Comparison of neural classifiers for vehicles gear estimation

Applied Soft Computing 11 (2011) 3580–3599

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Comparison of neural classifiers for vehicles gear estimation A. Wefky a , F. Espinosa a,∗ , A. Prieto b , J.J. Garcia a , C. Barrios c a

Department of Electronics, University of Alcala, Spain Automatic Department, Electrical Engineering Faculty, La Habana, Cuba c CIEMAT, Madrid, Spain b

a r t i c l e

i n f o

Article history: Received 7 July 2010 Received in revised form 19 November 2010 Accepted 7 January 2011 Available online 22 January 2011 Keywords: ANN classifier Vehicle gear estimation Multi-layer perceptron Radial basis function Linear vector quantization Probabilistic neural network

a b s t r a c t Nearly all mechanical systems involve rotating machinery with gearboxes used to transmit power or/and change speed. The gear position is an indication of the driver’s behavior and it is also dependent on road conditions. That is why it presents an interesting problem to estimate its value from easily measurable variables. Concerning individual vehicles, there is a specific relationship between the size of the tires, vehicle speed, regime engine, and the overall gear ratio. Moreover, there are specific ranges for vehicle speed and regime engine for each gear. This paper evaluates the use of neural network classifiers to estimate the gear position in terms of two variables: vehicle velocity and regime engine. Numerous experiments were made using three different commercial vehicles in the streets of Madrid City. A comparative analysis of the classification efficiencies of different neural classifiers such as: multilayer perceptron, radial basis function, probabilistic neural network, and linear vector quantization, is presented. The best results in terms of classification efficiency were obtained using multilayer perceptron neural network (92.7%, 91.5%, and 85.9% for a Peugeot 205, a Seat Alhambra, and a Renault Laguna respectively). The maximum likelihood classifier is used as a benchmark to compare with the neural classifiers. © 2011 Elsevier B.V. All rights reserved.

1. Introduction An internal combustion engine uses a multigear transmission to transmit its power to the drive wheels. There are two basic types of automotive transmission systems: manual gear transmission and hydrodynamic transmission. Manual gear transmission consists of a clutch, gearbox, final drive, and drive shaft. The gearbox provides a number of gear ratios ranging from three to five for passenger cars [1]. The Electronics Department at the University of Alcala (UAH), in collaboration with the Thermal Engines Group of the ETSIIUPM in Madrid and the Research Center for Environmental Energy and Technology (CIEMAT) in Madrid are developing the MIVECO project (Methodology and instrumentation on board the vehicles to assess real effects of traffic pollutant emissions). This research project, in which the authors are involved, is focused on the measurement of gases, particles and electromagnetic pollutant emissions of vehicles as a result of the driver activity, vehicle state, and road conditions. Consequently, the knowledge of the gear value and its relationship with the emissions measured is required. However, in most engine control systems it is difficult to find an easy sensor for monitoring the transmission

∗ Corresponding author. Tel.: +34 91 885 6545; fax: +34 91 885 6591. E-mail address: [email protected] (F. Espinosa). 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.01.030

gear selector position for a variety of different types of vehicles. Eq. (1) shows the formula relating the overall gear ratio (gro ), the size of the tires (Ct ), the vehicle speed (VV ), and the regime engine (RE ) [2]. The size of the tires is a constant depending on the type of the vehicle. The overall gear ratio includes a range of numbers for each gear. Consequently, the vehicle speed is linearly proportional to the regime engine for a specific gear. gro = Ct ∗

RE VV

(1)

Fig. 1 illustrates that linear relationship and shows that given the vehicle speed and the regime engine, the corresponding gear can easily be deduced. It shows that the areas defining each gear can be linearly separated [1]. However, unfortunately the situation in reality is different. According to many experiments done on different types of cars, the linear boundaries between different gear zones cannot be clearly distinguished. In other words, practically there are not specific clear values for VC1 , VC2 , and VC3 shown in Fig. 1. In many science branches, analysts are often faced with the task of classifying items based on measured data. A major difficulty faced by an analyst is that the data to be classified can often be quite complex, with numerous interrelated variables. Therefore, the time and effort required to develop a model to solve such classification problems can be significant. Neural networks are a proven

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Fig. 2. Block diagram of the proposed methodology.

Fig. 1. Vehicle speed versus regime engine for each gear.

and widely used technique to solve such complex classification problems [3–8]. Artificial neural networks (ANNs) are mainly geared to handle two broad categories of tasks, function approximation (regression) and pattern recognition (classification) [4]. In the former, the task is to learn a specific function of arbitrarily dependent and independent variables from a set of input-target pairs and then interpolate for unknown dependent variables. In the latter, the task is to learn to recognise several distinct generalisations of specifically presented patterns and then to use these learned categories to classify data from a set of unknown patterns. ANNs consist of simple parallel elements inspired by biological nervous system. The connection between the elements determines the network function. Nowadays there is a broad range of applications of ANNs in diverse fields, not just in engineering, science, and mathematics, but also in medicine, business, finance, aerospace, defense, robotics, telecommunications, transportation, etc. due to their computational speed, their ability to handle complex nonlinear functions and their robustness and great efficiency, even in cases where full information for the studied problems is absent. ANNs can be trained to perform different tasks such as pattern recognition, function fitting, classification, identification, control, and data clustering [2,8]. Neural networks have been commonly used for classification problems. For example, a Levenberg–Marquardt Multi-layer Perceptron (MLP) neural network was used for pattern classification [9]. Local expert neural classifiers have been used to enhance classification performance [10]. By combining the two concepts, the visual neural classifier provides excellent classification accuracy [11]. Besides, Multi-layer Perceptron, radial basis function, and networks based on Predictive Adaptive Resonance Theory (ARTMAP) have been utilized in pattern classification problems in remote sensing applications [12,13]. ANNs have also been exploited by the GIS community as an alternative tool for classification and feature extraction [14]. Moreover a Probabilistic Neural Network (PNN) classifier has been employed to evaluate seismic liquefaction potential [15]. In addition the Linear Vector Quantization network LVQ has already been exploited to develop an automotive occupant classification system (OCS) [16]. The pattern classification and recognition capabilities of ANNs have already been exploited in the field of automotive electronics. For example, the effectiveness of wavelet-based features for fault diagnosis of a gear box using artificial neural network and Proximal Support Vector Machines (PSVM) has been already investigated [17]. Moreover, a new on-line Fault Detection and Isolation (FDI) scheme for engines based on Radial Basis Function (RBF) neural networks was evaluated for a wide range of operational modes

