J. Software Engineering & Applications, 2010, 3: 374-383
doi:10.4236/jsea.2010.34042 Published Online April 2010 (http://www.SciRP.org/journal/jsea)
Time Series Forecasting of Hourly PM10 Using
Localized Linear Models
Athanasios Sfetsos, Diamando Vlachogiannis
Environmental Research Laboratory, INTR-P, National Centre for Scientific Research “Demokritos”, Attikis, Greece.
Received October 15th, 20009; revised November 4th, 2009; accepted November 8th, 2009.
ABSTRACT
The present paper discusses the application of localized linear models for the prediction of hourly PM10 concentration
values. The advantages of the proposed approach lies in the clustering of the data based on a common property and the
utilization of the target variable during this process, which enables the development of more coherent models. Two al-
ternative localized linear modelling approaches are developed and compared against benchmark models, one in which
data are clustered based on their spatial proximity on the embedding space and one novel approach in which grouped
data are described by the same linear model. Since the target variable is unknown during the prediction stage, a com-
plimentary pattern recognition approach is developed to account for this lack of information. The application of the
developed approach on several PM10 data sets from the Greater Athens Area, Helsinki and London monitoring net-
works returned a significant reduction of the prediction error under all examined metrics against conventional fore-
casting schemes such as the linear regression and the neural networks.
Keywords: Localized Linear Models, PM10 Forecasting, Clustering Algorithms
1. Introduction
Environmental health research has demonstrated that
Particulate Matter (PM) is a top priority pollutant when
considering public health. Studies of long-term expo-
sure to air pollution, mainly to PM, suggest adverse
long- and short-term health effects, increased mortality
(e.g. [1,2]), increased risk of respiratory and cardio-
vascular related diseases (e.g. [3]), as well as increased
risk of developing various types of cancer [4]. Hence,
the development and use of accurate and fast models
for forecasting PM values reliably is of immense inter-
est in the process of decision making and modern air
quality management systems.
In order to evaluate the ambient air concentrations of
particulate matter, a deterministic urban air quality
model should include modelling of turbulent diffusion,
deposition, re-suspension, chemical reactions and
aerosol processes. In recent years, an emerging trend is
the application of Machine Learning Algorithms
(MLA), and particularly, that of the Artificial Neural
Networks (ANN) as a means to generate predictions
from observations in a location of interest. The strength
of these methodologies lies in their ability to capture
the underlying characteristics of the governing process
in a non-linear manner, without making any predefined
assumptions about its properties and distributions.
Once the final models have been determined, it is then
a straight-forward and exceedingly fast process to gen-
erate predictions. However, ANN have also inherent
limitations. The main one is the extension of models in
terms of time period and location; this always requires
training with locally measured data. Moreover, these
models are not capable of predicting spatial concentra-
tion distributions.
Owing to the importance and significant concentra-
tions of PM in major European cities, there is an in-
creasing amount of literature concerned with the ap-
plication of statistical models for the prediction of
point PM values. For the purposes of the EU-funded
project APPETISE, an inter-comparison of different air
pollution forecasting methods was carried out in Hel-
sinki [5]. Neural networks demonstrated a better fore-
casting accuracy than other approaches such as linear
regression and deterministic models.
In [6], Perez et al. compared predictions produced by
three different methods: a multilayer neural network,
linear regression and persistence methods. The three
methods were applied to hourly averaged PM2.5 data for
the years of 1994 and 1995, measured at one location in
the downtown area of Santiago, Chile. The prediction
errors for the hourly PM2.5 data were found to range

