1.
APPLICATIONS OF GENERAL REGRESSION NEURAL NETWORKS FOR
PATH LOSS PREDICTION
Ileana POPESCU1
, Athanasios KANATAS3
, Philip CONSTANTINOU3
, Ioan NAFORNIŢǍ2
,
1
University of Oradea, Romania
Visitor Researcher at the Mobile Radiocommunications Laboratory,
National Technical University of Athens, Greece
Tel: +3010 772 3848 Fax: +3010 772 3851
Email: ipopes@cc.ece.ntua.gr
2
Department of Telecommunications
Technical University of Timisoara
3
Mobile Radiocommunications Laboratory
National Technical University of Athens
Abstract
This paper presents the results of the General Regression
Neural Networks applications for the prediction of
propagation path loss in a specific urban environment. We
have studied two neural network models; the first one is
used for path loss prediction while the second one is a
prediction model using error control. The performances of
the neural models are compared to the path loss values
measured in the city of Kavala, Greece, based on the
absolute mean error, standard deviation and root mean
square error between predicted and measured values.
Keywords: Neural Networks, Propagation loss models,
Channel Characterization
I. INTRODUCTION
The prediction of propagation path loss is a difficult
and complex task. The following classes of prediction
models have been identified: empirical, semi
deterministic and deterministic.
An alternative approach to a path loss prediction
model is based on the neural networks. The advantage of
using neural networks for field strength prediction is
given by the flexibility to adapt to arbitrary environments,
high speed processing and the ability to process a high
quantity of data.
Many authors have shown that neural networks
provide a good way of approximating functions using
neural networks [1] – [3].
The application of neural networks discussed in this
paper is considered as a function approximation problem
consisting of a nonlinear mapping from a set of input
variables containing information about potential receiver
onto a single output variable representing the error
between measured path loss and the path loss obtained by
applying the modified COST231WalfischIkegami
model (CWI) [4].
II. THE MEASUREMENTS
Field strength measurements used to design and test
the models were performed at 1890 MHz band in the city
of Kavala (Greece). A detailed description of the
measurement setup and procedure can be found in [5].
The fast fluctuations effects were eliminated by
averaging the measured received power over a distance of
6 m, that corresponds to approximately 40λ sliding
window. After converting the values from received power
to path loss versus distance, we compare the measured
path loss with the predicted path loss by the neural
network models and the empirical models, based on the
absolute mean error (µ), standard deviation (Std) and root
mean square error (RMS). The above statistical
parameters are used to investigate the suitability of neural
network models to describe the sitespecific environment.
Following the filtering process of the measured data
we have obtained more than 5000 measurement locations,
corresponding to the lineofsight (LOS) and nonlineof
sight (NLOS) cases.
The designed neural models and the examined
empirical models require parameters that describe the
propagation environment such as the street width, the roof
2.
top height and the building block spacing. Average values
were used since these variables change continuously
along one route. For the determination of these geometric
parameters a map with building database was used [5].
III. THE GRNN ARCHITECTURE
The General Regression Neural Network (GRNN) is a
neural network architecture that can solve any function
approximation problem. The learning process is
equivalent to finding a surface in a multidimensional
space that provides a best fit to the training data, with the
criterion for the “best fit” being measured in some
statistical sense. The generalization is equivalent to the
use of this multidimensional surface to interpolate the test
data.
1
k
K
i
x1
x2
φk
(x)
xM
w1
wk
wK
Input layer Hidden layer Output layer
φ1
(x)
xm
φK
(x)
y
Figure 1.General regression neural network
Figure 1 is the overall network topology implementing
the GRNN. As it can be seen from the figure, the GRNN
consists of three layers of nodes with entirely different
roles:
The input layer, where the inputs are applied,
The hidden layer, where a nonlinear
transformation is applied on the data from the
input space to the hidden space; in most
applications the hidden space is of high
dimensionality.
The linear output layer, where the outputs are
produced.
The most popular choice for the function ϕ is a
multivariate Gaussian function with an appropriate mean
and autocovariance matrix.
