A Novel Approach to Study House Rent Price Index of Taiwan Based on Hilbert-Huang Transform
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A Novel Approach to Study House Rent Price Index
of Taiwan Based on Hilbert-Huang Transform
*WANG Ming-Shu（王明舒）, *Wu Shaohua（吴绍华）, **Yu Tong（于桐）
* Department of Land Resource and Tourism Sciences, Nanjing University, Nanjing
210093, P.R. China
** Software Institute, Nanjing University, Nanjing 210093, P.R. China
Abstract: As housing price soars, renting house becomes a hot issue. To rightly
address the non-linear and non-stationary house rent price index is relevant to real
estate industry. This paper introduced a novel approach named Hilbert-Huang
Transform (HHT) to analysis the trends and turning points of the house rent price
index. The first step of HHT is Empirical Mode Decomposition (EMD), with which
any complicated data set can be adaptively decomposed into a finite number of
Intrinsic Mode Functions (IMFs). Then the reconstruction of IMFs indicates the
inherent characteristics in the house rent price index. After applying the Hilbert
transform (HT), instantaneous frequencies (IFs) give sharp identifications of the
changing points in the time-series. This approach was used to analysis the trend of
house rent price of Taiwan through HHT and EMD. The result shows the trends of
house rent price in ten-years, twenty-five years as well as in thirty-years. Additionally,
it self-adaptively discovered the three turning points in the real estate market which fit
close to the real estate cycle of Taiwan. Compared with traditional methods namely
Fourier Transform and Wavelet Transform which request priori, HHT is totally
self-adaptive, highly efficient and more applicability to analyze time-series in real
estate domain.
Key Words: House Rent Pricing, Hilbert–Huang Transform (HHT), Empirical Mode
Decomposition (EMD), Time-series Analysis, Taiwan
WANG Ming-Shu: No. 22 Hankou Road, Department of Land Resource and Tourism Sciences,
Nanjing University, Nanjing, P.R. China, 210093. Email:wangmingshu2010@gmail.com
11 Introduction
According to Maslow's hierarchy of needs(Maslow, 1946) , housing belongs to
the physiological needs, which is one of the fundamental requirements for human
survival. Since the global financial crisis sharpens the enduring contradiction between
man and earth, housing price soars. Therefore, in recent years, house-renting has
become a by-no-means ignorable issue (Diewert et al., 2009; Gallin, 2008). The term
house rent price is generally understood to mean the rental that a house owner obtains
by leasing use rights of the house. Then house rent price index is defined as a relative
number which reflects renting price fluctuates with time variation (as during a given
period). For governors and researchers, rightly analyzing house rent price index is
conducive to invigorate and regulate real estate markets, while for investors and real
estate agents, to appropriately study it is benefit to reduce information imbalance and
reach sensible assessments.
Apparently, house rent price index is both non-linear and non-stationary. To
accommodate the inherent non-linearity and non-stationarity of house rent price index,
a novel and powerful method- Hilbert–Huang Transform (HHT) (Huang et al., 1998;
Huang et al., 2003) is introduced in this paper. Through Empirical Mode
Decomposition (EMD), any complicated data set can be adaptively decomposed into a
finite number of Intrinsic Mode Functions (IMFs) which have a definite instantaneous
frequency(IF) and finally can be expressed in joint time-frequency-energy distribution
by Hilbert spectrum (HT). Throughout twelve years’ development, HHT has been
primarily applied to nature and engineering such as signal processing(Gan et al., 2008;
Peng et al., 2005), civil engineering(Calayir and Karaton, 2005; Han et al., 2007),
medical science(Ai and Li, 2008; Sadick et al., 2005) etc.. However, seldom has it
been implemented into real estate research.
This paper investigates house rent price index of Taiwan throughout forty years
and forecast the index by HHT and EMD. Section 2 presents a brief review of the
EMD and HHT algorithms. Section 3 exhibits the results of house rent price index
research of Taiwan by the proposed methods. Section 4 discusses the outputs. Finally,
section 5 gives some conclusions.
