An Algorithm for Assessment of Time Series Data Related to the Materials Used for Packaging in the Market ^†

Abstract

This article presents an algorithm for the assessment of time series data related to the materials used for packaging in the market in Bulgaria for the period 2010–2023. The considered elements include the quantities of the following types of materials: paper/cardboard, plastic, wood, metal, glass, and others. They are extracted from the built relational database and subsequently processed and summarized. In this regard, relevant criteria (rules) are formed and applied and certain variables are calculated. In addition, analysis of variance (Anova) and Tukey’s test are also used for these data. The results show that one of the materials (paper/cardboard) in 2010–2018 has relatively higher quantities compared to the rest materials. A similar situation occurs for the element plastic, in 2019–2023. The calculated relative shares of the respective quantities of packaging materials show that more than half of them in the market are made of paper/cardboard and plastic. Some dynamic changes are observed in the quantities for the materials metal and wood. The indicator values for the other materials are significantly lower than the rest. The developed algorithm can be applied to study other time series data in fields such as ecology, finance, etc.

1. Introduction

In the modern world, users from different fields in their respective organizations use large amounts of data [1,2] in their daily practice. These data are processed in order to be able to present relevant forecasts or conclusions [3,4] as well as to make adequate decisions [5] about the considered problem. According to Wang et al. [6] (p. 14322), “Time series data is common in data sets and has become one of the focuses of current research.” [6]. It is very important when examining relevant data to choose appropriate methods [7,8] and approaches [9]. The study of Esling and Agon [10] (p. 1) notes that “The purpose of time series data mining is to try to extract all meaningful knowledge from the shape of data.” [10]. Some of these problems are a subject of consideration in the current work.

This study examines time series data on the materials used for the production of packaging in the market in Bulgaria. The information related to these objects is presented on the website of the Bulgarian National Statistical Institute [11]. It is extracted from the indicated website and organized in a created relational database. At the same time, data concerning the produced output (BGN) of environmental goods and services in relevant economic activities are distributed and stored in separate tables of the database. Information about these listed objects is also extracted from the website of the National Statistical Institute [11]. The created relational database contains the following eight tables:

• Total _elements (Id, year, value (tons), id_name);
• Country (Id_ name, name);
• Materials (Id_material, material, Id_name);
• Values_tons (Id, year, value (tons), id_material);
• Economic activities (Id_activity, activity, Id_name);
• Production (Id, year, production (BGN), Id_ activity);
• Gross value added (Id, year, GVA (BGN), Id_ activity);
• Personnel (Id, year, number, Id_ activity).

The Country table contains the name of the country. The next two tables are Materials and Values_tons. The first of them stores the names of the packaging materials used, while the second presents the quantity of the relevant material for each year of the time period. The Total _elements table contains the total quantity of the materials used for the examined years. Information concerning the produced output (BGN) of environmental goods and services is stored in the other four tables. This leads to the expansion of the database. Two of the tables—Economic activities and Personnel—include information on economic activities and the number of employees in these activities for the considered years. The data on produced output (BGN) and gross value added (BGN) for the time period are stored in the tables Production and Gross value added. Some of the information presented in the database is studied in this work. The relationships created between the considered tables are of one-to-many type, as shown in Figure 1. The database is updated every year as new elements are entered into one of the fields of the listed tables.

The aim of this work is to present an algorithm for the assessment of time series data related to the materials used for packaging in the market in Bulgaria for the period 2010–2023. The investigated objects include the quantities of the following six types of materials: paper/cardboard, plastic, wood, metal, glass, and others. They are searched for in the built relational database. The extracted groups of elements are assessed and summarized. Through this connection, relevant criteria (rules) are formed and applied, and certain variables are calculated. The results of data processing for the examined types of materials are visualized mainly in tabular form.

In addition, analysis of variance (Anova) and Tukey’s test are also applied to the surveyed data regarding the quantities of the listed elements.

2. Materials and Methods

Algorithm description: The input data for the developed algorithm are stored in the built relational database, as can be seen from the scheme presented in Figure 2. The considered time period in this work covers 14 years, from 2010 to 2023.

The information related to the examined elements includes the following six types of packaging materials in the market in Bulgaria:

• Paper/cardboard;
• Plastic;
• Metal;
• Wood;
• Glass;
• Other.

The studied indicators for each of the listed objects include the relevant quantities as well as the time segment. Depending on the selected type of material, the search is carried out in four database tables. They are the following:

• Materials (Id_material, material, Id_name);
• Values_tons (Id, year, value (tons), id_material);
• Country (Id_ name, name);
• Total _elements (Id, year, value (tons), id_name).