The outputs of the hidden layer units are of the form
[ ] ( ) ( ) ( )
σ−−−=ϕ 2x
k
Tx
kk 2exp vxvxx (1)
when x
kv are the corresponding clusters for the inputs and
y
kv are the corresponding clusters for the outputs
obtained by applying a clustering technique of the
input/output data that produces K cluster centers [6].
y
kv is defined as
( )
( )
∑
∈
=
kclusterpy
y
k pyv (2)
Nk is the number of input data in the cluster center k, and
( ) ( ) ( )x
k
Tx
k
x
k,d vxvxvx −−= (3)
with
( )
( )
∑
∈
=
kclusterp
x
k p
x
xv (4)
The outputs of the hidden layer nodes are multiplied
with appropriate interconnection weights to produce the
output of the GRNN. The weight for the hidden node k
(i.e., wk) is equal to
( )∑
=
σ
−
=
K
1k
2
2x
k
k
y
k
k
2
,d
expN
v
w
vx
(5)
The selection of an adequate set of training examples
is very important in order to achieve good generalization
properties. The set of all available data is separated in two
disjoint sets: training set and test set. The test set is not
involved in the learning phase of the networks and it is
used to evaluate the performances of the models [7].
The configuration of the neural network model is
determined by the nature of the problem to be solved. The
dimension of the input vector used defines the number of
inputs neurons.
IV. PREDICTION MODEL
For the LOS case, the proposed neural network model
is trained with physical data, which includes the distance
between transmitter and receiver, the width of the streets,
the height of the buildings, the building separation and the
distance between base station and rooftop height. The
GRNN model has a single output which represents the
normalized propagation path loss. The dimension of the
training set is 1013 and the rest of 2026 examples were
used to test the model and to compare it with the
WalfischBertoni model (WB) [8], the single slope model
(SSM) [9] and the modified COST231WalfischIkegami
model (CWI) [4]. Table 1 represents the performance
achieved by each of the abovementioned models for the
entire test data.
Between the empirical algorithms, in LOS case, the
single regression model achieves the best performance.
However, this model is based on the distance between
transmitter and receiver, the frequency and the
propagation factor. It was found that in the LOS paths the
power decay factor ranges from minimum 1.56 up to 3.05.
Table 1.Comparison between the NN approach and the other
empirical models in LOS case
µ [dB] Std [dB] RMS [dB]
GRNN 5.04 4.54 6.78
SSM 5.43 4.80 7.24
CWI 7.04 4.06 8.23
WB 9.09 4.51 10.24
The better performance achieved by the neural
network model is due to the various inputs parameters
used to predict the propagation path loss and due to the
generalization properties of the network.
3.
Figure 2 represents the measured and predicted
propagation path loss by the generalized RBFNN model,
the SSM model and the CWI model for a specific route, in
LOS case.
115
105
95
85
75
65
55
13 21 30 39 48 57 67 76 86 95 105 114 124 133 143 152 155 159 164 169
Distance from transmitter [m]
Propagationpathloss[dB]
Measurements RBFNN CWI SSM
Figure 2.Measured and predicted path loss for LOS case, urban
environment
For the NLOS case, we have built three neural
network models:
1. The first one, called NN1, is trained with four
parameters: the distance between transmitter and
receiver (d), the width of the street (w), the
building separation (b) and the building height
(h).
2. The second model, called NN2, in addition to the
abovementioned data we also included the
distance between base station height and
building height (dhbs).
3. In the third neural model, named NN3, we have
included the street orientation (ψ) in addition to
the parameters used for the training of NN2.
A set of 420 examples was used for training purpose
while the rest of 1680 was used for test purpose.
Table 2 presents the results obtained by the three
different GRNN models, for the training set and for the
test set. R represents the correlation between predicted
values and the measurement data.
Table 2. GRNN models for NLOS case
Training set Test set
[dB] µ Std RMS µ Std RMS R
NN1 4.35 4.64 6.36 5.50 5.27 7.62 0.91
NN2 4.11 4.38 6.00 5.31 5.07 7.34 0.92
NN3 1.47 2.52 2.92 3.67 3.88 5.35 0.96
Table 3 Comparison between the proposed NN3 model and the
other empirical models in NLOS case
µ [dB] Std [dB] RMS [dB]
NN3 3.67 3.88 5.35
SSM 6.35 4.37 7.75
WB 6.08 4.14 7.40
CWI 6.96 4.62 8.38
Table 3 represents the comparison of the performance
achieved by the NN3 model and the Single Slope model
(SSM) [9], the WalfischBertoni model [8] and the
modified COST231WalfischIkegami model [4] for the
entire test patterns.
