Дослідження факторного впливу на однорідність помелу зерна кави методами статистичного аналізу
In order to assess the impact of each of the factors that affect the quality and uniformity of grinding coffee beans and to compare the impact of these factors, it is worth establishing a quantitative indicator of this impact. To solve this problem, dispersion analysis was used as a method of organi...
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Репозитарії
System research and information technologies| _version_ | 1867334437035835392 |
|---|---|
| author | Hryhorenko, Ihor Kondrashov, Serhii Hryhorenko, Svitlana Opryshkin, Oleksandr |
| author_facet | Hryhorenko, Ihor Kondrashov, Serhii Hryhorenko, Svitlana Opryshkin, Oleksandr |
| author_institution_txt_mv | [
{
"author": "Ihor Hryhorenko",
"institution": "Національний технічний університет “Харківський політехнічний інститут”, Харків"
},
{
"author": "Serhii Kondrashov",
"institution": "Національний технічний університет “Харківський політехнічний інститут”, Харків"
},
{
"author": "Svitlana Hryhorenko",
"institution": "Національний технічний університет “Харківський політехнічний інститут”, Харків"
},
{
"author": "Oleksandr Opryshkin",
"institution": "Національний технічний університет “Харківський політехнічний інститут”, Харків"
}
] |
| author_sort | Hryhorenko, Ihor |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2024-08-11T01:12:49Z |
| description | In order to assess the impact of each of the factors that affect the quality and uniformity of grinding coffee beans and to compare the impact of these factors, it is worth establishing a quantitative indicator of this impact. To solve this problem, dispersion analysis was used as a method of organizing sample data according to possible sources of dispersion. The chosen method made it possible to decompose the total dispersion into components caused by the influence of factor levels. Grinding time, geometric dimensions of the grain, moisture content of the grain, speed of rotation of the motor shaft were selected as factors influencing the homogeneity of grinding. The justification and assessment of the reliability of statistical conclusions about the informational significance of indicators affecting the homogeneity of coffee grinding was carried out to ensure the highest possible probability of the obtained result. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.2.10 |
| first_indexed | 2025-07-17T10:28:15Z |
| format | Article |
| fulltext |
I.V. Hryhorenko, S.I. Kondrashov, S.M. Hryhorenko, O.S. Opryshkin, 2024
Системні дослідження та інформаційні технології, 2024, № 2 137
UDC 519.237.4
DOI: 10.20535/SRIT.2308-8893.2024.2.10
STUDY OF THE FACTOR INFLUENCE ON THE UNIFORMITY
OF COFFEE GRAIN GRINDING
BY METHODS OF STATISTICAL ANALYSIS
I.V. HRYHORENKO, S.I. KONDRASHOV, S.M. HRYHORENKO,
O.S. OPRYSHKIN
Abstract. In order to assess the impact of each of the factors that affect the quality
and uniformity of grinding coffee beans and to compare the impact of these factors,
it is worth establishing a quantitative indicator of this impact. To solve this problem,
dispersion analysis was used as a method of organizing sample data according to
possible sources of dispersion. The chosen method made it possible to decompose
the total dispersion into components caused by the influence of factor levels. Grind-
ing time, geometric dimensions of the grain, moisture content of the grain, speed of
rotation of the motor shaft were selected as factors influencing the homogeneity of
grinding. The justification and assessment of the reliability of statistical conclusions
about the informational significance of indicators affecting the homogeneity of cof-
fee grinding was carried out to ensure the highest possible probability of the ob-
tained result.
Keywords: dispersion analysis, homogeneity of grinding, factor influence, model,
indicator of control, coffee bean.
INTRODUCTION
The problems of determining the factor influence on the quality and uniformity of
grinding and the creation of systems for controlling the grinding process are of
interest to both domestic scientists and the world scientific community. The paper
[1] states that one of the most common problems in the preparation of coffee
drinks in coffee machines is the unsatisfactory quality of coffee, which is associ-
ated with improper grinding of coffee beans, and the taste of the coffee drink de-
pends on the size of the ground particles and the uniformity of coffee grinding.
The paper [2] examines the influence of the origin of coffee beans and the tem-
perature during the grinding of roasted coffee. It is noted that the extraction de-
pends on the temperature, the chemical composition of the water, as well as the
available surface area of the coffee. The study [3] reported that some physico-
chemical characteristics such as extraction yield, total dissolved solids, total phe-
nol content, pH, and titrated acidity can strongly depend on the degree of grinding
of the coffee bean. The work [4; 5] is aimed at developing an experiment to study
the key factors affecting different methods of coffee preparation. The need to ob-
tain different degrees of grinding of coffee beans in order to be able to provide a
high flow rate of coffee in the preparation of espresso is discussed in the paper [6].
