A REVIEW ON OPTIMIZATION OF LEAST SQUARES SUPPORT VECTOR MACHINE FOR TIME SERIES FORECASTING

International Journal of Artificial Intelligence & Applications (IJAIA), Vol. 7, No. 2, March 2016
DOI: 10.5121/ijaia.2016.7203 35
A REVIEW ON OPTIMIZATION OF LEAST
SQUARES SUPPORT VECTOR MACHINE FOR
TIME SERIES FORECASTING
Yuhanis Yusof1
and Zuriani Mustaffa2
1
School of Computing, Universiti Utara Malaysia, Malaysia
2
Faculty of Computer Systems & Software Engineering, Universiti Malaysia Pahang,
Malaysia
ABSTRACT
Support Vector Machine has appeared as an active study in machine learning community and extensively
used in various fields including in prediction, pattern recognition and many more. However, the Least
Squares Support Vector Machine which is a variant of Support Vector Machine offers better solution
strategy. In order to utilize the LSSVM capability in data mining task such as prediction, there is a need to
optimize its hyper parameters. This paper presents a review on techniques used to optimize the parameters
based on two main classes; Evolutionary Computation and Cross Validation.
KEYWORDS
Least Squares Support Vector Machine, Evolutionary Computation, Cross Validation, Swarm Intelligence
1. INTRODUCTION
The Least Squares Support Vector Machines (LSSVM) [1] which is an extension of Support
Vector Machines (SVM) [2] has contributed significant impact in machine learning community.
LSSVM has been recognized as one of the tool in solving various data mining tasks which
include prediction, classification and many others [3].Nonetheless, besides its diversity in
application, the capability of LSSVM is highly dependent on the value of its hyper-parameters,
namely regularization parameter, γ and kernel parameter, σ2
[4].
Time series prediction is of interest in various fields. It is important due to prediction of future
values could be a vital input for current planning and decision making [5]. To realize the time
series prediction, historical data is employed to generate future values, i.e. to make prediction[6].
Due to the importance of time series prediction, numerous approaches have been presented, and
this includes in price prediction[7,8]. In literature, the price prediction modelswere presented in
various horizon or time scales; short term, mediumor long term[9]. Thus, different data frequency
has been employed depending on the suitability viz. daily, weekly, monthly or annually[10,11].
This paper reviews existing work that optimizes the aforementioned LSSVM parameters and the
discussion is divided into two sections; Optimization based on Cross Validation and Optimization
based on Evolutionary Computation.

36
2. OPTIMIZATION OF LSSVM USING CROSS VALIDATION
Researchers in [12] highlight the importance of primary energy consumption prediction by
presenting a prediction model utilizing LSSVM. Several factors that were chosen to be fed to the
prediction models include industrial index and population. Empirical data sets were from 1990 to
2008, which was limited for the case in China. For evaluation purposes, two performance criteria
were used namely Absolute Percentage Error (APE) and Mean Absolute Percentage Error
(MAPE). For parameter tuning, Cross Validation (CV) technique was employed. By using CV,
each training set, TR, is divided into X partitions in equal size (TR1, TR2, …,TRX). For each run,
the LSSVM is trained X times and one of X is set as testing set. Later, the trained model is used in
the testing set. Finally, by using the obtained parameters value, the average accuracy of X trials is
calculate to estimate the final accuracy. The CV-LSSVM technique achieved the predetermined
condition where the accepted error range is within 5%. In addition, as the researchers employed
yearly samples, it shows that LSSVM is capable to deal with small samples data sets as compared
to Artificial Neural Network (ANN) which frequently needs more data for training purposes [13].
In 2008, Liu and Wang presented a prediction technique that is based on LSSVM to predict
foreign exchange rate market [14]. In order to determine the optimized hyper-parameters of
LSSVM, CV approach was applied. For experimental purposes, four pairs of exchange rates time
series data were employed. The samples comprise of time series data from 2003-2007, which
consist of 1196 observations. By using 86:14 data proportion for training and testing respectively,
the efficiency of proposed technique is evaluated using Root Mean Square Error (RMSE), Mean
Absolute Error (MAE), and MAPE. However, even though the CV-LSSVM is capable to provide
promising results, the absence of comparison with other comparable prediction techniques made
the experiment incomplete.
