Forecasting OMX Vilnius stock index--a neural network approach/OMX Vilnius akciju indekso prognozavimas naudojant dirbtinius neuronu tinklus.
Dzikevicius, Audrius ; Stabuzyte, Neringa
1. Introduction
Stock market prediction brings a lot of discussion between
academia. First of all negotiations arise whether future prices can be
forecasted or not. One of the first theories against ability to forecast
the market is Efficient Market Theory (EMH). It states that current
prices "fully" reflect all available information so there is
no possibility to earn any excess profit (Fama 1970). Another important
statement was made several years later announcing, that stocks take a
random and unpredictable path, stock prices have the same distribution
and are independent from each other, so past movement cannot be used to
predict the future (Malkiel 1973). This idea stands for Random Walk
Theory. According to these statements no one investor could profit from
the market without additional unpublicized information or undertaking
additional risk. But these theories are facing critics and negotiations
that during the time prices are maintaining some trends so it is
possible to outperform the market by implementing appropriate
forecasting models and strategies.
Researchers provide many models for stock market forecasting. They
include various fundamental and technical analysis techniques.
Fundamental analysis involves evaluating all the economy as a whole,
analyzing exogenous macroeconomic variables, the root is based on
expectation. On the contrary, technical analysis is using historical
data, such as price and volume variables, preprocessing this data
mathematically and making future forecasts rooted in statistics.
Financial time series forecasting brings a lot of challenges
because of its chaotic, difficult, unpredictable and nonlinear nature.
The most traditional methods are made under assumption that relation
between stock price and certain variables is linear. There is evidence
that these techniques, such as moving average, do not have acceptable
accuracy (Dzikevicius et al. 2010). Most popular linear dependencies are
simple moving averages, exponential moving averages and linear
regression.
One of the newest approaches to forecast dynamic stock market
nature is looking for non-linear techniques such as artificial neural
networks (ANN). These methods, inspired by human brain, have an ability
to find non-linear patterns, to learn from past and generalize. Neural
networks are widely used in physical sciences but the popularity is
rising in the financial field as well. The main research paper target is
to evaluate the neural network ability to forecast stock market behavior
by implementing a multi-layer perceptron (MLP) model to predict stock
market index OMX Vilnius (OMXV) future movements (actual value and
direction of the index). The model's accuracy is compared with
several traditional linear models (moving average and linear
regression).
The organization of this paper is as follows. The second section
provides a brief review of previous researches, the third parts
describes data and chosen methodology, the fourth part presents
empirical results. The last section provides a brief summary and
conclusions.
2. Literature review
The born year of neural network method can be called the year 1958,
when the first neural network structure was defined. It was called
perceptron (Rosenblatt 1958). Another important date is the year 1986.
The authors introduced the 'back-propagation' learning
algorithm that still nowadays is the most popular and will be discussed
in a more detailed way in the next section (Rumelhart et al. 1986).
Nowadays modern ANN use of field is really wide: it includes
biological, physical science, industry, finance, etc. There are four
main reasons of such increasing popularity of use (Zhang et al. 1998).
First of them is that oppositely from the other traditional methods ANN
have very few assump tions, because they are learning from examples and
capturing functional relationships. The second advantage is
generalization --the ability to find the unseen part of population from
a noisy data. Thirdly, ANN are very good functional approximations and
the last one is non-linearity. On the other hand, these models also have
some weaknesses: they need training, a large data set and a time for
experimenting with the most suitable topology and parameters.
There is a really wide list of ANN applications in finance field.
ANN approach can be used to forecast inflation rate (Catik, Karacuka
2012), estimating credit risk (Boguslauskas, Mileris 2009), evaluating
foreign direct investment (Plikynas, Akbar 2005), etc. Zhang et al.
(1998) provides a detailed summary of modeling issues of ANN use in
forecasting.
The ability of forecasting stock market is also broadly discussed
and results are quite acceptable. A variety of stock market indexes is
analyzed using neural network approach. It includes such indexes as BEL
20 (Belgium stock market), BSE Sensex (Bombay stock market index),
S&P CNX Nifty 50 (India stock market index), ISE National 100
(Istanbul stock market index), KLCI (Malaysia stock market index), IGBM
(Madrid stock market index), TSE (Taiwan stock market index), Tepix
(Iran stock market index), etc. The summary of these researches is
provided in Table 1. As it is seen from Table 1, there is a list of
evidences that ANN can be used successfully in stock market prediction.
