Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

Comparative analysis of statistical and fuzzy integrated
time series and judgmental forecasting: an empirical
study of forecasting dry bulk shipping index
Okan Duruab, Shigeru Yoshidab（）

Contents

Abstract
1. Introduction
2. Methodology
3. Empirical results
4. Conclusion

Abstract
This paper investigates the accuracy of some statistical, judgmental and fuzzy timeseries methods for dry cargo freight market forecasting purposes. For statisticalextrapolation, the X12 ARIMA, TRAMO/SEATS and Holt-Winters exponential smoothing
methods are applied. In judgmental prediction, traditional expert opinion and Delphigroup decision are performed in an expert group. Additionally, the recent method, fuzzytime series, is used as Chen（1996）s procedure. Judgmental forecasting improved its
superiority among the most of the benchmark methods. Fuzzy time-series methodology
also provided relatively better accuracy than the statistical extrapolation.

１．Introduction
Forecasting shipping markets is attempted by many scholars and various methods areapplied. Most of the techniques are based on time series analysis and econometric causal
models. Especially in regular seasons of freight markets, such a statistical approachgive comparatively accurate results. However, freight markets had several unexpectedfluctuations due to war fare, sporadic economic boom/bust cycles, structural breaks
etc. In long term forecasting, causal models give a baseline and important drivers
like productivity or population can be quantitatively embedded to the models except1Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

technology cycles. Technology cycles should be judgmentally reviewed and predicted.
In short run, there are many judgmental factors effecting freight markets such aspolitical declarations, crowd feed etc.（Duru and Yoshida, 2008a, b）. Prediction of freightrates between weeks and months still has difficulties both in statistical methods andjudgmental methods.
The main research motivation of this paper is forecasting in weeks and months. For
statistical extrapolation, we choose automatic prediction tools such as the X12 ARIMA
and TRAMO/ SEATS, and an exponential smoothing method, Holt-Winters algorithm.
In judgmental forecasting, traditional expert opinion and Delphi group decision methods
are applied in a small expert group. Finally, an expert system method, Fuzzy integrated
time series methodology is also compared.
A new generation forecasting tool is the fuzziﬁcation of time series and forecasting withfuzzy logical rules. Rule based forecasting is suggested by Collopy and Armstrong（1992）
for structuring dynamics of systems by its features, and it presents an expert systemprocedure of prediction and combining of multiple methods under predetermined rulesof the series. Collopy and Armstrong（1992）proposed a procedure which mainly driven
by IF-THEN rules. IF-THEN modelling strategy is also employed by fuzzy logic type
expert systems similarly for computer intelligence integration. Rather than a complex
procedure, fuzzy time series is building rules for historical pattern. Fuzzy method is
also capable to analyze chaotic time series（Palit and Popovic, 2005）. As one of thebenchmark methods, fuzzy time series approach will be compared with judgmental
results.
Song and Chissom（1993a, b）first attempted to implement fuzzy set theory forforecasting task. The famous data set, the student enrolments of University of Alabama,is frequently used in the next studies to compare with previous fuzzy applications.
Shortly afterward, a stream of fuzzy time series existed to improve accuracy ofthe current approaches. Chen（1996）developed the initial study and also ensured
robustness of the forecasts. The main difference is presented that the study uses
simpliﬁed arithmetic operations instead of max-min composition operations of Song andChissom（1993a）. Huarng（2001）introduced heuristic fuzzy time-series model. Highorder fuzzy time series approach is also suggested by Chen（2002）. A time-invariant
fuzzy forecasting model is based on fuzzy logical relationships such as

〜

→

〜

, and

consolidates these logical connections as their rules for forecasting values. Chen（1996）s
methodology is one of the accurate model for prediction, and this paper purposes toimplement this method for freight rate task.
Forecasting practice is carried out on Baltic Dry Index（BDI）which is a composite
price of various cargoes, routes and ship sizes. BDI is ﬁxed for every trading day by
Baltic Exchange, London and it is also reﬂecting both voyage domain and time domain2Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

contracts.