2 HHT Approach
HHT consists of two parts: 1) EMD; 2) the Hilbert spectral analysis. Any
complicated data set can be decomposed into a finite and small number of intrinsic
mode functions (IMF) with EMD. An IMF is defined here as any function having the
same number of zero-crossing and extrema, and also having symmetric envelopes
defined by the local maxima, and minima respectively. The IMF also admits
well-behaved Hilbert transforms. EMD is adaptive and highly efficient. The IMF
contains instantaneous frequencies (IFs) as functions of time that give sharp
identifications of imbedded structures by Hilbert transform.
2.1 EMD Algorithm
Given a data set A, the process of EMD is as follows:
21) Initialize: r0= A (t), i=1.
2) Set hj-1= ri-1, j=1. Obtain all local maxima and minima of ri-1 and create the
upper envelop umax and lower envelope umin of hj-1.
3) Define: m= (umax+ umin)/2; then hj =hj-1-m.
4) Check the properties of hj. If hj is not an IMF, set j=j+1 and repeat procedure
(2-3).
5) Evaluate the residue ri= ri-1-IMFi, i=i+1. Repeat the sifting course (2-4) to
obtain the remaining IMFs. This loop (2-4) will not stop until the residue is below a
predetermined level or the residue has a monotonic trend.
EMD generates m IMFs: IMF1, IMF2… IMFm and a residual rm. The data set A
can be reconstructed as:
A(t) = ∑m
i=1 IMFi + rm ……(1)
2.2 Hilbert Spectral Analysis
For a given data X (t); the Hilbert transforms, Y (t) is defined as:
1 ∞ X(τ)
Y(t) = P ∫−∞ ……(2)
π t−τ
P represents the Cauchy principal value. Thus, X (t) and Y (t) combines Z (t):
Z(t) = X(t) + iY(t) = a(t)eiθ(t) ……(3)
Y
In equation (3), a(t) = √X 2 + Y 2 …… (4); θ(t) = tan−1 …… (5).
X
Instantaneous frequency is defined as:
dθ(t)
w(t) = ……(6)
dt
After applying the Hilbert transform, each IMF can be represented as:
IMFi = Re[ai (t)ei ∫ wi(t)dt ] ……(7)
So that the original equation is:
X(t) = Re ∑ni=1 ai (t)ei ∫ wi(t)dt ……(8)
3 Results
The monthly mean data of house rent price index of Taiwan from 1969 to 2007 is
received from Directorate-General of Budget, Accounting and Statistics, Executive
Yuan, Taiwan. The raw data (Figure 1) sets 1990 as the base period, in which the
average index is 100.
3Figure 1 Monthly Averaged House Rent Price Index of Taiwan: 1969 to 2007
After EMD, the raw data yields eight IMF components shown in Figure 2. Here
we can simply detect two features: 1) the amplitudes of the high frequency IMFs (i.e.
IMF1 and IMF2) suddenly increase around 1975, 1980 and 1990, which is reflected
by a series of obvious changes in the corresponding years; 2) there is a large
amplitude, in low frequency (i.e. IMF 5) with a period of approximately 25 years.
Figure 2 IMFs of data shown in Figure 1 by EMD
Figure 3 provides the data and steps of reconstruction of IMFs. Every sub-panel
plots the raw data in a dotted line and partial sum of the IMFs in a solid line. Figure
3(a) plots the raw data and the residual of the sifting process. Residual is actually the
‘residue’ after all possible oscillations are removed by EMD steps, which represents
4the overall trend of the raw data. Figure 3(b) indicates the residual adding the first two
longest oscillatory components, IMF7, and IMF6. This displays the smoothest trend
of data variation. Adding the IMFs step by step, we finally reached the sum of all the
IMFs (Figure 3(d)), which is almost identical to the raw data.
(a)
(b)
5(c)
(d)
Figure 3 Reconstruction of the data from IMFs
Figure 4 and Figure 5 presents some IFs of the corresponding IMFs. In this
time-frequency domain, turning points in the house rent price index marked more
significantly comparing with those in Figure 2. This will be discussed in depth in next
section.
