3 1 4 1 5 1
rx1 x2 1 2 rx3 x2 3 rx4 x2 4 rx5 x2 5 ry x2 ,
rxm xv rxv xm ,
x1 x3 1 x2 x3 2
r r r r r , (16)
3 x4 x3 4 x5 x3 5 y x3
ry xm rxm y ,
r
14
x x 1 r x x
2 4
2 rx x
3 4
3 4 r x x
5 4
5 r y x 4
,
r r r r r ,
x1 x5 1 x2 x5 2 x3 x5 3 x4 x5 4 5 y x5
that has a solution regarding the unknown coefficients m :
m m , (17)
1 rx1 x2 rx1 x3 rx1 x4 rx1 x5 ry x1 rx1x2 rx1x3 rx1x4 rx1x5
rx1 x2 1 rx3 x2 rx4 x2 rx5 x2 ry x2 1 rx3 x2 rx4 x2 rx5 x2
rx1 x3 rx2 x3 1 rx4 x3 rx5 x3 1 ry x3 rx2 x3 1 rx4 x3 rx5 x3 , 2 …. (18)
rx1 x4 rx2 x4 rx3 x4 1 rx5 x4 ry x4 rx2 x4 rx3 x4 1 rx5 x4
rx1 x5 rx2 x5 rx3 x5 rx4 x5 1 ry x5 rx2 x5 rx3 x5 rx4 x5 1
If the conditions ry xm rxm xv (4) are met, coefficients m can be calculated using the approximate
formula:
r r 0r
M
ry xm xm xv y xv xm xv
2
r r 0r
M
v 1,v m
m ry xm 2
(19)
M M xm xv y xv xm xv
1
m v ,v m
rxm xv 2 0 rxm xv 3 v 1,v m
To simplify the material presentation, we will consider the calculation of coefficients m to within the
summands of the order of smallness 0 rxm xv
m ry x 0 rx x
m
m v
(20)
Taking into account (12), we express the value of coefficients m through the value of coefficients m
Dy
m m (21)
Dxm
and substituting the values for the m (20) calculating, we obtain the linear regression equation in the
form (6)-(8) to within the summands of the order of smallness 0 rxm xv .
In the accepted approximation, the observed value depends only on the values of correlation
coefficient rx y between the model regressors and the explained value. The linear regression equation
m
can be used to preliminary estimate the severity of bronchial asthma. To improve the predicting
accuracy for coefficients m calculating, the formula (19) that takes into account the values of
correlation coefficients between the model regressors should be used.
6. Analysis of the Results
In this section, as an example, let us consider the building of five-factor linear regression
approximate models to within the summands of the order of smallness 0 rxm xv
a 1a1 2a 2 3a 3 4a 4 5a 5 (22)
The symbol “a” (approximate) in the model parameters means that to predict bronchial asthma
severity when calculating the coefficients ma the approximate formula (20) was used. As the
regressors, we will use the factors given in Table 2 and selected using the criterion (9). To calculate
coefficients ma (20), we will use Table 1 showing the values of correlation coefficients between the
model regressors and the explained value. The results of linear regression equations building for the
selected factors are shown in Table 3.