The majority of these researches described below select inputs
(variables) as lagged values of the dependant variable on different
periodicity. Some of them combine both historical data and
macroeconomic, fundamental data.
There are evidences that Lithuania's stock market index OMXV
was analyzed before. It includes implementing a set of GARCH models for
this index (Teresiene 2009), the effect of macroeconomic variables on
the index was analyzed by Tvaronaviciene and Michailova (2006), in 2009
by Pilinkus and one year later by Baranauskas (2010).
Traditional statistical methods of forecasting stock market using
moving averages were discussed by Dzikevicius and Saranda (2010) on the
OMX Baltic Benchmark and S&P 500 index. The results revealed that
every market is specific and needs a detailed analysis to find the most
appropriate parameters for every forecasting technique. In our research
ANN model forecasting accuracy is also compared with moving averages
approach.
3. MLP model, data and methodology
3.1. MLP structure
Artificial neural networks were inspired by biological science
--more exactly by human brain structure. Human brain and nervous system
is composed of small cells called neurons. First of all, human body
receives a signal from environment, and then the signal is transformed
by the receptors into an electric impulse and goes to neuron. Evaluation
of the signal is analyzed inside the network and impulse is sent out as
an effect. Neurons are connected through the synapses and are able to
communicate through them. Learning is a process of adjusting old
synapses or adding new ones by using some functions. A simplified
structure of this process is provided in Figure 1. The process of
transferring inputs to the outputs can be expressed mathematically.
Practically it can be implemented using various computer programs
software.
[FIGURE 1 OMITTED]
The basic ANN structure consists of artificial neurons that are
grouped into layers. A structure of one neuron (perceptron) is presented
in Figure 2.
[FIGURE 2 OMITTED]
Here [X.sub.1], [X.sub.2],...[X.sub.i], e called neuron's
inputs. Every connection has its weight attached [W.sub.ij] where j is
the number of the neurons and i stands for the i-th input. Weights can
be both positive and negative. The neuron sums all the signals it
receives by multiplying every input by its associated weight:
[h.sub.j] = [summation]([W.sub.ij] x [X.sub.i]). (1)
This [h.sub.j] is often called the summing node. This output
[h.sub.j] goes to the next step through an activation function
(sometimes it is called a transfer function f):
[O.sub.j] = f([h.sub.j]) = f([summation]([W.sub.ij] x [X.sub.i])).
(2)
Activation function f in most cases is non-linear. It gives the
final output [O.sub.j]. The activation function is chosen according to
specific needs: the most popular function is sigmoid (logistic)
function:
f(x) = 1/[1+[e.sup.-x]], (3)
and hyperbolic tangent function:
f(x) = [[e.sup.x] - [e.sup.-x]]/[[e.sup.x] + [e.sup.-x]]. (4)
These functions are most widely used because of their easy
differentiability but other variants are also possible. Linear
(identity) function can be used as well.
A structure of several perceptrons and their connections is called
a multi-layer perceptron model (MLP). It is typically composed of
several layers of neurons. In the first layer the external information
is received and it is called an input layer. The last layer is the
output layer where the answer of the problem is achieved. These two
layers are separated by one or more layers (called hidden layers). If
all the nodes are connected from lower to higher layers, the ANN is
called a fully connected network. The network is called a feedforward if
no cycles or loops of connections exist. There are other types of ANN,
but our research focuses on the most traditional feedforward MLP which
structure is provided in Figure 3.
[FIGURE 3 OMITTED]
The amount of neurons in every layer may have different specifics
and the number of hidden layers can be also from zero to tens or more.
The MLP structure depends on the nature of every specific data.
3.2. The Back-propagation algorithm
In order to find the most appropriate weights [W.sub.ij] the ANN
requires learning procedure. Supervised learning is the most common
type. The aim of it is to provide neural network with many previous
examples so that it could find the best approximation. The network must
be provided with input values together with corresponding output values.
Through the iterative process the network is adjusting values while the
acceptable approximation is reached. The process is called supervised
learning, and a set of examples--training set. The literature provides
more other types of learning algorithms, but this research focuses on
back-propagation learning algorithm.