２．Methodology
Forecasting of BDI index is conducted with three statistical methods, judgmental
forecasting and fuzzy time-series method. The brief description of methods used in thisresearch is as follows:

The present paper applied two automatic extrapolation method, the X12 ARIMA andTRAMO/SEATS, and an exponential smoothing method, Holt-Winters algorithm. The
X12 ARIMA seasonal adjustment algorithm is an improved version of the X11 ARIMA（Shiskin et.al., 1967; Findley et.al., 1998）. The method automatically detects seasonal
ARIMA conﬁgurations and applies for forecasting. TRAMO/SEATS program can treat
missing observations and outliers, and also estimate unobserved data. As in the X12
ARIMA, TRAMO/ SEATS also uses ARIMA-model-based methodology for extrapolation（Gómez and Maravall, 1996）.
Holt-Winters exponential smoothing is well known time series method which frequently
used in forecasting practice（Makridakis, Wheelwright, and Hyndman, 1998）. Holt
（1957）and Winters（1960）extended the classical form of exponential smoothing andconsolidated some improvements. Holt-Winters method can be applied to both additiveand multiplicative trended series.
Since the present paper attempts to use the mentioned statistical methods, thedetection of whether seasonal factors have unit roots has a critical role. Many scholars
indicated about the drawbacks of routine adjustment of seasonality（Barsky and Miron,
1989, Hyllberg et. al. 1990）. Barsky and Miron（1989）pointed out that the routine
elimination of the seasonal cycles is concluded by losing important information about theﬂuctuations. In existence of stochastic seasonality, Box-Jenkins type ARIMA modelling
can be proper for a univariate series. However, it is reported that spurious regression
results are probable if the series is not diﬀerenced at the seasonal frequency. Hyllberg
et. al.（1990）suggested a testing procedure for seasonal unit roots（HEGY test）, andthat method is also developed for monthly frequencies by Franses（1990）and Beaulieu
and Miron（1993）.
A seasonal unit root test is performed for the series of the BDI, and the results areprospective. The historical backgrounds of two empirical studies have seasonal unit
roots, and adjustment or seasonal diﬀerencing will be rational. The tests are conductedby monthly HEGY test for the models that is based on intercept only, intercept andtrend, and intercept, trend and seasonal dummies. Constant, trend and seasonal dummy

3Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

variables are found insigniﬁcant in most. The signiﬁcance tests of seasonal frequencies
are conducted by the reference -test and-test tables of Franses and Hobijn（1997）.

The single and joint existence of seasonal unit roots is indicated present in the applied
datasets. Therefore, smoothing and adjustment of seasonal diﬀerences will be consistent
as in the X12 ARIMA, Tramo/Seats and Holt-Winters methods.

After the development of fuzzy set theory, time-series forecasting is one of the good
example of fuzzy extended studies. Zadeh（1965）introduced fuzzy numbers, and it isused for many diﬀerent applications in the last half century. Fuzzy logic algorithms are
implemented for various engineering problems which mainly include some degree of
uncertainty. Uncertainty is one of the unique problems of several time series and it canbe improved by fuzziﬁcation of the data.
Fuzzy time-series methodology is appeared to implement the uncertainty of series andin some empirical studies, it provides higher accuracy. The basic principals of fuzzy set
theory and fuzzy time series are as follows:Ain a universe of discourse
which is a map:

is characterized by its membership function→［0, 1］. The value ofthe point θ in the fuzzy set（θ）is theof

. The fuzzy sets with boundaries that are not precise, it ispossible to take into account the uncertainties inherent in the systems.
Deﬁnition 1. （ ）（ =...,0,1,2,...）, is a subset of. Let （ ）be the universe of discoursedeﬁned by the fuzzy set μ（ ）. If （ ）consists of μ（ ）（ =1,2,...）, （ ）is called a fuzzytime series on （ ）.
Deﬁnition 2. If there exists a fuzzy relationship （ -1, ）, such that （ ）= （ -1） ◦（ -1, ）, where ◦ is an arithmetic operator, then （ ）is said to be caused by （ -1）. The
relationship between （ ）and （ -1）can be denoted by （ -1）→ （ ）.
Deﬁnition 3. Suppose （ ）is calculated by （ -1）only, and （ ）= （ -1）◦ （ -1, ）. For
any , if （ -1, ）is independent of , then （ ）is considered a time-invariant fuzzy timeseries. Otherwise, （ ）is time-variant.
Definition 4. Suppose （ -1）=
deﬁned as

〜

→

〜

where

〜

and

〜
〜

and （ ）=〜

, a fuzzy logical relationship can be

are called the left-hand side and right-hand side of the

fuzzy logical relationship, respectively.
The current fuzzy time series models（Song and Chissom, 1993a, b, 1994; Chen, 1996,
2002; Hwang et al., 1998; Chen and Hwang, 2000; Huarng, 2001a, b; Lee and Chou, 2004）
utilize discrete fuzzy sets to deﬁne their fuzzy time series. Fuzzy sets can be based on
triangular or trapezoidal shapes. The present work applies trapezoidal fuzzy numbers（Liu, 2007）.

