Fuzzy approach to forecasting the dynamics ofvegetation indices
Modern satellite monitoring technologies provide agricultural producers with valuable information on the health status of crops. The ability of remote sensors to detect slight variations in vegetation makes them a useful tool for quantitatively assessing variability within a given field, evaluating...
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Інститут кібернетики ім. В.М. Глушкова НАН України
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| Cite this: | Fuzzy approach to forecasting the dynamics ofvegetation indices / E.R. Aliyev, F.M. Salmanov // Проблеми керування та інформатики. — 2022. — № 2. — С. 53-68. — Бібліогр.: 8 назв. — англ. |
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| citation_txt | Fuzzy approach to forecasting the dynamics ofvegetation indices / E.R. Aliyev, F.M. Salmanov // Проблеми керування та інформатики. — 2022. — № 2. — С. 53-68. — Бібліогр.: 8 назв. — англ. |
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| description | Modern satellite monitoring technologies provide agricultural producers with valuable information on the health status of crops. The ability of remote sensors to detect slight variations in vegetation makes them a useful tool for quantitatively assessing variability within a given field, evaluating crop growth, and managing land based on current conditions. Remote sensing data, collected on a regular basis, allow producers and agronomists to create up-to-date vegetation maps reflecting the health and vigor of crops, analyze changes in plant condition over time, and forecast yield for a specific planting area.
Сучасні технології супутникового моніторингу поверхні Землі надають сільськогосподарським виробникам корисну інформацію — стан здоровʼя посівних культур. Здатність віддаленого датчика виявляти незначні відмінності у рослинності робить його корисним інструментом для кількісної оцінки мінливості в межах заданого поля, оцінки зростання сільськогосподарських культур та управління угіддями на основі поточних умов. Дані дистанційного зондування, що збираються на регулярній основі, дозволяють виробникам та агрономам складати поточну карту вегетації, що відображає стан та силу посівних культур, аналізувати динаміку зміни у стані рослин, а також прогнозувати врожайність на конкретній посівній площі.
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© E. ALIYEV, F. SALMANOV, 2022
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 53
МЕТОДИ КЕРУВАННЯ ТА ОЦІНЮВАННЯ
В УМОВАХ НЕВИЗНАЧЕНОСТІ
УДК 519.712.3
E. Aliyev, F. Salmanov
FUZZY APPROACH TO FORECASTING
THE DYNAMICS OF VEGETATION INDICES
Elchin Aliyev
Institute of Control Systems of Azerbaijan National Academy of Sciences,
elchin.aliyev@sinam.net
Fuad Salmanov
Institute of Control Systems of Azerbaijan National Academy of Sciences,
fuad.salmanli@sinam.net
Modern technologies for satellite monitoring of the Earthʼs surface provide agri-
cultural producers with useful information about the health status of crops. The
remote sensorʼs ability to detect subtle differences in vegetation makes it a use-
ful tool for quantifying variability within a given field, estimating crop growth,
and managing land based on current conditions. Remote sensing data, collected
on a regular basis, allows producers and agronomists to draw up a current vege-
tation map that reflects the condition and strength of crops, analyze the dynam-
ics of changes in plant condition, and predict yields in a particular area under
crops. To interpret these data, the most effective means are various vegetation
indices calculated empirically, that is, by operations with different spectral rang-
es of satellite monitoring multispectral data. Based on the time series of one of
these vegetation indices, the paper considers the annual dynamics of the develop-
ment of a plant culture in a particular field. The possibility of predicting the yield of
the given crop is considered based on fuzzy modeling of time series for the corre-
sponding spectral ranges of vegetation reflection obtained from satellite monitoring
images. The proposed fuzzy models of time series are investigated for adequacy and
suitability in terms of analyzing the features of the intra-annual of average long-term
dynamics of the vegetation index, typical for the given area under crop.
Keywords: crop, multispectral reflection of plants, vegetation index, fuzzy set, fuzzy
time series.
Introduction
Most agricultural crops are characterized by changes in the phases of development,
which is reflected in the dynamics of the spectral-reflective properties of plants. The
study of seasonal and long–term changes in the spectral-brightness characteristics of
crops is possible through the analysis and modeling of the dynamic series of vegetation
indices, which makes it possible to quantify the features of the vegetation cover and the
regularity of its temporal dynamics. At the same time, standard algorithms for solving
problems of predicting the dynamics of the spectral–reflective properties of plants work,
as a rule, with «crisp» or structured data from satellite sensing of the Earth, that is, with
data presented in the form of averaged numbers. Therefore, averaging the results of
measurements of spectral ranges for calculating vegetation indices is one of the most
common empirical operations in data collection systems for accurate agriculture. In par-
mailto:elchin.aliyev@sinam.net
54 ISSN 1028-0979
ticular, the achievement of the required accuracy in the process of averaging the values
of vegetation indices is achieved by multiple measurements, where the results of indi-
vidual measurements are partially compensated for by positive and negative deviations
from the exact value. At the same time, the accuracy of their mutual compensation im-
proves with an increase in the number of measurements, since the average value of neg-
ative deviations in modulus verge towards the average value of positive deviations.
Nevertheless, multispectral satellite monitoring data, for example, the values of spectral
ranges should be considered as weakly structured, that is, those that are known to be-
long to a certain interval [1]. For example, the region of maximum reflection of plant
cell structures is in the wavelength range from 750 nm to 900 nm, which is the near in-
frared region of the electromagnetic spectrum.
More adequate reflections of weakly structured spectral ranges can be evaluative
concepts of the type «HIGH», «LOW», etc., which can be formally described by the
corresponding fuzzy sets as the terms (values) of the linguistic variable «spectral reflec-
tivity of plants» [2, 3]. Based on this premise, it becomes obvious the importance and
relevance of studying methods for predicting seasonal and long–term changes in agri-
cultural crops in spectral–brightness characteristics using fuzzy time series (FTS) of sat-
ellite monitoring indicators relative to spectral ranges.
Problem definition
Existing approaches to the calculation of vegetation indices are usually based on
two independent parts of the electromagnetic spectrum of vegetation reflectivity [4]: on
the reflection in the red region of the spectrum in the range from 620 nm to 750 nm,
which accounts for the maximum absorption of solar radiation by chlorophyll of higher
vascular plants, and on the reflection in the near infrared region of the spectrum in the
range from 750 nm to 900 nm, where the region of maximum reflection of the cellular
structures of the leaf is concentrated. One of the wide-spread indicators for solving
problems regarding the assessment of vegetation cover is the NDVI (Normalized Dif-
ference Vegetation Index), which is calculated by the formula
NIR RED ,NDVI =
NIR RED
−
+
(1)
where NIR and RED are the reflection coefficients in the near infrared and red regions
of the electromagnetic spectrum, respectively. Both coefficients are calculated by map-
ping the red and infrared regions of the spectrum onto a unit segment using trivial
equalities:
1
1
620
RED , (620, 750),
750 620
−
=
−
2
2
750
NIR , (750, 900).
900 750
−
=
−
The NDVI value varies from 0 to 1: the higher its value is the higher the vegetation
intensity, and vice versa, the lower the index value, the sparser is the vegetation, and the
tendency to zero generally indicates open ground. So, it is necessary to adapt a fuzzy
method for forecasting seasonal and long-term changes in agricultural crops in the spec-
tral–brightness characteristics using FTS of spectral ranges of remote sensing data and re-
flecting certain vegetation parameters in a particular pixel of a satellite image. As an ex-
ample of testing fuzzy models, time series were selected that reflect the annual dynamics
of the coefficients of the spectral ranges RED and NIR (see Table 1 and Fig. 1), obtained
from images of a fixed pixel in the corresponding MODIS images (LPDAAC — Land
Processes Distributed Active Archive Center) (see Fig. 2) of crop area in Jonesboro (USA,
Arkansas) with geographic coordinates (– 90,1614583252562, 35.8135416634583) [4].
