Interval estimation of construction cost at completion using least squares support vector machine.
Cheng, Min-Yuan ; Hoang, Nhat-Duc
Introduction
In construction industry, project success has always foundered on
the high uncertainty in operational environment. Thus, it is not
surprising that construction projects frequently suffer cost overrun
(Nassar et al. 2005). In order to operate profitably, construction
companies must frequently evaluate project cost at completion to detect
deviations and to carry out appropriate responses. However, construction
firms typically focus on budget planning during the initial project
stage, which practically ignores the impact of engineering cost changes
and information updates during construction (Cheng et al. 2010). This
fact prevents effective project cost control and detection of potential
problems. Therefore, cost estimation is a crucial task and it needs to
be carried out at various stages of a project (Liu, Zhu 2007). Moreover,
the accuracy of construction cost estimation is a critical factor in the
success of the project (Kim et al. 2004). Poor cost estimation may
result in profit loss and occasionally leads to project failure.
Due to its importance, various predictive methods have been
proposed for cost estimation. Approaches that are applicable to cost
estimation range from statistics based multivariable regression analysis
to machine-learning techniques such as Classification and Regression
Trees (CART), M5 model tree (M5-MT), Artificial Neu ral Network (ANN),
Support Vector Machines (SVM), and Least Squares Support Vector Machine
(LS-SVM).
Multivariable regression analysis is a very powerful statistical
tool that can be used as both an analytical and a predictive technique
in assessing the contribution of potential new items to the overall
estimation, although it is limited in modeling non-linear relationships
(Kim et al. 2004). In addition, when the number of input variables
becomes considerably large, the prediction performance of this method
often deteriorates significantly.
CART (Breiman et al. 1984) is a classification method which
utilizes historical data to construct decision trees. A CART model that
forecasts the value of continuous variables from a set of input
variables is known as a regression-type model (Razi, Athappilly 2005).
One major advantage of the decision tree based model is its ability to
handle small-size data set. Moreover, CART can mitigate the negative
effect of outliers because the model is capable of isolating the
outliers in a separate node. However, one disadvantage of CART is that
it may produce unstable decision trees (Timofeev 2004). The reason is
that insignificant modification of learning sample could result in
radical changes in the decision tree. In addition, previous works (Razi,
Athappilly 2005) have indicated that prediction performance of CART can
be inferior to ANN.
A model tree (MT) is similar to a decision tree, but includes the
multivariate linear regression functions at the leaves and is able to
predict continuous numeric value attributes (Shrestha, Solomatine 2006;
Witten, Frank 2000; Kaluzny et al. 2011). The algorithm separates the
parameter space into subspaces and constructs a local linear regression
model in each of them. Thus, MT is, to some degree, similar to a
piecewise linear function. In the M5-MT, the nodes of the tree are
selected over the attribute that maximizes the expected error reduction
as a function of the standard deviation of output parameter (Bonakdar,
Etemad-Shahidi 2011). MT is proved to have the capability of learning in
an efficient manner and it can tackle regression tasks with high
dimensionality. Compared to other machine learning techniques, MT
training process is relatively fast and the results are interpretable
(Shrestha, Solomatine 2006).
ANN is a viable alternative for forecasting construction costs and
in practice, it has been used to construct various cost prediction
models (Hegazy, Ayed 1998; Zhu et al. 2010; Sonmez 2011). This method
eliminates the need to find a mapping relationship that mathematically
describes the construction cost as a function of input variables. When
the influence factors and the structure of ANN are all specified, the
task boils down to collecting a reasonable number of data to train the
ANN. However, the training process of ANN based models is often
time-consuming; and ANN also suffers from difficulties in selecting a
large number of controlling parameters such as hidden layer size,
learning rate, and momentum term (Bao et al. 2005).
Furthermore, one major disadvantage of ANN is that its training
process is achieved through a gradient descent algorithm on the error
space, which can be very complex and may contain many local minima
(Kiranyaz et al. 2009). Thus, the training of ANN is likely to be
trapped into a local minimum and this definitely hinders the forecasting
capability. To overcome such issue, evolutionary algorithms, such as
Genetic Algorithm (GA) and Particle Swarm Optimization (PSO), can be
used to train the ANN model (Nasseri et al. 2008; Zhang et al. 2007). It
is because these advanced optimization techniques can significantly
reduce the chance of getting trapped in local minima. Hence, the
training process possibly settles in an optimum solution; nevertheless,
this cannot be guaranteed (Kiranyaz et al. 2009).
In construction area, SVM has been utilized in cost estimation
(Cheng et al. 2010; Kong et al. 2008; An et al. 2007; Hongwei 2009). The
principles of SVM are based on the structural risk minimization and
statistical learning theory. The SVM based models also involve
identification of influence factors, collection of data sample, and
training/testing process. After the mapping function has been
established, the model is capable of predicting the future value of
project cost. The advantages of SVM are widely known including strong
inference capacity, excellent generalization, and accurate prediction
ability (Lam et al. 2009; Huang et al. 2004). Nevertheless, SVM training
process entails solving a quadratic programming problem subjected to
inequality constraint. This means that SVM's training process for
large data sets requires expensive computational cost (Guo, Bai 2009).
To overcome the drawback of SVM, LS-SVM has been proposed recently
by Suykens et al. (2002), Gestel et al. (2004), and Brabanter et al.
(2010). LS-SVM is a modified version of SVMs to alleviate the burden of
computational cost. In LS-SVM's training process, a least squares
cost function is proposed to obtain a linear set of equations in the
dual space. Consequently, to derive the solution, it is required to
solve a set of linear equations, instead of the quadratic programming as
in standard SVM. And, this linear system can be efficiently solved by
iterative methods such as conjugate gradient (Wang, Hu 2005). Studies
have been carried out to demonstrate the excellent generalization,
prediction accuracy, and fast computation of LS-SVM (Yu et al. 2009;
Samui, Kothari 2011; Chen et al. 2005). Despite of its superiority,
application of LS-SVM in construction cost estimation is still very
limited.
Additionally, when applying LS-SVM, it is recognizable that the
tuning parameters, namely regularization and kernel function parameters,
play an important role in establishing the predictive model (Yu et al.
2009; Suykens et al. 2002). These parameters control the model's
complexity, and they are needed to be determined properly via
cross-validation. In doing so, the main objective is to obtain an
optimal model that can explore the underlying input-output mapping
function and is capable of producing the best predictive performance on
new data (Bishop 2006). In this study, DE, a population-based stochastic
search engine proposed by Storn and Price (Price et al. 2005), is
employed in the cross-validation process to achieve such objective.
