Investigating the Effect of Noise Elimination on LSTM Models for
Financial Markets Prediction Using Kalman Filter and Wavelet
Transform
AMIN KARIMI DASTGERDI, PAOLO MERCORELLI*
Institute of Product and Process Innovation, Leuphana University of Luneburg,
Luneburg, GERMANY
Abstract:- Predicting financial markets is of particular importance for investors who intend to make the most
profit. Analysing reasonable and precise strategies for predicting financial markets has a long history. Deep
learning techniques include analyses and predictions that can assist scientists in discovering unknown patterns
of data. In this project, application of noise elimination techniques such as Wavelet transform and Kalman filter
in combination of deep learning methods were discussed for predicting financial time series. The results show
employing noise elimination techniques such as Wavelet transform and Kalman filter, have considerable effect
on performance of LSTM neural network in extracting hidden patterns in the financial time series and can
precisely predict future actions in these markets.
Key-Words: - Deep Learning, Time-series Forecasting, Financial Markets, LSTM, Wavelet Transform, Kalman
Filter
Received: June 2, 2021. Revised: December 4, 2021. Accepted: January 17, 2022. Published: January 18, 2022.
1 Introduction
Among time series predictions, Stock market
prediction is considered as one of the most
challenging problems, due to its noisy and unstable
features [1]. Since the early 70s, extensive efforts
have been commenced to predict stock prices by
using new mathematical methods, time series, and
more advanced tools including artificial intelligence
and many tests on price information and stock
indexes in countries such as United States, United
Kingdom, Canada, Germany, and Japan to show the
presence or absence of a specific structure in stock
price information [2]. To analyze this issue, the use
of deep learning has attracted a great deal of
attention in recent years and has been mentioned
much in the communities related to artificial
intelligence and machine learning. Deep learning
implies the set of machine learning-based
algorithms and methods that tries to discover
complex patterns in data and model the high-level
abstractions. Neural networks were first used in
1977 to predict exchange markets [3].
Between 1977 and 1995, a total of 213 academic
activities in the field of artificial neural networks in
the field of commerce. Of these, 54 were in the
financial sector and two were in forecasting and
analyzing other time series [4]. Most of the
researches in this field were conducted to look for
the ability of neural networks to identify non-linear
patterns in time series and unknown price
movements. Deep learning characteristics provided
significant success for these types of approaches in
machine learning.
Current prediction models suffer from
deterministic and stochastic noise which exists in
financial time series. Therefore, an appropriate
algorithm which could eliminate the noise from
financial time series without influencing the real
values seems to be necessary to enhance the
performance of these models. Wavelet transform
and Kalman filter are two technique which are
commonly used to eliminate the noise feature of
financial time series. These techniques are
considered for filtering and mining single-
dimensional signals [5]. These techniques have been
used in this project to eliminate the input financial
time series and then feed them into the LSTM neural
network.
Accordingly, this project report can be defined in
several sections. The second section includes
literature review and problem statement and its
analysis to enhance the structure of neural networks.
In the third section, the framework of the model
which used in this project is discussed by showing
its flowchart and describing different parts of it. The
fourth section is dedicated to explanations about the
project implementation and in the last part result of
the project are discussed and analyzed.
2 Literature Review
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The statistical methods based on analyzing the past
market changes are widely accepted and employed
in the methods that predict the financial markets.
These approaches use different linear and non-linear
methods to predict the market. Two views on linear
and non-linear methods are presented [6], which in
[7], the non-linear models have been considered to
be better than linear models; however, linear models
may work better than or as good as non-linear
methods [8]. Prediction can be divided into three
categories: short-term prediction, mid-term
prediction, and long-term prediction, each of which
works in different time ranges. In this regard, the
available methods for predicting the stock price can
be divided into three categories: fundamental
analysis, technical analysis, time series analysis.
These types of assessments can be described in three
sections: fundamental, technical, time series.