The idea of this algorithm is to adjust the weights in a way that
error between desired output (target) [d.sub.i] and actual output
[y.sub.i] ould be reduced:
E = 1/2 [summation][([y.sub.i] - [d.sub.i]).sup.2]. (5)
First of all, partial derivatives of the error according to the
weights are calculated: [??]E/[??][w.sub.ij] for all output neurons and
hidden neurons. The size of weight changes can be determined by learning
rate [alpha] (values between 0 and 1), and the weights are adjusted
according to the formula until the convergence of error function is
reached:
[W.sub.new] = [W.sub.old] - [alpha] x [??]E/[??][W.sub.old]. (6)
The learning rate controls the speed of the convergence: if it is
small the process becomes slower and using a large value of [alpha] the
error function E ay not converge.
3.3. Steps in implementing MLP model
Before implementing the MLP method several characteristics must be
decided:
--Input selection. It is necessary to choose such variables that
have a prediction power to the output. In time series forecasting the
most common inputs are various periods lagged values of the dependable
variable.
--The number of outputs. This selection is directly related to the
problem of what the object of forecasting is.
--Data preprocessing (normalization). In order to make the training
process easier, data can be scaled. If the network uses such activation
functions as sigmoid or hyperbolic tangent, it is necessary to make the
data from the same range. Data preprocessing can include logarithmic,
linear transformation or statistical normalization.
--The NN architecture. A number of hidden layers and a number of
neurons in each layer need to be decided. Previous researches reveal
that in most cases one hidden layer is sufficient. The amount of hidden
neurons is some kind of experiment to find the best results.
--Activation function. This function describes a relationship
between inputs and outputs in the one neuron and the whole network. The
literature reveals that the most common activation function used is
sigmoid function but it is advisable to test several of them.
--The NN training. Using back-propagation training algorithm
learning rates, momentum and number of iterations parameters can be
chosen. The best practice also comes from the experience by testing
several parameters combination of every specific data.
--The training and testing data. Once the learning process is done
by providing the network with training data (examples), the accuracy of
this model should be evaluated by providing it with new data. The
network makes forecasting using new inputs (testing set) and the
accuracy is evaluated by comparing the actual value with the output. All
the current data should be provided in two proportions. The most widely
used proportion is 90% of training data and 10% of testing data.
--Performance measures. The most frequently used tools for
evaluating forecasting accuracy are: Mean Absolute Deviation (MAD), the
Sum of Squared error (SSE), Mean Square error (MSE) and Mean Absolute
Percentage Error (MAPE):
MAD = [[summation][absolute value of [Y.sub.t] - [F.sub.t]]/N, (7)
SSE = [summation][([Y.sub.t] - [F.sub.t]).sup.2], (8)
MSE = [[summation][([Y.sub.t] - [F.sub.t]).sup.2]/N, (9)
MAPE = 1/N [summation][absolute value of [[Y.sub.t] -
[F.sub.t]/[Y.sub.t]]]. (10)
Where [Y.sub.t] is an actual value and [F.sub.t] value is
forecasted. These formulas can be used for evaluating the accuracy of
actual index forecasts. In order to evaluate the accuracy of forecasting
index direction, Sign Prediction correctness (SP) metric is involved:
SP [summation](Correct sign pre dictions)/Total predictions. (11)
The prediction sign is evaluated by taking a difference of two
consequent forecasted future values and comparing it to the actual
market index movement of the targets.
There are arguments in the literature that the prediction of market
index sign in the future is more important than the prediction of actual
value. In our research both cases are included.
3.4. Data and methodology
The research focuses on the predictability of OMX Vilnius Stock
Index. OMXV--a capitalization weighted index that includes all the
shares listed on the Main and Secondary lists. Daily historical data is
taken from the official stock exchange site NASDAX OMX (Vilnius Stock
exchange, http:// www.nasdaqomxbaltic.com). The period investigated is
01.01.2000-30.04.2012. The data is analyzed through two periodicities:
daily data (3605 of data points) and monthly data (150 data points).
Statistical information about OMXV is provided in Table 2.
