Abdullah Almaksour
email: abdullah.almaksour@irisa.fr

Eric Anquetil
email: eric.anquetil@irisa.fr

Fast Incremental Learning Strategy Driven by Confusion Reject for Online Handwriting Recognition

In this paper, we present a new incremental learning strategy for handwritten character recognition systems. This learning strategy enables the recognition system to learn "rapidly" any new character from very few examples. The presented strategy is driven by a confusion detection mechanism in order to control the learning process. Artificial characters generation techniques are used to overcome the problem of lack of learning data when introducing a new character from unseen class. The results show that a good recognition rate (about 90%) is achieved after only 5 learning examples. Moreover, the rate quickly rises to 94% after 10 examples, and approximately 97% after 30 examples. A reduction of error of 40% is obtained by using the artificial characters generation techniques.

Introduction

In the last decade, the world witnessed the emergence of touch screen devices, from Personal Digital Assistants (PDAs) to Tablet PCs and other instruments that use penbased interfaces. More and more efforts are needed to make online handwriting recognition systems more robust and adaptive in order to meet the increasing user requirements. One of these new requirements is to build a recognition system that allows the user to choose his own group of gestures and assign them to different interactive commands, e.g. "copy", "paste", "undo", etc. This application context imposes specific conditions on the used learning approach. In such approach, the system must be able to rapidly learn a new unseen form using only few data. The later is due to the fact that users would rarely be willing to wait and repeat each new gesture a dozen of times for training the system. Furthermore, such a learning process must be progressively repeated for each newly added data. For all these reasons, this study aims at introducing a fast incremental learning strategy for online handwriting recognition systems. In this study, the strategy was validated on handwritten characters. However, the long term objective is to use this strategy in any fast incremental gesture learning system.

The main difficulty is to build on-the-fly a handwriting classifier from scratch, with the constraint of few available learning examples, and then to adapt it incrementally in order to achieve high recognition rate as soon as possible.

Incremental learning algorithm is defined in [START_REF] Polikar | Learn++: An incremental learning algorithm for supervised neural networks[END_REF] as being one that meets the following criteria: it should be able to learn additional information from new data; it should not require access to the original data (i.e. data used to train the existing classifier); it should preserve previously acquired knowledge (it should not suffer from catastrophic forgetting, significant loss of original learned knowledge); and it should be able to accommodate new classes that may be introduced with new data.

Several incremental learning techniques have been presented in different application contexts and for different classification approaches. In the incremental learning approach presented in [START_REF] Polikar | Learn++: An incremental learning algorithm for supervised neural networks[END_REF], small batches over separated periods of time are used to add the new knowledge, so it can be considered as a slow and offline incremental learning process. Furthermore, the weak classifiers used in this approach can not be adapted after that they had been learned. Another incremental learning system was presented in [START_REF] Gary | An effective neuro-fuzzy paradigm for machinery condition health monitoring[END_REF]. The learning algorithm in this system fits the incremental learning setting described above, and it can be considered as fast learning strategy. However, it uses classifier with basic structure that can not cope with the complexity of handwritten character classification problem.

Indeed, our intention is to design a new strategy that meets both the instantly incremental learning mechanism and the ability of using a complex and powerful classifier structure. The first challenge in designing a prototype-based incremental learning system is to answer these two questions: when and how creating a new prototype?

We presented in previous work [START_REF] Almaksour | Fast online incremental learning with few examples for online handwritten character recognition[END_REF] an incremental learn-ing strategy with two distinct phases. A new prototype is created for each new example in the first phase, and no prototype creation is occurred during the second phase. In this paper, we present a homogeneous incremental learning strategy with two fundamental points. The first one is that the prototype creations are completely controlled, during the entire learning process, by the detection of confusion rejection. And the second point is using the original handwritten characters generation techniques, presented in previous work [START_REF] Mouchere | Synthetic on-line handwriting generation by distortions and analogy[END_REF], in order to create representative prototypes despite the lack of learning example. The rest of this article is organized as follows. In Section 2, we present the used classification approach (fuzzy inference system) and the basic principles of our incremental learning system. Then, we describe in Section 3 our incremental learning strategy. Section 4 then describes the method we used to generate artificial characters to accelerate the learning process. Finally, Section 5 shows the results of our experiments.

2 Principles of our approach

Fuzzy Inference System

We have chosen a Fuzzy Inference System (FIS) as a classification approach for several reasons: first of all, it is a light system, with few of parameters. And also, its flexible nature allows to adapt it easily to absorb a new knowledge.

The classifier is formalized by an order 0 Takagi-Sugeno FIS [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]. The fuzzy rules make a link between intrinsic models that describe the properties of the handwritten characters and the corresponding label. Each intrinsic model is defined by a set of fuzzy prototypes P i in n dimensions. For a K classes problem, a rule R i is built for each P i : IF X is P i THEN s i 1 = a i1 and ... and s i c = a ic and ... and s i K = a iK where X is the feature vector of the character X to recognize. As each prototype can participate to the description of each class, the rule R i has numeric conclusions connecting P i with each class c = 1..K by a prototype score s i c . The a ic is a weight that expresses the participation of P i in the description of the class c.

The fuzzy prototypes are learned separately on each class by using an unsupervised clustering algorithm. Fuzzy prototypes P i are defined by their membership function β i ( X) (eq. ( 1)). This one is an hyper-ellipsoidal Radial Basis Function (RBF) with a center μ i . Its shape is given by a covariance matrix Q i using the Mahalanobis distance d Qi ( X, μ i ):

β i ( X) = 1/(1 + d Qi ( X, μ i )). (1)
The conclusions a ic of each rule are computed with the pseudo-inverse method or back-propagation algorithm. It gives the optimum values to discriminate the classes.

To recognize an unknown character X, its membership degree to all fuzzy prototypes is computed (eq. ( 1)) and the sum-product inference is used to compute the system outputs which are scores s c for each class.

s c = N i=1 β i .s i c N i=1 β i , ( 2
)
where β i is the if-part activation of the rule R i for X, N is the number of rules and s i c is the prototype score given by the rule i for the class c. The decision is given by choosing the class having the best (maximum) score.

Incremental Learning

There are two manners by which a prototype-based recognition system can be incrementally learned. The first manner is by creating new prototypes for newly available data. The second is by modifying the existing prototypes according to new data.

Although prototype creation helps to improve significantly the classifier capacity (recognition rate), the complexity of the classifier (computing time and memory space) becomes unacceptable if prototypes number becomes sizable. For this reason, the use of prototypes adjustment is essential in order to get a dynamic and light system. Prototypes adjustment (or adaptation) can improve the capacity with same complexity of the classifier, but this technique has some limitations. It is difficult for it to change rapidly the decision borders in the case where the new knowledge (presented by the new data) is little represented by the existing prototypes. In such case, the decision of creating a new prototype is intuitively more reasonable.

We note that both techniques are complementary, and the first challenge to have an efficient incremental learning model is to find the best strategy to combine these two techniques. We suppose that learning and adapting processes are supervised: each example is correctly labeled. This labeling is possible by asking the user to check the recognition or by using self-supervised technique. We present below the techniques of prototype creation and adjustment used is our system.

Prototype creation

Adding a new prototype in a FIS is achieved by adding a new rule. This process is composed of two parts: the creation of the prototype and the calculation of associated conclusions. A prototype is created in the feature space by choosing a center that determines the position of the prototype, and a covariance matrix that represents the size and shape of the prototype.