In this respect, fundamental analysis is based on
the assumption that securities (generally financial
markets) possess intrinsic values that can be
estimated by investors. Technical analysis in
financial markets is a way of predicting the probable
behavior of a chart through past data, such as price
and its changes, trading volume, etc. In time series
analysis, the time series can be defined as a regular
sequence of observations for a selected variable in
many prediction problems. In financial markets, the
stock price is considered as the selected variable.
Fig. 1: Overview of variety of prediction methods for financial markets
So far, many researchers have tried to analyse non-
linear time series and find a technique to separate
noise of time series. One effective and powerful
algorithm to separate the true noise component of
original time series which has been proposed by
Broomhead and King [9] is singular spectrum
analysis (SSA). SSA tries to extract a series of
singular values that contains the independent
information of the original time series through
singular spectral decomposition (SVD) and then
analyse the features of the original time series over
different time scales by reconstructing these singular
values[10]. To explore the existing noise which lies
in financial markets, Hill et. al [11] performed
blockwise white noise tests. They realized that the
white noise hypothesis can be accepted for financial
markets of China and Japan which shows that those
markets are weak form efficient while the white
noise hypothesis can not be accepted for financial
markets of Unites States and United Kingdom.
Xiang Zhang [12] proposed a SVR-ARMA model
for stock price prediction based on soft thereshold
wavelet denoising which sym wavelet was selected
as the wavelet basis function and three vanishing
moments and three decomposition layers was
considered. The maximum-minimum criterion has
been employed to determine the threshold. Paisit
Khanarsa et al. [13] proposed a self-identification
ResNet-ARIMA order model to solve the issue of
identifying ARIMA order and automatically learn
the ARIMA order from known ARIMA time series
data via sample autocorrelation function, the sample
partial autocorrelation function and differencing
time series images. In another study, Brogaard et al.
[14] proposed a return variance decomposition
model to separate the role of different types of
information and noise in stock price movements.
They came to the conclusion that 31% of the return
variance is from noise. Xiaodan Liang et al. [15]
proposed a new multioptimal combination wavelet
transform (MOCWT) method which makes benefit
of threshold-denoising function to reduce the degree
of distortion in signal reconstruction to predict S&P
500 prices. Zargar et al. [16] proposed a model
called “opening noise trading model” in which the
opening price of the stock market was assumed to
contain a component of noise which is orthogonal to
the true price change caused by the arrival of new
information among all the Nifty stocks. They have
also estimated the share of noise in the opening
price.
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2.1 Problem Statement
In this research, Wavelet Transform and Kalman
Filter were used to eliminate the noise from
financial markets data which includes closing price,
volume, technical indicators and other factors
affecting the stock price. Also, a fully connected
dense LSTM was used for prediction. The actions
can be performed, considering the following
objectives:
• Improving the prediction ability of future
behaviors of financial markets using deep
learning techniques.
• Investigating the effect of noise elimination on
the accuracy of financial markets prediction.
3 Project Methology
As can be seen in Fig. 2. In this project, financial
time series which were the indices of five different
stock markets were used as inputs. The noise of
these financial time series was eliminated using
discrete wavelet transformation as well as Kalman
filter algorithms. Then denoised financial time series
were used as inputs of LSTM neural network. An
LSTM neural network was trained based on training
data and was used for prediction of the unseen data.
Finally, the accuracy of the LSTM model was
evaluated with comparison of predicted data and
real values.
Fig. 2: The flowchart of the proposed model
3.1 Wavelet Transform
Wavelet transform is used in this project because of
its ability to noise elimination of financial time
series data. In this project, the Haar function
considered as the basic wavelet function because
this function in addition to breaking down financial
time series into time and frequency ranges, has the
advantage of significantly reducing processing time
[5]. Time complexity of Wavelet transform with the
Haar function as the basic function is O(n) where n
is the size of the time series [17]. For continuous
Wavelet transform, the wallet function is defined as
follows:
(1)
Where a and τ are the scaling and translation
factors, respectively. is basic Wavelet which
follows a rule called the Wavelet Acceptance
Condition [18]:
Where φ (w) is a function of the frequency w as
well as Fourier transient. x (t) denotes a
complete square integral function. Continuous
Wavelet Transform to wavelet is defined as
follows:
Where (φ(t))  represents conjugate function. the
inverse of continuous wavelet transform is as
follows:
3.2 Kalman Filter
Kalman filter enables inference of unmeasured
variables from indirect and noisy measurement [19].
