Universidade de Aveiro Departamento deElectrónica, Telecomunicações e Informática, 2019
Ângelo Miguel
Raposo Almeida e
Sousa
PREVISÃO DE INDICADORES-CHAVE DE
DESEMPENHO DE REDES MÓVEIS
-APLICAÇÃO ÀS REDES MÓVEIS
CELULARES-FORECASTING KEY PERFORMANCE
INDICATOR OF MOBILE NETWORKS
-APPLICATION TO MOBILE CELLULAR
NETWORKS-Universidade de Aveiro Departamento deElectrónica, Telecomunicações e Informática, 2019
Ângelo Miguel
Raposo Almeida e
Sousa
PREVISAO DE INDICADORES-CHAVE DE
DESEMPENHO DE REDES MÓVEIS
-APLICAÇÃO ÀS REDES MÓVEIS
CELULARES-Dissertação apresentada à Universidade de Aveiro para cumprimento dos requesitos necessários à obtenção do grau de Mestre em Engenharia Elec-trónica e Telecomunicações 2019, realizada sob a orientação cientí ca de Professor Doutor Aníbal Manuel de Oliveira Duarte, Professor Catedrático do Departamento de Eletrónica, Telecomunicações e Informática da Univer-sidade de Aveiro e do Doutor João Afonso Bastos, Nokia, Aveiro
o júri / the jury
presidente / president Professora Doutora Susana Isabel Barreto de Miranda Sargento Professora Catedrática da Universidade de Aveiro
vogais / examiners committee Professora Doutora Isabel Maria Simões Pereira
Professora Auxiliar da Universidade de Aveiro(Arguente Principal) Professor Doutor Aníbal Manuel de Oliveira Duarte Professor Catedrático da Universidade de Aveiro(Orientador)
agradecimentos /
acknowledgements Gostaria de agradecer à minha família por ter proporcionado um apoio in-condicional ao longo de toda esta jornada. Um especial bem-haja ao meu orientador, Professor A. Manuel Duarte, bem como ao meu coorientador, Professor João Bastos, por terem estado sempre presentes e dispostos a ajudar.
Em último, mas não menos importante, gostaria de agradecer aos meus amigos e todas as pessoas que conheci e interagi durante a minha vida académica.
Palavras Chave Redes moveis · Previsão · Alisamento Exponencial · KPI · Séries Temporais ·ARIMA · Random-Walk
Resumo O aumento do tráfego de dados no mundo, aumentou a necessidade dos operadores de redes móveis terem um maior cuidado a planear e gerir as suas infraestruturas. Este trabalho explora o desempenho de vários mod-elos estatísticos de previsão aplicados a tráfego de voz e de dados. Os dados têm origem numa rede móvel Europeia. Relativamente aos modelos preditivos, foram aplicados modelos clássicos como alisamento exponencial, Holt-Winters, ARIMA, Random-Walk; bem como duas propostas de mod-elos mais recentes. Em suma, esta dissertação mostra o desempenho de alguns modelos estatísticos clássicos, e como estes se comparam a modelos recentemente propostos. Também mostra que operadores de redes podem usar métodos preditivos estatísticos para tentar obter informações de como a sua rede pode reagir no futuro, dando assim informações valiosas para que estes efetuem uma melhor gestão da sua rede.
Key Words Mobile networks · Forecasting · Exponential Smoothing · KPI · Time Series ·ARIMA · Random-Walk
Abstract The increase of data tra c in the world has increased the need for mo-bile network operators to take greater care in planning and managing theirs infrastructures. This work explores the performance of several statistical forecasting models aplied in voice and data tra c. This data was obtained from an European mobile network, and, regarding the predictive models, it was applied classic models like exponential smoothing, Holt-Winters, AR-IMA, Random-Walk; as well as two more recent model proposals. Regarding the daily data, the proposed model could predict values with higher preci-sion compared to the other models. For hourly data, depending on the time zone where the models were tested, the models with higher performance were Random-Walk and the second proposed model. In summary, this dissertation shows the performance of several classic statistical models, and how these compare to recently proposed models. It also shows that mobile network operator can use statistical forecasting methods to try to get information on how their network might react in future, giving valueable insights to perform a better managment of their network.
Contents
Contents i
List of Figures iii
List of Tables v
Acronym vi
1 Introduction 1
1.1 Motivation and Objectives . . . 2
1.2 Outline . . . 2
2 Mobile Networks 4 2.1 The Primitive Telecommunications Networks . . . 4
2.1.1 Key Attributes of Cellular Mobile Networks . . . 5
2.1.2 Territorial spatial partitioning, roaming and handover . . . 5
Multiplex Access Techniques . . . 7
2.2 Mobile Networks . . . 8
2.2.1 First Generation (1G) . . . 8
2.2.2 Second Generation (2G) . . . 9
Global System for Mobile Communications (GSM) . . . 10
System Architecture . . . 10
General Packet Radio Service (GPRS) . . . 12
2.2.3 Third Generation (3G) . . . 13
Universal Mobile Telecommunications System (UMTS) . . . 14
System Architecture . . . 14 2.2.4 Fourth Generation (4G) . . . 15 System Architecture . . . 16 2.2.5 Fifth Generation (5G) . . . 17 2.3 Key Indicators . . . 18 3 Forecasting 20 3.1 Time Series . . . 21
3.2 Time series components . . . 21
3.3 Basic Forecasting Methods . . . 26
3.3.1 Average Method . . . 26
3.3.3 Seasonal Naïve Method . . . 26
3.3.4 Random-Walk Method . . . 27
3.4 Exponential Smoothing Methods . . . 28
3.4.1 Simple Exponential Smoothing - Brown's Model . . . 28
Optimization . . . 28
Special Case . . . 29
3.4.2 Holt's linear trend method . . . 29
Damped trend methods . . . 30
3.4.3 Triple Exponential Smoothing - Holt-Winters' Model . . . 30
3.4.4 Double Seasonal Holt-Winters . . . 32
3.5 ARIMA models . . . 34
4 Data utilization strategy and model evaluation criteria 41 4.1 Accuracy Measure Process . . . 41
4.1.1 Accuracy measures . . . 41
4.2 Forecasting horizon accuracy . . . 43
5 Analysis and Applications of Forecasting - Case Study 46 5.1 Case Study - Daily data analysis . . . 46
5.1.1 Forecasting models . . . 51
Random Walk (RW) . . . 51
Exponential Smoothing (ES) . . . 51
ARIMA . . . 51
Recent proposed model (M4-Comp) . . . 52
5.1.2 Forecasting Results . . . 53
5.1.3 Accuracy Results . . . 54
5.2 Case Study - Hourly data analysis . . . 58
5.2.1 Sliding Window vs Expanding Window . . . 61
5.2.2 Forecasting models . . . 62
Random Walk model . . . 62
Exponential smoothing models . . . 62
5.2.3 Forecasting Results . . . 63
5.2.4 Accuracy Results . . . 63
5.2.5 Forecasting Results - 2 . . . 65
6 Conclusion and Future Work 69 6.1 Future Work . . . 70
Bibliography 71
List of Figures
1.1 International Telecommunication Union (ITU) estimations for global
informa-tion and communicainforma-tions technology subscripinforma-tions. Source: [1] . . . 1
2.1 Bell System international switchboard in 1943. Source: [2] . . . 5
2.2 Base station representation and their coverage area. . . 6
2.3 Two clusters represented as well as the roaming and handover process. . . 6
2.4 Frequency and time division duplex scheme. . . 7
