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Espacios. Vol. 37 (Nº 30) Año 2016. Pág. 5

Forecasts Combination from the Perspective of Linear Correlation: A Systematic Review

Combinación de Pronósticos dentro de la perspectiva de la correlación lineal: Una revisión sistemática

Vera Lúcia Milani MARTINS 1; Liane WERNER 2

Recibido: 28/05/16 • Aprobado: 22/06/2016

Contents

1. Introduction

2. Forecasts Combination

3. Method

4. Results and Discussions

5. Final considerations

Acknowledgements

References

1. Introduction

Forecasting methods are common themes in many researches in recent decades. The researcher's approach takes place since the application in several areas to propose new techniques. According to Egrioglu, Aladag & Yolcu (2013), it is clear that the forecasting activities play an important role in our daily life, what motivates these propositions. Furthermore, the research are motivated mainly due to computational advances and the need for improvements in corporate management, since the forecast demand generated through structured techniques often used to assist in decision-making process (Slack, 2007).

Over the years, many forecast techniques were developed. Each of these techniques has different ways to capture the information behavior of a data series. This way, it is natural to imagine that a prediction made ​​up of several of these techniques can represent more widely the characteristics of the data series. Thus, in 1969, Bates and Granger have submitted what is consider the initial model of forecasts combination (Wallis, 2011).

Since the submission of the combination model referred, almost five decades have passed. Approximately 20 years after the publication of this model, Clemen (1989) conducted an extensive literature review, comprising 209 publications on the subject. Simultaneously, in 1989, Granger published a reflection on the combination, emphasizing their evolution and perspectives. In 2011, Wallis published a study on an overview of the forecast combination, 40 years after the initial study. About three decades have passed since the first literature review on combinations, period which many proposals for combinations were done, also applications and comparative studies about the performance of the methods were performed. In some of these studies (Clemen, 1989; Makridakis & Hibon, 2000; Stock & Watson, 2004; Patton & Sheppard, 2009; Andrawis, Atyia, & El-Shishiny, 2011; Martins & Werner, 2012), several combinations of forecasts had on average, superior accuracy over the individual forecasts.

Some of these studies approach the correlation between the errors generated with individual forecasts. Some cases, the errors obtained through individual predictions are combined considering that these errors are independent events and assigning null value to the linear correlation, disregarding the effect of this in the calculation of the combinations weights (Werner, 2005; Elliott & Timmermann, 2005; (Andrawis, Atyia, & El-Shishiny, 2011) . In other studies, there are no reference to the type of relationship existing between the errors of the individual forecasts (Stock & Watson, 2004; Prudêncio & Ludermir, 2006; Patton & Sheppard, 2009). Considering the above, this study seeks to identify and understand the development over time of the studies using the linear correlation between the errors of the individual forecasts and their effects for forecast combination. The study presents a brief systematic review of the literature on this topic learned in specific scientific databases. For this review, are considered studies conducted after the literature review presented by Clemen (1989) and the reflection published by Granger (1989).

2. Forecasts Combination

To find a model that represents the reality and predict with efficiency it is the main objective of forecasters. For this purpose were developed different ways to obtain predictions. One of these forms congregate different predictions and is known as combined forecasts (Webby & O'connor, 1996).

According to Costantini and Pappalardo (2010), the forecasts combination is a method commonly used to improve forecast accuracy. The proposal to combine different forecasts initially presented by Bates and Granger (1969) and considered by Clemen (1989) an interesting method for forecasting. In addition, the literature indicates that the linear forecasts combination is generally more accurate than individual forecasts (Clemen, 1989; Makridakis & Hibon, 2000; Stock & Watson, 2004; Patton & Sheppard, 2009; Costantini & Pappalardo, 2010; Martins & Werner, 2012).

Many studies have been motivated by the initial proposal of the combination method. In 1974 Newbold and Granger published a comparative study of the techniques of individual predictions and combinations obtained by the method presented in 1969. This study also showed a method’s extension, its results indicate that there was a gain in accuracy when univariate forecasts were combined. In 1989, Clemen presented an extensive literature review on the subject and Granger also revisited the subject publishing a reflection on combinations twenty years after. Most recently in 2011 Wallis published a study on the scenario of the forecasts combination forty years after the seminal article. Chan, Kingsman & Wong (1999) presents a comparative study of combination methods applied to real data. The combination of continuous forecasts was the theme of the work of Yang (2004), with focus on meeting the theoretical assumptions of the models. Wang & Chang (2010) used the fuzzy neural network to combine forecasts for a panel manufacturing. Chen (2011) proposes a combined approach using both linear model and the nonlinear model, to the tourism demand forecasting.