As a result, it is arguably one of the most important
discoveries in the field of mathematical engineering
and has been used to solve various engineering
problems in the area of monitoring and control of
complex dynamic systems such as manufacturing
processes, aircraft navigation, ships, and spacecrafts.
While the Kalman filter is used to estimate of a state
vector in a linear model in a dynamical system, the
extended Kalman filter (EKF) is used in order to
estimate a non-linear model [20].
The Kalman filter estimates the state of a
dynamic system having certain types of random
behavior [21]. The system must be described in a
state space form:
x (n x 1) is called the state vector. It is composed
of any set of variables sufficient to completely
describe the unforced motion of a dynamic system. z
(l x 1) is called the observation vector. It concerns
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data that can be known through measurements. wk
and vk are the state and measurement white noise
with known covariance matrices Qk and Rk,
respectively. They are mutually not correlated. is
the state transition matrix, Hk the observation
matrix. The Kalman filter is based on a recursive
algorithm [21]. At time tk, the optimum combination
of measured and estimated results is given by:
where denotes the a priori estimate [21]. The
“hat” denotes an estimate and the superscript minus
indicates that this is the best estimate prior to
assimilating the measurement at tk. The Kalman
filter gain Kk can be written as:
where the error covariance matrix Pk associated
with the optimal estimate is obtained from:
To recursively compute the Kalman filter gain
for the next step, the predictions for the state
estimate and covariance at the next step are given
by:
The derivation of the extended Kalman filter
allows the estimation of a non-linear system state.
To separate noise from signal, Kalman filter uses
state vector. The state vector consists of all the
features describing the models of the random
processes. Additional state variables are appended to
account for either non-white state or measurement
noise. and Qk are straightforwardly deduced
from the models. The observation vector is just the
measured signal that is a mix of signal and noise.
Therefore, the observation matrix is usually a
combination of unit matrices and zeros matrices
according to the state vector. The algorithm of the
discrete Kalman filter will be used to estimate the
signal, Once the system is properly designed [21].
3.3 LSTM
LSTM networks are a type of recurrent neural
networks, with the advantage of having feedback
links attached to some layers of the network which
enables them to learn long-term dependencies.
LSTM networks can easily remember information
for long periods of without any learning struggle.
They were introduced by Hochreiter &
Schmidhuber [22], and were developed and refined
by many researchers and have been applied to solve
many problems such as robot control, time series
prediction, speech recognition, rhythm learning,
music composition, grammar learning, handwriting
recognition, text-based language translation [23] and
so on.
All recurrent neural networks have the form of a
chain of repeating modules of neural network. In
standard RNNs, this repeating module will have a
very simple structure, such as a single tanh layer.
Unlike recurrent neural networks, LSTM is well-
designed to learn from experience to predict time
series when there are time steps with no certain size.
In addition, thanks to having the memory unit, it can
solve the problem of the vanishing gradient by
having the memory unit keep the time related
information for an uncertain time [24]. The change
in the memory unit has made the LSTM network to
be capable of remembering long-term dependencies.
A memory cell consists of four components: input
gate, target gate, forget gate, and one recurrent
neuron. The gates control the interactions between
the memory cell and its neighboring memory cells.
The input gate determines whether an input signal
can alter the status of a memory cell. On the other
hand, the target gate determines whether the status
of one memory cell can change the status of another
memory cell. The target gate also selects to
remember or forget the previous status [25].
Cell state is the most important part of LSTM. It
is shown in Fig. 3. (The horizontal line running
through the top). It runs straight down the whole
chain, with only some minor linear interactions.
