Bull. Hiroshima Inst. Tech. Research Vol.48（2014）87-91

ArticleRevisit to Modelling Global Annual Average Temperature - A Parametric Approach

Satyabrata PAL*, Arunava GHOSH**, Subhabaha PAL*** and Sanpei KAGEYAMA****（Received August 22, 2013）

Abstract
Global annual average temperature is regarded as a precise indicator of the warming of the globe
over centuries and if we track the available record on it, it is observed that the available period of
133 years (1980-2012) can be partitioned into three distinct phases, 1880-1936, 1937-1980 and 19812012, based on temperature variation pattern (Pal et al., 2013). The content of the said paper deals
with the development of models, parametric and nonparametric, which produce high precision levels (as measured by value of R2 coefﬁcient) as high as 0.84. In this paper, a class of parametric models has been developed which is a combination of three mathematical functions, namely, cubic, exponential and trigonometric, respectively. An extensive search of models belonging to the abovementioned class identiﬁes four different models whose R2 values lie within the range, 0.826 to 0.860
as a distinct improvement in the value of R2 vis-a-vis the precision level, over that obtained in Pal etal. (2013).
Key Words: Global warming, Parametric models

having greater precision levels, as measured by R2values) to the models available in the literature so far
surveyed. After wide search in the quest for superior
models it is found that a combination of three models (all distinct in nature), namely, cubic, exponential
and trigonometric, produces higher values of R2 coefﬁcient as high as 0.86. The identiﬁed class (whichis, indeed, a combination) contains four models denoted by M1, M2, M3 and M4, respectively. The model equations, predicted values under different models, standard errors of estimates and 95%conﬁdence
limits are given in Tables 1, 2 and 3, respectively.
Table 4 contains the values of different precision criteria in respect of the above four models. The graphplots of the four models along with the data set are

1. Introduction

The aspect of global warming is the most dreadfulenvironmental phenomenon the impact of which islooming large as time advances. The concern is thathuman activities are the principal factors triggering
this alarming state to which the future of the earth isalmost bound to reach unless appropriate measures
are devised to annihilate the underlying effects ofglobal warming.
Recalling the partitions mentioned in Pal et al.
(2013), it is observed that the third phase (1981 2012) is the most critical period. The present study
is referred to this phase, and efforts have been madeto develop models which are superior (in terms ofAgricultural Statistics, Bidhan Chandra Krishi Viswavidyalaya, Faculty of Agriculture, Mohanpur, Nadia, West Bengal, 741252, Indiae-mail: satyabrata.pal@hotmail.com＊＊＊＊
Uttar Banga Krishi Viswavidyalaya, Department of Agricultural Statistics, Pundibari, Coochbehar, West Bengal, 736165, India＊＊＊＊e-mail: arunava_ubkv@yahoo.co.in＊＊＊＊
SAP Development Lead, Manipal Global Education Services, Bangalore, Karnataka, 560002, Indiae-mail:subhabaha@msn.com＊＊＊＊
Department of Environmental Design, Hiroshima Institute of Technology, Hiroshima, 731-5193, Japan＊＊＊＊e-mail:s.kageyama.4b@it-hiroshima.ac.jp
＊＊＊＊
＊＊＊＊

― 87 ―

Satyabrata PAL, Arunava GHOSH, Subhabaha PAL and Sanpei KAGEYAMA

where ε’s are errors and follow NID (0, σ 2), the values of u and v are ﬁxed at 5 and 3 in case of the models, M1, M2 and M3, and the values of u and v areﬁxed at 3 and 4 in case of the model M4 respectively.
Also, t means years.
Remark: In case of each model, various values forthe pair, (u, v), are tried but the optimum value forthe pair (u, v), as given above, maximizes R2 value
obtained by fitting the respective model. For eachsuch model, the remaining coefﬁcients are estimatedtill the global convergence with respect to each suchis reached after successive iterations.

given in Annexure.
Section 2 is devoted to a description of thesource of data and the method employed in the paper. Section 3 presents the results and subsequentdiscussions on the findings evolved in the paper.
The last section, Annexure, displays the four graphplots (along with plots of distribution of residuals) inrespect of the four models respectively.
Over all, it can be mentioned that the two models, M1 and M2, have been identified and these twomodels are superior to the models obtained in Pal etal. (2013), diagnostic check results (on independence and normality of the estimated model errors)
being satisﬁed in case of these two models.