Time Series Forecasting of Hourly PM10 Using Localized Linear Models375
from 30% to 60% for the neural network, from 30% to
70% for the persistence approach, and from 30% to 60%
for the linear regression, concluding however that the
neural network gave overall the best results in the predic-
tion of the hourly concentrations of PM2.5.
In [7], Gardner undertook a model inter-comparison
using Linear Regression, feed forward ANN and Classi-
fication and Regression Tree (CART) approaches, in
application to hourly PM10 modelling in Christchurch,
New Zealand (data period: 1989-1992). The ANN
method outperformed CART and Linear Regression
across the range of performance measures employed. The
most important predictor variables in the ANN approach
appeared to be the time of day, temperature, vertical
temperature gradient and wind speed.
In [8], Hooyberghs et al. presented an ANN for fore-
casting the daily average PM10 concentrations in Bel-
gium one day ahead. The particular research was based
upon measurements from ten monitoring sites during the
period 1997-2001 and upon the ECMWF (European
Centre for Medium-Range Weather Forecasts) simula-
tions of meteorological parameters. The most important
input variable identified was the boundary layer height.
The extension of this model with further parameters
showed only a minor improvement of the model per-
formance. Day-to-day fluctuations of PM10 concentra-
tions in Belgian urban areas were to a larger extent
driven by meteorological conditions and to a lesser
extent by changes in anthropogenic sources.
In [9], Ordieres et al. analyzed several neural-network
methods for the prediction of daily averages of PM2.5
concentrations. Results from three different neural net-
works (feed forward, Radial Basis Function (RBF) and
Square Multilayer Perceptron) were compared to two
classical models. The results clearly demonstrated that
the neural approach not only outperformed the classical
models but also showed fairly similar values among dif-
ferent topologies. The RBF shows up to be the network
with the shortest training times, combined with a greater
stability during the prediction stage, thus characterizing
this topology as an ideal solution for its use in environ-
mental applications instead of the widely used and less
effective ANN.
The problem of the prediction of PM10 was ad-
dressed in [10], using several statistical approaches such
as feed-forward neural networks, pruned neural networks
(PNNs) and Lazy Learning (LL).The models were de-
signed to return at 9 a.m. the concentration estimated for
the current day. The forecast accuracy of the different
models was comparable. Nevertheless, LL exhibited the
best performances on indicators related to average good-
ness of the prediction, while PNNs were superior to the
other approaches in detecting the exceedances of alarm
and attention thresholds.
In view of the recent developments in PM forecasting,
the present paper introduces an innovative approach
based on localized linear modelling. Specifically, two
alternative localized liner modelling approaches are de-
veloped and compared against benchmark models such
as the linear regression and the artificial neural networks.
The advantage of the proposed approach is the identifica-
tion of the finer characteristics and underlying properties
of the examined data set through the use of suitable clus-
tering algorithms and the subsequent application of a
customized linear model on each one. Furthermore, the
use of the target variable in the clustering stage enhances
the coherence of the localized models. The developed
approach is applied on several data sets from the moni-
toring networks of the Greater Athens Area and Helsinki,
during different seasons.
2. Modelling Approaches
Time series analysis is used for the examination of a data
set organised in sequential order so that its predominant
characteristics are uncovered. Very often, time series
analysis results in the description of the process through
a number of equations (Equation (1)) that in principle
combine the current value of the series, yt, to lagged val-
ues, yt-k, modelling errors, et-m, exogenous variables, xt-j,
and special indicators such as time of the day. Thus, the
generalized form of this process could be written as fol-
lows:
yt = f (yt-k, xt-j, et-m | various k,j,m and special indicators)
(1)
2.1 Linear Regression
This approach uses linear regression models to determine
whether a variable of interest, yt, is linearly related to one
or more exogenous variable, xt, and lagged variables of
the series, yt. The expression that governs this model is
the following:
   
j
jtj
k
ktkt ycy x (2)
The coefficients c, β, γ are usually estimated from a least
squares algorithm. The inputs should be a set of statisti-
cally significant variables, defined under Student’s t-test,
estimated from the examination of the correlation coeffi-
cients or using a backward elimination selection proce-
dure from a larger initial set.
2.2 Artificial Neural Networks (ANN)
The multi-layer perceptron or feed-forward ANN [11]
has a large number of processing elements called neurons,
which are interconnected through weights (wiq, vqj). The
neurons expand in three different layer types: the input,
the output, and one or more hidden layers. The signal
flow is from the input layer towards the output. Each
neuron in the hidden and output layer is activated by a

