Transcript phylab.fudan.edu.cn

Adaptability, stationarity and phase
transition: a bottom-up approach for
business cycle modeling
Xiaohui Li, Guang Yang, Kenan An, and Jiping Huang
Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China
Abstract
To probe into the intrinsic emergence
mechanism of business cycles, we construct an
agent-based model about resource allocation
market for reproducing them in a complex
adaptive system from a different perspective.
Common relative economic researches are
inclined to collect data for statistics while few are
associated with ABM. Our work finds that the
adaptive capacities affects real expectations.
Behaviors of suppliers in the free-competition
market are leading factors for business cycles
because they know well about the market and
choose best time to make a change. Interestingly,
through analyses of agents’ adaptabilities we find
a critical line separating two opposite phases.
Methods


Discrete fourier transform
The fundamental idea of the spectral analysis is that,
the economic time series are thought as composition
of a large number of fluctuated sine components with
different cycles.
𝑓 𝑘 =
1
𝑦𝑡 =
𝑁



Model
Result
𝑁−1
𝑡=0
𝑁−1
𝑘=0
2𝜋
𝑦𝑡 𝑒 𝑁 𝑖∙𝑡𝑘 , 0
𝑓(𝑘) 𝑒
,0 ≤ 𝑡 ≤ 𝑁 − 1
Discrete fourier transform converts the appearances of
a time series from time domain into frequency domain.
Periodogram method
Periodogram method converts a series from time
domain into frequency domain. For a power-law noise,
such as fractional Brownian motion (fBm) and
fractional Gaussian noise (fGn), its power spectrum
density is proportional to 1 𝑓 𝛽 , 𝛽 stands for the color
of a signal as an exclusive exponent.
1
𝐴 𝑓 =
2𝜋𝑁

2𝜋
− 𝑁 𝑖∙𝑘𝑡
≤𝑘 ≤𝑁−1
𝑁−1
𝑡=0
 Fig. 3: This picture depicts the dependence
between the Arbitrage Index and the exponent.
The phase transition points vary from each
other. For low arbitrage index, the series comes
more close to a fBm; oppositely high arbitrage
index brings a fGn more possibly.
2
𝑦𝑡 𝑒 𝑖∙𝑡𝑓
It explains the frequencies density distribution, where
𝑓 stands for frequency and 𝐴(𝑓) for its amplitude.
Actually, 𝐴(𝑓) is proportional to 1 𝑓𝛽 indicating a
power law, so we should use a linear fitting for a loglog periodogram to get the coefficient 𝛽.
Results
•
The market is isolated without any interference
•
Perfect competition
•
Only two alternative commodities in the
market
•
Two kinds persons, one for suppliers, noted as
M, the other for consumers, noted as N
•
Agent have adaptive expectations, make
choices according to historic records.
•
Minority game
Materials


Human experiment: 51 students attend on 127
rounds after several warming-up rounds.
Several errors caused by artificial problems,
some person join the game half way, are all
not affecting the participants as well as the
experiment analyses.
 Fig. 4: Contour map shows the relation between
periodogram' exponent and model parameters S
and P. There is a clear cutting line separating two
different phases (black solid line). Above the
critical line is a fBm phase where agents are
relatively cleverer. Under the critical line is a fGn
phase where agents are inclined to make a
random choice because of short of information.
 Fig. 1: (a) and (b) show the local linear kernel
regression results for N and M percentage
series (blue). The smooth kernel part (red) and
noisy residual part (green) are separated from
the original series. (B) and (D) show the
spectral analyses. The prominent peaks 𝑓2 , 𝑓3 ,
𝑓4 are considered to be corresponding to the
response time. The smallest frequency 𝑓1
seems to be an intrinsic frequency.
Simulation: Use 100 M-type and 100 N-type
agents for model simulation. Each initial
parameter group of S and P runs 5000 rounds
long, repeated more than 20 times
Summary


Methods


Local linear kernel regression
A logistic kernel function for linear estimating
is used. The regression process can be seen as
minimizing a formula, where the optimal
bandwidth is determined by minimizing MSE,
that means 𝑀𝑆𝐸 𝑓 = 𝐵𝑖𝑎𝑠 2 𝑓 + Variance(𝑓)
 Fig. 2: With different initial parameters S and P
the simulated time series are different. (a) and
(c) are the direct results of N-type agents'
𝑘
𝑡𝑗 − 𝑡0
percentage in room 1. Series in (a) and (c) are
min
𝐾ℎ (
)(𝑦𝑗 − 𝛼 𝑡0 − 𝛽 𝑡0 ∙ 𝑡𝑗 )
𝛼 𝑡0 ,𝛽 𝑡0
ℎ
𝑗=1
representing for noise signals of different
natures, the former for fGn and the latter for
After obtaining the optimal bandwidth, then
fBm. (b) and (d) show the periodograms of the
substitute it into the former minimizing
two series respectively.
problem to get the result smoothed series.
Reference:
[1] XH Li, G Yang, KN An, JP Huang, Adaptability, stationarity and phase transition: a bottom-up approach for business cycle modeling, in preparation.

Through human experiment and an agent-based
model, we achieve the reason of business cycle
on basis of a free-competition market with only
two substitutes. The imbalance of demands and
supplies in a short term leads to the generation
of local fluctuations. Nevertheless, the primary
reason for occurrence of business cycle comes
from the conversion of resources allocation
between the two commodity markets.
The cycles information is seen from spectral
analyses. We find that the response time for
suppliers decides cycles with a certain length.
Response time is determined by suppliers
themselves on basis of Arbitrage Index. Suppliers
will make changes until no chances for a profit in
the market.
Besides, we also explore the performances of
the business cycle model as a function of S and P,
where 𝛽 in periodograms can be deemed to a
robust indicator for time series. When increasing
arbitrage index, it will experience a crash. A
critical line that separates two phases of the
system is clearly depicted in the contour map.
The dependence between S and P are different
in some domains.