3. Results and Discussions

This section contains the results obtained after ﬁtting
the models to the data. These are given in four tables, Table 1 to Table 4 respectively. Also given are
the results on diagnostic checking (Draper andSmith, 1998) on the errors obtained after fitting themodels (vide Table 5). It is evident from the p-values (Shapiro Wilk test and Kolmogorov Smirnovtest) and z-values (Run Test) listed in Table 5 that theerrors (after ﬁtting the model in each case) are independent and also are normally distributed, implyingthat the results on diagnostic checks are satisﬁed. In
fact, it can be seen from Table 5 that the probability
values/levels are greater than .05/.025 as the casemay be, and hence the null hypotheses concerning
independence and normality can’t be rejected, or inother words the model errors are independent and arenormally distributed.

2. Materials and MethodsThe data sources are:
IPCC Report 2007 - http://www.ipcc.ch/publications_and_data/publications_and_data_reports.shtml#1
Temperature Data Source - http://data.giss.nasa.
gov/gistemp/tabledata_v3/GLB.Ts+dSST.txt
This paper has taken into consideration the parametric and nonparametric models considered in Pal
et al. (2013). Different structures of models createdas combinations of mathematical functions of various kinds are here explored.
The structure of the combination which is foundas the best is of the type: Cubic + Exponential +Trigonometric, and the model expression is givenbelow.
y = a + bt + ct2 + dt3 + het + qsinu(wt) － rcosv(zt) + ε,Table 1: Estimated model equations

ModelEquations

M1: (u=5, v=3)

y = 14.20 － 0.01t + 0.002t － 0.00004t － 103E-17et + 0.057sinu(887.7t) + 0.057cosv(117.7t) + ε

M2: (u=5, v=3)

y = 14.24 － 0.02t + 0.003t2 － 0.00007t3 + 1.05E-15et － 0.076sinu(－54.9t) － 0.046cos (－0.957t) + ε

M3: (u=5, v=3)

y = 14.19 － 0.009t + 0.002t2 － 0.00004t3 － 147E-17et － 0.04sinu(－45.47t) + 0.06cosv(0.774t) + ε

M4: (u=3, v=4)

y = 14.16 － 0.006t + 0.002t2 － 0.00004t3 － 142E-17et － 0.06sinu(－0.83t) + 0.04cosv(－1.24t) + ε23v

For example, 103E-17et = 103.10-17et― 88 ―

Revisit to Modelling Global Annual Average Temperature - A Parametric Approach
Table 2: Predicted values under different models

Year

Observed values

1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
20112012
2013

14.28
14.09
14.27
14.12
14.08
14.15
14.28
14.35
14.24
14.39
14.38
14.18
14.20
14.28
14.43
14.32
14.45
14.61
14.39
14.40
14.52
14.60
14.60
14.52
14.65
14.59
14.62
14.49
14.59
14.66
14.55
14.56

NA

Predicted values
M2
M3
14.13514.25714.20614.18714.30414.12914.212
14.12614.13614.19914.16914.207
14.21314.21614.21614.29214.22314.28914.30614.27414.29514.26614.27714.22814.27114.29614.36414.37614.37814.38514.39414.42314.45514.45814.512
14.48614.45614.457
14.39214.387
14.52214.47014.627
14.57114.60414.55814.50514.591
14.57914.626
14.61214.621
14.56914.603
14.532
14.545
14.611
14.56214.645
14.611
14.56814.59614.55414.55014.66914.321

