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.