Application of random walk model to fit temperature in 46 gamma world cities from 1901 to 1998

doi:10.4236/ns.2010.212174

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Vol.2, No.12, 1425-1431 (2010)

doi:10.4236/ns.2010.212174

Natural Science

Application of random walk model to fit temperature in

46 gamma world cities from 1901 to 1998

Shaomin Yan1, Guang Wu1,2*

1State Key Laboratory of Non-food Biomass Enzyme Technology, National Engineering Research Center for Non-food Biorefinery,

Guangxi Key Laboratory of Biorefinery, Guangxi Academy of Sciences, Nanning, China

2DreamSciTech Consulting, Shenzhen, China; *Corresp onding Author: [email protected]

Received 2 October 2010; revised 3 November 2010; accepted 6 November 2010.

ABSTRACT

Very recently, we have applied the random walk

model to fit the global temperature anomaly,

CRUTEM3. With encouraging results, we apply

the random walk model to fit the temperature

walk that is the conversion of recorded tem-

perature and real recorded temperature in 46

gamma world cities from 1901 to 1998 in this

study. The results show that the random walk

model can fit both temperature walk and real

recorded temperature although the fitted results

from other climate models are unavailable for

comparison in these 46 cities. Therefore, the

random walk model can fit not only the global

temperature anomaly, but also the real recorded

temperatures in various cities around the world.

Keywords: Gamma World Cities; Global W ar mi ng ;

Modeling; Random Walk; Tempe ratu re Cha ng e

1. INTRODUCTION

During our recen t studies on the potential influen ce of

global warming on the evolution of proteins from influ-

enza A virus [1-5], we noticed that the temperature fluc-

tuates around the temperature trend. On the one hand,

these fluctuations can be easily attributed to random ef-

fects; on the other hand, these fluctuations might imply

that we might need a random model to describe the

temperature pattern because the output of any determi-

nistic model is generally a smooth curve.

Without smooth temperature trend, the temperature

behaves irregularly either to increase or to decrease for

some period of time, from where we could not find out a

clear pattern in temperature change along the time

course. The irregular ups and downs appear similar to

the random walk, which firstly came into our mind. The

random walk comes from the observation of tossing a

single coin: although theoretically each side of coin has

0.5 chances to be up or down, the tossing of coin se-

quentially results in either head up or tail up for several

times continuously rather than each tossing generates

alternative result. The addition of sequentially tossed

results would be a random walk [6]. Computationally

each tossing of coin can be done with the generation of

random numbers, which can be classified as 1 or –1 if a

random number is larger or smaller than its previous one.

As the generation of random numbers is through the

Monte-Carlo simulation with different seeds, thus the

random walk is a model with its own model parameter,

seed.

Therefore, we can consider the random walk model as

alternative model to fit the temperature because it can

produce a really fluctuated curve. Following that, we

applied the random walk model to fit the global tem-

perature anomaly, CRUTEM3, and got a very encourag-

ing result [7]. Since then, an intensive literature search

indicates that Gordon noticed the similarity between

global warming and random walk in 1991 [8].

Likely, the random walk could provide an alternative

model to describe the temperature, however, two ques-

tions raised here are whether the random walk model can

be applied only to the global temperature anomaly not to

local temperature and whether the random walk model

can be applied only to the anomaly n o t to really reco rd ed

temperature.

In order to answer these questions, we need to use the

random walk model to fit the real recorded temperature

in different places around the world. In this study, we

apply the random walk model to fit the temperature in 46

gamma world cities from 1901 to 1998.

2. MATERIALS AND METHODS

2.1. Data

Forty-six gamma world cities are chosen according to

S. M. Yan et al. / Natural Science 2 (2010) 1425-1431

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Wikipedia [9]. However, the number of these cities and

their order are changed frequently due to the characteris-

tics of Wikipedia, thus the cities were dated in February,

2010.

The temperatures recorded in these 46 cities from

1901 to 1998 based on 0.5˚ by 0.5˚ latitude and longi-

tude grid-box basis cross globe are obtained from the

website of Oak Ridge National Laboratory [10].

The latitudes an d longitudes of these 46 gamma world

cities are determined using Get Lat Lon [11].