Time Series Forecasting of Hourly PM10 Using Localized Linear Models 376
nonlinear function that relies on a weighted sum of its
inputs and a neuron-specific parameter, called bias, b.
The response of a neuron in the output layer as a function
of its inputs is given by Equation (3), where f1 and f2 can
be sigmoid, linear or threshold activation functions.
12
11
(())
lm
iiqqjjj
qj
yfwf vxbb





 q

(3)
The strength of neural networks lies in their ability to
simulate any given problem from the presented example,
which is achieved from the modification of the network
parameters through learning algorithms. In this study, the
Levenberg-Marquardt [12] algorithm is applied because
of its speed and robustness against the conventional
back-propagation.
The most important issue concerning the introduction
of ANN in time series forecasting is “generalization”,
which refers to their ability to produce reasonable fore-
casts on data sets other than those used for the estimation
of the model parameters. This problem has two important
parameters that should be accounted for. The first is data
preparation, which involves pre-processing and the se-
lection of the most significant variables. The second em-
braces the determination of the optimum model structure
that is closely related with the estimation of the model
parameters. Although, there is no systematic approach,
which can be followed [13], some useful insight can be
found using statistical methods such as the correlation
coefficients.
The second aspect can be jointly tackled under the
cross-validation training scheme. The data set is split into
three smaller sets the training (TS), the evaluation or
validation (ES) and the prediction or testing (PS) sets.
The model is initialized with a few parameters. The next
step is to train the model using data from the training set
and when the error of the evaluation set is minimized, the
model parameters and configuration are stored. The
number of parameters is then increased and a new net-
work is trained from the beginning. If ES error is lower
compared to the previously found minimum, then the
parameters of this new model are stored. This iterative
process is terminated when a predefined number of itera-
tions are reached (Figure 1).
In this study, ES was formed using a Euclidean metric
withholding a percent value (here 25% is used) of the TS
data that are located nearest to other data. The strength of
this approach lies in the fact that TS covers more distinct
characteristics of the process, thus, allowing for the de-
velopment of a model with better generalization capabili-
ties.
2.3 Nearest Neighbours
This class of hybrid models includes a local modelling
and a function approximation to capture recent dynamics
Figure 1. Iterative cross-validation training
of the process. The underlying aim of these predictors is
that segments of the series neighbouring under some dis-
tance measure may correspond to similar future values.
This claim was endorsed by the work of Farmer and Si-
dorowich [14] that showed that the chaotic time-series
prediction is several orders of magnitude better using
local approximation techniques rather than universal ap-
proximators. The tricky part in these models is the selec-
tion of the embedding dimension, which effectively de-
termines segments of the series, and the number of
neighbours. Initially, it is required to estimate the em-
bedding dimension d and time delay τ of the attractor as
follows:

 j)(t),)1((),...,()(


x


dtytytY (4)
In this study, a value of τ = 1 was used and Y(t) had the
same parameters as the linear regression model. The
number of neighbours was not pre-determined but was
set to vary between predefined limits. A small number of
neighbours increase the variance of the results whereas a
large number can compromise the local validity of a
model and increase the bias of results. Once the nearest
neighbours to Y(t) have been identified, an averaging
procedure is followed in the present study to generate
predictions.
2.4 Local Models with Clustering Algorithms
(LMCA)
The idea behind the application of clustering algorithms
in time series analysis is to identify groups of data that
share some common characteristics. On each of these
groups, the relationships amongst the members are mod-
elled through a single equation model. Consequently,

Time Series Forecasting of Hourly PM10 Using Localized Linear Models377
l

each of the developed models has a different set of pa-
rameters. The process is described in the following steps:
1) Selection of the input data for the clustering algo-
rithm. This can contain lagged and/or future characteris-
tics of the series, as well as other relevant information.
C(t) = [yt, yt-k, xt-j]. Empirical evidence suggests that
the use of the target variable yt is very useful to discover
unique relationships between input-output features. Ad-
ditionally, higher quality modelling is ensured with the
function approximation since the targets have similar
properties and characteristics. However, this occurs to
the expense of an additional process needed to account
for this lack of information in the prediction stage.
2) Application of a clustering algorithm combined
with a validity index or with user defined parameters, so
that ncl clusters will be estimated.
3) Assign all patterns from the training set to the ncl
clusters. For each of the clusters, apply a function ap-
proximation model, , so that
ncl forecasts are generated.
(,) 1...
titktj c
yfyi n

x,
Successful application of this method has been re-
ported on the prediction of locational electricity marginal
prices [15], Mckay Glass and daily electric load peak
series [16], the A and D series of the Santa Fe forecasting
competition [17] and hourly electric load [18].
In this study, the k means clustering algorithm was se-
lected [19]. It is a partitioning algorithm that attempts to
directly decompose the data set into a set of groups
through the iterative optimization of a certain criterion.
More specifically, it re-estimates the cluster centres
through the minimization of a distance-related function
between the data and the cluster centres. The algorithm
terminates when the cluster centres stop changing.
The optimal number of clusters is determined using a
modified cluster validity index, CVI, [20], which is di-
rectly related to the determination of the user-defined
(here the number of clusters) parameters of the clustering
algorithm. Two indices are used for showing an estimate
of under-partitioning (Uu) and over-partitioning (Uo) of
the data set:
min
1
1
d
c
U
MD
c
U
o
c
i
iu