In non-incremental learning systems, where the system is built using learning dataset with sufficient number of examples, non-supervised classification methods are used to calculate the prototypes (centers and covariance matrices) for each class. As mentioned in the introduction, one of the fundamental criteria in an incremental learning system is to eliminate the need to use any prior learning data. For this reason, when adding a new rule, a hyper-spherical prototype is creating in the feature space using the only one current example. A first simple solution is to initialize the prototype's center by the current example, and the covariance matrix by a matrix proportional to identity matrix (thus setting the initial size of the prototype). These prototypes are then distorted to have hyper-ellipsoidal shape thanks to the adjustment technique briefly presented below.

Prototypes adaptation

We use in our system the ADAPT method [START_REF] Mouchere | Writer style adaptation in on-line handwriting recognizers by a fuzzy mechanism approach : The adapt method[END_REF]. The Adaptation by Adjustment of ProtoTypes (ADAPT) method allows to modify all the prototypes of FIS by re-centering and reshaping them for each new example. This is done according to their participation in the recognition process. The principle of ADAPT is inspired by the LVQ algorithm. The method is fully detailed in [START_REF] Mouchere | Writer style adaptation in on-line handwriting recognizers by a fuzzy mechanism approach : The adapt method[END_REF].

Our incremental learning strategy driven by confusion reject

We presented in previous work [START_REF] Almaksour | Fast online incremental learning with few examples for online handwritten character recognition[END_REF] an incremental learning strategy with two distinct phases. During the first phase, that we called fast incremental learning phase, a significant number of prototypes are added in order to rapidly enhance the classifier's performance. Then in the second phase, called adaptation phase, prototype creation is minimized and a classifier adaptation technique (ADAPT) is then used to continue the learning process.

The new strategy that we present in this paper is based on the detection of a confusion reject in order to trigger the new prototype creation process. We aim to limit prototypes number in the system by avoiding creating prototypes in places where there is no likelihood of confusion. This eliminates adding "unnecessary" prototypes for a specific class where this class was already dominant.

The advantage of this strategy is that for "easy to learn" classes, prototype creation will be early limited by the absence of confusions with other classes. However, the system can continue to add new prototypes in order to learn "difficult to learn" classes as long as confusions are triggered. Before presenting the used algorithm, we describe the notion of confusion reject on which this strategy is based.

Confusion reject

The purpose of the confusion rejection is to assess the reliability of the classifier by detecting patterns for which the classifier is likely to misclassify. These errors are near the decision boundaries because scores of at least two classes are nearly equal.

Confusion reject can be realized by defining a reject zone on each side of decision boundaries. Each pattern within one of these zones is considered as potential error and is therefore rejected. Figure 1 shows an example in two dimensions of confusion reject zones for three classes. To formalize the confusion reject we use the notion of reliability functions as defined in [START_REF] Mouchere | A unified strategy to deal with different natures of reject[END_REF]. Here, the reliability function ψ(X) represents the degree of confusion in classifying an sample X :

ψ(X) = (Sc 1 (X) -Sc 2 (X)/Sc 1 (X) (3)
where Sc 1 (X) is the score obtained for the best class and Sc 2 (X) is the score obtained for the second best class. A sample X is then rejected when the degree of confusion is below a specific threshold. The threshold value represents the width of the reject zones around decision boundaries.

Algorithm

The principle of this strategy is based on the idea of favoring the creation of prototypes at the beginning, and then progressively reducing the number of creations relying on the adaptation technique to keep up the learning. The Adjustment of the number of created prototypes is provided by changing the threshold value of confusion reject. We use a less strict threshold for a given class each time we add a new prototype for this class. Algorithm 1 describes our incremental learning strategy.

Incremental learning acceleration using artificial data generation

Due to lack of knowledge in the beginning of learning a new class, generating synthetic handwritten characters can improve the quality of created prototypes to be more representative of user writing style. Thus, a new prototype is created by calculating the center and the covariance matrix As presented in [START_REF] Mouchere | Synthetic on-line handwriting generation by distortions and analogy[END_REF], several online handwriting generation techniques can be used to build artificial learning dataset, in respecting the user writing style. We use in our system two techniques of characters generation. The first one uses classical image distortions, such as scaling and slanting. The second one is based on the particularity of on-line handwriting; it applies two online distortions on the character: speed variation and curvature modification. Thus, several variations of the same class are generated from a single example of this class. Distortions limits are carefully chosen in order to avoid generating an artificial character that does not look like any more the original one, so it will not help to learn the user writing style.

Results

The experiments are based on the recognition of the 26 isolated lower case Latin letter. The writer specific datasets were written on a PDA by 18 writers. Each writer has randomly inputted 40 times each character, i.e 1040 characters per writer. In order to estimate the performance of the incremental learning strategy for each writer, we proceed by a 4-fold cross-validation technique. Three quarters of the dataset (780 letters) are used to incrementally learn the system, and one quarter (260 letters) is used to estimate the evolving of system capacity during the learning process. The presented results in the figures are the average of the 18 tests (18 writers). Each pattern in our system is described by a set of 21 features. A new example of each class is presented to the system in each learning cycle.

We call "strategy 1" the two phases incremental learn- ing strategy that we presented in previous work [START_REF] Almaksour | Fast online incremental learning with few examples for online handwritten character recognition[END_REF], and "strategy 2" the new incremental learning strategy driven by confusion reject. In the strategy 1, the fast incremental learning phase for each class ends after introducing 10 examples of that class. The system is then switched to the adaptation phase. For the strategy 2, the policy of reducing the confusion reject threshold during the learning process is illustrated in figure 2.

We compare in these experiments the performance of the two incremental learning strategies in terms of the complexity of the classifier, and the quality of the classifier. The complexity of the classifier is related to the required memory space and the computing time required for recognizing one character. In our system, memory space and computing time depend totally on the number of prototypes created in the system. The quality of the classifier is evaluated by the recognition rate obtained with the test datasets.

Moreover, we evaluate in these experiments the impact of using the artificial characters generation on the quality and the complexity of the classifier in our incremental learn- ing system. Figure 3 shows the evolution of total prototypes number in the FIS for the 26 classes during the learning process. While figure 4 shows the evolution of the mean recognition rate for the 18 writers. Through these two figures, we note that the strategy 2 results in a classifier's quality equal to or greater than that obtained with the strategy 1, with creating fewer prototypes. We find that using the strategy 2 with artificial characters generation, a recognition rate about 90% is reached after only 5 learning examples, and such rate rapidly improves reaching 94% after 10 examples, and about 97% after 30 examples. We note also that an error reduction about 40% is achieved using the artificial characters generation.

Figure 3 shows the impact of the characters generation on the classifier's complexity. Since prototype creation in strategy 1 is systematic during the phase 1 and inhibited in the phase 2, characters generation has no impact on the number of created prototypes. In contrast, in the strategy 2, the more the system is efficient the less prototype creations are needed. This is confirmed in figure 3 where we note that the use of characters generation with strategy 2 helps to reduce the total prototypes number in the system.

With disregarding the axis of time represented by the increased number of learning examples in figures 3 and 4, we illustrate the quality/complexity ratio of the classifier obtained by each strategy (figure 5). We conclude that strategy 2 achieves a better recognition rate for the same number of prototypes comparing to strategy 1. It can be also noted that the quality/complexity ratio of the classifier is enhanced thanks to the artificial characters generation.

Conclusions

In the context of online handwritten character recognition systems, we introduce in this paper a new fast incre- mental learning strategy driven by confusion reject mechanism. Using this strategy, the system -which is based on fuzzy inference classifier -is able to learn from scratch new forms with few learning data. It also improves and adapts to each newly available data. Artificial characters generation has been used in the system in order to accelerate the learning by improving the recognition rate. Results have shown that a good recognition rate is achieved after introducing more than 5 examples per class. A special emphasis for a possible future work is placed on reducing the number of prototypes in the system either by deleting the "useless" prototypes or by merging redundant ones.