Table 1 also shows the corresponding NDVI calculated using formula (1).
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 55
Table 1
N Date NIR RED NDVI N Date NIR RED NDVI
1 18.02.2000 0,2036 0,0958 0,3599 16 25.06.2000 0,8565 0,1452 0,7101
2 26.02.2000 0,3175 0,1445 0,3745 17 02.07.2000 0,8651 0,1453 0,7124
3 05.03.2000 0,3639 0,1523 0,4099 18 11.07.2000 0,8702 0,1455 0,7135
4 15.03.2000 0,3623 0,1623 0,3812 19 20.07.2000 0,3357 0,1256 0,4554
5 21.03.2000 0,2219 0,1025 0,3680 20 27.07.2000 0,1125 0,0678 0,2479
6 29.03.2000 0,1717 0,0835 0,3457 21 02.08.2000 0,3666 0,1348 0,4623
7 06.04.2000 0,1676 0,0845 0,3296 22 12.08.2000 0,6051 0,1245 0,6587
8 15.04.2000 0,1407 0,0765 0,2957 23 20.08.2000 0,5828 0,1463 0,5987
9 22.04.2000 0,1106 0,0659 0,2535 24 28.08.2000 0,4628 0,1354 0,5473
10 29.04.2000 0,1214 0,0689 0,2759 25 03.09.2000 0,3492 0,1158 0,5019
11 08.05.2000 0,1502 0,0815 0,2966 26 13.09.2000 0,3523 0,1233 0,4815
12 15.05.2000 0,1529 0,0813 0,3058 27 20.09.2000 0,3450 0,1389 0,4259
13 24.05.2000 0,1664 0,0855 0,3211 28 29.09.2000 0,2457 0,1125 0,3719
14 09.06.2000 0,4084 0,1324 0,5104 29 07.10.2000 0,2173 0,1045 0,3505
15 15.06.2000 0,5890 0,1356 0,6257 30 15.10.2000 0,2058 0,1056 0,3217
Time series of the NDVI index against the background of the dynamics of the RED and
NIR coefficients obtained for a fixed pixel of MODIS images (LPDAAC) are in Fig. 1.
Fig. 1
FTS: Main Stages of Predictive Modeling
The existing approaches to fuzzy modeling of time series involve the sequential
implementation of the following main stages (procedures): 1) establishing the coverage
of the entire set of historical data in the form of a universal set (universe); 2) fuzzifica-
tion of weakly structured historical data of time series; 3) establishing internal relation-
ships in the form of fuzzy relations and dividing them into groups; 4) finding fuzzy out-
puts (predicts) of the applied model and their defuzzification.
One of the ways to establish the universe and calculate the optimal number of
evaluative concepts for fuzzy evaluation of the historical data of the time series was
proposed in [6], the essence of which is to perform sequentially the following steps.
Step 1. Assorting the time series data 1{ }n
t tx = into an ascending sequence ( ){ },p ix
where p is a permutation that sorts the data values in ascending order: ( ) ( 1) .p i p ix x +
Step 2. Calculation of the average value for all pairwise distances id =
( ) ( 1)p i p ix x += − between any two consecutive values ( )p ix and ( 1)p ix + and standard
deviation by formulas:
1
1 2 ( ) ( 1)
1
1
( , ,..., ) ,
1
n
n p i p i
i
AD d d d x x
n
−
+
=
= −
−
(2)
15.10
27.07
11.07
25.06
09.06
18.02
1,0
0,8
0,6
0,4
0,2
0,0
1
8
.0
2
.2
0
0
0
0
8
.0
5
.2
0
0
0
0
5
.0
3
.2
0
0
0
2
4
.0
5
.2
0
0
0
2
1
.0
3
.2
0
0
0
1
5
.0
6
.2
0
0
0
0
6
.0
4
.2
0
0
0
0
2
.0
7
.2
0
0
0
2
2
.0
4
.2
0
0
0
2
0
.0
7
.2
0
0
0
0
2
.0
8
.2
0
0
0
2
0
.0
8
.2
0
0
0
0
3
.0
9
.2
0
0
0
2
0
.0
9
.2
0
0
0
0
7
.1
0
.2
0
0
0
1
5
.1
0
.2
0
0
0
NDVI NIR RED
56 ISSN 1028-0979
1
2
1
1
( ) .
1
n
AD i
i
d AD
n
−
=
= −
−
(3)
Step 3. Detection and elimination of anomalies — outliers that need to be reset.
For this, both the mean distance AD and the standard deviation ,AD established in
the previous step, are used. In this case, the values of pairwise distances that do not sat-
isfy the following condition are subject to outlier
.AD i ADAD d AD− + (4)
Step 4. Recalculation of the mean distance AD for the set of remaining values .id
Step 5. Establishing the universe U in the form min max[ , ]U D AD D AD= − + =
1 2[ , ],D D= where minD and maxD are the minimum and maximum values, respective-
ly, on the entire data set.
Step 6. Finding the optimal number of evaluative concepts as criteria for evaluat-
ing the historical data of the time series. It is carried out based on the formula
2 1 .
2
D D AD
m
AD
− −
=
(5)
Considering the above step-by-step data fuzzification procedure FTS models of
both the NDVI and the corresponding reflection coefficients NIR and RED are pro-
posed below.
Forecasting the fuzzy time series of the NDVI: method №1
There are various ways to describe the qualitative criteria for assessing the values
of historical data using fuzzy sets. One of them is quite trivial, initially implying a set of
qualitative criteria, for example, of the form:
too low (value): 1
1 2 3 4 5 6 7 8
1 0,5 0 0 0 0 0 0
;A
u u u u u u u u
= + + + + + + +
very low: 2
1 2 3 4 5 6 7 8
0,5 1 0,5 0 0 0 0 0
;A
u u u u u u u u
= + + + + + + +
more than low: 3
1 2 3 4 5 6 7 8
0 0,5 1 0,5 0 0 0 0
;A
u u u u u u u u
= + + + + + + +
low: 4
1 2 3 4 5 6 7 8
0 0 0,5 1 0,5 0 0 0
;A
u u u u u u u u
= + + + + + + +
high: 5
1 2 3 4 5 6 7 8
0 0 0 0,5 1 0,5 0 0
;A
u u u u u u u u
= + + + + + + +
more than high: 6
1 2 3 4 5 6 7 8
0 0 0 0 0,5 1 0,5 0
;A
u u u u u u u u
= + + + + + + +
very high: 7
1 2 3 4 5 6 7 8
0 0 0 0 0 0,5 1 0,5
;A
u u u u u u u u
= + + + + + + +
too high: 8
1 2 3 4 5 6 7 8
0 0 0 0 0 0 0,5 1
,A
u u u u u u u u
= + + + + + + +
where each historical data of the time series is interpreted considering the belonging of
the interval of its localization ( 1 8)ju j = to one or another fuzzy set jA with a rather
trivial membership function.