The structure or scheme of an experiment is described, by the factors in-
volved in it and the ways in which different levels of different factors are com-
bined [7]. The variance is used here as the simplest measure of dispersion, provid-
I.V. Hryhorenko, S.I. Kondrashov, S.M. Hryhorenko, O.S. Opryshkin
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 138
ing an opportunity to compare the influence of the factor under study and the fac-
tor of chance [8]. If the dispersion is due to the joint action of random causes and
the change in the levels of the factors, then by obtaining an estimate of the total
response variance and the estimation of the variances of the factors, one can find
an estimate of the residual variance, and then, using statistical variance compari-
son criteria, rank the factors according to the degree of their effect on the response
dispersion.
PRESENTATION OF THE TASK
The quality of grinding coffee beans is influenced by a number of factors that
negatively affect the uniformity of grinding, which are difficult to stabilize to re-
duce the impact on the original value. As previously mentioned, it is necessary to
carry out a procedure for randomizing factors in order to make their impact ran-
dom and to be able to use statistical criteria.
Suppose that Н — is the parameter of the control object characterizing the
homogeneity of coffee grinding, which needs to be determined; F1, … Fn — fac-
tors affecting the quality of the grinding process (e.g. grinding time, speed of rota-
tion of the motor shaft, temperature at the engine stator, distance between mills,
grain moisture, geometry of the bean: width, thickness, length). The result that
takes into account the action of each of the factors can be written in the form of a
mathematical model in which the influencing factors are Н and (n – 1) factors due
to the variability of the remaining control indicators. This statement is due to the
fact that the remaining indicators characterize quantitatively (n – 1) the physical
properties of the control object and can be directly measured.
To assess the homogeneity of coffee grinding, consider the model of the ef-
fect on the measurement result of the control index F taking into account the ef-
fects of four factors (H and factors whose levels are quantified by the values of
the three control indicators (grinding time, geometric grain sizes, grain moisture).
Data in cross-classification are denoted by symbols with four indices δγβα ,,, .
Since the control indicators are not additive, it is necessary to introduce compo-
nents into the model characterizing the interaction between the indicators. Thus,
the mathematical model has the form:
βγαδαγαβδγβα )()()()( ABfCfBfACBAfFF i
αγδαβδαβγγδβδ )()()()()( fBCfACfABBCAC
.)()( αβγδαβγδβγδ ifABCABC (1)
where δγ,β,α, are the number of factor levels; F — is the general aver-
age; f — is the deviation of the measurement result of the control indicator F
from its average value F , which is due to the influence of the parameter H;
δγβ ,, CBA — deviation of the measurement result F from F , due to three
factors; γδβγαδαγαβ )(,)(,)(,)(,)(,)( BCACABfCfBfA — deviations due to
pairwise interactions of all influencing factors; ,)(,)(,)( αγδαβδαβγ fBCfACfAB
βγδ)(ABC — deviations due to the interaction of three influencing factors;
αβγδ)( fABC — deviation, which is due to the interaction of four influencing fac-
Study of the factor influence on the uniformity of coffee grain grinding by methods …
Системні дослідження та інформаційні технології, 2024, № 2 139
tors; αβγδi — random residue; і — is the number of multiple measurement at
fixed levels δγ,β,α, .
The initial conditions for model (1) will be:
1. ;0;0;0;0
δ
δ
γ
γ
β
β
CBAf
2.
;0)(;0)(
;0)(;0)(;0)(;0)(;0)(
;0)(;0)(;0)(;0)(;0)(
δ
γδ
γ
γδ
δ
βδ
β
βδ
γ
βγ
β
βγ
δ
αδ
α
αδ
γ
αγ
αβ
αβ
α
αβ
BCBC
ACACABABfC
fCfBfBfAfA
3.
;0)()()(
;0)()()(
;0)()()(
;0)()()(
δ
βγδ
γ
βγδ
β
βγδ
αγδ
γ
αγδ
α
αγδ
α β
αβδαβδαβδ
α β
αβγαβγαβγ
ABCABCABC
fBCfBCfBC
fACfACfAC
fABfABfAB
4. ;0)()()()(
δ
αβγδ
γ
αβγδ
β
αβγδαβγδ fABCfABCfABCfABC
α
5. .0
α β γ δ
αβγδ
i
i
In addition to these conditions, restrictions are imposed on the random balance:
1) all iαβγδ are mutually independent;
2) 22 ][ abcqdМ 22
αβγδ ][ σM i ;
3) random variables iαβγδ are distributed according to the normal law.