Figure 1 .

 1 Figure 1. Confusion reject zones.

Figure 2 .Figure 3 .

 23 Figure 2. Policy of reducing the confusion reject threshold.

Figure 4 .

 4 Figure 4. Evolution of recognition rate during the learning process.

Figure 5 .

 5 Figure 5. Classifier quality/complexity ratio obtained by each strategy.

Algorithm 1 :

 1 Incremental learning algorithm driven by confusion reject.

	foreach new example e do
	if e is the first example of the class C then
	create a new prototype around e;
	apply adaptation technique;
	reject threshold[C] = initial value;
	else
	apply adaptation technique;
	calculate the confusion degree;
	if confusion reject then
	create a new prototype around e;
	reduce the value of reject threshold[C];
	end
	end
	end
	from synthetic characters, which results in hyper-ellipsoidal
	prototypes instead of hyper-spherical prototypes (created
	using the original character only).


Abdullah Almaksour
email: abdullah.almaksour@irisa.fr

Harold Mouchère
email: harold.mouchere@irisa.fr

Eric Anquetil
email: eric.anquetil@irisa.fr

Fast Online Incremental Learning with Few Examples For Online Handwritten Character Recognition

Keywords: Incremental learning, online handwritten character recognition, fuzzy inference system, supervised adaptation, handwritten character generation

An incremental learning strategy for handwritten character recognition is proposed in this paper. The strategy is online and fast, in the sense that any new character class can be instantly learned by the system. The proposed strategy aims at overcoming the problem of lack of training data when introducing a new character class. Synthetic handwritten characters generation is used for this purpose. Our approach uses a Fuzzy Inference System (FIS) as a classifier. Results have shown that a good recognition rate (about 90%) can be achieved using only 3 training examples. And such rate rapidly improves reaching 96% for 10 examples, and 97% for 30 ones.

Introduction

In the last decade, the world witnessed the emergence of touch screen devices, from Personal Digital Assistants (PDAs) to Tablet PCs and other instruments that use penbased interfaces. Online handwriting recognition systems specialists have to keep up with the advancement of such innovations. More and more efforts are needed to make these systems more robust and adaptive in order to meet the rapidly increasing user requirements. One of these new requirements is to build a recognition system that allows the user to choose his/her own group of gestures and assign them to different interactive commands, e.g. "copy", "paste", "undo", etc. This application context imposes specific conditions on the used learning approach. Such approach must be capable to be rapidly learned using only few data. The later is due to the fact that users would rarely be willing to wait and repeat each new gesture a dozen of times in training the system. Further more, such a learning process must be progressively repeated for each newly added data. For all these reasons, this study aims at proposing an original fast incremental learning strategy for online handwriting recognition systems. In this study, the strategy was validated on alphabetical handwritten characters. However, the long term objective is to use such this technique in any fast incremental gesture learning system.

The main difficulty in the proposed problem is to build an on-the-fly handwriting classifier from scratch, with the constraint of little available knowledge (few learning examples), and then to adapt it incrementally in order to achieve high recognition rate as soon as possible.

In this paper, we propose an incremental learning process with two distinct phases. In order to overcome the problem of lack of examples, and to rapidly increase the system recognition rate, we integrate the original online synthetic handwritten characters generation techniquespresented in previous works [START_REF] Harold | Synthetic on-line handwriting generation by distortions and analogy[END_REF] -in the learning process.

Generally, for each new learning data in an incremental learning approach, either a new knowledge must be added to the system, or the already existing knowledge has to be adapted to represent this data. In our approach, prototype-based classifier is used for knowledge representation. In the proposed strategy and during the first phase that we call fast incremental learning phase, a significant number of prototypes must be added in order to rapidly maximize the capacity of the classifier, using the few available data. Then in the second phase, called adaptation phase, prototypes creation is minimized, and a classifiers adaptation technique is then used to enhance the recognition rate.

The rest of this paper is organized as follows. In section 2, we provide an overview of several approaches suggested for incremental learning algorithms. We present our approach in section 3. Next, section 4 describes the fast incremental learning phase; and section 5 describes the adaptation phase. Then, our experimental protocol and results are presented in section 6.

Related works

In many practical applications of machine learning systems, acquisition of representative set of training data is considered to be expensive and time consuming. In our context, in which a new classifier must be created from scratch, the time needed to build representative and complete dataset is relatively long and unacceptable from the user's viewpoint. Therefore, such an application scenario requires the classifier to be rapidly trained from a few examples, and then get incrementally updated by any newly available examples.

Incremental learning algorithm is defined in [START_REF] Polikar | Learn++: An incremental learning algorithm for supervised neural networks[END_REF] as being one that meets the following criteria:

1. it should be able to learn additional information from new data, 2. it should not require access to the original data (i.e. data used to train the existing classifier), 3. it should preserve previously acquired knowledge (it should not suffer from catastrophic forgetting, significant loss of original learned knowledge), 4. it should be able to accommodate new classes that may be introduced with new data.

These four points, which apply for any general incremental learning problem, perfectly match the characteristics of our special problem, a fast learning algorithm for online handwritten characters system. Several incremental learning techniques have been proposed in different application contexts and for different classification approaches, some of which are described below. Learn++ [START_REF] Polikar | Learn++: An incremental learning algorithm for supervised neural networks[END_REF] is an incremental algorithm based on synergistic performance of an ensemble of weak classifiers. It generates multiple hypotheses using training data sampled according to carefully chosen distributions of training dataset into smaller training sub-sets. In this approach, small batches over separated periods of time are used to integrate the new knowledge, so it can be considered as a slow and offline incremental learning process. Further more, the created classifiers can not be adapted after that they had been learned. Therefore, the only method to be used by the incremental learning process in order to add new knowledge is to add new classifiers.

ILFN [START_REF] Gary | An effective neuro-fuzzy paradigm for machinery condition health monitoring[END_REF] is an Incremental Learning Fuzzy Neural network. The ILFN classifier uses gaussian radial basis functions to structure decision boundaries. It employs a hybrid of both a supervised and an unsupervised learning schemes to generate its prototypes. This algorithm fits the incremental learning setting described above, and it can be considered as fast learning strategy. However, it uses a basic prototype structure that can not cope with the complexity of handwritten characters classification problem.

Fuzzy ARTMAP [START_REF] Carpenter | Fuzzy artmap: A neural network architecture for incrementalsupervised learning of analog multidimensional maps[END_REF] is based on generating new decision clusters in response to new patterns that are sufficiently different from previously seen instances. This suf-ficiency is controlled by a user-defined vigilance parameter. Fuzzy ARTMAP is a versatile algorithm; however, it is sensitive to selection of the vigilance parameter, to the noise levels in the training data and to the order in which the training data is presented to the algorithm.

Indeed, our intention is to design a new strategy that meets both, the instantly incremental learning mechanism as proposed in ILFN, and the ability of using a complex and powerful classifier structure as in Learn++. And so, we can get a fast incremental learning method with a good classification capacity that can solve complex problem, such as handwritten character recognition problem.

Our approach

As mentioned above, there are two manners by which the recognition system can be incrementally learned. The first manner is by creating a new prototype for each new example of the character class, the second one of learning is by modifying the existing prototypes that represent the character class according to the characteristics of the new example. In our proposed strategy, we proposed these two methods as complementary techniques to achieve fast and stable increment learning. On one hand, although that prototypes creation helps to improve significantly the recognition rate, its cost can be unacceptable -in term of time and memory -if we continue to create for each new example. On the other hand, prototypes adaptation can slowly enhance the recognition rate, without charging the system by creating new prototypes. As classification mechanism, we have chosen a Fuzzy Inference System (FIS). FIS is a light prototype-based classifier with few parameters. The flexibility of FIS meets our need for an adaptive classification approach.

