An Approach to Prediction Using Networked Multimedia ^†

Abstract

One of the tasks of statistical analysis is related to the development of forecasts with different horizons. The results of modeling the development trend can also be used for prognostic purposes. At the same time, the assumption is made that during the forecast period, the phenomenon under study will exhibit the same patterns of development that it exhibited during the base period. Network multimedia is a unifying link in the parallel development of multimedia and communication technologies. The integrated interaction of technological solutions in the field of multimedia and computer networks is a condition for achieving a greater final application effect in the presentation of information. Experimental studies of modern network multimedia in operational conditions are important for revealing bottlenecks in their functioning. On this basis, recommendations can be made to improve performance indicators, such as performance, reliability, mode of service, etc. This publication is devoted to the experimental study of the trend and the possibility of predicting network multimedia with time series. The implemented algorithm for automated trend determination examines pre-set different trends–linear, quadratic, cubic, hyperbolic, fractional-rational, logarithmic, exponential, exponential, combined–and chooses the most effective of them.

1. Introduction

Forecasting is a process in which the development of a given scalar or complex quantity over time can be fully or partially predicted. In the modern world, driven by economics, the predictability of various economic parameters provides a sense of security, the ability to reduce losses and better plan activities and necessary resources, as well as plan the overall development of a given system.

Dynamic time series in telecommunications, energy and business can be obtained by measuring characteristics of technical, natural, social, economic and other systems [1,2]. With modern trends, time series analysis tasks to detect patterns and trends in various phenomena and processes are becoming increasingly important.

The objectives of time series analysis are as follows [3,4]:

• Description—display of summarized statistics and graphical representation.
• Analysis and interpretation—finding a model that describes the time dependence in the data.
• Prediction—given a sample from the series, the next value or the next few values are predicted.

Network multimedia is a unifying element in the parallel development of multimedia and communication technologies. The integrated interaction of technological solutions in the field of multimedia and computer networks is a condition for achieving a greater final applied effect in the presentation of information [5].

Experimental studies of modern network multimedia in operational conditions are important for revealing the bottlenecks in their functioning. On this basis, recommendations can be made for improving operational indicators, such as performance, reliability, serviceability, etc.

This article is dedicated to the experimental study of the trend and forecasting capability of network multimedia with time series.

2. Methodology

Factors influencing network multimedia change over time. These changes can be described as a function of the respective factor from the moment of its measurement.

y = y(t).

(1)

The following methods are used to study the sustainable development trend:

• Moving the average method—Averages are calculated for large adjacent data for a certain number of members of the series and the averaging process is successively moved.
• Method of average growth and average rate—This is applied when there are sufficient grounds to assert that development has the character of arithmetic or geometric progression. The data can be aggregated in advance.
• Graphical method—This is used for illustration and analysis. At the discretion of the researcher, a line is drawn closest to the empirical one and its values are considered on the y-axis scale, which are taken as smoothed (equalized) values for the time series.
• Analytical method (least squares method)—This is the most advanced method for aligning dynamic series. It includes the following:

(a)

Choosing a pattern expressing the trend.

(b)

Determination of the symbols of the periods.

$${\mathbf{t}}_{\mathrm{i}} (\mathrm{i}=1,2,\dots ,\mathrm{n})$$

(2)

(c)

Determination of the mathematical expression of the function by which the main trend will be described. Depending on the specific case, the following basic models can be applied.

(d)

Determination of the parameters of the equations, the model of which is chosen in specific cases.

(e)

Determination of the values of the trend model, i.e., the estimates, by substituting the symbols in the equation expressing the main trend ${\widehat{y}}_i{t}_i$

(f)

Verification of the correctness of the calculation procedures.

3. Linear Trend Pattern

The trend of different quantities often follows a linear model, and the forecast function is in the following form [6]:

$$\widehat{y}={\beta }_1+{\beta }_{2}t.$$

(3)

Its coefficients are determined by the linear system ${\beta }_1{\beta }_2$

$$\left|\begin{array}{c}{\mathit{\beta }}_{1}n+{\mathit{\beta }}_2\sum _{\mathit{i}=1}^{\mathit{n}}{\mathit{t}}_{\mathit{i}}=\sum _{\mathit{i}=1}^{\mathit{n}}{\mathit{y}}_{\mathit{i}}\\ {\mathit{\beta }}_1\sum _{\mathit{i}=1}^{\mathit{n}}{\mathit{t}}_{\mathit{i}}+{\mathit{\beta }}_2\sum _{\mathit{i}=1}^{\mathit{n}}{\mathit{t}}_{\mathit{i}}^2=\sum _{\mathit{i}=1}^{\mathit{n}}{\mathit{t}}_{\mathit{i}}{\mathit{y}}_{\mathit{i}}\end{array}\right.,$$

(4)

in which n is the number of measurements.