2.5 Multiple Access Techniques, Frequency Division Multiple Access (FDMA) -Time Division Multiple Access (TDMA) - Code Division Multiple Access (CDMA). 8 2.6 Example of rst mobile generation. Source: [3] . . . 9
2.7 Global System for Mobile Communications (GSM) architecture . . . 10
2.8 GPRS architecture. . . 13
2.9 Universal Mobile Telecommunications System (UMTS) Architecture. . . 14
2.10 Long Term Evolution (LTE) architecture. . . 16
2.11 Road map for 5G. Source: [4] . . . 17
2.12 Innovations for 5G mobile communications compared to 4G. Source: [5] . . . . 18
3.1 Quarterly Australian Electricity production. . . 22
3.2 Trend example. . . 22
3.3 Seasonality example. . . 23
3.4 Remainder example. . . 24
3.5 Quarterly Australian Beer production. . . 25
3.6 Monthly sales of anti-diabetic drugs in Australia. . . 25
3.7 Total quarterly gas production in Australia(in petajoules). . . 27
3.8 Forecasting example comparing Holt, Damped Holt and SES methods. . . 30
3.9 Forecasting with Holt-Winters additive and multiplicative seasonality methods . Source: Adapted from package fpp2 in R . . . 32
3.10 Hourly total number of voice calls. Source: Tier 1 Europe mobile network operator. . . 33
3.11 Forecasting example with double seasonal Holt-Winters method . . . 34
3.12 Flowchart of ARIMA process. . . 38
3.13 Quarterly retail trade: Euro area . . . 38
3.14 Forecast of European retail trade index using ARIMA(1,0,0)(1,1,0)[4] model and ARIMA(0,0,0)(0,1,0)[4] model. . . 39
3.15 Block diagram of ARMA model. . . 40
4.2 Illustration of expanding window validation for one-step-ahead forecast. . . 43
4.3 Illustration of sliding window validation for one-step-ahead forecast. . . 44
5.1 Daily data tra c usage, as percentage of total capacity, generated by four Radio Network Controllers in a 3G mobile network, and covering the period from October 10, 2014 to August 13, 2015. . . 47
5.2 Seasonal plots for data tra c. . . 48
5.3 Periodogram of RNC1 . . . 48
5.4 Daily voice tra c usage, as percentage of total capacity, generated by four Radio Network Controllers in a 3G mobile network, and covering the period from October 10, 2014 to August 13, 2015. . . 49
5.5 Seasonal plots for voice tra c . . . 50
5.6 Periodogram of RNC1 . . . 50
5.7 Comparison of root mean squared error for voice tra c . . . 54
5.8 Comparison of mean absolute percentage error for voice tra c . . . 55
5.9 Comparison of root mean squared error for data tra c. . . 56
5.10 Comparison of mean absolute percentage error for data tra c. . . 57
5.11 Hourly voice tra c usage, for postpaid and prepaid service, and total number of SMS sent, obtained from a Mobile Switching Center. Covering a period of time from 07:00 UTC 03-05-2017 to 12:00 UTC 29-11-2017. . . 58
5.12 Spectrum analysis of KPI 1, 2 and 3. . . 59
5.13 Seasonal plots of KPI's, weekly seasonality represented on left and daily on the right side . . . 60
5.14 Expanding Window vs Sliding Window, MAPE and RMSE plots using Double Seasonal Holt's Winter Forecasting method. . . 62
5.15 Comparison of mean absolute percentage errors for voice tra c postpaid, pre-paid and total volume of SMS. . . 63
5.16 Comparison of root mean squared errors for voice tra c postpaid, prepaid and total volume of SMS. . . 64
5.17 Hourly voice tra c usage, for prepaid and postpaid service, and total number of SMS sent, obtained from a Mobile Switching Center. Covering a period of time from 07:00H UTC 03-05-2017 to 05:00H UTC 05-09-2017. . . 66
5.18 Comparison of mean absolute percentage errors for voice tra c postpaid, pre-paid and total volume of SMS. . . 67
5.19 Comparison of root mean squared errors for voice tra c postpaid, prepaid and total volume of SMS. . . 68
List of Tables
3.1 Weights attached to observations for dierent values of α [29]. . . 29 3.2 Special cases of ARIMA models . . . 36 5.1 ARIMA candidate models for RNC4 data tra c . . . 51
Acronym
1G First Generation. 2G Second Generation. 3G Third Generation.
3GPP Third Generation Partnership Project. 4G Fourth Generation.
5G Fifth Generation.
AMPS Advanced Mobile Phone Service. AR Autoregressive.
ARIMA Autoregressive Integrated Moving Average. AUC Authentication Center.
BSC Base Station Controller. BSS Base Station System. BTS Base Transceiver Stations.
CDMA Code Division Multiple Access. CN Core Network.
CS Circuit Switch.
D-AMPS Digital Advanced Mobile Phone System. DF Dickey-Fuller.
eNB Evolved NodeB. EPC Evolved Packet Core.
EIR Equipment Identity Register. ES Exponential Smoothing.
ETSI European Telecommunications Standards Institute. FDD Frequency Division Duplex.
FDMA Frequency Division Multiple Access. GGSN Gateway GPRS Support Node. GMSK Gaussian Minimum Shift Keying. GPRS General Packet Radio Service.
GSM Global System for Mobile Communications. HLR Home Location Register.
HSS Home Subscriber Server. iid identically distributed.
IMEI International Mobile Equipment Identity. IMS IP Multimedia Subsystem.
IMT-2000 International Mobile Telecommunications 2000. ISDN Integrated Service Digital Network.
ITU International Telecommunication Union. KPI Key Performance Indicators.
KPSS Kwiatkowski-Phillips-Schmidt-Shin. LTE Long Term Evolution.
MA Moving Average.
MAPE Mean Absolute Percentage Error. ME Mobile Equipment.
MME Mobility Management Entity. MS Mobile Station.
MSC Mobile Switching Center. NMT Nordic Mobile Telephony.
NSS Network Swtiching System.
NTT Nippon Telegraph & Telephone Co.
OFDMA Orthogonal Frequency Division Multiple Access. OSS Operations Suport System.
PCRF Policy and Charging Rules Function. PCU Packet Control Unit.
PDN Packet Data Network.
PDN-GW Packet Data Network Gateway. PS Packet Switch.
PSTN Public Switched Telephone Network. RMSE Root Mean Squared Error.
RNC Radio Network Controller. RW Random Walk.
S-GW Serving Gateway.
SC-FDMA Single Carrier Frequency Division Multiple Access. SGSN Serving GPRS Support Node.
SIM Subscriber Identity Module. SMS Short Message Service. SSE Sum of Squared Errors.
TACS Total Access Communication System. TDD Time Division Duplex.
TDMA Time Division Multiple Access. UE User Equipment.
UMTS Universal Mobile Telecommunications System. USIM Universal SIM.
UTRAN UMTS Terrestrial Radio Access Network. VLR Visitor Location Registers.
Chapter 1
Introduction
The mobile phone is becoming a more and more indispensable tool for the daily life of many people. It passed from a device that was just used for voice communication, to a tool that is the joining of various equipment's, such as television, computer, camera, console game, among others.
Figure 1.1: ITU estimations for global information and communications technology subscrip-tions. Source: [1]
Figure 1.1 shows the global subscription of several communication technologies, from 2001 till 2018. We see an increasing trend on the usage of mobile-cellular telephone subscriptions, showing that there is more subscriptions than people by the year 2018. There is also an in-creasing trend of individuals using internet, as well as the mobile-broadband subscriptions. On the other hand, the xed-telephone subscriptions is decreasing.