On every periodicity the data is divided into two sets: historical
data (or training data) and forecasting data (testing data) that is
known and used to evaluate the accuracy of predictions. The
predictability is evaluated for the actual value forecasting and for the
future movement (sign) forecasting. The proportion of historical and
forecasting data is approximately taken as 90% and 10%. So, daily data
consists of 3245 historical and 360 testing data sets. Monthly data: 135
historical data and 15 testing data.
We use such forecasting tools: Simple Moving Average (SMA),
Multiple regression and several structures of MLP (learning algorithm is
back-propagation) in order to make daily and monthly predictions for the
actual value and the movement of the index. As the input or variables
from the previous 4 period lagged values of the index are used (4 lagged
daily values and 4 lagged monthly values). The accuracy is compared
using (7)-(11) formulas.
The prediction using Simple Moving Average (SMA) is done under
assumption, that forecasted value of the fifth period is equal to the
average of last four periods:
SMA = [[[summation].sup.n.sub.t=1][Y.sub.t]]/n = [A.sub.t], (12)
[F.sub.t+1] = [A.sub.t]. (13)
Multiple regression forecast is calculated under assumption that
the fifth period index value is dependent variable from the previous
lagged four periods values.
As in some MLP cases sigmoid and hyperbolic tangent functions are
used, the data is preprocessed using linear transformation to the
interval [a,b]. In the sigmoid case: [0,1], hyperbolic tangent case:
[-1;1].
[X.sub.n] = [[(b-a) x ([x.sub.0] - [X.sub.min])]/[([X.sub.max] -
[X.sub.min])]] + a. (14)
Calculations are done with MS Excel 2007 and MATLAB R2009B, Neural
Network Toolbox.
4. Empirical results
4.1. Daily forecasting
The accuracy of forecasting future daily values was evaluated by
traditional multiple regression, simple moving average and 12 MLP models
containing different structures and transfer functions. The forecasting
ability was evaluated for both the prediction of actual value and also
the sign of future index movement. All MLP structures contained 1 hidden
layer and the number of hidden neurons varied from 1 to 6. Learning rate
a was chosen to be 0,1 because other combinations did not improve the
results. Every MLP model used 3000 epochs (iterations) to train. The
empirical results are provided in Table 3. As it is seen from the
results, the lowest forecasting error for the actual index value was
achieved by using MLP model with one hidden layer and 2 hidden neurons
with the selection of log-sigmoid transfer function. The results are
very similar to the multiple regression results. Nevertheless, this MLP
model outperforms multiple regression method according to all types of
errors calculated, the difference is very slow. For this reason standard
deviation of absolute percentage error is calculated. For the multiple
regression case the standard deviation of absolute percentage error is
0.01030. and for the MLP model (1 hidden layer with 2 hidden neurons and
log sigmoid transfer function) it is 0.01028. So, the second case
provides a little bit more stable forecasts. The graph of the absolute
percentage error for this case is provided in Figure 4 showing the error
for every of 350 predictions of the future index actual value.
As the results reveal, the simple moving average was the least
accurate forecasting technique for future daily values.
The best accuracy for prediction of future index direction achieved
was 53.06 % using MLP with one hidden layer and selection of 1 neuron
(both transfer functions) and also with selection of the hyperbolic
tangent transfer function and 5 neurons in 1 hidden layer. All
forecasting techniques for index direction provide approximately 50% of
correct predictions and that is quite a poor result.
4.2. Monthly forecasting
Monthly forecasting results are found to be really unacceptable.
The main reason could be a cause of the lack of training set (historical
data used to construct the model). Only 135 historical samples were used
to predict 15 future values. The results are provided in Table 4.
The empirical results reveal that all MLP structures were
unsuccessful in predicting future index values. The lowest forecast
error for the actual value was achieved by using multiple regression but
the results are really poor. The highest accuracy for predicting index
direction is approximately 53% by using MLP with 1 hidden layer, 4
hidden neurons and selection of log sigmoid transfer function. It is
quite the same result as for making predictions for daily future index
movements, but other forecasting techniques provided less than 50% of
correct predictions.
[FIGURE 4 OMITTED]
5. Conclusions and further investigation
In comparison of forecasting future OMXV index using daily and
monthly basis, the daily predictions are several times more accurate. In
making daily predictions the lowest forecasting error was achieved by
using MLP with 1 hidden layer and 2 hidden neurons with the selection of
log sigmoid transfer function. The best index direction movement
forecasting was also made by using several MLP model's topologies
-53.06%. In this case MLP model outperformed both traditional multiple
regression and simple moving average methods.