[18]. Moreover, a comparative analysis between MLP, feature map, RBF, and LVQ neural networks for the classification of faults in electrical systems has been developed [19]. A diagnostic tool for faults in engine intake systems was designed based on MLP neural classifier using normalized process variables [20]. An auto-associative neural network was used to build a diagnostic model to distinguish between sensor and process faults [21]. The advantage of the neural classifiers as compared to traditional techniques is the easy adaptation to different types of data and the use of its complex configuration to find the best nonlinear function relating the input and output data [22]. Moreover, they do not depend on the statistical aspects of the data set, and they can be used with ordinal and nominal data types together, which can be trained with relatively few training samples as it is not necessary to select a data distribution model, unlike techniques such as Maximum Likelihood Classifiers (MLC). However, the price to pay to gain these merits is the lengthy training phase [23]. This paper proposes neural network models to estimate the gear position based on the regime engine and the corresponding vehicle speed. The models have been trained, validated, and tested with experimental tests using three commercial vehicles: a Peugeot 205, a Seat Alhambra, and a Renault Laguna. This paper is arranged as follows. Section 2 shows the methodology followed for data collection and the visualization of the measured data from sensors. Section 3 illustrates the different types of neural network classifiers chosen in this work. Section 4 describes the results obtained from testing the proposed classifier models. Finally, conclusions are included in Section 5. 2. Methodology and experimentation The methodology followed is to adjust the estimated value of the gear position (GE ) as determined by the neural classifier under study in terms of the agreement degree (minimum error εG ) with the actual gear position (GA ) acquired during the experiments with different drivers and commercial vehicles (see Fig. 2). As stated earlier, the other variables of the ANN classifier are RE and VV. Among the included sensors to register the vehicle state are the contact and sensor-less automotive RPM measurement sensor RPM-8000 from KMT shown in the left side of Fig. 3 [24], and the ultra high sensitive GPS receiver HI204III-USB, shown in the right side of Fig. 3, utilized for measuring the vehicle speed as well as the instantaneous global position of the vehicle every second [25]. The regime engine and vehicle speed of the Peugeot 205, shown in Fig. 4, are measured in different zones in Madrid City with different driving conditions. The instantaneous values of the gear ratio were calculated using Eq. (1). Consequently the corresponding gear positions were deduced using a specific range of the overall gear ratio information for each gear as illustrated in Table 1. The class distributions of the datasets used in this work are illustrated in Table 2 where each dataset has been divided randomly to training (70%) and testing (30%) subsets. Fig. 5 illustrates the feature space or the normalized regime engine versus the normalized vehicle speed showing the different gear data with different markers and colours. It shows the manually made linear boundaries as well

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Fig. 3. GPS (right) and RPM sensor (left) used in the MIVECO project.

Table 1 Ranges of gro with the corresponding gear for P205. Range of the overall gear ratio (gro ) From

Gear To

20 1,187,628 674,544 487,968 383,916 312,156

– 20 1,187,628 674,544 487,968 383,916

Neutral First Second Third Fourth Fifth

Fig. 4. Regime engine and vehicle speed of the Peugeot 205.

as their equations separating the different gear zones. This figure clearly supports the proposal of a classifier to deduce the gear from the vehicle speed and regime engine. The regime engine and vehicle speed of the Seat Alhambra and Renault Laguna are shown in Figs. 6 and 7. Both of them are measured, as part of the tasks included in the MIVECO project, in real experiments in Madrid City. The corresponding position of the gear is recorded manually employing the software GUI designed for the project. In Figs. 8 and 9 the normalized engine speed is plotted versus the normalized vehicle speed of both the Seat Alhambra and Renault Laguna respectively. Different colours and markers are used to easily distinguish between the different gear positions.

Fig. 5. Normalized regime engine versus vehicle speed of the Peugeot 205.

Table 2 Class distributions of the 3 databases used in this work. Neutral

Seat Peugeot Renault a b

Training. Testing.

First

Tra

Tsb

Tra

2361 1871 3962

1523 1264 2593

1657 205 1453

Second Tsb 1090 152 962

Third

Fourth

Fifth

Tra

Tsb

Tra

Tsb

Tra

Tsb

Tra

Tsb

2762 770 2596

1891 453 1756

799 1328 1759

548 943 1188

925 821

588 520

1703

1176

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Fig. 9. Normalized regime engine versus vehicle speed of the Renault Laguna. Fig. 6. Regime engine and vehicle speed of the Seat Alhambra.

It can be concluded from the feature space of the available three data sets that the most prepared dataset for classification results is that of the Peugeot 205. On the contrary, the datasets of the Seat Alhambra and Renault Laguna are relatively noisy. This means that the classification results of the classifiers tested with the testing dataset of the Peugeot 205 would give the best results. 3. Neural classifiers

Fig. 7. Regime engine and vehicle speed of the Renault Laguna.

Fig. 8. Normalized regime engine versus vehicle speed of the Seat Alhambra.

There are two broad categories of neural classifiers, supervised and unsupervised. This is similar to the statistical pattern recognition techniques, where such methodologies like k-means clustering and maximum likelihood classifiers are examples of unsupervised and supervised classifiers respectively. An unsupervised classifier is completely free and separates a given feature space into classes based on the similarities distinguished between the data samples. Therefore, the user cannot specify how to classify the data or how many classes exist in the feature space. On the other hand, a supervised classifier learns by examples provided in the training data set. Consequently, the user has full control of the classifier functionality. This paper has focused on supervised neural classifiers only because the output classes are well defined and previously known. According to the state of the art of neural classification techniques, the neural classifiers used in this paper include two types of the multilayer perceptron (MLP) network, radial basis function network (RBF), linear vector quantization (LVQ), and probabilistic neural networks (PNN). The architectures of those networks are described in the following section. As it is illustrated in Fig. 2, the

Fig. 10. Multilayer perceptron network with hard-limit transfer functions.

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Fig. 11. Example illustrating the design methodology of the MLP1 network.

inputs to each type of neural classifiers are the regime engine and vehicle speed; and the output is the gear. 3.1. Multilayer perceptron (MLP) network MLP is a fully connected feed-forward network with hidden layer(s) of nodes between inputs and outputs. It is probably the most widely used architecture for practical applications. The specification of the transfer (activation) function is a critical design issue in the application of MLP neural networks in classification. Two types of transfer functions have been used in the present study. The first MLP network, called MLP1, uses a hard-limit transfer function. The other MLP network, called MLP2, uses a hyperbolic tangent transfer function. 3.1.1. MLP1 The first mathematical model of a neuron, in which a weighted sum of input signals is compared to a threshold to determine whether or not the neuron is activated, was presented by McCulloch and Pitts [26]. The topology of the MLP1 network is shown in Fig. 10 where it consists of one hidden and one output layers. The transfer function of both layers is a hard-limit function. Due to its simplicity and straightforward design process, the MLP1 network has been adopted in this work to accomplish the classification process. A detailed description of the design strategy of such network is included in Ref. [27].

