Bull. Hiroshima Inst. Tech. Research Vol. 47 (2013) 149 – 152

ArticleModelling the Global Mean Temperature

Subhabaha PAL*, Satyabrata PAL** and Sanpei KAGEYAMA***(Received Oct. 31, 2012)

Abstract
Reality has it been that over the last decade a substantial rise in population along with the corresponding energy demand has increased the concentration of green-house gases in the atmosphere.
The above two phenomena have left their major cumulative adverse consequences, being reflected, interms of an increase in the concentration of the direct (CO2) and indirect (NO, SO2, CO) green-housegases over the years. It is needless to mention that these above-mentioned systematic changes haveultimately manifested themselves in terms of the menacing global warming phenomenon. The phenomena, “the melting of land ice and the simultaneous thermal expansion,” have left their subsequent
marks to play the prime cause of the global sea-level rise, and, indeed, rigorous analytical studies/
researches on this aspect have confirmed that the average rise in sea-level globally stands at the rateof 2mm per year in conjunction with an increase in the global surface temperature of 0.74 ± 0.18°C(1.33 ± 0.32°F). The above message, however, follows from the content of the 2007 fourth assessment
report by the Intergovernmental Panel on Climate Change (IPCC) (see Section 2). This paper presents
research findings in regard to the global warming phenomenon, based on the development of theparametric and non-parametric models providing medium to high precision levels, so to say, to enrich
our wisdom on the extent of change in the global average temperature over the recent years.
Key Words: Global warming, Green-house gases, Parametric and non-parametric modellingincrease in the absolute mean global temperature com-

1. Introduction

puted at different periods comprising varying number

It is no denying that the global warming is the mostof years over the above entire time span. To be more

dreaded phenomenon of the present time being cre-specific, the said period of 132 years can be partitioned

ated by the injudicious actions perpetrated to interfere

basically into three main time-periods, 1880–1936,in (or disturb) the natural ways governing the balance

1937–1980 and 1981–2011, based on the temperatureof the environment essential for the sustenance of thevariation pattern. An analysis of the temperature-

life in this abode, called the mother earth. A peep into

record reveals the following: (a) during the time-the data on the annual global average temperature will

period, 1880–1936, the mean absolute global tempera-

reveal the existence of the unerring presence of ature was around 13.77 degree centigrade which

steady increasing trend in it. Indeed, a thorough exam-

marginally increased to 14 degree centigrade duringination of the data on the global average temperature

the time-period 1937–1980; (b) the standard deviations

over the last 132 years (1880–2011), data source men-of annual mean global temperature during both thetioned in Section 2, has clearly exhibited a steadytime-periods, 1880–1936 and 1937–1980, were almost* SAP Development Lead, Manipal Global Education Services, Bangalore, Karnataka, India** Department of Agricultural Statistics, BCKV, Mohanpur-741252, West Bengal, India*** Department of Environmental Design, Hiroshima Institute of Technology, Hiroshima 731-5193, Japan─ 149 ─Subhabaha PAL, Satyabrata PAL and Sanpei KAGEYAMAhigh.

the same (0.09913 for 1880–1936 during 57-year period
and 0.09314 for 1937-1980 during 44-year period); and2. Materials and Methods

however, (c) the change was remarkable during theshort time period 1981–2011 (last 31 years), wherein

The data sources are:the mean absolute temperature rose up to 14.41 degree
centigrade, and the variation in the mean absolute tem-IPCC Report 2007 –perature observed during this period was also found

http://www.ipcc.ch/publications_and_data/publications

to be very much noteworthy (with standard deviation_and_data_reports.shtml#10.155). The above revelation has been summarised inTemperature Data Source –Table 1Time period

Mean absolute temperatureStandard deviation

1880–1936

13.77390.09913

1937–1980

14.0086

0.09314

1981–200914.4150.15500

http://data.giss.nasa.gov/gistemp/tabledata_v3/
GLB.Ts+dSST.txt
Among the parametric models, polynomial models (up to order 8, graphs being differentiated using 8

Table 1.

distinct colours, indeed, graph with respect to each

From what has been mentioned above (also Tableorder is displayed by a particular distinct colour) have1), it is clear that the extent of variation existing in thebeen tried but as revealed from Table 2, the values ofglobal mean absolute temperature is substantially

R-square coefficient exhibit only tiny increase (of thegreater in the third period (31 years from 1981–2011),

order, less than 0.03), when models involving orders

and thus it appears most appropriate to model thehigher than the quadratic are used. Thus in the final

global temperature in the said period. Towards thisTable 3, results corresponding to polynomials, linear

effort we have called upon the parametric, non-para-and quadratic only, have been given. The graph plot ismetric and semi-parametric models to represent thealso presented below.

pattern existing in the global mean absolute temperature in the third period. As revealed from Table 3, thePOLYNOMIAL FIT

Spline and Loess models provide relatively more precise fits (as the values of the R-square coefficient forthese models are much more than those of other models examined in this paper). In fact, it is well-known
from statistical literature that the value of the R-squarecoefficient indicates the level of precision with respect
to a given tested model. Indeed, it also speaks about
the degree of closeness of the tested model with thedata on which the model is fitted. A high value ofR-square (very close to one) indicates a very good fit,
so to say, implying that the level of precision is very
Table 2

─ 150 ─Modelling the Global Mean Temperature

As will be mentioned below, Table 3 includes thevalues of R-square coefficient corresponding to thebetter fits as precision-levels for these models arehigher.

quadratic model only, apart from the non-parametric

Table 3models, Kernel and Loess and the semi-parametric

Model

model, Spline. Indeed, the non-parametric and semiparametric models are widely known, however,detailed ramifications of such models can be found in,
Thisted (1988), Simonoff (1996), Hardle et al. (2004),
very brief reviews in Kageyama et al. (2006), and Palet al. (2012). Following the general usage of represen-Degrees of freedom

Mean square error

R-squareLinear29

0.00940.7372

Quadratic

28

0.00940.7450Kernel3.417

0.00950.7468

Loess9.376

0.00770.8402

Spline8.750

0.00800.8290tation, “Y” and “X” denote the “Yearly mean absolute

The graphs with regard to non-parametric andsemi-parametric fits have been presented in the Annex-global temperature” and “Year” respectively.

ure given below. Each graph exhibits (a) plots of tem-

3. Results and Discussion

perature records (dots), (b) the fitted straight line (redThe following Table 3 records the values of theline), and (c) the specific nonparametric/semi-para-R-square coefficient for the precise models presented

metric model fitted (in pink colour), mentioned in thehere. It is observed that Loess and Spline models giveheading with respect to each graph.

AnnexureSpline Fit

Kernel Fit

─ 151 ─Subhabaha PAL, Satyabrata PAL and Sanpei KAGEYAMA

Loess Fit

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─ 152 ─ing. Chapman and Hall, New York.