Information can easily flow along it without being
changed. The LSTM can remove or add information
to the cell state using gates, which are carefully
regulated structures. Gates are a way to optionally
let information through. They are composed out of a
sigmoid neural net layer and a pointwise
multiplication operation. The sigmoid layer outputs
numbers between zero and one, describing how
much of each component should be let through. A
value of zero means “let nothing through,” while a
value of one means “let everything through!”.
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Fig. 3: Structure of a LSTM unit
The first step in LSTM is to decide what
information is going to be thrown away from the
cell state. This decision is made by a sigmoid layer
called the “forget gate layer.” It looks at and
xt, and outputs a number between zero and one for
each number in the cell state . A one represents
“completely keep this” while a zero represents
“completely get rid of this.”
(10)
The next step is to decide what new information
should be stored in the cell state. This has two parts.
First, a sigmoid layer called the “input gate layer”
decides which values should be updated. Next, a
tangent hyperbolic layer creates a vector of new
candidate values, , that could be added to the
state. In the next step, these two will be combined to
create an update to the state [26].
(11)
(12)
Then the old cell state, , will be updated into
the new cell state . The old state will be multiplied
by , forgetting the things we decided to forget
earlier. Then will be added. This is the new
candidate values, scaled by how much it has been
decided to update each state value.
(13)
Finally, the output will be based on the cell state,
although a filtered version. First, a sigmoid layer
will be run which decides what parts of the cell state
is going to be the output. Then, in order to scale the
values between -1 and 1 the cell state is put through
tanh and then multiply it by the output of the
sigmoid gate, so that only output the parts it has
been decided.
(14)
4 Implementation
All data which has been used in this project is from
WIND database that developed by Shanghai Wind
Observation Company as well as the CSMAR
database that developed by GTA Technical and
Education Organization Shen Jen and another
financial database are located at
(http://www.investing.com). data are from July 1,
2008 to September 30, 2016.
4.1 Software
In this project we propose to implement and
evaluate the model based on Jupiter Notebook
version 6.2.0 and Python programming language as
well as Python interpreter version 3.8. All tests were
implemented on a computer with 8-core processor at
2.21 GHz and 6 GB main memory by Windows
operating system 10, 64-bits.
Some Important Python libraries which were
used in this project are: numpy, pandas, sikit-learn,
pywt, pykalman, matplotlib and plotly.
4.2 Data Description
The five indexes of the selected exchange include
financial markets of China (CSI300), Hong Kong
market (Hang Seng), Tokyo market (Nikkei 225),
New York exchange markets (S&P500 and DJIA).
As a rule, in financial markets around the world, the
market situation may affect the credibility of the
neural network; hence, the datasets of some markets
with different conditions were used to overcome this
issue. The data of the S&P500 and DJIA index are
known as the most advanced financial markets in
the world. On the other hand, CSI300 is considered
as a rookie market and has been used as a
developing market. In addition to the
aforementioned markets, the Hang Seng index and
the Nikkei 225 index were used as relatively-
developed markets. Therefore, these markets
provide a natural condition for evaluating the
robustness and efficiency of models in markets with
different conditions.
Three sets of variables were used as input. first
set of variables which are considered as basic are
opening price, highest price, lowest price, last price
and volume of transactions. Second set of input
variables is included twelve commonly used
indicators are technical indicators for each indicator.
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the last set of input variables is included
macroeconomic variables. macroeconomic
conditions with no doubt have a great impact on the
performance of the stock markets between different
regions [27]. So macroeconomic variables can
transfer some information to the predictive neural
network. In this project two types of
macroeconomic indices of exchange rate and
interest rate are used. Dollar index was used as a
conversion rate. In addition, the interbank interest
rate in each country was used as the interest rate. In
table 1 definitions of all the variables which were
used in this project are given.