Furthermore, we suppose here that learning and adapting processes are supervised: each example is labeled correctly. This labeling is possible by asking the user to check the recognition or by using self-supervised technique as in [START_REF] Oudot | Self-supervised writer adaptation using perceptive concepts : Application to on-line text recognition[END_REF]. In the rest of this section, we present firstly our learning strategy, and then we explain the principals of the used FIS.

Fast incremental adaptive learning strategy

In the beginning of the learning process, creating new prototypes is indispensable, because the character classes are not yet sufficiently modeled by the existing prototypes. However, even when creating new prototypes for the new examples, the adaptation technique is still useful to adjust the previous prototypes according to these examples. Because of memory and recognition time constraints, prototypes creation must not be continued infinitely. The adaptation algorithm will be the alternative learning methods. After creating sufficient number of prototypes for a specific class, i.e. after introducing N examples of that class, the system is switched to the adaptation phase. During this phase, prototypes adaptation process will play the main role when introducing a new example. But at the same time, prototypes creation is allowed when a specific number of miss-classified examples are reached. Algorithm In order to enhance prototypes creation and adaptation processes, the last M examples (about 1 or 2 per class) are stored in a specific buffer. Then, this buffer is used as small training dataset during conclusions calculation and prototypes adaptations, as described later.

Fuzzy Inference System Principles

We have choose a fuzzy inference system as a classification approach for several reasons: first of all, it is a light system, with few of parameters. And also, its flexible nature allow to adapt it easily to absorb a new knowledge.

Description

The classifier is formalized by an order 0 Takagi-Sugeno FIS [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]. The fuzzy rules make a link between intrinsic models that describe the properties of the handwritten characters and the corresponding label. Each intrinsic model is defined by a set of fuzzy prototypes P i in n dimensions. For a K classes problem, a rule R i is built for each P i : IF X is P i THEN s i 1 = a i1 and ... and s i c = a ic and ... and s i K = a iK , where X is the feature vector of the character X to recognize. As each prototype can participate to the description of each class, the rule R i has numeric conclusions connecting P i with each class c = 1..K by a prototype score s i c . The a ic is a weight that expresses the participation of P i in the description of the class c.

Learning

The fuzzy prototypes are learned separately on each class by using an unsupervised clustering algorithm. Fuzzy prototypes P i are defined by their membership function β i ( X) (eq. ( 1)). This one is an hyper-ellipsoidal Radial Basis Function (RBF) with a center µ i . Its shape is given by a covariance matrix Q i using the Mahalanobis distance d Qi ( X, µ i ) [START_REF] Krishnapuram | A possibilistic approach to clustering[END_REF]:

β i ( X) = 1/(1 + d Qi ( X, µ i )). (1)
The conclusions a ic of each rule are computed with the pseudo-inverse method or back-propagation algorithm. It gives the optimum values to discriminate the classes.

Recognition

To recognize an unknown character X, its membership degree to all fuzzy prototypes is computed (eq. ( 1)) and the sum-product inference is used to compute the system outputs which are scores s c for each class.

s c = N i=1 β i .s i c N i=1 β i , (2)
where β i is the if-part activation of the rule R i for X, N is the number of rules and s i c is the prototype score given by the rule i for the class c. The decision is given by choosing the class having the best (maximum) score.

Fast incremental learning phase

In this phase, the incremental learning process consists of two principal algorithms: prototypes creation and conclusion calculation. As mention above, artificial handwriting generation techniques are used in order to achieve high recognition rate by few learning examples.

Artificial handwriting generation

In order to achieve fast incremental learning and to overcome the lack of data available at the beginning of learning process, there are two strategies to artificially build a training dataset from few samples of writer:

Generating noised-features data. In this strategy, after computing the n-dimensions features which represents the original character, we add noise to each feature value in order to create new points in the classification space. This basic technique can be applied to generate a sufficient learning dataset for the classification models which require a minimum number of examples to represent the new knowledge, for example MLP and SVM based systems. In our system, there is no pre-condition on the size of data in order to create a new prototype; that is because we can create one simple round prototype around the original character point. So technically, this type of data generation is not useful. On the other hand, this mathematical approach of data generation may not help to create a representative prototype, and it may not be capable to model user writing style.

Generating synthetic handwritten characters. As proposed in [START_REF] Harold | Synthetic on-line handwriting generation by distortions and analogy[END_REF], different online handwriting generation techniques can be used to build artificial handwriting dataset, in respecting the user style representing by the original example. This kind of generation help the classifier not only to build a sufficiently large data set, but also to improve the recognition rate faster, and finally to help the recognition system to models the writer style without need of a large number of real examples. In [START_REF] Harold | Synthetic on-line handwriting generation by distortions and analogy[END_REF], three possible strategies to generate synthetic handwritten characters are proposed. The first one uses classical image distortions, such as scaling and slanting. The second one is based on the particularity of on-line handwriting; it applies two online distortions on the character: Speed Variation and Curvature Modification. The third one is based on the use of analogical proportion on handwriting.

Prototypes creation

In the used FIS, several fuzzy prototypes have to be created to represent each class (character). In this phase, we start with an empty set of prototypes. So we need to create as much as possible and as soon as possible of prototypes to cover the classification space. Due to the lack of knowledge in the beginning of learning, generating synthetic handwritten characters is an important and indispensable source of knowledge, in order to enhance prototypes creation, in term of quality and quantity.

One prototype will created for each new example of the character during the fast incremental learning phase, until introducing N examples per class. Then the system switches to the adaptation phase, where the prototypes are moved and deformed to accommodate the knowledge acquired by the new examples.

The fuzzy prototypes are learned by using unsupervised clustering algorithm based on the possibilistic Cmeans. In the fast incremental learning problem, we need to represent and to benefit from all the available knowledge. The unsupervised clustering algorithm neglects the isolated points considering them as noise. Indeed, in our context, these points can help to model user writing style since they result by using online handwriting generation algorithm that model handwriting deformation. To meet this need, we added an enhancement on the clustering algorithm to get it capable to cover the whole cloud of points by the minimum number of possible prototypes .

Unsupervised Classification with Reject Mechanism

The reject mechanism proposed by [START_REF] Mouchere | A unified strategy to deal with different natures of reject[END_REF] is used in our system to know how many points are covered and wellrepresented after creating the prototype. A given point is considered rejected if the activations of all created proto-types in the system by this point are less than the reject threshold. The algorithm is presented as following:

Generate np artificial points (characters). 2. Apply C-means over original and artificial points to

build the principal hyper-ellipsoidal prototype.

Conclusions calculation

For each newly created prototype, the conclusions are initialized by 1 for the conclusion that corresponds to the character class, and 0 for all other classes. Then, these conclusions are adjusted by applying a back-propagation algorithm over the examples stored in the buffer.

Adaptation phase

The writer adaptation is done during the use of the system, even during the fast incremental learning phase. So the presented approach is iterative, i.e. it uses just the last example written by the user and stored in the buffer.

In the used FIS, the adaptation can be made in several ways in order to better discriminate the classes. The prototypes used in the if-parts can be re-centered, distorted, removed. It is also possible to add new ones to take into account the specificities of the writer. The conclusions in the then-parts can also be optimized in order to reestimate the participation of prototypes in the description of each class. The following section presents in more details the approach used to make the premises adaptation by re-centering and re-shaping prototypes.

Premise adaptation: ADAPT

The ADaptation by Adjustment of ProtoTypes (ADAPT) method allows to modify all the prototypes of the FIS by re-centering and re-shaping them for each new example that is inputted. This is done according to their participation in the recognition process. We note that conclusions adaptation is done using back-propagation algorithm over the examples stored in the buffer.