The network design methodology is as follows. Because of the hard-limit transfer function, the MLP1 network is able to separate the different gear regions using only linear decision boundaries. The number of the decision boundaries equals that of the hidden neurons. That is the rule used to choose the number of hidden neurons. In addition, the number of output nodes equals the number of the gear classes. In this way, the hidden layer is responsible for constituting the decision boundaries, while the output layer is used to combine the decision boundaries together. Finally, the network weights and biases have been estimated using hand calculations. Each linear decision boundary shown in Figs. 4, 7 and 8 has a linear algebraic equation relating the normalized regime engine and vehicle speed. The constants of this equation determine the network weights and biases [8,27]. Fig. 11 shows a simple straightforward example of the design methodology followed with the MLP1 network. It shows a feature space consisting of two variables p1 and p2. Moreover, it shows that the problem includes 3 linearly separable classes. Two decision boundaries are needed in order to distinguish between the three classes. Consequently, the MLP1 architecture for this example consists of 2 hidden neurons (number of boundaries) and 3 output neurons (number of classes) as shown in the topology of the network in Fig. 11 (right side). In order to estimate the weights and biases of the hidden and output layers, the design strategy consists of two phases. Firstly the weights and biases of the hidden layer are calculated, and then those of the output layer are determined. In order to calculate the weights and biases of the hidden layer, the equations of the decision boundaries are obtained as shown in Fig. 11 given any two points on each boundary. The decision boundary between the categories is determined by the equation, Wp + b = 0

(2)

where W is the weight matrix of the hidden layer, p is the input vector, and b is the bias vector. Eq. (2) can be expanded for the hidden layer of our example as follows: 1 1 p1 + w12 w11 p2 + b11 = 0

1 1 w21 p1 + w22 p2 + b12 = 0

where the superscripts denote the layer index and the subscripts denote the start and end points of the weights. Comparing these equations with those of the decision boundaries shown in Fig. 11 (left side) yields, Fig. 12. Multilayer perceptron network with hyperbolic tangent transfer functions.

1 1 1 1 = 4, w12 = −1, b1 = −4, w21 = 1, w22 = −1, b2 = −2 w11

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Table 3 RMSE of different training cycles with MLP2 for different vehicles to account for initial conditions. RMSE of different training cycles of MLP2 Peugeot Seat Laguna

0.501 0.466 0.354

0.183 0.664 0.172

0.360 0.231 0.175

0.432 0.313 0.198

In this way, the weights and biases of the hidden layer are already determined. The second phase is to calculate the weights and biases of the output layer. If the input vector p lies in the region of class 1, the output vector y should be (1, 0, 0), and the two hidden neurons would give zeros because the input vector lies at the left of both of the boundaries. In this way, b21 > 0, b22 < 0 and b23 < 0. Therefore we can choose the following values, b21 = 1, b22 = −1 and b23 = −1. Similarly, if the input vector p lies in the region of class 2, the output vector y should be (0, 1, 0), and the upper hidden neuron would give one whereas the lower one would give zero because the input vector lies at the right of the first boundary 2 + b2 < and at the left of the other boundary. In this way, w11 1 2 2 2 2 0, w21 + b2 > 0 and w31 + b3 < 0. Given the previous results, we 2 = −2, w 2 = 2 and w 2 = 0. can obtain, w11 21 31 In the same way, if the input vector p lies in the region of class 3, the output vector y should be (0, 0, 1), and the hidden neurons would give one because the input vector lies at the 2 + w 2 + b2 < 0, w 2 + right of both boundaries. In this way, w11 12 1 21 2 + b2 < 0 and w 2 + w 2 + b2 > 0. Given the previous results, w22 2 31 32 3 2 = 0, w 2 = −2 and w 2 = 2. In this manner all we can obtain, w12 22 32 the parameters of the network have been already determined. In this work, the linear decision boundaries have been developed manually. Then the equation of each boundary is determined. Consequently, the weights and biases of the corresponding perceptron neuron can be calculated. In this way, the network design phase is fast, simple, and reliable relative to the other methods. 3.1.2. MLP2 The architecture of the MLP2 network is shown in Fig. 12. The MLP2 network trained with Levenberg–Marquardt algorithm has been chosen to analyze the data because it has many properties useful for the gear estimation problem such that it can efficiently learn large data sets and develop nonlinear decision boundaries. To obtain a good generalization, the entire data set has been divided into training (70%) and testing (30%) groups.

Fig. 13. Structure of a radial basis function network.

0.295 0.224 0.282

0.480 0.468 0.226

0.361 0.587 0.279

0.134 0.209 0.472

0.318 0.228 0.282

0.224 0.316 0.284

The MLP2 network parameters have been chosen as follows, the transfer function of the hidden and output layers of the MLP network is hyperbolic tangent, the number of output neurons equals the number of classes, and the number of hidden neurons equals the number of boundaries required to separate the classes, where each hidden neuron along with its associated weight connections to the inputs represents a hyperplane in the feature space. The position as well as the orientation of a particular separating boundary varies by changing the values of the associated weights and biases going to the corresponding hidden node during training [23]. The Levenberg–Marquardt algorithm updates the network weights and biases (x vector) using the formula described in Eq. (3), where J is the Jacobian matrix containing first derivatives of the network errors with respect to the weights and biases. Vector e represents the network errors, and  is a damping parameter. The training phase using the Levenberg–Marquardt algorithm by means of Matlab has been repeated ten times in order to account for the

Fig. 14. Architecture of a probability density function network.

Fig. 15. Topology of a learning vector quantization network.

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Fig. 16. (a) Decision boundaries of the MLP2 network with Peugeot 205. (b) Classification regions of the MLP2 network with Peugeot 205.

Fig. 17. (a) Decision boundaries of the RBF network with Peugeot 205. (b) Classification regions of the RBF network with Peugeot 205.

effects of initial conditions as shown in Table 3 [27]. The columns of this table show the testing root mean squared error (RMSE) of different trained MLP2 models, whereas the rows describe different vehicles. The highlighted elements of Table 3 indicate the minimum RMSE values. The variances of the RMSE are 0.0152, 0.0272, and 0.0082 for Peugeot 205, Seat Alhambra, and Renault Laguna respectively. This analysis shows that the initial conditions really affect the performance of the neural classifier as usual with the backpropagation algorithm.

by Eq. (4), between the real gear values and the model estimated ones. Where O is the vector of observed (real) values, P is the vector of model-estimated (predicted) values, and N is the number of samples in the testing subset.

xk+1 = xk − [J T J + I]

−1 T

J e

(3)

The early stopping criterion has been used for the Levenberg–Marquardt algorithm as follows. After each training iteration, the network is tested for its generalization performance. The training process stops when the generalization performance reaches the maximum on the test data set. In other words, the network is trained iteratively based on the number of training epochs. Each training epoch decides the value of synaptic weight and bias of the network. Thereafter, the trained network is tested on the testing data set, which gives the prediction error. If the prediction error exceeds the goal, the network is again trained with increased number of epochs and so the process is repeated. The best backpropagation MLP model has been chosen based on the minimum generalization (testing) RMSE error, calculated

  N 1 (Pi − Oi )2 RMSE =  N

(4)

i=1

3.2. Radial basis function (RBF) network The RBF network is the main practical alternative to the multilayer perceptron for various applications. It belongs to the most recent networks. It can be used for approximating functions as well as recognizing patterns. Radial basis networks can require more neurons than multilayer perceptron networks, but often they can be designed in a fraction of the time it takes to train multilayer perceptron networks. They work best when many training vectors are available. The RBF network is a two-layer feed forward network where each hidden unit implements a radial transfer function. The output nodes implement a weighted sum of the hidden unit outputs. However the net input of the hidden nodes, i.e. radial basis neurons, is the vector distance between its weight vector W and the

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Table 4 Confusion matrix of the MLC for Peugeot 205. Real gear

Estimated gear Na 1st 2nd 3rd 4th a

Na

1st

2nd

1169 (92.5%) 41 54 0 0

0 127 (83.6%) 25 0 0

0 14 368 (83.7) 71 0

3rd

4th

0 0 162 754 (80%) 27

0 0 14 0 574 (97.6%)

Neutral.