In related work, an Empirical Mode Decomposition (EMD) was integrated with LSSVM in
dealing with exchange rate prediction [15].The EMD was utilized to decompose the foreign
exchange rates into several International Monetary Fund (IMF) components and one residual
component. The LSSVM then performs the prediction of the IMFs and residual value
independently. Comparing the result against those produced by EMD-ARIMA, single LSSVM
and single ARIMA, the EMD-LSSVM outperforms other experimented prediction techniques
based on several criteria, which includes MAE, MAPE, Correct Uptrend (CP) and Correct
Downtrend (CD). For parameter tuning, the CV approach was employed.
Progressing further, in [16], short term prediction of meteorological time series data has been
presented using CV-LSSVM. Prior to training task, three pre-processing techniques were
performed namely data filtering which is to remove redundant components from the data sets,
removing outliers from the data sets and finally data normalization. The evaluation of the
presented technique is based on several statistical metrics which includes MAPE and RMSE.
Comparison againts various ANN based predictor showed that the performance of CV-LSSVM is
at par with Probabilistic Neural Network (PNN).
Meanwhile, [17] employed LSSVM to predict short-term power load in Inner Mongolia. The case
study used one year load series from that region, and later divided them into 90:10 for training
and testing respectively. For parameter tuning, CV technique was utilized. The results derived
from this approach is regarded as efficient for the problem under study. However, no comparison
with other technique was conducted.

37
A study in [18] integrated the Empirical Mode Decomposition (EMD) with LSSVM in predicting
future wind speed and output power. The purpose of applying EMD is similar as described in [15]
which is to decompose the data sets utilized into a series of different scales of IMF. By employing
daily average wind speed from March to May 2006, the result obtained by the proposed model is
compared against several techniques which include single LSSVM. The hyper-parameters of
LSSVM are tuned based on CV approach. The comparison was made by taking into account the
value produced in terms of MAPE and RMSE by each identified prediction model. Final results
indicated that the EMD-LSSVM outperformed the other experimented techniques by producing
lower MAPE and RMSE.
In related work [19], demonstrated a CV-LSSVM for air temperature prediction. This study
utilized monthly average temperature from 1951 to 2003. From that sample, only year 2003 data
is allocated for testing set. The findings of the experiment conducted were evaluated using
RMSE, Relative Error (RE) and prediction accuracy and involved another two prediction
techniques, namely single LSSVM and Radial Basis Function Neural Network (RBFNN). Final
results indicated that RMSE produced by EMD-LSSVM was 0.7202, which was the smallest
among the experimented prediction techniques.
Meanwhile, in [20], the CV-LSSVM was presented in prediction of flashover voltage of
insulators. In the study, several comparisons were performed between different types of kernels,
which include RBF kernel and Polynomial kernel. The results indicated that Radial Basis
Function (RBF) kernel yield lower RMSE. Meanwhile, the comparison against ANN is in favor
to CV-LSSVM.
Previously, the application of LSSVM in a Bayesian evidence framework was examined by [21].
In the presented study, two different cases were investigated, namely prediction for U.S. short
term interest rate and daily index closing price return. Prior to training, normalization technique
was performed to all data sets. Upon completing the experiments, the development of proposed
model is proven to be successful as an improvement is made in MSE, MAE and Negative Log
Likehood (NLL) as compared to traditional methods, which includes autoregressive model.In
[22], a hybridization of Group Method Data Handling (GMDH) with LSSVM was presented to
predict electrical energy consumption. GMDH is the ANN based model which finds the optimal
model based on inductive sorting out approach. For LSSVM parameter selection, CV technique
was utilized. When evaluated based on MAPE, the presented technique offers lower MAPE as
compared to the comparison algorithms, which includes single GMDH and LSSVM.
