Expert Systems with Applications 39 (2012) 6168–6173
Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications
journal homepage: www.elsevier.com/locate/eswa
A dynamic meta-learning rate-based model for gold market forecasting
Shifei Zhou a,⇑, Kin Keung Lai a, Jerome Yen b
a
b
Department of Management Sciences, College of Business, City University of Hong Kong, Kowloon, Hong Kong
Department of Finance, Hong Kong University of Science and Technology, Kowloon, Hong Kong
a r t i c l e
Keywords:
EMD
BPNN
Meta-learning
Forecasting
i n f o
a b s t r a c t
In this paper, an improved EMD meta-learning rate-based model for gold price forecasting is proposed. First,
we adopt the EMD method to divide the time series data into different subsets. Second, a back-propagation
neural network model (BPNN) is used to function as the prediction model in our system. We update the
online learning rate of BPNN instantly as well as the weight matrix. Finally, a rating method is used to identify the most suitable BPNN model for further prediction. The experiment results show that our system has a
good forecasting performance.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
get trapped (Orr & Leen, 1992), affecting the prediction precision
adversely.
1.1. Motivation
Forecasting gold price is becoming increasingly important. For
long in history, gold has been traded actively on international markets. Many derivatives of gold trading in international gold markets are also traded, such as gold futures, gold options, gold
forward contracts, and so on (Grudnitski & Osburn, 1993; Shafieea
& Topalb, 2010). Remarkably, since the price of gold varies within a
limited range, gold is able to reduce the effect of inflation, control
the rise of price and help carry out constrictive monetary policy
(Atsalakisa & Valavanisb, 2009). Hence, gold becomes an important
tool for risk hedging as well as an investment avenue. Therefore,
predicting the price of gold has become very significant and important to investors.
However, the current forecasting algorithm offers poor precision in solutions for non-linear problems. We mainly focus on
the artificial neural network (ANN) forecasting algorithm. First,
the traditional ANN prediction algorithms employ a global fixed
learning rate to change the weight matrix. This causes two problems. On one hand, if the learning rate is too small, the weight matrix changes very slowly.
Then, the network training process takes a long time to converge (Orr & Leen, 1996). On the other hand, if the learning rate
is too large, we may miss the minimal optimal and cause the algorithm to diverge (Fig. 1). Second, unlike the polynomial function,
most non-linear problems have complex error surfaces. Hence,
there are many local minima in which the training process may
⇑ Corresponding author. Tel.: +852 5344 8097; fax: +852 3442 0189.
E-mail addresses: [email protected], [email protected] (S. Zhou).
0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.eswa.2011.11.115
1.2. Contributions
In this paper, we propose an improved empirical mode decomposition model (IEMD) to forecast the trend and price of gold market. The IEMD model can make efficient use of historical data to
predict the future price. During the back-propagation neural network training process, we dynamically change the learning rate
by using a global meta-learning rate. Our main contributions are
listed as follows:
(1) High convergence speed. When the trend of historical data is
downward, we change the learning rate to increase the weight
matrix in the same direction to address the decreasing speed
of the error function. Similarly, when the trend is upslope, the
gradient will be increased, to ensure the process will ‘‘climb
the hill’’ of error function effectively. Both methods accelerate
convergence speed of the neural network training process.
(2) High prediction precision. By calculating the gradient of the
error function, we minimize the mean square error by
changing the weight matrix of the prediction model. The
novel system we propose is able to update the learning rate
of back-propagation neural network instantly, and capture
the trend of data series. As a result, our system is able to
yield a high prediction precision for gold market.
(3) Significant application. Gold plays an important role in international markets. It is receiving increasing attention from
investors. Gold offers a way of risk hedging and it can also
be stored without depreciation. A precise prediction of price
of gold can help investors save money. This, therefore, shows
the potential significance of our system.