6Figure 4 Time-IF of IMF1
Figure 5 Time-IF of IMF2
4 Discussions
Unlike the traditional methods of transform, namely Fourier transform and
Wavelet transform, HHT is a novel approach designed to handle non-linear and
non-stationary data sets. Fourier transform is only applicable to linear and stationary
data, while Wavelet transform can work with linear but non-stationary data. Moreover,
a priori is the common basis of both Fourier transform and Wavelet transform.
However, one of the most shining characteristics of HHT is self-adaptive. As house
rent price index is inherently non-stationary and non-linear, it is crucial to adopt a new
method to better analyze such process. In the following parts, as an empirical study,
the trends and turning points of house rent price index of Taiwan are discussed.
4.1 The trends
Within the given data span, the trend is an intrinsically fitted monotonic function,
or a function in which there can be at most one extremum. (Huang et al, Proc. Roy.
Soc. Lond., 1998; Wu et al. PNAS 2007) Furthermore, the trend should be determined
by the same mechanism that generates the data, which means it should be intrinsic and
local property of the data. Being local implies it has to cope with a local length scale,
7and be valid only within that length span. Being intrinsic signifies the method for
defining the trend has to be adaptive. In brief, trend should not be determined by
regressions, but should be determined by successively removal of oscillations.
In Figure 2, the Residual is not an outcome of averaging process; rather it is the
residuum after removing all the possible oscillations through EMD. In figure 3(a), the
solid line (Residual) represents the general trend of the house rent price index of
Taiwan. The slope of Residual is approximately 2.39 per year across the total period.
In Figure 3(b), as the first two longest period oscillations were added to the
Residual, the solid line expressed the trend of the house rent price index of Taiwan
about thirty years. With adding a shorter period oscillation (i.e. IMF5), Figure 3(c)
displays the trend of that around twenty-five years. Since a more short period
oscillation (i.e. IMF4) was added, we figured out an approximately ten-year-trend of
the house rent price index of Taiwan. Figure 6 shows the trends in different
timescales.
Figure 6 Trends of the House Rent Price Index of Taiwan in Different Timescales
4.2 The turning points
As is mentioned in Section 3, according to the IMF1 and IMF2 in Figure 2, there
are conspicuous changes in amplitudes around the year of 1975, 1980 and 1990. To
state it more clearly, in Figure 4, the instantaneous frequency of IMF1 expands
suddenly and sharply in 1973, 1977 and 1989. In Figure 5, the instantaneous
frequency of IMF2 balloons evidently in 1977 and 1988.
Regardless the enduring hot issue whether the house prices fluctuate with the
vibration of rents or rents varies with the mutation of house prices; the consensus is
that rents are interrelated with house prices. The turning points detected
self-adaptively by EMD and HHT coincide with the three times when real estate
8market of Taiwan raised sharply: 1) from 1973 to 1974; 2) from 1978 to 1980; 3) from
1987 to 1990. Many scholars have reached the consensus that the first two periods
when house prices soar were affected by the boosting price of international crude oil.
For the third period, it is because the increase of total currency supply to surpassed the
demand of the general economic growth, so that financial institutions released more
mortgages to stimulate house demand. Moreover, since 1987, the atmosphere of
gambling which was generated from lottery ticket swapped to real estate market. In
consequence, house price went upward at this time.
5 Conclusions
As the house rent price index is mostly inherent non-linear and non-stationary,
this paper introduced a novel approach especially compatible with non-linear and
non-stationary data set. Empirical study of the house rent price index of Taiwan by
EMD and HHT manifested the overall trend and trends in ten-years, twenty-five years
as well as in thirty-years. In addition, utilizing EMD and HHT, this paper
self-adaptively discovered the three turning points in the real estate market of Taiwan.
Changing points are vital in most business domains, hence a further study regarding
puny turning points detection in real estate market is expected to explore with EMD
and HHT.
Acknowledgement
The authors would like to thank the Research Center for Adaptive Data Analysis
of National Central University of Taiwan for providing the open source code of EMD.
Meanwhile, we appreciate the anonymous reviewers and editors for their hard work.
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