Bull. Hiroshima Inst. Tech. Research Vol.49（2015）89-95

ArticleGlobal Temperature-Quest for Models with Superior Fitting

Satyabrata PAL*, Arunava GHOSH**, Sanpei KAGEYAMA*** and Kazuki MATSUBARA****（Received September 22, 2014）

Abstract
The record on the data on global annual average temperature over the last thirty two years presents a continuous increase (though very mild) in temperature over years and this warming of theglobe is, indeed, a matter of deepest concern to all of us at the moment. It is required to develop appropriate measures to contain the temperature-rise and when such measures (which perceptibly andmost satisfactorily address the problem) are created, it is our mandate to implement those holistically in order to save the human civilization from a perilous future. Global temperature models expose
the inherent trend existing in the temperature-record. A literature-survey on modelling of global
temperature reveals the following articles, Pal and Pal, 2011, Pal et al. (2013) and Pal et al. (2014),
wherein the precision levels (in terms of R2 values) of the models did not exceed 0.89.This paper
presents superior global annual average temperature models (parametric based on mathematicalfunctions) with precision levels (in terms of R2 values) varying in the range, 0.875 to 0.940.
Key Words: Greenhouse gases, Global warming, Parametric models

tions of the climate system in combination with theoretical understanding of its physics and general circulation models themselves based on the fundamental
laws of physics. Sophisticated statistics are used todemonstrate the signiﬁcance of recent changes in theclimate system. In understanding the climate change afaithful analysis of the record of global temperaturespartially accounts for the impact of such changes.

1. Introduction

The phenomenon of global mean warming observed
annually since the past 19thcentury is far outside therange of observational uncertainty in global temperature datasets, and there is therefore no shade of doubtthat the world has warmed. A wide range of observed
climate indicators continues to show changes that areconsistent with a globally warming world, and ourunderstanding of how the climate system responds tothe rising greenhouse gas levels needs to be enrichedand development of appropriate measures is the needof the hour. The analysis of the nature and causes ofclimate change is based on comprehensive observa-

This paper is devoted to a formulation of mostsuperior models, the premise and caveat are that thedata set on global average annual temperature available on NASA’s site is regarded as the prime base on
which an extensive modelling work has been under-

Sampling and Official Statistics Unit, Indian Statistical Institute, Kolkata, 700108, India,
e-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＊＊＊＊
Department of Environmental Design, Hiroshima Institute of Technology, Hiroshima, 731-5193, Japan,＊＊＊＊
e-mail: s.kageyama.4b@it-hiroshima.ac.jp＊＊＊＊
Matsunaga High School, Fukuyama, 729-0112, Japan＊＊＊＊e-mail: additivestructure@gmail.com
＊＊＊＊
＊＊＊＊

― 89 ―

Satyabrata PAL, Arunava GHOSH, Sanpei KAGEYAMA and Kazuki MATSUBARA

taken. Recalling from the partitions (in entire timetrajectory of 132 years, 1880-2012, mentioned in Pal
and Pal, 2011, Pal et al., 2013 and Pal et al., 2014), it
is observed that the third phase (1981-2012) is themost critical period; reasons had been explained in
the above papers appeared in 2011 and 2013 respectively. The study undertaken in this paper refers tothis phase, and most superior models, which arestructured by taking into account combinations offourth-degree polynomial functions, exponentialfunctions and trigonometric functions on the time
variable‘t’, have been presented. Such models aresuperior (in terms of having greater precision levels,
as measured by R2 values) to the models available in
the literature, so far surveyed. The developed models(ﬁve in numbers and represented by the symbols, M1,M2, M3, M4 and M5) have their precision levels (interms of R2 values) varying in the range, 0.875 to0.940.The diagnostic test results concerning the builtup of the models conﬁrm that the assumptions on independence and on normality hold unequivocally.

