=Paper= {{Paper |id=Vol-3611/paper3 |storemode=property |title=Grey wolf optimizer combined with k-nn algorithm for clustering problem |pdfUrl=https://ceur-ws.org/Vol-3611/paper3.pdf |volume=Vol-3611 |authors=Katarzyna Prokop |dblpUrl=https://dblp.org/rec/conf/ivus/Prokop22 }} ==Grey wolf optimizer combined with k-nn algorithm for clustering problem== https://ceur-ws.org/Vol-3611/paper3.pdf

Grey wolf optimizer combined with k-nn algorithm
for clustering problem
Katarzyna Prokop
Faculty of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44100 Gliwice, Poland

Abstract
The clustering problem is an important task in machine learning. Clustering algorithms allow for the division of the set
into individual clusters on the basis of a specific measure. Such an idea is used for undescribed data, where its label must
be automatically assigned. One of the most popular algorithms is the 𝑘-nearest neighbors. In this paper, we propose a
modification of this algorithm by combining it with heuristics, i.e. the grey wolf optimizer. The idea assumes that individuals
in heuristic will be understood as a sample and unknown classes as victims in heuristic. Then the heuristic operation is used
for analyzing the set. The proposition was described in terms of original algorithms and proposed hybridization of them.
Then it was tested on Iris Flower Dataset and obtained results were discussed in terms of its advantages.

1. Introduction extracted information can be used for describing an ob-
ject and used in further classification.
Machine learning algorithms are known as data-hungry. In this paper, a hybridization of 𝑘-nn and selected
It means, that many of them need a large number of heuristic algorithm was proposed. It is an alternative
samples to fit/train the model. However, in many cases, way that indicates that these two solutions can be com-
collected data are not labeled and cannot be used in the bined and result in good accuracy. For the research the
supervised training process. Therefore, the clustering Grey Wolf Optimizer was chosen. This is a fairly young
method can be used to split them into some classes/clusters. method [5], uses the hierarchy of units in the herd. This
An example of clustering visual features was presented in heuristic algorithm shows competitive results compared
[1]. Another solution is modifying a k-means algorithm to other known metaheuristics for the function optimiza-
by introducing some dynamic changed conditions [2, 3]. tion problem. The Gray Wolf Optimizer can be success-
Moreover, different approaches to clustering are mod- fully used, for example, in industry [15] or smart home
eled and it can be seen in the example of deep spectral solutions [16].
clustering that uses an auto-encoder network [4].
Moreover, the optimization task is important in the
area of machine learning. Therefore, many newer algo- 2. Methodology
rithms are modeled as an alternative and accurate ap-
proach [5]. Except for new models, the hybridization of The main idea is to combine the operation of k Nearest
them is introduced. One such example is a cooperative Neighbors classifier with Grey Wolf Optimizer and test
idea of many such algorithms [6]. The application of the effectiveness of the method obtained as a result.
these algorithms shows that it is a promising approach
and can help in different areas of artificial intelligence. 2.1. k Nearest Neighbors Algorithm
For instance, meta-heuristic algorithms were used in the
The k Nearest Neighbors classifier (𝑘-nn) is an example of
federated training process of convolutional neural net-
an algorithm that is used for finding the 𝑘 most closely
works [7, 8]. Also, it was combined to create a neuro-
related items to the one that is considered, which makes
swarm heuristic for dynamics of covid19 [9]. Interesting
classifying this object enabled. This algorithm is used
solution was to use nature-inspired algorithms in im-
for clustering, interpolation and even classification tasks
age analysis [10, 11]. Heuristic algorithms were used for
[17, 18]. Therefore this algorithm determines the similar-
motion planning of aircraft [12], or others engineering
ity between two objects using selected measures. Based
problems. In most cases, engineering problems can be
on the results a group of the items with the least differ-
presented as an optimization task, where the best coeffi-
ence i s created. This s et contains eponymous 𝑘 elements.
cients must be found to reach the best results [13]. Except
Objects reminding the considered item are called “neigh-
using heuristics in hybridization and optimization, these
bors”. By the “voice of majority”, they are responsible
algorithms are also used for feature extraction [14]. The
for assigning the tested object to the appropriate class.
This means that obtained class is the most frequently
IVUS 2022: 27th International Conference on Information Technology
$ ktn.prokop@gmail.com (K. Prokop)
appearing label among neighbors.
© 2022 Copyright for this paper by its authors. Use permitted under Creative The algorithm is also used in a clustering problem.
Commons License Attribution 4.0 International (CC BY 4.0).
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ISSN 1613-0073 It is possible to create groups of elements with similar