Regarding the type of deviations αf , δγβ ,, CBA we note the following:
1) αf — is a random variable because it reflects the effect of a priori uncer-
tain levels of the parameter of the control object Н;
2) δγβ ,, CBA — are parameters by virtue of the metrological values of the
control indicators.
Since due to the randomness of the levels of the parameter Н the resulting
model (1) is not exclusively parametric, but can be attributed to mixed models.
TRANSITION TO THE MOST COMMON FACTOR FUEL MODEL
It is a well-known fact that the multifactor model (1) requires for its study a sam-
ple size of rk 4 , where k — is the number of levels for each of the factors, and
r — is the number of multiple observations for all possible combinations of levels of
the influencing factors. In order for the results obtained during the variance analysis to
be statistically significant, the value must satisfy the conditions 4k , 1r .
I.V. Hryhorenko, S.I. Kondrashov, S.M. Hryhorenko, O.S. Opryshkin
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 140
Thus, at 5k , and 2r the minimum volume of the number of four-
dimensional observations of the control index F should be 1250254 values. It
turns out that it is very difficult, and sometimes almost impossible to provide such
a large number of non-standard samples of coffee beans while maintaining the
complete uniformity of the measurement experiment.
In order to avoid this complexity, we simplify model (1), leaving only the main
deviations αf , δγβ ,, CBA and the deviations caused by pairwise interactions
αδαγαβ )(,)(,)( fCfBfA . The resulting model will contain an increased residual
αβγδνψ , which also includes a random residual iαβγδε , and deviations caused by the
action of three: βγδαγδαβδαβγ )(,)(,)(,)( ABCfBCfACfAB and four — αβγδ)( fABC ,
influencing factors, as well as pairwise deviations ,)( βγAB ,)( AC γδ)(BC :
αβγδναδαγαβδγβα ψ)(,)(,)( fCfBfACBAfFF (2)
In order to reduce the complexity of the model (1) it can be reduced to three
two-factor simplified cross-classification models:
νδγ;)()( αβναββα iAfAAfFF ; (3)
νδα;)()( αγναγγα iBfBBfFF ; (4)
νiγfCCfFF C β;)( αδν)(αδδα . (5)
where — is the number of multiple measurements of the F indicator in the ta-
ble cell with the original model data (3), (4) і (5); αβν)(ψ A , αγν)(ψ B , αδν)(ψ C —
random residues due to three factors: grinding time, geometric grain size, grain
moisture, respectively.
A comparison of the residuals of the presented models with the residuals of
models (1) and (2) shows that simplifying the model obviously reduces its accu-
racy, because these residuals are more than αβγδνψ , а αβγδνψ > iαβγδε .
Let’s analyze the condition that allows us to synthesize model (2) on the ba-
sis of models (3), (4), and (5). For each of these models, there is the same main
deviation s, additional deviations δγβ ,, CBA , and deviations due to the effects of
pairwise interaction.
The first and most important condition is to ensure that the standard devia-
tions in models (3), (4), and (5) are equal to each other. To do this, the number of
groups of observations of the control indicator F must be the same for all models,
which corresponds to the same number of rows in the original data table. At the
same time, the number of values of the control indicator F in each group should
be the same b, and the value of αβγδF in the middle of each group should remain
unchanged (where N — is the number of measurements) for any of the models (3),
(4), and (5). The grouping of iF values should be carried out according to the
specified groups of values of the parameter Н.
It should be borne in mind that the method of forming columns (subgroups)
in the table of the established source data should be determined by the established
additional influencing factor, and provide a single procedure for selecting αβγδF
values for each of the subgroups of the group with a fixed number bαα, ,1 . The
Study of the factor influence on the uniformity of coffee grain grinding by methods …
Системні дослідження та інформаційні технології, 2024, № 2 141
number of subgroups in each of the groups must also be the same m, for each of
the models (3), (4) and (5). Each of the subgroups of any of the groups (cells of
the source data table) will have the same bmNg — the number of observations.