Prototype re-centering

The principle is inspired by the LVQ algorithm [START_REF] Kohonen | The self-organizing map[END_REF]. The simplest supervised version (LVQ1) brings the nearest prototype closer to a correctly classified example and moves it away if the example is misclassified. This method can not be used directly in our FIS. The first reason is that our prototypes are not crisply labeled as they participate in the recognition of all classes. Further more, all prototypes are taken into account to recognize an entry.

We propose an adaptation methods called ADAPT, in which all prototype's centers are updated for each new example, and the update of each center µ i of the prototype P i must improve the score of each class. In this way, there are three reasons to have a significant displacement ∆µ i : the class score s c is different from the one expected; the participation s i c of the prototype to the final decision is high; and the activation of the premise is high. Equations [START_REF] Backer | Texture segmentation by frequency-sensitive elliptical competitive learning[END_REF][START_REF] Gary | An effective neuro-fuzzy paradigm for machinery condition health monitoring[END_REF] gives the prototype updating with the proposed method:

µ i ⇐ µ i + λ * δ ′ * ( X -µ r ) (3)
δ ′ = β i * C c=1 (b c -s c ) * s i c , (4)
with b c the expected class score for s c : 1 if c is the example class and 0 otherwise. The adaptation parameter λ lies between 0 and 1. It controls the amplitude of the displacement and thus the adaptation rate. In order to increase the speed and the robustness of the adaptation, we use a classical mechanism [START_REF] Kohonen | The self-organizing map[END_REF] consisting in decreasing the λ value. A high value allows a fast adaptation but makes it unstable. A low one allows a stable and robust adaptation but slower. So a decreasing λ allows a fast and robust adaptation. Here λ decreases from λ max to λ min . The same technique is used for α from eq. ( 5) of the prototype reshaping strategy presented in the next section.

Prototype re-shaping

The re-centering of the prototypes allows to fit the new localization of the writer data in the input space. To better represent the repartition of these new data, the shape of the prototypes must also be adapted. The shape of the prototype P i is given by the associated covariance matrix Q i . So, re-shaping the prototypes corresponds to the reevaluation of these matrices [START_REF] Backer | Texture segmentation by frequency-sensitive elliptical competitive learning[END_REF].

ADAPT proposes an iterative formula to estimate the inverse covariance matrix :

Q -1 i ⇐ Q -1 i 1 -αδ ′ - αδ ′ 1 -αδ ′ • (Q -1 i m) • (Q -1 i m) T 1 + αδ ′ ( m T Q -1 i m) , (5)
with m = X -µ r and α is the learning rate.

Prototypes creation in adaptation phase

Based on the reject mechanism used in the unsupervised classification, we can divide the character during adaptation phase into four categories: correctly classed and accepted; correctly classed and rejected; incorrectly classed and accepted; incorrectly classed and rejected.

For each class, the characters that belong to the last category are saved. Therefore, when the number of errors for one class exceeds a specific threshold E, a new prototype is created for that class using the miss-classified characters. The artificial character generation is used for the prototype creation. In order to limit the number of prototype creations, the threshold E is incremented after each creation.

Experimentation

Experimental protocol

The experiments are based on the recognition of the 26 isolated lower case Latin letters, without any constraints for the writer. The writer specific databases were written on a PDA by 11 users. Each writer has inputted 40 times each characters i.e. 1040 characters per writer. To estimate the adaptation performances for each writer, we proceed by a 4-fold cross-validation technique (called 4f c-v). 3/4 of the writer database (780 letters) is used to learn and adapt the system and 1/4 (260 letters) is used to evaluate the results of the learning process to this personal handwriting style. The presented results are the average of these 11 tests.

The ADAPT method uses the following parameter values that were found experimentally: λ decreases from 0.05 to 0.005 and α decreases from 0.005 to 0.001. The data buffer has a length of M = 2. The characters are described by a set of 21 features similar to those used in RESIFCar [START_REF] Anquetil | Integration of an on-line handwriting recognition system in a smart phone device[END_REF]. The class description is made by defining few hyper-ellipsoidal prototypes, and several round prototypes for the rejected points. The fast incremental learning phase for each class finishes after introducing 10 examples of that class. Then, the adaptation technique with more strict prototype creation method is used. For each character, the handwriting generation strategy is used to generate 300 synthetic characters. Image distortions and on-line distortions are used for the generation.

Four different experiments has been made to show the effect of using synthetic characters generation, the reject mechanism for the unsupervised classification, and the adaptation process on the evolution of the average recognition rates.

Experimental results

Figure 1 illustrates the average recognition rate evolution for the 11 writers. The effect of using the adaptation process and the reject mechanism on this evolution are presented in this figure. Also, this figure represents a comparison between using synthetic handwritten characters generation with regards to noised-features points generation. Table 1 summarizes the experimental results by comparing the recognition rates at different positions during learning process. The number of created prototypes is also presented in this table (in round brackets).

An error reduction of about 25% can be achieved in using synthetic handwritten characters generation, instead of noised-features points generation. It can be noted that eventhough the role of adaptation process is more impor- tant during the adaptation phase, it is also useful during the fast incremental learning phase. For instance, after indroducing 3 examples of each class, 10% of recognition errors are avoided by using the adaptation. It can also be mentioned that using the reject mechanism does indeed improve the recognition rate, especially at the early stage of learning. By example, for also 3 characters per class, using the reject reduces the global recognition errors by 11%. We can note that the fact of minimizing prototypes

Conclusion and future work

In the context of online handwritten character recognition, we have presented a new strategy for fast online incremental learning based on fuzzy classifier. This approach is able to learn a new character class with few learning data. Results have shown that high recognition rate can be achieved after introducing more than 5 examples per class.

Future works can focus on two points. The first one is the inhibition of some prototypes over the time. At the early stage of learning, a significant number of prototypes are created to rapidly maximize the recognition rate. Some of these prototypes can be later inhibited to enhance system performance. The second point is to use the proposed approach to build a dynamic online handwritten gestures recognition system for real context application.

Figure 1 .

 1 Figure 1. Evolution of the recognition rate during the two learning phases(%)

Table 1 .

 1 Recognition rates (%) and prototypes count adaptation phase has no negative effect on the recognition evolution rate, providing the moment of changing the phase is well chosen.

	Nb. characters
	per class	1	3	10	30
	Adaptation &	70.0	84.1	91.7	94.8
	Noised features	(26)	(52)	(260)	(266)
	Adaptation &	73.1	87.9	95.3	97.1
	Synth. characters	(26)	(52)	(260)	(262)
	Reject &	74.6	88.1	94.6	95.3
	Synth. characters (198) (580) (1834) (1914)
	Reject &	75.1	89.2	95.7	97.3
	Adaptation &	(200) (584) (1850) (1872)
	Synth. characters
	creation during the


Keywords: Eddy-currents, power losses, skin effect. Permanent magnet machines, finite-element methods

Rare Earth Permanent magnet (PM) synchronous machines with the pulse width modulated (PWM) supply are exposed to magnets' heating due to harmonic fields which rotate in relation to rotor. In cases when skin effect is negligible, the corresponding eddy-current losses in magnets can be reduced by segmentation of magnets. In this paper we propose an analytical method to evaluate the effect of both circumferential and axial segmentations on losses' reduction in various conditions concerning the skin effect. With the skin effect operating, the magnets' division may lead to increase rather than decrease of the eddy-current losses. A criterion to determine range of parameters leading to this unexpected situation is presented. Finite element calculation confirms the results of the analysis.