2. Materials

The 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
www.nature.com/news/earth-summit-rio-reportcard-1.10764
3. Methods

The general structure of the model equation isyt = a + bt + ct2 + dt3 + pt4 + fgt+ het +q sinu(wt)－ r cosv(zt) +εt,where a, b, c, d, p, f, h, q, u, w, r, v and z are thecoefﬁcients involved, and εt＇s are errors and follow
NID (0, σ 2).
The estimation of the parameters and developmentof the models are done using SAS (9.2) Software developing programmes on PROC NLIN module.
PROC NLIN module (SAS) is a set of procedural
codes for the parameter estimation in case of nonlinear model in Statistical Analysis System (SAS) Software. These Codes have been used to obtain the ﬁvemodel equations by applying SAS package.(It isneeded to mention that extensive search in regard tomodel coefﬁcients and also in regard to initial values have been made). The methods adopted here are
distinctly different from the other papers cited inSection 1. This optimum choice has contributed undoubtedly to increasing the precision (in terms of R2values) of the models.

Section 2 is devoted to a description of thesource of data, the sources being available on net as
mentioned. Section 3 presents the method employedin the paper. Section 4 presents the results and subsequent discussions in details on the ﬁndings evolvedin the paper. Extensive simulation works with respect to inclusion of the mathematical function in themodel equation and also works with respect tochoice of initial values have been made. Table 1
presents the expressions of the estimated model
equations of the superior models. Table 2 contains
the observed data-set (on NASA’s site as mentionedabove) the predicted values corresponding to theabove five Models M1 to M5. Table 3 presents thestandard errors of the estimates and 95% conﬁdence
limits. Table 4 includes the results on the diagnostic
checks for each of the ﬁve models developed and Table 5 lists the values in respect of the alternative precision criteria (including R2 values) for each of theﬁve models. The last section, Annexure, displays theﬁve graph-plots (with observed values and the corresponding ﬁtted values) in respect of the ﬁve models.
In ﬁne, it can be mentioned that the developed models have afﬁrmed their superiority and importance.

4. Results and Discussions

In Table 1, ﬁve models M1, M2, M3, M4 and M5 (withmodel expressions) are given. These models are obtained by simulation (employing extensive trial anderror procedure) keeping in mind the basic objectives(to minimise MSE and to maximise R2 values) so as
to generate superior models. It is evident that thegenerated models represent superior models.

― 90 ―Global Temperature-Quest for Models with Superior Fitting
Table 1: Estimated model equations

ModelsM 1:(u = 3, v = 6)
M 2:
(u = 3, v = 6)
M 3:
(u = 7, v = 2)
M 4:
(u = 3, v = 6)
M5:
(u = 7, v = 3)

Equationsy = 13.91 + 0.105t － 0.011t + 0.00057t － 9.58E-6t4 － 373(－0.00115)t + 5.59E-15et +23

0.106 sin3(60430t) － 0.117 cos6(－29514.8t) + ε

y = 14.10 + 0.015t － 0.00089t2 + 0.000091t3 － 2.03E-6t4 － 0.6298(－0.3239)t － 193E-18et －0.107sin3(60433.3t) + 0.1013 cos6(－19461.8t) + ε

y = 14.05 + 0.028t － 0.0026t2 + 0.000175t3 － 3.48E-6t4 － 0.688(－0.3254)t + 2.25E-15et +0.1098sin7(619659t) + 0.093cos2(－150331t) + ε

y = 14.08 + 0.045t － 0.00435t2 + 0.00025t3 － 4.48E-6t4 － 0.2208(－0.5397)t + 6.97E-16et －0.0674 sin3(60429.9t) － 0.0973 cos6(－29415.5t) + ε

y = 13.909 + 0.0447t + 0.00252t2 － 0.00028t3 + 5.705E-6t4 － 0.2363(－0.6405)t － 123E-17et －0.0670 sin7(619647t) － 0.1961cos3(－219465t) + ε

For example, 123E-17 means 123.10-17.

