
Modelling and Analysis on Noisy Financial Time Series
OPEN ACCESS JCC
Section 4, the experimental results are specified and
evaluated with the sample financial time series available
in book (Analysis of Financial Time Series) [3]. Section
5 concludes the paper.
2. Filters
Normally, the financial time series is embedded with
high level of noise (random trading behaviors), such as
white noise and colored noise. It is very difficult to de-
termine the level of such unknown noise and find the
appropriate filtering techniques that can separate the de-
terministic time series and random events. If the raw
noisy time series is less denoised, the prediction model
performs poorly due to the high level noise; if the noisy
time series is over denoised, the filtered time series loses
some genuine features of raw time series. The conven-
tional time series cannot filter such high level noise such
as financial time series. Indeed, the effective filtering is
dependent on the several factors, e.g., the ability to re-
move the noise, the types of noise, and the thresholds
estimation, etc.
In this paper, two different types of filtering tech-
niques are utilized in this paper: one is the traditional
non-linear low-pass filter with forward and backward
filtering (FBF) processes; another is the wavelet based
denoising method (WLD) for which the time series is
projected into orthogonal basis. Also, the measure crite-
rion known as approximate entropy (ApEn) is considered
to evaluate the performance of proposed filters.
2.1. Forward and Backward Filter
The forward-backward filter (FBF) actually is a matrix
with no-linear processing networks. It utilizes the
second-order matrix SOS and the scale vector G, by
conducting the forward and reverse the filtering pro-
cesses [5]. The scale G defines the weights of input sam-
ples. The SOS and G are defined by:
01 1121011121
02 1222021222
012 012
123456
G[ ]
LL LLLL
bbbaaa
bbbaaa
SOS
bbbaaa
wwwwww
=
=
FBF filters the time series X with the SOS filter de-
scribed by the matrix SOS and the vector G. After filter-
ing in the forward direction, the filtered sequence is then
reversed and run back through the filter. In this project,
the Butterworth second-order filtering is used for filter-
ing the time series.
The example of FBF filtering process with the finan-
cial time series is illustrated in Figure 1.
2.2. Wavelet-Based Denosing
Wavelet theory is an emerging new signal processing
technique in recent two decades [6,7], which is called the
mathematical microscope due to it high recognition ac-
curacy in both time domain and frequency spectrum.
With the scaling factor a (dilation factor) and translation
parameter b, a, b ∊ R, and a ≠ 0. The prototype wavelet
is scaled and translated. The wavelet function can be
expressed as:
is the normalized factor, so as to make sure for
all a, b, Ψ() has the unit energy.
The concept of multiresolution was proposed by Mal-
lat and Meyer in 1989 [8], meaning that one signal can
be decomposed into the orthogonal projections and can
also be fully reconstructed. The components of the de-
composition are divided into the approximation (a) and
details (d) at different levels. The approximation repre-
sents the major feature of the signal and the details de-
scribe the detailed changes and noise. The time series can
be denoised by removing some ingredients from the pro-
jections in details.
The example of wavelet denoising is given in Figure s
2 and 3.
2.3. Performance Measurement
To evaluate two filters proposed above, two measure-
ments are introduced: 1). One indicator is the fit rate of
autoregression model (AR), the details about AR model
are available in Chapter 3. 2). Another criterion known as
the approximate entropy (ApEn) [9] is also introduced.
The major ability of ApEn is to evaluate the time se-
ries by quantifying the amount of regularity and the un-
predictability of fluctuations. The successful applications
have been found in EEG signal diagnosis [10] and in
financial time series [11]. In [11], it was reported that the
uncertainty events such as the Asian financial crisis can
be detected by analyzing Hang Seng index. There are
two parameter m and r in ApEn. The value of m is be-
tween 2 - 3, and the value of r is about 0.2 × σ (σ is the
value of standard deviation of the time series).
The smaller of ApEn, the better regularity and trends
of the time series.
Here, an example is given to assess the performance of
two filters using AR model and ApEn. Firstly, I compare
the quality of filtering with time series L0 using AR(p)
model, detailed as below (FPE—final prediction error,
MSE—mean square error):
• For original signal L0, the results with AR (6) are: Fit
to estimation data—0.3754%, FPE—0.0002323, MSE
—0.0002308. Even AR with order of 30, the results