In order to model additional factor injection (based on all vortex data) on
display F, you need to carry out the following operations:
– rank the intragroup values of the control indicator Fp, you need to carry out
the following operations;
– break the ranked (for all b groups) series of values of the Fp indicator into
m subgroups;
– in each of the subgroups, select the g values of the information index F
corresponding to the g values of the Fp indicator and enter them into the source
data cell.
The resulting mb table of observation results of control measure F values
with g multiple observations in each of mb cells can now be used for variance
analysis of any of the models (3), (4) and (5) cross classifications corresponding
to a given additional factor influencing Fp, 3,1 p .
We will introduce the designation of these three factors through uts FFF ,, .
In general, we will denote any of these factors through Fp. Therefore, any of the
models (3), (4), and (5) can be represented in the form:
zzzz fppfFF ψ)( αα , (6)
where zψ — is a random residue.
The complete decomposition of the sum of the squares of deviations of the
values zF from F , under the initial conditions and constraints of the model
(1), will have the following form:
pfppf WWWWW . (7)
The results of the variance analysis of the model (6) are presented in Table 1,
where zzz FFFF ,, — are the average values in rows, columns and in cells.
T a b l e 1 . Results of dispersion analysis
Source of variability Number of degrees
of freedom k Sum of squares
The main factor H b – 1
b
f FFgpW
1
2)(
Additional factor Fp m – 1
m
z
zp FFgbW
1
2)(
Interaction between
H and Fp
(b–1)(m–1)
b m
z
zzfp FFFFgW
1 1
2)(
Remainder
(in the middle of the cell)
bm (g–1)
b m
z
g
zzp FFW
1 1 1
2)(
General N – 1
b m
z
g
z FFW
1 1 1
2)(
I.V. Hryhorenko, S.I. Kondrashov, S.M. Hryhorenko, O.S. Opryshkin
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 142
Unfortunately, for models (3), (4) that (5), the same F and the sums of W
and fW is represented by the sum of squares from F from F model (2) as the
union of the sum of (7), 3,1 p with the fictitious residual sum *
W , which char-
acterizes the influence of factors not taken into account in the model:
*
WWWWWWWWW fuftfsutsf . (8)
The resulting ratio (8) makes it possible to simplify the previous model (2)
and represent it in the following form:
,ψ)()()( *
αβγδναδαγαβδγβ fCfBfACBAfFF α (9)
where *
αβγδνψ — is a random residue due to the action of three factors: grinding
time, geometric grain size, grain moisture. The values of the sums of the right part
of the expression (8), in addition to *
W are calculated according to the equations
of the sums of the squares of the tables of the results of the variance analysis of
the models (3), (4) and (5), similar to Table 2, replacing the Fp factor with a spe-
cific additional factor — uts FFF ,, . The sum of *
W can be calculated from any
of the equations:
;
;
;
*
*
*
sftfstu
ufsfust
uftfuts
WWWWWW
WWWWWW
WWWWWW
(10)
WWWWWW futs 22* . (11)
Table 2 presents the results of the variance analysis of the simplified model
(9), the random residue of which will be fictitious (determines the sum of *
W in
equation (8)).
T a b l e 2 . Results of the variance analysis with the model (8)
Source of variability Source of variability Sum of the squares
of deviations
The main factor Н fk = b - 1 fff kWW
Factor sF sk = m - 1 sss kWW
Factor tF tk = m - 1 ttt kWW
Factor uF uk = m - 1 uuu kWW
Interaction sHF )1()1( mbk fs fsfsfs kWW
Interaction tHF )1()1( mbk ft ftftft kWW
Interaction uHF )1()1( mbk fu fufufu kWW
Remainder )23(( mbNk kWW *
General k = N-1 kWW
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Системні дослідження та інформаційні технології, 2024, № 2 143
We will conduct a study of the simplified model (9). This model, occupies
an intermediate position between model (1) and models (3), (4) and (5). From
(10) it follows that the residual sum of the simplified model (11) is less than the
residual sums of the models (3), (4) and (5). This indicates the increased accuracy
of this model compared to the cross-classification models (3), (4), and (5). Based
on the results of Table. 2, we conclude that the number of degrees of freedom of
the residual sum *
W decreases with an increase in the number of b groups (by the
level of parameter H) and the number of m subgroups (by the level of additional
factors uts FFF ,, ).
Let’s write k from Table 2 in the following form:
m
b
mbNk 3 . (12)
From the resulting expression (12) it follows that the number of degrees of
freedom is the greater, at constmb , the greater the ratio mb .
