Volume 15 Issue 04, April 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
70
PROCESSING EXPERIMENTAL RESULTS AND CREATING AN OPTIMAL
MATHEMATICAL MODEL
Asraev Z.R.
PhD, Associate Professor of Bukhara State Technical University
Annotation:
The scientific research focuses on the systematic approach to analyzing
experimental data and deriving an effective mathematical model that best represents the
underlying phenomena. This process involves data collection, pre-processing, statistical analysis,
and model selection. By using optimization techniques and mathematical tools, researchers aim
to identify the most accurate and efficient model, ensuring that it not only fits the experimental
data but also generalizes well to future experiments. The work typically includes model
validation, error analysis, and refinement to ensure the model’s robustness and predictive power.
This process is crucial in various scientific and engineering fields where accurate modeling of
complex systems is required for prediction and decision-making.
Keywords:
experimental data, data processing, mathematical model, optimization, model
selection, statistical analysis, prediction, regression analysis, machine learning, system
identification
Introduction
. In order to automate production and achieve scientifically sound results in a
market economy, it is important to create adequate mathematical models of objects, processes,
systems or phenomena and conduct scientific research based on them. Such adequate models
directly depend on the accuracy and reliability of the experimental data. Today, experiments are
not only a modern means of obtaining scientifically sound knowledge in the relevant fields of
natural and technical sciences, but also a necessary component of the implementation of
innovative technologies in economics, sociology, politics, the military sphere, as well as in
production.
There are two methods for studying the objects, systems, processes, or phenomena mentioned
above. The first is the transition from an object to a model, and the second is the transition from
a model to an object. One of the main stages of the transition from an object to a model is the
processing of experimental data. Such experimental data serve as the basis for analyzing the
subject area, developing physical or econometric models, as well as studying the state of the
object or predicting it, and determining alternative parameters. Therefore, in recent years,
scientists have paid increased attention to the development of new statistical methods for
processing experimental data, improving them and increasing their adequacy, and widely
introducing them into various sectors of the economy. This, in turn, places demands on the
quality of experimental work and the introduction of cost-effective methods and tools.
Such experiments are conducted in three parts: studying the object, analyzing the conditions, and
exploring the possibilities of conducting the experiment.
Research shows that it is advisable to conduct experiments in the following stages:
Volume 15 Issue 04, April 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
71
studying the essence of the object or process being studied and expressing it
mathematically based on the information collected, analyzing and determining the conditions
and means of conducting the experiment;
creating conditions that ensure the most efficient conduct of experiments and research of
the object under study;
collection of experimental results, their registration and mathematical processing and
description of the processed results in a convenient form;
experience the results detailed analysis and lighting ;
experience the results application refers to the state of the object decisions
acceptance q ilish , as well as from the model when making decisions , predicting , avoiding
the object in the process or use in its optimization .
Experiments can be divided into several classes according to the type of research object. They
are physical , engineering , medical , biological , social , sociological and other classes are
included . Nowadays, it is scientifically proven by scientists and engineering
their
experiences of the transmission general The rules are scientifically based out That's it in
the experiments , he gave an idea of processes natural and artificial physical objects ( structures )
are studied in detail . According to the rules of conducting experiments , researcher them start
eating physical skills in situations measuring processes must be repeated several times .
Later, it will be possible to give the required values of the input variables, vary them over a large
range, and remove or, conversely, include parameters whose dependence is not studied during
this time interval. Such opportunities are limited in other class problems. Some methods used in
conducting physics and engineering experiments, for example, methods of statistical data
processing, can also be successfully applied to non-technical problems.
It is appropriate to divide research experiments into classes according to the following
characteristics:
by the degree of proximity of the experiment to the object being used directly, that is, the
object from which new information needs to be obtained (natural, demonstration or polygonal,
model, computational experiments);
for the purposes of conducting the experiment (research, testing or control, management-
optimization, guidance);
according to the degree of influence on the conditions of conducting the experiment
(passive and active experiments);
according to the level of direct human participation in experiments (non-automated -
automatic, automated tools that change and create the conditions for conducting experiments,
Volume 15 Issue 04, April 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
72
collect and process experimental data).