In practice, cost estimation in construction industry is often
stated in the form of a point forecast (Trost, Oberlender 2003;
Iranmanesh et al. 2007; Cheng et al. 2010; Zhu et al. 2010). However,
decision makers require not only accurate forecasting of certain
variables but also the uncertainty associated with the forecasts. Point
estimation does not take into account the various sources of uncertainty
that stem from the model itself, input variables, and tuning parameters.
Thus, incorporating prediction uncertainty into deterministic forecasts
can help improve the reliability and the credibility of the model
outputs (Shrestha, Solomatine 2006).
Various approaches (Wonnacott, T. H., Wonna cott, R. J. 1996;
Heskes 1997; Mencar et al. 2005; Bra banter et al. 2011) have been
introduced to achieve interval estimation. However, existing methods
also have many limitations such as requiring the prior distributions of
the uncertain input parameters or data and involving certain assumptions
about the data and error distribution. The accuracy and the credibility
of those approaches rely significantly on the precision of prior
information and their assumptions. Another class of methods for deriving
prediction interval (PI) is relied on re-sampling or bootstrap. Although
bootstrap based methods (Sonmez 2011; Stine 1985; Lam, Veall 2002) can
yield accuracy prediction result, this method is notably characterized
by high computational cost.
Recently, a new framework for estimation of PI which is based on
machine learning technique has been established by Shrestha and
Solomatine (2006). The proposed method does not require any assumption
and prior knowledge of input data or model error distribution. Moreover,
it also does not demand intensive computational cost as in bootstrap
based methods. In their research work (Shrestha, Solomatine 2006), the
superiority of machine-learning based interval estimation (MLIE) over
traditional methods is exhibited.
Therefore, this study aims to propose an artificial intelligence
model, namely as EAC-LSPIM, that hybridizes various advanced techniques
including LS-SVM, MLIE, and DE to help project manager in construction
cost prediction. The newly built model incorporates the strengths and
mitigates the weaknesses of each individual technique. The research goal
is to build a model that can operate automatically without human
intervention and can deliver accurate and reliable forecasting results.
Equipped with this tool, it is expected that the tasks of cost control
and cost planning in construction industry can be carried out
effectively.
The remaining part of this paper is organized as follows. The
second section of this paper reviews related research works on
estimating of construction cost at completion, LS-SVM, techniques for
achieving prediction intervals, and DE. In the third section, the
DE-based cross-validation process is introduced. The fourth section
describes the framework of the newly proposed model in detail.
Simulation and result comparison of the model are demonstrated in the
fifth section.
1. Review of pertinent literature
1.1. Estimate of project cost at completion
In construction management, estimating cost of work at completion
is vitally important for project success. To achieve this, project
managers often rely on Earned Value Management (EVM) methodology. EVM is
widely known as a management technique that relates resource planning
and schedule usage and technical performance requirement (Abba 1997).
EVM comprises of three essential components that support project
control: Plan Value (PV) or Budgeted Cost of Work Schedule (BCWS),
Earned Value (EV) or Budgeted Cost of Work Performed (BCWP), and Actual
Cost (AC) or Actual Cost of Work Performed (ACWP). In the construction
industry, project managers emphasize the application of EVM as it
provides a tool for tracking project status and for measuring project
performance.
EVM is a systematic approach to forecast Estimate at Completion
(EAC). The role of EAC is accentuated due to the fact that managers or
planners can appraise the total project cost based on the estimated
value of EAC. Iranmanesh et al. (2007) point out that the correct and
the on time EAC is essential for preventive response during the project
execution. If EAC indicates an overrun in cost, the project managers can
use proper strategies to adjust construction cost. In the situation of
cost overrun, project managers arguably carry out a value engineering
program for cost reduction in which scope or quality in some sections of
project is decreased. Another option is to require additional budget to
offset overrun cost.
At every completion period, managers can extract data from progress
report, calculate project Earned Value (EV) and predict EAC. The EAC can
be computed by formula using cost management data provided by the
contractor in the Cost Performance Report or the Cost/Schedule Status
Report. The reliability of these reports depends on the degree to which
the contractor adheres to internal controls involving measuring
performance on a contract (Christensen 1993).
According to previous works by Christensen (Christensen 1993;
Christensen et al. 1995) and Chen (2008), determining an appropriate
estimation of EAC is an arduous task. To obtain EAC, managers need to
collect voluminous cost management data provided by the contractor in
progress report, usually monthly report. For the contractors, in order
to form the periodic report to the owner, their site-engineers must
gather sufficient data summarized in the daily man-hours summary, daily
material summary and daily equipment summary. Finally, one can use
various formulas to compute EAC based on the combination of several data
elements presented in the report: BCWS, BCWP, and ACWP.
To forecast the EAC, numerous index-based formulas have been
utilized. Those formulas are divided into three categories:
non-performance method, performance method, composite method
(Christensen et al. 1995; Chen 2008; Cheng et al. 2010). Based on a
survey carried out by Christensen et al. (1995), the accuracy of
index-based formulas depends significantly on the type of system, and
the stage and phase of project. This interprets why performance of a
particular formula might be quiet acceptable in a certain case, while it
could be much worse in other cases (Cheng et al. 2010). Project planners
must employ their own judgments to ascertain a most appropriate EAC or a
range of reasonable EACs. Currently, there is no official guidance on
how to choose an amenable EAC calculation according to a specific
setting.
Besides index-based formulas, other EAC prediction methods are
based on regression analysis (Iranmanesh et al. 2007; Christensen et al.
1995). The regression-based formulas are typically derived using linear
or nonlinear univariate regression analysis (Christensen 1993). However,
methods based on traditional regression analysis also have disadvantages
such as their limitations in describing nonlinear relationships (An et
al. 2007). In addition, the number of influence factors for construction
cost estimate can be appreciable (Trost, Oberlender 2003; Cheng et al.
2010) and the underlying regression function is possibly very intricate.
That fact explains why EAC estimation based on traditional regression
analysis is not widely used in the industry (Christensen 1993).
Needless to say, EAC prediction problem is complicated since it
involves voluminous construction data, considerable number of influence
factors, and complicated regression function. Thus, it is reasonable for
planners or managers to resort to more advanced methods, specifically
Artificial Intelligence (AI) methods, such as Artificial Neural Network
(ANN) and Least Squares Support Vector Machine (LS-SVM).