Table 3
Linear Regression Models in Dimensionless Form
regressor exact dimensionless approximate dimen- approximate model
Missing
element
model (22) sionless model (23) error (24)
e 1e1 2e2 a 1a1 2a2 a a e
3e3 4e 4 5e5 3a3 4a 4 5a5
Sheepwool a 0.1521
e 0.1851 a 0.3371
Allergic
Rabbit hair
rhinitis
(0.110) 2 0.0353
Bronchial asthma 0.114 2 0.3813 0.224 2 0.4163
Domestic dust (0.185)5 0.0346
0.1275 0.3436 0.3125 0.3776
Atopic dermatitis
Sheepwool e 0.2501 a 0.3371 a 0.0881
Bronchial
Rabbit hair
asthma
0.773 2 0.260 4
Allergic rhinitis 0.224 2 0.322 4 0.549 2 0.062 4
Domestic dust 0.1815 0.2846
0.3125 0.3776 (0.131)5 0.0936
Atopic dermatitis
Sheep wool a 0.1081
e 0.2231 a 0.3371
Sheep wool Rabbit hair
Bronchial asthma
(0.05)3 0.08 4
Allergic rhinitis 0.3613 0.223 4 0.4163 0.322 4
Domestic dust (0.223)5 0.0326
0.0895 0.3426 0.3125 0.3776
Atopic dermatitis
Rabbit hair e 0.138 2 a 0.224 2 a 0.086 2
Bronchial asthma
0.3903 0.222 4
Allergic rhinitis 0.4163 0.322 4 (0.026)3 0.1 4
Domestic dust 0.1125 0.3516
0.3125 0.3776 (0.2)5 0.0266
Atopic dermatitis
Sheep wool e 0.2101 a 0.3371 a 0.1281
Domestic
Rabbit hair
0.085 2 0.3853 (0.139) 2 0.0313
0.224 2 0.4163
dust
Bronchial asthma
Allergic rhinitis 0.247 4 0.3476 (0.076) 4 0.036
0.322 4 0.3776
Atopic dermatitis
Sheepwool e 0.2331 a 0.3371 a 0.1041
dermatis
Rabbit hair
Atopic
0.127 2 0.3173 (0.098) 2 0.0993
Bronchial asthma 0.224 2 0.4163
Allergic rhinitis 0.250 4 0.1465 (0.073) 4 0.1655
0.322 4 0.3125
Domestic dust
We inject an accurate linear regression model
e 1e1 2e 2 3e 3 4e 4 5e 5 (23)
where, in parameters, the symbol “e” (exact) means that coefficients 1e are calculated in accordance
with the formula (17) without the introduction of any assumptions concerning the values of correlation
coefficients rx x smallness between the model regressors. The results of linear regression equations
m v
building are also given in Table 3.
We subtract the equation (23) from the equation (22) and obtain an expression for the approximation
error estimation
5
a a e
m 1
ma me m (24)
associated with replacing the exact model with an approximate one. Results of a comparative
assessment of the models (22) and (23) are given in Table 3
Table 4
Results of Analysis of the Linear Regression Approximate Model Use
approximate model
missing exact model approximate model error
regressor
element (26) (25)
Ya Yе Ya
Sheep wool Ye 0.095 0.09 X1 Ya 0.227 0.164X1 Ya 0.133 0.074X1
Allergic
Rabbit hair 0.04 X 2 0.485X 3
rhinitis
0.079X 2 0.53X 3 (0.039) X 2 0.045X 3
Bronchial asthma
0.035X 4 0.471X 5 0.066X 4 0.501X 5 (0.03) X 4 0.3 X 5
Domestic dust
Atopic dermatitis
Sheep wool Ye 0.175 0.121X 1 Ya 0.284 0.164X 1 Ya 0.109 0.942X1
Bronch.
Rabbit hair 0.027X 2 0.165X 4 0.079X 2 0.205X 4 (0.052) X 2 0.04 X 4
asthma
Allergic rhinitis
0.051X 5 0.389X 6 0.066X 5 0.501X 6 (0.015) X 5 0.113X 6
Domestic dust
Atopic dermatitis
Sheep wool Ye 0.126 0.108X 1 Ya 0.275 0.164X 1 Ya 0.149 0.056X1
Rabbit hair
Bronchial asthma 0.460X 3 0.155X 4 0.530X 3 0.205X 4 (0.07) X 3 0.05X 4
Allergic rhinitis
0.025X 5 0.469X 6 0.066X 5 0.501X 6 (0.041) X 5 0.032X 6
Domestic dust
Atopic dermatitis
Rabbit hair Ye 0.108 0.05X 2 Ya 0.234 0.08X 2 Ya 0.126 0.03X 2
Sheep wool
Bronchial asthma 0.497X 3 0.141X 4 0.53X 3 0.205X 4 (0.33) X 3 0.064X 4
Allergic rhinitis
0.031X 5 0.482X 6 0.066X 5 0.501X 6 (0.035) X 5 0.02 X 6
Domestic dust
Atopic dermatitis
Sheep wool Ye 0.087 0.102X 1 Ya 0.173 0.164X 1 Ya 0.086 0.062X1
Domes-tic
Rabbit hair 0.03X 2 0.491X 3 0.079X 2 0.53X 3 (0.049) X 2 0.039X 3
dust
Bronchial asthma
0.157X 4 0.475X 6 0.205X 4 0.501X 6 (0.048) X 4 0.026X 6
Allergic rhinitis
Atopic dermatitis
Sheep wool Ye 0.161 0.113X 1 Ya 0.291 0.164X 1 Ya 0.13 0.05X1
Rabbit hair
dermat.