Cisco also estimated that mobile data tra c would grow by 7-fold from 2017 to 2022, which re ects an increase of 40%, as average, an annual growth rate. Cisco also forecasted that there would be an increase of 5% in the mobile user's worldwide. [6].
This increase in data tra c services and usage, is putting pressure on the network ca-pacity of mobile network operators. To continue to oer service of quality, is mandatory the improvement and increase in the capacity of network where and when is needed.
To maximize the pro ts, it is necessary to make appropriate usage of the resources that mobile network operators already have and being able to allocate them where they are needed, when required. To achieve these objectives, it is of great importance to have an idea about how the networks might behave in the future. One way to know how the network might be-have, is by extracting key indicators from the network past behavior, and use them to attempt forecasting its future values. By doing this, it can give valuable insights where and when those resources are required and should be allocated.
1.1 Motivation and Objectives
It's impossible to predict, precisely, the future and be sure of when and where a network upgrade is needed. However, by extracting information of the current network and applying forecasting models, it's possible to have a reasonable estimation of what can happen in a close period. In this way, it is possible to have enough reliable information that can be used to optimize investment while maintaining adequate levels of network service.
The aim of this dissertation is to explore and evaluate the accuracy of several statist-ical forecasting models on key performance indicators of mobile networks.
1.2 Outline
This work starts with a brief introduction of some of the current problems that network operators are facing and how forecasting techniques can solve or improve those questions. This is the main motivation of this work, it involves the prediction of indicators, collected from mobile networks, using statistical forecasting models and measuring their accuracy per forecast horizon.
Chapter 2 presents an overall introduction of mobile network technologies generations, the objectives in their development and the architecture. It ends with a subchapter about the importance of KPI's to mobile network operators.
Chapter 3 starts with a brief explanation about forecasting principles, followed by how the data, in this case, KPI's, can be aggregated. Then it's presented some of the most used state-of-art statistical forecasting methods, starting by the most basic ones, like Random-Walk, and followed by some more complex, like ARIMA and exponential smoothing.
Chapter 4 presents several strategies that can be used on available data to measure the accuracy of forecasting models. It also presents two accuracy measures.
Chapter 5 presents two main case studies, each case study has dierent KPI's collec-ted dierently from 3G mobile networks.
The last chapter, chapter 6, concludes this dissertation with the work conclusions and possible future work.
Chapter 2
Mobile Networks
2.1 The Primitive Telecommunications Networks
The telecommunication world that we know today can be said that has its roots on the invention of the telegraph. This device was invented and patented in the early 1800s by Samuel Morse, and it permitted the high-speed transmission of signals through wire at rates and distances which, at the time were considered impressive.
This invention opened the world to the transmission of information over long distances. However, as a downside, it needed an expert in Morse code to use it. Each channel could only communicate one way at the same time. However, this problem could be solved by having two circuits [7].
Preceding the invention of the telegraph, in 1876, Bell patented the telephone. This invention enabled the transmission of voice between devices. Therefore, no longer was need an expert to transmit messages, which in other words, made the technology accessible to anyone that had enough monetary power(it was a costly service). Nevertheless, the technology was able to gain the attention of a much larger group of possible users.
In these early stages of communication, the xed telephones were all connected through wires to a central o ce. This was because it would be too expensive and impractical to extend and connect wires from every telephone to all others. It was in the central o ce that someone would switch the connection to the desired destination, manually. The operator that made this manual switch was known as "switchboard operator".
Figure 2.1: Bell System international switchboard in 1943. Source: [2]
Figure 2.1 dates from 1943, and it shows the Bell system international switchboard. It's possible to see several "switchboard operators" answering calls and doing the manual switch to the desired destination.
Due to the high costs of having these infrastructures, the expanding di culty, and natural problems that are associated with having people working, the telecommunications operators started looking for other switching alternatives.
The solution found was the automatic switches, which started to replace the "switchboard operators" in the mid 1920s. This permitted a reduction of operations costs and easier man-agement of the provided service [8].
2.1.1 Key Attributes of Cellular Mobile Networks
Before starting introducing the several existing generations, there are certain characterist-ics that are present in each one. Therefore, for a better understanding, some of these concepts are going to be presented before. For example the geographical position of the base stations, the frequency re-utilization, as well as access techniques to have multiple subscribers using the same channel at the same time.
2.1.2 Territorial spatial partitioning, roaming and handover
It's important to mention that the spectrum used for communications is limited. Therefore, there is a need to reuse the existing frequencies. Network operators can reuse the same frequency if the range of a speci c station don't interfere with another. The geographical radio coverage area of a base station is designated as cell, and usually represented by an hexagonal. The goal of these cells is to be able to handle as many communications as possible. Depending on the types of antennas, the station can be situated in middle of a cell, or, it can be positioned in each corner, having three directional antennas that covers three dierent cells.
Figure 2.2: Base station representation and their coverage area.
Figure 2.2 illustrates the territorial spatial partitioning of base stations and their coverage areas. The coverage areas, cells, are represented by the hexagons, whereas the base stations are positioned in the corners. In this case the base stations need to have three directional antenna, each covering a range of 120 degrees, pointing to opposite directions from each others. Each cell has it's own band of frequencies, this is so that there is no radio interference's between subscribers and dierent cell. However, there is a certain range from the antennas, where their power is already low enough and there is no more radio interference's. There-fore, the same band of frequencies can be reused. In summary, adjacent cells have dierent frequency bands, but the frequencies can be reused if there is enough distance between cells.
The term, Cluster, is used for a group of cells that don't reuse the same frequency. A subscriber is not bind to one cell. It can move freely, and when this process happens, it shouldn't happen an interruption of the communication.
Figure 2.3: Two clusters represented as well as the roaming and handover process. Figure 2.3 shows the two possible scenarios that can happen when a user moves from one cell to another.
There are two possible scenarios. If the subscriber moves to an adjacent cell within the same cluster, then this process is called Handover, represented by a orange color in the gure. During this process there should be a switch-over of the communication channel without the user noticing.
When the subscriber moves away from it's cell and enters another cluster that belongs to other network operator, in other words, another area where he is registered, it's denoted as Roaming. This process, represented as blue in the gure, is usually associated to the process when a person moves to another country, which has a dierent mobile network operator with their own cluster of cells.
Multiplex Access Techniques
The spectrum used for communications is very limited. To increase the e ciency of it's usage, it was created and proposed several access techniques. These techniques permit multiple subscribers to transmit simultaneously in a certain geographical area, and to have bidirectional communication, (in other words, using one channel to transmit and receive at same time).
Figure 2.4: Frequency and time division duplex scheme.
Figure 2.4 shows the duplex scheme that can be applied to transmit and receive at same time. The rectangles in orange, DL, represents the radio frame of the downlink. The blue rectangles UL, represent the radio frame of the uplink.
In the Frequency Division Duplex (FDD), the downlink(transmission path from base sta-tion to the mobile equipment) and uplink (transmission path from mobile equipment to the base station) have dierent frequencies, enabling the transmission and reception at the same time. In the Time Division Duplex (TDD), both downlink and uplink use the same frequency, there is a designated time-slot for each. So the transmission of data occurs in determined moments [9].
These multiplexing techniques, in addition to being used to enable bidirectionality as shown above, can also be used to enable multiple access to the same media. Therefore, in-creasing the overall capacity of total number of subscribers accessing the network at the same time.