This finding makes it possible to further plot the number of groups and sub-
groups in the tables of the original data of the models (3), (4), and (5). In this
case, the number of b groups (subranges of measurement of the control parameter
H (homogeneity) is desirable to increase, and the number of subgroups should be
reduced, reducing m to a minimum. For example, take m = 2. This will increase
the number of degrees of freedom of the residual sum *
W . Thus, it is obvious
that the main advantage of the simplified model is the possibility of simultaneous
testing of the Н0, hypothesis, that is, when the influence of factors Н, uts FFF ,, ,
on the information measure of control F is absent. In this way you can write Н0:
0... bp ff .
This is the main hypothesis and its components have the following form:
0...:0 mp
s ssH ; 0...:0 bmp
sf fsfsH ;
0...:0 mp
t ttH ; 0...:0 bmp
tf ftftH ;
0...:0 mp
u uuH ; 0...:0 bmp
uf fufuH .
The test of the hypotheses presented is done in relation to the corresponding
mean squares )...,,,,,( fuutsf WWWWW to the mean residual square W followed
by comparison of the obtained F — statistics with the corresponding percentage
points for F — distributions.
This advantage of the simplified model (9) makes it possible to estimate the
amount of expected information about the levels of parameter H for the informa-
tion measure F when considering the levels of both influencing factors and their
interactions:
2
1log
F
FI ,
where fF Wσ 2 ;
I.V. Hryhorenko, S.I. Kondrashov, S.M. Hryhorenko, O.S. Opryshkin
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 144
2
1
2 )(
)1(
1
FF
n
W
n
i
ifF
,
where
n
i
iF
n
F
1
1
; 2
Fσ — is a function of the sum of the squares of deviations
)...,,,,,( fuutsf WWWWW in Table 1 and Table. 2.
Table 3 presents the equation for calculating 2
F for the simplified model (9)
with different combinations of factors affecting the information indicator F.
T a b l e 3 . Calculated Ratios for Parameter 2
F
Factorial influences
taken into account
2
F
Additional factor sF )()( *
kkkkkkWWWWWW fuftfsutfuftfsut
Additional factor tF )()( *
kkkkkkWWWWWW fuftfsusfuftfsus
Additional factor uF )()( *
kkkkkkWWWWWW fuftfstsfuftfsts
sF , tF )()( *
kkkkkWWWWW fuftfsufuftfsu
sF , uF )()( *
kkkkkWWWWW fuftfstfuftfst
tF , uF )()( *
kkkkkWWWWW fuftfssfuftfss
sF , tF , uF )()( *
kkkkWWWW fuftfsfuftfs
sF , tF , uF , sHF )()( *
kkkWWW fuftfuft
sF , tF , uF , tHF )()( *
kkkWWW fufsfufs
sF , tF , uF , uHF , )()( *
kkkWWW ftfsftfs
sF , tF ,
uF , sHF , tHF , uHF W
If we do not take into account all the influencing factors, and this corre-
sponds to obtaining information under multifactorial influence, then
2
*
2
bN
WWWWWWW fuftfsuts
F .
In order to perform the verification of statistical findings, we present model
(9) as a one-factor model of one-sided classification, i.e., when the influence
of additional factors is determined solely by the magnitude of the random
residue αi :
,αiii fFF , (13)
where bα ,1 ; ni ,1 ; bNn .
In this case, the magnitude of n multifactorial observations in each of the b
groups is the same.
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Model (13), due to the uncertainty of the levels of the control parameter H,
refers to the variance component models.
We will use the formal analysis of the model (13) and write the expression
of the full sum of the squares of deviations W through the sum of the two terms
b
i
f FFnW
1
2)( ,
where F — group mean values of the performance trait; F — is the overall
average; n — is the number of units of the population in each group.
Residual variance (random) is the sum of the group sums of the squares of
deviations of all variant of the resultant trait in groups from the mean values of
the trait in them:
g
i
i
b
FFW
1
2
1
)( .
Consider now the ratio of the mean squares for the recorded sums fW and W :
)(/
)1(/
bNW
bW
F f
.
F-statistics can be used to test one of two hypotheses:
][...][: 10 bFMFMH ][...][: 11 bFMFMH .
A rule follows from the theory: if statistical conclusions indicate the validity
of the main hypothesis 0H , then the parameter Н does not affect the change in the
control indicator F. That is, the indicator F does not carry information about the
change in the levels of the control parameter Н, and if the hypothesis 1H , is valid,
then the indicator F is informative in relation to the control parameter Н. Deci-
sions 0μ (valid hypothesis 0H ) and 1 (valid hypothesis 1H ) are accepted, com-
paring the F statistic with the critical value FK.