It should be noted that a model created on the basis of collecting and processing experimental
data is undoubtedly one of the necessary, but not sufficient, conditions for a positive impact on
the development of the relevant field.
Today, the rapid development of computer technologies has created opportunities for the
automatic collection of experimental data and their processing, as well as the development of
many technical and software packages for production automation and their successful
implementation.
The main goal of our scientific research is to study one-factor processes and create a regression
equation for such processes. An adequate mathematical model can be selected based on the data
of experimental research processes using computer technologies. Instrumental software makes
the work of scientific researchers much easier. It frees researchers from manually calculating the
regression equation based on the data obtained from the experimental results. Also, the software
allows to choose an alternative mathematical model in the management of technological
processes. By entering the results of scientific research into the database of the program, it
creates opportunities to choose a regression equation based on computer modeling, build
appropriate graphs, and make alternative decisions.
Taking into account that creating an automated software system for selecting a regression
equation for one-factor processes based on experimental research data is a complex process, we
will develop them conditionally in several stages.
Phase 1. Include experimental data or observational results.
the influencing factors
X and the resulting factors Y
, as well as its physical meaning in the
development of the process under study. One of the necessary factors for the effective use of
statistical methods in the construction of empirical relationships is the preliminary statistical
processing of experimental data or observation results.
The main content of initial data processing is to eliminate gross errors and suspicious values that
may arise due to measurement of indicator values or some unexpected reasons.
During initial processing, specific issues are usually resolved: errors and outliers of the observed
quantity and other indicators are eliminated, analyzed, omitted measurements are restored,
measurement data are condensed (homogeneity is checked, univariate data is combined, data is
grouped, parameters of the measured data are estimated), and distribution laws are studied.
Any model should allow you to simulate the observed phenomenon or process (in any language -
mathematical, graphical, algorithmic, conversational, etc.). The specific goals set also determine
the language in which the model will be written. Today, most technical and physical models are
written in the language of mathematics.
Let x
1
,x
2
,... ,x
N
be the results of observing the input of an object, and y
1
,y
2
,... y
N
be the
observations of the output results 1,2,…N in discrete time units, respectively. Such observations
Volume 15 Issue 04, April 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
73
associate the object with an unknown operator F
0 :
Y
i
=F
0
(x
i
) (i=1,2,...,N)
is to synthesize such a model operator
F , where F
0
is the estimation of
x
i
and
y
i
from
observations . Naturally, any one model operator by criterion b '
F is
the object operator
F
0
ga
ya q in b ' death demand will be done , that is
F ~ F
0
.
Stage 2. Calculation of statistical indicators.
To effectively use statistical methods in constructing empirical relationships, preliminary
statistical processing of experimental data or observation results is required. This involves
calculating the arithmetic mean, central moments, dispersion, root mean square difference, and
coefficients of variation.
Stage 3. Calculation of the correlation coefficient.
The essence of this analysis is to determine the degree of probability of a relationship (usually
linear) between two or more random variables. The set of such random variables includes the
initial random variables (
X
) and the resulting random variable (
Y
). Correlation analysis allows
you to select factors or regressors (in a regression model) that have a significant impact on the
resulting factor and assess the degree of agreement with experimental data.
Correlation connections
different densities level b can die .
If (
X )
factor k '
indicator for the price close to each other , the resulting factor (
U )
of average price around
dense located values
suitable if it comes , such a part is dense is calculated . In a linear
relationship, as the
X
indicator increases (decreases),
the U
indicator also increases (decreases),
and the relationship is expressed by a straight-line equation. If the
U indicator decreases
(increases) as the
X
indicator increases (decreases), then the relationship is inverse and is
expressed by some kind of curved line equation.
The closer the correlation coefficient value is to |1|, the stronger the linear relationship between
X
and
Y
, while the closer it is to 0, the stronger the linear relationship. If the correlation coefficient
is 0, there is no linear relationship, but it can produce a nonlinear correlation.
Step 4. Determination of the optimal regression equation.