Table 1. Definitions of input variables
Definition
Variable name
First set of input variables (basic information)
First price and last traded price of a share in day
Open/Close Price
The highest and lowest quoted price in single day
High/Low Price
volume of shares traded in day
Trading Volume
Second set of input variables (technical indicators)
It means convergence and divergence of the moving average. It is used in
technical analysis to gain ability, direction and acceleration in a process
MACD
The acceleration power of the oscillation in a share trend is well represented. It
helps us find the beginning and the end of process.
CCI
It shows the fluctuations in the market well in the share
ATR
Provides a relative definition of the highest and lowest prices that help us
identify precise patterns.
BOLL
Exponential moving average is a type of moving average that gives more weight
to the latest data.
EMA 20
Moving Average 5 /10 days
MA5/MA10
Monthly rate of movement. Helps us find the end or the downtrend or uptrend.
MTM6/ MTM12
Price change rates. Shows the rate of change in the price of share.
ROC
Random motion index. Indicates the proximity of the end price to the midpoint
in that range.
SMI
Measures the amount of pressure to buy or sell in share.
WVAD
Third set of input variables (macroeconomic variables)
US dollar index
Exchange Rate
Interbank interest rates
Interest Rate
4.3 Model Training
In this project, dataset is divided into three parts:
training data, validation data and test data and the
back propagation algorithm is used to train the
LSTM network. learning rate, batch size and
frequency of training set was set to 0.05, 32, and
100, respectively. Convergence rate is controlled
by learning rate which is a descending function of
time. When convergence is achieved but the
combinations of parameters are varied It can be
concluded that the experimental result are stable. In
Fig. 4. Training loss and validation loss of LSTM
model for different markets are depicted. As can be
seen from different plots training loss and
validation loss converge after 100 epochs which
means the prediction error of the LSTM model is
relatively small.
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Fig. 4: Training loss and validation loss of LSTM model for different markets
4.4 Model Evaluation
4.4.1 MAPE
MAPE measures the accuracy of predicting a
method. For instance, in the process of predicting
trends, this measurement is used as an error
function for regression problems in machine
learning. MAPE indicates the accuracy in
percentage and is obtained from the Eq. (10).
(15)
4.4.2 Theil U
Theil U is a relative measurement of differences
between two variables. Theil U reduces deviations
to the power of two to give more weight and
importance to more significant errors. Theil U is
derived from the Eq. (11).
(16)
4.4.3 RMSE
RMSE is the difference between the actual value
and the value predicted by the model or statistical
estimator. RMSE is obtained from the Eq. (12).
(17)
4.4.4 R: in investing, R is generally interpreted as
the percentage of a fund or security's movements
that can be explained by movements in a
benchmark index. For example, an R-squared for
a fixed-income security versus a bond index
identifies the security's proportion of price
movement that is predictable based on a price
movement of the index. R is derived from the Eq.
(13).
(18)
In the equations above, represents the
actual value of data at time t, the variable
implies to the value predicted by the model at time
t, and n represents the amount of data.
5 Results
In this section results are given. First Fig. 5.
presents the plot depicting the performance of
Wavelet transform and Kalman filter for S&P500
index market.
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Fig. 5: Results of Wavelet transform and Kalman filter for S&P500 index
In order to evaluate each model, four key error metrics were considered which are root mean squared error
(RMSE), mean absolute percentage error (MAPE), R and Theil U. Tables 2 to 5 contains performance of
LSTM model which has used data that has been smoothed by both Wavelet transform and Kalman filter as well
as raw data. “LSTM” refers to the model which used the raw data, “WLSTM” refers to the model which used
the data that has been denoised by Wavelet transform and “KLSTM” refers to the model which used data that
has been smoothed by Kalman filter.