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 57
In [5], to describe the qualitative evaluation criteria by appropriate fuzzy sets, the
following trapezoidal membership functions (TMF) are used
1
1
1 2
2 1
2 3
4
3 4
4 3
4
0,
, ,
( ) 1, ,
, ,
0,
k
k
k
k k
k k
A k k
k
k k
k k
k
x a
x a
a x a
a a
x a x a
a x
a x a
a a
x a
−
−
=
−
−
(6)
whose parameters satisfy the conditions:
2 1 3 2 4 3k k k k k ka a a a a a− = − = − ( 1 ).k m=
So, on the entire data set of the NDVI time series (see Table 1), using formulas (2)
and (3) the mean value 0,0161AD = and the standard deviation 0,0140AD = are es-
tablished, respectively. After resetting the pairwise distances id that do not satisfy con-
dition (4) or, more specifically, the condition 0,0161 0,0140 0,0161 0,0140,id− +
based on the remaining set of pairwise distances the final value of the mean value was
obtained as 0,0132.AD = In this case, the desired universe is constructed as a segment
[0, 2479 0,0132; 0, 2479 0,0132] [0, 2347; 0,7267],U = − + = where 0,2479 and 0,7135
are the minimum and maximum values of the NDVI index, respectively. At the same
time, the number of fuzzy subsets of this universe, describing the qualitative criteria for
evaluating NDVI indices, is calculated by equality (5) as follows:
0,7267 0,2347 0,0132
18,1424 18.
2 0,0132
m
− −
= =
Based on the use of the TMF (6) with parameters
1ka summarized in Table 2,
where ( 1 4)i = and ( 1 18),k = the corresponding fuzzy sets Ak are established (Fig. 2).
Table 2
Fuzzy
set
Parameters of the TMF Fuzzy
set
Parameters of the TMF
1ka
2ka
3ka
4ka
1ka
2ka
3ka
4ka
A1 0,2347 0,2479 0,2611 0,2743 A10 0,4722 0,4854 0,4986 0,5118
A2 0,2611 0,2743 0,2875 0,3007 A11 0,4986 0,5118 0,5250 0,5382
A3 0,2875 0,3007 0,3139 0,3271 A12 0,5250 0,5382 0,5514 0,5646
A4 0,3139 0,3271 0,3403 0,3535 A13 0,5514 0,5646 0,5778 0,5910
A5 0,3403 0,3535 0,3667 0,3799 A14 0,5778 0,5910 0,6042 0,6174
A6 0,3667 0,3799 0,3931 0,4062 A15 0,6042 0,6174 0,6306 0,6438
A7 0,3931 0,4062 0,4194 0,4326 A16 0,6306 0,6438 0,6570 0,6702
A8 0,4194 0,4326 0,4458 0,4590 A17 0,6570 0,6702 0,6834 0,6965
A9 0,4458 0,4590 0,4722 0,4854 A18 0,6834 0,6965 0,7097 0,7267
Fuzzification of NDVI indices by the presented trapezoidal membership functions
is carried out according to the principle: NDVI is described by the fuzzy set to which its
value belongs with the highest degree. When the NDVI value belongs to the interval
2 3
[ , ],k ka a the appropriate fuzzy analog is easily determined. In other cases, clarifica-
tions are needed. According to (6) for NDVI 0,5019= we have:
11
(0,5019)A =
0,2491= and
10
(0,5019) 0,7509A = (see Fig. 3). Then the fuzzy set 10A is the ana-
58 ISSN 1028-0979
log of NDVI, since the value of the corresponding membership function at the
point 0,5019 is greater. The fuzzy analogs of all NDVIs are summarized in Table 3.
Fig. 2
Fig. 3
Table 3
N Date NDVI Fuzzy set N Date NDVI Fuzzy set
1 18.02.2000 0,3599 A5 16 25.06.2000 0,7101 A18
2 26.02.2000 0,3745 A6 17 02.07.2000 0,7124 A18
3 05.03.2000 0,4099 A7 18 11.07.2000 0,7135 A18
4 15.03.2000 0,3812 A6 19 20.07.2000 0,4554 A9
5 21.03.2000 0,3680 A5 20 27.07.2000 0,2479 A1
6 29.03.2000 0,3457 A4 21 02.08.2000 0,4623 A9
7 06.04.2000 0,3296 A4 22 12.08.2000 0,6587 A16
8 15.04.2000 0,2957 A3 23 20.08.2000 0,5987 A14
9 22.04.2000 0,2535 A1 24 28.08.2000 0,5473 A12
10 29.04.2000 0,2759 A2 25 03.09.2000 0,5019 A10
11 08.05.2000 0,2966 A3 26 13.09.2000 0,4815 A10
12 15.05.2000 0,3058 A7 27 20.09.2000 0,4259 A7
13 24.05.2000 0,3211 A4 28 29.09.2000 0,3719 A5
14 09.06.2000 0,5104 A11 29 07.10.2000 0,3505 A5
15 15.06.2000 0,6257 A15 30 15.10.2000 0,3217 A4
As is known, FTS modeling is based on the analysis of internal cause and effect rela-
tions, which are presented in the form of implication «If <…>, then <…>». Identified inter-
nal relations are grouped according to the principle: if the fuzzy set kA or the bunch of
1,0
0,8
0,6
0,4
0,2
0,0
0,4722 0,4854 0,4986 0,5118 0,5250 0,5382
0,7590
0,2941
0,5019
A10
A11
M
em
b
er
s
h
ip
f
u
n
ct
io
n
v
al
u
es
1,0
0,8
0,6
0,4
0,2
0,0
0
,2
3
4
7
0
,3
0
0
7
0
,2
4
7
9
0
,3
1
3
9
0
,2
6
1
1
0
,3
2
7
1
0
,2
7
4
3
0
,3
4
0
3
0
,2
8
7
5
0
,3
5
3
5
0
,3
6
6
7
0
,3
7
9
9
0
,3
9
9
1
0
,4
0
6
2
0
,4
1
9
4
0
,4
3
2
6
Indeces NDVI
0
,4
5
9
0
0
,4
7
2
2
0
,4
8
5
4
0
,4
9
8
6
0
,5
1
1
8
0
,5
2
5
0
0
,5
3
8
2
0
,5
5
1
4
0
,5
6
4
6
0
,5
7
7
8
0
,5
9
1
0
0
,6
0
4
2
0
,6
1
7
4
0
,6
3
0
6
0
,6
4
3
8
0
,6
5
7
0
0
,6
7
0
2
0
,6
8
3
4
0
,4
4
5
8
0
,6
9
6
5
0
,7
0
9
7
0
,7
2
6
7
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 59
sets 1,i iA A + relates to one or several sets, then a group of the 1st order is localized relative
to kA or the group of the 2nd order is localized relative to 1,i iA A + (see Table 4 and Table 5).
Table 4
A1
A2
A3
A1
A4
A3
A5
A4
A6
A5
A7
A5
A10
A10
A12
A10
A16A14
A1
A9
A3
A7
A4
A11
A5
A5
A7
A6
A9
A1
A10
A7
A14
A12
A18
A18
A2
A3
A4
A4
A5
A6
A6
A7
A7
A4
A9
A16
A11
A15
A15
A18
A18 A9
Table 5
A5, A6
A7
A5, A4
A4
A1, A2
A3
A4, A11
A15
A18, A9
A1
A16, A14
A12
A10, A7
A5
A6, A7
A6
A4, A4
A3
A2, A3
A7
A11, A15
A18
A9, A1
A9
A14, A12
A10
A7, A5
A5
A7, A6
A5
A4, A3
A1
A3, A7
A4
A15, A18
A18
A1, A9
A16
A12, A10
A10
A5, A5
A4
A6, A5
A4
A3, A1
A2
A7, A4
A11
A18, A18
A18, A9
A9, A16
A14
A10, A10
A7
Within the fuzzy time series of the NDVI index, internal relationships are grouped
according to the principle: if a fuzzy set kA or a bunch of sets of the form 1,i iA A+ are
connected to one or several sets at once, then a group of the 1st order is localized rela-
tive to kA or, respectively, a group of 2nd order is localized relative to 1, .i iA A+ The
2nd order relationships for 27 groups are already presented in Table 5, and the 1st order
relationships are divided into 15 groups, which are summarized in Table 6.
