S. M. Yan et al. / Natural Science 2 (2010) 1425-1431
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1426
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.