1.2. Least squares support vector machine for regression analysis
This section is dedicated to describing the LS-SVM's
mathematical formulation. Consider the following model of interest,
which underlies the functional relationship between a response variable
and one or more independent variables (Suykens et al. 2002; Wang, Hu
2005):
y(x) = [w.sup.T] [phi](x) + b, (1)
where: x [member of] [R.sup.n], y [member of] R, and [phi](x):
[R.sup.n] [right arrow] [R.sup.nh] is the mapping to the high
dimensional feature space. In LS-SVM for regression analysis, given a
training dataset [{[x.sub.k],[y.sub.k]}.sup.N.sub.k = 1], the
optimization problem is formulated as follows:
minimize [J.sub.p (w,e) = 1/2 [w.sup.T] w + [gamma] 1/2
[N.summation over.(k = 1)] [e.sup.2.sub.k]; (2)
subjected to [y.sub.k] = [w.sup.T][phi]([x.sub.k]) + b + [e.sub.k],
k = 1,...N,
where: [e.sub.k] [member of] R are error variables; [gamma] > 0
denotes a regularization constant.
In the above optimization problem, it is noted that the objective
function includes a sum of squared fitting error and a regularization
term. This cost function is similar to standard procedure in training
feedforward neural networks and is related to a ridge regression (Wang,
Hu 2005). However, when w becomes infinite dimensional, one cannot solve
this primal problem. Therefore, it is necessary to construct the
Lagrangian and derive the dual problem (Suykens et al. 2002).
The Lagrangian is given by:
L(w, b, e; [alpha]) = [J.sub.p] (w, e) - [N.summation over.(k =
1)][[alpha].sub.k]{[w.sup.T][phi]([x.sub.k]) + b + [e.sub.k] -
[y.sub.k]}, (3)
where: [[alpha].sub.k] are Lagrange multipliers. The conditions for
optimality are given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)
After elimination of e and w, the following linear system is
obtained:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)
where: y = [y.sub.1] ,..., [y.sub.N], [1.sub.v] = [1;...;1] and
[alpha] = [[[alpha].sub.1]...; [[alpha].sub.N]].
And the kernel function is applied as follows:
[omega] = [phi][([x.sub.k]).sup.T][phi]([x.sub.l]) =
K([x.sub.k,[x.sub.l]). (6)
The resulting LS-SVM model for function estimation is expressed as:
y(x) = [N.summation over (k =
1)][[alpha].sub.k]K([x.sub.k],[x.sub.l]) + b, (7)
where: [[alpha].sub.k] and b are the solution to the linear system
(5). The kernel function that is often utilized is Radial Basis Function
(RBF) kernel. Description of RBF kernel is given as follows:
K([x.sub.k], [x.sub.l]) = exp([parallel][x.sub.k] -
[x.sub.l][[parallel]].sup.2]/2[[sigma].sup.2]), (8)
where: [sigma] is the kernel function parameter.
In the case of the Radial Basis Function kernel, there are two
tuning parameters ([gamma], [sigma]) that are needed to be determined in
LS-SVM. The regularization parameter ([gamma]) controls the penalty
imposed to data points that deviate from the regression function.
Meanwhile, the kernel parameter ([sigma]) affects the smoothness of the
regression function. It is worth noticing that proper setting of these
tuning parameters is required to ensure desirable performance of the
prediction model (Suykens et al. 2002).
1.3. Regression analysis with prediction intervals
1.3.1. Background
Regression analysis is the study of the function that underlies the
relation between the dependent variable Y and a vector x as the
independent variable (Olive 2007). A typical regression model can be
expressed as follows:
[Y.sub.i] = m([x.sub.i]) + [e.sub.i], i = 1, ..., (9)
where: m denotes a function of x and [e.sub.i] is the prediction
error.
Various methods are used to find the estimate [??] of m. These
methods range from traditional techniques, such as multiple linear
regression model and many time series, nonlinear, nonparametric, and
semiparametric models (Olive 2007), to various machine learning
techniques, such as M5-MT (Bhattacharya, Solomatine 2005; Jekab sons
2010), ANN (Zhu et al. 2010; Wong et al. 1997), SVM (Cheng et al. 2010;
Lu et al. 2009), and LS-SVM (Suykens et al. 2002; Brabanter et al.
2010).
Once the mapping function is obtained, the primary task is to
predict the future value of Y when a specific input x is presented to
the system. In point estimation, Y is expressed as a single value. On
the contrary, in interval estimation, the prediction result is given in
the form of an interval of possible values. In many situations, interval
estimation draws more attention than point estimation. The reason is
that the requirement of decision makers not only resides in an accurate
forecasting but also in the inherent uncertainty of the forecasts
(Shrestha, Solomati ne 2006).
[FIGURE 1 OMITTED]
Interval estimation includes the upper and lower limits between
which a pointwise value of response variable is expected to lie with a
certain level of confidence (usually 95%). The range restricted by those
limits is known as prediction interval (PI) (Fig. 1). Prediction
intervals as outputs are desirable since they provide a range of values
that most likely include the point estimation of the predicted variable.
In addition, one can employ prediction intervals to discern the accuracy
of the estimation provided by the model, and then decide to keep or
reject the result (Mencar et al. 2005).
1.3.2. Evaluating performance of prediction interval
Once the output with interval has been obtained, the Prediction
Interval Coverage Probability (PICP) can be utilized for performance
evaluation (Shrestha, Solomatine 2006; Khosravi et al. 2010). PICP
measures the proportion of data point lying within the PI. In some
cases, the empirical PICP can be much less than the pre-specified level
of confidence. This phenomenon indicates that the derived PIs are not
reliable (Khosravi et al. 2011). Hence, PICP is oftentimes expected to
be equal or greater than the level of confidence, since this reflects
the reliability of the prediction results.
However, PICP is not the only metric for evaluating PIs. The reason
is that one can simply construct a very large PI to achieve the maximum
reliability of the prediction outcomes (e.g. 100%). Nevertheless,
extremely large PIs, in practice, reduce the usability of forecasting
results because the interval estimation does not convey any valuable
information for the decision-makers (Khosravi et al. 2011). Hence, to
guarantee the usability of the interval estimation, Mean Width of
Prediction Interval (MPI) (Khosravi et al. 2010; Shrestha, Solomatine
2006), which measures the average width of the PIs, is also needed to be
considered. Accordingly, a well-constructed PI should achieve the
balance between reliability and usability. Put differently, it is
desirable to obtain a high PICP corresponding to a narrow MPI (Khosravi
et al. 2010, 2011). Nevertheless, these two criteria oftentimes conflict
with each other and this makes interval estimation a challenging
problem. Due to its importance and challenge, studies have dedicated in
establishing PIs for variety of prediction models.
2.3.3. Previous works on prediction interval estimation
Sonmez (2011) integrated neural networks with bootstrap prediction
intervals for range estimation of construction costs. In this approach,
neural networks are used for modeling the mapping function between the
influence factors and costs. Bootstrap method is utilized to quantify
the level of variability included in the estimated costs. However, to
construct interval estimates based on the bootstrap, which possibly
produces accurate intervals, requires heavy computational expense
(Brabanter et al. 2011).