Table 2: Predicted values under different models

Predicted values

Year

Observed values

M1

M2

M3

M4

M5

1981

14.28

14.2780

14.2774

14.2783

14.2935

14.2995

1982

14.09

14.0802

14.0773

14.081914.0882

14.0966

1983

14.27

14.2447

14.2640

14.2668

14.1830

14.201

1984

14.12

14.1884

14.1449

14.1321

14.1318

14.1053

1985

14.08

14.1120

14.0809

14.074214.2271

14.1339

1986

14.15

14.1586

14.1963

14.2013

14.1754

14.1926

1987

14.28

14.2563

14.321114.3223

14.2568

14.2725

1988

14.35

14.2591

14.289614.2843

14.2344

14.3148

1989

14.24

14.2349

14.2382

14.2421

14.2601

14.3498

1990

14.39

14.3338

14.2860

14.2898

14.2839

14.292

1991

14.38

14.3785

14.2931

14.311014.3599

14.2711

1992

14.18

14.2572

14.2266

14.2621

14.2434

14.1884

1993

14.2

14.2003

14.2710

14.2532

14.2640

14.2641

1994

14.28

14.3598

14.3866

14.322214.3743

14.3023

1995

14.43

14.4186

14.3969

14.3876

14.4377

14.3994

1996

14.32

14.3626

14.3766

14.4168

14.3072

14.4431

1997

14.45

14.4262

14.4578

14.4528

14.400914.4637

1998

14.61

14.5199

14.5261

14.4879

14.520214.4769

1999

14.39

14.410714.4342

14.4690

14.4885

14.4196

2000

14.4

14.3479

14.3515

14.431214.3633

14.4742

2001

14.52

14.5333

14.4795

14.4665

14.5143

14.4776

2002

14.6

14.6786

14.6142

14.5418

14.6067

14.5541

2003

14.6

14.6101

14.5639

14.5488

14.560714.5548

2004

14.52

14.5224

14.5324

14.5399

14.5174

14.5889

2005

14.65

14.5834

14.6431

14.6334

14.6492

14.609

2006

14.59

14.630714.6916

14.7163

14.6354

14.5803

2007

14.62

14.5780

14.5746

14.6180

14.5752

14.6144

2008

14.49

14.5069

14.5013

14.4893

14.5627

14.5839

2009

14.59

14.5884

14.5908

14.5499

14.6150

14.6215

2010

14.66

14.6638

14.6337

14.6439

14.5718

14.5594

2011

14.55

14.5585

14.5586

14.5680

14.5792

14.573

2012

14.56

14.5564

14.560214.5573

14.5588

14.5619― 91 ―

Satyabrata PAL, Arunava GHOSH, Sanpei KAGEYAMA and Kazuki MATSUBARA
Table 3: Standard errors of estimates and 95% conﬁdence limits
Model
coefﬁcients

SE

LCL

UCL

SE

LCL

UCL

SE

LCL

UCL

SE

LCL

UCL

SE

LCL

UCL

a

0.15

13.60514.2230.112

13.86814.3350.127

13.79214.3220.112

13.852

14.32

0.127

13.64

14.17

b

0.05

－0.005

0.216

0.044

－0.077

0.108

0.051

－0.077

0.133

0.046

－0.051

0.14

0.046

－0.051

0.14

c

0.005

－0.024

0.001

0.005

－0.012

0.01

0.006

－0.0150.01000.006－0.016

0.007

0.007

－0.0120.016

d0.00030.00001

0.001

0.0002－0.0004

0.00060.0003－0.0004

0.00070.0003－0.00030.00080.0004－0.0010.0005

p4.052E-6 －0.00002 －1.13E-6 3.877E-6 －0.00001 6.056E-6 4.361E-6 －0.00001 5.616E-6 4.265E-6 －0.00001 4.415E-6 6.674E-6 －8.22E-6 0.00002