Monthly predictions are found to be really poor. Multiple
regression method outperforms moving average and all MLP structures in
forecasting the actual value but nevertheless the results are
inaccurate. The best direction forecast is done by MLP with 1 hidden
layer, 4 neurons and log sigmoid transfer function -53.33%.
Further investigations may be improved by adding more variables,
using not only consequent lagged historical values. For the reason that
this research uses only one type of neural network--feedforward network,
also, more neural network types should be discussed including different
algorithms of learning.
doi: 10.3846/btp.2012.34
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Audrius Dzikevicius (1), Neringa Stabuzyte (2)
Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223
Vilnius, Lithuania
E-mails: (1) audrius.dzikevicius@vgtu.lt (corresponding author);
(2) n.stabuzyte@gmail.com
Received 15 June 2012; accepted 10 September 2012
Vilniaus Gedimino technikos universitetas, Sauletekio al. 11,
LT-10223 Vilnius, Lietuva
El. pastas: (1) audrius.dzikevicius@vgtu.lt; (2)
n.stabuzyte@gmail.com
Iteikta 2012-06-15; priimta 2012-09-10
Audrius DZIKEVICIUS is an Associate Professor at the Department of
Finance Engineering of Vilnius Gediminas Technical University (VGTU),
defended doctoral dissertation "Trading Portfolio Risk Management
in Banking" (2006) and was awarded the degree of Doctor of Social
Sciences (Economics). In 2007 he started to work as an associate
professor at the Department of Finance Engineering of VGTU. His research
interests cover the following items: portfolio risk management,
forecasting and modeling of financial markets, valuing a business using
quantitative techniques.
Neringa STABUZYTE was awarded Bachelor's degree (Statistics)
in 2010. At the present time she continues her postgraduate studies on
"Investment Management" in Vilnius Gediminas Technical
University. The research interests cover the following items:
investment, forecasting of financial markets.
Table 1. Summary of previous stock market index modeling
issues using ANN forecasting
Author Publishing Index Object
data ticker
(Year)
Lendasse, De Bodt, 2000 BEL 20 Forecasting
Wertz, Verleysen the tendency
of BEL 20
Thenmozhi 2006 BSE Forecasting
SENSEX daily returns
of BSE SENSEX
Desai, Joshi, 2011 S&P CNX Forecasting
Juneja, Dave Nifty 50 the daily
direction of
S&P CNX
Nifty 50
Kara, 2011 ISE Forecasting
Boyacioglu, National the daily
Baykan 100 direction of
ISE National
100 index
Aris, Mohamad 2008 KLCI Forecasting
the direction
of KLCI index
Fernandez-Rodriguez, 2000 IGBM Forecasting
Gonzalez-Martel, future IGBM
Sossvila-Rivero index value
and sign
prediction
Chen, Leung, Daouk 2003 TSE Forecasting
the direction
of TSE index
after 3, 6,
and 12 months
Panahian 2011 TEPIX Forecasting
future trend
of TEPIX
index
Author Data set Method Results
applied
Lendasse, De Bodt, 2600 daily MLP with Average
Wertz, Verleysen index data 1 hidden 65.30%
layer and accurate
5 hidden approximations
neurons of the sign
Thenmozhi 3667 daily MLP with 96.6% accuracy
returns of 1 hidden of testing data
the index layer and
4 hidden
neurons
Desai, Joshi, Daily index MLP with ANN based
Juneja, Dave data 01.09. 1 hidden investment
2009-30.04. layer and strategy
2011 20 hidden outperformed
neurons "buy and hold"
strategy
Kara, 2733 daily MLP with Average
Boyacioglu, index data 1 hidden accuracy
Baykan layer with 75.74%
various
numbers
of hidden
neurons
Aris, Mohamad 5254 daily MLP with ANN model
index data 1 hidden outperforms
layer and moving
1,2 or 4 average
hidden
neurons
Fernandez-Rodriguez, Daily index Feedback Sign
Gonzalez-Martel, data 02.10. network predictions
Sossvila-Rivero 1991-15.10. range 54-58%,
1997 trading
strategy based
on ANN
outperforms
"buy and hold"
strategy in
"bear" and
stable market
episodes
Chen, Leung, Daouk Daily Probabilistic PNN outperforms
index data neural GNN and random
1982-1992 network, walk
Generalized
methods of
moments,
random walk
Panahian Daily MLP with 1 ANN model
index data hidden layer outperformed
2007-2010 and 3 hidden multiple
neurons and regression
multiple model
regression
Table 2. Statistics of OMXV
N Min Max Mean St.dev. Kurt. Skew.