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features applying 𝑘-nn. There are many different ways Algorithm 1: k Nearest Neighbors Algorithm.
of dividing the same dataset so various methods and their
Input: dataset with 𝑚 vectors 𝑥𝑖 , unclassified
variable elements can be customized for this purpose.
vector 𝑦, neighbors number 𝑘
Records in a given database and tested elements can
Output: 𝑦 group
be treated as vectors. Then the similarity between them
1 𝑖 := 1;
may be calculated by a distance function. In this paper
2 for 𝑖 ≤ 𝑚 do
Euclidean metric and Manhattan metric were applied. As-
3 Calculate the distance from 𝑦 to 𝑥𝑖 using
suming 𝑎 and 𝑏 are two records being compared, where
measure 𝑑;
each of them consists of 𝑛 attributes, the following vec-
4 𝑖 + +;
tors are obtained:
5 end

𝑎 = [𝑎1 , ..., 𝑎𝑛 ], 6 Sort the records in the database in ascending
order relative to the calculated distances;
𝑏 = [𝑏1 , ..., 𝑏𝑛 ], 7 𝑗 := 1;
8 for 𝑗 ≤ 𝑘 do
which means that attributes should take numerical values.
9 Make a note of the assigned group for 𝑥𝑗 ;
Distance function between 𝑎 and 𝑏 for the Euclidean
10 𝑗 + +;
metric is defined as below:
11 end
12 Select the most popular group;
⎯
⎸ 𝑛
(1) 13 return 𝑦 group;
⎸∑︁
𝑑(𝑎, 𝑏) = ⎷ (𝑎𝑖 − 𝑏𝑖 )2 .
𝑖=1

For the Manhattan metric, distance function can be de-
scribed by: takes command of the herd when the leader is indisposed.
Further two groups of wolves are distinguished: the third
∑︁𝑛
level in the hierarchy are individuals who are doing fairly
𝑑(𝑎, 𝑏) = 𝑎𝑖 − 𝑏𝑖 . (2) well and the last group consists of old and sick wolves.
𝑖=1
The tasks undertaken by the pack include mainly search-
Let mark every 𝑖 database’s element with 𝑛 attributes ing for food, i.e. hunting mammals. In nature, wolves
𝑡ℎ