2.2. Temperature Walk

At first, we use the simplest random walk model,

which starts at zero and at each step moves by ±1 with

equal probability [6]. In other words, the simplest ran-

dom walk can be considered as a sequential result of

tossing a fair coin, by wh ich we record the head as 1 and

the tail as –1, and then we add th e results along the time

course.

For this purpose, we need to convert the temperature

into the temperature walk as shown in Table 1. When

the temperature at certain year is higher than its previous

one, we classify it as 1, otherwise we classify it as –1,

and then we add them as the random walk does.

2.3. Generation of Random Walk

We use the SigmaPlot [12] to generate random se-

quence for the random walk. Technically, the generation

of random walk is quite simple: we generate random

number either ranged from –1 to 1 or without limit, and

then we classify a generated random number as 1 if it is

larger than its previous one and as –1 if it is s maller than

its previous one. Thereafter we add these values as a

random walk.

2.4. Searching for Seed

To find a random walk that is very approximate to the

temperature walk is to find a seed that can generate such

a random walk. To the best of our knowledge, there is no

algorithm for searching seeds by converging the differ-

ence between observed curve and the curve produced by

random walk. Therefore the so-called fitting, which tra-

ditionally searches the optimum according to various

algorithms, becomes to search all possible seeds in order

to find out the seed that produces the random walk with

the least squares between random walk and temperature

walk.

2.5. Fitting Recorded Temperature

Thereafter, we use a more complicated random walk

model [13] to fit the recorded temperature, which is in

decimal format. In plain words, the simplest random

walk comes from tossing of double-sided coin, while

this random walk can be regarded as tossing of dice,

which cannot be only six-sided but as many as the deci-

mal data. In such a way, we generate random numbers,

and add them to construct the random temperature, and

the fitting is again to search the best seed that generates

the best fit.

2.6. Comparison

We use the least squares between temperature walk

and random walk, and between recorded temperature

and random temperature to evaluate which seed is the

best.

3. RESULTS AND DIS CUSSION

Ta b l e 1 shows how we construct a temperature walk

in Panama City. Its recorded temperature in 1901 was

18.8250 (cell 2, column 2), which corresponds to the

starting point of temperature walk, 0, (cell 2, column 4).

The temperature in 1902 was 19.825 0 (cell 3, co lumn 2),

which was higher than the temperature in 1901, 18.8250,

thus the temperature step was 1 (cell 3, column 3), and

the temperature walk is 1 (0 + 1) (cell 3, column 4). In

this manner, we construct the temperature walk from

1901 to 1998.

Similarly, Table 1 also shows how we construct a

random walk with generated random numbers. A good

seed that we found is 0.48531. The f irst random number

generated by this seed was 0.2629 (cell 2, column 5),

which corresponds to the starting point of random walk,

0, (cell 2, column 7). The second random number gener-

ated was 0.8817 (cell 3, column 5), which is larger than

the first random number, 0.2629 (cell 2, column 5). Thus

the random step was 1 (cell 3, column 6), and the ran-

dom walk is 1 (0 + 1) (cell 3, column 7).

The last column (column 8) in Table 1 is the differ-

ence between temperature walk and random walk (ran-

dom walk-temperature walk), whose squared sum is our

standard to find the best fit among seeds.

Figure 1 shows the fitted results in 12 cities repre-

sented differently geographic locations around the world.

As can be seen, the random walk (gray curve) mimicked

the temperature walk (black curve) with very small dif-

ference. Theoretically, a completely perfect fit would

have an extremely s mall probab ility. In the simplest case

of random walk, this probability would be (1/2)98.

Meanwhile, the total number of our fittings were one

million, which is a fraction of (1/2)98. Thus the fact that

we can find a relatively good fit within one million fit-

tings suggests that the random walk can describe the

temperature pattern from 1901 to 1998 in these cities.

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Figure 1. Comparison of temperature walk (black) with random walk (gray) in 12 cities from 1901 to 1998.

Openly accessible at

S. M. Yan et al. / Natural Science 2 (2010) 1425-1431

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Ta ble 1 . Conversion of recorded temperature in centigrade degree into temperature walk and generation of random walk for tem-

perature in Panama City from 1901 to 1998 (The random number is generated by SigmaPlot with the seed of 0.48531).