Figure 2.5: Multiple Access Techniques, FDMA - TDMA - CDMA.
Figure 2.5 illustrates the three most known and basic multiple access techniques.
FDMA, in this technique, each subscriber is attributed with a certain frequency. This way there is a continue transmission of data.
TDMA, in this technique, there is a time-slot attributed to each subscriber, here, the communication is not done in a continued way, but by burst of information.
CDMA, it uses the spectral scattering. Each subscriber is associated with a code, and can communicate simultaneously in the same frequency [10].
2.2 Mobile Networks
2.2.1 First Generation (1G)
In the early 1900s started to popularize the rst wireless mobile devices. It wasn't like the telephones that we know today, but more like moving radios, which could communicate both ways. However, due to the size, it was impractical for anyone to carry it. They were transported in vehicles and used mainly by taxi drivers, military and emergency vehicles, i.e., ambulances.
This technology had already started to be used by the military, starting in 1940 with World War II. It helped with the production and "push" the need of better devices. Transform the big mobile radios, into a more portable, energy e cient, and less costly device [11].
Figure 2.6: Example of rst mobile generation. Source: [3]
Figure 2.6 shows an example of the rst generation of the mobile telephone. At this time they weighed around 1.1Kg and took almost half a day to charge. Nevertheless, it marked the beginning of mobile telecommunications that we know today.
This rst generation of mobile telephones was based on analog transmission techniques, which means that they were very susceptible to interference's, the spectral e ciency was limited, and it had poor voice quality.
These telephones were also known as cell phones. This name derives from the fact that each region where there was a base station was denoted as a cell. A cellular system of base stations was made to increase the coverage and overall capacity of an operator.
At this time, there was still no standard, and each company tried to develop its own system. In Europe the Nordic Mobile Telephony (NMT), Advanced Mobile Phone Service (AMPS) in USA, Total Access Communication System (TACS) in UK, and Japan with Nippon Telegraph & Telephone Co (NTT). Without a standard, there was a problem of compatibility, where each brand of the mobile phone could only communicate within the same peers, in other words, if the desired destination telephone was not from the same operator, then they couldn't communicate [12].
2.2.2 Second Generation (2G)
The rst generation had several limitations, and could not aord the grow of subscribers. Analog systems had reduced range and capacity. The voice quality was also weak, and there was still no communication standard implemented.
To cross the rst generation problems, other systems were deployed in the early 1990s. Thus, marking the second generation of mobile networks.
This second generation was mainly known for the shift from analog to digital technologies, which improved the weak quality of voice, range, and scalability of the network. These systems were also implemented, having in mind the existing standards of landline requirements for Integrated Service Digital Network (ISDN). The ISDN was the landline technology that could provide voice and data at the same time by the use of two dierent channels.
There were also other services that were implemented in this generation. For example, the prepaid calling, the international roaming, and the most known one, Short Message Service (SMS). This last service was so popular that by the year 2000 almost 15 billion messages were sent every month worldwide [12].
Two of the most known implemented systems were the GSM, developed and introduced mainly in Europe, and Digital Advanced Mobile Phone System (D-AMPS) IS-54 in the USA. Global System for Mobile Communications (GSM)
The GSM was known for its digital transmission and the use of multiple access techniques like the FDMA and TDMA. It modulated the signals using Gaussian Minimum Shift Keying (GMSK), this technique increased the e ciency of the radio power ampli ers, which, as a result, allowed the cellphone battery to last longer. It also implemented a low-cost service like text messages (the SMS was the most popular service at that time, since for a low price people could send a capped number of characters).
It was the most popular and widely implemented system and suered several improvements along the years since the released date. Continuing until this days to have a large share of users compared to the other existing network generations.
As a note, most of GSM systems had the uplink frequency band between 935-960MHz and the downlink between 890-915MHz.
GSM standard was the result of a work group set up under the framework of European Telecommunications Standards Institute (ETSI) [13].
System Architecture
The network of this system can be considered to have two main components. The network, which is a xed infrastructure, and the subscribers, which represents the mobile service users of the network.
Figure 2.7 represents the several components and sub-systems that make part of a GSM mobile network. The sub-systems are respectively the Base Station System (BSS), Network Swtiching System (NSS), Operations Suport System (OSS) and the Mobile Station (MS). •Mobile Station (MS)
The MS is composed by a Mobile Equipment (ME) as well as a Subscriber Identity Mod-ule (SIM) card. The ME represents the hardware and software for radio transmission, in other words, the cellphone. The SIM card makes the ME operational and gives access to the subscribed services, it also contains the subscriber personal identities. With this card the subscribers can permute the ME as pleased while maintaining all the contracted services with the operator.
•Base Station System (BSS)
This system is responsible for making the connection between the MS and the NSS. It is responsible for managing all aspects regarding radio communication with mobile termina-tions, in addition to intervening in the management of subscriber mobility regarding dierent Base Transceiver Stations (BTS) areas, in other words, the "handover", as well as "roaming" process.
This system is composed by a BTS and a Base Station Controller (BSC).
The BTS is responsible for controlling the transmission aspects. It replaces the wired connections to the subscribers that are seen in xed-networks to a wireless connection.
The BSC controls a group of BTS. It is responsible for the establishment, release, and maintenance of all connections of cells.
•Network Swtiching System (NSS)
This subsystem is responsible for managing the communications between mobile sub-scribers, and subscribers with other networks, xed or mobile. It also contains the databases that store all information of subscribers.
The mains elements of this subsystem are: Mobile Switching Center (MSC)
The Mobile Switching Center (MSC) executes all of switching functions and also is respons-ible for establishing the connection to other networks, for example public xed networks, like Public Switched Telephone Network (PSTN) and ISDN. A single MSC can control several BSS.
Home Location Register (HLR)
The Home Location Register (HLR) is a database that contains the information about users in the subscription area controlled by a MSC. This information includes the actual localization, as well as the access services of the subscribers.
Visitor Location Registers (VLR)
The Visitor Location Registers (VLR) is a dynamic database associated to a certain MSC. It contains information of the subscribers that visit the area of that MSC. When a subscriber enters an area that is cover by another MSC, dierent of its original one, the information relative to this subscriber is transferred from his HLR to the VLR of the area that he visited. When this happens, in the HLR the new localization of that subscriber is also updated.
Authentication Center (AUC)
The Authentication Center (AUC) is the register that provides the necessary paramet-ers for authentication functions and encryption. These parametparamet-ers are used to verify user authenticity.
Equipment Identity Register (EIR)
The Equipment Identity Register (EIR) is a register that contains information on mobile equipment through a list International Mobile Equipment Identity (IMEI), of the mobiles re-gistered in the network. This way is possible to "cancel" calls from unauthorized or stolen equipment.
•Operations Suport System (OSS)
This subsystem interconnects with the NSS and BSS subsystems in order to control and monitor the whole system. Some of these functions include the tra c monitoring and pro-duction of reports regarding the several entities of the system and charging. It helps network managers to control and manage their network [14].
General Packet Radio Service (GPRS)
The 2G networks were designed with the primary focus on voice services. It was designed as a circuit-switch, which means that it has a constant bandwidth, this fact is suitable for a phone call, that has a regular data rate. However, with data services like web browsing, that have a variable bandwidth usage, the network doesn't perform well.