 (5)
MDi is the mean intra-cluster distance of the i-th clus-
ter. Here, dmin is the minimum distance between cluster
centres, which is a measure of intra-cluster separation.
The optimum number is found from the minimization of
a normalized combinatory expression of these two indi-
ces.
2.5 Hybrid Clustering Algorithm (HCA )
The hybrid clustering algorithm is an iterative procedure
that groups data, based on their distance from the hy-
per-plane that best describes their relationship. It is im-
plemented through a series of steps, which are presented
below:
1) Determine the most important variables.
2) Form the set of patterns H(t) = [yt, yt-k, xt-k].
3) Select the number of clusters nh.
4) Initialize the clustering algorithm so that nh clusters
are generated and assign patterns.
5) For each new cluster, apply a linear regression
model to yt using as explanatory variables the remaining
of the set Ht.
6) Assign each pattern to a cluster based on their dis-
tance.
7) Go to 5) unless any of the termination procedures is
reached.
The following termination procedures are considered:
a) the maximum pre-defined number of iterations is
reached and b) the process is terminated when all pat-
terns are assigned to the same cluster as in the previous
iteration in 6). The selection of the most important
lagged variables, 1), is based on the examination of the
correlation coefficients of the data.
The proposed clustering algorithm is a complete time
series analysis scheme with a dual output. The algorithm
generates clusters of data, the identical characteristic of
which is that they “belong” to the same hyper-plane, and
synchronously, estimates a linear model that describes
the relationship amongst the members of a cluster.
Therefore, a set of nh linear equations is derived (Equa-
tion (6)).
hjtjiktkiioit niXbyaay 1 ,
ˆ,,,, 


 (6)
Like any other hybrid model that uses the target vari-
ables in the development stage, the model requires a
secondary scheme to account for this lack of information
in the forecasting phase. For HCA and LMCA, the only
requirement is the determination of the cluster number,
nh and ncl respectively, which is equivalent to the estima-
tion of the final forecast.
The optimum number of HCA clusters is found from a
modified cluster validity criterion. An estimate of un-
der-partition (Uu) of the data was formed using the in-
verse of the average value of the coefficient of determi-
nation (Ri
2) on all regression models. Uo indicates the
over-partitioning of the data set, and dmin is the minimum
distance between linear models (Equation (7)). The op-
timum number is found from the minimization of a nor-
malized combinatory expression of these two indices.
min
1
2
1
1
d
c
U
R
h
U
o
h
i
i
u



 (7)

Time Series Forecasting of Hourly PM10 Using Localized Linear Models 378
2.6 Pattern Recognition
A pattern recognition scheme with three alternative ap-
proaches was then applied to convert the LMCA and
HCA output to the final predictions. Initially, a conven-
tional clustering (k-means) algorithm was employed to
identify similar historical patterns in the time series. The
second was to determine ncl / nh at each time step, using
information contained in the data of the respective cluster.
(p1) Select a second data vector using only histori-
cal observations Pt = [yt-k, xt-k]
(p2) Initialize a number of clusters nk
(p3) Apply a k-means clustering algorithm on Pt.
(p4) Assign data vectors to each cluster, so that
each of the nk clusters should contain km, m = 1,…, nk
data.
To obtain the final forecasts the following three alter-
natives were examined:
(M1) From the members of the k-th cluster find the
most frequent LMCA / HCA cluster, i.e. ncl / nh number.
(M2) From the members of the k-th cluster estimate
the final forecast as a weighted average of the LMCA/
HCA clusters. Here pi is the percentage of appearances of
the LMCA / HCA cluster in the k-th cluster data.
clh
i
ntitnnnandkiypy or ,...,1 ,...,1 , 
(8)
(M3) From the members of the k-th cluster estimate
the final forecast as a distance weighted average of the
HCA clusters.
2 and
or ,...,1 ,...,1
,