Table 6
G1: A1 A2,
A9
G4: A4 A3, A4,
A11
G7: A7 A4, A5,
A6
G10: A11 A15 G13: A15 A18
G2: A2 A3
G5: A5 A4, A5,
A6
G8: A9 A1,
A16
G11: A12 A10 G14: A16 A14
G3: A3 A1,
A7
G6: A6 A5, A7
G9: A10 A7,
A10
G12: A14 A12
G15: A18 A9,
A18
Denoting by ix the value of the NDVI index on the i-th day, and by 1ix + the value
of the NDVI index on the next ( i+1)-th day, a fuzzy relation of the 1st order, for exam-
ple, 2 3A A (from the group G2) can be interpreted as a fuzzy implicative rule «If
2 ,ix A= then 1 3ix A+ = ». Or, say, a fuzzy relation of the 1st order of the form
4 3 4 11, ,A A A A (from the group G4) can be interpreted as a fuzzy implicative rule:
«If 4 ,ix A= then 1 3ix A+ = or 1 4ix A+ = or 1 11ix A+ = ». Accordingly, the fuzzy rela-
tion of the 2nd order, for example, 5 6 7,A A A can be interpreted as «If 5ix A= and
1 6 ,ix A+ = then 2 7ix A+ = », or relationship 18 18 18 9, ,A A A A can be interpreted as
«If 18ix A= and 1 18,ix A+ = then 2 18ix A+ = or 2 9ix A+ = ».
Various approaches are used to determine «crisp» (defuzzified) predicts. The es-
sence of one of them is as follows [7]. If the value of the NDVI index for the i-th day is
described as a fuzzy set Aj, which forms only one relationship within the fuzzy time se-
ries, say ,j kA A then the fuzzy predict for the next ( i+1)-th day will be the set kA .
If there is a group of fuzzy relations, for example, of the form
1 2
, ,..., ,
pj k k kA A A A
then the union of fuzzy sets
1 2
...
pk k kA A A is the fuzzy predict for the ( i+1)-th day.
60 ISSN 1028-0979
In our case, the numerical interpretation of fuzzy predicts is based on the applica-
tion of the S. Chen rule (see [7]). As a numerical estimate of the fuzzy predict jA (the
output of the fuzzy model of the NDVI time series), the abscissa of the middle of the
upper base of the j-th trapezoid is considered (see Fig. 2). For example, for the fuzzy
predict 4 ,A described by the trapezoidal membership function with the parameters indi-
cated in Table 2, the numerical analog is the abscissa of the middle of the upper base:
(0,3271 0,3403) / 2 0,3337.+ = Really, according to the fuzzy set point estimate rule
(see [3]), the following formula is used to defuzzify a fuzzy set :A
max
max 0
1
( ) ( ) ,F A M A d
=
(7)
where { ( ) , }AA u u u U = is the -level set ( [0;1]);
1
1
( )
n
k
k
M A u
n
=
=
( )ku A is the cardinal number of the corresponding -level set. For a fuzzy predict
4
0 1 1 0
0,3139 0,3271 0,3403 0,3535
A = + + + (see Table 2) we have:
0 1, 1, = 4 {0,3139; 0,3535},A =
4( ) (0,3139 0,3535) / 2 0,3337.M A = + =
Then, according to (7), the point estimate of 4A or the defuzzified output of the 1st
order model is the following number:
1
4 4 4
0
1
( ) ( ) ( ) 0,3337.
1
F A M A d M A = = =
For relation , , ,i j t pA A A A where iA is the fuzzy analog of the NDVI for the i-
th day, the numerical predict for the next (i+1)-th day is calculated as the arithmetic
mean of the abscissas of the midpoints of the upper bases of trapezoids corresponding to
the fuzzy sets ,j tA A and .pA In particular, the fuzzy predict for the date 29.09.2000 is
the union 4 5 6A A A with the numerical estimate obtained as following:
0,3271 0,3403 0,3535 0,3667 0,3799 0,3931
2 2 2 0,3601.
3
F
+ + +
+ +
= =
Thus, considering the internal relationships of the 1st and 2nd orders for the fuzzy
time series of the NDVI index, the corresponding predictive models were obtained,
which are summarized in Table 7. Their geometric interpretations are shown in Fig. 4.
Table 7
N Date NDVI
1st order model 2nd order model
Fuzzy output Predict Fuzzy output Predict
1 18.02.2000 0,3599
2 26.02.2000 0,3745 A4, A5, A6 0,3601
3 05.03.2000 0,4099 A5, A7 0,3865 A7 0,4128
4 15.03.2000 0,3812 A4, A5, A6 0,3601 A6 0,3865
5 21.03.2000 0,3680 A5, A7 0,3865 A5 0,3601
6 29.03.2000 0,3457 A4, A5, A6 0,3601 A4 0,3337
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 61
Continuation Table 7
7 06.04.2000 0,3296 A3, A4, A11 0,3865 A4 0,3337
8 15.04.2000 0,2957 A3, A4, A11 0,3865 A3 0,3073
9 22.04.2000 0,2535 A1, A7 0,3337 A1 0,2545
10 29.04.2000 0,2759 A2, A9 0,3733 A2 0,2809
11 08.05.2000 0,2966 A3 0,3073 A3 0,3073
12 15.05.2000 0,3058 A1, A7 0,3337 A7 0,4128
13 24.05.2000 0,3211 A4, A5, A6 0,3601 A4 0,3337
14 09.06.2000 0,5104 A3, A4, A11 0,3865 A11 0,5184
15 15.06.2000 0,6257 A15 0,6240 A15 0,6240
16 25.06.2000 0,7101 A18 0,7031 A18 0,7031
17 02.07.2000 0,7124 A9, A18 0,5844 A18 0,7031
18 11.07.2000 0,7135 A9, A18 0,5844 A18, A9 0,5844
19 20.07.2000 0,4554 A9, A18 0,5844 A18, A9 0,5844
20 27.07.2000 0,2479 A1, A16 0,4524 A1 0,2545
21 02.08.2000 0,4623 A2, A9 0,3733 A9 0,4656
22 12.08.2000 0,6587 A1, A16 0,4524 A16 0,6504
23 20.08.2000 0,5987 A14 0,5976 A14 0,5976
24 28.08.2000 0,5473 A12 0,5448 A12 0,5448
25 03.09.2000 0,5019 A10 0,4920 A10 0,4920
26 13.09.2000 0,4815 A7, A10 0,4524 A10 0,4920
27 20.09.2000 0,4259 A7, A10 0,4524 A7 0,4128
28 29.09.2000 0,3719 A4, A5, A6 0,3601 A5 0,3601
29 07.10.2000 0,3505 A4, A5, A6 0,3601 A5 0,3601
30 15.10.2000 0,3217 A4, A5, A6 0,3601 A4 0,3337
MSE 0,0066 0,0017
MAPE (%) 14,4730 4,6533
MPE (%) – 0,0488 – 0,0205
At the end of Table 5, the values of the MSE (Mean Squared Error), the MAPE
(Mean Absolute Percentage Error) and the MPE (Mean Percentage Error) are presented,
which reflect the quality of the constructed predictive models, their adequacy and accu-
racy. Errors according to these criteria are calculated by the formulas [7]:
2
1
1
MSE ( ) ,
m
t t
j
F A
m =
= −
1
1
MAPE 100 %,
m
t t
tt
F A
m A=
−
=
1
1
MPE 100%,
m
t t
tt
F A
m A=
−
=
where m is the length of the time series; tA is the value of the NDVI index at time ;t
tF is the predict of .tA
Fig. 4
0,8
0,6
0,4
0,2
0,0
1
8
.0
2
.2
0
0
0
2
8
.0
4
.2
0
0
0
0
3
.0
3
.2
0
0
0
1
2
.0
5
.2
0
0
0
1
7
.0
3
.2
0
0
0
1
3
.1
0
.2
0
0
0
3
1
.0
3
.2
0
0
0
0
9
.0
6
.2
0
0
0
1
4
.0
4
.2
0
0
0
2
3
.0
6
.2
0
0
0
0
7
.0
7
.2
0
0
0
2
8
.0
7
.2
0
0
0
1
1
.0
8
.2
0
0
0
2
5
.0
8
.2
0
0
0
0
8
.0
9
.2
0
0
0
2
2
.0
9
.2
0
0
0
NDVI Time Series 1-st order model 2-nd order model
0
6
.1
0
.2
0
0
0
62 ISSN 1028-0979
The MSE criterion is most often used in choosing the optimal predictive model and
emphasizes large forecast errors. As can be seen from the forecasting results, the error
value for this criterion is quite low. The MAPE indicator shows how large the forecast
errors are compared to the real values of the time series. MPE is a more informative cri-
terion for assessing the adequacy of the predictive model, which determines the «bias»
of the constructed predict, that is, its permanent underestimation or overestimation.