M1
14.23614.13514.17814.24614.182
14.13414.24514.27014.19514.25814.37114.30514.22114.35214.44014.37914.39414.51014.47914.40514.511
14.59914.53314.51014.63214.64314.561
14.57914.633
14.59014.54614.56714.417M4

14.18714.23914.19614.167
14.17614.131
14.20614.24314.28514.33414.27414.29014.28114.306
14.39614.40914.47714.48114.45314.47214.44314.512
14.57314.60114.643
14.59014.583
14.57514.562
14.613
14.59214.55114.346NA-Not available

Table 3: Standard errors of estimates and 95% conﬁdence limits

M1
M2
M3
Modelcoefﬁcients
SE
LCL
UCL
SE
LCL
UCL
SE
LCL
UCL
a0.07
14.06
14.34
0.07
14.1
14.3
0.07
14.0
14.3
b0.02
－0.05
0.03
0.02
00
0.02
00c
0.00
－0.00
0.01
0.00
000.00
00d
0.00
－0.00
0.00
0.00
000.00
00h
0.00
0.00
0.00
0.00
00
0.00
00q
0.04
－0.02
0.13
0.03
00
0.03
00r
0.03
－0.12
0.01
0.03
00
0.03
00w
0.02
887.7
887.8
0.01
－54
－54
0.02 －45
－45
z
0.02
117.6
117.7
0.02
－1
0
0.02
0.74
0.81
SE = standard error; LCL = lower conﬁdence limit; UCL = upper conﬁdence limit

SE
0.08
0.02
0.00
0.00
0.00
0.03
0.05
0.02
0.03

M4
LCL
14.01
－0.05
0.00
0.00
0.00
－0.13
－0.15
－0.88
－1.29

Table 4: Values of different precision criteria for four models

Models(e.d.f. = 23)

MSE (meansquare error)

R2MAE (meanabsolute error)

Fo

Pr. (F greater than Fo)

M1: (u=5, v=3)0.00599

0.860

0.052817.71> 0.0001

M2: (u=5, v=3)0.006100.858

0.053017.35> 0.0001

M3: (u=5, v=3)0.006840.840

0.059315.14> 0.0001

M4: (u=3, v=4)0.007420.826

0.063213.73> 0.0001

e.d.f. = error degrees of freedom― 89 ―

UCL
14.32
0.04
0.01
0.00
0.00
－0.00
0.07
－0.79
－1.18

Satyabrata PAL, Arunava GHOSH, Subhabaha PAL and Sanpei KAGEYAMA
Table 5: Diagnostic checking for four models

Test for Normality
Models

Shapiro-Wilk test
Statistic (W)Test for Independence

Kolmogorov-Smirnov test

p-value

Statistic (D)p-value

Run test
z-valuep-valueM1: (u=5, v=3)

0.979

0.76

0.098

> 0.15

0.822

0.411

M2: (u=5, v=3)

0.971

0.543

0.121

> 0.150.00

1.00

M3: (u=5, v=3)

0.961

0.30

0.116

> 0.15

1.286

0.199

M4: (u=3, v=4)

0.966

0.396

0.091

> 0.15

0.639

0.523

W: Value of Shapiro-Wilk test statistic. D: Value of Kolmogorov-Smirnov test statistic

ANNEXURE
Here, the subscript, ‘t’ denotes year, e.g., t = 1, 2, 3, …

Oi represents the observed value of temperature at the i-th
point of time, i.e., yi.

mean ﬁrst, second, third, … years, respectively.

M1

M2

M3

M4

― 90 ―

Revisit to Modelling Global Annual Average Temperature - A Parametric Approach

M1 : (u=5, v=3)

M2 : (u=5, v=3)

M3 : (u=5, v=3)

M4 : (u=3, v=4)

References
Draper, N. R. and Smith, H. (1998). Applied Regression

Pal, S., Pal, S. and Kageyama, S. (2013). Modelling theAnalysis, John Wiley and Sons.

Global Mean Temperature. Bull. Hiroshima Inst.

Tech. Research 47, 149-152.

― 91 ―