As discussed above, even the utilized CV has provided promising results for the context of study,
however, without comparing CV with other techniques in the study conducted [14,23,24],
proving the efficiency of the presented method is therefore difficult. In addition, CV is revealed
to be time consuming [25] and tends to produce high error rates[26]. Due to that matter, an
application of Evolutionary Computation (EC) algorithm, which includes Evolutionary Algorithm
(EA) and Swarm Intelligence (SI) techniques, become research interest among academic
community in optimizing LSSVM hyper-parameters which is presented in the next section. A
summary of this section is tabulated in Table 1.

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2.1 Optimization of LSSVM using Evolutionary Computation
The Evolutionary Computation (EC) comprises of Evolutionary Algorithm (EA) and SI
techniques[27]. The EA is inspired by the natural evolution phenomenon while the imitation of
insects or animals is featured in SI technique[28]. Figure 1 illustrates the EC classes.
Figure 1: Classes of Evolutionary Computation
Evolutionary Computation (EC)
Evolutionary Algorithm (EA) Swarm Intelligence (SI)
Genetic Algorithm (GA),
Evolutionary Programming (EP)
Differential Evolution (DE)
Artificial Bee Colony (ABC)
Fish School (FS)
Particle Swarm Optimization (PSO)
Artificial Fish Swarm Algorithm (AFSA)
Ant Colony Optimization (ACO)
Fruit Fly Optimization (FFO)

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Categorized under EA group, Genetic Algorithm (GA) is the most prominent and widely used
technique as compared with other techniques within the same domain[28,29]. It was introduced
by John Holland and later was popularized by David Goldberg[30]. It models genetic selection
and imitate the natural evolution phenomenon to solve numerical optimization problems. The GA
consists of five main components viz. random number generator, fitness evaluation, reproduction,
crossover and mutation operations. The last three components are categorized as genetic operator.
In GA, the possible solutions are represented by chromosomes.
The hybrid technique of GA-LSSVM has been demonstrated to predict transmission loss in
deregulated power system [31]. In the study, the application of GA-LSSVM is used to learn
which generator is responsible for the losses by determining the optimal value for LSSVM hyper-
parameters. The data was broken down into three sets; 57% of samples are used for training, 14%
are set for validation and the rest 29% is reserved for testing. Simulations were performed and the
results obtained showed satisfactory result. Similar approach also has been tested by the authors
to predict reactive power which is compared against result produced by ANN[32]. The results
obtained suggested that both methods produced similar outputs; nonetheless GA-LSSVM
possesses an advantage in terms of computational time.
Researchers in [26] presented the GA-LSSVM for stock market prediction. In this study, the first
GA was utilized for feature selection while the second GA is for parameter tuning. The proposed
model is compared against the results produced by several techniques which include conventional
BPNN, ARIMA, SVM and variants LSSVM. From the experiment conducted, the GA-LSSVM is
proven to be able to present some selection models that are easier to clarify by applying small
number of features. However, for certain cases, the presented model was unable to outperform the
other comparison algorithms.
In the maritime field, ship motion prediction by utilizing Adaptive GA (AGA) with LSSVM has
been presented by [33]. In the study, by referring to the fitness values of individuals, the
probability of crossover and mutation are adaptively adjusted. To prove the efficiency of the
AGA-LSSVM, comparison with the BPNN and LSSVM were made. From the results obtained, it
clearly shows that AGA-LSSVM excels by yielding lower MAE as compared to the
experimented techniques. However, it would be appropriate if the comparison against GA-
LSSVM is considered as to see the difference between the newly GA with the original one.
Meanwhile, in [34], they examined the capability of GA-LSSVM in power spot prices prediction.
Prior to training, the data was pre-processed by applying Min Max Normalization. The
performance of the model was measured based on MAE and MAPE. After completing
experiment process, the average of MAE and MAPE produced by GA-LSSVM yielded lower
value as compared to value produced by ANN. In [3], the comparison between the GA-LSSVM,
standard SVM and LSSVM to predict dissolved gas concentration was performed. The findings
of the study indicated that the GA-LSSVM provided better results than the other two approaches.