S. Zhou et al. / Expert Systems with Applications 39 (2012) 6168–6173
2. Related works
2.1. Empirical mode decomposition
The empirical mode decomposition method was first proposed
by Huang et al. (1998). This time series data decomposition technique applies the Hilbert transform to nonlinear and nonstationary
time series data. The main idea of EMD is to decompose a time series into a sum of oscillatory functions.
The EMD can make effectively use historical data. Since the
EMD is able to divide the original time series into several subsets
of data by using the intrinsic mode function (IMF), characteristics
of different subsets of series can be identified by training different
BPNNs. We can use these trained BPNNs to predict the future trend
of time series and select one of the network models for future
prediction.
2.2. Online learning algorithm
In the online learning algorithm, weight vectors of input data
are updated immediately after presentation of each data point
(Saad, 1999). Therefore, the online learning algorithm is able to adjust weights of input data and capture the trend of time series
instantly.
Online learning algorithm has a number of advantages. As the
weight matrix is updated recursively, this algorithm can be used
when there is no fixed training set and new data keeps coming
in. Besides, local minimum is a problem for gradient descent in
nonlinear models whereas online learning can easily escape from
local minima when a noisy data is input. In this paper, we use on-
6169
line learning algorithm to update the weight matrix. At the same
time, we also update the learning rate to make sure the network
model can be trained at a fast convergence speed. To achieve this
goal, we apply a meta-learning rate to the model. Meta-learning
means learns from historical data, which studies how to choose
the right bias dynamically rate rather than a fixed base-learning
(Abraham, 2004; Ricardo & Youssef, 2002).
3. Improved empirical mode decomposition model (IEMD)
3.1. IEMD model structure
Yu et al. proposed an EMD-based neural network ensemble
learning model (Yu, Lai, Wang, & He, 2007; Yu, Wang, & Lai,
2008). In this model, they use forward neural network as the prediction model. The time series data is divided into several subsets
by using IMF (Huang et al., 1998). Each subset is used as an input
data for the FNN model for training. The FNN model is finally used
to predict future outcome. All predicted results are assigned a
weight and are combined together. This model is quite effective
for dealing with historical data. However, prediction results are
greatly affected by the weight of each FNN model. Besides, if one
of the FNN models has made a precise prediction, the final prediction may be imprecise because all predicted results are summed
together.
To overcome these problems, we propose an improved EMD online learning-based back-propagation neural network model. The
structure of the IEMD–BPNN–PMR model is shown in Fig. 2.
To make efficient use of the historical data, we adopt the EMD
method to partition the data into several subsets. Each subset of
data is used to train the back-propagation neural network. The
time series are decomposed according to the sifting procedure proposed by Yu et al. (2008). This procedure is repeated until all data
are divided and each subset of data has only one local minimal or
maximal. At the end of the sifting procedure, data series x(t) can be
expressed as follows:
xðtÞ ¼
n
X
ci ðtÞ þ r n ðtÞ
ð1Þ
i¼1
where n is the number of IMCs, namely number of data subsets, ci(t)
is the ith IMF, and rn(t) is the final residue of the procedure. Thus,
any time series data can be decomposed by using this EMD method.
Frequency components contained in each frequency band are different. They also change the variance of data series x(t), while rn(t) represent the trend of data series x(t).
3.2. Back-propagation neural network
Back-propagation neural network (BPNN) is a type of artificial
neural network (White, 1990) which is a class of typical intelligent
learning paradigm and is widely used in the field of data forecast-
Fig. 1. Effect of learning rate on algorithm convergence.
Fig. 2. An improved EMD online learning-based BPNN model.
6170
S. Zhou et al. / Expert Systems with Applications 39 (2012) 6168–6173
ing. In this paper, we use a three-layer neural network with
error-back-propagation algorithm. This network model is used as
the predicting model of our system. There are many nodes in the
hidden layer. Hence, there are multiple combinations of weights
and data points.
BPNN is able to provide flexible mapping between inputs and
outputs. Hornik, Stinchocombe, and White (1989) had proved that
a three-layer of feed-forward neural network with an identity
transfer function in the output unit and logistic functions in the
hidden layer can well approximate any arbitrary continuous function. Therefore, in this paper, we use the three-layer BPNN as the
forecasting model.
