Statistical tool to estimate and optimize the intensity of the dependence
between the parameters of a dynamic system
CARDEI PETRU
Computerized engineering
INMA
6, Ion Ionescu de la Brad Blvd., Sector 1, Bucharest
ROMANIA
Abstract: - The article presents a method of investigating the intensity of the connection between the
parameters measured in experiments for different dynamic processes. The method is based on a function
constructed from the correlation between the rows of records, called variables. The result is addressed to all
specialists who try to obtain relationships between the parameters of a process, starting from the results of
experimental research. It is considered that the relationships sought between the variables of the researched
process do not have a theoretical foundation or do not have a corresponding one. The tool for assessing the
intensity of the connection between the main parameter and the secondary parameters of a dynamic process is
tested on soil tillage processes in agriculture. A hierarchy of the intensity of the connection between the main
parameter and the secondary parameters or combinations thereof is provided. Using these hierarchies,
researchers can create regressions based on the priorities offered by these hierarchies.
Key-Words: - dynamic, process, parametric, statistics, estimators, intensity, connection
Received: August 9, 2021. Revised: March 27, 2022. Accepted: April 24, 2022. Published: June 3, 2022.
1 Introduction
The fundamental problem from which the research
whose results are given in this article starts is an old
one, over a century old. It consists in trying to
predict and minimize the soil tillage draft force
generated by agricultural machinery intended for
soil work. The literature is very rich in this field. We
have given the main references in this field of
research in [1], [2], and [3]. Prediction is one of the
most important goals of science, which is generally
aimed in any field of science. The purely theoretical
forecast, although not impossible, is rare and the
accuracy is debatable. In general, the experiment is
the one that provides answers for the construction of
models capable of predictions and optimizations,
and the whole experiment is the one that can
validate the model thus constructed. The
experimental program is decisive in the generation
of the mathematical model and its exploitation, [11-
12]. Over time, the original problem has taken on
enormous dimensions, covering virtually the entire
scientific field.
The experimental method is integrated into a more
general field of research, in which the results
presented in this article have their place: System
identification. The field of systems identification
uses statistical methods to construct mathematical
models of dynamical systems, [13]. The field of
systems identification also includes the optimal
design of experiments, which aims to generate
efficient information, able to provide mathematical
models appropriate to achieve the proposed goals.
One of the most common approaches to identifying
systems is to start experimenting and then try to
determine the mathematical relationships between
the parameters of the system, without going into
details about what is going on inside the system.
This approach to the problem is called identifying
the black box system, [14].
A dynamic mathematical model in this context is a
mathematical description of the dynamic behaviour
of a system or process, either in terms of time or
frequency. Examples include in this field are
physical processes, such as the movement of a
falling body under the influence of gravity, the
working process of an agricultural machine, the
processes of interaction of plants with the soil, etc.
One of the many possible applications of system
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Cardei Petru
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identification is the problem of control systems. For
example, it is the basis for modern data-driven
control systems, in which the concepts of system
identification are integrated into the design of the
controller and lay the groundwork for formal
evidence of controller optimization, [14].
From all that is described regarding the
identification of systems in the paragraph above,
this article provides a method and a statistical tool
for assessing the intensity of the connection between
the parameters of a dynamic system or a dynamic
process. The statistical tool consists of a function
based on the correlation coefficient between two
variables (series of experimental records of
numerical values of some process parameters)
which have one or more parameters, in relation to
which the optimal mathematical model of the
relationships will be calculated. These results are
compared in order to rank the intensities of the
dependencies between the process parameters. The
method based on this function elaborates the
strategy of using the constructed function to
evaluate the intensity of the connection between the
variables (parameters) of the experimental research
process and the formation of a hierarchy capable of
suggesting to researchers how to form regressions
capable of predictions and / or optimizations.
The proposed method and tool are simple and easy
to understand alternatives to the sophisticated
statistical methods used in the field of systems
identification: descriptive statistics, regressions,
machine learning, hypothesis testing, etc. [15-16].
The method can be extended using other estimators
of descriptive statistics, to achieve optimal statistical
models of dynamic processes. This article is an
advanced version of the preprint (not reviewed
and unpublished) archived in Research Gate
networking, [17].
2 Problem Formulation
It is considered several experimental data strings,
called variables, each with components
(characterizing, for example, a dynamic process). In
general, is supposed that there are no theoretically
or empirically established relationships between the
variables considered. In the current research, in such
situations, mathematical statistics are used to obtain
a measure of the relationship between various
variables (correlation coefficient) and
characterization of dependence through various
types of regressions.
Through experimentation, data series (variables) are
obtained for which the intensity of the connection
and the form of the dependence are investigated.