In [29], the GA-LSSVM was proposed for conductivity and tensile strength prediction for carbon
fiber composites. In this study, the predictability of proposed model was guide based on two
indices, namely Learning Mean Square Error (LMSE) and Generalization Mean Square Error
(GMSE). To further assess the superiority of GA-LSSVM, comparisons against Principal
Component Analysis-GABPNN (PCS-GABPNN) was made. Based on the results obtained, the
GA-LSSVM is capable to produce better generalization performance without having to use any
data pre-treatment as required in the identified competitor. In [35], the application of GA-LSSVM

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was proposed to predict the condition of hydro turbine generating units (HGUs). Good prediction
model for the context study is beneficial in providing backups to predictive maintenance system
in mechanical equipment fault diagnosis. As compared to CV-LSSVM and BPNN, GA-LSSVM
performed slightly better than CV-LSSVM while BPNN performed dismally due to over fitting
problem.
In 2005, LSSVM optimized by GA has been used to deal with prediction of chaotic time series
data[36]. Tested on benchmark time series data, both training and testing set were divided
equally. The comparison against CV-LSSVM suggested that GA-LSSVM is capable to produced
lower error rate, relative to the RMSE. Further development of GA-LSSVM has been presented
by [37] (2013) in solving network flow prediction. In the study, the modification is put forward to
GA, where Self-adaptive GA is introduced. The modification involved the standard operation of
crossover and mutation. Using hourly network access flow, the data sets were first normalized
using Min Max Normalization. The conducted experiment indicated that Self-adaptive GA-
LSSVM is superior over the other identified approach. Nevertheless, the comparison with the
standard GA-LSSVM was absent in the study.
Introduced by Kennedy and Eberhart in 1995 [38], PSO is considered as a well-known SI
technique which was inspired by social behavior of birds flocking or fish schooling. The
application of PSO can be seen broadly utilized in solving various optimization issues [28]. In
this technique, it is initialized with a group of random particles which represent possible
solutions, and then searches for global optimal by updating generations. As compared to GA,
there is no evolution operator such as crossover and mutation in PSO, which make it simpler in
implementation. However, the choice in parameter selection such as acceleration constant and
weight still need to be considered[39].
The efficiency of PSO-LSSVM has been demonstrated by [4] in critical heat flux prediction
(CHF). Applied in nuclear science field, accurate prediction of CHF is vital not only for safety
but also in designing nuclear reactors. For that matter, several relevant inputs are considered for
training and later, the samples of data sets are divided into training and testing. The performance
of the proposed method is guided based on three indices, viz. the R2
, Mean Relative Error (MRE)
and RMSE. Compared against ANFIS, the result obtained is in favor to the PSO-LSSVM.
Prediction for water quality utilizing hybrid of PSO with LSSVM was studied by [40]. In this
study, 3256 data points of water figures from Lixiu River water database were employed as
empirical data. The results obtained by the presented technique are compared against the results
produced by using CV-LSSVM, BPNN, ARIMA and GML. The incapability of CV in this study
supports the results obtained in [41] and confirms the fact stated in previous the studies[25].
However, while the PSO-LSSVM outperforms CV-LSSVM, BPNN and ARIMA, it loses to
GML as prediction accuracy presented by the GML technique is higher.
Unlike the previous study of PSO-LSSVM [4, 40], in [42], the dynamic inertia weight PSO
(WPSO) is employed to optimize LSSVM hyper-parameters. The optimal value of parameters
obtained is then used by LSSVM for prediction task. Upon completing simulation task, the RE,
which is selected as performance evaluation metric of the study indicated that the PSO-LSSVM
outperformed both experimented prediction techniques, viz. BPNN and LSSVM. Since the PSO
used in this study has been modified, a comparison with the standard PSO needs to be performed.

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While the work in [3] demonstrated GA-LSSVM for gas prediction, [43] examined the capability
of PSO-LSSVM for the similar task. In this study, dissolved gas data from several electric power
companies in China were employed as empirical data. As to improve the generalization of
LSSVM, the raw data were first normalized before training task. For comparison purposes, four
single prediction models were chosen, including three ANN based prediction models and SVR.