Practically, the individual sets of elements regarding the surveyed types of packaging materials are searched for in fields located in several related tables (Figure 3). Each of these fields contains the corresponding values of the considered indicator. As a result of performing these operations, the extracted groups of elements for the six types of materials are stored in a separate file.

The obtained data are assessed and then summarized. In this regard, the relevant criteria (rules) are formed and applied, and certain variables are calculated. For this purpose, it is necessary to find the following:

• The quantity of the type of material used for packaging production, which is the highest (${\mathrm{m}}_{{\mathrm{h}}_{\mathrm{j}}\mathrm{i}}$, $1<=\mathrm{j}<=6,$ $1<=\mathrm{i}<=14$) and the smallest (${\mathrm{m}}_{{\mathrm{l}}_{\mathrm{j}}\mathrm{i}}$) for each year of the time period;
• A subset of consecutive years of the studied period where the values of the quantities of a given packaging material (${\mathrm{m}}_{\mathrm{j}\mathrm{i}}$, ${\mathrm{m}}_{\mathrm{j}\mathrm{i}+1}$,…,${\mathrm{m}}_{\mathrm{j}\mathrm{i}+\mathrm{p}-1}$; $\mathrm{p}>=2$, $\mathrm{i}+\mathrm{p}<=15$) are higher than those for the rest of the materials. Here, the number of elements (p) included in this subset is also found;
• A subset of consecutive years of the considered time interval in which the values of the quantities of a given packaging material (${\mathrm{m}}_{\mathrm{j}\mathrm{i}}$, ${\mathrm{m}}_{\mathrm{j}\mathrm{i}+1}$,…,${\mathrm{m}}_{\mathrm{j}\mathrm{i}+\mathrm{q}-1}$; $\mathrm{q}>=2$, $\mathrm{i}+\mathrm{q}<=15$) are significantly smaller than the others;
• The relative share of a given quantity of packaging material used in the market to the total quantity of the studied packaging materials used for a given year:

  $$k_{\mathrm{j}\mathrm{i}}={\displaystyle \frac{{\mathrm{m}}_{\mathrm{j}\mathrm{i}}}{{\displaystyle \sum _{\mathrm{b}=1}^6{\mathrm{m}}_{\mathrm{b}\mathrm{i}}}}}·100,$$

  (1)

  where $\mathrm{j}\in \overline{1;6}$, $\mathrm{i}\in \overline{1;14}$;
• The percentage change in a given quantity of material used for packaging in the market (${\mathrm{f}}_{\mathrm{j}\mathrm{i}+1}$) for the current year compared to the previous year:

  $${\mathrm{f}}_{\mathrm{j}\mathrm{i}+1}=\left({\displaystyle \frac{{\mathrm{m}}_{\mathrm{j}\mathrm{i}+1}}{{\mathrm{m}}_{\mathrm{j}\mathrm{i}}}}·100\right)-100,$$

  (2)

  where $\mathrm{j}\in \overline{1;6}$, $\mathrm{i}\in \overline{1;14}$.

The results of the processing of the data related to the packaging materials used are mainly visualized in tabular form. They could be used in future periods when it is necessary to make the relevant comparisons and analysis [12,13] regarding the studied elements. In addition, it should be pointed out that analysis of variance (Anova) [14,15] and Tukey’s test [16] are also applied to the investigated data in this work.

The presented algorithm can be applied to study other time series data in fields such as ecology, finance, etc.

3. Results and Discussion

The surveyed subsets of objects concerning the materials used for packaging in the market in Bulgaria are extracted from the presented database. In this regard, the search for the relevant groups of elements is carried out in two sequences of related tables—Country, Materials, and Values_tons (Figure 3), as well as Country and Total_elements.

The results of the data processing show that one of the packaging materials used (in this case paper/cardboard) in the period 2010–2018 has a relatively higher quantity than the other five types of materials. Here, the formed subinterval includes nine consecutive years. In the first five, a variation (decrease or increase) in the quantity of the studied material (paper/cardboard) is observed (Figure 4). The situation is radically different in the time period 2015–2018. The change in this indicator is significantly smoother. In this case, the calculated increase in the quantity of the considered material is about 1.2 times. The quantity of another packaging material (plastic) are comparatively higher than those for the remaining materials in the next subinterval from 2019 to 2023 of five consecutive years. It should be noted here that a reduction of about 18.06% in the values of the indicated material is calculated for the last three years of the time segment. Invariably, over the whole 14-year period, the lowest values of the quantities are established for other packaging materials, as can be seen from Figure 4. By comparing the quantity of the type of material used for packaging production during the studied time interval, it can be seen that the values for plastic in 2020 are the highest (Figure 4).