Table 5 Confusion matrix of the MLP1 classifier for Peugeot 205. Real gear

Estimated gear N 1st 2nd 3rd 4th

N

1st

2nd

3rd

4th

1246 (98.6%) 18 0 0 0

8 129 (84.9%) 15 0 0

0 12 327 (72.2%) 114 0

0 0 14 912 (96.7%) 17

0 0 0 7 581 (98.8%)

N

1st

2nd

3rd

4th

1246 (98.6%) 18 0 0 0

5 128 (84.2%) 19 0 0

0 0 32 891 (94.5%) 20

0 0 0 2 586 (99.7%)

Table 6 Confusion matrix of the MLP2 classifier for Peugeot 205. Real gear

Estimated gear N 1st 2nd 3rd 4th

0 7 385 (85%) 61 0

Table 7 Confusion matrix of the RBF classifier for Peugeot 205. Real gear

Estimated gear N 1st 2nd 3rd 4th

N

1st

2nd

3rd

4th

1239(98%) 21 0 0 4

14 122 (80.3%) 16 0 0

1 12 337 (74.4%) 103 0

0 0 2 912 (96.7%) 29

0 0 0 5 583 (99.1%)

Table 8 Confusion matrix of the PNN classifier for Peugeot 205. Real gear

Estimated gear N 1st 2nd 3rd 4th

N

1st

2nd

3rd

4th

1176(93%) 77 11 0 0

0 137 (90.1%) 15 0 0

0 169 257 (56.7%) 27 0

0 186 101 635 (67.3%) 21

0 32 0 0 556 (94.6%)

Table 9 Confusion matrix of the LVQ classifier for Peugeot 205. Real gear

Estimated gear N 1st 2nd 3rd 4th

N

1st

2nd

3rd

4th

1214 0 50 0 0

11 0 141 0 0

31 0 294 128 0

5 0 20 898 20

0 0 0 52 536

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Fig. 18. (a) Decision boundaries of the PNN network with Peugeot 205. (b) Classification regions of the PNN network with Peugeot 205.

Fig. 19. (a) Decision boundaries of the LVQ network with Peugeot 205. (b) Classification regions of the LVQ network with Peugeot 205.

input vector P, multiplied by the bias b, as shown in Eq. (5), where a is the net input to the radial basis transfer function and * denotes the dot product. Each neuron in the radial basis layer will give an output value that depends on how close the input vector is to each of the weight vectors of the given neurons. In other words, the RBF neurons whose weight vectors are a bit different from input vector will have an output of about one. On the contrary, the RBF neuron whose weight vector is close to the input vector will have a value of about zero.

Table 10 Classification efficiencies of different neural classifiers for P205.

2

˛ = e−((W ∗p),b)

(5)

Its main advantage is the speed of learning, where the training procedure consists of two stages. In the first stage, the parameters of the basis function are set to model the unconditional data density. Then, in the second stage, the output layer weights are determined by solving a quadratic optimization problem [28]. In order to use an RBF network, the following parameters should be determined: the hidden layer transfer function, the number of hidden units, the neurons width (spread), and the training algorithm for finding the network parameters. The way in which the network is used for data modelling is different when approximating time-series and using pattern classification. In the first case, the network inputs represent the data samples at certain past time instants, while the network has only one output representing a

Neural classifier type

Classification efficiency (ANR %)

MLC MLP1 MLP2 RBF PNN LVQ

86.97 90.23 92.4 89.7 80.36 69.47

signal value. In a pattern classification application, the inputs represent feature entries (regime engine and vehicle speed in this paper), while each output corresponds to a class. In this case, the hidden units correspond to clusters (subclasses) [29]. Different activation functions have been tested for the RBF network. In pattern classification applications, the Gaussian function is preferred [29,30]. In time series modelling, the most used activation function is the thin-plate spline [31]. In this paper, the RBF network is created, trained, and tested using the neural networks toolbox of Matlab. The radial basis function has been chosen as the kernel function of the hidden units while a linear transfer function has been used for the output layer as shown in Fig. 13. The number of the output neurons equals that of the gear classes. The network has been designed using the built-

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Fig. 20. (a) Decision boundaries of the MLP2 network with Seat Alhambra. (b) Classification regions of the MLP2 network with Seat Alhambra.

in function “newrb” that iteratively creates one neuron at a time. The process of adding hidden neurons continues until the mean squared error defined in Eq. (6), where O is the vector of observed (real) values, P is the vector of model-estimated (predicted) values, and N is the number of samples in the training subset, falls below an error goal or a maximum number of neurons have been reached. The training and testing processes have been repeated with various combinations of neuron width (spread) and number of hidden neurons. The classification is realized on the maximum membership principle. In other words, an input vector belongs to the class with the maximum output neuron. 1 (Pi − Oi )2 N N

MSE =

(6)

i=1

3.3. Probabilistic neural network (PNN) The probabilistic neural network is one of the most widely used neural networks for classification purposes. It is based on the Bayesian classification theory and the estimation of probability density functions. It is actually a parallel implementation of a standard Bayesian classifier [32]. It is a feed-forward net-

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Fig. 21. (a) Decision boundaries of the RBF network with Seat Alhambra. (b) Classification regions of the RBF network with Seat Alhambra.

work consisting of a radial basis layer followed by a competitive layer as shown in Fig. 14. The functionality of the network is as follows: when an input is presented, distances from the input vector to the training prototype vectors are calculated by the radial basis layer as discussed earlier. Then, the competitive layer sums these contributions for each class of inputs to produce as its net output a vector of probabilities. Finally, a competitive transfer function picks the maximum of these probabilities, and produces a 1 for that class and a 0 for the other classes. The first layer weights are set to the transpose of the matrix formed from the input-target training pairs like Hamming network. However, the second-layer weights are set to the matrix of target vectors [33]. Its main advantage is the quick training compared with other types of networks like MLP and RBF, where only one pass through the data is required to build the network and there is only one smoothing factor, spread value of the RBF neurons. On the other hand, the main criticism of the PNN algorithm is the very rapid increase in memory and computing time when the dimension of the input vector and the quantity of training samples increase. This is because the number of the RBF hidden neurons equals the number of input-target training vectors [15,34].

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Fig. 22. (a) Decision boundaries of the PNN network with Seat Alhambra. (b) Classification regions of the PNN network with Seat Alhambra.