When a subscriber wants to browse a web page, it wants it to load fast, and after this load is complete, it doesn't need to continue to exchange more data, because the subscriber will be visualizing the packed data for some time. The bandwidth requirement during this time is zero, which means that resources are being wasted.
For this kind of bursty data services, a packet-switching approach is much more e cient. What this does is gather the necessary data in packets before it is sent over the network, in other words, there is only a transmission of data when there's a request.
The General Packet Radio Service (GPRS) was designed as a packet switch service added to the GSM network. As a consequence, this system could increase the maximum transfer speed of data. The subscriber is also only charger by the used data volume, instead of the connection duration.
Figure 2.8: GPRS architecture.
Figure 2.8 shows the architecture of a GPRS network. This upgrade added three elements. •Packet Control Unit (PCU)
This element is an upgrade to the previous BSC. It redirects the data from the several BSC in a packet and transfers it to the new node, Serving GPRS Support Node (SGSN). The voice tra c still ows from BSC to MSC like in the GSM.
•Serving GPRS Support Node (SGSN)
The GPRS divided the voice that data tra c. The voice tra c continued to be processed by the previous architecture, GSM, but the data tra c is now processed by the GPRS architec-ture. For the SGSN, the main responsibilities are the mobility management and authentication of users. This includes detecting new subscribers in the service area and registering them, the handover process as well as payment methods within the network.
•Gateway GPRS Support Node (GGSN)
This element connects the GPRS network with other networks, like Packet Data Network (PDN). It contains the routing information and is responsible for payment methods of other network services [15].
In summary, this upgrade is considered as the 2.5 Generation. It came as an improvement of the existing network which was only designed in the form of circuit switch.
2.2.3 Third Generation (3G)
The number of people using internet services in xed networks was rising steadily. There-fore, the new generation of mobile networks needed to be designed and planned in order to accommodate this tendency.
GPRS, enable a faster and more e cient way of data communication compared to previous network generations. However, data transfer rates were still meager and not enough.
It was also around the year 2000, that organizations like ITU, Third Generation Partner-ship Project (3GPP), which are responsible for issues or releases that concern communication technologies, that released the International Mobile Telecommunications 2000 (IMT-2000), which de ned as Third Generation (3G), systems that could provide speeds from 144kbps to 2Mbps or more, as well as improved spectrum e ciency. [16] [17].
Universal Mobile Telecommunications System (UMTS)
3GPP Release 99 contains all the speci cations related to UMTS. This network aimed to merge the characteristics of both systems, circuit-switch, and packet-switch.
In UMTS it was introduced the concept of Wideband Code Division Multiple Access (WCDMA), which is a dierent access method compared to the previously used, TDMA and FDMA. In this method, there is a unique code assigned to each user and the bandwidth of a single carrier was also increased. This way of multiplexing enable the reuse of the same frequencies in neighborhood cells, which, simpli ed the planing of the network. Unwanted noises can also be ltered when decoding.
Compared to the GPRS, which for communication needed to nd a channel that was empty. WCDMA, attributes a code to data and voice tra c, it doesn't need to wait for an empty channel.
This fact increased the data transfer rates, from 53.6Kbps to 383Kbps(theoretical values), and increased the security of the transmission of data (by the introduction of codes).
System Architecture
UMTS reuses most of the previous structures used by GPRS and GSM.
Figure 2.9: UMTS Architecture.
Figure 2.9 shows the architecture of a UMTS network. This network can be divided in three subsystems into a similar way to the previous GSM and GPRS. These subsystems are the Core Network (CN), UMTS Terrestrial Radio Access Network (UTRAN), and the User
Equipment (UE).
• Core Network (CN) The CN has the main function of switching and routing the in-formation of subscribers.
It divides into two domains. One for Circuit Switch (CS) that consists of the WMSC and GMSC gateway for switching circuits networks. The other eld is for Packet Switch (PS), based on GPRS. It consists of a GPRS support node, called SGSN, and in a gateway for packet switching networks called GGSN.
Some other elements like HLR, EIR, VLR, and AUC are shared by both domains. The transmission on the core network is Asynchronous Transfer Mode (ATM), with adapt levels dedicated to switches and packets modes.
•UMTS Terrestrial Radio Access Network (UTRAN)
The UTRAN is responsible for the management of the radio interface with the mobile equipment. It has two elements. The base station, that can also be called Node B, and the Radio Network Controller (RNC), which is the element that controls Node B (it can control more than one).
Node B is responsible for transmission and reception of radio signals; modulation and demodulation; it uses the WCDMA combined with FDD.
The RNC has the function of control and management of radios resources; channel alloc-ation; ciphering; admission control; control of handover mechanisms.
•User Equipment (UE)
The UE is composed of two elements, the ME, which is the equipment acquired by the user, and the Universal SIM (USIM), it's function is the same of the SIM card, but it has a universal standard feature.
In the UMTS network, the UE can operate in three dierent modes.
In the mode PS/CS, where it can provide both services, PS and CS domains. In the PS mode where it only provides the packet switch service. In the CS mode where it only provides the circuit switch service [18] [17].
2.2.4 Fourth Generation (4G)
In 2004, 3GPP proposed the LTE. This standard was marketed as a Fourth Generation (4G) and was only implemented in 2009.
In a similar way to the change between the Second Generation (2G) and the 3G, from 3G to the 4G the aim was to improve the data transfer rates, increase the capacity of the network and reduce latency (which is the total time for a packet to travel between nodes).
The solution in the LTE was to use two dierent modulation techniques. For uplink (ter-minal to base station) it used Single Carrier Frequency Division Multiple Access (SC-FDMA), or downlink (base station to terminal) it used Orthogonal Frequency Division Multiple Access (OFDMA), this techniques can be used together with FDD or TDD. It had exible band-widths, which, with the addition of multiple antennas enable higher data transfer rates, up to 100Mbps (theoretically) and higher capacity compared to previous network generations [19].
System Architecture
The new architecture of the network re ects the implementation of services based on IP packages, as well as performance optimization and an improvement in the ratio, cost-e ciency for the operators.
Figure 2.10: LTE architecture.
Figure 2.10 illustrates the several components and systems that constitute the LTE net-work.
It consists in a radio access network, Evolved Radio Access Network (E-UTRAN), and a core network, Evolved Packet Core (EPC). This technology permits the compatibility with other network generations and a reduction in the operation and maintenance costs [20]. •Evolved Radio Access Network (E-UTRAN)
E-UTRAN handles all the communications between the UE and the core network, EPC. It only has one element, the Evolved NodeB (eNB).
eNB has two functions. The rst is communication with all mobiles, and the second is to control the low-level operations of them. The eNB's are interconnected with each other and communicate with mobiles via wireless.
•Evolved Packet Core (EPC)
EPC, an architecture that replaces packet switch domains of previous network generations. The circuit switch domain was withdraw. However, it has the means to handle calls by transferring them to the older generation of mobile networks, or by using IP Multimedia Subsystem (IMS).
It's constituted by Serving Gateway (S-GW), Packet Data Network Gateway (PDN-GW), Mobility Management Entity (MME), Policy and Charging Rules Function (PCRF) and Home Subscriber Server (HSS).
S-GW - It makes the transaction the radio access network and the EPC network. All the IP packages from the user are transferred by it. It makes the control of data from each user. PDN-GW - This Gateway is responsible for the IP attribution to the UE. It lters the
packages for each user based on tra c ow models. It is used as an input and output of data tra c from the UE and the interface between LTE networks or other networks.