Mencar et al. (2005) proposed a method for estimating prediction
interval for neuro-fuzzy network such that the system provides an
estimate of the uncertainty associated with predicted output values.
This method does not require any strict assumption on the unknown
distribution of data. However, the derived intervals are constant
throughout the input domain. This feature might not reflect the true
phenomenon happening in real-world time series data. In these cases,
inherent uncertainty may distribute unequally in different periods of
time (Cheng, Roy 2011).
[FIGURE 2 OMITTED]
Another method for constructing PI, which is based on machine
learning approach, is established by Shrestha and Solomatine (2006). In
their study, the authors presented a method to estimate PI via
uncertainty of the model output. The crucial idea herein is the
historical residuals between the model outputs and the corresponding
observed data can be the quantitative indicators of the difference
between the model and the modeled real world system and provide the
valuable information to evaluate the model uncertainty.
The machine learning based interval estimation (MLIE) approach
(Shrestha, Solomatine 2006) can be divided into five main steps (Fig.
2). First, the point estimation process is carried out. A regression
technique is employed to learn the underlying mapping function between
input data and outputs. Second, the input data points are separated into
different clusters that have similar historical residuals, which are
obtained from point estimation process, using fuzzy c-means clustering.
In the third step, prediction limits (PLs) for each cluster are computed
based on empirical distributions of the errors associated with all data
points of one cluster. In the next step, PLs for each training data
point is then calculated according to their membership grades in each
cluster. In the final step, a machine learning (ML) technique can be
deployed to learn the underlying functions between the input data and
the computed PLs for training data. PLs for testing data can be inferred
using those underlying functions.
Another advantage of MLIE method is its independence on the machine
learning technique. However, this approach lacks a mechanism for
selecting tuning parameters of the regression machine appropriately.
Additionally, the performance of the proposed MLIE, in term of
prediction accuracy and of computational cost, can be enhanced
significantly by using more superior technique such as LS-SVM.
1.4. Differential Evolution optimization algorithm
This section describes the standard algorithm of Differential
Evolution (DE) proposed by Storn and Price (Price et al. 2005; Storn,
Price 1997). The algorithm (Fig. 3) consists of five main stages:
initialization, mutation, crossover, selection, and stopping condition
verification. Given that the problem at hand is to minimize a cost
function f(X), where the number of decision variables is D, we can
describe each stages of DE in details.
1.4.1. Initialization
DE commences the search process by randomly generating NP number of
D-dimensional parameter vectors [X.sub.i,g] where i = 1, 2, NP and g
denotes the current generation. In original DE algorithm, NP does not
change during the optimization process (Storn, Price 1997). Moreover,
the initial population (at g = 0) ought to cover the entire search space
in a uniform manner. Thus, we can simply generate these individuals as
follows:
[FIGURE 3 OMITTED]
[X.sub.i,0] = LB + rand [0,1] x (UB - LB), (10)
where: [X.sub.i0] is the decision variable i at the first
generation. rand [0,1] denotes a uniformly distributed random number
between 0 and 1. LB and UB are two vectors of lower bound and upper
bound for any decision variable.
1.4.2. Mutation
A vector in the current population (or parent) is called a target
vector. Hereafter, the terms parent and target vector are used
interchangeably. For each target vector, a mutant vector is created
according to the following equation (Storn, Price 1997):
[V.sub.i,g + 1] = [X.sub.r1, g] + F([X.sub.r2,g] - [X.sub.r3,g]),
(11)
where: [r.sub.1], [r.sub.2], and [r.sub.3] are three random indexes
lying between 1 and NP. These three randomly chosen integers are also
selected to be different from the index i of the target vector. F
denotes the mutation scale factor, which controls the amplification of
the differential variation between [X.sub.r2,g] and [X.sub.r3,g.]
[V.sub.ig + 1] represents the newly created mutant vector.
1.4.3. Crossover
The crossover stage aims to diversify the current population by
exchanging components of target vector and mutant vector. In this stage,
a new vector, named as trial vector, is created. The trial vector is
also called the offspring. The trial vector can be formed as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)
where: [U.sub.t,g + i] is the trial vector; j denotes the index of
element for any vector; randj is a uniform random number lying between 0
and 1; Cr is the crossover probability, which is needed to be determined
by the user; rnb(i) is a randomly chosen index of {1, 2, NP} which
guarantees that at least one parameter from the mutant vector
([V.sub.j,i,g + i)] is copied to the trial vector ([U.sub.j,i,g + i]).
1.4.4. Selection
In this stage, the trial vector is compared to the target vector
(Price et al. 2005). If the trial vector can yield a lower objective
function value than its parent, then the trial vector replaces the
position of the target vector. The selection operator is expressed as
follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)
1.4.5. Stopping criterion verification
The optimization process can terminate when the stopping criterion
is met. The user can set the type of this condition. Commonly, maximum
generation ([G.sub.max]) or maximum number of function evaluations (NFE)
can be used as the stopping condition. When the optimization process
terminates, the final optimal solution is readily presented to the user.
2. Differential evolution based cross-validation
As mentioned earlier, in machine learning, one important objective
is to construct a prediction model that can deliver the best
generalization. The reason is that the performance on the training data
set is not necessarily a good indicator of predictive performance on the
testing data due to the problem of over-fitting (Bishop 2006).
Over-fitting arises when a regression model fits the training set very
well, but performs poorly on the new data set.
[FIGURE 4 OMITTED]
Hence, to build a desirable prediction model, one commonly used
technique is S-fold cross-validation (Bishop 2006; Samarasinghe 2006;
Suykens et al. 2002). The training data is divided into S folds and this
allows a proportion (S -1) / S of the available data to be used for
training while other portion of the data is for assessing model
performance. However, one major disadvantage of cross-validation is that
the number of training runs that must be performed is increased by a
factor of S, and this can impose difficulty for models with high
computational expense in the training process (Bishop 2006). Moreover,
another challenge is that there might be infinite combinations of
model's parameters. Thus, it is problematic and time-consuming when
designing the combinations of parameters for the cross-validation
process.
Since our study employs LS-SVM as the regression machine, there are
two parameters needed to be determined, namely regularization parameter
[gamma] and RBF kernel parameter [sigma]. To avoid over-fitting and
drawbacks of traditional cross-validation approach, the new model
utilizes DE (Price et al. 2005) to automatically explore the various
combinations of ([gamma], [sigma]) and to identify the optimal set of
these tuning parameters. In the following section, the DE-based
cross-validation (Fig. 4) is described in details.
In the step of data processing, the training data set is divided
into S folds (e.g. 5 folds). In each run, one fold is used as a
validating set; meanwhile, the other folds are used for training the
model (Fig. 5). Tuning parameters of LS-SVM is initialized randomly
using Eqn (10). The lower bounds for [gamma] and [sigma] are both 0.001.