MPE indicators for predictive models of the 1-st and 2-nd orders are ( 0,0488)− and
( 0,0205),− respectively, which reflects a slight bias, that is, having percentage values
close to zero, not exceeding 5 %. Otherwise, if there were a large negative MPE per-
centage, then the predictive models would be consistently overestimating. If the MPE
indicator showed a large positive percentage value, then the constructed predictive
models would be consistently underestimating.
Forecasting the fuzzy time series of the NDVI: method №2
Based on the above procedures (Step 1–Step 5) for the time series relative to NIR
and RED reflectance (see Table 1), the final average distance NIR 0,0144AD = and
RED 0,0022AD = were established. In accordance with these values, the required uni-
verses for each time series are obtained in the form:
NIR [0,1106 0,0144; 0,8702 0,0144] [0,0962; 0,8846],U = − + =
where 0,1106 and 0,8702 are the minimum and maximum values of the NIR reflection
coefficient, respectively.
RED [0,0659 0,0022; 0,1623 0,0022] [0,0637; 0,1645],U = − + =
where 0,0659 and 0,1623 are the minimum and maximum values of the RED reflection
coefficient, respectively.
Based on the application of formula (5), the optimal values of the number of evalu-
ation criteria for estimation the NIR and RED reflection coefficient were obtained as
follows:
NIR
0,8846 0,0962 0,0144
26,9158 27,
2 0,0144
m
− −
= =
RED
0,1645 0,0637 0,0022
22,2444 22.
2 0,0022
m
− −
= =
To describe the qualitative evaluation criteria by fuzzy sets, trapezoidal member-
ship functions in the form of (6) are also used. As a result, to evaluate NIR time series
data (see Table 8 and Fig. 5) the appropriate fuzzy sets ( 1 27)iA i = were formed, and
to evaluate RED time series data (see table 9 and Fig. 6) the appropriate fuzzy sets
( 1 22)jB j = were formed.
Table 8
Fuzzy
set
Membership function parameters Fuzzy
set
Membership function parameters
1i
a
2i
a
3i
a
4i
a
1i
a
2i
a
3i
a
4i
a
A1 0,0962 0,1106 0,1250 0,1394 A15 0,4988 0,5132 0,5276 0,5419
A2 0,1250 0,1394 0,1537 0,1681 A16 0,5276 0,5419 0,5563 0,5707
A3 0,1537 0,1681 0,1825 0,1969 A17 0,5563 0,5707 0,5851 0,5994
A4 0,1825 0,1969 0,2112 0,2256 A18 0,5851 0,5994 0,6138 0,6282
A5 0,2112 0,2256 0,2400 0,2544 A19 0,6138 0,6282 0,6426 0,6570
A6 0,2400 0,2544 0,2688 0,2831 A20 0,6426 0,6570 0,6713 0,6857
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 63
Continuation Table 8
A7 0,2688 0,2831 0,2975 0,3119 A21 0,6713 0,6857 0,7001 0,7145
A8 0,2975 0,3119 0,3263 0,3406 A22 0,7001 0,7145 0,7288 0,7432
A9 0,3263 0,3406 0,3550 0,3694 A23 0,7288 0,7432 0,7576 0,7720
A10 0,3550 0,3694 0,3838 0,3982 A24 0,7576 0,7720 0,7864 0,8007
A11 0,3838 0,3982 0,4125 0,4269 A25 0,7864 0,8007 0,8151 0,8295
A12 0,4125 0,4269 0,4413 0,4557 A26 0,8151 0,8295 0,8439 0,8582
A13 0,4413 0,4557 0,4700 0,4844 A27 0,8439 0,8582 0,8726 0,8846
A14 0,4700 0,4844 0,4988 0,5132
Trapezoidal membership functions of fuzzy sets ( 1 27).iA i =
Fig. 5
In Table 9 fuzzy sets as qualitative criteria for RED reflection evaluation are presented.
Table 9
Fuzzy
set
Membership function parameters Fuzzy
set
Membership function parameters
1i
a
2i
a
3i
a
4i
a
1i
a
2i
a
3i
a
4i
a
B1 0,0637 0,0659 0,0681 0,0703 B12 0,1125 0,1147 0,1169 0,1191
B2 0,0681 0,0703 0,0726 0,0748 B13 0,1169 0,1191 0,1213 0,1235
B3 0,0726 0,0748 0,0770 0,0792 B14 0,1213 0,1235 0,1258 0,1280
B4 0,0770 0,0792 0,0814 0,0836 B15 0,1258 0,1280 0,1302 0,1324
B5 0,0814 0,0836 0,0859 0,0881 B16 0,1302 0,1324 0,1346 0,1368
B6 0,0859 0,0881 0,0903 0,0925 B17 0,1346 0,1368 0,1391 0,1413
B7 0,0903 0,0925 0,0947 0,0969 B18 0,1391 0,1413 0,1435 0,1457
B8 0,0947 0,0969 0,0992 0,1014 B19 0,1435 0,1457 0,1479 0,1501
B9 0,0992 0,1014 0,1036 0,1058 B20 0,1479 0,1501 0,1524 0,1546
B10 0,1036 0,1058 0,1080 0,1102 B21 0,1524 0,1546 0,1568 0,1590
B11 0,1080 0,1102 0,1125 0,1147 B22 0,1568 0,1590 0,1612 0,1645
Trapezoidal membership functions of fuzzy sets ( 1 22).jB j =
Fig. 6
1,0
0,8
0,6
0,4
0,2
0,0
0
,0
9
6
2
0
,1
6
8
1
0
,1
1
0
6
0
,1
8
2
5
0
,1
2
5
0
0
,1
9
6
9
0
,1
3
9
4
0
,2
1
1
2
0
,1
5
3
7
0
,2
2
5
6
0
,2
4
0
0
0
,2
5
4
4
0
,2
6
8
8
0
,2
8
3
1
0
,2
9
7
5
0
,3
1
1
9
NIR reflection
0
,3
4
0
6
0