The network design phase consists of two steps. The first step is to find a good training set that gives satisfactory testing results. The second step is to find the optimum value for the spread parameter of the RBF layer [35]. In the present work, a trial-and-error process has been followed to select the best training group and the optimum spread value. The Matlab built-in function “newpnn” has been used to create the PNN network. 3.4. Learning vector quantization (LVQ) Vector quantization (VQ) is a standard statistical clustering technique. It typically maps a large set of N-dimensional training data vectors into a small set of binary representative points (symbols) corresponding to K classes, thus achieving a significant data compression. LVQ is a pattern classification method in which each output unit represents a particular class or category. The weight vector for an output node is called the reference or codebook vector. It is the mission of the training algorithm to design the code book vectors. The LVQ neural network is one method that can be used to design the codebook vectors. During training the output units are arranged by adjusting their weights through supervised training. After training, when an input vector is introduced, the LVQ network assigns it to the same class as the output unit

Fig. 23. (a) Decision boundaries of the LVQ network with Seat Alhambra. (b) Classification regions of the LVQ network with Seat Alhambra.

that has a weight vector (codebook) closest to the input vector [36]. An LVQ network has a first competitive layer and a second linear layer as can be seen in Fig. 15. The number of hidden neurons is more than or equal to the number of output neurons. The linear layer transforms the competitive layer’s subclasses into target classes. Both the competitive and linear layers have one neuron per (sub or target) class [37]. In this work, the LVQ neural classifier has been created using the built-in function “newlvq”, trained, and tested with Matlab. The number of hidden neurons, as well as the learning rate, has been varied and the classification efficiencies have been registered to determine the best number of hidden neurons. LVQ learning in the competitive layer is based on a set of input/target pairs {p1 , t1 }, {p2 , t2 }, . . . {pk , tk }. To train the network, an input vector P is presented, and the distance from P to each row of the input weight matrix W (weights of the competitive hidden layer) is computed. Then the hidden neurons compete. When the resulting vector of the competitive layer is multiplied by the weights of the output linear layer, the single 1 in the resulting vector of the competitive layer selects the class associated with that input. Then the ith row of W is adjusted in such a way as to move this row closer to the input vector P if the assignment is correct,

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Table 11 Confusion matrix of the MLC classifier for Seat Alhambra. Real gear

Estimated gear N 1st 2nd 3rd

N

1st

2nd

3rd

1344 (88.2%) 99 74 6

185 840 (77.1%) 55 10

10 74 1782 (94.2%) 25

0 0 36 512 (93.4%)

N

1st

2nd

3rd

1513 (9.3%) 10 0 0

253 826 (75.8%) 9 2

201 47 1606 (84.9%) 37

27 0 1 520 (94.9%)

N

1st

2nd

3rd

1399 (91.9%) 57 49 18

128 923 (84.7%) 27 12

46 39 1770 (93.6%) 36

0 0 21 527 (96.2%)

Table 12 Confusion matrix of the MLP1 classifier for Seat Alhambra. Real gear

Estimated gear N 1st 2nd 3rd

Table 13 Confusion matrix of the MLP2 classifier for Seat Alhambra. Real gear

Estimated gear N 1st 2nd 3rd

Table 14 Confusion matrix of the RBF classifier for Seat Alhambra. Real gear N Estimated gear N 1st 2nd 3rd

1422 (93.4%) 56 21 24

1st 222 834 (76.5%) 22 12

2nd 78 46 1740(92%) 27

3rd 0 0 18 530 (96.7%)

Table 15 Confusion matrix of the PNN classifier for Seat Alhambra. Real gear

Estimated gear N 1st 2nd 3rd

N

1st

2nd

3rd

1425 (93.6%) 49 49 0

314 375 (34.4%) 395 6

73 15 1365 (72.2%) 438

N

1st

2nd

3rd

1473 (96.7%) 4 46 0

568 469(43%) 53 0

83 46 1742 (92.1%) 20

0 0 146 402 (73.4%)

0 0 133 415 (75.7%)

Table 16 Confusion matrix of the LVQ classifier for Seat Alhambra. Real gear

Estimated gear N 1st 2nd 3rd

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Fig. 24. (a) Decision boundaries of the MLP2 network with Renault Laguna. (b) Classification regions of the MLP2 network with Renault Laguna. Table 17 Classification efficiencies of different neural classifiers for Seat Alhambra. Neural classifier type

Classification efficiency (ANR %)

MLC MLP1 MLP2 RBF PNN LVQ

88.24 88.7 91.5 89.65 68.97 80.96

and to move the row away from P if the assignment is incorrect. Such corrections move the hidden neuron toward vectors that fall into the class for which it forms a subclass, and away from vectors that fall into other classes. The iterative equation according which the weights of the competitive hidden layer W are updated is shown below where k is the iteration index and ˛ is the step size parameter, W (k) = W (k − 1) ± ˛(P(k) − W (k − 1))

(7)

4. Results The major objective of this paper is to compare and evaluate the classification accuracy of the MLP1, MLP2, RBF, PNN, and LVQ neural

Fig. 25. (a) Decision boundaries of the RBF network with Renault Laguna. (b) Classification regions of the RBF network with Renault Laguna.

classifiers, against the benchmark, maximum likelihood classifier (MLC). Classification accuracy is normally measured by presenting a testing set of data to the network under study and reporting the number of correctly classified samples. The method of presenting classification accuracy depends on the relative number of samples in each class and the requirements of either overall or class-by-class accuracy [23]. However, the most common way of expressing the neural classifier efficiency is to give the percentage of total number of correctly classified samples in the testing dataset [38]. Because this figure, called percent correct (PCC) calculated by means of Eq. (7), can be very misleading if the data set is unbalanced, i.e. the class sizes are not the same, (see Table 2 to know the class distributions in the datasets used in this paper), a better approach is to report the average per-class accuracy (the Average Normalized Response, ANR) as shown in Eq. (9) where ni denotes the number of correctly classified samples in the ith class, ti denotes the total number of samples in the ith class, and T denotes the number of classes. This can be easily understood by a simple example: if a data set of 1000 samples contains 990 samples from class A and 10 samples only from class B. If the classifier classifies all the samples as class A, the PCC accuracy, according to Eq. (8), will be 99%. Of course, this is not a good indication of the classifier’s true performance. The classifier has 100% hit rate for class A, but a 0% hit rate for class B. However, calculating the ANR accuracy using Eq. (9) gives 50%.

A. Wefky et al. / Applied Soft Computing 11 (2011) 3580–3599

Fig. 26. (a) Decision boundaries of the PNN network with Renault Laguna. (b) Classification regions of the PNN network with Renault Laguna.