3605 63.18 591.44 249.15 151.22 -1.22 0.39
Table 3. Accuracy results for different daily forecasting
techniques
Transfer function--hyperbolic tangent
1 hidden layer--1 hidden neuron
MAD SSE MSE MAPE Correct
sign (SP)
2.6718 7070.8355 19.6412 0.7451% 53.0556%
1 hidden layer--2 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
2.6618 7032.1291 19.5337 0.7414% 51.1111%
1 hidden layer--3 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
2.7500 7330.4234 20.3623 0.7672% 52.2222%
1 hidden layer--4 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
2.8248 8313.9507 23.0943 0.7883% 50.8333%
1 hidden layer--5 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
2.8966 13886.7201 38.5742 0.8173% 53.0556%
1 hidden layer--6 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
2.9082 10485.9830 29.1277 0.8167% 50.8333%
Multiple regression
MAD SSE MSE MAPE Correct
sign (SP)
2.6635 7054.4705 19.5958 0.7420% 51.1111%
Transfer function--log sigmoid
1 hidden layer--1 hidden neuron
MAD SSE MSE MAPE Correct
sign (SP)
2.6718 7070.8355 19.6412 0.7451 % 53.0556%
1 hidden layer--2 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
2.6607 7028.5305 19.5237 0.7412 % 51.6667%
1 hidden layer--3 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
2.9236 11212.0085 31.9778 0.8225% 51.1111%
1 hidden layer--4 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
4.7524 53881.3056 1496.8925 1.4120% 51.1111%
1 hidden layer--5 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
2.9042 10427.2433 28.9646 0.8140% 52.5000%
1 hidden layer--6 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
2.9393 10701.1447 29.7254 0.8255% 50.2778%
Simple moving average (4)
MAD SSE MSE MAPE Correct
sign (SP)
3.9180 14640.9342 40.67 1.09% 49.17%
Table 4. Accuracy results for different monthly forecasting
techniques
Transfer function--hyperbolic tangent
1 hidden layer--1 hidden neuron
MAD SSE MSE MAPE Correct
sign (SP)
17.2318 9213.7714 614.2514 5.0885% 40.0000%
1 hidden layer--2 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
20.3298 9379.2678 625.2845 8.8388% 46.6667%
1 hidden layer--3 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
19.1361 10211.3395 680.7560 5.5538% 46.6667%
1 hidden layer--4 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
25.5821 14978.8270 998.5885% 7.5084% 33.3333%
1 hidden layer--5 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
31.8335 29343.0403 1956.0403 8.8157% 40.0000%
1 hidden layer--6 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
32.0707 26318.8587 1754.5906 9.1244% 46.6667%
Multiple regression
MAD SSE MSE MAPE Correct
sign (SP)
15.2200 6309.3059 420.6204 4.4712% 46.6667%
Transfer function--log sigmoid
1 hidden layer--1 hidden neuron
MAD SSE MSE MAPE Correct
sign (SP)
17.2318 9213.7714 614.2514 5.0886% 40.0000%
1 hidden layer--2 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
20.3298 9379.26.76 625.2845 5.8388% 46.6667%
1 hidden layer--3 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
26.6590 17170.8006 1144.7200 8.0470% 46.6667%
1 hidden layer--4 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
27.8724 18768.8735 1251.2582 7.9776% 53.3333%
1 hidden layer--5 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
31.0626 21620.1964 1441.3464 8.8771% 33.3333%
1 hidden layer--6 hidden neurons
MAD SSE MSE MAPE Correct
sign (SP)
35.3390 31104.7844 2073.6523 9.9803% 40.0000%
Simple moving average (4)
MAD SSE MSE MAPE Correct
sign (SP)
19.9212 11502.7355 766.8490 5.9461% 46.6667%