Assume that yi represent the ith hidden node, then
yi ¼
n
X
wi xi þ w0
ð2Þ
tion will be accordingly increased. Therefore, this relationship helps
modify the weight matrix to control the errors.
Then, following Saad (1999), we can calculate the increment of
weight as follows, using Eq. (6).
Dwij ¼ lij
@Ej
¼ lij ðt j yj Þf 0 ðyi Þxi
@wij
ð7Þ
where lij represents the online learning rate. We can see that the
increment of weight will depend on the learning rate. If l ij is large,
each change of weight will be similarly large.
The online weight update that uses a local and time-varying
learning rate for each weight can be defined as follows.
wij ðt þ 1Þ ¼ wij ðtÞ þ lij ðtÞDwij ðtÞ
ð8Þ
where Dwij(t) is the increment of weight, defined in Eq. (7).
i¼1
where xi is the data input and wi is the corresponding weight. Meanwhile, yi is also called a net input, used in latter calculations. After
the hidden layer’s nodes are generated, we use the activation function f() to transform the net input into output, namely,
yj ¼ f ðyi Þ
ð3Þ
where j = 1, . . . , m and m represents the number of nodes in the hidden layer. By using this network model, we finally make the prediction for the time series.
3.3. Prediction model rating
1
ðt j yj Þ2
2
ð4Þ
where tj is the target value of the time series; yj is the predicted
outcome from the jth BPNN model; and j ranges from 1 to m. Then,
we compare all these Ej and find the minimal one, say Ej, and increase the rate of jth model by 1. This procedure is repeated
throughout the training process. Finally, we get a BPNN prediction
model with the largest rate for prediction.
4. Improved online learning algorithm
4.1. Online weight update
In this section, an improved online learning algorithm is proposed. We start with the mean squared error and formulate some
mathematical methods to deduct the relationship between weight
and MSE.
First, we use the chain rule to decompose the gradient into two
factors:
@Ej
@Ej @yj
¼
@wij @yj @wij
ð5Þ
where wij represents the weight value from the ith data input to the
jth hidden node. By combining Eqs. 2, 3 and 5, we have the following result:
G¼
@
@Ej
¼
@wij
1
ðt
2 j
2
yj Þ @yj
¼ ðyj t j Þf 0 ðyi Þxi
@yj
@wij
As discussed above, the online learning algorithm has the ability
to escape from the local minimum. The noise in the stochastic error
surface bounces the network out of local minimums as long as they
are not too severe.
When the weight matrix is modified, we simultaneously change
the local learning rate by gradient descent method. In order to reduce error function, we change the learning rate before changing
the weights. Then, the gradient can be defined as follows.
nij ðt þ 1Þ ¼
We have to ensemble the prediction results from different
BPNN models. Thus, the prediction model rating (PMR) method is
proposed to handle this problem. In this section, we calculate the
mean squared error E(MSE) of every output of the BPNN model.
Ej ¼
4.2. Online learning rate update
ð6Þ
where G represents the gradient of MSE. G is not necessarily a positive value since f0 (yi) may be a negative value. Here, we assume G is
positive. Then, if there is an increment in weight wij, the error func-
@Eðt þ 1Þ
@Eðt þ 1Þ @wij ðt þ 1Þ
¼
@ lij ðt þ 1Þ @wij ðt þ 1Þ
@ lij ðtÞ
¼ ðyj tj Þf 0 ðyi Þxi Dwij ðtÞ
ð9Þ
By applying the result of Eq. (9) and introducing a meta-learning
rate, we can define the online learning rate as follows.
lij ðtÞ ¼ lij ðt 1Þ kðyj tj Þf 0 ðyi Þxi Dwij ðt 1Þ
ð10Þ
where k is a global meta-learning rate.