Meanwhile, the upper bounds for [gamma] and [sigma] are specified to be
10000 and 100, respectively.
[FIGURE 5 OMITTED]
In LS-SVM training, LS-SVM is utilized to learn the regression
function between input and output for each run. These regression
functions can be described in the form of Eqn (7). After the training
process, LS-SVM is applied to predict the output of the validating sets.
In order to determine the optimal set of tuning parameters, the
following objective function is used in the step of fitness function
evaluation:
[F.sub.fitness] = [5.summation over (k = 1)] [E.sup.k.sub.tr]/5 +
[5.summation over (k = 1)] [E.sup.k.sub.va]/5, (14)
where: [E.sup.k.sub.tr] and [E.sup.k.sub.va] denotes the training
error and validating error, respectively, for kth run. The training and
validating errors herein are Root Mean Squared Error calculated as
follows:
RMSE = [square root of [N.summation over (j = 1)] [([Y.sup.j.sub.P]
- [Y.sup.j.sub.A]).sup.2]/N], (15)
where: [Y.sup.j.sub.P] and [Y.sup.j.sub.A] denote predicted and
actual value for output jth. In addition, N is the number of training
data in each run.
The fitness function, in essence, represents the trade-off between
model generalization and model complexity. It is worth noticing that
well-fitting of the training set may reflect the model complexity.
However, complex model tends to suffer from over-fitting (Bishop 2006;
Suykens et al. 2002). Thus, incorporating the error of the validating
data can help identify the model that features the balance of minimizing
training error and generalization property.
In each generation, the DE optimization carries out mutation,
crossover, and selection process to guide the initial population to the
final optimal solution. The search terminates when the current
generation g achieves the maximum number of generation [G.sub.max].
After being optimized, the prediction model is ready to be used in the
next step.
3. Interval estimation of construction cost at completion using
LS-SVM inference model (EAC-LSPIM)
Figure 6 provides the overall picture of the model EAC-LSPIM.
Before describing the model in detail, it is noted that our study
benefits from previous research works of Chen (2008) and Cheng et al.
(2010) in identifying the influence factors for EAC prediction (Table
1), and of Shrestha and Solomatine (2006) in establishing the MLIE.
3.1. Input data
Historical data sets (Table 2) used in this paper were collected
from 13 reinforced concrete building projects executed between 2000 and
2007 by one construction company headquartered in Taipei City, Taiwan.
Building heights ranges from 9 to 17 stories (including underground
floors). Contract values ran from NT$80 million to NT$1.1 billion. Total
floor areas for the projects ranged from 2,094 [m.sup.2] to 31,797
[m.sup.2]. Besides, construction durations varied between 15 to 63
months. Historical data sets were separated into training sets (from 1
to 11) and testing sets (12 and 13). The training and testing data sets
consist of 262 and 44 data cases respectively. Table 3 provides
descriptive statistics of influencing factors as well as desired output
of the historical data. In Table 4, the sample of 10 input variables
from project 2, which had 24 completion periods, is used to illustrate
the data set.
[FIGURE 6 OMITTED]
3.2. LS-SVM for point estimation of Estimate to Completion
Herein, LS-SVM is employed to learn the mapping function between
model's input and output. Each 1 x 10 vector of influence factor
acts as input for LS-SVM. Input vectors and observed values of Estimate
to Completion (ETC) take the role of training data to obtain the
prediction model. LS-SVM uses regularization parameter ([gamma]) and RBF
kernel parameters ([sigma]), which are chosen by DE-based
cross-validation process. After the training process, the model is
capable of inferring unknown ETC value whenever new input information is
presented.
3.3. Estimate at Completion calculation
In this step, Actual cost percentage (AC) of completed jobs is
added to the Estimate to Completion (ETC) in order to obtain the
Estimate at Completion (EAC) values, as defined in Eqn (16):
[EAC.sub.P] = A[C.sub.P] + ET[C.sub.P], (16)
where: [EAC.sub.P] denotes point estimation of EAC; ET[C.sub.P]
represents point estimation of ETC; A[C.sub.P] is actual cost
percentage.
ETC is a value used to determine forecasted expenditures necessary
to complete remaining project work. AC percentage is a known value
defined as the ratio of actual construction cost (AC) value to the
Budget at Completion (BAC). It is noted that the BAC itself is the cost
of the project when all contracted works are completed. EAC replaces BAC
for the predicted total cost of the project at a specific period during
construction.
3.4. Estimation of EAC prediction interval
Prediction interval (PI) estimation is carried out in four steps.
First, the input dataset is separated into a certain number of clusters
corresponding to distributions of historical residuals, obtained from
the point estimation, using fuzzy c-means clustering algorithm (FCMC)
(Bezdek 1981). The FCMC is an unsupervised machine learning technique
employed to separate data into different clusters. Notably, using this
technique, a data point can belong to many clusters, and the degrees of
belonging are quantified by fuzzy membership grades. In FCMC, the number
of clusters needs to be specified by the user. Commonly, the optimal
number of clusters can be selected so that it results in the clustering
performance corresponding to the smallest Xie-Beni index. For details of
FCMC and selecting cluster number, readers are guided to the previous
works of Xie and Beni (1991), and of Oliveira and Pedrycz (2007).
The next step is to compute the lower and upper PIs for each
cluster. Given a certain level of confidence (e.g. 95% or [alpha] is
5%), the PIs for each cluster is calculated from empirical distributions
of the corresponding historical residuals (e). To construct (100 -
[alpha])% PI, the ([alpha]/2) x 100 and (1-([alpha]/2)) x 100 percentile
values are taken from empirical distribution of residuals for lower and
upper PI, respectively (Fig. 7). The mathematical expression for
calculating lower and upper PIs for cluster i ([PI.sup.L.sub.ci] and
[PI.sup.U.sub.ci]) is given as follows:
[PI.sup.L.sub.ci] = [e.sub.j] j : [j.summation over (k = 1)]
[[mu].sub.i,k] < [alpha]/2 [n.summation over (j = 1)] [[mu].sub.i,j];
(17)
[PI.sup.U.sub.ci] = [e.sub.j] j : [j.summation over (k = 1)]
[[mu].sub.i,k] > (1 - [alpha]/2) [n.summation over (j = 1)]
[[mu].sub.i,j]; (17)
where: j is the index of the sorted data point that satisfies the
corresponding inequalities; [e.sub.j] denotes historical residuals of
sorted data point j; and [[mu].sub.i,j] is membership grade of data
point j to cluster i.