as 𝑥𝑖 = [𝑥𝑖1 , ..., 𝑥𝑖𝑛 ], 𝑖 = 1, ..., 𝑚. Thus 𝑚 is the number hunt in various configurations: alone, in pairs, or as a
of all records in the dataset. Obviously, the number 𝑘 is whole pack.
less than or equal to 𝑚. Tested item can be presented as Grey Wolf Optimizer uses a group hunting strategy.
𝑦 = [𝑦1 , ..., 𝑦𝑛 ]. By the pseudocode 1, with the selected The different levels of the wolf hierarchy can be repre-
distance function 𝑑, the 𝑘 Nearest Neighbors Algorithm sented by the symbols: 𝛼 – the male wolf, 𝛽 – the second
is shown. strongest wolf, 𝛿 – the third level of the hierarchy, and
𝜔 – old and sick wolves. In particular, the first three
2.2. Grey Wolf Optimizer levels have an impact on the operation of the algorithm.
Firstly, the pack consisting of a fixed number of wolves
The method proposed in this paper besides the classifier is initiated. A wolf is treated like a vector 𝑥:
also uses an optimizer. In detail, Grey Wolf Optimizer
was applied. This algorithm is an example of a heuristic 𝑥 = [𝑥1 , ..., 𝑥𝑛 ],
method of optimization which means that its aim is to
find an approximate solution to a given problem but the values of which determine the wolf’s location. The
there is no guarantee of its correctness. Heuristics are number 𝑛 defines the dimension of a given problem that
useful in case of high resource cost or high computational is a number of variables of the function 𝑓 (·) wanted to
complexity of classic methods. optimize. In the initial pack, coordinates 𝑥1 , ..., 𝑥𝑛 are
Grey Wolf Optimizer was developed in 2014 [5] based drawn from a given interval. Next, the three strongest
on the behavior of a pack of wolves. Wolves are predators individuals are selected: 𝛼, 𝛽, 𝛿 (the best wolf in the third
and live in herds where a hierarchy occurs. Every pack is level of hierarchy). This operation takes place by compar-
led by a leader, the so-called male wolf. This individual ing the values of the function 𝑓 (·) for all individuals in
is responsible for launching attacks. The male wolf is the herd. When a hierarchy is established, wolves move
also the strongest wolf in the pack and initiates all pack’s around in relation to the victim they are hunting. Since 𝛼
actions. It is selected from the herd by victories in direct is always the leader in the hunt, followed by 𝛽 and 𝛿, the
battles with other wolves. An important role in the pack position of the other wolves depends on the movements
is also played by the second strongest wolf. With the of the strongest individuals (because they are closest to
male wolf, they complement each other. This individual the victim).
Assuming being in the 𝑗 𝑡ℎ time step, the location of Algorithm 2: Grey Wolf Optimizer.
the wolf 𝑥 in the next moment can be defined by equation Input: wolves number 𝑤, iterations number
3: 𝑗𝑚𝑎𝑥 , range of the arguments, range 𝑎
𝑋𝐴 + 𝑋𝐵 + 𝑋𝐷
𝑥𝑗+1 = , (3) Output: individual 𝛼
3
1 Generate initial pack with 𝑤 individuals;
where 2 𝑗 := 0;
𝑋𝐴 = 𝑥𝛼 − 𝐴𝑗 · 𝐷𝛼 , (4) 3 for 𝑗 < 𝑗𝑚𝑎𝑥 do

𝑋𝐵 = 𝑥𝛽 − 𝐴𝑗 · 𝐷𝛽 , (5) 4 Calculate 𝑎𝑗 according to the equation (8);
5 Calculate 𝐴𝑗 according to the equation (7);
𝑗
𝑋𝐷 = 𝑥𝛿 − 𝐴 · 𝐷𝛿 . (6)
6 Calculate 𝐶 𝑗 according to the equation (12);
𝑋𝐴 , 𝑋𝐵 , 𝑋𝐷 are the coefficients depending on the posi- 7 Find the best individuals 𝛼, 𝛽, 𝛿;
tions of the best wolves at the moment (denoted as 𝑥𝛼 , 8 ℎ := 0;
𝑥𝛽 , 𝑥𝛿 ), 9 for wolves do
𝐴𝑗 is a parameter that updates in each iteration 𝑗: 10 Calculate distances from the best
individuals 𝐷𝛼 , 𝐷𝛽 , 𝐷𝛿 according to
𝐴𝑗 = 2 · 𝑎𝑗 · 𝑝, (7) the equations (9), (10), (11);
11 Calculate coefficients 𝑋𝐴 , 𝑋𝐵 , 𝑋𝐷
and 𝑝 is a random value from the range [0, 1]. The
according to the equations (4), (5), (6);
value 𝑎𝑗 depends on the interval [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ] set in
12 Update wolf’s location according to the
the beginning and is calculated for each moment 𝑗 =
equation (3);
0, 1, . . . , 𝑗𝑚𝑎𝑥 according to the formula:
13 ℎ + +;
𝑎𝑚𝑎𝑥 − 𝑎𝑚𝑖𝑛 14 end
𝑎𝑗 = 𝑎𝑚𝑎𝑥 − · 𝑗. (8) 𝑗 + +;
𝑗𝑚𝑎𝑥 15
16 end
Usually, it is assumed that the 𝑎𝑗 value decreases from 2 17 Find the best individuals 𝛼, 𝛽, 𝛿;
to 0 [15]. Then 𝑎𝑚𝑖𝑛 = 0 and 𝑎𝑚𝑎𝑥 = 2. The values 𝐷𝛼 , 18 return 𝛼;
𝐷𝛽 , 𝐷𝛿 evaluate the distance from the given individual
𝑥 to the best adapted wolves:

𝐷𝛼 = 𝐶 𝑗 · 𝑥𝛼 − 𝑥, (9) where each 𝑖𝑡ℎ unit (wolf) stores information about the
𝑖𝑡ℎ 𝑛-dimensional record from the specified database.
𝑗
𝐷𝛽 = 𝐶 · 𝑥𝛽 − 𝑥, (10) Naturally, the pack is the same size of 𝑚 as the consid-
ered dataset. Next the strongest individuals 𝛼, 𝛽, 𝛿 have
𝑗
𝐷𝛿 = 𝐶 · 𝑥𝛿 − 𝑥, (11)
to be selected. Due to the necessity of hierarchy estab-
where lishment, the values of the function 𝑓 (·) for individual
𝐶 𝑗 = 2 · 𝑟, (12) wolves have to be compared. The function 𝑓 (·) corre-
sponds to the Euclidean distance function (1) or distance
which is recalculated for each iteration 𝑗 and 𝑟, like 𝑝, is
for Manhattan metric (2). The strongest units are the
a random value in the range [0, 1].
wolves closest to the victim at the moment. When a hi-
The above operations are performed a certain number
erarchy in the herd is established, the positions of the
of times (𝑗𝑚𝑎𝑥 ) to finally select the best-adapted wolf (𝛼)
wolves are updated. Being in the 𝑗 𝑡ℎ time step, the loca-
that is closest to the victim, i.e. the wanted solution. The
tion of appropriate wolf in the next moment is described
scheme of the algorithm is presented in the pseudocode
by the formula (3), which uses coefficients 𝑋𝐴 , 𝑋𝐵 , 𝑋𝐷
2.
represented by the equations (4), (5), (6). Each of these
coefficients is a difference between location from one of
2.3. Hybridization the best wolves (𝛼, 𝛽 or 𝛿) and product of the parameter
𝐴𝑗 defined by the formula (7) and corresponding 𝐷 co-
Let 𝑦 = [𝑦1 , ..., 𝑦𝑛 ] be an 𝑛−dimensional vector of un-
efficient (𝐷𝛼 , 𝐷𝛽 or 𝐷𝛿 described by the equations (9),
known class as in the 𝑘 Nearest Neighbors classifier
(10), (11)), respectively. The parameter 𝐴𝑗 is modified
model. Then 𝑦 is identified with the victim that wolves
in each iteration due to the variability of the parame-
hunt, while the pack consists of records from the database
ter 𝑎𝑗 (described by the equation (8)). Additionally, the
with 𝑛 attributes, the values that are stored by wolves.
value of the 𝐷 coefficient is influenced by the value of
Therefore, the population consists of units of the follow-
the changing 𝐶 𝑗 parameter defined by the formula (12).
ing form:
The above steps are repeated 𝑗𝑚𝑎𝑥 times. Then, the
𝑥𝑖 = [𝑥𝑖1 , ..., 𝑥𝑖𝑛 ], (13)
best wolf from the pack is selected (𝛼). It is the record 3. Experiments
that after 𝑗𝑚𝑎𝑥 iterations is at the shortest distance from
the considered vector 𝑦 out of the entire pack. Searching The hybrid method using Grey Wolf Optimizer and 𝑘-nn
for such an individual takes place in total 𝑘 times in was tested in a process of matching the class to the object
order to identify 𝑘 closest neighbors of the 𝑘 Nearest from a database. The database of iris flowers was used
Neighbors algorithm. Finally, according to the concept for experiments. The author of this dataset, the British
of the 𝑘 Nearest Neighbors algorithm, the appropriate biologist and statistician Ronald Fisher [19], shared data
class for the 𝑦 vector is selected based on the occurrence in 1936.
of individual classes among neighboring records. The iris flowers dataset consists of 150 records describing
The pseudocode 3 shows the structure of solving the the appearance of these plants. Every record stores infor-
classification problem using the Grey Wolf Optimizer. mation about 5 attributes: the length of the plot of the
flower cup, the width of the plot, the length of the petal
and its width, and also the name of the species. Thus
Algorithm 3: Combination of 𝑘-nn with Grey
the first four characteristics are expressed by numerical
Wolf Optimizer.
value and the fifth one constitutes a class label presented
Input: dataset of 𝑚 records (𝑥𝑖 ), unclassified in a text form. Three species of iris are included in the
vector 𝑦, nearest neighbors number 𝑘, collection: Iris setosa, Iris virginica, Iris versicolor.
iterations number 𝑗𝑚𝑎𝑥 , range of 𝑎 In order to test the designed method, a program was
Output: 𝑦 class implemented. 20% of all records were checked whether
1 Generate initial pack consists 𝑚 records; the appropriate class was matched. For calculations, the
2 Create array 𝑐𝑙𝑎𝑠𝑠𝑒𝑠 of length 𝑘; program retrieved the first four attributes of each record
3 𝑖 := 0; omitting the class labels. However, labels were stored
4 for 𝑖 < 𝑘 do for later comparison to obtaining results with the actual
5 𝑗 := 0; state. As it was earlier described, every record was treated
6 for 𝑗 < 𝑗𝑚𝑎𝑥 do like a vector to use the created method. To determine
7 Calculate 𝑎𝑗 according to the formula effectiveness of the method, the following coefficient 𝑎𝑐𝑐
(8); was defined:
𝑛𝑐
8 Calculate 𝐴𝑗 according to the formula 𝑎𝑐𝑐 = · 100%, (14)
𝑛𝑡
(7);
9 Calculate 𝐶 𝑗 according to the formula where 𝑛𝑐 is the number of correct matches and 𝑛𝑡 is
(12); equal to the number of all tested records. The final result
10 Find the best individuals 𝛼, 𝛽, 𝛿; was rounded to two decimal places.
11 ℎ := 0; The program was tested for four variants of iterations
12 for wolves do number. A range of 𝑎 in all cases was assumed to [0; 2].
13 Calculate distances from the best For each option effectiveness of the method was checked
individuals 𝐷𝛼 , 𝐷𝛽 , 𝐷𝛿 according for Euclidean distance function 1 and also for distance in
to the equations (9), (10), (11); Manhattan metric 2 by launching the program five times
14 Calculate coefficients 𝑋𝐴 , 𝑋𝐵 , 𝑋𝐷 in both cases and calculating the arithmetic mean of the
according to the equations (4), (5), obtained results. These operations were performed for
(6); 15 different values of 𝑘 parameter: 𝑘 = 1, ..., 15.
15 Update wolf’s location according to The first v ariant o f i terations n umber w as 𝑗 𝑚𝑎𝑥 =
the equation (3); 100. For the Euclidean metric, the highest arithmetic
16 ℎ + +; means of 𝑎𝑐𝑐 value was obtained for 𝑘 = 2 and reached
17 end 71.33%. This value did not fall below 34.00%. In the case
18 𝑗 + +; of Manhattan distance, the arithmetic means of the 𝑎𝑐𝑐
coefficient assumed values between 26.00% and 44.00%.
19 end
Detailed results are placed in Table 1.
20 Find the best individuals 𝛼, 𝛽, 𝛿;
In the second test, the number of iterations was modi-
21 Assign to 𝑐𝑙𝑎𝑠𝑠𝑒𝑠[𝑖] the class of individual
fied to 25. The obtained results are presented in the table
𝛼;
2. It turned out that for Euclidean distance significant
22 𝑖 + +;
improvement of effectiveness was achieved regardless of
23 end
the value of 𝑘. In this case, the arithmetic mean of 𝑎𝑐𝑐
24 Choose the most frequently repeated class in
was greater than or equal to 90.00%. It means that nearly
the array 𝑐𝑙𝑎𝑠𝑠𝑒𝑠;
all of the tested records were correctly classified. For the
25 return 𝑦 class;
Manhattan distance, the results were similar to the first
Table 1 Table 3
Arithmetic mean of the 𝑎𝑐𝑐 coefficient Arithmetic mean of the 𝑎𝑐𝑐 coefficient
for 𝑗𝑚𝑎𝑥 = 100, 𝑎 ∈ [0; 2]. for 𝑗𝑚𝑎𝑥 = 50, 𝑎 ∈ [0; 2].
k Euclidean distance Manhattan distance k Euclidean distance Manhattan distance
1 55.33% 31.33% 1 65.33% 35.33%
2 71.33% 33.33% 2 61.33% 34.67%
3 62.00% 43.33% 3 70.67% 30.67%
4 44.00% 34.67% 4 72.00% 38.67%
5 34.00% 37.33% 5 59.33% 36.00%
6 41.33% 28.67% 6 71.33% 36.00%
7 64.67% 38.67% 7 62.66% 34.00%
8 70.67% 28.67% 8 67.33% 36.66%
9 47.33% 34.66% 9 70.00% 31.33%
10 53.33% 35.33% 10 64.00% 30.67%
11 65.33% 27.33% 11 66.67% 34.67%
12 53.33% 41.33% 12 67.33% 36.00%
13 58.00% 44.00% 13 62.67% 36.00%
14 58.67% 30.67% 14 72.00% 34.00%
15 48.00% 26.00% 15 60.00% 30.67%