The ANR can be utilized as a comparison criterion between different classifiers on the same dataset. But using a single figure like ANR does not inform the user about the accuracy in any specific area of the feature space. Reporting each specific class accuracy by means of the confusion matrix in addition to the ANR figure reflects more information to the user concerning class-by-class and overall accuracies. PCC =

total number of correctly classified samples × 100 total number of testing samples

(8)

ANR =

(n1 /t1 ) + (n2 /t2 ) + · · · + (nm /tm ) × 100 T

(9)

The confusion matrix, also called matching matrix or table of confusion, is a visualization tool typically used in supervised learning in the field of artificial intelligence. Each row represents the instances in a predicted (estimated) class, while each column represents the instances in an actual (real) class. In this work, the rows of the confusion matrix represent the estimated gear positions, while the columns denote the real ones. The objective of presenting the confusion matrices in this work is to show the class by class accuracies, as can be seen in the highlighted diagonal elements of the confusion matrices in Tables 4–9, 11–16, and 18–23 as well as to check if the classifier is confusing two gear positions (i.e. mislabeling one gear as another).

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Fig. 27. (a) Decision boundaries of the LVQ network with Renault Laguna. (b) Classification regions of the LVQ network with Renault Laguna.

In this section the ANR metric is exploited to assess the classification efficiency of the proposed models by simply comparing the target and model-estimated classes and providing the overall accuracy. The classification confusion matrices provide the class by class accuracies. 4.1. Peugeot 205 The number of gear positions (classes) realized in this work using Peugeot 205 is 5, refer to Fig. 5. Therefore it is necessary to use 4 separating boundaries in order to distinguish between the different gears using MLP1 or MLP2. Consequently both MLP1 and MLP2 neural network classifiers contain 4 hidden neurons. Moreover, the number of hidden neurons of the PNN classifier has been set to 5, equal to number of classes. However, the number of hidden neurons has been varied as a control parameter and 29 and 11 neurons have been chosen for RBF and LVQ classifiers respectively. The spread parameter of the hidden neurons of both RBF and PNN classifiers have been varied as a control parameter also and 0.1 has been chosen for both types of classifiers as can be shown in Tables A1 and A2 in the appendix. The learning rate of the LVQ classifier has been set to 0.1 as a result of doing multiple trials with different learning rates as shown in Table A3 in the appendix. Figs. 16–19 show the decision boundaries as well as the classification regions established by the

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Table 18 Confusion matrix of the MLC classifier for Renault Laguna. Real gear

Estimated gear N 1st 2nd 3rd 4th 5th 6th

N

1st

2nd

3rd

4th

5th

6th

2277 (87.8%) 48 191 20 57 0 0

432 450 (46.8%) 80 0 0 0 0

108 26 1498 (85.3%) 71 52 1 0

0 0 208 874 (73.6%) 102 2 2

0 0 0 3 499(96%) 9 9

0 0 0 0 170 976(83%) 30

0 0 0 0 32 146 3936 (95.7%)

Table 19 Confusion matrix of the MLP1 classifier for Renault Laguna. Real gear

Estimated gear N 1st 2nd 3rd 4th 5th 6th

N

1st

2nd

3rd

4th

5th

6th

2537 (97.8%) 3 6 20 21 6 0

670 146 (15.2%) 119 27 0 0 0

293 39 952 (54.2%) 390 56 20 6

23 0 50 955 (80.4%) 125 30 5

0 0 0 3 460 (88.5%) 31 26

0 0 0 0 9 1117(95%) 50

0 0 0 0 4 147 3963 (96.3%)

Table 20 Confusion matrix of the MLP2 classifier for Renault Laguna. Actual (true) gear

Estimated gear N 1st 2nd 3rd 4th 5th 6th

N

1st

2nd

3rd

4th

5th

6th

2419(93%) 75 60 38 1 0 0

275 614 (63.8%) 73 0 0 0 0

70 82 1473 (83.9%) 106 9 10 6

8 0 111 1001(84%) 50 16 2

3 0 8 7 453 (87.1%) 39 10

0 0 0 0 9 1086 (92.3%) 81

0 0 0 0 4 138 3972 (96.5%)

Table 21 Confusion matrix of the RBF classifier for Renault Laguna. Actual (true) gear

Estimated gear N 1st 2nd 3rd 4th 5th 6th

N

1st

2nd

3rd

4th

5th

6th

996 (91.9%) 77 11 0 0 0 0

151 430 (67.8%) 53 0 0 0 0

65 44 586 (79.1%) 33 13 0 0

9 0 61 337 (81.2%) 8 0 0

1 0 36 85 24 (16.4%) 0 0

0 0 58 1 243 0(0%) 0

0 27 354 0 298 0 0(0%)

Table 22 Confusion matrix of the PNN classifier for Renault Laguna. Actual (true) gear

Estimated gear N 1st 2nd 3rd 4th 5th 6th

N

1st

2nd

3rd

4th

5th

6th

2128 (82.1%) 187 262 9 7 0 0

261 346 (36%) 285 70 0 0 0

31 93 756 (43.1%) 784 83 9 0

0 0 131 757 (63.7%) 265 11 24

0 0 11 54 239 (46%) 156 60

0 0 0 0 113 724 (61.6%) 339

0 0 0 0 0 969 3145 (76.4%)

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Table 23 Confusion matrix of the LVQ classifier for Renault Laguna. Actual (true) gear N Estimated gear N 1st 2nd 3rd 4th 5th 6th

1st

2556 (98.6%) 0 22 15 0 0 0

774 0 (0%) 188 0 0 0 0

2nd

3rd

4th

5th

6th

638 0 805 (45.8%) 287 3 0 23

71 0 67 910 (76.6%) 93 3 44

6 0 0 81 340 (65.4%) 6 87

0 4 0 27 110 855 (72.7%) 180

0 1 0 0 0 141 3972 (96.5%)

neural classifiers. These figures clearly point out the capability of the MLP2 network to successfully classify the gear position of the Peugeot 205 as compared to the other types of classifiers. Tables 4–9 describe the confusion matrices of the neural classifiers used with the Peugeot 205. Table 10 shows the classification accuracies of the neural classifiers used for the Peugeot 205.

4.2. Seat Alhambra Four separating boundaries, or 4 hidden neurons, have been employed to classify the feature space using MLP1, MLP2, or PNN classifiers. However, the spread parameter of the hidden neurons of both RBF and PNN classifiers has been changed as a smoothing parameter as can be seen in Tables A4 and A5. As a result of multiple experiments, the best classification rate was found when the spread parameter was set to 0.1. The best number of hidden neurons for RBF and LVQ classifiers is 27 and 11 respectively. The best learning rate of the LVQ classifier was found to be 0.1 as shown in Table A6. The decision boundaries as well as the resulting classification regions developed by the neural classifiers are depicted in Figs. 20–23. These figures clearly point out the capability of the MLP2 network to successfully classify the gear position of the Seat Alhambra as compared to the other types of classifiers. Tables 11–16 describe the confusion matrices of the neural classifiers used with the Seat Alhambra. Table 17 shows the classification accuracies of the neural classifiers used for the Seat Alhambra.