The conditional densities of the probability distribution of F-statistics are a
linearly transformed random variable with central bNbF ,1 , the distribution:
bNbFHFf ,10 )( ,
bNb
f FgHFf
,12
2
1 1)( , (14)
where the variances 2
f and 2 refer, respectively, to the random deviations of
f і i in the model (13).
The variance of 2
f can be represented as the sum:
222
rHf ,
where 2
H — the variance of the control parameter H, which is due to the uncer-
tainty of its values in the measurement range HD ; 2
r — the variance of the
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ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 146
measurement result of the values of the control parameter H, which is due to the
uncertainty of the reproduction of the values of H by the technical means of con-
trol (in fact, this is the variance of the measurement result).
Testing hypotheses about the significance of the effects of factors and their
interactions is carried out using the Fisher criterion. To do this, calculate the ratio
of the corresponding mean squares to the remaining middle square. The obtained
values are compared with the KF values found in the F-distribution tables for the
accepted significance level 05,0 , or higher (0,01 – 0,001) and the number of
degrees of freedom.
Hence, if the control measure F is sensitive to a change in the level of the
control parameter H over the entire range of its measurements, then the F-statistic
will be characterized by the distribution density (14) for all measured levels of the
indicator H.
Now consider the problem of one-sided testing of the parameter H within the
implementation of the alternative hypothesis 1H . This hypothesis must be repre-
sented as a complex one, which will check the correspondence between the real
value f of the control parameter H and the control value 0f :
0
1H : 0ff (parameter H is normal);
1
1H : 0ff (parameter H is not normal).
Such situations must correspond to the choice of one of two solutions:
0 : KFF ,
where F — calculated value of the F-criterion; KF — tabular value of the
F-criterion. Then the influence of the factor on the control parameter is not prov-
en, but the absence of influence of the factor is not proven.
If:
1 : KFF ,
then statistical observation proves with a given probability the influence of the
factor on the control parameter.
Enter the following designations: 1 and 2 — probabilities of errors of
the first and second kind:
)( 011 ffP ; )( 002 ffP .
Then the probabilities of choosing solutions 0 , 1 are respectively deter-
mined by the expressions:
21][ KFFP , (15)
1][ KFFP . (16)
The fragment F has distribution (14), then from the expressions (15) and
(16) it flows
,])1([
;1])1([
1
12
0,1
2
12
1,1
nFFP
nFFP
KbNb
KbNb (17)
Study of the factor influence on the uniformity of coffee grain grinding by methods …
Системні дослідження та інформаційні технології, 2024, № 2 147
where
.
;
2
2
2
0
2
22
2
1
r
rH
(18)
Considering the system of equations (17), one can find the value of KF that
satisfies both of these expressions. To do this, it is necessary to ensure that the
condition is fulfilled
2
0
2
1
,,1
1,,1
1
1
2
1
n
n
F
F
bNb
bNb . (19)
The numerator and denominator of the left side of the inequality (19) are
)1( 1 and 2 — are the percentage points of the central F — the distribution
with )1( b and )( bN degrees of freedom.
It can be shown now that the expression that (19) corresponds to the expression:
2
0
2
1
2
,1
2
1,1
2
1
b
b
χ
χ
, (20)
where 2
1,1 1b , 2
,1 2b — percentage points of the central 2 distribution with
)1( b degrees of freedom.
From formula (20) it follows that given number of levels b of control pa-
rameter H (groups of results of observations of parameter F of model (13)) and
given ratio 2
0
2
1 it is possible to estimate the reliability of decision-making
0 and 1
2
1A 21
,
fixed, for example, the value of 1 (Naiman – Pearson criterion) and calculate
2 , as the interval:
2
z
0
1
2
1
2
12
min
2
1
2
1
e
b
b
b
,
where
2
1,1min 1bz ;
2
1b
— gamma function.
The relation is determined from the formula (18):
2
2
1
r .
