Stochastic Modeling: A Thorough Guide to Evaluate, Pre-Process, Model and Compare Time Series with MATLAB Software

By Hossein Bonakdari and Mohammad Zeynoddin

Stochastic Modeling: A Thorough Guide to Evaluate, Pre-Process, Model and Compare Time Series with MATLAB Software allows for new avenues in time series analysis and predictive modeling which summarize more than ten years of experience in the application of stochastic models in environmental problems. The book introduces a variety of different topics in time series in the modeling and prediction of complex environmental systems. Most importantly, all codes are user-friendly and readers will be able to use them for their cases. Users who may not be familiar with MATLAB software can also refer to the appendix.

This book also guides the reader step-by-step to learn developed codes for time series modeling, provides required toolboxes, explains concepts, and applies different tools for different types of environmental time series problems.

• Provides video tutorials on the use of codes
• Includes a companion site with 3,000 lines of programming, 70 principal codes and 100 pseudo codes
• Highlights multiple methods to Illustrate each problem

• Language: English
• Publisher: Elsevier Science
• Release date: Apr 13, 2022
• ISBN: 9780323972758

Hossein Bonakdari

Dr. Bonakdari obtained his PhD in Civil Engineering from the University of Caen Normandy (France). He has worked for several organizations and most recently as an Associate Professor at the Department of Civil Engineering of the University of Ottawa (Canada). He is one of the most influential scientists in the field of developing novel algorithms for solving practical problems through the decision-making abilities of AI. His research also focuses on creating comprehensive methodologies in the areas of simulation modeling, optimization, and machine learning algorithms. The results obtained from his research have been published in international journals and presented at international conferences. He was included in the list of the world's top 2% scientists, published by Stanford University, and is on the Editorial board of several journals.

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Chapter 1

Introduction

1.1 Time series

Many phenomena naturally have variation with time; these phenomena encompass a wide variety of fields of study. One such example is water quality data from a stream. The water quality parameters (alkalinity, turbidity, organic content, etc.) may change throughout the year and interannually. The approach used to study these phenomena of interest is to collect data regarding the parameters that have affected them in the past and those that will influence them in the future. Following this approach, time series may be considered the best tools for analyzing the collected past and present data to be able to make future decisions (Akhbari et al., 2019). In fact, time series are very important in the development of planning and management policies. With the help of time series, you can see the trend of changes in phenomena from the past to the present along with the differences between the observed and expected values due to fluctuations in the phenomena. A clear picture of the behavior of phenomena of interest can be developed through time series modeling. Periodic fluctuations and seasonal changes can be observed in phenomena, allowing their behavior to be understood and related to other influencing phenomena. If we consider the example of water quality data once more, well-developed time series models may allow one to understand how seasonal differences influence water quality indicators, such as turbidity. Not only time series allow us to explain the influence of seasonal changes on water quality indicators, but they also allow us to understand interannual changes for the same season. This information can allow municipal officials to develop watershed management policies or allow for the optimization of drinking water production for example. The management and planning decisions that are made are based on the prediction of future conditions from past and present time series data. Forecasts are constantly needed, and over time, the effects of these predictions on actual performance are measured. From the constant measurement of performance, the predictions are regularly updated, and the decisions corrected. This cycle continues in an iterative fashion in order to achieve the desired conclusion (Azari, Soori, and Bonakdari, 2017; Langridge, Gharabaghi, McBean, Bonakdari, and Walton, 2020).

Over the recent years, vast improvements in technology have made recording data for time series modeling more practical and reliable. As a result, time series modeling is one of the most practical tools for investigating a variety phenomenon in science, engineering, and economics (Azimi, Bonakdari, Ebtehaj, Gharabaghi, and Khoshbin, 2018; Binesh and Bonakdari, 2014; Bonakdari, Ebtehaj, Gharabaghi, Vafaeifard, and Akhbari, 2019; Hui Pu, Bonakdari, Lassabatère, Joannis, and Larrarte, 2010). The concept of time series modeling allows researchers to assess the outcomes of a variety of phenomena at any time with minimum costs and efforts. Using forecasted data, they can plan possible solutions to problems, make decisions, and implement control measures.

1.1.1 Time series in environmental epidemiology

One of field that has widely benefited from time series concept is environmental epidemiology (Bhaskaran, Gasparrini, Hajat, Smeeth, and Armstrong, 2013; Bonakdari, Pelletier, and Martel-Pelletier, 2020a, 2020b; Corcuera Hotz and Hajat, 2020; Tejasvini, Amith, and Shilpa, 2020). Environmental epidemiology (Fig. 1.1) allows researchers to forecast the outcomes of any phenomenon like health field or assess the impact of environmental exposures such as weather, air pollutants, and other contributing factors impacting health condition. Using study data to forecast future events enables decision makers to plan solutions and implement control measures in a way that is much simpler and much more cost-effective when compared to other methods, such as randomized control tests (Bonakdari et al., 2021; Bonakdari, Pelletier, and Martel-Pelletier, 2020c; Bonakdari, Tardif, Abram, Pelletier, and Martel-Pelletier, 2020).

Figure 1.1 Schematic of environmental epidemiology.