MME - It is equivalent to the HLR and VLR in UMTS network. It's the control node which process the signaling between the UE and the core network, with the interaction of the HSS, which has all the information of each user. It enables the easy optimization of the networks and their total exibility regarding the capacity.
PCRF - Is responsible for the policy and charging control element of services [21].
2.2.5 Fifth Generation (5G)
With almost 10 years passing since the implementation of the 4G. A new generation is being planned and developed. Terms like smart cities, smart homes, Industry4.0 are in trend, and all of them have something in common. Vast amounts of devices connected and a massive amount of data that can be extracted and processed. All of this falls on the next generation of mobile networks.
To accommodate the all requirements of this "new" era, it's a must that the new network needs higher data transfer rates, low latency, and exibility.
Figure 2.11: Road map for 5G. Source: [4]
Figure 2.11 shows a simple planned roadmap for 5G technology. The development and research project started in 2011, two years later, after the implementation of the 4G. The fth generation is still under construction, and it's only planned to be operational around 2021.
Figure 2.12: Innovations for 5G mobile communications compared to 4G. Source: [5] Figure 2.12 shows several parameters speci cations from 4G and Fifth Generation (5G) networks.
While in the 4G network could reach data transfer speeds up to 100Mbps. The fth generation will have speeds up to 10Gbps, that's an increase of speed of 100x. Another aim was the reduction of latency. There should be a reduction from 10ms, 4G, to just 1ms. This will enable better e ciency and increase the performance of communication services.
The 5G technology will operate in a dierent frequency spectrum. It will have the 700MHz spectrum for Internet of Things(IoT), then a 3.4-3.8GHz spectrum as well as 24.25-27.5GHz for other services.
2.3 Key Indicators
The biggest goal of a company can be said to have the highest returns and the best service. For mobile network operators, the aim is the same. However, it's not so easy to have positive returns at the end of each year. Right decisions must be taken that can lead the company to this end.
A way to help decision making is by gathering and analyzing data from the company. This data can give valuable information or insights, if extracted and analyzed correctly.
The data usually consists of several indicators, each representing a part of the whole system. It's up to a company to choose its own indicators. Therefore, the decision on which indicator is relevant is extremely important, after all, the company will make use of it to make decisions.
The most critical performance indicators are denoted as Key Performance Indicators (KPI). In the case of mobile network operators, by monitoring these indicators, they can increase the performance and e ciency of the network. By allocating resources when and where they are needed [22].
Even with the fact that it is up to each company to choose their owns KPI's, the 3GPP has already emitted Releases which mention their classi cation. The main groups were access-ibility (which is related to the access phase), retainability (which is associated with the stage after the access phase is completed), integrity (related to the transmission and retransmission errors and quality), mobility (all KPI's related to the handover and location updates), and
nality the resources, which are related to the tra c utilization of the interfaces.
KPI must be well documented and with de ned units. In the case of mobile networks, the KPI's are expressed in a variety of units or, sometimes, unit less. In telecommunication, most of the units are expressed as percentage, time interval, Erlang, and bits per second (bps). Regarding the type, it can be a mean (based on a several sample results), a ratio (percentage of a particular occurrence), a cum(a cumulative measurement), etc [23].
Chapter 3
Forecasting
It is the very existence of time that leads humans to try to predict the future. Knowing something beforehand can give valuable insights into what decisions must be taken to point to the desired goal.
Along the years, several methods were tested and developed that could produce predictions. Being these methods associated with a structured hypothesis in scienti c contexts, the term "forecast" started to be used [24].
Forecasting methods are divided and classi ed into two main "branches", Qualitative or Quantitative.
Qualitative methods consist of the opinion and judgment of a specialist; they don't manip-ulate nor make use of data, they are not statistics oriented. Nevertheless, they are handy in a case where there is not enough information, or there is a time limit to develop quantitative methods, thus often used in practice, mainly in long term forecasts.
Quantitative methods are used to forecast future data as a function of past data. Usually, these methods are based on the analysis of data and used in the short and medium term[25]. When lunching a new product or service in the market, where past data is not yet avail-able, other approaches are used, namely those based on diusion of innovation or imitation mechanisms [26].
3.1 Time Series
Being the focus of this work on quantitative forecast methods, one needs to have sets of data to be analyzed and processed. These sets are usually referred to as Time Series and, as the name suggests, it constitutes a sequence of observations that are recorded in time.
Such sets of observations are done and recorded in several elds, such as economics, market-ing, natural sciences, among many others, i.e., the hourly number of page views for a website, the daily temperature in a particular region or the annual newborn babies in a country.
There are two possible ways to take observations in time. Continuous or Discrete. It's considered continuous when the observations are taken continuously in a certain period, i.e., the electrical signals from sensors. It's discrete when the observations are made in speci c time points or periods. Regarding these periods, they can be equidistant, where the observations are taken with xed spaced time intervals, or they can be non-equidistant, where the length of time intervals is not constant.
In practice, when one talks about time series, it is usually referring to discrete and equidistant time series. This is because it makes it easier to process and analyze the data.
Regarding time series, there are also speci c characteristics that can dierentiate them. When the values of the observations can be represented entirely by a mathematical function, and future values predicted exactly at any instance of time, it is said that we have a determ-inistic process. When the set of values of observations are random, making it impossible to predict the exact future values, it is said that we have stochastic process [27][28].
A time series can be classi ed as stationary, when the statistical properties like mean value, variance are constant or almost constant over time, oating around the same value and not time dependent. White Noise is a case of a stationary process, which is a sequence of independent and identically distributed (iid) random variables, with zero mean and variance σ2. If one of the proprieties that de ne a stationary process is not constant, then it will be considered a non-stationary time series.
For a stochastic time series to be considered stationary, the random factor associated needs to have a constant mean and a nite variance.
3.2 Time series components
The rst thing that should be done before thinking of forecasting is to plot and visualize the time series. Graphs enable the visualization of many features such as patterns, unusual observations, changes over time, among others. These features can give valuable information that can be used to make a better approach when forecasting.
Regarding patterns, one should look for trend and seasonality. Therefore, they can also be considered time series components. There is also another component that can be referred to as remainder or random, which is usually associated with the random factor that is present in most of the time series.
Figure 3.1: Quarterly Australian Electricity production.
Figure 3.1, is a time series example, the observations represent the total quarterly elec-tricity production in Australia (in billion kWh) from 1980 till 2010. In this case, there is a positive trend along the time. The production increased from less than 2.5 to more than 5.5 billion kWh in a range of 30 years. There is also a seasonality that repeats itself every year, the possible peak corresponding to the summer season, probably due to the high temperatures and the extensive use of air conditioners, increasing the consumption/demand of electricity.
• Trend: It is denoted as a trend when there is a continuing increase or decrease in the level of the data; it can change over time and doesn't need to be always linear.
Figure 3.2: Trend example. Figure 3.2 is the trend that resulted from from Figure 3.1.
One method that can be used to extract the trend is by using Moving Averages. The Moving average process assumes the concept that near observations in time are likely to be close in value. In other words, each observation corresponds to the average of its neighbor observations within a precisely de ned range. By taking averages, the magnitude of the variation that can occur is going to be penalized, and the time series will be "smoothed". The
range length can be dierent depending on the time series. However, in the case where there's a seasonal pattern, the distance used to take averages should be the same as the seasonal pattern period.
There is one problem with this method, and that is, if the value that is taking averages is situated in the middle, then, the nal trend will have fewer values, because it can't calculate the initial and nal values of the time series. One way to solve this problem is by taking averages of the values only on the right side or left, depending on if we want to smooth the beginning or the end of the time series.