Table 2. Predictive accuracy based on MAPE metric
DJIA
S&P 500
Nikkei
Hang
CSI
0.007
0.007
0.025
0.011
0.019
LSTM
0.005
0.006
0.018
0.009
0.019
WLSTM
0.009
0.008
0.015
0.007
0.014
KLSTM
Table 3. Predictive accuracy based on Theil U metric
DJIA
S&P 500
Nikkei
Hang
CSI
0.004
0.005
0.017
0.007
0.019
LSTM
0.003
0.003
0.013
0.006
0.018
WLSTM
0.005
0.004
0.011
0.005
0.014
KLSTM
Table 4. Predictive accuracy based on R metric
DJIA
S&P 500
Nikkei
Hang
CSI
0.933
0.934
0.891
0.972
0.970
LSTM
0.961
0.963
0.930
0.982
0.972
WLSTM
0.906
0.947
0.949
0.987
0.982
KLSTM
Table 5. Predictive accuracy based on RMSE metric
DJIA
S&P 500
Nikkei
Hang
CSI
172
21
598
354
126
LSTM
130
15
478
280
122
WLSTM
203
18
407
240
96
KLSTM
It is clear from tables 2 to 5 that almost in all
cases LSTM models which used noised eliminated
data have better performance and lower error in
predicting the actual value. It can also be interpreted
from the results that the performance of LSTM
neural networks in the datasets of developed
markets is better than its performance in the datasets
of developing and relatively-developed markets.
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The reason of this issue is related to efficient market
hypothesis (EMH) [28] which states that the
efficiency of a market affects the predictability of its
asset. In other words, the more advanced markets in
which more traders participate and therefore more
transactions are recorded, the patterns created in the
stock price chart are more reliable and the stock
trend and movements become more predictable.
6 Conclusion
Stock price prediction is considered to be difficult
since stock data is a random walk time series which
contains high level of noise. Hence, employing a
noise reduction technique to process the underlying
noise in stock data is inevitable. However, not all
denoising techniques are beneficial for all stock
markets. In other words, if we use an inappropriate
technique to preprocess and denoise the data it will
destroy the integrity and authenticity of the data
which leads to miss some usefulness of the price
information. Therefore, choosing the right denoising
technique is an important task that should be
considered with caution.
In order to achieve a more superior prediction
results for the LSTM neural network, In this project,
authors used two different algorithms of noise
elimination which were Wavelet transform and
Kalman filter. These algorithms have been used in
order to investigate the effect of noise elimination
from noisy financial time series on enhancing the
LSTM model performance in comparison with
Wavelet transform. With a glimpse on results, it is
obviously clear that using a noise elimination
technique such as Kalman filter or Wavelet
transform can enhance the results of predicting
financial markets and employing a noise elimination
technique is absolutely necessary for anyone who is
trying to predict the financial markets.
It can be concluded from the results that since
Kalman Filter marginally outperforms Wavelet
transform in denoising developing and relatively
developed stock indexes, it is going to be a better
choice for denoising systems that are less mature and
have higher volatility. On the other hand, Wavelet
Transform presents a better performance than
Kalman filter in developed stock indexes which
implies that it is more powerful in processing noise
data and could be beneficial in denoising more
advanced markets. These findings imply that, to
better preserve the original information, processing
noisy time series should be done with careful
consideration of the stock market’s level of maturity.
Employing a noise eliminator algorithm such as
Wavelet Transform or Kalman Filter is an effective
factor to enhance the performance of a LSTM model,
which allows the neural network to better identify
the available patterns in the price chart and make
more precise predictions. According to the results of
this project, deep learning approaches are capable of
identifying and extracting hidden patterns in the
financial time series and can accurately predict the
future behavior of such markets.
One of the most demanding and time-consuming
tasks in training deep learning models is to find an
optimal value for hyperparameters which are
commonly set with try and error such as number of
LSTM units, regularization parameters, dropout
percentage and so on. Future works could be the
study of these hyperparameters and proposing
methods which are capable of effectively finding the
optimal values and enhance the performance of
these networks. Also, the change in structure and the
number of layers of LSTM neural networks can be
studied to improve its performance by using
professional noise elimination algorithms to
eliminate noise from input data. Furthermore,
profitability can be modeled by incorporating
trading commissions into input variables.
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WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2022.19.39
Amin Karimi Dastgerdi, Paolo Mercorelli
E-ISSN: 2224-2899
441
Volume 19, 2022