,3
5
5
0
0
,3
6
9
4
0
,3
8
3
8
0
,6
4
2
6
0
,6
5
7
0
0
,6
7
1
3
0
,6
8
5
7
0
,7
0
0
1
0
,7
1
4
9
0
,7
2
8
8
0
,7
4
3
2
0
,7
5
7
6
0
,7
7
2
0
0
,7
8
6
4
0
,8
0
0
7
0
,8
1
5
1
0
,8
2
9
5
0
,3
2
6
3
0
,8
4
3
9
0
,8
5
8
2
0
,8
8
4
6
0
,3
9
8
2
0
,4
1
2
5
0
,4
2
6
9
0
,4
4
1
3
0
,4
5
5
7
0
,4
7
0
0
0
,4
8
4
4
0
,4
9
8
8
0
,5
1
3
2
0
,5
2
7
6
0
,5
4
1
9
0
,5
5
6
3
0
,5
7
0
7
0
,5
8
5
1
0
,5
9
9
4
0
,6
1
3
8
0
,6
2
8
2
0
,8
7
2
6
M
em
b
er
s
h
ip
f
u
n
ct
io
n
v
al
u
es
1,0
0,8
0,6
0,4
0,2
0,0
0
,0
6
3
7
0
,0
7
4
8
0
,0
6
5
9
0
,0
7
7
0
0
,0
6
8
1
0
,0
9
6
9
0
,0
7
0
3
0
,0
9
9
2
0
,0
7
2
6
0
,1
0
1
4
0
,1
0
3
6
0
,1
0
5
8
0
,1
0
8
0
0
,1
1
0
2
0
,1
1
2
5
0
,1
1
4
7
RED reflection
0
,1
1
9
1
0
,1
2
1
3
0
,1
2
3
5
0
,1
2
5
8
0
,1
2
8
0
0
,1
3
0
2
0
,1
3
2
4
0
,1
3
4
6
0
,1
3
6
8
0
,1
3
9
1
0
,1
4
1
3
0
,1
4
3
5
0
,1
4
5
7
0
,1
4
7
9
0
,1
5
0
1
0
,1
5
2
4
0
,1
5
4
6
0
,1
5
6
8
0
,1
1
6
9
0
,1
5
9
0
0
,1
6
1
2
0
,1
6
4
5
M
em
b
er
s
sh
ip
f
u
n
ct
io
n
v
al
u
es
0
,0
7
9
2
0
,0
8
1
4
0
,0
8
3
6
0
,0
8
5
9
0
,0
8
8
1
0
,0
9
0
3
0
,0
9
2
5
0
,0
9
4
7
64 ISSN 1028-0979
According to the above principle, fuzzification of NIR and RED reflections is car-
ried out by trapezoidal membership functions, the corresponding parameters of which
are presented in Table 8 and Table 9. Obtained fuzzy analogs for all NIR and RED data
are summarized in Table 10.
In Table 10 Fuzzy time series of reflection coefficients NIR and RED are shown.
Table 10
N
NIR time series RED time series
N
NIR time series RED time series
NIR Fuzzy set RED Fuzzy set NIR Fuzzy set RED Fuzzy set
1 0,2036 A4 0,0958 B7 16 0,8565 A27 0,1452 B19
2 0,3175 A8 0,1445 B18 17 0,8651 A27 0,1453 B19
3 0,3639 A10 0,1523 B20 18 0,8702 A27 0,1455 B19
4 0,3623 A10 0,1623 B22 19 0,3357 A9 0,1256 B14
5 0,2219 A5 0,1025 B9 20 0,1125 A1 0,0678 B1
6 0,1717 A3 0,0835 B5 21 0,3666 A10 0,1348 B16
7 0,1676 A3 0,0845 B5 22 0,6051 A18 0,1245 B14
8 0,1407 A2 0,0765 B3 23 0,5828 A17 0,1463 B19
9 0,1106 A1 0,0659 B1 24 0,4628 A13 0,1354 B16
10 0,1214 A1 0,0689 B1 25 0,3492 A9 0,1158 B12
11 0,1502 A2 0,0815 B4 26 0,3523 A9 0,1233 B14
12 0,1529 A2 0,0813 B4 27 0,3450 A9 0,1389 B17
13 0,1664 A3 0,0855 B5 28 0,2457 A5 0,1125 B11
14 0,4084 A11 0,1324 B16 29 0,2173 A4 0,1045 B9
15 0,5890 A17 0,1356 B16 30 0,2058 A4 0,1056 B10
In the framework of the fuzzy time series NIR and RED, the identified internal re-
lationships of the 1st and 2nd orders are summarized in Table 11 and Table 12, respectively.
Table 11
NIR time series RED time series
G1: A1 A2, A10, A11 G9: A11 A17 G1: B1 B1, B4, B16 G9: B14 B1, B17, B19
G2: A2 A1, A2, A3 G10: A13 A9 G2: B3 B1
G10: B16 B12, B14,
B16, B19
G3: A3 A2, A3, A11 G11: A17 A13, A27 G3: B4 B4, B5 G11: B17 B11
G4: A4 A4, A8 G12: A18 A17 G4: B5 B3, B5, B16 G12: B18 B20
G5: A5 A3, A4 G13: A27 A9, A27 G5: B7 B18 G13: B19 B14, B16, B19
G6: A8 A10 G6: B9 B5, B10 G14: B20 B22
G7: A9 A1, A5, A9 G7: B11 B9 G15: B22 B9
G8: A10 A5, A10, A18 G8: B12 B14
Table 12
NIR time series RED time series
G1: A1, A1 A2 G15: A9, A5 A4 G1: B1, B1 B4 G15: B14, B17 B11
G2: A1, A2 A2 G16: A9, A9 A9, A5 G2: B1, B4 B4 G16: B14, B19 B16
G3: A1, A10 A18 G17: A10, A5 A3 G3: B1, B16 B14 G17: B16, B12 B14
G4: A2, A1 A1 G18: A10, A10 A5 G4: B3, B1 B1 G18: B16, B14 B19
G5: A2, A2 A3 G19: A10, A18 A17 G5: B4, B4 B5 G19: B16, B16 B19
G6: A2, A3 A11 G20: A11, A17 A27 G6: B4, B5 B16 G20: B16, B19 B19
G7: A3, A2 A1 G21: A13, A9 A9 G7: B5, B3 B1 G21: B17, B11 B9
G8: A3, A3 A2 G22: A17, A13 A9 G8: B5, B5 B3 G22: B18, B20 B22
G9: A3, A11 A17 G23: A17, A27 A27 G9: B5, B16 B16 G23: B19, B14 B1
G10: A4, A8 A10 G24: A18, A17 A13 G10: B7, B18 B20 G24: B19, B16 B12
G11: A5, A3 A3 G25: A27, A9 A1 G11: B9, B5 B5 G25: B19, B19 B19, B14
G12: A5, A4 A4 G26: A27, A27 A27, A9 G12: B11, B9 B10 G26: B20, B22 B9
G13: A8, A10 A10 G13: B12, B14 B17 G27: B22, B9 B5
G14: A9, A1 A10 G14: B14, B1 B16
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 65
As a result, for each value of the NIR and RED reflections within the time series
the corresponding crisp predicts were obtained by applications of obtained predictive
models and the rule of defuzzification. Desired results are summarized in Table 13 and
Table14, respectively.