• Seasonality: A seasonal pattern is the type of variation when the patterns are similar in their behavior, and they repeat each other in xed periods, usually daily, weekly, monthly, or annually.
Figure 3.3: Seasonality example.
Figure 3.3 is the nal seasonal pattern that was extracted from Figure 3.1, for the interval of 2006 till 2010.
After extracting the trend of a time series, (this process is denoted as de-trending), the next step is to remove the seasonality. To obtain the seasonality, one can average all "seasons". For example, if a time series as a weekly seasonality, we can average all the observations of Mondays, then Tuesdays... This way, it's possible to obtain a reasonable estimation of the seasonal pattern that can be extracted from the de-trended time series.
• Remainder: After one identi es the components such as trend and seasonality and extracts them from the time series. One obtains what is called the remainder or residuals.
Figure 3.4: Remainder example.
Figure 3.4 is the result of extracting both trend and seasonal components from Figure 3.1. The remainder should only be composed of a stochastic factor that has zero mean, a nite variance and null autocorrelations. It's vital to analyze and make sure whether the remainder shows these characteristics. Otherwise, it will say that the extraction process of the trend and seasonality was not done correctly and there is still space for further improvements. A possible way to achieve an improvement is by changing the "window" when using the moving average technique to de-trend [29].
Time series decomposition
It was already shown in sub-chapter 3.2 that a time series can be separated into three com-ponents and how to obtain them. If a time series is composed of these three comcom-ponents, then we can also represent it as a function of them. The decomposing process is usually referred to as time series analysis.
yt= f (St, Tt, Rt), (3.1)
where yt is the value of the time series, St the seasonal component, Tt the trend component
and Rtthe remainder component, all at period t.
It is useful to represent the time series as a function of its components because it's easier to model the trend and the seasonal patterns separately. With the assumption that the patterns are composed by a deterministic and a non-deterministic part, one can easily nd the deterministic component and forecast its future values.[29]
Depending on the time series, it's considered two types of decomposition, additive or mul-tiplicative.
Additive Decomposition
The additive decomposition is considered when the magnitude of the seasonal component of the time series doesn't depend on the level and stays constant along with the time series.
It can be mathematically described as:
The following image is a good example of an additive decomposition:
Figure 3.5: Quarterly Australian Beer production.
Figure 3.5 represents the total quarterly beer production in Australia(in megalitres) from 1956 till 1975. It's a perfect example of an additive decomposition case where it is composed by a trend, a seasonal and a remainder, and where the seasonal component variation stays almost constant along with the level of the time series.
Multiplicative Decomposition
In a case where the seasonal component depends on the level and, i.e., increases as the level of the time series increases, in other words, the variance is proportional to the trend, then it should be considered a multiplicative decomposition.
It can be expressed as:
yt= St∗ Tt∗ Rt. (3.3)
As example of a multiplicative decomposition:
Figure 3.6: Monthly sales of anti-diabetic drugs in Australia.
Australia. In this case, one should consider the multiplicative decomposition because, as it can be seen in the gure, the magnitude of the variation of the seasonal component is increasing, doesn't stay constant over time.
As a note, there is an alternative way to transform the multiplicative decomposition into an additive decomposition, this is done by taking the logarithmic values. Therefore, yt =
St∗ Tt∗ Rt is equivalent to log(yt) = log(St) + log(Tt) + log(Rt).
3.3 Basic Forecasting Methods
In this subchapter, it's going to be mention some of the most basic and used forecasting methods that should be included. Even if they are considered basic and simple, it's proven the di culty of nding better methods that can outperform the basic ones, in most cases, some are even used as a benchmark for comparison. This fact was made proven during the M4-Competitions, which is a global competition organized with the intent to evaluate and compare the accuracy of dierent forecasting methods. [31]
3.3.1 Average Method
A straightforward method that consists of what the name says. In this method, future values are equal to the average of all observations. It can be mathematically represented by:
ˆ
yt+h|t= y = (y1+ ... + yt)/t, (3.4)
where y1 + ... + yt is the historical data or observed values and ˆyt+h|t the estimation of h
forecast horizon.
3.3.2 Naïve Method
Naïve method assumes that future values are equal to the value of the last observation. This method performs well when the time series doesn't have seasonal patterns.
It can be mathematically represented by: ˆ
yt+h|t= yt, (3.5)
where ˆyt+h|tis the forecast for t+h, ytthe actual value for t period, and h the forecast horizon.
3.3.3 Seasonal Naïve Method
Following the same logic as the Naïve method, the seasonal version consists on the as-sumption that future values are the same as the ones observed in the last "season", i.e., for a time series with yearly seasonality, the next future value of January, will be the same as the previous value observed in January.
For seasonal time series, this method is one of the simplest and reliable forecasting methods, it's easy to implement and understand, but most importantly, challenging to be outperformed in terms of accuracy by more sophisticated techniques, enabling him being a good benchmark when comparing models.
The following formula is the mathematical representation of this method: ˆ
yt+h|t = yt+h−m(k+1), (3.6) where m is the seasonal period, h the forecasting horizon, and k = (h − 1)/m (represents the number of seasonal periods over which the forecast is being made).
The following Figure can give a better understanding of the practical results of forecasting using the previously mentioned methods.
Figure 3.7: Total quarterly gas production in Australia(in petajoules).
Figure 3.7 represents the total quarterly gas production in Australia (in petajoules) from the rst quarter of 2000 until the second quarter of 2010. It was performed a forecast with a horizon of 2 years.
This time series has a "strong" seasonal pattern, with its seasonal peak being in the winter followed by it's lower being in the summer, this veri es with the assumption with the gas demanding for heating. For this time series, the seasonal naïve method is a good choice that can probably give an estimation close to reality.
3.3.4 Random-Walk Method
Random-walk consists of the assumption that future steps or directions in a time series are unpredictable based on past actions. The process is mathematically described as:
yt= yt−1+ t, (3.7)
where tis a random process with zero mean and nite variance. It's this random factor that
makes the process unpredictable, whereas the level can move away from the mean either up or down. If we consider as 0, then the Random-Walk works the same way as the Naïve methods. Also, the mathematical process presented was only to describe non-seasonal data. Nevertheless, the same logic can be applied for seasonal data, where each forecast is equal to the last observed value from the same "season" [32][33] .
The Random-Walk method should be used in time series that are stationary. In case where there are deviations in the mean value (there is a trend), there is an alternative process called Random-Walk with drift and is mathematically described as:
yt= c + yt−1+ t, (3.8)
where c = E(yt− yt−1) and is a random process with zero mean and nite variance.
3.4 Exponential Smoothing Methods
In the late 1950s, appeared the rst exponential smoothing methods(Brown, 1959; Holt, 1957; Winters, 1960). These proposed methods consisted of weighted averages of observations that aimed to describe the time series. Usually, the weight applied is higher in the most recent observations, meaning that it gives them a higher in uence compared to older observations.
This forecasting method is in use even nowadays and thrives in many elds, i.e., business, economic, and nance. It's a forecasting concept that is easy to understand and usually used to perform forecasts [29].
3.4.1 Simple Exponential Smoothing - Brown's Model
It started when Brown developed a model to describe the process where there is no trend nor seasonality. Taking only into consideration the observed data but not treating it as equal, instead of with dierent weights, where the latest values have a higher in uence compared to previous ones.
