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HOME | ÍNDICE POR TÍTULO | NORMAS PUBLICACIÓN
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
ABSTRACT:
Several forecast combination methods were
proposed since 1969 when the initial technique
was presented. Some studies approach the
correlation between the errors generated with
individual forecasts, being null or not, mainly as a
way to assign weights to the forecasts combined.
Considering the above, this study seeks to identify
and to follow the development over time of the
studies using the linear correlation between the
errors in the forecast combinations. For this, the
study presents a brief systematic review of the
literature, using online form databases of journals
available between 1989 and 2013. The analysis of
the articles found contemplates the counting
publications, pages and authors, the ratio of
publications per year and per application area with
a focus on those that mention linear correlation and
a brief description about the methods used in some
articles. In the search were found 72 articles that
after reading resulted in 32 articles that composes
this study. In these articles, it was found that of the
91 authors, only 4 had more than one publication
on the subject. It was observed also concentration
of studies in the area of Natural Sciences.
Regarding the approach 15 papers accounted for
applying the methods of combination, one
RESUMEN:
Varios métodos de combinación de pronósticos
fueron propuestos desde 1969 cuando se presentó
la técnica inicial. Algunos estudios abordan la
correlación entre los errores generados con
pronósticos individuales, sean nulas o no,
principalmente como una forma para asignar pesos
a los pronósticos de combinadas. Teniendo en
cuenta lo anterior, este estudio busca identificar y
seguir el desarrollo en el tiempo de los estudios
utilizando la correlación lineal entre los errores en
las combinaciones de pronóstico. Para ello, el
estudio presenta una breve revisión sistemática de
la literatura, usando bases de datos de formulario
en línea de revistas disponibles entre 1989 y 2013.
El análisis de los artículos encontrados contempla
contar con publicaciones, páginas de los autores, la
relación de publicaciones por año y por área de
aplicación con un enfoque en que mención
correlación lineal y una breve descripción sobre
los métodos utilizados en algunos artículos. En la
búsqueda fueron encontrados 72 artículos que
después de la lectura resultó en 32 artículos que
componen este estudio. En estos artículos, se
encontró que de los 91 autores, sólo 4 tenían más
de una publicación sobre el tema. Se observó
también concentración de estudios en el área de
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conducting a review of approaches to the topic and
16 were descriptions, adaptations, comparisons or
proposing methods of combining forecasts.
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,
especially in to 2013.
Keywords: Forecasts Combination, Correlation
Errors, Systematic Review.
Ciencias naturales. Respecto a los 15 documentos
de enfoque contabilizados aplicando los métodos
de combinación, una realizando una revisión de
enfoques para el tema y 16 fueron descripciones,
adaptaciones, comparaciones o proponer métodos
de combinación de pronósticos. Observando estos
planteamientos relacionados con la línea de
tiempo, hay una carencia de publicaciones en la
década de 1990 y la reanudación de los estudios de
mediados de la década de 2000, especialmente en a
2013.
Keywords: Combinación de pronósticos,
correlación de errores, revisión sistemática.
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).
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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
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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.
Journal Articles Authors Pages
Published
Average of
Published
Pages
Applied Mathematical Modeling, 1 1 9 9.00
Computers & Industrial Engineering 1 1 11 11.00
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Computers & Operations Research 1 1 21 21.00
Energy 1 2 12 12.00
Energy and Buildings 1 4 12 12.00
European Journal of Operational Research 2 3 26 13.00
Expert Systems with Applications 1 2 8 8.00
Fisheries Research 1 5 13 13.00
International Journal for Numerical Methods and Fluids 2 7 26 13.00
International Journal of Climatology 1 1 14 14.00
International Journal of Energy Research 1 2 12 12.00
International Journal of Forecasting 1 2 20 20.00
International Transactions in Operational Research 1 1 12 12.00
Journal of Forecasting 2 5 23 11.50
Journal of Geophysical Research: Atmospheres 1 13 20 20.00
Journal of Hydrology 3 5 41 13.67
Journal of International Money and Finance 1 3 38 38.00
Journal of Natural Gas Science and Engineering 1 4 12 12.00
Journal of Statistical Planning and Inference 1 2 28 28.00
Procedia - Social and Behavioral Sciences 1 2 5 5.00
Quarterly Journal Of The Royal Meteorological Society 1 6 10 10.00
TELLUS (A and B) 3 18 40 13.33
The American Journal of Emergency Medicine 1 2 4 4.00
Tourism Economics 1 1 16 16.00
Water Resources Research 1 2 14 14.00
Total 32 95 447
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.
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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.
Areas of
Knowledge Articles Authors Pages
Published
Average of
Published
Pages
Health Sciences
1 2 4
4.00
3% 2% 1%
Mathematics
3 6 75
25.00
9% 6% 17%
Natural Sciences
13 56 176
13.54
41% 59% 39%
Social Sciences
1 2 5
5.00
3% 2% 1%
Engineering
5 14 61
12.20
16% 15% 14%
Operations 9 15 126 14.00
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Research 28% 16% 28%
Total
32 95 447
13.97
100% 100% 100%
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.
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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.
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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
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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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22/11/2018 Revista ESPACIOS | Vol. 37 (Nº 30) Año 2016
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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
Revista Espacios. ISSN 0798 1015
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