I. INTRODUCTION

HEN designing high-speed permanent magnet (PM) machine, an effort to obtain eddy-current losses reduction in the magnets is necessary in order to improve their performances and diminish the risk of demagnetization.

Circumferential segmentation [START_REF] Polinder | Eddy-current losses in the permanent magnets of a PM machine[END_REF]- [START_REF] Toda | Rotor eddy-current loss in permanent magnet brushless machines[END_REF] of PM has been considered as efficient methods of reduction of the eddycurrent losses, in cases when pole-arc width and radial dimension of magnet segments are less than the skin depth of interest. Axial segmentation, initially considered inefficient [START_REF] Altallah | Rotor loss in Permanent-magnet brushless AC machines[END_REF], has also been proved beneficial for reduction of losses [START_REF] Ede | Effect of Axial segmentation of permanent magnets on rotor loss in modular permanent-magnet brushless machines[END_REF], [START_REF] Yamazaki | Loss Investigation of interior permanentmagnet motors considering carried harmonics and magnet eddy currents[END_REF].

An analytical equation of the eddy-current losses in magnets, due to time harmonics in stator currents, has been developed [START_REF] Polinder | Eddy-current losses in the permanent magnets of a PM machine[END_REF] to account for effect of PM circumferential segmentations. This equation was based on assumption that the pole-arc dimension of magnet is so small that flux density over it is uniform. Performance of this equation has been proved by comparing calculated and measured impedances of PM machine for different harmonics at 1~100kHz.

In [START_REF] Altallah | Rotor loss in Permanent-magnet brushless AC machines[END_REF][START_REF] Toda | Rotor eddy-current loss in permanent magnet brushless machines[END_REF][START_REF] Ede | Effect of Axial segmentation of permanent magnets on rotor loss in modular permanent-magnet brushless machines[END_REF], analytical method for predicting the eddy-current losses in magnets, due to space harmonics of magneto motive force (MMF), has been presented and used to quantify the efficiency of one dimensional (circumferential or axial) segmentations of PM when looking for loss reduction. The reported analytical techniques to evaluate the eddy-current losses, for the cases when skin depth at the frequencies of interest is greater than both pole-arc and radial dimensions of magnets, were validated by 2-D time-stepping finite element (FE) analysis [START_REF] Altallah | Rotor loss in Permanent-magnet brushless AC machines[END_REF][START_REF] Toda | Rotor eddy-current loss in permanent magnet brushless machines[END_REF], and by 3-D magneto-static FE analysis [START_REF] Ede | Effect of Axial segmentation of permanent magnets on rotor loss in modular permanent-magnet brushless machines[END_REF].

However, the condition that "skin depth is greater than both pole-arc width and radial height", is not always satisfied. Out of the concerned range of parameters, PM segmentation may increase rather than reduce the eddy-current loss. We will call the unexpected deviation of this rule the "anomaly of segmentation".

Recently, the anomaly with respect to axial segmentation has been shown in [START_REF] Yamazaki | Loss Investigation of interior permanentmagnet motors considering carried harmonics and magnet eddy currents[END_REF] for Interior PM motor, using 3-D timestepping finite element nonlinear analysis with very important calculating time. To explain these results, authors introduced complicate theoretical solutions of the eddy-current losses in thin conductor, when the uniform magnetic field is applied. When the skin depth δ is large enough in relation to the width w and the length l of magnet segments, an approximation was made to obtain a simpler equation of losses, which represents the reduction of losses with respect to axial magnet divisions. However, we estimate that it's still rather complicated and we need numerical calculation to comprehend the effect of magnet segmentations.

On the other hand, when δ is much smaller than w and l, another approximation was given to represent the increase of losses with respect to magnet divisions. By observing the results, authors determine that the eddy-current losses become maximum when the length of conductor is nearly twice of the skin depth.

In our paper, a surface-mounted PM synchronous machine with 72 stator slots fed by pulse width modulated (PWM) voltage supply is designed with quasi sinusoidal winding In case of non-sinusoidal supply the eddy-current losses cannot be given by scalar summation of separately predicted loses for each harmonic. As a matter of fact, the "relevant harmonics", e.g. the 5 th and the 7 th (or the 11 th and the 13 th ) time harmonics induce in rotor the eddy-currents with the same 6 th (or 12 th ) harmonic frequency but in opposite directions (-6ω and +6ω). Superposition of individual loss may differ substantially from losses dissipated by composed fields [START_REF] Kaczmarek | Magnetic Losses Evaluation in PM SM Drive with Non Sinusoidal Supply by Finite Elements Analysis and Superposition Principle[END_REF].

Instead of superposition of losses separately resulting from the two relevant harmonics, in this paper we will firstly calculate the composed flux density and then evaluate the corresponding losses. The aim of this paper is to determine by simple criterion the range of parameters within which the anomaly of PM segmentation can take place. In other words, only out of this range we can try to optimize the PM segmentation in both the circumferential and axial directions. The analytical predictions of the eddy-current losses with effect of magnet segmentations are validated by 3-D FE harmonic analysis with locked rotor technique, taking about 1 h for PC calculation, the time amount largely inferior when compared to that reported with time-stepping method.

II. EDDY-CURRENT LOSSES IN CASE OF NEGLIGIBLE SKIN EFFECT

A. Traditional Analytical Model

An analytical model (1) of the eddy-current loss per unit volume P v [START_REF] Polinder | Eddy-current losses in the permanent magnets of a PM machine[END_REF], [START_REF] Leander | Electromagnetic and electromechanical machines[END_REF]- [START_REF] Kirtley | Eddy Currents, Surface Impedance and loss Mechanisms[END_REF], often applied to laminated steel, can be used to explain the idea of PM pole-arc segmentation for reducing eddy-current loss.

ρ π 6 2 2 2 2 w B f P m v = (1)
where ρ is PM resistivity, w is pole-arc width of PM segmentation, B m is maximum instantaneous flux density in magnets and f is its frequency. It shows that the eddy-current losses per unit volume are proportional to square of the width w. The losses diminish when the magnets are divided into smaller segments. In this equation, following assumptions are used:

1) The width w is less than the skin depth so that the skin effect can be neglected.

2) The magnetic flux density B is homogeneous over the PM width.

3) The width w is much smaller than the PM segmentation length l, so that end effect can be neglected.

However, all these three assumptions may be controversial in general cases. For example, the widths of magnet segments between 10 mm and 70 mm may be greater than the skin depth

θ 0 θ 0 + p N θ π 3 2 α 1 α 2 α k α Nθ ∆θ w/2 l/2 dy=l/w*dx h dx dy B 1 2 3 4
x y Fig. 1. Cross section of one pole magnet and one of its segmentation of interest, and the flux density resulting from stator currents can't be considered homogeneous over a large width of magnet (e.g. 70mm, covering about 1/3 of one pole), even if the skin effect could be neglected. In fact, the flux density is sum of sinusoidal functions of time and mechanical angle θ m :

( )

∑ ± + = m h h p t h B t ( θ φ ω θ cos ) , B (2)
where B h is magnitude of harmonic flux density, hω and φ h are its angular frequency and phase angle, and p is pole pairs number. Besides, magnets being rarely in elongated form, the widths are often comparable to the lengths. We should note that the assumption of homogeneous B can be valid, in any case, only when the skin effect is negligible. Therefore, we should, at first, determine the range of parameters, in which not only the skin effect can be neglected but also the B can be considered as homogeneous.

B. Conditions for Assumption of Homogeneous Flux Density

With both half the width and half the length of segmentation (Fig. 1) less than the skin depth of interest δ (w/2 <δ and l/2 <δ), or with only half the width less than the skin depth and the width much less than the length (w/2 <δ and w<<l), we can consider the skin effect negligible in magnet segments.