[FIGURE 7 OMITTED]
In the third step, the PI for each tcraining data point is
calculated using the weighted mean of PI of each cluster:
[PI.sup.L.sub.j] = [c.summation over (i = 1)] [[mu].sub.i,j] x
[PI.sup.L.sub.ci]; (19)
[PI.sup.U.sub.j] = [c.summation over (i = 1)] [[mu].sub.i,j] x
[PI.sup.U.sub.ci], (20)
where: [PI.sup.L.sub.j] and [PI.sup.U.sub.j] are lower and upper
prediction intervals for data point j.
Finally, prediction limits of EAC for each data point are computed
as follows:
[EAC.sup.L.sub.j] = [y.sub.i] + [PI.sup.L.sub.j] ; (21)
[EAC.sup.U.sub.j] = [y.sub.i] + [PI.sup.U.sub.j] ; (21)
where: [EAC.sup.L.sub.j] and [EAC.sup.U.sub.j] are lower and upper
prediction limits of EAC for input data j.
3.5. LS-SVM for inference of EAC prediction limits
Once the prediction limits for each training data point are
obtained, LS-SVM is utilized to establish two regression functions that
model the relationship between the input data and its corresponding
prediction limits. Tuning parameters of LS-SVM in this step are also
selected via DE-based cross-validation. When the training process
finishes, the model is then capable of estimating lower and upper
prediction limits for new instances of input data.
3.6. Interval estimation of project cost at completion
In this step, the final model outputs ([EAC.sub.p],
[EAC.sup.L.sub.j], and [EAC.sup.U.sub.j]) are presented. The interval
estimation of total cost is available for decision-making process. The
planners or managers can anticipate the cost required to complete the
project associated with uncertainty described in the form of prediction
intervals.
4. Simulation result and comparisons
After the training process, the proposed model, EAC-LSPIM, is
utilized to predict two testing projects (12 and 13). Projects 12 and 13
consist of 27 and 17 completion periods, respectively. To achieve
interval forecast of EAC, the level of confidence is set as 95%, which
is corresponding to a of 5%. In order to evaluate the accuracy of EAC
point estimation, Root Mean Square Error (RMSE), Mean Absolute
Percentage Error (MAPE), and Mean Absolute Error (MAE) are employed. In
addition, to assess the performance of EAC interval estimation, PICP and
MPI are utilized.
Prediction results of EAC-LSPIM for two testing projects are
illustrated in Tables 5 and 6, and Figures 8 and 9. In these tables and
figures, EACA denotes the actual EAC. Meanwhile, EACL, EACP, and EACu
represent lower prediction limit, point estimation and upper prediction
limit of EAC, respectively. Deviation is the error between point
estimate of EAC and the actual EAC.
In the experiment in which projects 12 and 13 serve as testing
cases, the RMSE, MAPE, and MAE of point estimate are 0.044, 3,741, and
0.034, respectively. The PICP and the MPI derived from EAC-LSPIM are
97.73% and 19.22, respectively. Since the level of confidence is set as
95%, the derived PICP is desirable; and this demonstrates the
reliability of the prediction results. Meanwhile, it can be observed
that the width of PIs yielded by the proposed model is acceptable. On
average, the range of predicted EAC is 19.22%, and this is relatively
satisfactory in an operational environment of construction industry,
which is often hazarded by uncertainty.
In order to validate the superiority of EAC-LSPIM, its performance
is compared to other benchmarked approaches. It is noted that the newly
developed model is composed of LS-SVM, MLIE, and DE-based
cross-validation. In order to validate the superiority of the proposed
prediction model, various machine learning techniques, namely M5-MT,
ANN, and LS-SVM, has been integrated with MLIE and are applied to for
interval prediction of EAC. For LS-SVM, the selection of tuning
parameters is achieved via the grid search approach (Suykens et al.
2002; Shu et al. 2010). Utilizing this approach, various pairs of
([gamma] and [delta]) are tried and the one with the best
cross-validation accuracy is chosen. Accordingly, the values of [gamma]
and [sigma] obtained from the grid search method, for point estimation
of EAC, are 256 and 2.8, respectively. Meanwhile, the optimal values of
[gamma] and [sigma] found by DE are 251.4 and 3.9, respectively.
The result comparison is shown in the Table 7. From Table 7, it is
observable that the proposed model, EAC-LSPIM, has achieved the best
result in point estimate of EAC having the smallest RMSE, MAPE, and MAE
of testing data. Moreover, the model also yields the most desirable
performance in interval estimation of project cost. Its prediction
interval has the highest PICP value (97.73%) with relatively narrow MPI
(19.22) compared to other outcomes.
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
To better demonstrate the performance of each benchmark model, the
results of different combinations of 13 projects for training and
testing have been added. In this experiment, 13 cases of experiment are
carried out. In each case, a project serves as a testing set, the rest
are training sets. As shown in Table 8, based on the average prediction
results, the proposed model achieves the most desirable outcome. For
point estimation, the RMSE, MAPE, and MAE of EAC-LSPIM are 3.812, 0.035,
and 0.042, respectively. Meanwhile, for interval estimation, the PICP
and MPI of the proposed model are 98.43% and 21.89. Observably,
EAC-LSPIM is the most accurate model in point forecast of project cost.
Moreover, it also achieves the highest PICP value corresponding to a
relatively small value of MPI. These facts strongly proved the
superiority of the new model over other benchmark approaches.
Conclusion
This study proposes a new prediction model, namely EAC-LSPIM, to
assist project manager in construction cost planning and monitoring. To
address the uncertainty in construction cost forecasting, this study
incorporates LS-SVM, MLIE, and DE to achieve interval forecasting of
construction project cost.
In EAC-LSPIM, the utilization of LS-SVM is twofold. First, LS-SVM
is used to infer the underlying function between input data and point
estimation of ETC. Second, it is employed to model the mapping
relationship between the input data and the prediction limits of EAC.
Moreover, by using MLIE, the new model derives the prediction
interval by evaluating the uncertainty inherent in the data set, without
any assumption or prior knowledge about model's error distribution.
In order to avoid over-fitting, our study employs DE search engine
in the cross-validation process. The DE-based cross-validation
successfully identifies the most appropriate set of tuning parameters
and eliminates the need of expertise or trial-and-error process in
parameter setting.
Consequently, the proposed model has the capacity to operate
automatically without human intervention and domain knowledge. In
addition, simulation and performance comparison have demonstrated the
accuracy, the reliability, and the usability of EAC-LSPIM prediction.
Therefore, the newly established model has a great potential to assist
decision-makers in the field of construction management.