Over the years, different combination methods have been proposed (Newbold & Granger, 1974; Makridakis & Winkler, 1983; Granger & Ramanathan, 1984; Lobo, 1991; Chan, Kingsman & Wong, 2004). However, one of the most popular methods of combining individual forecasts is still the arithmetic mean (Flores & White, 1989; Taylor & Bunn, 1999). Some results of comparative studies of different combination methods indicate that when the forecasting process is stable, the results are satisfactory, but when there is no stability, should be consider a change in the forecasts weights (Deutsch, Granger, & Teräsvirta, 1994.; Chan, Kingsman & Wong, 2004; Timmermann, 2006).

The minimum variance method proposed by Bates and Granger (1969) consists in to realize the linear combination of two previsions with different weights. In this method, the forecasts objective should be non-biased and the forecasts combination is obtained by assigning a weight to each of the individual forecasts to be combined. Its structure is shown as Equation (1).

Despite the evolution motivated by the method, the literature lacks studies which focus on the type of correlation between forecast errors or even if there is a correlation. The method of minimum variance is based on the variability and in the linear relationship between the forecasts errors therefore neglect this information can change the quality of the combined forecast.

 In this study, was sought to highlight the search opportunity related to the use of linear correlation between the forecast errors in the structures of the combinations. For cases in which non-stability is checked in the process, current situation demand data, several authors suggest considering a change in the weights of each individual forecast in combination (Deutsch, Granger, & Teräsvirta, 1994; Chan, Kingsman & Wong, 2004; Timmermann, 2006). A possible alternative to assign different weights in forecasts combination is to use the linear correlation, as the study presented in 1969.

3. Method

To realize this study it was performed a literature review in a systematic way. In this step, the goal is to list the methods of combining prediction and identify among existing methods, which one uses the linear correlation coefficient in its structure. To do this review, were used online form databases of several journals available.

The databases journals queried were: Scopus, J-STOR, Web of Knowledge, Scielo, Open Science Directory, Biblioteca do Conhecimento, Directory of Open Access Journals (DOAJ), NCBI PubMed, Science Direct, and Wiley Online Library. The survey was conducted by searching in the titles, abstracts and keywords through combinations of expressions: i) Combining forecasts and Linear Correlation; ii) Combining forecasts and Error Correlated; iii) Forecast combination and Linear Correlation; iv) Forecast combination and Error Correlated; v) Combined Forecasts and Linear Correlation; vi) Combined Forecasts and Error Correlated; vii) Combine Forecasting and Linear Correlation; viii) Combine Forecasting and Error Correlated.

To set a period for the search criteria, publications from the year 1989 until 2013 were consider. The year 1989 marks the limit of how far were covered the research on combined forecasts addressed by Clemen (1989) in their revision and notes on this subject. This study is considered referenced by the authors of the area for represent a complete revision to date, covering 209 research articles and books.

The analysis of the articles founded includes a count of the number of posts, pages and authors, the ratio of publications per year and per application area. Among the related articles were selected those that mention the linear correlation between the forecast errors, in order to identify research gaps and to direct research lines of future.

4. Results and Discussions

The research covers the queries held in databases of journals available online and performed by search keywords. Words and expressions searched were: combining forecasts, forecast combination, combined forecasts and combine forecasting. These expressions were related with the terms: linear correlation and error correlated. The search was limited to exploration for these words in the titles, abstracts and articles keywords. The publication period is also restricted to the period from 1989 to 2013.

At first, the search in all databases returned 141 articles, of which 69 were identified more than once, after exclusion 72 articles remaining. Even using the filters described above, some of these articles do not address the issue of combining forecasts and the correlation between the errors. After reading each article, it was identified a total of 32 works that really approach the theme.

The 25 journals, which were found 32 papers related to the themes are present in Table 1, with the publication’s number. It is clearly seen that there is no concentration of publications in the journals, indicating that the theme is approached in different areas.

Table 1. Journals, authors and pages numbers.

Regarding the number of publications found and their annual distributions, is possible to see that since the year 2005 there was an increase in the number of publications addressing the specific issues of combining forecasts, correlated errors and linear correlation. In the period between 1989 and 2004, the average it is only 0.56 publications per year, and in the years of 1990, 1991, 1993, 1994, 1998, 2000, 2001, 2002 and 2004 were not related publications regarding this topic. Figure 1 shows the number of publications per year, and the percentage corresponding to these publications from 1989 until 2013. It was noticed a growth of publications specifically in 2013.