a
d
d
t
PPd
nnnandkiyty
i
a
i
a
i
i
iti
clh
i
ntit
(9)
The optimal number of clusters for the pattern recog-
nition stage was determined using the modified com-
pactness and separation criterion for the k-means algo-
rithm discussed previously in section “Local Models with
Clustering Algorithms”.
3. Data Description and Results
The previously described forecasting methodologies
were applied to eight different data sets both univariate
and multivariate. The data sets were hourly PM10 con-
centration values from the monitoring network in the
Greater Athens Area and in the cities of Helsinki and
London, spanning over different seasons. It should be
clarified that meteorological data were available only
from the Helsinki station. The results returned by the
applied algorithms for each station are discussed sepa-
rately in the following sections.
In addition to the combined LMCA / HCA – PR meth-
odology, the ideal case of a perfect knowledge of the ncl /
nh parameter is also presented. This indicates the predic-
tive potential, or the least error that the respective meth-
odology could achieve. Also, the base-case persistent
approach (yt = yt-1) is presented as a relative criterion for
model inter-comparison amongst different data sets. The
ability of the models to produce accurate forecasts was
judged against the following statistical performance met-
rics:
Root Mean Square Error




k
i
iiPO
k
RMS
1
2
1 (10)
Normalized RMS








k
i
i
k
i
ii
OO
PO
NRMS
1
2
1
2
(11)
Mean Absolute Percentage Error




k
ii
ii
O
PO
k
MAPE
1
1
100 (12)
Index of Agreement







 k
i
ii
k
i
ii
OOPO
PO
IA
1
2
1
2
)(
1 (13)
Fractional Bias
)(5.0
)(
PO
PO
FB 

 (14)
3.1 Greater Athens Area – Aristotelous Str
The selected station from the Greater Athens Area moni-
toring network was Aristotelous Str. It is located at
23°43΄39΄΄ North and 37°59΄16΄΄ West, at an elevation
height of 95 m above ground level. It is characterized as
an urban station, positioned in the city centre with traffic
dominated emissions. The training and the prediction sets
covered the periods from 1/7/2001 to 14/8/2001 and
15/8/2001 to 31/8/2001, respectively.
The analysis revealed that the most influential vari-
ables were PMt-1, PMt-2, PMt-24, PMt-25 and an indicator
for the time of the day. This data set was used for the
development of all methodologies and the input set for
the pattern recognition scheme. The results on Table 1
indicate that with the exception of NN, all other conven-
tional approaches demonstrate a reduction of the predic-
tion error by approximately 6% on the basis of the RMS
error compared to the base case persistent method. The
difference between LR and ANN was not found to be

Time Series Forecasting of Hourly PM10 Using Localized Linear Models 379
statistically significant, although the later was marginally
better under all criteria.
The application of the local linear models was able to
reduce the predictive error by an order of magnitude de-
pending on the pattern recognition scheme that was ap-
plied. Both LMCA and HCA are capable of reaching
exceedingly lower prediction error, with IA above 0.98,
if all ncl/nh clusters are predicted correctly at each time
step. Figure 2 presents a graphical description of the
prediction error of the HCA-perfect cluster forecast. The
HCA coupled with the M3 scheme returned the overall
best prediction error that was approximately 8% lower
than that of the persistent approach.
3.2 Greater Helsinki Area – Kallio
The data from the Helsinki monitoring network were
from the suburban station of Kallio, with co-ordinates
25°52΄92΄΄ W and 66°75΄47΄΄ N and elevation height of
21 m above sea level. The training set was from 3/9/2003
to 9/11/2003, whereas the unknown prediction set
spanned from 10/11/2003 to 30/11/2003.
The developed models for the prediction of PM10 val-
Table 1. Prediction results from Aristotelous
RMS NRMS MAPE d FB Nu of
clusters
Persistent 9.5596 0.3112 13.006 0.9223 –0.0002
LR 9.0193 0.277 12.6536 0.9007 0.0052
ANN 8.9311 0.2716 12.3984 0.9152 0.0037
NN 10.117 0.3485 14.3699 0.892 –0.0094 24
LCMA ncl = 4 nk = 32
Perfect 4.6355 0.0732 7.2108 0.9813 –0.0043
M1 9.6748 0.3187 13.3351 0.8999 –0.0107
M2 9.0637 0.2797 12.434 0.9121 –0.0052
M3 9.0559 0.2793 12.3804 0.9108 –0.009
HCA ncl = 8 nk = 13
Perfect 2.1522 0.0158 2.857 0.9961 –0.0002
M1 9.6085 0.3144 12.5104 0.9105 –0.0134
M2 8.8787 0.2684 12.3668 0.915 0.0048
M3 8.8153 0.2646 12.3368 0.9178 0.0046
Figure 2. HCA perfect cluster forecast for the Aristotelous station (Athens)