Upon completing the simulation task, the PSO-LSSVM was concluded to be superior than the
identified techniques relative to MAPE and R2
. However, despite the good performance of the
model, the author admitted that the produced results might be unreliable due to the insufficient
size of actual data.
Experimented on chaotic time series data, [44] presented the use of PSO-LSSVM. For empirical
purposes, two benchmark data sets were employed namely Mackey–Glass time series data and
Lorenz time series data. As compared to fuzzy prediction based on SVM, the PSO-LSSVM
showed superior performance for both cases.
As [33] presented AGA-LSSVM for ship motion prediction, [45] demonstrated the hybridization
of PSO with LSSVM for the similar context of study. Taking the similar approach as in [33], the
optimization technique, which is PSO, is modified as to enhance the generalization performance
of LSSVM in prediction. The modification of PSO is done by integrating it with Simulated
Annealing (SA), termed ad Hybrid Particle Swarm Optimization (HPSO). Tested on a real data,
the performance of proposed method which was guided by RMSE produced an encouraging
result. However, the evaluation does not involve any other prediction techniques.
Carbon price prediction using hybrid ARIMA and LSSVM has been demonstrated by [46]. For
optimization of hyper-parameters of interest, PSO algorithm was employed. However, the PSO
was only used to tune the LSSVM hyper-parameters, while the ARIMA parameters are derived
empirically. The research data employed was daily carbon price which covers the period from
2005 to 2010. The performance of proposed model was compared against several single and
hybrid techniques, which includes ARIMA-ANN and also LSSVM and the evaluation is based on
RMSE. Findings of the study suggested that the presented model demonstrate better predictive
capability by producing lower error.
Besides GA and PSO, another EC algorithm which includes Artificial Fish Swarm Algorithm
(AFSA) [47,48], Ant Colony Optimization (ACO) [49], Differential Evolution (DE) [50] and
Gravitational Search Algorithm (GSA)[51] were also presented in literature as one of the
optimization tool for LSSVM.
AFSA which was first proposed by [48] and is inspired by the intelligent swarming behavior of
fish. In AFSA, after initialization of population, the algorithm will go through the selection of the
best behavior to be executed, which involved between swarm and follow behavior. The behavior
which produced the minimum result will be chosen for the next stage. Even though AFSA
possesses an advantage in avoiding local minimum, it suffers from imbalance between
exploitation and exploration process, which would affect the optimization process[52]. In
literature, [53] examined the hybridization of LSSVM with AFSA. Using electrical load data set
as an empirical data, the produced results indicated that AFSA-LSSVM is capable to yield
smaller average RE as compared to PSO-LSSVM.
On the other hand, ACO [49] imitates the foraging behavior of ants, which acts as agents in
searching the shortest path between the colony and the food source. This algorithm is reported to

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guarantee the convergence; however the time to converge is inconsistent [54]. Besides, as this
algorithm is developed based on routing problems, the theoretical is reported to be quite difficult
to be applied in other applications. In optimizing LSSVM hyper-parameters, [55] demonstrated
the ACO which has been applied for share price prediction. As to eliminate the fluctuant
component in raw data, Wavelet Transform (WT) was employed which served as preprocessor to
LSSVM. The proposed model is measured based on MSE and the comparison was made against
single SVM and ACO-SVM without WT. The results obtained indicated that ACO-LSSVM
offered promising results by producing lower MSE.
Meanwhile, DE [50] is inspired by the mechanism of natural selection which considered as
extension of GA. The difference between DE and GA is, in DE, all possible solutions have an
equal chance in evaluation task, while in GA, the chance of updating the solution is depends on
fitness value. The application of DE involved the predefined of several control parameters which
affect the optimization process of DE and is revealed to have tendency of falling into local
minimum [56]. This situation leads to the modification of DE (termed as CDEK) which has been
demonstrated for optimizing the LSSVM hyper-parameters [56]. In the study, CDEK-LSSVM
was tested on estimation of thermal process. Empirical experimentation suggested that the
proposed model demonstrate better predictive power than the standard DE-LSSVM.