The intensity of the link between two experimental
variables can be conveniently estimated using the
correlation coefficient between the two variables,
according to, for example, [4]. Because the main
function with which we will operate in this research
works using the operator of the correlation
coefficient provided by [5], in writing the function
we will keep the name of that operator. Therefore, is
considering, for example, two experimental
variables: ,
, , data numerically and
graphically in fig. 1.
Fig. 1. Two one-dimensional variables with ten
components each, numerical values and graphical
representation.
The basic function is defined through formula (1):
(1)
where is a set of vector variables, and is the set
of real numbers, the domain of definition of the
function
being the Cartesian product of with
itself and with . By we mean the variable
obtained from the variable by raising the power
of each component (which models the
hypothetical form of the dependency between the
two variables) , obviously if the operation is
allowed. For the correlation coefficient function
we used the abbreviated notation:
(2)
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where the covariance function is defined by the next
formula:
(3)
and
being the average values of the vectors
, and , and the function:
(4)
is the standard mean deviation of the variable . For
two fixed variables (for example those in fig. 1), the
function
becomes a function of a real variable.
The variation of this function is difficult to study
theoretically, and for our immediate purposes, it is
easier to study this function numerically. Please
note that when researchers look for exponential
dependency laws, in general, the exponents are not
very large, so we will represent the function
over a reasonable length of time for this application.
By selection (scanning several intervals for ), we
found the interval between 1.5 and 2.5, the interval
in which the function
has a minimum equal to -1
for the value = 2 of the argument.
Fig. 2 Dependence of the function
, on , for
between 1.5 and 2.5.
As an immediate interpretation, the function
indicates an inverse (in the sense that if
increases, then decreases) dependence between
the variables and , of the form:
(5)
If we specify that we chose the two variables as the
temporal and the spatial variable of a vertical fall of
a point body, with zero initial velocity:
(6)
where is the current height of the body, is the
height at the initial moment, = 0, = 9.81m / s2 is
the local gravitational acceleration, and is the time,
then the indication of the function
, is not a
surprise but seems more like a promise to obtain a
tool to complete statistical investigations performed
on experimental data. At a theoretical level, there
remain, for the function
and, in general for the
research method described, some problems that
need to be solved:
D1) the theoretical study of the variation of the
function
must be tried (even if it will not be
possible to solve until the end by analytical
methods), to find out if there is always an extreme
point, or if there are cases in which it has several
points of extreme.
D2) if there are cases in which the function
has
several extreme points, hypotheses must be
advanced on the choice of a physically acceptable
variant for the relationship whose model is being
tried.
D3) If the function
does not show extreme values
at physically acceptable intervals, can it be
concluded that the variables are independent?
D4) The construction of generalized functions of
type (1), such as:
(7)
where is a function defined on ,
and can be inspired by additional information on the
phenomenon studied or by the experience of the
statistical operator. These problems remain for
specialized theoretical studies. In the next chapter,
we will focus on some experimental results from the
activity of researchers in the field of agricultural
tillage machines.
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3 Results and comments
In this chapter, the proposed research method will
be applied in order to estimate the intensity of the
soil tillage draft force connection for a machine
designed for soil tillage, with a series of parameters
of the process interaction with the soil, in
experiments, as well as with their combinations. For
this purpose, the experimental data published in [6],
obtained for a plough with two mouldboards, will be
used. The data is voluminous, it is in the public
domain and that is why we do not reproduce it. The
authors [6] vary the humidity, depth and working
speed and measure the soil tillage draft force. The
behaviour of function (1) will be investigated, for
variable is the soil tillage draft force, and
variable is specified in table 1, first column. For
convenience in writing, denote by , the soil
moisture, by working depth, respectively by , the
working speed, and by , the soil tillage draft force.
Table 1 Evaluations of the connection between the
soil tillage draft force and other parameters of the
soil processing with the plough with two
mouldboards from [6].
Variable
extreme
Extreme of correlation
coefficient
-0.00003
0.124
3.547
0.871
1.228
0.423
1.774
0.871
1.547
0.928
1.237
0.829
0.774
0.928
The results in Table 1 show that the parameters with
the greatest influence on the tensile strength, , are
the combinations and , for which the
correlation coefficient takes values greater than 0.9.
The working depth is dominant in these products.
This working depth, , at the first or second power,
gives a maximum correlation coefficient of 0.871,
while the working speed, , at the first power
reaches a maximum correlation coefficient with the
value 0.423, all with the soil tillage draft force, .