Table 2 Table 4
Arithmetic mean of the 𝑎𝑐𝑐 coefficient Arithmetic mean of the 𝑎𝑐𝑐 coefficient
for 𝑗𝑚𝑎𝑥 = 25, 𝑎 ∈ [0; 2]. for 𝑗𝑚𝑎𝑥 = 1000, 𝑎 ∈ [0; 2].
k Euclidean distance Manhattan distance k Euclidean distance Manhattan distance
1 96.67% 37.33% 1 36.67% 38.67%
2 93.33% 42.67% 2 32.00% 37.33%
3 91.33% 37.33% 3 38.00% 30.67%
4 94.67% 33.33% 4 42.67% 35.33%
5 92.67% 38.66% 5 42.00% 32.00%
6 91.33% 32.67% 6 47.33% 33.33%
7 94.67% 36.67% 7 30.67% 39.33%
8 90.00% 27.33% 8 34.00% 26.67%
9 92.00% 42.67% 9 57.33% 29.34%
10 92.00% 37.33% 10 44.67% 31.33%
11 94.67% 37.33% 11 71.33% 30.00%
12 94.67% 37.33% 12 61.33% 33.33%
13 92.00% 40.00% 13 60.67% 33.33%
14 97.33% 35.33% 14 56.00% 34.00%
15 93.33% 38.00% 15 40.66% 33.33%

