S. M. Yan et al. / Natural Science 3 (2011) 430-435

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2.3. Generating Random Walk

We use the SigmaPlot [7] with different seeds to gen-

erate random sequences ranged from –1 to 1, and then

we classify a random number as 1 if it is larger than its

previous one and as –1 if it is smaller than its previous

one. Thereafter we add the classified ±1 as random walk.

2.4. Searching for Seed

To the best of our knowledge, there is no algorithm

available to find the right seed, which produces the best

fit between random walk and observed data. However,

this is not a problem with current computational tech-

nique, because we can simply search all the seeds in

searching space and compare their outcomes.

2.5. Fitting Recorded Precipitation

Hereafter, we use a more complicated random walk

model [8] to fit the recorded precipitation, which is in

decimal format. In plain words, the simplest random

walk comes from tossing of double-sided coin, while

this random walk could be regarded as tossing of dice,

which can be not only six-sided but as many as the de-

cimal data. In this way, we generate random numbers,

and add them to construct the random walk, and the fit-

ting is again to search the best seed that generates best

fit.

2.6. Comparison

For determining the best seed, we compare the least

squared errors between precipitation walk and random

walk, and between recorded precipitation and random

precipitation generated from different seeds.

3. RESULTS AND DISCUSSION

Table 1 shows how we construct a precipitation walk

and its corresponding random walk. For the precipitation

walk, we have the follows: 1) the starting point is the

annual precipitation in 1901, 847 mm (cell 2, column 2),

and this starting point corresponds zero in sense of pre-

cipitation walk (cell 2, column 4); 2) the annual precipi-

tation in 1902 is 737 mm (cell 3, column 2), which is

smaller than the first one, 847 mm (cell 2, column 2), so

we assign –1 as precipitation step (cell 3, column 3), 3)

the precipitation walk is –1 (0 + (–1)) (cell 3, column 4),

and 4) the similar computation is applied to all the data

in columns 2, 3, and 4.

For the random walk, we have the follows: 1) a good

seed we found is 6.98078, and this seed generates a se-

ries of random numbers (column 5), 2) the first random

number, 0.36795 (cell 2, column 5), is considered as the

starting point corresponding to 0 in random walk (cell 2,

column 7), 3) the second random number, –0.74132 (cell

3, column 5), is smaller than the first random number,

0.36795 (cell 2, column 5), so we assign –1 (cell 3,

column 6), 4) the random walk is –1 (0 + (–1)) (cell 3,

column 7), and 5) the similar procedure is applied to all

the data in columns 5, 6, and 7. In the same manner, we

construct the precipitation walk and random walk.

The figures in the left side of Figure 1 show the fit-

tings of precipitation walk in 7 European capitals using

random walk model. As can be seen, the curve generated

by random walk generally passes through the precipita-

tion walk. Theoretically, the chance for a completely

perfect fitting of precipitation walk is an extremely rare

event. In our case, there are 98 annual precipitations,

thus the completely perfect fitting has the chance of

(1/2)98 theoretically, which is extremely small. Clearly

this probability is very difficult to achieve in limited

time because the space of our search is limited to one

million of seed. So the fitting results in the left side of

Figure 1 suggest that a good seed can be relatively eas-

ily found, thus we consider that the random walk can

describe the precipitation walk, although we cannot

compare our results with other results because the other

models do not set an equal-sized step.

Actually we can view the precipitation walk, which is

the conversion from its annual precipitation, as the trend

of recorded precipitation. This is so because this trend

answers the very basic question of whether the precipita-

tion at certain year is larger (1) or smaller (–1) than its

previous year.

Yet, the cities in Figure 1 cross the whole Europe,

thus there would be uncountable factors affecting the

precipitations, but the random walk still can fit them.

This furthermore suggests that the random walk can de-

scribe the precipitation patterns in terms of precipitation

walk.

Table 2 shows how we use a random walk model to

fit the recorded annual precipitation, here we only need

to construct the random precipitation: 1) the starting

point is the first recorded annual precipitation, which is

847 mm (cell 2 in column 2 and column 4), 2) the seed

for Table 2 is 1.31923, 3) the first random number gen-

erated by the seed is 64.17878 (cell 3, column 3), 4) we

add this value to the previous precipitation datum (847)

resulting in 911.17878 mm (cell 3, column 4), and 5)

along this procedure, we get the random precipitation in

column 4.

The figures in the right side of Figure 1 display the

fittings of recorded precipitation with random precipita-

tion in 7 European capitals. In these figures, the precipi-

tation demonstrates very remarkable fluctuations along

the time course, which do not show any clear sign of

visible pattern. This is the basis for conducting random-