Table 13
N Date NIR
Fuzzy ana-
log
1st order model 2nd order model
Fuzzy
output
Predict
Fuzzy
output
Predict
1 18.02.2000 0,3599 A4
2 26.02.2000 0,3745 A8 A4, A8 0,2616
3 05.03.2000 0,4099 A10 A10 0,3766 A10 0,3766
4 15.03.2000 0,3812 A10
A5, A10,
A18
0,4053 A10 0,3766
5 21.03.2000 0,3680 A5
A5, A10,
A18
0,4053 A5 0,2328
6 29.03.2000 0,3457 A3 A3, A4 0,1897 A3 0,1753
7 06.04.2000 0,3296 A3
A2, A3,
A11
0,2424 A3 0,1753
8 15.04.2000 0,2957 A2
A2, A3,
A11
0,2424 A2 0,1465
9 22.04.2000 0,2535 A1
A1, A2,
A3
0,1465 A1 0,1178
10 29.04.2000 0,2759 A1
A2, A10,
A11
0,3095 A1 0,1178
11 08.05.2000 0,2966 A2
A2, A10,
A11
0,3095 A2 0,1465
12 15.05.2000 0,3058 A2
A1, A2,
A3
0,1465 A2 0,1465
13 24.05.2000 0,3211 A3
A1, A2,
A3
0,1465 A3 0,1753
14 09.06.2000 0,5104 A11
A2, A3,
A11
0,2424 A11 0,4053
15 15.06.2000 0,6257 A17 A17 0,5779 A17 0,5779
16 25.06.2000 0,7101 A27 A13, A27 0,6641 A27 0,8654
17 02.07.2000 0,7124 A27 A9, A27 0,6066 A27 0,8654
18 11.07.2000 0,7135 A27 A9, A27 0,6066 A9, A27 0,6066
19 20.07.2000 0,4554 A9 A9, A27 0,6066 A9, A27 0,6066
20 27.07.2000 0,2479 A1
A1, A5,
A9
0,2328 A1 0,1178
21 02.08.2000 0,4623 A10
A2, A10,
A11
0,3095 A10 0,3766
22 12.08.2000 0,6587 A18
A5, A10,
A18
0,4053 A18 0,6066
23 20.08.2000 0,5987 A17 A17 0,5779 A17 0,5779
24 28.08.2000 0,5473 A13 A13, A27 0,6641 A13 0,4629
25 03.09.2000 0,5019 A9 A9 0,3478 A9 0,3478
26 13.09.2000 0,4815 A9
A1, A5,
A9
0,2328 A9 0,3478
27 20.09.2000 0,4259 A9
A1, A5,
A9
0,2328 A3, A9 0,2616
28 29.09.2000 0,3719 A5
A1, A5,
A9
0,2328 A3, A9 0,2616
29 07.10.2000 0,3505 A4 A3, A4 0,1897 A4 0,2041
30 15.10.2000 0,3217 A4 A4, A8 0,2616 A4 0,2041
66 ISSN 1028-0979
Table 14
N Date NIR
Fuzzy
analog
1st order model 2nd order model
Fuzzy
output
Predict
Fuzzy
output
Predict
1 18.02.2000 0,0958 B7
2 26.02.2000 0,1445 B18 B18 0,1424
3 05.03.2000 0,1523 B20 B20 0,1512 B20 0,1512
4 15.03.2000 0,1623 B22 B22 0,1601 B22 0,1601
5 21.03.2000 0,1025 B9 B9 0,1025 B9 0,1025
6 29.03.2000 0,0835 B5 B5, B10 0,0958 B5 0,0847
7 06.04.2000 0,0845 B5 B3, B5, B16 0,0980 B5 0,0847
8 15.04.2000 0,0765 B3 B3, B5, B16 0,0980 B3 0,0759
9 22.04.2000 0,0659 B1 B1 0,0670 B1 0,0670
10 29.04.2000 0,0689 B1 B1, B4, B16 0,0936 B1 0,0670
11 08.05.2000 0,0815 B4 B1, B4, B16 0,0936 B4 0,0803
12 15.05.2000 0,0813 B4 B4, B5 0,0825 B4 0,0803
13 24.05.2000 0,0855 B5 B4, B5 0,0825 B5 0,0847
14 09.06.2000 0,1324 B16 B3, B5, B16 0,0980 B16 0,1335
15 15.06.2000 0,1356 B16
B12, B14,
B16, B19
0,1302 B16 0,1335
16 25.06.2000 0,1452 B19
B12, B14,
B16, B19
0,1302 B19 0,1468
17 02.07.2000 0,1453 B19
B14, B16,
B19
0,1350 B19 0,1468
18 11.07.2000 0,1455 B19
B14, B16,
B19
0,1350 B19, B14 0,1357
19 20.07.2000 0,1256 B14
B14, B16,
B19
0,1350 B19, B14 0,1357
20 27.07.2000 0,0678 B1
B1, B17,
B19
0,1173 B1 0,0670
21 02.08.2000 0,1348 B16 B1, B4, B16 0,0936 B16 0,1335
22 12.08.2000 0,1245 B14
B12, B14,
B16, B19
0,1302 B14 0,1246
23 20.08.2000 0,1463 B19
B1, B17,
B19
0,1173 B19 0,1468
24 28.08.2000 0,1354 B16
B14, B16,
B19
0,1350 B16 0,1335
25 03.09.2000 0,1158 B12
B12, B14,
B16, B19
0,1302 B12 0,1158
26 13.09.2000 0,1233 B14 B14 0,1246 B14 0,1246
27 20.09.2000 0,1389 B17
B1, B17,
B19
0,1173 B17 0,1379
28 29.09.2000 0,1125 B11 B11 0,1113 B11 0,1113
29 07.10.2000 0,1045 B9 B9 0,1025 B9 0,1025
30 15.10.2000 0,1056 B10 B5, B10 0,0958 B10 0,1069
Thus, predictive model for NDVI time series is restored by application of the em-
pirical formula (1) for each day according to the corresponding forecast data NIR and
RED. The desired models are interpreted in the form of Table 15 and Fig. 7.
Table 15
N Date NDVI
Model
N 1
Model
N 2
N Date NDVI
Model
N 1
Model N
2
1 18.02.2000 0,3599 16 25.06.2000 0,7101 0,6722 0,7099
2 26.02.2000 0,3745 0,2951 17 02.07.2000 0,7124 0,6360 0,7099
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 67
Continuation Table 15
3 05.03.2000 0,4099 0,4269 0,4269 18 11.07.2000 0,7135 0,6360 0,6343
4 15.03.2000 0,3812 0,4337 0,4034 19 20.07.2000 0,4554 0,6360 0,6343
5 21.03.2000 0,3680 0,5964 0,3887 20 27.07.2000 0,2479 0,3301 0,2748
6 29.03.2000 0,3457 0,3287 0,3482 21 02.08.2000 0,4623 0,5356 0,4765
7 06.04.2000 0,3296 0,4240 0,3482 22 12.08.2000 0,6587 0,5138 0,6591
8 15.04.2000 0,2957 0,4240 0,3177 23 20.08.2000 0,5987 0,6626 0,5948
9 22.04.2000 0,2535 0,3724 0,2748 24 28.08.2000 0,5473 0,6622 0,5523
10 29.04.2000 0,2759 0,5356 0,2748 25 03.09.2000 0,5019 0,4553 0,5005
11 08.05.2000 0,2966 0,5356 0,2920 26 13.09.2000 0,4815 0,3026 0,4724
12 15.05.2000 0,3058 0,2795 0,2920 27 20.09.2000 0,4259 0,3301 0,3094
13 24.05.2000 0,3211 0,2795 0,3482 28 29.09.2000 0,3719 0,3530 0,4028
14 09.06.2000 0,5104 0,4240 0,5045 29 07.10.2000 0,3505 0,2985 0,3314
15 15.06.2000 0,6257 0,6323 0,6247 30 15.10.2000 0,3217 0,4638 0,3124
MSE 0,0137 0,0021
MAPE (%) 25,5716 5,9490
MPE (%) – 0,1161 – 0,0182
Fig. 7
As can be seen from the results of NDVI forecasting by the 2nd method, the error
value according to the MSE criterion is quite low (0,0137 for Model N 1, and 0,0021
for Model N 2), which cannot be said about the error value according to the MAPE cri-
terion (25,5716 for Model N 1 and 5,9490 for Model N 2). The MPEs for the 1st and
2nd order predictive models are ( 0,1161)− and ( 0,0182),− respectively, reflecting a
slight bias below the 5 % threshold.