There are two disadvantages of using fixed global meta-learning
rate. Since the meta-learning rate is fixed, the increment of learning rate is also fixed from Eq. (10). This fixed increment may sometimes speed up the movement of error cost towards the minima
when the movement is down-slope. A fast accelerating speed
causes the training process to pass by the local minimal and diverge from data trend, the process of which is shown on
Fig. 1(b). Besides, a fixed learning rate increment cannot adaptively
adjust the converge speed of the training process. It can only increase or decrease the speed at a fixed acceleration. This, therefore,
can not improve the learning speed effectively. Hence, we mainly
focus on finding an adaptive meta-learning rate in the remaining
part.
In general, the error cost is consisted of k output nodes. And
each individual output node is corresponding to different estimated output. According to the different output, we can calculate
the error cost. Hence, we define the error cost function as follows.
Eðt þ 1Þ ¼
1
eðt þ 1ÞT eðt þ 1Þ
2
ð11Þ
where e(t + 1) is the difference between the target output and prediction results, namely
ej ðtÞ ¼ yj ðtÞ t j ðtÞ
ð12Þ
Since there exist k output nodes, there exists an output error vector,
which is defined as e(t) = [e1(t), e2(t), . . . , ek(t)]T 2 Rk1, and yj(t) is the
jth entry of estimated result at time t.
As mentioned above, even though the learning rate is updated
instantly, the fixed global learning rate may cause problems and
have poor performance for training convergence. Meanwhile, in
S. Zhou et al. / Expert Systems with Applications 39 (2012) 6168–6173
order to make the neural network to re-learn from new input data,
the neural network learning process should be recursive and
iterative.
From Yu et al. (2007), the error cost vector can be rewritten as
follows.
eðt þ 1Þ ¼ eðtÞ þ lðtÞnðtÞ
ð13Þ
By combining Eqs. (11) and (13), we get the following equation.
1
½eðtÞ þ lðtÞnðtÞT ½eðtÞ þ lðtÞnðtÞ
2
Eðt þ 1Þ ¼
ð14Þ
The error cost at time (t + 1) is the function of error cost at time t
and learning rate and gradient of learning rate on error cost. This
equation shows that the history data is used to compute the error
cost function.
When using online learning algorithm, we consider the increment process of the error cost as a continuous one. This means that
we can take derivative of the error cost function and find the local
minimal data point. For a continuous function, we obtain a local
optimal when the first derivative of this function is equal to 0.
Namely,
dEðt þ 1Þ
1
1
¼ nT ðtÞ½eðtÞ þ lðtÞnðtÞ þ ½eðtÞ þ lðtÞnðtÞT nðtÞ ¼ 0
dl l¼l ðtÞ 2
2
ij
ð15Þ
where lij ðtÞ is the local optimal learning rate, at which we obtain a
minimal square error. To continue the process, we take the second
derivative of error cost function. We find that the derivative result is
larger than 0, which means that we obtain a lower bound local minimal at lij ðtÞ.
2
d Eðt þ 1Þ
dl2 ¼ nT ðtÞnðtÞ > 0
ð16Þ
l¼lij ðtÞ
By solving Eq. (15), we obtain the expression of learning rate,
namely
lðtÞ ¼ nðtÞeðtÞ
ð17Þ
nT ðtÞnðtÞ
From the definition of l(t), we find that the learning rate is affected
by the estimated output and the change of weight matrix at the last
epoch training. Therefore, we can draw a relationship among the
meta-learning rate, estimated output and change of weight matrix
through combining the Eqs. (10) and (17). Then, we can obtain
the meta-learning rate as follows.
k¼
nT ðtÞeðtÞ þ lðt 1ÞnT ðtÞnðtÞ
nT ðtÞnðtÞ
n1 ðt 1Þ
ð18Þ
6171
From the above definition of k, we know that this meta-learning
rate changes with time data series. At different data point, we obtain different meta-learning rate. When considering the definition
of n(t) and l(t), we find that the change of k is proportional to the
change of weight matrix Dw(t).
Like what is shown in Fig. 3, when the error cost moves in the
direction of down slope, Dw(t) will decrease, which accordingly
causes the meta-learning rate to decrease.