Time Series Forecasting of Hourly PM10 Using Localized Linear Models 380
Table 2. Linear regression model details for Helsinki 1
Variable Coef. St. Error t-stat. VariableCoef. St. Error t-stat.
c 4.7626 1.1361 4.1921 T
t-1 1.0627 0.2753 3.8601
PM t-1 0.7611 0.0247 30.8584 T
t-2 –1.0446 0.274 –3.8128
PM t-2 0.0622 0.0246 2.5319 u
t-1 –0.749 0.2213 –3.3847
PM t-24 0.0232 0.0136 1.7008 u
t-2 0.6094 0.2216 2.7493
RH t-1 0.2055 0.0547 3.7582 v
t-1 0.6673 0.2242 2.9767
RH t-2 -0.2361 0.0547 -4.317 v
t-2 –0.4508 0.2257 –1.9968
Table 3. Prediction Results from Helsinki 1
RMS NRMS MAPE d FB Nu of
clusters
Persistent 5.1208 0.2793 33.3564 0.9301 0.0001
LR 4.9654 0.2626 36.4317 0.9073 –0.0139
ANN 5.1722 0.2849 39.5785 0.9085 –0.0591
NN 5.6876 0.3446 43.8667 0.857 –0.0489 13
LCMA ncl = 3 nk = 61
Perfect 3.033 0.098 18.1484 0.9724 0.0038
M1 5.1044 0.2775 37.1176 0.9295 –0.0119
M2 4.892 0.2549 37.5676 0.9193 0.0008
M3 4.8416 0.2497 36.905 0.9229 0.0049
HCA ncl = 7 nk = 19
Perfect 1.5653 0.0261 8.9912 0.9932 –0.0051
M1 5.2179 0.29 42.6351 0.9072 0.021
M2 4.8139 0.2468 37.4036 0.9203 –0.0018
M3 4.7612 0.2415 36.7128 0.9239 –0.0006
ues from Helsinki contained meteorological parameters
that were identified using a combination of statistical
correlation properties and stepwise linear regression,
discarding all those that were judged statistically as not
significant under Student’s t-test. The finally selected
parameters and their estimation from the least squares fit
are shown on Table 2.
The prediction results (Table 3) demonstrate that the
forecasting ability of the conventional models is some-
what similar to that of the base-case persistent approach.
The large prediction error of the ANN can be partly ex-
plained by the linear nature governing process that relates
PM10 values to lagged values and from the over-fitting of
the applied training scheme. The introduction of the
LMCA and HCA localized models coupled with the M3
pattern recognition scheme returned the least overall pre-
diction error that was approximately 5.5% and 7% re-
spectively lower on the RMS criterion and double under
NRMS. Figure 3 shows the values of the prediction error
of the LMCA-M3 modelling approach.
m3)
Figure 3. Prediction and error with LCMA – M3 approach
3.3 Greater London Area – Bloomsbury
The data from the Greater London Area were from the