4

Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

The research steps of the fuzzy time series extrapolation of Chen（1996）can be
described as follows:
Step 1. Collect the historical data.

Step 2. Deﬁne the universe of discourseminamong all. For easy partioning of

The universe of discourse
=[

min

-

,

1

+

max

2

. Find the maximum

max, two small numbers

1

and the minimumand2

are assigned.

is then deﬁned by:]

Step 3. Determine the appropriate length of interval . Here, the average-based length
method（Huarng, 2001b）can be applied to determine the appropriate . The length ofinterval is computed by the following steps:
a．Calculate all the absolute diﬀerences between the values

-1

and

as the ﬁrst

diﬀerences, and then compute the average of the ﬁrst diﬀerences.
b．Take one-half of the average as the length.
c．Find the located range of the length and determine the base（Table 2）.
d．According to the assigned base, round the length as the appropriate .
Step 4. Define fuzzy numbers. The number of intervals（fuzzy numbers）,, is

computed by
=（

max

+

-

2

min

Thus, there are
〜

+

1

）/ .

Eq. 4

intervals andfuzzy numbers, which are, respectively. Assume that the= [ 4, 5],...,
numbers,

-3
〜

,

1

〜

-3

,

〜

, ...,-2],=（ 0, 1, 2, 3）,2=（ 1, 2, 3, 4）,

3=（ 2, 3, 4, 5）,

〜

...

= [

-2

-2

1

,

-1

-1

],

-1

= [

= [ 1, 2],,], and2,..., m, and2= [ 2, 3],
= [

,

3
+1

〜

,

1

〜

2

, ...,= [ 3, 4],4]. The fuzzy

can be deﬁned as follows:

2

1
〜

...

〜

-1
〜

= [

〜

intervals are

,

1

=（

=（

-2
-1

,

,

-1

,

,

,
+1

+1

,

+2

）
,

）
,

Step 5. Fuzzify the historical data. If value of
belongs to fuzzy number

〜

. All

is located in the range of, then it

must be classified into the corresponding fuzzy

numbers.
Step 6. Generate the fuzzy logical relationships. For all fuzziﬁed data, derive the fuzzy
logical relationships based on Deﬁnition 4. The fuzzy logical relationship is likewhich denotes that if the-1

value of time -1 is〜, then that of time is

〜

〜

→

〜

,

.

Step 7. Establish the fuzzy logical relationship groups. The derived fuzzy logicalrelationships can be arranged into fuzzy logical relationship groups based on the same

5Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

fuzzy numbers on the left-hand side of the fuzzy logical relationships.
The fuzzy logical relationship groups are like the following:
〜

〜

→

〜

→

〜

,

1
〜

,

2
〜

→

,

3

...
〜

→

〜

,

Step 8. Calculate the forecasted outputs. The forecasted value at time ,, isdetermined by the following three heuristic rules. Assume the fuzzy number of-1〜time -1 is.

Rule 1. If the fuzzy logical relationship group ofis〜, which is（

-1

, ,

+1

,

of

is

, which is（

3

,...,

〜

→

〜

1

＝
＝

where

〜

＋

, ,

,

-1

＋...＋

＋...＋

1

=（

〜

is one-to-one;

〜

→

〜

, then the value

+2

〜

〜

is one-to-many;

→

〜

,

〜

1

→

〜

,

2

〜

→

is calculated as follows:

〜

2

1-1

〜

→ φ, then the value of）.

+1, and then the value of〜〜is empty;）.

Rule 3. If the fuzzy logical relationship group of
〜

〜

+2

Rule 2. If the fuzzy logical relationship group of
〜

at

-1

,

1-1

＋...＋

1

,

1

,

1+1

,

）,

1+2

1+1

,

〜

=
（

2

＋...＋

+1

,

2+1

2-1

,

2

,

1+2

,

2+2

＋...＋

）
,...,and

〜

+2

.

=
（

-1

,

,

+1

,

+2

）
.