When k is decreasing, it will decrease the learning rate in turn
and, therefore, lower down the learning speed and converge to
the local minimal. Conversely, when the error cost function makes
upslope motions, the weight matrix will increase.
With a positive change of weight matrix, the meta-learning rate
is also increased. Since the moving process of error cost function is
trying to escape from the local minimal, the increment of metalearning rate will accelerate the divergence process.
5. Experiments
5.1. Evaluation criteria
In this part, three classes of measurement criteria including level
prediction, convergence speed and directional forecasting are employed to evaluate the performance. First, unlike Bo, Wang, and
Lai (2007), we use the mean absolute error (MAE) (Willmott &
Matsuura, 2005) rather than the criteria of root mean squared error
(RMSE) because RMSE varies with variability within the distribution of error magnitudes and MAE is a more natural measure for
the average error.
Thus, we use MAE to compare the prediction precision of different forecasting models with our novel model. The definition of
MAE is given by Eq. (11).
MAE ¼
N
1 X
^ i yi j
jy
N i¼1
ð19Þ
^i is the predicwhere N represents the total number of data points; y
tion result of ith data point; and yi is the target value of ith data
point. The smaller the value of MAE, the more precise the prediction
model will be.
Second, we compare our model with other prediction models in
terms of convergence speed. Though an algorithm can run at a high
speed, this does not mean the algorithm is good because if the prediction result has a large value of MAE, the algorithm will be useless to investors. Therefore, we consider the running time of the
models together with the prediction precision.
Third, in order to measure the direction of movement of time
series accurately, we use a directional statistic (Dstat) (Yu, Wang, &
Lai, 2005), which is defined by the following equation.
Dstat ¼
N
1X
ai 100%
N i¼1
ð20Þ
^iþ1 yi Þðyiþ1 yi Þ > 0, otherwise, ai = 0.N reprewhere ai = 1, if ðy
sents the total number of testing samples. The larger this factor is,
the more proper direction the forecasting model will have.
5.2. Experimental results
Fig. 3. Effect of meta-learning rate on convergence.
In this section, we use daily gold future price data from the
database of global financial data (GFD). The database provides
the most extensive time series data on commodities available
anywhere. It includes historical data of about 100 different commodities. In this paper, we use data spanning from 1 Jan 2000
to 13 Jul 2010, a total of 2636 observations. And we use the rest
of the time series spanning from 29 Dec 2009 to 14 Jul 2010 for
6172
S. Zhou et al. / Expert Systems with Applications 39 (2012) 6168–6173
evaluation of prediction performance, which contains a total of
136 observations.
To begin the experiment, we firstly use the EMD technique to
handle the time data series. Before we use this technique, there
are two requirements which should be satisfied.
(1) The number of extreme points should be equal to zero points
of the series or differ by only one;
(2) The mean value of envelope of local minimal and local maximal must be zero.
With these two requirements, the IMCs contained in EMD may
become more harmonics when we apply EMD to decompose the
data series (Huang, Shen, & Long, 1999). The decomposition procedure is descried as follows:
(1) Identify all the extreme points in the data series, including
local maximal and local minimal;
(2) Generate upper and lower envelops xup(t) and xlow(t) for the
all local extrema;
(3) Compute the mean envelope m(t) for all the extrema from
xup(t) and xlow(t);
(4) Calculate the difference between mean envelope at point t
and x(t)
dðtÞ ¼ xðtÞ mðtÞ
ð21Þ
(5) Check d(t): (a) if d(t) meets the above two requirements, an
IMC is identified and replace x(t) with r(t) = x(t) d(t); (b) if
d(t) does not meet, then replace x(t) with d(t);
(6) Repeat (1)–(5) until the following stopping condition:
T
X
½dj ðtÞ djþ1 ðtÞ2
2
t¼1
dj ðtÞ
< SC
ð22Þ
where SC is the stopping condition, usually it is set between 0.2 and
0.3.