In the publications observed, were related 91 different authors, including 4 that had more than one publication on the topic, they are, D. Ridley, G.Grell, J. Wilczak and S. McKeen. D. Ridley presented three papers in 1995, 1997, 1999, one of these studies showed a new way of combining forecasts and the main focus of their work is the method developed. G. Grell, J. Wilczak and S. McKeen presented two joint projects in 2007 and 2008 with primary focus on adaptation and application of methods of the natural sciences.

The areas of knowledge related by the journal discussed in this review were: Health Sciences, Mathematics, Natural Sciences, Social Sciences, Engineering and Operations Research Sciences. Table 2 shows the number of authors in each area of knowledge, the number of publications and the number of pages published.

Figure 1. Number of publications per year and the publications percentage.

According to Table 2 is possible to see that the highest number of publications focuses on the area of ​​Natural Sciences. The publications in this area are mainly related to phenomena of nature and mostly represent applications of methods of combining. Also noteworthy is the number of authors in this area which encompasses 59% of all referenced in this study and the number of published pages that representing 39% of the total compared to the other scientific areas. In the area of mathematics, it was observed that the publications had on average 25 pages, nearly twice the overall average that was 13.97 pages.

Table 2. Amounts and Percentages of Publications by Areas of Knowledge.

Regarding the approach used in the 32 articles selected, 15 accounted for applying the combination methods, 1 conducted a review and define guidelines for selecting forecast techniques and 16 were descriptions, adaptations, comparisons or proposing methods of combining forecasts. These were classified respectively as: Application, Review and Method. The time relation with the approach presented in the study and the knowledge area are presented in Figure 2.

According to Figure 2, the area of ​​Operational Research presents publications since 1992. While the fields of Engineering and Health Sciences presents publications since 1997. For the area of Natural Sciences, in the data bases surveyed, was found the first publication in 1999. The area of Mathematics presents its publications on this topic from 2003. Recently, in 2012, there was a publication in the field of Social Sciences, indicating an expansion of areas to investigate methods of combining forecasts observing the correlation between the errors.

Referring to Figure 2 is possible to visualize scarcity of publications relating to combination of forecasts and correlation between errors in the final period of the 1980s, 1990s and early 2000s. During this period, there were few publications, getting clear resumption of interest of the authors in of this topic publication from mid-2000s.

Figure 2. Timeline.

The articles considered in this study with the approach classified as method are presented only in the areas of Natural Science, Engineering and Operational Research. Observing these 16 articles, it was verified that the authors use with a higher frequency the follow theories: Artificial Neural Network (ANN), Least Square, Linear Regression and Linear Correlation. According to study presented by Paliwal & Kumar (2009), some of the commonly used traditional statistical techniques applied for prediction are multiple regression and logistic regression, most recently the ANNs have been used as an alternative to these techniques. The Figure 3 presents a brief theory description used in each article and how they were evaluated, besides presents the instruments used.

5. Final considerations

The review presented was performed digitally, included the search for keywords in different journal databases. It was obtained 141 items. Duplicated outcomes were eliminated resulting in 72 items. After reading the articles it was found that many did not address the theme effectively, these were eliminated remaining the 32 articles that comprise this study.

In these studies there was diversity of authors. Among the 91 authors identified, only 4 had more than one publication on the theme. The authors who have published more articles just one presented a new method and is the only author of his papers. While the other three authors presented two joint works with approach in the Natural Sciences. These results preclude the identification of a research center on combination of forecasting and correlated errors.

Figure 3. Description Articles.

About the articles approach observed just one presented a literature review on the application of methods, the remaining were about methods (16) and applications (15). Observing these approaches related to the timeline, there is a lack of publications in the 1990s and the resumption of studies from the mid-2000s. Also observed concentration of studies in the area of knowledge of Natural Sciences, especially in studies applied to the combination methods.

The number of articles found referencing the theme is relatively low, focusing mainly on applications and unfolding or proposed methods. There are few publications in journals related to the areas of Health Sciences, Social Sciences and Mathematics, as well as other areas of knowledge not detected by this research. It was not possible to identify a core of research on combined forecasts and correlated errors. In subsequent studies can detect the motivations related to growth of the application of this research line, particularly in the area of Natural Sciences.

Acknowledgements

This work was supported by CNPq, National Council for Scientific and Technological Development - Brazil.

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1. Industrial Engineering Department of the Federal University of Rio Grande do Sul, Brazil. Email: vlmmartins@yahoo.com.br
2. Industrial Engineering Department of the Federal University of Rio Grande do Sul, Brazil. Email: liane@producao.ufrgs.br