Under this condition, the flux density over the width of magnet is a sinusoidal function [START_REF] Altallah | Rotor loss in Permanent-magnet brushless AC machines[END_REF]. The flux, which penetrates the k th magnet segment in Fig. 1, can be estimated either by integration:

∫ ∆ + ⋅ ⋅ ⋅ = Φ θ θ θ θ θ 0 0 ) , ( d r l t B ( 3
)
or by multiplication between an "average" flux density value at center of segment and segmentation surface area:

θ θ θ θ ∆ ⋅ ⋅ ⋅ ∆ + = = Φ r l t B ) , 2 1 ( 0 (4) with p N θ π θ 3 2 = ∆
, N θ : number of pole-arc segments, and the PM covering only 2/3 of one pole. The equation ( 4) gives good evaluation, compared with (3), when N θ is greater than 2. This approximation has been verified both for unique harmonic +6ω or -6ω, and for the composed one (+6ω ∧ -6ω).

We can note that the condition to neglect skin effect is more strict than N θ >2 for our studied machine. In other words, when the skin effect is negligible, the assumption of homogeneous flux density over magnet segment can also be established. Under this condition, we can evaluate the effect of PM segmentations on the reduction of losses.

C. Reduction of Eddy-Current Losses by Circumferential and Axial Segmentation

Consider a PM segmentation with width w and length l (Fig. 1). We assume that the eddy-currents circulate in paths in rectangular form with a fixed ratio between width and length, e.g. w /l. Time variant flux density B induces voltage e in one of these paths, 1234 (Fig. 1):

( ) ( ) dt t d x w l dt S t d e k k , 4 , 2 α θ α θ = - = ⋅ = - = B B (5)
where S being area closed by the path 1234 is given by 2x*2y=4l/wx² with x/y=w/l, the flux density B is vector sum of all its harmonic components (2) and α k is centre of the k th segment. Resistance R of this path is given by:

hdx w l l w x hdx y hdy x R       + =         + = 4 4 4 ρ ρ (6)
where h is the radial height of magnets and dy=l/w*dx. Thus, differential power losses are:

( ) ( ) dx dt t d hx l w w l R e dP k ind 2 3 2 2 , 1 1 4       = + ⋅ ⋅ = = α θ ρ B ( 7
)
The instantaneous eddy-current losses in magnet segment can be obtained by integrating x from 0 to w /2:

( ) ( ) 2 2 2 2 0 _ , 1 16         = ⋅ + ⋅ = = ∫ dt t d l w w VOL dP P k w ind k ind α θ ρ B ( 8
)
where VOL being volume of the magnet segment is given by w*l*h. In order to obtain the average losses in one period, we must calculate the rms value of deviation of the flux density B.

) + ( + + ( + ) + ( Β + ) + = + = θ ω θ ω θ ω θ ω (9) we obtain 2 _ 12 2 _ 6 2 2 2 2 ) cos( 2 1 2 1 ) cos( 2 1 2 1 rms rms D C B A rms f f CD D C AB B A f + =       - + + +       - + + = θ θ θ θ (10)
It means that we can separately calculate the rms value of the components with different frequencies (e.g. f 6 and f 12 ) and then, get the integral rms value by root square summation. However, this does not work for the "relevant harmonic" components rotating with the same frequency in opposite direction. The integral rms value of f 6 , composed of two components with the same frequency (e.g. f 6A and f 6B , the indices A and B for opposite direction of rotation), dependent upon their phase difference (e.g. ABcos(θ A -θ B )), can't be obtained simply from their own individual rms values (e.g. 1/2A² et 1/2B²).

In such cases, we should calculate the vector sum of the two flux density components with the same frequency, and then evaluate losses due to the composed flux density, as has been shown in [START_REF] Kaczmarek | Magnetic Losses Evaluation in PM SM Drive with Non Sinusoidal Supply by Finite Elements Analysis and Superposition Principle[END_REF]. Taking the harmonics 6ω for example, the composed flux density is:

( ) ( ) k k k m p t B p t B t B α ϕ ω α ϕ ω α θ + + + - + = = - - + + - ∧ + 6 6 6 6 6 6 6 cos 6 cos ) , ( (11)
where B +6 is flux density magnitude and φ +6 is phase angle of the +6ω component, B -6 and φ -6 are those of the -6ω component.

The time average eddy-current losses in single magnet segment due to the relevant time harmonics become: (  ) with w=W/N θ and l=L/N Z , W: total pole-arc width of magnets on one pole, L: total axial length of magnets on one pole, N θ : number of pole-arc segments, N Z : number of axial segments

( ) ( )                         - - ⋅ + - ⋅ + ⋅ = - + - + + -
                - +       - - ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ = + - = - + - + ∑ 2 2 6 6
We have proved that the function F/N θ in ( 13) is quasiindependent of N θ for N θ >2 (Fig. 2) but it depends on the phase difference between two relevant harmonics ( ) 2 Comparing to the traditional analytical model ( 1) with only w taken into account, we consider magnets in any rectangular form, with both dimensions w and l explicitly engaged (Fig. 1).

In the contrary to the equation proposed in [START_REF] Yamazaki | Loss Investigation of interior permanentmagnet motors considering carried harmonics and magnet eddy currents[END_REF], this relationship is simple and clearly represents the effect of circumferential and axial segmentation on the eddy-current losses.

In short, to exploit equation ( 13) for eddy-current losses prediction, we should:

(a) determine skin depth δ (b) determine the parameter range where skin effect is negligible : (w/2 <δ and l/2 <δ) or (w/2 <δ and w <<l) (c) in these cases, apply equation (13) to evaluate eddycurrent loss reduction due to PM segmentations Whenever the skin effect is operating, the analytical model ( 1) and (13) are inappropriate. The skin effect leads to nonhomogeneous flux density and lowers the total flux passing through the magnets. We will discuss in next chapter influence of the skin effect on the efficiency of PM segmentation to reduce the eddy-current losses. 13) is independent of Nθ, which implies that the term between parentheses is also independent of Nθ.

III. EDDY-CURRENT LOSSES IN CASES OF THE SKIN EFFECT OPERATING

A. High Frequency Eddy-Current Losses

A useful formula (14) for high frequency losses (10kHz with δ=0.078mm, for lamination thick 0.356mm) has been reported in [START_REF] Brauer | Laminated steel eddy-current loss versus frequency computed using finite elements[END_REF] to evaluate the eddy-current losses in laminated steel when skin depth δ is much less than (actually at least 5 times smaller) the material thickness. where H tan is the peak tangential incident magnetic intensity and R s is the surface impedance in ohm per square given by 1/δσ. It shows that specific eddy-current losses per tangential surface are independent of material dimensions. We apply this formula for losses in magnets, S being given by 2*(l+w)*h. Thus, the eddy-current losses per unite volume are:

( )       + ∝ + ⋅ = = w l l w wl S P VOL P P v 1 1 2 (15)
When diminishing the width or/and the length of magnet segments, the eddy-current losses increase. These are conditions of unexpected deviation from the classical rule of segmentation which states that the eddy-current losses in magnet diminish as we proceed with its division. Actually, they can rise. We call it the "anomaly of segmentation".

B. Any Frequency Eddy-Current Losses

A general formula has been presented in [START_REF] Bertotti | Hysteresis in magnetism-For physicists, materials scientists, and engineers[END_REF] for any frequency, but only for conductor in elongated form with w<<l:

δ δ δ δ µ ρπ / cos / / sin / 1 2 3 w w ch w w sh B w f P s V + - = (16)
where B s is peak of surface flux density and µ is magnet permeability.