These results motivate the consideration of the
multiplicative combinations between depth and
working speed, in all the calculation formulas
proposed for the soil tillage draft force, listed in [1],
[2] and [3]. All calculation formulas for soil tillage
draft force, examined in [1], [2] and [3] have integer
exponents for working depth, working speed or
combinations thereof. If taking into account the
results of the investigation method presented in this
paper, the exponents would become numbers that
are no longer integers, then, in order to save the
dimensional correctness of the formulas, it is
proposed that these combinations be replaced by
dimensionless reports, which can be raised to any
real powers without creating dimensional problems
for formulas. A correctly dimensional solution
which uses exponents that are not integers is
suggested in [7]. Results somewhat similar to those
in Table 1, are obtained by processing other
experiments. For example, for the experimental
results in [8], the estimates in Table 2 are obtained.
Table 2 Evaluations of the connection between the
soil tillage draft force and other parameters of the
soil processing with the plough with two
mouldboards from [8].
Variable
extreme
Extreme of correlation
coefficient
0.00007
0.933
0.415
0.339
0.000035
0.933
0.941
0.824
1.339
0.666
0.471
0.824
The experimental results given in [9], for a system
of physical simulation of the working process of
some chisel working body, are described in table 3.
Table 3 Estimates of the relationship between soil
tillage draft and other parameters of soil processing
with the plough with two mouldboards from [9].
Variable
extreme
Extreme of
correlation
coefficient
1.039
0.689
1.659
0.038
0.519
0.689
0.512
0.424
9.581
0.332
0.256
0.424
Rake angle
1.048
0.631
Cut angle
-0.00001487
0.111
a2·sin(rake
angle)
1.25
0.971
av·sin(rake
angle)
0.982
0.727
The experiments whose data are published in [10]
bring in the set of measured data some parameters
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rarely considered: soil density, resistance to soil
cone index, soil moisture, cohesion and shear stress
of the soil. An analysis similar to the above of these
records leads to the results in Table 4.
Table 4 Estimates of the relationship between
soil tillage draft force and other parameters of
soil processing with the plough with two
mouldboards from [10].
Variable v2
x extreme
Extreme of
correlation
coefficient
a (working
depth)
1.088
0.659
Soil density
18.307
0.812
Resistance to
vertical
penetration of
the soil
1.808
0.842
Soil moisture
91.387
-0.387
Soil cohesion
0.0000001
0.871
Ground shear
stress
4.495
0.925
The results obtained using the research method
described are supported by the function of
investigating the intensity of the link between the
process parameters, (1), allowing the extraction of
useful observations for the use of this technique or
research methods.
First of all, it is noted that the working depth is the
process parameter most intensely correlated with the
soil tillage draft force (or which has the greatest
influence on the soil tillage draft force). It is also
observed that at the working speeds characteristic of
soil tillage works, the influence of the working
speed parameter on the soil tillage draft force is
small in relation to the influence of other parameters
working depth, the tilt angle of the working body,
shear stress and soil cone index. It can be suspected
that at high speeds, exceeding a certain threshold
whose value depends on the mechanical and
structural characteristics of the soil, the influence of
the working speed on the soil tillage draft force
becomes higher.
It is also possible to observe the usefulness of
finding the intensity of the connections between the
process parameters, using intermediate functions: in
table 3 it is observed that the soil tillage draft force
depends on the angle of inclination, but more
intensely on its sine.
4 Conclusion
This article proposes a tool for researching the
intensity of the connection between the parameters
of a process, studied experimentally. The final goal
is to obtain sufficiently intense parametric
combinations related to the main parameter of the
process so that an acceptable regression formula can
be proposed. The regression formula will be
physically adjusted so that the requirement that the
arguments of the transcendent functions are
dimensionless is met.
The proposed research method and the function
which is the main tool have been tested and provide
the intensity of the link between the main parameter
of the tillage process in agriculture (soil tillage draft
force) and the other process parameters, some input,
others command and control (depth, width and
working speed, in particular).
The proposed investigation method produces a
hierarchy of the influence of parameters or
parametric combinations on the main parameter of
soil processing, the soil tillage draft force. This
ranking is then used to create possible regressions
for predicting the soil tillage draft force.
Although it is tested on the process of tillage in
agriculture, the proposed tool can be used similarly
for other processes in agriculture or other fields in
which experimental research is carried out to find
relationships between characteristic parameters,
relationships that cannot be substantiated
theoretically.
The tool can be further refined until optimal
regressions are obtained, at least in the first stage,
obviously only if there will be requests.
References:
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Relative ordering tests for draft force models in
soil tillage, ERDEV Proceedings conference,
Jelgava, 2021.
[2] Cardei P., Muraru S., Muraru C., Sfiru R.,
Validation and ordering test for an formula of
soil tillage draft force, extended to the
dependence on soil moisture, preprint, DOI:
10.13140/RG.2.2.26955.54569, 2019.
[3] Cardei P., Condruz P., Sfiru R., Muraru C.,
Test for physical laws of draft force generated
in tillage operations, ERDEV Conference
Proceedings, 2020.
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Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
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