4.3. Renault Laguna The testing dataset of the Renault Laguna was applied to different neural classifiers. Seven separating boundaries, or 7 hidden neurons, were used in case of MLP1, MLP2, or PNN classifiers. In addition, the best spread parameter of the hidden neurons of both RBF and PNN classifiers was found to be 0.2 as can be seen in Tables A7 and A8. Moreover, the best classification accuracies were obtained when the number of hidden neurons for RBF and LVQ classifiers is 29 and 11 respectively as shown in Tables A7 and A9. Besides, the optimum learning rate of the LVQ classifier was found to be 0.01. The decision boundaries and the corresponding classification regions developed by the neural classifiers are drawn in Figs. 24–27. These figures clearly point out the capability of the MLP2 network to successfully classify the gear position of the Renault Laguna as compared to the other types of classifiers. Tables 18–23 describe the confusion matrices of the neural classifiers used with the Renault Laguna. Table 24 shows the classification accuracies of the neural classifiers used for the Renault Laguna.

Table 24 Classification efficiencies of different neural classifiers for Renault Laguna. Neural classifier type

Classification efficiency (ANR %)

MLC MLP1 MLP2 RBF PNN LVQ

81.2 75.3 85.9 72.96 58.40 65.09

Fig. 28. Bar plot of all the classifier accuracies versus different vehicles.

Table 25 Parameters of the RBF, PNN and LVQ classifiers. Training parameters

Learning rate (LVQ)

Number of hidden neurons

Peugeot 205 Seat Renault

Spread of hidden neurons

RBF

LVQ

RBF

PNN

29 27 29

11 11 11

0.1 0.1 0.2

0.1 0.1 0.1

0.1 0.1 0.01

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5. Conclusions A set of conclusions can be drawn from the results discussed in the previous section. Fig. 28 shows a bar plot of the classifiers accuracies (ANR %) for the three studied cases. Firstly, the MLP2 (multilayer perceptron neural network with hyperbolic tangent hidden and output transfer functions) classifier has already exceeded the MLC bench mark classifier and achieved the best classification results for all vehicles. But of course, the price to be paid is the time consumed during the training phase. Secondly, the effect of the transfer function of the multilayer perceptron classifier has been studied by introducing the simple MLP1 (multilayer perceptron neural network with hard-limit hidden and output transfer functions) classifier that has achieved relatively good results. The advantage of that type of classifier is the ease of design relative to the MLP2 backpropagation network. By fine tuning the separating boundaries, the results of the MLP1 may be further improved. Thirdly, the performance of the RBF classifier is reasonable relative to the reference MLC classifier. It is advisable to use that type of classifier when the designer needs a well known classifier topology in a short time relative to the backpropagation MLP classifier (MLP2 network). The price to pay to shorten the training cycle is to use a relatively large number of hidden RBF neurons. Fourthly, the performance of the LVQ classifier is better than that of the PNN classifier in case of noisy feature spaces, such as the case of Alhambra and Laguna, and vice versa for noiseless feature space, like the case of 205. Finally, the training parameters used in the three cases are relatively similar as can be inferred from Table 25. Considering the comparative analysis, the authors propose the MLP2 classifier (multilayer perceptron neural network with hyperbolic tangent hidden and output transfer functions) to estimate the gear position of a vehicle in terms of the regime engine and the vehicle speed. The decision boundaries as well as the classifica-

tion regions of the MLP2 network showed its excellent capability of distinguishing the gear position values as depicted previously in Figs. 16, 20 and 24. Concerning future works, the authors intend to deal with the same problem of gear classification but with other machine learning tools such as fuzzy clustering techniques and support vector machines in order to compare with the classification efficiency of the artificial neural networks. On the other hand, the authors intend to extend the experiments to different models of the same vehicle types as well as with different drivers of the same vehicles in order to try to obtain a universal model valid for a wide range of vehicles. Acknowledgment This work has been supported by the Spanish Ministry of Environment through the MIVECO Project (MMA 071/2006/3-13.2). Appendix A. This section details the classification results by introducing some details concerning the training process of some models. The highlighted values in Tables A1–A9 represent the case (state) of the maximum classification efficiency. 1. Peugeot 205 • RBF: See Table A1. • PNN: See Table A2.

Table A1 Ha

1

0.5

0.4

0.3

0.2

0.15

0.1

0.05

0.01

69.12 70.91 72.26 72.37 71.83 71.58 71.6 72.55 76.06 75.58 77.94 78.76 80.69 79.43 79.54 78.74 78.87 79.55 79.29 79.36 79.96 80.01 79.62 79.37 79.37 79.37

70.12 70.4 70.4 72.1 72.29 72.29 73.89 73.89 80.7 80.8 80.67 80.94 81.19 81.12 80.9 80.31 79.63 79.96 80.13 80.07 79.69 80.38 80.26 80.77 80.86 81.36

68.8 71.24 71.05 72.16 72.83 72.87 73.24 79.38 80.38 80.34 80.37 79.24 80.4 79.92 79.62 79.72 79.87 79.76 80.77 80.57 80.82 81.54 81.44 82.57 83.32 82.15

72.36 71.73 72.3 73 76.54 75.82 78.12 77.63 76.47 76.56 77.14 76.76 77.58 77.5 76.87 80.54 81.28 81.5 81.57 81.66 81.46 81.62 81.5 81.37 81.5 81.61

38.39 38.39 48.74 60.92 64.68 69.93 74.46 77.16 77.14 75.65 75.85 76.6 78.37 81.04 81 81.33 81.33 81.33 81.33 81.33 81.66 81.21 81.25 81.79 81.98 82.94

65.3 68.75 70.09 70.32 72.16 80.93 77.4 77.61 77.85 79.98 82.5 84.49 83.82 83.83 84.8 85.02 85.04 85.17 85.02 85.3 85.51 85.46 85.7 87.18 87.04 86.98

56.2 60.17 65.34 67.4 69.8 72.71 71.42 73.61 74.09 76.34 80.85 83.52 84.57 85.29 84.97 85.29 85.71 87.06 87.75 88.4 88.4 88.95 89.04 89.58 89.71 89.7

58.85 61.35 63.37 66.75 67.61 70.68 73.22 74.11 75.72 77.4 78.61 80.49 81.5 82.6 83.32 83.39 83.78 83.92 84.13 85.2 84.49 85.05 85.56 85.94 86.54 86.71

42 43.29 43.29 44.7 47.6 47.72 49.2 50.77 51.66 51.81 52.93 53.75 54.62 54.7 55.35 56.38 57.16 57.26 57.49 57.75 58.02 58.81 59.11 59.72 60.00 60.49

b

S 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a b

Number of hidden neurons. Spread.