According to the way in which the factor t and the parameters are involved, in the regression model, the models are linear and nonlinear with respect to the ${\mathit{\beta }}_1{\mathit{\beta }}_2{\mathit{\beta }}_{\mathit{i}}$ factor t and linear and nonlinear with respect to the parameters. For example, the model (3) is linear with respect to both the factor t and the parameters and ${\mathit{\beta }}_1{\mathit{\beta }}_2$.

4. Nonlinear Trend Patterns: Converting Nonlinear Models to Linear Models

For various specific quantities that change over time, it is possible that other nonlinear models [7] such as quadratic, cubic, hyperbolic, fractional–rational, logarithmic, exponential, exponential or combined (

Table 1. Trend patterns used.

Trend Type	Mathematical Model
Linear	$\widehat{y}={\beta }_1+{\beta }_{2}t$
Quadratic	$\widehat{y}={\beta }_1+{\beta }_{2}t+{\beta }_3{t}^2$
Cubic	$\widehat{y}={\beta }_1+{\beta }_{2}t+{\beta }_3{t}^2+{\beta }_4{t}^3$
Hyperbolic	$\widehat{y}={\beta }_1+{\displaystyle \frac{{\beta }_2}{t}}, t\ne 0$
Fractional–rational	$\widehat{y}={\displaystyle \frac{t}{{\beta }_1+{\beta }_{2}t}}, t\ne -{\displaystyle \frac{{\beta }_1}{{\beta }_2}}$
Logarithmic	$\widehat{y}={\beta }_1+{\beta }_{2}ln\left(t\right)$
Exponential	$\widehat{y}={\beta }_1{e}^{{\beta }_{2}t}$
Power model	$\widehat{y}={\beta }_1{t}^{{\beta }_2}$
Combined	$\widehat{y}={\displaystyle \frac{t}{{\beta }_1+{\beta }_2{e}^{-t}}}, t\ne ln{\displaystyle \frac{{\beta }_1}{{\beta }_2}}$

) are closer to reality (Table 1). Although some of them seem unlikely as a predictive function for the considered quantities, we use all of them in the prediction algorithm, as it can be applied in other situations.

For quadratic regression, an approximating function is sought as a polynomial of the second degree. The following coefficients are found by the linear system:

$$\widehat{\mathrm{y}}={\mathsf{\beta }}_1+{\mathsf{\beta }}_2\mathrm{t}+{\mathsf{\beta }}_3{\mathrm{t}}^2{\mathsf{\beta }}_1{\mathsf{\beta }}_2{\mathsf{\beta }}_3;$$

(5)

$$\left|\begin{array}{c}{\mathit{\beta }}_{1}n+{\mathit{\beta }}_2{\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}+{\mathit{\beta }}_3{\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}^2={\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{y}}_{\mathit{i}}\\ {\mathit{\beta }}_1{\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}+{\mathit{\beta }}_2{\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}^2+{\mathit{\beta }}_3{\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}^3={\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}{\mathit{y}}_{\mathit{i}}\\ {\mathit{\beta }}_1{\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}^2+{\mathit{\beta }}_2{\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}^3+{\mathit{\beta }}_3{\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}^4={\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{t}}_{\mathit{i}}^2{\mathit{y}}_{\mathit{i}}\end{array}\right.$$

(6)

The basic forecasting algorithm is obtained through a time series, in which models are created on input–output data and which, ultimately, must be used to predict additional already known data and evaluated according to errors.

The idea we apply to building nonlinear trend patterns is to reduce them through substitution (substitution) to a linear one [8]. Then, the linear model is calculated using the least squares method and the reverse substitution is performed until the originally sought nonlinear model is constructed.

Nonlinear models are divided into intrinsically linear and intrinsically nonlinear models. The first group (intrinsically linear) includes models that become linear with respect to the parameters after appropriate transformations of t$\widehat{y}$, where

$$\widehat{y}={\beta }_1{e}^{{\beta }_{2}t},$$

(7)

which, after logarithm of both sides, takes the following form:

$$ln\left(\widehat{y}\right)=ln\left({ \beta }_1{e}^{{\beta }_{2}t}\right)=ln\left({\beta }_1\right){+\beta }_{2}t.$$

(8)

The following expression can be obtained:

$$u=ln\left(\widehat{y}\right), v=t,$$

(9)

where

$$u=a+bt, a=ln\left({\beta }_1\right), b={\beta }_2.$$

(10)

After compiling this relationship, we can find the optimal linear model (10) by the method of the least squares with the smallest error to the values of the sample, i.e., we determine the coefficients and b. After performing the reverse substitution, in our case, it follows from (10) that

$${\beta }_1{=e}^a, {\beta }_2=b,$$

(11)

which define the exponential trend pattern describing the sample data, as shown below:

$$\widehat{y}={\beta }_1{e}^{{\beta }_{2}t}.$$

(12)

5. Results

The experimental studies were conducted at the University of Ruse with the following hardware and software shown in

Table 2. Hardware and software used.