Every forecasting work is subject to managerial review, and some of them will berevised and improved to damp the current factors, and also include judgmental factors.
In managerial level or expert group level, judgmental forecasting is one of the uniquesolutions for sporadic situations and unexpected market movements. When ﬂuctuations
depend on judgmental factors, crowd psychology, or political issues, expert judgments
can reﬂect market psychology to the results.
For judgmental forecasting, expert opinion and Delphi group consensus methods areproposed to maintain the subjective exercise of the forecasting study. These methods
are implemented in many judgmental forecasting studies. Expert opinion is a basic
method based on individual decisions, and it is a single iteration application. Expert
decisions are composed of individual forecast of the BDI. Result of expert judgments
is consolidated by simple average method. A multi-iteration method, Delphi, is also6Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

performed as a second judgmental practice（Rowe and Wright, 1999）. The advantagesof Delphi exist from its anonymous, multi-session, and feedback structure. A group of
experts is required to predict BDI in one and two period ahead horizons. The expert
group has no interaction with each other. The group is required to predict twice and in
the middle of two iterations, a summary report of the ﬁrst round is provided.
The judgmental study is performed in a small group（8 experts in expert opinion study
and 9 experts in Delphi work）and experts are selected from freight negotiation ﬁeld
such as shipbrokers or chartering managers etc. There are many judgmental works
based on a small group（Rowe and Wright, 1999）, but the number of experimental
work can be considered limited. In practice, organization of such an empirical study has
several diﬃculties. Although we guaranteed to keep information anonymous, most of theexperts do not accept to share their opinions. However, most of the shipping companies
use their judgments in managerial level in practice. Experts of the present study aremainly from Turkey, but also some of them are from Singapore and U.K.

３．Empirical results
This study attempts to compare statistical extrapolation, fuzzy logic extrapolation andexpert judgments for forecasting of the BDI data series. The forecasting accuracy of themethods is assessed with the absolute percentage error（APE）for single predictions
and the mean absolute percentage error（MAPE）and root mean squared error（RMSE）metrics for overall accuracy comparison. Table 1 shows the accuracy resultsof proposed methods. Expert judgments exposed superiority over statistical and fuzzy
logical forecasting practice. Statistical methods particularly resulted incapable for shortterm. Fuzzy logical time-series prediction improved inferior, but adjacent output versus
judgmental forecasting. Delphi group consensus practice could not improve prediction
accuracy rather than a single iteration expert opinion study.

４．Conclusion
The present study pointed out relative accuracy of judgments and fuzzy time seriesrather than univariate time series methods. In spite of small number of repetitive works,
the study indicates possibility of superior performance in decision methods.
Market intentions have a key role particularly in short term decisions such as trading
between spot and time charter markets, assessment of sale and purchase of older ﬂeet,
short term asset management. For this type of decisions, managerial decision making
strategy and developing company prediction are also crucial. For future extensions of
this study, design of company judgmental prediction system or judgmental adjustment ofstatistical results is considered to be beneﬁcial to improve practical applications.

7Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

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9

10January 08
March 08
June 08

July 08
August 08

1 month
3 months
6 months

1 month
2 months

60.54
42.85
0.70
25.64
62.38
38.42
3179.94

17.52
50.85
41.18
3428.06

17.95
54.49
36.35
3057.47

15.27
39.14
23.73
2059.41

0.99
24.30
16.31
1535.34

31.23
23.19
1.83

72.40
54.46
10.65

Tramo/ Seats

63.16
45.20
0.96

Holt-Winters

38.08
22.79
3.37

Table. Judgmental, statistical and fuzzy time-series forecast errors（MAPE & RMSE）
Forecast horizon
Date
Judgmental
Fuzzy time-series
X12 ARIMA

Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

Comparative analysis of statistical and fuzzy integrated time series and judgmental forecasting: an empirical study of forecasting dry bulk shipping index

ProﬁlesOkan Duru is a candidate for doctor's degree at the Graduate School of Maritime
Sciences, Kobe University, Japan. He received Master Degree in Supply-demand analysis
of Turkish coal import shipping from Istanbul Technical University, Turkey. He iscurrently member of International Association of Maritime Economists and International
Institute of Forecasters. The present studies are judgmental and statistical forecastingof shipping markets and industry, and composite forecasting structures for shipping
markets. He got Master Degree in Freight Forecasting of Dry Bulk Market.
Shigeru Yoshida is a professor at the Graduate School of Maritime Sciences, Kobe
University, Japan. He specializes in shipping and transport economics and management
and his main published work is "Growth and Competitiveness of Modern Japanese
Shipping Industry". He is the current vice-president of Japan Society of Logistics andShipping Economics（JSLSE）and former chief editor of Journal of JSLSE.

11