Time Series Forecasting of Hourly PM10 Using Localized Linear Models381
Bloomsbury station located in the city centre of London
(51°31'24" N, 0°7'54" W), characterised as an urban
background station. The training set was selected to
cover the period from 1/9/2005 to 22/10/2205, whereas
the unknown prediction set comprised data ranging from
23/10/2005 to 6/11/2005.
The stepwise regression with a threshold value for the
t-statistic of  1.96, corresponding to the 95% confidence
interval, revealed as the most significant values PMt-1,
PMt-2, PMt-24. Additionally, an indicator for the time of
the day was utilized. That data set was used for the de-
velopment of all methodologies while the input set for
the pattern recognition scheme. The analysis of the re-
sults (Table 4) indicated that none of the conventional
forecasting approaches managed to return consistently
lower prediction errors than the base case persistent ap-
proach. The least prediction error was returned from the
ANN that was 3.6% lower than the persistent approach
on the basis of the RMS error.
The developed localized linear model (HCA) has sig-
nificant forecasting potential, as it can be observed in
Figure 4, under the assumption of a perfect knowledge
of the future cluster in the pattern recognition stage. The
percentage improvement over the bench-mark persistent
approach ranged from 40-70%. Similar results were
found for the other two data sets
4. Discussion
The development and application of accurate models for
forecasting PM concentration values in a rather fast and
efficient manner is of primary concern in modern air
quality management systems. The applied LR and ANN
are nowadays mature approaches that have been inte-
grated in many operational systems and could be used for
the benchmarking of novel methodologies. The results of
this work yielded that for the majority of the examined
data sets, the linear approach marginally outperforms
ANN. This indicates that the underlying process could
possess predominantly linear characteristics.
The main focus of this work was the development and
application of novel localized linear models. These were
based on clustering algorithms as a means to identifying
Figure 4. Index of agreement for HCA – perfect cluster
forecast
Table 4. Prediction results from London Bloomsbur y
RMS NRMS MAPE d FB Remarks
Persistent 4.4165 0.272 16.4202 0.9282 –0.0007
LR 4.3119 0.2593 17.281 0.9257 0.0206
ANN 4.256 0.2526 16.9101 0.9266 0.0221
NN 5.1193 0.3655 22.466 0.8933 0.0075 14
LCMA ncl = 4 nk = 16
Perfect 4.1665 0.2421 17.0711 0.9324 0.0193
M1 4.4947 0.2817 17.9071 0.9137 –0.0136
M2 4.2704 0.2543 17.051 0.9228 0.0069
M3 4.3005 0.2579 17.9051 0.9229 0.021
HCA ncl = 7 nk = 29
Perfect 1.2401 0.0214 5.1061 0.9945 0.0064
M1 4.3246 0.2608 16.8142 0.8877 –0.0188
M2 4.2795 0.2554 16.8375 0.9163 0.0097
M3 4.2513 0.2521 16.9179 0.8812 0.0046

Time Series Forecasting of Hourly PM10 Using Localized Linear Models 382
similar properties of the time series. The LMCA identi-
fied clusters based on their proximity on the embedding
space, whereas HCA identified grouped points that were
described by the same linear model. As both approaches
included the target variable in the model development
stage, a pattern recognition scheme was needed to ac-
count for this lack of information in the prediction stage.
The final prediction model was reached with the use of
the modified CVI coupled with a pattern recognition
scheme. The results suggested M3 as the most effective
choice, because it produced consistently the least predic-
tion error, under all metrics. For the RMS and MAPE
errors, the improvement over the persistent approach
ranged from 3.5% (London) to 7.7% (Athens and Hel-
sinki). This value was almost doubled for NRMS and IA
for the respective data sets. The HCA produced the least
prediction error on every single examined data set, com-
pared both to conventional approaches and the LCMA.
5. Conclusions
This paper introduced the application of localized linear
models for forecasting hourly PM10 concentration values
using data from the monitoring networks of the cities of
Athens, Helsinki and London. The strength of this inno-
vative approach is the use of a clustering algorithm that
identifies the finer characteristics and the underlying re-
lationships between the most influential parameters of
the examined data set and subsequently, the development
of a customized linear model. The calculated clusters
incorporated the target variable in the model develop-
ment phase, which was beneficial for the development of
more coherent localized models. However, in order to
overcome this lack of information in the prediction stage
a complementary scheme was required. For the purposes
of this study, a pattern recognition scheme based on the
concept of weighted average distance (M3) was devel-
oped that consistently returned the least error under all
examined metrics. The calculated results show that the
proposed approach is capable of generating significantly
lower prediction error against conventional approaches
such as linear regression and neural networks, by at least
one order of magnitude.
6. Acknowledgements
The assistance of members of OSCAR project (funded
by EU under the contract EVK4-CT-2002-00083), for
providing the Helsinki data for this research, is gratefully
acknowledged. The Department of Atmospheric Pollu-
tion and Noise Control of the Hellenic Ministry of Envi-
ronment, Physical Planning and Public Works is also
acknowledged for the provision of the Athens data. The
data from London are from the UK air quality archive
(http://www.airquality.co.uk).
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