Previously, in 2004, the hybridization of DE-LSSVM has been presented which experimented on
three benchmark functions [57]. Empirical results showed that the DE-LSSVM offered good
convergence performance, however, in terms of prediction accuracy, the results obtained is
unsatisfying.
Progressing further, [58] investigated the usefulness of online LSSVM in prediction of turbine
heat rate. In the study, the GSA[59] was utilized to optimize the hyper-parameters of LSSVM.
GSA is a population-based algorithm which is motivated from Newtonian gravity. Apart from
population size and number of iteration, several control parameters need to be predefined by the
user, which includes gravitational constant and inertia mass. Upon completing the experiment, the
experimental results suggested that GSA-LSSVM offered better generalization capability as
compared to the other identified prediction techniques.
In related work, [41] presented a hybrid of LSSVM with Bacterial Colony Chemotaxis (BCC) for
short term load prediction. Categorized as bionic algorithm, BCC is colony-intelligent
optimization which is based on bacterium’s reaction to chemo attractants to find the optimum.
Upon completing the experiment, lower MAPE was produced by BCC-LSSVM as compared to
several identified techniques, which includes BPNN and CV-LSSVM. In terms of training time,
the demerits of CV that was highlighted in [25] was proven as ineffective in the study where it
took much longer time to complete the training task as compared to time recorded using BCC-
LSSVM.
Meanwhile, in [60], the LSSVM is hybridized with Fruit Fly Optimization (FFO)[61] for electric
load prediction. The FFO is inspired from the food searching behavior. In the study, the FFO is
employed as an optimizer to LSSVM and the FFO-LSSVM is compared against several identified
techniques which include single LSSVM and regression technique. Final results suggested that
the FFO-LSSVM is capable to produce lower error rate relative to MAPE, MSE and Average
Absolute Error (AAE).

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As discussed above, both GA and PSO are widely utilized in optimizing the LSSVM hyper
parameters. However, despite its extensive use of GA in literature, several deficiency of this
technique has been reported. There is high possibility of GA to face with stagnation in the local
minimum, premature convergence, low convergence speed and difficulty in parameters
selection[53].As for PSO, it is also revealed that the PSO tends to fall into local minimum[62].
Besides GA and PSO, the application of other EC algorithms which includes AFSA, ACO, DE
and GSA are also presented in literature. However, the applications of these techniques are less
obtrusive except for Artificial Bee Colony (ABC)[63]. A study on the prediction of non-
renewable commodity such as crude oil and gold has been reported in 2011using the ABC-
LSSVM [64]. Results showed that the ABC-LSSVM possess higher accuracy than existing
prediction models that includes Backpropogation Neural Network (BPNN) and Differential
Evolution – Least Squares Support Vector Machines (DE-LSSVM).The work was later extended
by the researchers to improve the search mechanism of ABC and this was presented as eABC-
LSSVM [65,66]. The extension includes two sub algorithms termed as lvABC-LSSVM [67] and
cmABC-LSSVM. The lvABC algorithm was introduced to overcome the local optima problem by
enriching the searching behaviour using Levy mutation. On the other hand, the cmABC algorithm
that incorporates conventional mutation addresses the over-fitting or under-fitting problem faced
by ABC. Experiments on energy crude oil daily prices showed that eABC-LSSVM produced
promising MAPE, which is 6.3310%. With that, the PA obtained is 93.6690%. This is followed
by ABC-LSSVM with 7.1769% of MAPE. The summary of literature based on EC algorithms is
presented in Table 2.
Table 2: Time Series Prediction Technique using LSSVM Optimized by EC Algorithm
Authors
Research
Area
Independent
Variable
Method
Data
Frequency
Evaluation
Metric
Mustafa, et al.
(2011)
Transmission
loss
prediction
Load curve
and load
demand
GA-
LSSVM
Hourly MSE
Mustafa, et al.