The one-step-ahead forecast for time t + 1 is a weighted average of all the observations. ˆ
yt+1|t= αyt+ (1 − α)ˆyt|t−1, (3.9)
where α is the weight parameter, also called a smoothing parameter. This parameter ranges values between [0, 1].
An alternative representation of simple exponential smoothing is the component form. This representation is composed of a forecast equation and a smoothing equation for each of the components included in the method.
The simplest way to transmit the history of a process is through a single state, lt, called
the level. Thus, simple exponential smoothing it is composed by: Forecast equation: ˆyt+h|t= lt,
Level equation: lt= αyt+ (1 − α)lt−1. (3.10)
Optimization
It is essential to de ne proper smoothing parameters for every time series.
For better understanding about the impact of the smoothing parameter α upon the way observations are taken into account, its illustrated in the following table.
Table 3.1: Weights attached to observations for dierent values of α [29]. Observations α = 0.2 α = 0.4 α = 0.6 α = 0.8 yt 0.200 0.400 0.60 0.800 yt−1 0.160 0.240 0.260 0.160 yt−2 0.128 0.114 0.096 0.032 yt−3 0.102 0.008 0.038 0.006
With the Table 3.1 is easy to understand that the weight given to the observations decreases exponentially (thus the model name of exponential smoothing), and given enough observations, the sum of the smoothing parameter α nears the value 1, also, the higher the value of α, the faster that this can be seen.
When α is close to 1, we consider as "fast learning" because the algorithm gives more weight to the most recent observations and adapts much faster to changes. If α is close to 0, we consider as "slow learning" because the algorithm gives previous observations more weight and therefore, the model is less reactive to high variations in the newest observation data.
One way to estimate suitable parameters is by taking into consideration the observed data and then minimize the Sum of Squared Errors (SSE), for dierent parameter values.
SSE :
T
X
t=1
(yt− ˆyt|t−1)2 (3.11)
The example given was SSE, but one can apply other error measures, (the logic is always the same, minimizing the error, which is the dierence between real and estimated value) [29]. Special Case
There is a particular case that needs to be referred when dealing with simple exponential smoothing, and that is when α = 1, in this case, the simple exponential smoothing will be the same as the Random-walk method.
3.4.2 Holt's linear trend method
In 1957, Holt extended the simple exponential smoothing model to allow the forecasting of data with a trend, either upward or downward. This method involves a forecast equation and two smoothing equations(one for the level and one for the trend, using two smoothing parameters, α and β for each equation respectively). The h-step-ahead forecast is given by combining the level ltand trend bt at time t.
Forecast equation: ˆyt+h|t= lt+ hbt
Level equation: lt= αyt+ (1 − α)(lt−1+ bt−1)
Trend equation: bt= β(lt− lt−1) + (1 − β)bt−1. (3.12)
Like the simple exponential smoothing, the parameters α and β take a value between 0 and 1. The level, ltis a weighted average of the actual value at time t and the sum of the level
plus the trend at t − 1, the trend is a weighted average of the trend in the previous period and the more recent information on the change in the level.
Damped trend methods
Holt's method produces linear forecasts with a consistent trend. Because of its in exibility, Gardner & McKenzie in 1985 introduced a parameter that "dampens" the trend to a at line given a big enough forecast horizon.
There are several ways to describe the model mathematically. However, the original one from (Gardner and McKenzie) is written as:
Forecast equation: ˆyt+h|t= lt+ (φ + φ2+ ... + φh)bt
Level equation: lt= αyt+ (1 − α)(lt−1+ φbt−1)
Trend equation: bt= β(lt− lt−1) + (1 − β)φbt−1, (3.13)
where the damped parameter is φ, with a range of values between 0 and 1. Nevertheless, in practice, this value is most likely to be between 0.8 and 0.98. The high values in the parameter are since the damping has a powerful eect for smaller values [34].
The following image serves as an example of forecasting value using the referred models:
Figure 3.8: Forecasting example comparing Holt, Damped Holt and SES methods. Figure 3.8 illustrates the total annual air passengers(in millions) of air carriers registered in Australia between 1970 and 2016. The forecast horizon is 18 years. The value of φ was bound with a value of 0.95 to exaggerate the damp eect for comparison. However, as usual, proper parameters can be estimated by minimizing scale errors.
Since the damped method gives more exibility to the forecasts, it should be preferred against a "normal" model.
3.4.3 Triple Exponential Smoothing - Holt-Winters' Model
The previous methods could only capture the level and trend. However, there are a lot of real-world examples where time series have seasonal patterns. This fact led Winter, in 1960, to extend Holt's method to capture such behavior. The result was the Holt-Winters
seasonal method, which is composed of an additional equation that accommodates/includes the seasonality. The seasonal component is designated by st, and the smoothing parameter is
γ. The frequency of the seasonality is denoted by m.
The seasonal component can be considered as additive or multiplicative, and there are dierent equations for each case. Because the Holt-Winters method is expressed by three types of exponential smoothing equations, in some literature is also called by triple exponen-tial smoothing.
Additive Seasonality
The basic equations that constitute Holt-Winters' additive method are:
Forecast equation: ˆyt+h|t= lt+ bth + st+h−m(k+1)
Level equation: lt= α(yt− st−m) + (1 − α)(lt−1+ bt−1)
Trend equation: bt= β(lt− lt−1) + (1 − β)bt−1
Seasonal equation: st= γ(yt− lt−1− bt−1) + (1 − γ)st−m, (3.14)
where m is the length of seasonality or frequency, ltrepresents the level of the series, btdenotes
the growth/trend, st is the seasonal component, ˆyt+h|t is the forecast for h periods ahead, k
is the integer part of (h − 1)/m, which ensures that the estimates of the seasonal indices used for forecasting come from the nal period of the sample. The smoothing parameters, (α, β, γ) ∈ [0, 1]. As with all exponential smoothing methods, initial values and smoothing parameters can be calculated by minimizing the forecasting error.
The level equation represents a weighted average between the seasonally adjusted obser-vation and the non-seasonal forecast for the time t. The trend equation is identical to Holt's linear method. The seasonal equation shows a weighted average between the current seasonal index and the seasonal index of the same season of the previous period [34].
Multiplicative Seasonality
The basic equations for Holt-Winters' multiplicative method are as follows:
Forecast equation: ˆyt+h|t= (lt+ bth)st−m+h+ m Level equation: lt= α yt st−m + (1 − α)(lt−1+ bt−1) Trend equation: bt= β∗(lt− lt−1) + (1 − β∗)bt−1 Seasonal equation: st= γyt lt−1+ bt−1 + (1 − γ)st−m. (3.15)
Figure 3.9: Forecasting with Holt-Winters additive and multiplicative seasonality methods . Source: Adapted from package fpp2 in R
Figure 3.9 is an illustration to give a better understanding of the forecasting results of both methods. The same example was given in 3.3, where the black line represents the total quarterly gas production in Australia (in petajoules) from 2000 till 2010, then the red and blue lines represent the forecasting value, for a horizon of 2.5 years, using Holt-Winters additive and multiplicative seasonality methods respectively.
As a note, there is also a Holt-Winters method with a damped trend, improving the exibility of the method when forecasting the trend of a time series and therefore, improving the overall accuracy when forecasting dierent time series.
3.4.4 Double Seasonal Holt-Winters
With the evolution of technology, the possibility to extract more information was made. Nowadays, one can obtain information with time spawns of hours, minutes, or even less. These high-frequency time series usually exhibit more than one seasonal pattern. Therefore, more improvements needed to be done to previous models.