test. The arithmetic mean between 27.33% and 42.67% of iterations significantly increased. Table 4 presents
was obtained. the arithmetic mean of the 𝑎𝑐𝑐 coefficient in this vari-
Another test was performed for 50 iterations. As in the ant. Again, the distance function for Manhattan brought
previous step, results for the Manhattan metric did not results similar to other tests. None of the values of param-
improve noticeably. The highest arithmetic mean 38.67% eter 𝑘 causes results markedly better than others. If the
can be observed for 𝑘 = 4. 30.67% is the lowest obtained Euclidean metric is applied, results depend on 𝑘 value.
value. For Euclidean metric results are not as good as For example, when 𝑘 = 7, the arithmetic mean of 𝑎𝑐𝑐
in the previous test but there were much more correctly is equal to 30.67% and it is the lowest result. Simultane-
classified records than for 100 iterations, for example ously, for 𝑘 = 11 it is 71.33%. Thus the range of these
when 𝑘 = 4 it was 72.00% in this variant (𝑗𝑚𝑎𝑥 = 50) and results is quite big – almost 40%.
44.00% when 𝑗𝑚𝑎𝑥 = 100. Other values are presented in
table 3.
The last test assumed 𝑗𝑚𝑎𝑥 = 1000 so the number
4. Conclusions heuristic with interior-point for nonlinear sitr
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