The results of the calculations show that the share ($k_{\mathrm{j}\mathrm{i}}$) of one of the materials (paper/cardboard) to the total quantity of the considered materials changes in range from 43.19% to 25.03% for the studied years of the time segment. High values of the variable ($k_{\mathrm{j}\mathrm{i}}$) are obtained for another of the surveyed elements (plastic). The indicated variable reaches values between 25.36 and 32.19% (

Table 1. Results concerning calculated shares of the materials used during investigated years.

Year	Plastic	Paper/ Cardboard	Metal	Wood	Glass	Other
2010	25.52%	43.19%	4.90%	5.83%	19.91%	0.64%
2011	30.18%	35.05%	4.26%	6.82%	22.05%	1.64%
2012	29.23%	37.19%	4.44%	6.12%	21.45%	1.57%
2013	27.58%	38.36%	4.54%	7.06%	21.15%	1.31%
2014	26.96%	34.22%	4.22%	12.87%	20.66%	1.06%
2015	25.36%	34.55%	6.41%	11.62%	20.89%	1.17%
2016	25.70%	35.20%	7.57%	12.20%	18.38%	0.94%
2017	26.47%	33.81%	7.41%	13.15%	18.43%	0.73%
2018	26.40%	33.94%	7.56%	13.07%	18.28%	0.75%
2019	29.38%	25.03%	5.63%	14.36%	22.37%	3.22%
2020	32.19%	31.30%	4.12%	14.74%	15.71%	1.94%
2021	29.48%	26.52%	5.51%	17.86%	18.30%	2.33%
2022	28.36%	25.47%	5.60%	18.29%	19.85%	2.43%
2023	27.91%	27.36%	5.83%	16.42%	20.41%	2.07%

Source: Own calculations on the basis of data from National Statistical Institute [11].

). The calculated shares of the quantities for two of the examined materials (glass and wood) remained relatively low in comparison with those for the above considered materials for each year of the time period. Their values vary between 15.71 and 22.37% and 5.83 and 18.29%, respectively. The relative shares for the remaining two materials (metal and others) are significantly lower. Their values vary from 7.57% to 4.12% for the first of the listed materials and from 0.64% to 3.22% for the second, respectively. Moreover, the relative shares of the respective quantities of packaging materials show that more than half of them in the market are made from paper/cardboard and plastic.

This work calculates and analyzes the percentage change (${\mathrm{f}}_{\mathrm{j}\mathrm{i}+1}$) in a given quantity of material used for packaging in the market for the current year compared to the previous year. The decline in the quantities of metal and glass is calculated for 3 years of the period. These years are 2011, 2019, and 2020 for the first of the listed elements and 2016, 2020, and 2023 for the second, respectively. The same process is observed for the elements plastic and wood in 4 non-consecutive years, and for glass and other materials in 5 non-consecutive years, as shown in

Table 2. Presentation of the calculated percentage changes in the studied elements.

Year	Plastic	Paper/ Cardboard	Metal	Wood	Glass	Other
2011	15.84%	−20.51%	−14.80%	14.42%	8.46%	151.77%
2012	1.22%	10.88%	8.75%	−6.17%	1.65%	0.00%
2013	0.44%	9.81%	8.91%	22.88%	4.96%	−11.16%
2014	5.74%	−3.49%	0.70%	97.07%	5.72%	−12.53%
2015	−2.51%	4.68%	57.19%	−6.39%	4.81%	14.32%
2016	8.76%	9.28%	26.81%	12.69%	−5.60%	−13.90%
2017	10.82%	3.36%	5.34%	15.93%	7.87%	−16.12%
2018	9.50%	10.20%	12.00%	9.10%	8.90%	11.70%
2019	24.03%	−17.79%	−16.99%	22.52%	36.36%	381.96%
2020	6.23%	21.28%	−29.13%	−0.50%	−31.88%	−41.66%
2021	−12.59%	−19.15%	27.71%	15.63%	11.13%	14.57%
2022	−1.92	−2.10%	3.70%	4.42%	10.65%	6.35%
2023	−4.41	4.34%	1.15%	−12.77%	−0.12%	−17.32%

Source: Own calculations on the basis of data from National Statistical Institute [11].

.