f65344.2－1366791359330.906－2.519

1.26

1.121

－3.026

1.65

0.332

－0.912

0.47

0.212

－0.678

0.206

g

0.202

－0.423

0.42

0.342

－1.038

0.39

0.379

－1.1180.467

0.454－1.487

0.408

0.262

－1.186－0.095hM1M2M3M4M5

2.09E-15 1.23E-15 9.95E-15 2.21E-15 －48E-16 4.42E-15 2.51E-15 －298E-17 7.47E-15 2.55E-15 －462E-17 6.01E-15 3.25E-15

q0.019

0.067

0.145

0.023

－0.155－0.059

0.038

0.031

0.189

0.027

－0.123－0.011－8E-15

5.55E-15

0.034－0.13810.0045

r0.031

0.056

0.182

0.037

－0.179－0.0240.039－0.175－0.011

0.041

0.011

0.184

0.09

－0.384－0.008w0.00960430.060430.1

0.008

60433.3

60433.3

0.008

6196596196590.015

60429.860429.9

0.127

13.65

14.17

z

0.006

－29514.8 －29514.8

0.008－19461.8 －19461.8

0.011－150331 －150331

0.009－29415.5 －29415.5

0.046

－0.05

0.14

SE = standard error; LCL = lower conﬁdence limit; UCL = upper conﬁdence limit[Note: Approximate 95% confidence limits (upper and lower) with respect to any estimator of a parameter may
sometimes be equal. It happens when the precision level of the estimator (here any coefﬁcient in the model equation)is extremely high. In such cases the length of the conﬁdence interval may collapse to zero].

Table 4: Diagnostic checking for ﬁve models

Test for Normality
Shapiro-Wilk test
Kolmogorov-Smirnov test

ModelsStatistic
M1: (u = 3, v = 6)
M2: (u = 3, v = 6)
M3: (u = 7, v = 2)
M4: (u = 3, v = 6)
M5: (u = 7, v = 3)

p-value

Statistic

p-value> 0.1500> 0.1500> 0.1500> 0.1500> 0.1500

0.957389

0.23260.116880.979969

0.79870.106910.984552

0.91560.104930.977162

0.71390.09738

0.985271

0.92990.07678

Test for Independence
Run test
Asymp.Sig.
z-value(2-tailed)
0.000
1.000
0.203
0.839
0.639
0.523
0.203
0.839
0.000
1.000

Table 5: Values of different precision criteria for four models

Models (edf = 20)
M1: (u = 3, v = 6)
M2: (u = 3, v = 6)
M3: (u = 7, v = 2)
M4: (u = 3, v = 6)
M5: (u = 7, v = 3)

MSE (meansquare error)
0.00597
0.003880.004810.00576
0.00615

R2values
0.940
0.921
0.902
0.883
0.875

MAE (meanabsolute error)
0.03155
0.037197
0.041204
0.045301
0.049199

Fo

Pr (F>F0)28.39< 0.0001< 0.0001< 0.0001< 0.0001< 0.0001

21.29
16.83
13.75
12.77

edf = error degrees of freedom

This section contains the results obtained afterﬁtting the models to the data. These are given in ﬁve
tables, Tables 1 to 5 respectively. Tables 1 to 3 display the model equations, observed and predictedvalues, standard errors of the estimates and confidence limits respectively. Tables 4 and 5 present theresults on the diagnostic checks (Draper and Smith,1998) and values of the alternative precision criteria
respectively in respect of each model. The models

developed in this communication are the most precise models (each of four models has its precisionlevel more than 0.88, each of three models has its
precision level more than 0.90, and each of two models has its precision level more than 0.92 and there is
1 (one) model which has its precision level as highas 0.94, the precision levels are determined on thebasis of R2 values) existing in the literature (so farsurveyed by us).
― 92 ―

Global Temperature-Quest for Models with Superior Fitting

Annexure
Here, the subscript, “ t ”, denotes year, e.g., t = 1,2,3, ...
mean first, second, third, ... years, respectively. O

represents the observed value of temperature at the i-thi

point of time, i.e., yi.

M1

M2

M3

M4

― 93 ―

Satyabrata PAL, Arunava GHOSH, Sanpei KAGEYAMA and Kazuki MATSUBARA
M5M1M2

M3M4― 94 ―

Global Temperature-Quest for Models with Superior Fitting
M5References
Draper, N.R. and Smith, H. (1998). Applied Regression

Pal, S., Pal, S. and Kageyama, S. (2013). Modelling theGlobal Mean Temperature. Bull. Hiroshima Inst. Tech.

Analysis, John Wiley and Sons.

Pal, S. and Pal, S. (2011). A Revisit to the Global Warming

Research 47, 149-152.

Pal, S., Ghosh, A., Pal, S. and Kageyama, S. ( 2014 ).

Phenomenon. Proceedings (published online) of the58thWorld Statistics Congress organised by theRevisit to Modelling Global Annual Average

International Statistical Institute, Hague, held at

Temperature - A Parametric Approach. Bull. Hiroshima

Dublin, Ireland.

Inst. Tech. Research 48, 87-91.

― 95 ―