Figure 3 represents the measured and predicted
propagation path loss by the GRNN model, CWI model
and measurements for one route characterized by a base
station antenna located below rooftop
140
130
120
110
100
90
80
70
60
50
19 29 40 54 71 89 108 127 145 164 183 202 221 240 259 278 297 316
Distance [m]Pathloss[dB]
Measurements Neural Model CWI Model
Figure 3. The comparison between the measured and predicted
path loss, in case of a particular route, NLOS case, urban
environment
V. ERROR CORRECTION MODEL
Our purpose is to build an error correction model: the
neural network is used to compensate for the errors
obtained by applying COSTWalfischIkegami (CWI)
model [4]. In Figure 4 is represented the training phase of
the neural network structure and in Figure 5 is represented
the schematic diagram of the prediction phase. The inputs
of the neural network consist of the available physical
parameters and the output is represented by the difference
error between the path loss computed by CWI model and
measurements.
Physical data
Computed PL
Measured PL
Neural Network
Training Algorithm
Neural Network
PLcomputed
PLmeasured
E

+
Epredicted
PLcomputed
PLcorrected = PLcomputed  Epredicted
+

Figure 4. The schematic diagram of the training process of the
neural network
Since our purpose is to train the neural network to
perform well for all the routes, we should build the
training set including points from the entire set of
measurements data. For training and test purpose we have
4.
used the same number of patterns as in the prediction
model. The test set was used to test the models and to
compare them to each other. Also, a comparison between
the “best” neural model, measurements values and the
CWI model is presented.
Physical data
Neural Network
PLcomputed
Epredicted

PL = PLcomputed  Epredicted
+

COSTWalfisch
Ikegami Model
Figure 5. The schematic diagram of the prediction
We have studied the following GRNN models:
NN4 with four inputs: distance between
transmitter and receiver (d), the width of the
street (w), the height of the building (h) and the
building separation (b).
NN5 with 5 inputs: distance between transmitter
and receiver (d), the width of the street (w), the
height of the building (h), the building separation
(b) and the distance between base station height
and the building height (dhbs).
NN6 with 5 inputs: distance between transmitter
and receiver (d), the width of the street (w), the
height of the building (h), the building separation
(b) and the street orientation (ψ).
NN7 with 6 inputs: distance between transmitter
and receiver (d), the width of the street (w), the
height of the building (h), the building separation
(b), the difference between base station height
and building height (dhbs), and the street
orientation (ψ) and the difference between base
station height.
Table 4.Hybrid neural network models
Training set Test set
[dB] µ Std RMS µ Std RMS R
NN4 4.79 5.50 7.30 6.12 6.13 8.66 0.77
NN5 4.35 4.74 6.44 5.68 5.59 7.97 0.81
NN6 1.57 2.66 3.10 3.79 4.07 5.57 0.91
NN7 1.47 2.49 2.90 3.65 3.85 5.30 0.92
The comparison between the results obtained by the
GRNN models for the training and test patterns are
presented in Table 4.
Table 5 represents the comparison between the hybrid
model NN7 and CWI model for the entire test set.
Table 5. Comparison between the proposed neural model and
the CWI model
[dB] µ Std RMS
NN7 3.65 3.85 5.30
CWI 6.97 4.62 8.38
140
130
120
110
100
90
80
70
60
50
19 29 40 55 71 89 108 127 145 164 183 202 221 240 259 278 297 316
Distance [m]
Pathloss[dB]
Measurements Neural Model CWI Model
Figure 6. Comparison between the prediction made by the
hybrid model, CWI model and measurements, in case of a
particular route, urban environment
VI. CONCLUSIONS
In this paper we have developed applications of the
General Regression Neural Networks for the prediction of
propagation path loss and we have compared them with
measurements and with the prediction made by different
empirical models. It is noticed a significant improvement
in the prediction made by neural models due to their
generalization property. Another advantage of the use of
neural networks is the fact that they are trained with
measurements, so the included propagation effects are
more realistic.
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