Compared with the traditional Fourier analysis, this EMD technique owns several advantages. First, this method is robust to handle the nonlinear and nonstationary data since it can automatically
and adaptively select the fluctuations from the original time series.
Second, EMD can divide the time series into several independent
IMCs and a residual component. The behavior of these decomposed
components only depends on the nature of the original data. Final-
Table 1
The MAE and running time comparisons.
Forecasting model
MAE
Rank
Time (s)
Rank
BPNN
FNN
EMD–FNN–ALNN
IEMD–BPNN–PMR
9.8292
8.9748
3.3549
2.1677
4
3
2
1
83
72
124
79
3
1
4
2
ly, this method is easy to understand and use and owns wide
applications.
The decomposition results are shown on Fig. 4. Each of the subfigures represents an IMC. These sub-data series are then used for
the BP neural network training. Since the IMC have extracted the
characteristics from the original data, the training of neural network will have much high convergence speed by using these data
series.
In many literatures, researchers use the logistic sigmoid
f ðuÞ ¼ 1þe1 u as activation function for units in the hidden layer. This
function has a small asymmetric range from 0 to 1. Then, this logistic sigmoid function worsens the condition of the network. Accordingly, in our prediction model, we choose tanh as the activation
function f(), which has a symmetric range from 1 to 1 and is
more suitable as activation function for hidden units.
Since tanh0 (u) = 1 tanh (u)2, then we have f 0 ðyi Þ ¼ 1 y2i .
From Table 1, MAE values of different models are different.
BPNN has large MAE of 8.9748 and ranks the last. Our model,
IEMD–BPNN–PMR, ranks at the first place with MAE value of
2.1677 and is followed by EMD–FNN–ALNN model, BPNN model
and FNN model. This sequence represents a descending order of
prediction precision. In terms of running time, we can see that
the FNN model spends less time to converge. It takes only 72 s,
which is only 20.11% of the total time of all models. Our model
ranks at the second place and takes 79 s to converge.
The FNN model is a simple type of neural network. For a
three layer FNN, the model consists of a linear combination of
input data and the weight matrix and an activation function to
transfer the combined results. Hence, the FNN model will take
the least running time to converge. However, the precision of
this model is also the worst. Conversely, our model adopts a
BPNN model as the prediction model. Although it takes more
time to get converged, prediction results are more precise than
the FNN model. Our model runs faster than the individual BPNN
model. This is because we have employed an EMD method to
decompose the input data, which accelerates the convergence
speed to some extent. Besides, the usage of improved online
learning algorithm makes sure the weight matrix is changed instantly at each epoch. As a result, the training process has a high
convergence speed.
Table 2 shows results of comparison of different models on the
aspect of directional forecasting. Ranking at the first place is the
IEMD–BPNN–PMR model, which is followed by EMD–FNN–ALNN
model.
The reason why our model can predict correctly the trend of
data series most of the time is that we make some changes to
the original BPNN model and use tanh as the activation function,
Table 2
The Dstat (%) comparisons.
Fig. 4. The decomposition of gold market price.
Forecasting model
Dstat (%)
Rank
BPNN
FNN
EMD–FNN–ALNN
IEMD–BPNN–PMR
50.74
59.56
66.91
86.76
4
3
2
1
S. Zhou et al. / Expert Systems with Applications 39 (2012) 6168–6173
which has a symmetric range from 1 to 1. This helps the activation function to capture the series trend flexibly and accurately.
6. Conclusions
In this paper, we propose an improved EMD online learningbased model for gold price forecasting. We first use the EMD method
to partition the time series data into several subsets of data series,
which are used for training the neural network. Then, back-propagation neural network is employed to function as the prediction model
in our novel system since BPNN is one of widely studied and used
learning algorithms in the academia. By studying the essential process of BPNN mathematically, we formulate a deduction process to
show that online learning can be updated instantly. As the learning
rate is updated online, the weight is also updated after every data input. This ensures that the BPNN model is able to escape from the local minimum of the error function and, therefore, is able to show a
good performance in terms of quick convergence speed. The experiment results show that our system has offers good prediction results as well as running time.