One of the problems in constructing a regression equation is to determine the type of
analytical function that reflects the mechanism of the relationship between the outcome
and factor indicators. In order to show a certain relationship with this or that equation, the
researcher must put forward a working hypothesis that can be subsequently confirmed or
refuted.
In this situation, the following analytical method of determining the relationship is used: Let
Y be
a function of
the variable
X
with parameters
"a"
and
"b"
. Then we choose the empirical
relationship from the following set of functions:
1)
b
ax
y
+
=
- linear function;
Volume 15 Issue 04, April 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
74
2)
b
x
a
y
=
-
level function;
3)
y
b
a
y
=
- k ' index function;
4)
b
x
a
y
+
=
- hyperbolic function;
5)
b
ax
y
+
=
1
- fraction rational function;
6)
b
ax
x
y
+
=
- fraction rational function;
7)
b
x
a
y
+
=
lg
is a logarithmic function.
that represents an alternative representation
(
)
b
a
x
f
y
,
,
=
of the constructed graph, we perform
the following steps:
x
1
and
x
n )
on the cross section where the experimental results are presented and find the
following intermediate values:
a)
2
2
1
1
n
n
y
y
арифметик
Y
x
x
арифметик
X
+
=
+
=
,
b)
n
n
y
y
геометрик
Y
x
х
геометрик
X
=
=
1
1
,
c)
n
n
n
n
y
y
y
y
гармоник
Y
x
x
x
x
гармоник
X
+
=
+
=
1
1
1
1
2
2
,
2) We determine the values of the function
X
"Y"
(
Y*)
corresponding to the calculated values of
" " . If the calculated
X
value is equal to one of the given values of X, then the corresponding
given value is
Y*
. If the calculated
X
value does not correspond to the distribution values, then
X
the intermediate points (
i
and
i+1) at which these values lie
are determined:
X
i
<
X
< X
i+1
and
Y*
is found using the interpolation formula:
)
(
1
1
*
i
i
i
i
i
i
X
X
X
X
Y
Y
Y
Y
+
=
,
The arithmetic,
geometric
, and
harmonic values of Y * are determined
, respectively , and the
following differences are calculated:
Volume 15 Issue 04, April 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
75
ap
aр
Y
Y
P
=
*
1
,
ap
гaр
Y
Y
P
=
*
4
,
геo
геo
Y
Y
P
=
*
2
,
гap
aр
Y
Y
P
=
*
5
,
геo
aр
Y
Y
P
=
*
3
,
гap
гaр
Y
Y
P
=
*
6
,
ap
гео
Y
Y
P
=
*
7
Each
P
k
(k=1,2,…,7)
corresponds to a certain dependency function:
P
1
y=ax+b,
b
x
a
y
P
+
=
4
,
b
ax
y
P
=
2
,
b
ax
y
P
+
=
1
5
,
x
ab
y
P
=
3
,
b
ax
x
y
P
+
=
6
,
P
7
y=algx+b,
the calculated
P
k is then
determined. The relationship to which the smallest value corresponds is
considered a regression equation representing the distribution of the experimental results, and its
unknown parameters a and b are determined using the least squares method.
( )
=
2
x
x
n
y
x
xy
n
a
i
,
( )
=
2
2
2
x
x
n
x
xy
x
y
b
Step 5. Graphical analysis.
At this stage, the correlation graph of the values obtained as a result of the research and the
graphs of all seven calculated equations (functions) are built in one coordinate system.
Among them, if the graph of the corresponding equation is close to the correlation graph (main
graph) of the values obtained based on the research results, then this equation is considered valid.
References
1.
1. Lvovsky E.N. “Statistical methods for constructing empirical formulas”, M., Nauka.
Volume 15 Issue 04, April 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
76
2012.
2.
2.2. Kabulov A.B., Kenjaboev O.T. Economic mathematical methods and models in
valuation. – Tashkent: “Fan”, 2006.
3.
3. Ermakov, S.M., Mikhailov G.A. Statistical modeling / S.M., Ermakov, G.A. Mikhailov.
- M.: Nauka, 2002. - 296 s.