0.044X 2 0.404X 3 0.079X 2 0.53X 3
Atopic
(0.035) X 2 0.126X 3
Bronchial asthma
0.158X 4 0.041X 5 0.205X 4 0.066X 5 (0.046) X 4 0.025X 5
Allergic rhinitis
Domestic dust
Using relation (21) between the parameters m and m , we proceed from the dimensionless model
(22) to its analogue (6), where, while calculating coefficients m and m , the expression (20) was
used
Ya 0a 1a X 1 2a X 2 3a X 3 4a X 4 5a X 5 (25)
The analysis of the research results given in Table 3 demonstrates a satisfactory accuracy of the
linear regression approximate model. The error of replacing an exact model with an approximate one
corresponds to the expected value of the error, defined as 0 rxm xv .
Results of proceeding from the linear regression dimensionless approximate model (22) to the model
(25) are given in Table 4. Also, let us inject an accurate linear regression model
Yе 0е 1e X 1 2e X 2 3e X 3 4e X 4 5e X 5 (26)
that corresponds to the model (23). Results of proceeding from the model (23) to the model (26) as well
as an approximation error of this proceeding Ya Yе Ya are given in Table 4.
This approximation is used to demonstrate the technique of approximate models building for
preliminary prediction of bronchial asthma severity. The value of the error that occurs when replacing
an accurate linear regression model with an approximate model can be estimated using this formula
M
rx x ry x
0 rxm xv m v v
(27)
v 1,v m
In more detailed studies, a linear regression approximate model where the coefficients m and m
are determined to within the summands of the order of smallness 0 rxm xv 2 , should be used. The error
of replacing an exact model with an approximate one significantly depends on the value of a correlation
coefficient between model regressors. Therefore, analysis of the approximate model area of application
is of utmost importance. The cases when the parameters of the model satisfy both condition (9) and the
requirement ry xm rxm xv are considered as the most successful use of approximate models.
To calculate the coefficients of the regression model, specialized software was developed using an
object-oriented programming language Java (java 1.8; jdk 1.8.0_162). Mathematical operations on
matrices were carried out using the open source mathematical package org.apache.commons commons-
math3 (version 3.2) [25], available under license: The Apache Software License, Version 2.0.
Computing resources are involved in the calculation: processor Intel®Core™ i7-4790CPU 3.6GHz;
RAM 32GB; OS: Windows 10, 64-bit.
7. Conclusion
In this paper, the application of technique of a five-parameter approximate model building to predict
bronchial asthma severity is discussed. The fact that, when predicting bronchial asthma severity, the
observed value depends on a large number of factors usually weakly related to each other is the ground
of the proposed technique. Depending on the region, age, living conditions, more than a hundred of
factors causing the disease can be revealed. A preliminary examination of a patient usually reveals from
5 to 7 factors characterizing the disease severity. This explains the existence of publications discussing
the building of a regression model with 3-5 factors. In each case, the specified set of factors is different.
This requires a huge number of linear regression models, the building of which is associated with
significant computational difficulties. The developed technique allows us to simplify the process of
five-factor linear regression models building significantly. The fact that this technique doesn't require
any massive computational resources to build linear regression multiparameter models is one of its
important advantages. It is a good way to preliminary analyze the disease severity. In this paper, not
only the technique of linear regression approximate model building but also the area of its application
as well as an error of proceeding from an accurate model to an approximate one are analyzed. The
presence of strong correlations between the model regressors has a significant effect on the specified
error. Two approximations that can be used to build linear regression models are discussed in details.
As a shallow analysis, an approximation without correlations between the model regressors can be used.
For a detailed analysis, an approximation with linear dependence of model coefficients on correlation
coefficients between model regressors taken into account while calculating these model coefficients
should be used. Approximations with nonlinear dependences taken into account when calculating the
model coefficients are a prospect for further studies. Building of linear regression approximate models
containing 10-15 regressors as well as the analysis of the proceeding error from an exact model to an
approximate one dependence on the number of regressors in the model is of particular practical interest.
This article discusses the type of linear multivariate regression model. The choice of the type of
model and the number of factors in the model is an urgent issue in predicting the severity of bronchial
asthma disease. This circumstance determines the prospects for further research: 1) comparative
analysis of the accuracy of predicting the severity of bronchial asthma disease depending on the type of
regression model with the same number of model regressors; 2) comparative analysis of the accuracy
of predicting the severity of bronchial asthma disease depending on the number of factors used in the
model; 3) selection of a criterion for assessing the quality of prediction.
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