Server	Meet1	Meet
CPU	2x AMD EPYC 7402 24 Core Processor	4x Intel(R) Xeon(R) CPU E7-4820 @ 2.00 GHz Total Cores 8
RAM	64 GB Registered ECC DDR4	128 GB DDR3 ECC Registered memory
Ethernet	2 1 GbE LAN via Intel® i350-AM21	Intel® 82576 Dual-Port Gigabit Ethernet Controller
Video	ASPEED AST2500 BMC graphics	Matrox G200 eW 16 MB DDR2
CPU	2× AMD EPYC 7402 24 Core Processor	4× Intel(R) Xeon(R) CPU E7- 4820 @ 2.00 GHz Total Cores 8
RAM	64 GB Registered ECC DDR4	128 GB DDR3 ECC Registered memory
Ethernet	2 1 GbE LAN via Intel® i350-AM21	Intel® 82576 Dual-Port Gigabit Ethernet Controller
Video	ASPEED AST2500 BMC graphics	Matrox G200 eW 16 MB DDR2
Storage	SATA3 (6 Gbps) (2× NVMe on PCi)	6 Gb/s SAS/SATA ports (1× Samsung SSD 850)
	Ubuntu 18.04.6 LTS	Ubuntu 18.04.6 LTS
Software	BigBlueButton Virtual Classroom (v.2.4.4)	BigBlueButton Virtual Classroom (v.2.4.4)

. The BigBlueButton platform is installed on both servers and is used by engineering faculties at the following addresses: https://meet.uni-ruse.bg/b/ (accessed on 21 September 2024) and https://meet1.uni-ruse.bg/b/ (accessed on 21 September 2024).

The period of the study includes the last 11 consecutive weeks of the winter semester in the 2024–2025 academic year (

Table 3. Period of the study.

Measurement Moment (t)	Week	Dates in 2024
1	41	07.10	13.10
2	42	14.10	20.10
3	43	21.10	27.10
4	44	28.10	03.11
5	45	04.11	10.11
6	46	11.11	17.11
7	47	18.11	24.11
8	48	25.11	01.12
9	49	02.12	08.12
10	50	09.12	15.12
11	51	16.12	22.12

). We have the average data for “Active Audio Streams” (AAP). For convenience, we designate the weeks as moments of measuring the time factor with t1 = 1, t2 = 2, etc.

We use the first 10 measurements, i.e., the moments of t = 1, … 10, to create a linear model and make a prediction for the 11th element (11). The linear model created during the implementation has the form $\widehat{y,}$:

$$\widehat{\mathrm{y}}=56.2+42.75\mathrm{t},$$

(13)

and for t = 11, an estimated value is obtained, where $\widehat{y}=526.50$.

The results for the trend type are presented in

Table 4. Predict the change of active audio streams.

Trend Type	Sample Period/Forecast Period	Estimated Value	Real Value	Absolute Error
Linen	07.10–15.12/ 16.12–22.12	526.50	522	4.50
Quadratic	07.10–15.12/ 16.12–22.12	530.36	522	8.36
Cubical	07.10–15.12/ 16.12–22.12	517.48	522	4.52
Hyperbolic	07.10–15.12/ 16.12–22.12	385.81	522	136.18
Fractional–rational	07.10–15.12/ 16.12–22.12	365.22	522	156.78
Logarithmic	07.10–15.12/ 16.12–22.12	456.06	522	65.94
Exponential	07.10–15.12/ 16.12–22.12	1,747,574.94	522	1,747,052.94
Steepened	07.10–15.12/ 16.12–22.12	499.11	522	22.89
Combined	07.10–15.12/ 16.12–22.12	310.24	522	211.76

. The real value of AAP in week 11 is 522 and the error is about 0.86% or an absolute error MAE = |522 − 526.50| = 4.50 (Figure 1).

They show that in short-term forecasts, the general trend finds a place in the methods for effective forecasting. The errors are quite acceptable for the first three models. In this case, the linear trend pattern followed by the cubic one turned out to be the most suitable. The most erroneous forecast is obtained with an exponential pattern. It is noticed that considering the values of factors as a time series is an effective means of predicting them in the short term.

Indeed, in some cases, long-term forecasts can be inaccurate and impractical, as actually noted in the literature. Therefore, to improve the accuracy of forecasting in actual modeling, we chose prediction models that work well with a relatively small sample size in the context of case trend research.

6. Conclusions

Forecasting through time series has its advantages in short-term forecasts of quantities that have clearly distinguishable trends.

The implemented algorithm for automated trend determination examines pre-set different trends—linear, quadratic, cubic, hyperbolic, fractional–rational, logarithmic, exponential, exponential, combined—and chooses the most effective of them.

When predicting the values for a longer period, an increase in errors is observed as they move away from the boundaries of the values in the sample.