1.1.2 Engineering and sequential data

Another domain of application of time series is in engineering. One example of an emerging application of time series modeling is in water management. As communities continue to expand and grow, so do their water requirements, whether it be for energy production, drinking water supply, or agricultural practice (Fig. 1.2). Hydrological time series data are critical in order to allow policy makers to make effective decisions regarding water management and resource sustainability (Kazemian-Kale-Kale et al., 2020; Lotfi et al., 2019; Lotfi et al., 2020; Zaji, Bonakdari, and Gharabaghi, 2019). With the help of these time series, costs associated with the implementation of water management policies (including operation and maintenance of water utilities) can be greatly reduced, and the management of water resources efficiently conducted (Soltani et al., 2021; Zinatizadeh, Pirsaheb, Bonakdari, and Younesi, 2010). The example of water resources is one of several applications of time series in engineering, which is more exorbitant than other examples due to the global water crisis. Time series are also used in other engineering sections, for instance, estimating and forecasting the amount of energy consumed by industries and home consumers, surveying natural cycles in these uses and managing energy consumption, modeling the amount of traffic in a corridor and forecast future needs, study the production and harvest of agricultural products and compare with the needs of society and many other examples.

Figure 1.2 Cycle of data curing, analyzing, and decision making.

1.1.3 Historical data for forecasting future economy

Further application of time series modeling can be found in the study of economics (Fig. 1.3). With this tool, producers, sellers, and industry owners can observe and interpret markets in order to identify supply and demand requirements (Larsson and Nossman, 2011; Qiu, Ren, Suganthan, and Amaratunga, 2017). Moreover, by using the appropriate methods, they can forecast future demands in order to prepare themselves for potential downturns or periods of sustained economic growth. From the above discussion, we can see that time series play a significant role in almost all scientific and management fields. Therefore, an intimate knowledge of how to implement time series modeling and the required pre- and post-processing steps is vitally important (Ebtehaj, Zeynoddin, and Bonakdari, 2020; Moeeni, Bonakdari, and Fatemi, 2017; Zeynoddin, Ebtehaj, and Bonakdari, 2020).

Figure 1.3 Preprocessing, analyzing, and modeling data in economy.

1.2 Stochastic and stochastic with exogenous variables

1.2.1 Stochastic models

The ability to model, implement those models, and analyze model outcomes are fundamental skills that are required of many real-world applications. These applications span a diverse range of sectors from medical to civil and military (Moazamnia and Bonakdari, 2014; Mojtahedi, Ebtehaj, Hasanipanah, Bonakdari, and Amnieh, 2019; Momplot et al., 2012). In practice, in order to achieve an effective plan, appropriate modeling should be done with intelligible analysis and review, at the lowest cost.

Different approaches are needed to model phenomena and predict parameters. One of the approaches used to model phenomena and analyze time series is to use models, which, based on mathematical concepts and relationships describe systems and make it possible to predict future parameters and conditions. These models can be classified into three categories: (1) statistical models; (2) artificial intelligence (AI) models; (3) and integrated models (Fig. 1.4). Integrated models are formed by combination of statistical and AI models (Azimi, Bonakdari, and Ebtehaj, 2017; Bonakdari, 2011; Moeeni, Bonakdari, and Ebtehaj, 2017b; Moeeni, Bonakdari, Fatemi, and Zaji, 2017; Wang, Hu, Ma, and Zhang, 2015). Each category of models may be further sub-divided into several categories, each of which may be employed in the resolution of different types of problems. In this text, stochastic modeling, a sub-class of statistical methods, is presented. These models are lauded amongst industry members and the academic community alike as they have an easily comprehensible structure, can be readily applied to a variety of problems, and have high precision in short-term forecasts (Zeynoddin, Bonakdari, Ebtehaj, Azari, and Gharabaghi, 2020).

Figure 1.4 Abstract model categorization.

1.2.2 Stochastic model structure

Stochastic models are tools for estimating the probabilistic distributions of results using random variables by one or more inputs. These random variables are based on the fluctuations, observed in historical data for a given period using data standardization techniques. These models can be used to predict time series, such as the amount of precipitation, number of patients, customers, or any other parameter at a given time in the future. In these models, it is assumed that the data are stationary and normal (Brockwell and Davis, 2016; Zeynoddin and Bonakdari, 2019). Therefore, the datasets used for modeling must be prepared and preprocessed before modeling.

1.2.3 Model classifications

The first statistical stochastic model to be introduced was the auto-regressive (AR) model, which was able to establish a correlation between current and previous values in the series. This model quickly gained popularity due to the simplicity of its structure and is still used in many annual or seasonal modeling applications. The AR model works well for modeling phenomena whose parameters are relatively stable and exhibit relatively small changes without dramatic fluctuation. If the series changes exhibits dramatic fluctuations under certain conditions, such as an outbreak of a disease, a sudden growth in market share, or flood conditions in a river, the AR model will not perform well. In order to address this deficiency of the AR model, a new model was developed through the addition of moving average algorithm creating the auto-regressive moving average (ARMA) model. If there are significant seasonal fluctuations, seasonal ARMA models can be