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Table A5

Table A2

Spread

Classification efficiency (%)

0.001 0.005 0.01 0.05 0.1 1

32.83 59.58 78.32 80.36 80.36 80.36

Spread

Classification efficiency (%)

0.001 0.005 0.01 0.05 0.1

31.04 53.19 67.96 68.97 68.97

Table A6 Table A3

H

Number of hidden neurons

LRa = 0.001

LRa = 0.01

LRa = 0.1

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

45.08 57.93 61.58 60.29 61.52 63.05 63.1 63.49 63.78 64.58 64.45 64.61 64.11 64.98 65.21 65.53

53.13 65.3 64.91 64.96 66.4 66.1 65.21 64.52 65.86 67.4 66.59 65.03 66.86 65.47 64.87 64.53

55.05 64.9 66.69 65.37 66.85 66.74 69.47 67.31 66.6 67.49 68.11 67.89 67.15 67.51 67.53 67.69

a

0.001

0.01

0.1

73.34 72.68 73.83 70.83 69.37 71.73 69.69 71.61 71.56 72.71

73.92 73.06 76.89 71.43 73.67 75.36 73.69 75.01 80.25 74.76

74.37 73.62 80.48 73.9 80.07 79.85 80.94 80.96 80.28 80.47

S 4 5 6 7 8 9 10 11 12 13

• RBF: See Table A4. • PNN:

Learning rate.

See Table A5. • LVQ:

• LVQ See Table A6.

See Table A3.

1. Renault Laguna

1. Seat Alhambra Table A4

H

0.01

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

49.53 50.92 51.01 53.09 54.54 54.67 55.45 56.09 56.52 57.28 57.33 57.77 58.5 58.87 58.96 59.51 59.86 60.22 60.57 60.71 60.71 61.3 61.35 61.56 61.97 62.47 62.97

69.39 71.01 75.77 79.9 82.01 87.05 86.47 87.39 86.7 87.66 87.6 88.27 88.46 89.36 89.48 89.54 89.38 89.46 89.53 89.55 89.64 89.25 89.52 89.65 89.65 89.54 89.63

76.95 77.73 83.02 84.71 85.36 87.52 87.95 88.13 88.43 89.05 88.99 88.88 89.16 89.08 89.15 89.26 89.38 89.22 89.07 88.92 89.04 89.32 89.36 89.36 89.36 89.25 89.22

78.4 82.92 84.05 85.65 87.67 87.67 86.31 86.65 86.92 87.78 87.88 88.39 88.45 88.45 88.45 88.40 89.2 88.63 88.66 88.64 88.7 88.74 88.75 88.87 89.02 89.03 89.05

79.14 83.25 84.19 85.78 86.20 85.73 86.18 87.94 87.91 88.4 88.39 88.36 88.48 88.43 88.28 88.13 88.74 88.34 88.6 88.69 88.56 88.71 88.86 89.25 89.1 88.91 88.96

79.67 82.5 83.69 86.41 87.02 87.73 87.47 87.71 87.9 87.85 88.28 88.28 88.28 88.52 88.54 88.74 88.24 88.9 88.77 88.66 88.69 88.69 88.7 88.7 88.73 88.76 89.19

82.41 82.41 83.48 84.04 87.02 86.95 86.86 86.99 86.99 86.99 87.61 87.85 87.91 88.31 88.31 88.31 88.68 88.44 88.29 88.31 88.15 88.34 88.43 88.47 88.57 88.59 89.03

81.97 82.64 82.95 82.95 82.95 85.33 85.19 86.63 87.56 87.65 87.65 87.65 87.41 87.82 87.7 87.66 88.40 88.02 88.16 88.28 88.2 88.34 88.33 88.85 88.85 88.84 88.95

81.22 82.15 83.11 85.48 85.59 85.77 85.63 86.46 86.57 87.57 87.66 88.18 88.14 88.04 88.1 88.24 88.05 88.21 88.71 88.86 88.76 88.74 88.81 88.83 88.86 88.8 88.87

80.82 82.5 83.65 84.75 84.75 84.97 86.53 87.05 87.72 87.72 87.72 87.56 87.56 87.29 87.29 87.29 88.18 87.90 88.65 88.53 88.27 88.22 88.02 88.11 88.07 88.10 88.10

80.18 81.56 83.41 85.16 85.21 85.27 86.03 86.78 87.20 86.85 87.65 88.30 88.36 88.40 88.37 88.21 87.79 88.21 88.2 88 88.26 88.18 88.25 88.25 88.25 88.26 88.53

S 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

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Table A7

H

S=1

S = 0.5

S = 0.4

S = 0.35

S = 0.3

S = 0.25

S = 0.2

S = 0.1

S = 0.01

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

51.36 48.36 52.83 54.23 54.42 56.57 60.45 56.74 60 61.41 62.21 62.16 63.4 63.4 63.4 63.24 64.91 67.07 67.07 66.35 66.44 66.44 66.04 66.04 65.84

59.67 63.11 62.76 64.13 63.9 60.87 60.87 61.61 64.78 65.74 65.87 66.47 68.3 68.58 73.98 74.30 74.44 71.59 74.49 69.23 69.19 69.42 69.42 67.12 67.39

52.62 49.77 55 56.66 61.35 64 63.96 64.34 64.34 66.60 64.38 65.27 65.47 66.27 67.37 67.01 69.27 66.8 66.65 66.63 67.24 68.33 67.41 65.76 66.37

50.03 51.89 58.11 56.96 56.88 63.6 63.53 63.53 64.81 66.20 66.47 65.78 64.75 65.45 65.27 64.98 64.37 64.45 65.39 65.25 67.03 66.85 67.86 68.42 68.33

49.3 49.54 54.86 58.38 64.68 66.07 65.86 68.36 68.64 68.8 68.79 67.98 67.89 68.13 67.8 68.48 68.80 68.24 68.50 68.08 67.79 68.51 67.77 68.69 69.4

53.53 56.46 59.54 59.54 62.97 62.06 62.54 66.05 67.73 67.62 68.3 68.82 68.46 68.28 68.01 67.72 67.94 68.85 68.91 68.36 68.82 68.88 67.42 67.42 67.33

46.23 53.52 57.71 60.03 60.29 61.04 61.04 62.24 64.28 65.07 68.56 68.31 69.43 69.5 69.56 69.35 68.89 69.57 70.04 70.48 70.46 70.54 68.68 72.96 72.51

49.01 55.66 55.66 55.66 59.85 60.41 61.82 62.9 63.49 63.49 66.24 66.48 66.03 65.55 65.26 67.03 68.33 67.81 68.30 68.34 68.82 68.5 68.22 68.61 68.61

32.79 33.11 35.59 36.05 37.12 38.1 39.14 39.61 41.5 42.30 42.81 43.23 43.47 43.91 44.01 44.07 44.27 44.90 45.23 45.64 45.62 45.98 46.14 46.33 46.84

Table A8

References

Spread

Classification efficiency (%)

0.001 0.005 0.01 0.05 0.2

23.39 54.36 58.14 58.4 58.4

Table A9

Number of hidden neurons

LR = 0.001

LR = 0.01

7 8 9 10 11 12 13 14 15 16 17 18 19 20

48.26 58.18 51.31 55.97 64.54 56.11 63.64 63.84 63.55 57.88 64.29 64.51 57.19 64.82

36.24 59.10 53.18 57.28 65.78 55.92 65.02 64.94 64.6 66.53 68.2 65.67 58.39 67.99

• RBF See Table A7. • PNN See Table A8. • LVQ See Table A9.

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