The growth in the quantity of one of the materials (in this case glass) was relatively smoother in the time segments 2011–2015, 2017–2018, and 2021–2022. It should also be noted that significant growth was established in 2019. Compared to the year 2018, the change in the variable (${\mathrm{f}}_{\mathrm{j}\mathrm{i}+1}$) was about 36.36%. But, a sharp decline was calculated in 2020. It was about 31.88%. Some dynamic changes were observed in the quantity of the material metal. During 2015 compared to 2014, the quantity of the considered element significantly increased by about 57.19%. The situation was similar in 2016, where the increase was about 26.81%. The reverse process was observed in 2011, 2019, and 2020, where the decline was relatively large. It was about 14.80% for the first of the listed years, 16.99% for the second, and 29.13% for the third. The changes were also quite intense for another studied element (wood). It should be noted here that a significant increase (97.07%) was obtained in 2014. Large increases were observed in 2013 and 2019. The values of the variable (${\mathrm{f}}_{\mathrm{j}\mathrm{i}+1}$) were 22.88% for the first of the listed years and 22.52% for the second. But, a certain decline was calculated in 2023. Compared to the year 2022, the change of ${\mathrm{f}}_{\mathrm{j}\mathrm{i}+1}$ was about 12.77%. In addition, the growth in the quantities of other materials was quite fast in 2011. Compared to the year 2010, the variable ${\mathrm{f}}_{\mathrm{j}\mathrm{i}+1}$ increased by about 151.77% (Table 2). The situation was similar in 2019. Here, the studied indicator increased by over 4.8 times.

A statistical method such as analysis of variance (Anova) is also used in this study. The results of data processing show that there are statistically significant differences between the quantities of the used materials. Moreover, by applying Tukey’s test, the following four groups are formed (Table 3):

• Two types of materials (plastic and paper/cardboard) are included in one group. The values of their quantities are relatively higher. In this case, it can be noted that these materials are more often used for the production of packaging;
• One of the considered materials (glass) is presented in the next group;
• Another studied element (in this case wood) forms a separate group;
• The remaining two considered materials (metal and other) are presented in one group. In this case, the values of their quantities are the lowest.

Table 3. Results of the assessment of the studied data on the packaging materials.

Material	Std. Deviation	Assessment of the Packaging Materials Used (Tons)
Other	4674.615	7146.821 a
Metal	8175.697	24,605.856 a
Wood	27,488.32	56,075.283 b
Glass	16,334.321	85,716.406 c
Plastic	29,168.282	122,018.624 d
Paper/cardboard	15,930.922	139,751.774 d

Means with the same letter are not significantly different. Source: Own calculations on the basis of data from National Statistical Institute [11].

Table 3. Results of the assessment of the studied data on the packaging materials.

Material	Std. Deviation	Assessment of the Packaging Materials Used (Tons)
Other	4674.615	7146.821 a
Metal	8175.697	24,605.856 a
Wood	27,488.32	56,075.283 b
Glass	16,334.321	85,716.406 c
Plastic	29,168.282	122,018.624 d
Paper/cardboard	15,930.922	139,751.774 d

Means with the same letter are not significantly different. Source: Own calculations on the basis of data from National Statistical Institute [11].

4. Conclusions

This work presents an algorithm for the assessment of time series data related to the materials for packaging used in the market in Bulgaria from 2010 to 2023. The considered elements include the quantities of the following types of materials: paper/cardboard, plastic, wood, metal, glass, and others. The surveyed objects are searched for in the built database. The extracted sets of elements are assessed and summarized. Statistical methods such as analysis of variance (Anova) and Tukey’s test are also applied to the studied data.

The results of the calculations show that one of the packaging materials used (paper/cardboard) in the segment 2010–2018 has a relatively higher quantity than the other five types of materials. A similar situation occurs for plastic in the next subinterval of 2019–2023. In addition, the lowest values of the quantities are established for the other packaging materials for the whole period studied. The calculated relative shares of the respective quantities of packaging materials show that more than half of them in the market are made from paper/cardboard and plastic. Some dynamic changes are observed in the quantity for metal. During 2015 compared to 2014, the quantities of the considered element significantly increased by about 57.19%. The reverse process was observed in 2011, 2019, and 2020, where the decline was relatively large. The changes were also quite intense for another studied element—wood. A significant increase (97.07%) was calculated in 2014. But, a certain decline was observed in 2023. Compared to the year 2022, the change in the values was about 12.77%.

The results from the statistical evaluation of the quantities of packaging materials during the considered interval show four groups with statistically significant differences.

The presented algorithm can be applied to study other time series data in fields such as ecology, finance, etc.