The results obtained already make it possible to move on to the practical use
of the simplified model. At the same time, it must be remembered that in the prac-
tice of variance analysis, there may be cases when the number of replicates ob-
tained for each combination of levels of the factors under study is different and
I.V. Hryhorenko, S.I. Kondrashov, S.M. Hryhorenko, O.S. Opryshkin
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 148
equal ijm ; ijm numbers can have any value, including zero, but each row and
column must have at least one, and some have two non-zero values. In this case,
there are possible situations in which it is impossible to obtain unbiased estimates
for all parameters of the model. The need to introduce a system of weight coeffi-
cients into the calculated ratios complicates the calculations, the analysis of such
plans involves, as a rule, the use of a computer.
SUMMARIES
When executing the partition, the following results are obtained:
– to assess the homogeneity of coffee grinding, a mathematical model of the
influence of four factors on the result of measuring the control indicator has been
developed;
– a simplified model of cross-classifications was proposed for further use
and investigated, which took into account the effects of simultaneous interaction
of four factors (grinding time, geometric grain size, grain moisture, speed of rota-
tion of the motor shaft) on the result of measurement of the unit control indicator
(uniformity of coffee grinding);
– obtained equations that allow to evaluate the reliability of statistical con-
clusions in relation to the informational significance of control indicators for the
proposed simplified model of cross-classification;
– analytical ratios are obtained, which allow to estimate the amount of in-
formation on each of the control indicators under the factor influence on the pro-
posed linear function of the transformation of these indicators.
– the advantages of the chosen approach are that it makes it possible to as-
sess the validity of multiparameter control results of three or more levels of the
control indicator and to select the most informative subsets of these values.
REFERENCES
1. V.P. Misyats, M.M. Rubanka, and S.A. Demishonkova, “System of adaptive control
of the drive of automatic coffee machines,” Bulletin of the Khmelnytskyi National
University, no. 1, pp. 151–159, 2021 (293). doi: 10.31891/2307-5732-2021-293-1-151-159.
2. E. Uman et al., “The effect of bean origin and temperature on grinding roasted cof-
fee,” Sci. Rep., vol. 6, 24483, 2016. doi: https://doi.org/10.1038/srep24483.
3. Nancy Cordoba, Laura Pataquiva, Coralia Osorio, Fabian Leonardo Moreno Moreno,
and Ruth Yolanda Ruiz, “Effect of grinding, extraction time and type of coffee on
the physicochemical and flavor characteristics of cold brew coffee,” Sci. Rep., 9,
8440, 2019. doi: https://doi.org/10.1038/s41598-019-44886-w.
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sign on Coffee Extraction Factors Impacting the Measurable Percent of Total Dis-
solved Solids in Solution,” Asia-Pacific Journal of Management Research and Inno-
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A. Fagundes, “Estimation of percentage of impurities in coffee using a computer
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1929/agriambi.v26n2p142-148.
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2022. doi: https://doi.org/10.1007/s00217-021-03868-x.
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7. Ihor Hryhorenko, Elena Tverytnykova, Svitlana Hryhorenko, and Viktoria Krylova,
“Temperature sensor research as a part of a microprocessor system by statistical
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2022, Kharkiv, Ukraine, pp. 102–107. doi: 10.1109/ХПИНеделя57572.2022.9916478.
8. І. Hryhorenko, S. Kondrashov, and O. Opryshkin, “Formation of test impacts for the
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pp. 19–26, 2023. doi: https://doi.org/10.20998/2413-4295.2023.01.03.
Received 20.07.2023
INFORMATION ON THE ARTICLE
Ihor V. Hryhorenko, ORCID: 0000-0002-4905-3053, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: grigmaestro@gmail.com
Serhii I. Kondrashov, ORCID: 0000-0002-5191-8562, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: Serhii.Kondrashov@khpi.edu.ua
Svitlana M. Hryhorenko, ORCID: 0000-0003-0150-4844, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: sngloba@gmail.com
Oleksandr S. Opryshkin, ORCID: 0009-0008-7094-5129, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: Aleksandr.Opryshkin@cit.khpi.edu.ua
ДОСЛІДЖЕННЯ ФАКТОРНОГО ВПЛИВУ НА ОДНОРІДНІСТЬ ПОМЕЛУ
ЗЕРНА КАВИ МЕТОДАМИ СТАТИСТИЧНОГО АНАЛІЗУ / І.В. Григоренко,
С.І. Кондрашов, С.М. Григоренко, О.С. Опришкін
Анотація. Для оцінювання впливу кожного із факторів, які впливають на
якість та однорідність помелу зерна кави, і порівняння впливу цих факторів
варто встановити кількісний показник цього впливу. Для цього використано
дисперсійний аналіз як метод організації вибіркових даних відповідно до мож-
ливих джерел розсіювання. Обраний метод дозволив розкласти загальне роз-
сіювання на складові, зумовлені впливом рівнів факторів. Факторами, що
впливають на однорідність помелу, обрано: час помелу, геометричні розміри
зерна, вологість зерна, швидкість обертання валу двигуна. Проведено обґрун-
тування та оцінювання достовірності статистичних висновків про інформацій-
ну значущість показників, що впливають на однорідність помелу кави для за-
безпечення максимально високої вірогідності отриманого результату.