Year Recorded

Temperature Temperature Step Temperature WalkRandom NumberRandom Step Random Walk Difference

1901 18.8250 0 0.2629 0 0

1902 19.8250 1 1 0.8817 1 1 0

1903 19.2250 –1 0 0.2996 –1 0 0

1904 19.8000 1 1 0.9392 1 1 0

1905 19.7750 –1 0 -0.1281 –1 0 0

… … … … … … …

1991 20.7083 –1 4 0.4440 1 4 0

1992 20.0333 –1 3 0.5661 1 5 2

1993 20.0750 1 4 -0.8129 –1 4 0

1994 20.4750 1 5 0.0808 1 5 0

1995 20.2250 –1 4 -0.8109 –1 4 0

1996 19.6917 –1 3 -0.1161 1 5 2

1997 19.6750 –1 2 -0.9795 –1 4 2

1998 20.7333 1 3 -0.3824 1 5 2

On the other hand, the temperature walk in fact an-

swers the simplest and very basic question of whether

the temperature in this year is higher (1) or lower (–1)

than that in the previous year, which could arguably be

the first human concept in comparison of temperature

between two time points. The answer to this simplest

question for years would construc t the temperature walk.

Therefore, we consider that the temperature walk has a

very sound basis.

In fact, not only the selected cities in Figure 1 show a

good fit, but also the rest of cities were found no excep-

tion (Ta b le 2), here the exception can be understood as

the fitted curve has a totally opposite direction with re-

spect to its temperature curve. We therefore consider that

the random walk model can fit not only the global tem-

perature anomaly, but also the temperatures in different

locations around th e world.

Figure 2 shows the fittings for the recorded tempera-

ture. The difference between Figures 1 and 2 is that the

recorded temperature was in decimal format while the

temperature walk was in ±1 format. As seen in Figure 2,

the random walk model can fit these recorded tempera-

tures even the temperatures in these locations have dif-

ferent patterns. Again, there is no exception for the fit-

tings in all 46 gamma world cities (columns 3 and 5,

Table 2), which furthermore demonstrate that the ran-

dom walk model can fit not only the global temperature

anomaly, but also the temperatures in different locations

around the world.

In this study, we u sed the su m of least squares to com-

pare which seed produces a better fitting. Thus, a ques-

tion raised here is how small the sum of least squares is

that we can say the random walk model fits the tem-

perature well. We could consider this question in three

ways: 1) We have already mentioned that the p robability

to find the best fit for the temperature walk is (1/2)98,

and this chance cannot be found in limited time, how-

ever, some better fits can be found among our total one

million fits; 2) On the other hand, the sum of least

squares can be mainly used for model comparison, i.e. to

use our sum of least squares to compare with the sum of

least squares obtained from other models in fitting the

temperature of this group of cities although no other

results are available for comparison; 3) from modeling

viewpoint, the random walk model has an advantage

over other models because the random walk model has

only one model parameter, whereas other models have

numerous model parameters, thus theoretically the ran-

dom walk model has a larger chance than other models

to find out the best fit because each model parameter has

a certain space to search the optimal parameter.

The random walk actually has a very deeply physical

mechanism: weather is a highly multidimensional at-

tractor of Navier-Stokes Equations of atmosphere with

highly complicated boundary conditions. Worse, it is a

non-autonomois system with at least two-periodic forc-

ing, diurnal and annual. And annual measurement is a

result of averaging of diurnal Poincare Maps. Given

what we know about weather, this entire attractor is

highly chaotic, its annual maps are likely of very fractal

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Figure 2. Comparison of recorded temperature (black) with random temperature (grey) in 12 cities from 1901 to 1998.

S. M. Yan et al. / Natural Science 2 (2010) 1425-1431

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Ta b le 2. Model parameters (seeds) and fitted results for fitting temperature change in 46 gamma world

cities from 1901 to 1998 using random walk model.