Although we apply the novel algorithm to the gold market and
obtain some impressive results, there is still some improvement
we can make. The global meta-learning rate can be further studied
to see whether there exists a relationship between MAE and data
samples. Besides, our model can also be applied to stock and warrants market (Huang, Nakamori, & Wang, 2005; Lu, 2010). There
will be more interesting and practical results.
References
Abraham, A. (2004). Meta learning evolutionary artificial neural networks. Neuro
Computing, 56, 1–38.
Atsalakisa, G. S., & Valavanisb, K. P. (2009). Suveying stock market forecasting
techniques- Part II: Soft computing methods. Expert Systems with Applications,
36(3, Part 2), 5932–5941.
6173
Bo, W., Wang, S., Lai, K. K. (2007). A hybrid ARCH-M and BP neural network model
for GSCI futures price forecasting. In International conference on computational
science (3) (pp. 917–924).
Grudnitski, G., & Osburn, L. (1993). Forecasting S& P and gold futures prices: an
application of neural networks. Journal of Futures Markets, 13(6), 631–643.
Hornik, K., Stinchocombe, M., & White, H. (1989). Multilayer feedforward networks
are universal approximators. Neural Networks, 2, 359–366.
Huang, W., Nakamori, Y., & Wang, S.-Y. (2005). Forecasting stock market movement
direction with support vector machine. Computers & Operations Research.,
32(10), 2513–2522.
Huang, N. E., Shen, Z., & Long, S. R. (1999). A new view of nonlinear water waves:
The Hilbert spectrum. Annual Review of Fluid Mechanics, 31, 417–457.
Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., et al. (1998). The
empirical mode decomposition and the Hilbert spectrum for nonlinear and
nonstationary time series analysis. Proceedings of the Royal Society A:
Mathematical Physical & Engineering Sciences, 454, 903–995.
Lu, C.-J. (2010). Integrating independent component analysis-based denoising
scheme with neural network for stock price prediction. Expert Systems with
Applications., 37(10), 7056–7064.
Orr, G. B., & Leen, T. K. (1992). Learning in neural networks with local minima.
Physical Review A, 46(8), 5221–5231.
Orr, G. B., & Leen, T. K. (1996). Using curvature information for fast stochastic
search. Neural Information Processing Systems, 606–612.
Ricardo, V., & Youssef, D. (2002). A perspective view and survey of meta-learning.
Artificial Intelligence Review, 18(2), 77–95.
Saad, D. (Ed.). (1999). On-line learning in neural networks. New York, NY, USA:
Cambridge University Press.
Shafieea, S., & Topalb, E. (2010). An overview of global gold market and gold price
forecasting. Resources Policy, 35(3), 178–189.
White, H. (1990). Connectionist nonparametric regression: multilayer feedforward
networks can learn arbitrary mappings. Neural Networks, 3, 535–549.
Willmott, C. J., & Matsuura, K. (2005). Advantages of the mean absolute error (MAE)
over the root mean square error (RMSE) in assessing average model
performance. Climate Research, 30(3), 79–82.
Yu, L., Lai, K. K., Wang, S., He, K. (2007). Oil price forecasting with an emd-based
multiscale neural network learning paradigm. In International conference on
computational science (3) (pp. 925–932).
Yu, L., Wang, S., & Lai, K. K. (2005). A novel nonlinear ensemble forecasting model
incorporating GLAR and ANN for foreign exchange rates. Computers & OR, 3,
2523–2541.
Yu, L., Wang, S. Y., & Lai, K. K. (2007). An on-line learning algorithm with adaptive
forgetting factors for feedforward neural networks in financial time series
forecasting. Nonlinear Dynamics and Systems Theory, 7(1), 97–112.
Yu, L., Wang, S., & Lai, K. K. (2008). An EMD-based neural network ensemble
learning model for world crude oil spot price forecasting. Soft Computing
Applications in Business, 261–271.