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Min-Yuan CHENG (a), Nhat-Duc HOANG (b)
(a) Department of Construction Engineering, National Taiwan
University of Science and Technology, Taiwan
(b) Faculty of Building and Industrial Construction, National
University of Civil Engineering, Hanoi, Vietnam
Received 28 Oct 2011; accepted 08 Jun 2012
Corresponding author: Nhat-Duc Hoang
E-mail:
[email protected],
[email protected]
Min-Yuan CHENG is currently a Professor at the Department of
Construction Engineering, National Taiwan University of Science and
Technology. He holds lectures in Construction Automation and
Construction Process Re-engineering. He has published many papers in
various international journals such as Automation in Construction,
Journal of Construction Engineering and Management, and Expert Systems
with Applications. His research interests include management information
system, applications of artificial intelligence, and construction
management process reengineering.
Nhat-Duc HOANG is currently a lecturer at Department of Technology
and Construction Management, Faculty of Building and Industrial
Construction, National University of Civil Engineering, Hanoi, Vietnam.
He got the MSc and PhD degrees at National Taiwan University of Science
and Technology, Taipei, Taiwan. His research focuses on applications of
Artificial Intelligence in Construction engineering and management. His
research articles have been published in Journal of Computing in Civil
Engineering (ASCE), Engineering Applications of Artificial Intelligence,
and Automation in Construction.
Table 1. EAC prediction's influencing factors
No. Influence Index
Factor (IF)
IF1 Construction Construction
duration progress (%)
IF2 Actual cost A[C.sub.p]
IF3 Planned cost E[V.sub.p]
IF4 Cost management CPI
IF5 Time management SPI
IF6 Subcontractor Subcontractor
management billed index
IF7 Contract payment Owner billed
index
IF8 Change order Change order
index
IF9 Construction CCI
price
fluctuation
IF10 Number of Climate effect
rainy day index
No. Definition
IF1 Duration to date/
revised contract duration
IF2 Actual Cost/
Budget at Completion
IF3 Earned Value/
Budget at Completion
IF4 Earned Value/Actual Cost
IF5 Earned Value/Planned Value
IF6 Subcontractor billed amount/
Actual Cost
IF7 Owner billed amount/
Earned Value
IF8 Revised contract amount/
Budget at Completion
IF9 Construction material price
index of that month/
construction material price
index of initial stage
IF10 (Revised project duration--number
of rainy day)/
revised project duration
Table 2. Project information
Projed Total area Underground Ground Buildings Start date
([m.sup.2]) floors floors
1 12622 2 9 1 2003/12/1
2 4919 3 11 1 2003/12/13
3 19205 5 8 1 2000/5/20
4 5358 3 9 1 2000/11/15
5 27468 2 11 3 1999/12/16
6 31797 2 9 4 2001/7/4
7 7707 2 14 1 2001/11/24
8 10087 3 14 1 2002/6/18
9 3479 1 10 1 2003/6/2
10 7289 2 8 1 2005/6/15
11 6352 4 11 1 2004/3/5
12 4774 2 11 1 2004/2/21
13 3094 2 7 1 2005/10/1
Projed Finish date Duration Contract Prediction
(days) amount (NTD) periods
1 2005/8/22 630 289,992,000 29
2 2005/11/10 689 149,300,000 24
3 2002/5/19 729 332,800,000 20
4 2002/11/14 729 199,600,000 25
5 2001/12/3 718 1,142,148,388 26
6 2003/3/31 635 530,000,000 20
7 2003/10/20 695 153,500,000 22
8 2004/7/6 749 216,000,000 27
9 2004/9/30 486 85,714,286 18
10 2006/9/15 457 190,844,707 20
11 2006/2/18 715 202,241,810 31
12 2006/2/20 730 145,377,589 27
13 2007/2/28 515 102,500,000 17
Table 3. Descriptive statistics of historical data
IF1(%) IF2(%) IF3(%) IF4 IF5 IF6 IF7
Mean 65.48 51.62 60.66 1.20 1.00 1.06 0.89
Median 65.30 49.25 58.05 1.16 1.00 1.08 0.91
Minimum 2.40 0.00 0.00 0.34 0.40 0.00 0.00
Maximum 130.60 132.70 141.60 2.33 1.62 1.90 1.57
Std. Dev. 32.92 31.50 35.72 0.23 0.09 0.26 0.22
IF8 IF9 IF10 ETC(%) EAC(%)
Mean 1.03 1.05 0.89 42.93 93.87
Median 1.00 1.03 0.90 42.00 91.70
Minimum 0.87 0.97 0.70 0.00 73.20
Maximum 1.42 1.20 1.00 108.70 132.70
Std. Dev. 0.10 0.06 0.07 30.81 14.84
Table 4. Input data for project 2 with 24 completion periods
IF1 IF2 IF3 IF4 IF5 IF6 IF7 IF8 IF9 IF10
1
2.4 0 0 1 1.0 1.0 1.0 1.0 1.0 1.0
6.9 2.5 0 1.8 1.0 1.2 1.0 1.0 1.0 1.0
11 6.3 11 1.3 1.0 1.8 1.0 1.0 1.1 1.0
15.5 9.6 12.7 1 1.0 1.3 1.0 1.0 1.1 1.0
19.9 12.6 14.8 1.1 1.0 1.0 0.9 1.0 1.1 1.0