(2012)
Reactive
power
prediction
Load curve
and load
demand
GA-
LSSVM
Hourly MSE
Yu, et al. (2009) Stock market
prediction
S&P 500
index, Dow
Jones
Industrial
Average
(DJIA) index
and New
York Stock
Exchange
(NYSE)
index
GA-
LSSVM
Monthly Hit ratio
Fu, et al. (2010) Ship motion
prediction
Heading
angle
AGA-
LSSVM
Seconds MAE
Sun & Zhang
(2008)
Power spot
prices
prediction
Power spot
prices
GA-
LSSVM
Daily MAE and
MAPE
Xie, et al. (2009) Dissolved
gas
concentration
Dissolved
gas analysis
data
GA-
LSSVM
- Absolute
error

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prediction
Yang, et al. (2010) Conductivity
and tensile
strength
prediction
modeling for
carbon fiber
composites
Temperature,
average
fiber, fiber
content,
conductivity,
and several
others
GA-
LSSVM
- LMSE and
GMSE
Wang, et al.
(2005)
Chaotic time
series data
prediction
Henon map
time series
data
GA-
LSSVM
- RMSE
Liao &Balzen
(2013)
Network
flow
prediction
Network
access flow
Self-
Adaptive
GA-
LSSVM
Hourly RMSE and
MAPE
Jiang &
Zhang(2013)
Critical heat
flux
prediction
Mass flow
rate,
pressure, the
heated length
to tube
diameter
ratio and
equilibrium
quality
PSO-
LSSVM
- Relative Error
Xiang & Jiang
(2009)
Water quality
prediction
Water
figures of
Lixiu River
PSO-
LSSVM
- MAPE
Shen, et al. (2009) Stock return
prediction
Stock return
of 5 days
backward
and closing
prices of 3
days
forward, and
several
others
WPOS-
LSSVM
Daily Error
Liao, et al.(2011) Dissolved
gases
prediction
Dissolved
gas
data110kV
and 220kV
PSO-
LSSVM
- MAPE and R2
Liu & Yao (2009) Chaotic time
series
prediction
Mackey-
Glass time
series data
and Lorenz
time series
data
PSO-
LSSVM
- RMSE
Zhou & Shi
(2010)
Ship motion
prediction
A series of
roll and pitch
time
series data of
a real ship
during her
seakeeping
trial
HPSO-
LSSVM
- RMSE

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Zhu & Wei (2013) Carbon price
prediction
Carbon price
data sets
PSO-
ARIMA-
LSSVM
Daily RMSE, Dstat
Chen, et al. (2008) Electricity
load
prediction
Electricity
load data
AFSA-
LSSVM
Hourly RE
Fang & Bai
(2009)
Share price
prediction
Share price
data
ACO-
LSSVM
Daily MSE
Dos Santos, et al.
(2012)
Thermal
process
prediction
Temperature,
relative
humidity
CDEK-
LSSVM
Second R2
Feoktistov&Janaqi
(2004)
Prediction Benchmark
functions
DE-
LSSVM
- -
Zhang, et al.
(2013)
Turbine heat
rate
prediction
Heat rate GSA-
LSSVM
Hourly R2
, APE and
MAPE
Li (2009) Short-term
load
prediction
Load value
of last three
hours, the
load value of
the very hour
in last three
days, and
several
others
BCC-
LSSVM
Hourly MAPE and
computational
time
Li, et al. (2012)
Mustaffa, et al.
(2013)
Electric load
prediction
Crude oil
price
prediction
Electricity
consumption
Daily spot
price and the
variance
FFO-
LSSVM
eABC-
LSSVM
Annually
Daily
MAPE, MSE
and AAE
MAPE,MSE
3. CONCLUSION
As the success of LSSVM relies on the tuning of its parameters, optimizing its hyper-parameters,
which are the regularization parameter, γ and kernel parameter, σ2
is important. It is learned from
existing work that optimizing the hyper-parameter is best implemented using the evolutionary
algorithms. And, this includes the employment of swarm intelligence algorithms such as the
Particle Swarm Optimization, Fish School Algorithm, Fruit Fly Optimization and Artificial Bee
Colony.
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