Fitting of Temperature Walk Fitting of Recorded Temperature

City Seed Sum of Squares Seed Sum of Squares

Panama City 0.48531 132 3.18016 36.99532

Casablanca 0.65260 132 2.49415 17.83066

Chennai 5.36599 190 2.47712 9.76773

Brisbane 0.79213 112 2.66817 17.87435

Quito 0.69948 142 1.03352 19.37610

Stuttgart 0.58342 134 0.34619 57.06779

Denver 3.89146 148 1.59888 70.45055

Vancouver 4.02676 120 0.76050 55.00014

Zagreb 1.90779 116 8.66967 51.31698

Guatemala City 0.24543 152 1.83630 16.95153

Cape Town 0.89001 161 0.76454 1 4. 07663

San Jose 1.02726 97 0.14315 27.59630

Ljubljana 1.9037 148 3.84243 42.53592

Minneapolis 0.35895 152 0.36478 109.71508

Santo Domingo 4.29052 132 1.74614 12.86316

Seattle 1.42289 116 0.25169 39.26277

Manama 2.02165 124 0.68838 27.38144

Shenzhen 4.83125 148 5.95931 16.49588

Guadalajara 2.04735 146 0.36390 23.99288

Antwerp 6.65174 140 2.48303 46.55537

Kolkata 9.56581 134 5.27468 11.90250

Rotterdam 0.11644 132 2.48303 44.69400

Lagos 1.77962 144 1.69805 23.25883

Philadelphia 1.16765 120 3.33953 53.14891

Perth 0.03095 131 0.75289 26.84818

Amman 1.41563 136 0.58696 30.47234

Manchester 2.64776 113 2.66954 32.74658

Riga 3.64228 112 2.83735

144.92674

Detroit 0.00136 124 0.01219 85.23191

Guayaquil 0.72301 139 3.97256 25.55697

Wellington 2.20855 142 3.87540 25.28364

Portland 0.28335 132 0.07853 49.35406

Porto 1.92792 133 0.38211 20.36737

Edinburgh 0.77095 104 0.97533 28.76731

Tallinn 0.16784 132 2.83735 147.22377

San Salvador 1.06139 184 3.05414 16.39708

St. Petersburg 3.09342 120 2.09755 170.32610

Port Louis 0.08480 136 3.02959 6.53177

San Diego 2.18277 250 0.12773 44.65777

Calgary 1.37753 100 1.11564 139.44530

Almaty 2.39794 164 9.93285 93.91648

Birmingham 0.45373 142 0.95604 53.31933

Islamabad 5.62339 110 0.91978 33.98368

Doha 0.64310 135 0.70169 27.9417

Vilnius 0.33912 116 0.53868 130.0170

Colombo 7.06029 184 2.74576 5.83075

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nature, and invariant measure of these maps are likely a

fractal-dimensional sieves with a hyperbolic point at

every location. The annual iteration of this map is like a

coin tossing [14].

Openly accessible at

Still, the random walk could be viewed as the Drunk-

ard’s walk [15] as the climate change seems to be a

complex process, and the drunkard walk is also used to

describe complex processes. It should be obvious that

climate change is a complex process, for the climate

change involves many factors that are not reducible to

exactly one thing. And the climate affects the ecosystem

that is a complex process itself.

On the other hand, the random mechanism in random

walk model is different from other random factors in

modeling, where the random factors were mainly con-

sidered as a minor factor [16].

At this moment, we have no way to know whether or

not the random walk model works better than other cli-

mate models, simply because the results of temperature

fitting in these cities using other climate models are not

available for comparison.

In conclusion: we use the random walk model to fit

the temperature change in 46 gamma world cities from

1901 to 1998, and the r esults show that the random walk

model can fit both the temperature walk and recorded

temperature in different cities worldwide. This study

confirms that the random walk model can be used to

analyze the temperature change, which suggests that the

random mechanism could be a factor driving the tem-

perature change in these cities. The use of random walk

to fit the temperature change is still at very early stage

nevertheless more studies are needed in order to better

understand the temperature change and its modeling.

4. ACKNOWLEDGEMENTS

This study was partly supported by Guangxi Science Foundation

(07-109-001A, 08-115-011, 09322001 and 2010GXNSFA013046). The

authors wish to thank Dr Hong Zhang at Biyee SciTech Inc., MA, USA

for helpful discussion. The authors also wish to thank Dr Alexei A.

Predtechenski at Standard Microsystems Corporation, Austin, TX,

USA for his discussion and su pport in internet discuss io n g roup.

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