According to this equation, when diminishing the magnet's width w, the total eddy-current loss will increase at first and then decrease. This confirms both "anomaly of segmentation" and the classical reduction of the losses with magnet segmentation.

Using this equation, we can detect maximum eddy-current losses with respect to PM circumferential segmentation, i.e. "peak of eddy current losses". This peak separates two situations: (i) segmentation of PM increases the eddy-current losses; (ii) segmentation of PM reduces the eddy-current losses. However, under the condition w<<l, only the circumferential segmentation effect can be taken into account.

C. Peak of Eddy-Current Losses versus Circumferential and Axial Segmentations

The aim of this section is to find out a simple criterion to detect the peak of eddy current losses with respect to circumferential and axial segmentations. We begin with analyzing distribution of flux in conduction area with skin effect.

The flux density in the magnet's volume (Fig. 1) can be represented as a function of the pole-arc coordination x with Bs being the flux density on the magnet's surface:

( ) ( ) δ δ 2 ) 1 ( cos ) 1 ( cos ) ( d j x j B x B s + + = (17)
The flux passing through the surface (w*l) of the conducting area is given by:

( ) ( )       + + ⋅ ⋅ ⋅ = ⋅ ⋅ = Φ ∫ δ δ 2 1 tan 1 1 2 ) ( 2 / 0 w j j w B l w dx l x B s w (18)
With the ratio w/2δ much less than 1 (skin effect negligible), we can consider the flux density within the magnet's volume as homogeneous and equal to the surface flux density Bs. On the contrary, with w/2δ higher than 1 (skin effect operating), the skin effect prevents homogeneous penetration of magnetic flux into the conduction area. The flux density is higher in proximity of two edges at x= w/2 and x=-w/2 than in the center at x=0 (Fig. 3.a). For w/2δ ≥ 2, we can approximate the total flux by: ( ) Considering model of Fig. 3, we can predict the existence of a simple criterion to estimate location of the peak of eddycurrent losses with respect to circumferential and axial segmentations of PM:

        ⋅ ⋅ = ⋅ ⋅ ⋅ ≈ Φ s s B l w B l w 2 2 2 1 2 δ δ
2 2 ≈ δ w and 2 2 ≈ δ l (20)
This is the generic condition to assure "absolute nonhomogeneity" of the flux density, which permits to model the skin effect by two bands of homogeneous flux density. In other words, it's the limit to authorize the approximation of (19).

IV. FINITE ELEMENT SIMULATION

3D harmonic finite element (FE) analysis has been used to validate the proposed prediction technique. Only one of the six pole-pairs in studied PM synchronous machine has been modeled, with periodic boundary conditions in θ direction (Fig. 4). In Z direction, with aid of periodic boundary conditions, we can model part of machine corresponding to one magnet segment length, (e.g. 25mm when number of axial segments N z =10), instead of the entire length of machine (250mm). The PM covers only 2/3 of a pole pair arc.

Since the analytical method does not consider losses resulting from variation of magnet working point due to stator slot opening, its results can only be compared with those from FE calculation with the magnets unmagnetized and modeled as simple conductor. The ignored contribution however can frequently be considered as negligible.

To overcome time consuming problem of 3D time-stepping FE analysis, the locked rotor technique is applied. We determine the harmonic frequency of stator currents so that harmonic MMF rotates around the locked rotor at correct relative angular velocity, e.g. 6ω (1800Hz), 12 ω (3600Hz), 18 ω (5400Hz)… etc. number of pols arc segments Nθ eddy-current losses (W) Fig. 5. Variation of the eddy-current losses with respect to number of pole arc magnet segments at f=5400Hz and with only one segment in Z direction. Three groups of relevant harmonics, inducing eddy-currents at 1800, 3600 and 5400Hz, are discussed. At 5400Hz, the skin depth δ being 6.2mm, the peak of losses versus the pole-arc segmentations, by keeping Nz=1, locates at N θ =4 where w/2δ=1.4 (Fig. 5). Circumferential segmentations rise at first and then reduce the eddy-current losses.

Similar phenomena can be found with axial segmentations. The peak of losses versus the axial segmentations, by keeping Nθ=1, locates at Nz=11 where l/2δ=1.8 (Fig. 6).

Within this range of parameters, N θ ≤ 4 and Nz ≤ 11 at 5400Hz, circumferential and axial segmentations can not lead to the expected reduction of losses in magnets.

At lower frequencies, the skin effect being less remarkable, the range of parameters, within which the anomaly of PM segmentation takes place, is reduced. For example, at 3600Hz (Fig. 7) the range is: N θ ≤ 3 (w/2δ=1.5) and Nz ≤ 10 (l/2δ=1.6) and at 1800Hz (Fig. 8), it is: N θ ≤ 2 (w/2δ=1.6) and Nz ≤ 7 (l/2δ=1.7).

In Fig. 8, we can note that the segmentation, in direction of magnet's minimum dimension over the surface perpendicular to flux direction, i.e. circumferential (pole arc) direction, is more efficient than the axial one.

In cases when skin effect can be neglected, e.g. at 1800Hz for N θ =(4,5,6,10) and N z =1, or for N θ =10 and N z =(1,3,6), we can compare prediction of the effect of circumferential and axial segments by the ratio I). Taking the case with four pole-arc segments and one axial segment as base value of the eddycurrent losses, the effect of segmentations for reduction of losses is represented by a ratio less than 1.

V. CONCLUSION

In cases when the skin effect is negligible any circumferential or axial segmentations of magnets always leads to reduction of eddy-current losses. Otherwise we should count with enhancement of the eddy-current dissipation for certain schemes of magnets' division, followed by classical reduction of the eddy-current losses when proceeding with further subdivision. A simple criterion has been proposed to detect the geometries of PM segments corresponding to the point of maximum eddy-current losses.

The analysis concerning one frequency on the rotor side, it can be applied to one pair of stator harmonics which generate this frequency -e.g. 5 th and 7 th generating the 6 th in the rotor's magnets. Whenever higher harmonics are also relevant, we can easily conclude on a compromise on actual subdivision rule.

An efficient way of the finite element verification is the locked-rotor method with its simpler modeling and less computational strains comparing to the time-stepping procedures.

  time average eddy-current losses in magnets of one pole are:( )

Fig. 2 .

 2 Fig. 2. Function F/Nθ in the equation (13) is independent of Nθ, which implies that the term between parentheses is also independent of Nθ.

2 δFig. 3 .

 23 Fig. 3. (a). The skin effect leads to non-homogeneous flux distribution in magnets. This can be modeled by two bands with homogeneous flux. (b). Segmentation doubles number of bands, as well as eddy-current losses.

Fig. 4 .

 4 Fig. 4. Finite element model of one pole-pair of PM synchronous machine, with 6 PM segments in circumferential direction and 10 in axial direction. The dimension in Z direction of this model corresponds to 1/10 of the entire machine length, i.e. 250/10=25mm.

Fig. 6 .Fig. 7 .

 67 Fig.6. Variation of the eddy-current losses with respect to number of pole arc and of axial magnet segments, at f=5400Hz

Fig. 8 .

 8 Fig. 8. Variation of the eddy-current losses with respect to number of pole arc and of axial magnet segments, at f=1800Hz

  When calculating the rms value of a sum of several sinusoidal functions with different frequencies:

	f	f	6	f 12	4 4 3 4 2 4 1 A t A 6 sin(	4 4 3 4 2 4 1 B t 6 sin
					f	6	A	f	6	B
					4 4 3 4 2 4 1 C t C ) 12 sin	4 4 3 4 2 4 1 D t D 12 sin
							f 12	C	f	12	D

and concerns the modeling, the design and the optimization of these equipments for specific applications.