24.2 14.3 16.4 1 1.0 0.9 0.8 1.0 1.1 1.0
28.7 16.5 16.4 1 1.0 1.4 1.4 1.0 1.1 0.9
33 19.4 18.8 1.2 1.0 1.2 1.2 1.0 1.1 0.9
37.4 22.3 26.5 1.2 1.0 1.4 1.2 1.0 1.1 0.9
41.8 25.3 30.4 1 1.0 1.4 1.1 1.0 1.1 0.9
46.1 29.2 29.7 1 1.0 1.3 1.3 1.0 1.1 0.9
50.6 32.6 33.1 1 1.0 1.2 1.2 1.0 1.1 0.9
54.9 36.4 37 1 1.0 1.3 1.2 1.0 1.1 0.9
59.3 40.6 41.5 1 1.0 1.1 1.1 1.0 1.1 0.9
63.5 43.8 44.8 1 1.0 1.0 1.0 1.0 1.1 0.9
72.2 54.6 55.8 1 1.0 1.1 1.1 1.0 1.1 0.9
76.5 61.5 62.9 1 1.0 1.1 1.0 1.0 1.1 0.8
81 66.1 67.6 1 0.9 1.0 1.0 1.0 1.1 0.8
85.2 72.5 82.5 1.1 1.0 1.1 1.0 1.0 1.1 0.8
89.7 78.5 92 1.2 1.0 1.1 1.0 1.0 1.1 0.8
94.1 79.6 94.4 1.2 1.0 1.2 1.0 1.0 1.1 0.8
98.4 81.6 99 1.2 1.0 1.2 1.0 1.0 1.1 0.8
102.9 84.5 100.5 1.2 1.0 1.2 1.0 1.0 1.1 0.8
107.2 91.2 100.5 1.1 1.0 1.1 1.0 1.0 1.1 0.8
Table 5. Interval Estimation of [EAC.sub. for project 12
No. [EAC.sub.A] [EAC.sub.L] [EAC.sub.P] [EAC.sub.U] Deviation
1 91.69 74.45 84.79 93.77 6.90
2 91.69 69.06 79.41 88.39 12.28
3 91.69 79.11 89.33 98.31 2.36
4 91.69 78.35 88.74 97.85 2.95
5 91.69 76.39 86.75 95.73 4.94
6 91.69 73.08 83.34 92.39 8.35
7 91.69 73.86 84.01 93.11 7.68
8 91.69 73.02 82.82 91.71 8.87
9 91.69 78.20 87.94 96.78 3.75
10 91.69 82.36 92.59 101.57 0.90
11 91.69 80.73 91.00 99.98 0.69
12 91.69 80.89 91.26 100.29 0.43
13 91.69 80.14 90.55 99.57 1.14
14 91.69 84.09 94.48 103.46 2.79
15 91.69 80.85 91.25 100.22 0.44
16 91.69 83.75 94.12 103.09 2.43
17 91.69 80.40 90.74 99.72 0.95
18 91.69 80.69 91.03 100.01 0.66
19 91.69 75.33 85.69 94.68 6.00
20 91.69 77.65 88.02 97.00 3.67
21 91.69 78.10 88.48 97.47 3.21
22 91.69 80.08 90.47 99.46 1.22
23 91.69 77.24 87.62 96.61 4.07
24 91.69 82.37 92.73 101.71 1.04
25 91.69 82.57 92.48 101.29 0.79
26 91.69 82.12 91.70 100.44 0.01
27 91.69 83.60 93.24 102.06 1.55
Table 6. Interval Estimation of EAC for project 13
No. [EAC.sub.A] [EAC.sub.L] [EAC.sub.P] [EAC.sub.U] Deviation
1 92.49 83.98 94.33 103.31 1.84
2 92.49 86.85 97.42 106.39 4.93
3 92.49 78.87 89.27 98.25 3.22
4 92.49 78.16 88.54 97.52 3.95
5 92.49 76.97 87.32 96.30 5.17
6 92.49 83.88 94.24 103.22 1.75
7 92.49 90.91 101.26 110.24 8.77
8 92.49 91.51 101.82 110.79 9.33
9 92.49 86.05 95.73 104.49 3.24
10 92.49 81.05 90.32 98.91 2.17
11 92.49 80.64 90.69 99.54 1.80
12 92.49 84.49 94.79 104.04 2.30
13 92.49 79.70 90.23 99.39 2.26
14 92.49 78.55 89.39 98.61 3.10
15 92.49 77.98 88.68 97.90 3.81
16 92.49 81.06 91.12 100.08 1.37
17 92.49 84.62 94.86 103.97 2.37
Table 7. Prediction result comparison for 2 testing projects
Prediction model M5-MT ANN LS-SVM EAC-LSPIM
Training
MAPE 3.16 3.11 1.81 2.36
MAE 0.03 0.03 0.02 0.02
RMSE 0.04 0.05 0.03 0.03
Testing
MAPE 7.63 3.83 3.90 3.74
MAE 0.07 0.04 0.04 0.03
RMSE 0.09 0.06 0.05 0.04
Interval
Estimation
PICP 81.8 90.9 93.2 97.7
MPI 27.9 23.0 20.8 19.2
Table 8. Prediction result comparison for 13 projects
Model Project
1 2 3 4 5 6 7
M5-MT MAPE 4.3 10.6 9.7 7.3 1.7 6.3 5.0
MAE 0.0 0.1 0.1 0.1 0.0 0.1 0.0
RMSE 0.1 0.1 0.1 0.1 0.0 0.1 0.1
PICP 93.1 79.2 90.0 88.0 100.0 95.0 86.4
MPI 21.2 22.1 27.5 26.4 14.0 24.1 21.1
ANN MAPE 7.2 7.7 6.4 4.5 4.1 6.5 3.2
MAE 0.1 0.1 0.1 0.0 0.1 0.1 0.0
RMSE 0.1 0.1 0.1 0.0 0.1 0.1 0.0
PICP 89.7 100.0 100.0 100.0 84.6 85.0 100.0
MPI 28.5 31.6 24.1 23.1 25.6 28.6 30.4
LSSVM MAPE 4.0 4.0 4.0 6.4 5.3 7.3 6.6
MAE 0.0 0.0 0.0 0.1 0.1 0.1 0.1
RMSE 0.1 0.0 0.0 0.1 0.1 0.1 0.1
PICP 96.6 100.0 100.0 100.0 84.6 100.0 95.5
MPI 21.6 22.0 26.2 34.0 19.7 36.4 32.1
EAC- MAPE 3.5 3.7 4.3 6.5 2.8 6.7 1.7
LSPIM MAE 0.0 0.0 0.0 0.1 0.0 0.1 0.0
RMSE 0.0 0.0 0.0 0.1 0.0 0.1 0.0
PICP 96.6 100.0 100.0 100.0 92.3 100.0 100.0
MPI 17.8 24.3 16.6 25.1 17.7 29.1 20.2
Model Average
8 9 10 11 12 13
M5-MT 4.9 8.1 8.0 10.3 6.8 8.2 7.0
0.0 0.1 0.1 0.1 0.1 0.1 0.1
0.0 0.1 0.1 0.1 0.1 0.1 0.1
96.3 94.4 87.1 80.7 85.2 82.4 89.1
25.2 36.5 26.4 27.4 28.4 25.0 25.0
ANN 7.6 5.6 3.1 8.8 4.7 5.7 5.8
0.1 0.0 0.0 0.1 0.0 0.1 0.1
0.1 0.1 0 0.1 0.1 0.1 0.1
92.6 88.9 100 93.6 92.6 88.2 93.5
22.7 25.4 27.1 30.7 23.4 21.5 26.4
LSSVM 3.9 5.5 6.8 5.6 5.0 4.3 5.3
0.0 0.0 0.1 0.1 0.0 0.0 0.1
0.0 0.1 0.1 0.1 0.1 0.1 0.1
100.0 83.3 87.1 96.8 92.6 94.1 94.7
35.6 22.8 24.6 21.4 22.4 22.1 26.2
EAC- 4.2 3.9 2.2 2.5 3.6 3.9 3.8
LSPIM 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0 0 0.0 0.0 0.0 0.0
100.0 94.4 100 100.0 96.3 100.0 98.4
28.1 20.8 21.2 25.4 19.0 19.3 21.9
