Data processing: measuring – calibrating – or testing – Measurement system – Statistical measurement
Reexamination Certificate
2000-04-27
2002-11-05
Hilten, John S. (Department: 2863)
Data processing: measuring, calibrating, or testing
Measurement system
Statistical measurement
C702S181000, C702S183000, C703S011000
Reexamination Certificate
active
06477481
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a device and a method for outputting stochastic processes, and a data recording medium. More particularly, the present invention relates to a device and a method for outputting stochastic processes, and a data recording medium which realize high speed simulation by utilizing a chaotic map based on an addition formula of a tangent function, especially for simulating Lévy's stable processes which represent fluctuation of stock prices or exchanges, or transmission times in the World Wide Web (WWW) traffic.
2. Description of the Related Art
Not only physical phenomena, chemical phenomena, biological phenomena, and the like, various activities in the human society such as merchandizing, money transfer, and value shifts have been simulated with utilizing models of stochastic processes for analyzing them to find out the best solution.
In the conventional simulations, it has been assumed that shift stationary increment during a predetermined period in the stochastic process obey an explicit density function for distribution such as Gaussian distribution. In other words, random values in the stochastic process have been obtained by utilizing the von Neumann inverse function method to obtain random variables in accordance with the density function after generating uniform random variables.
It has been known that transmission times in the WWW traffic, stock price fluctuation, price fluctuation in the foreign exchange market, or the like during a predetermined period (for example, several seconds or several minutes) shifts in accordance with the Lévy distribution (stable distribution) (R. N. Mantegna & H. E.
Stanley, Nature, vol. 376, pp. 46-49, 1995). The Lévy distribution is a distribution having parameters of indexes &agr; and &bgr; (equation 1). By the Lévy distribution, it is impossible to obtain an analytic density function unless the indexes &agr; and &bgr; show specific values. On the contrary, a Normal (Gaussian) distribution is able to show an analytic density function.
P
(
x
; &agr;, &bgr;)={fraction (1/2&pgr;)}∫
−∞
∞
exp(
izx
)&psgr;(
z
)
dz
EQUATION 1
where
&psgr;(z)=exp{−i&ggr;z−&eegr;|z|
&agr;
[1+i&bgr;sgn(z)&ohgr;(z, &agr;)]}
0<&agr;≦2
−1≦&bgr;≦1
&ggr;≧0
&ohgr;(z, &agr;)=tan(&pgr;&agr;/2) for &agr;≠1
&ohgr;(z, &agr;)=(2/&pgr;) log |z | for &agr;=1
In case of obtaining the Lévy's stable processes with utilizing the von Neumann inverse function method, Fourier integration must be carried out once as shown by equation 1 to obtain a density function, further, time-integration of random variables obtained by the density function is required in order to obtain Lévy's stable processes. A result after these computationally heavy steps is, however, an approximation.
The Cauthy distribution is a distribution by which its density function is obtained explicitly. The Cauthy distribution corresponds to the Lévy distribution at &agr;=1 and &bgr;=0. It has been observed that the Lévy distribution but not the Cauthy distribution appears in the empirical data of distribution of transmission times in WWW traffic traffic, stock price fluctuation, price fluctuation in the foreign exchange market, or the like.
Accordingly, integrating process must be carried out many times in the conventional technique, because it requires time-integration of the stationary increments obtained by the von Neumann inverse function method after generating uniform random variables and Fourier integration. As a result, it requires very long process time for calculation. Moreover, the result of the simulation is not reliable due to the truncation of the limits of integration.
Demands for efficient simulations such as technical simulation for developing improved communication protocol, industrial simulation for evaluating financial risks corresponding to fluctuation of stock prices or exchanges have been developing. In other words, great demand for simulating the Lévy's stable processes has been raised in the industry field.
SUMMARY OF THE INVENTION
The present invention have made in consideration of the above. It is an object of the present invention to provide a device and a method for outputting stochastic processes, and a data recording medium. More particularly, it is an object of the present invention to provide a device and a method for outputting stochastic processes, and a data recording medium which realize high speed simulation by utilizing a chaotic map based on an addition formula of a tangent function, especially for simulating Lévy's stable processes which represents fluctuation of stock prices or exchanges, or transmission times in the WWW traffic.
The present invention for accomplishing the above objects will now be disclosed in accordance with its principle.
FIG. 1
is a block diagram schematically showing a device for outputting stochastic process according to the present invention. As shown in
FIG. 1
, an output device
101
for outputting stochastic process according to the present invention comprises a plurality of random variable output units
102
, a normalizing unit
103
, and a result output unit
104
.
Each of the random variable output units
102
outputs random variable sequence
105
which obeys an analytical density function of limit distribution. The normalizing unit
103
normalizes the sum of the random variables in the same row over the plurality of sequences
105
output by the plurality of random variable output units
102
. The result output unit
104
time-integrates normalized values
106
sequentially supplied by the normalizing unit
103
, and sequentially outputs the time-integrated value as result values in the stochastic process.
The result values
107
sequentially output by the result output unit
104
represent the stochastic process such as Lévy's stable processes.
Calculations carried out by the random variable output units
102
are not related to each other. Therefore, the random variable output units
102
can perform concurrent calculation or parallel calculation independently. An SIMD (Single Instruction Multi Data) parallel computer is available for simulating stochastic process in a case where the plurality of the random variable output units
102
do the same calculations but handling different data. The SIMD parallel computer realizes high speed simulation.
The density function of limit distribution in the sequence output by each random variable output unit
102
may be a function &pgr;(·) having the characteristics shown in equation 2.
&rgr;(
x
)≅
c
−
|x|
−(1+&agr;)
for
x→−∞
&rgr;(
x
)≅
c
+
|x|
−(1+&agr;)
for
x→+∞
EQUATION 2
where 0<&agr;≦2
In a case where the number of the random variable output units
102
is N and the i-th (1≦i≦N) random variable output unit
102
outputs a sequence (x(i,
0
), x(i,
1
), x(i,
2
), . . . ) of random variables, the normalizing unit
103
outputs a sequence “(v(
0
), v(
1
), v(
2
), . . . )” of normalized values obtained by equation 3, and the result output unit
104
outputs a sequence “(L(
0
), L(
1
), L(
2
), . . . )” of result values obtained by equation 4.
v
(
t
)
=
∑
i
=
1
N
x
(
i
,
t
)
-
A
B
A
=
0
,
B
=
N
1
/
α
for
0
<
α
<
1
A
=
0
,
B
=
N
for
α
=
1
,
x
_
=
0
A
=
N
2
sin
x
N
_
,
B
=
N
for
α
=
1
,
x
≠
0
A
=
N
x
_
,
B
=
N
1
/
α
for
1
<
α
<
2
EQUATION 3
where {overscore (x)} is the expectation value of the random variable output x by the random variable output units
L
(
t
)
=
∑
j
=
0
t
v
(
j
)
EQUATION 4
It has been proved that the normalized value sequence “v(
0
), v(
1
), v(
2
), . . . ”
Communications Research Laboratory, Ministry of Posts and Teleco
Hilten John S.
Le John
Luce Forward Hamilton & Scripps LLP
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