Ключові слова: дисперсійний аналіз, однорідність помелу, факторний вплив,
модель, показник контролю, зерно кави.
|
| id | journaliasakpiua-article-285190 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:15Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/22/37a62fb8a51a135cbd4a16895f76c022.pdf |
| spelling | journaliasakpiua-article-2851902024-08-11T01:12:49Z Study of the factor influence on the uniformity of coffee grain grinding by methods of statistical analysis Дослідження факторного впливу на однорідність помелу зерна кави методами статистичного аналізу Hryhorenko, Ihor Kondrashov, Serhii Hryhorenko, Svitlana Opryshkin, Oleksandr dispersion analysis homogeneity of grinding factor influence model indicator of control coffee bean дисперсійний аналіз однорідність помелу факторний вплив модель показник контролю зерно кави In order to assess the impact of each of the factors that affect the quality and uniformity of grinding coffee beans and to compare the impact of these factors, it is worth establishing a quantitative indicator of this impact. To solve this problem, dispersion analysis was used as a method of organizing sample data according to possible sources of dispersion. The chosen method made it possible to decompose the total dispersion into components caused by the influence of factor levels. Grinding time, geometric dimensions of the grain, moisture content of the grain, speed of rotation of the motor shaft were selected as factors influencing the homogeneity of grinding. The justification and assessment of the reliability of statistical conclusions about the informational significance of indicators affecting the homogeneity of coffee grinding was carried out to ensure the highest possible probability of the obtained result. Для оцінювання впливу кожного із факторів, які впливають на якість та однорідність помелу зерна кави, і порівняння впливу цих факторів варто встановити кількісний показник цього впливу. Для цього використано дисперсійний аналіз як метод організації вибіркових даних відповідно до можливих джерел розсіювання. Обраний метод дозволив розкласти загальне розсіювання на складові, зумовлені впливом рівнів факторів. Факторами, що впливають на однорідність помелу, обрано: час помелу, геометричні розміри зерна, вологість зерна, швидкість обертання валу двигуна. Проведено обґрунтування та оцінювання достовірності статистичних висновків про інформаційну значущість показників, що впливають на однорідність помелу кави для забезпечення максимально високої вірогідності отриманого результату. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-06-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/285190 10.20535/SRIT.2308-8893.2024.2.10 System research and information technologies; No. 2 (2024); 137-149 Системные исследования и информационные технологии; № 2 (2024); 137-149 Системні дослідження та інформаційні технології; № 2 (2024); 137-149 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/285190/301166 |
| spellingShingle | дисперсійний аналіз однорідність помелу факторний вплив модель показник контролю зерно кави Hryhorenko, Ihor Kondrashov, Serhii Hryhorenko, Svitlana Opryshkin, Oleksandr Дослідження факторного впливу на однорідність помелу зерна кави методами статистичного аналізу |
| title | Дослідження факторного впливу на однорідність помелу зерна кави методами статистичного аналізу |
| title_alt | Study of the factor influence on the uniformity of coffee grain grinding by methods of statistical analysis |
| title_full | Дослідження факторного впливу на однорідність помелу зерна кави методами статистичного аналізу |
| title_fullStr | Дослідження факторного впливу на однорідність помелу зерна кави методами статистичного аналізу |
| title_full_unstemmed | Дослідження факторного впливу на однорідність помелу зерна кави методами статистичного аналізу |
| title_short | Дослідження факторного впливу на однорідність помелу зерна кави методами статистичного аналізу |
| title_sort | дослідження факторного впливу на однорідність помелу зерна кави методами статистичного аналізу |
| topic | дисперсійний аналіз однорідність помелу факторний вплив модель показник контролю зерно кави |
| topic_facet | dispersion analysis homogeneity of grinding factor influence model indicator of control coffee bean дисперсійний аналіз однорідність помелу факторний вплив модель показник контролю зерно кави |
| url | https://journal.iasa.kpi.ua/article/view/285190 |
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