Conclusion
The article uses one of the methods of precision farming, associated with the ap-
plying of the NDVI vegetation index, which allows to predict crop volumes and most
accurately assess the real state of growing plants. The last statement is relative since the
index itself does not reflect the absolute values of plant volumes. Nevertheless, accord-
ing to the obtained multispectral data, it is possible to evaluate the development of crops
and predict its future yield. It should be considered that the value of the NDVI index
changes throughout the entire growing season, that is, during the initial growth, the pe-
riod of flowering and maturation its indicators differ significantly, in fact, as this is
demonstrated by the dynamics of NDVI using the example of one pixel (see Fig. 1). The
practice of precision farming has shown that the most active increase in the NDVI oc-
curs during the growing season, during the flowering period, the growth of the crop
slows down and halts, and in the process of crop maturation, the index gradually de-
creases.
0,8
0,6
0,4
0,2
0,0
1
8
.0
2
.2
0
0
0
2
9
.0
4
.2
0
0
0
0
5
.0
3
.2
0
0
0
1
5
.0
5
.2
0
0
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2
1
.0
3
.2
0
0
0
1
1
.1
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.2
0
0
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.0
3
.2
0
0
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9
.0
6
.2
0
0
0
1
5
.0
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.2
0
0
0
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.0
6
.2
0
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.0
7
.2
0
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7
.0
7
.2
0
0
0
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.0
8
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.0
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.0
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0
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NDVI Time Series 1-st order model 2-nd order model
0
7
.1
0
.2
0
0
0
68 ISSN 1028-0979
The fuzzy approaches to predicting the annual dynamics of the NDVI proposed in
the article can be easily projected to process multispectral data obtained from all pixels
of the corresponding vegetation maps. If vegetation maps used in precision farming al-
low only visually determining differences in the state of plants, then owing to digitaliza-
tion it becomes possible to interpret the color range of vegetation — from light tones
with a low index to dark color with high NDVI index.
Е.Р. Алієв, Ф.М. Салманов
НЕЧІТКИЙ ПІДХІД ДО ПРОГНОЗУВАННЯ
ДИНАМІКИ ВЕГЕТАЦІЙНИХ ІНДЕКСІВ
Алієв Ельчин Рашид огли
Інститут систем керування Національної Академії Наук Азербайджана,
elchin.aliyev@sinam.net
Салманов Фуад Мухтар огли
Інститут систем керування Національної Академії Наук Азербайджана,
fuad.salmanli@sinam.net
Сучасні технології супутникового моніторингу поверхні Землі надають сільсько-
господарським виробникам корисну інформацію — стан здоровʼя посівних культур.
Здатність віддаленого датчика виявляти незначні відмінності у рослинності робить
його корисним інструментом для кількісної оцінки мінливості в межах заданого по-
ля, оцінки зростання сільськогосподарських культур та управління угіддями на ос-
нові поточних умов. Дані дистанційного зондування, що збираються на регулярній
основі, дозволяють виробникам та агрономам складати поточну карту вегетації, що
відображає стан та силу посівних культур, аналізувати динаміку зміни у стані рос-
лин, а також прогнозувати врожайність на конкретній посівній площі. Для інтерпре-
тації цих даних найефективнішими засобами є всілякі вегетаційні індекси, що ро-
зраховуються емпірично, тобто шляхом операцій із різними спектральними діапа-
зонами мультиспектральних даних супутникового моніторингу. На основі часового
ряду одного з таких вегетаційних індексів у статті розглядається річна динаміка ро-
звитку посівної культури на конкретному полі. Можливість прогнозування врожай-
ності цієї посівної культури розглядається на основі нечіткого моделювання часо-
вих рядів за відповідним спектральним діапазоном відображення вегетації, отрима-
ним зі знімків супутникового моніторингу. Запропоновано нечіткі моделі часових
рядів, вивчені на адекватність і придатність з погляду аналізу особливості внут-
рішньорічної середньорічної динаміки індексу, типової для цієї посівної площі.
Ключові слова: сільскогосподарська культура, мультиспектральне відобра-
ження рослин, вегетаційний індекс, нечітка множина, нечіткий часовий ряд.
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Submitted 13.04.2022
mailto:elchin.aliyev@sinam.net
https://micasense.squarespace.com/atlasflight
https://goo.gl/maps/%1fYAddomuoXsD4QQN36
https://goo.gl/maps/%1fYAddomuoXsD4QQN36
|
| id | nasplib_isofts_kiev_ua-123456789-210875 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | English |
| last_indexed | 2026-03-13T18:17:53Z |
| publishDate | 2022 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Aliyev, E.R. Salmanov, F.M. 2025-12-19T17:22:09Z 2022 Fuzzy approach to forecasting the dynamics ofvegetation indices / E.R. Aliyev, F.M. Salmanov // Проблеми керування та інформатики. — 2022. — № 2. — С. 53-68. — Бібліогр.: 8 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/210875 519.712.3 10.34229/2786-6505-2022-2-4 Modern satellite monitoring technologies provide agricultural producers with valuable information on the health status of crops. The ability of remote sensors to detect slight variations in vegetation makes them a useful tool for quantitatively assessing variability within a given field, evaluating crop growth, and managing land based on current conditions. Remote sensing data, collected on a regular basis, allow producers and agronomists to create up-to-date vegetation maps reflecting the health and vigor of crops, analyze changes in plant condition over time, and forecast yield for a specific planting area. Сучасні технології супутникового моніторингу поверхні Землі надають сільськогосподарським виробникам корисну інформацію — стан здоровʼя посівних культур. Здатність віддаленого датчика виявляти незначні відмінності у рослинності робить його корисним інструментом для кількісної оцінки мінливості в межах заданого поля, оцінки зростання сільськогосподарських культур та управління угіддями на основі поточних умов. Дані дистанційного зондування, що збираються на регулярній основі, дозволяють виробникам та агрономам складати поточну карту вегетації, що відображає стан та силу посівних культур, аналізувати динаміку зміни у стані рослин, а також прогнозувати врожайність на конкретній посівній площі. en Інститут кібернетики ім. В.М. Глушкова НАН України Проблемы управления и информатики Методи керування та оцінювання в умовах невизначеності Fuzzy approach to forecasting the dynamics ofvegetation indices Нечіткий підхід до прогнозування динаміки вегетаційних індексів Article published earlier |
| spellingShingle | Fuzzy approach to forecasting the dynamics ofvegetation indices Aliyev, E.R. Salmanov, F.M. Методи керування та оцінювання в умовах невизначеності |
| title | Fuzzy approach to forecasting the dynamics ofvegetation indices |
| title_alt | Нечіткий підхід до прогнозування динаміки вегетаційних індексів |
| title_full | Fuzzy approach to forecasting the dynamics ofvegetation indices |
| title_fullStr | Fuzzy approach to forecasting the dynamics ofvegetation indices |
| title_full_unstemmed | Fuzzy approach to forecasting the dynamics ofvegetation indices |
| title_short | Fuzzy approach to forecasting the dynamics ofvegetation indices |
| title_sort | fuzzy approach to forecasting the dynamics ofvegetation indices |
| topic | Методи керування та оцінювання в умовах невизначеності |
| topic_facet | Методи керування та оцінювання в умовах невизначеності |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210875 |
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