Mich Ele Sebag
email: fmichele.sebag@polytechnique.fr

Caroline Ravis
email: caroline.ravise@polytechnique.fr

Marc Schoenauer
email: marc.schoenauerg@polytechnique.fr

Controlling Evolution by means of Machine Learning

A safe control of evolution consists in preventing past errors of evolution to be repeated, which could be done by keeping track of the history of evolution. But maintaining and exploiting the complete history is intractable. This paper therefore investigates the use of machine learning (ML), in order to extract a manageable information from this history. More precisely, induction from examples of past trials and errors provides rules discriminating errors from trials. Such rules allow to a priori estimate the opportunity of next trials; this knowledge can support powerful strategies of control.

Several strategies of ML-based control are experimented on the Royal Road, a GAdeceptive and a combinatorial optimization problem. The control of mutations unexpectedly compares to that of crossovers.

Introduction

Control of evolution aims at keeping some balance between the exploitation and the exploration tasks devoted to evolutionary search 7]. This control involves both the selective pressure (the average number of o spring allowed for the best individual(s)) and the disruptiveness of evolution operators, which must be su cient to discourage premature convergence 3]. Simply put, the disruptiveness of operators is evaluated from the chances for o spring not to convey the same relevant information than parents. Only boolean search spaces and crossover and mutation operators are considered throughout this paper.

Controlling the disruptiveness of crossover and mutation can be done at three levels:

The search space can be designed as to decrease the disruptiveness of operators regarding relevant schemas; e.g. allowing don't care zones, termed introns, decreases the disruptiveness of both mutation and crossover [START_REF] Lee | Dynamic control of genetic algorithms using fuzzy logic techniques[END_REF]12]. Disruptiveness is directly concerned by the crossover and mutation rates. These may be adjusted by means of brute force 23] (still the most usual way), as well as through statistical estimations 9], or adaptation 8, 2], or evolution itself 13].

Last, the e ects of crossover and mutation can be adjusted by evolution itself [START_REF] Scha Er | A study of control parameters a ecting online performance of genetic algorithms for function optimization[END_REF][START_REF] Sebag | Controlling crossover through inductive learning[END_REF][START_REF] Schoenauer | Constrained ga optimization[END_REF][START_REF] Dejong | A formal analysis of the role of multi-point crossover in genetic algorithms[END_REF]: this only requires to include the control choices in the search space. Evolution can thereby optimize the type of crossover 29], or the mask of crossover 24], or the variance of mutation [START_REF] Schoenauer | Constrained ga optimization[END_REF][START_REF] Dejong | A formal analysis of the role of multi-point crossover in genetic algorithms[END_REF] most suited to an individual, for free.

The control of evolution presented in this paper aims at adjusting the e ects of crossover and mutation, and is originated from a common sense remark: what has been done with bad results in the past (e.g., give birth to an individual that was not to be retained in the population), needs not be repeated. Preventing evolution from repeating its past errors constitutes a safe control, i.e. a control that cannot mislead evolution. However, maintaining and exploiting the list of past errors of evolution is intractable. Therefore, the history of evolution needs be summarized and put in some tractable way. This paper investigates the use of machine learning (ML) to this end; more precisely, induction from examples [START_REF] Michalewicz | Genetic Algorithms+Data Structures=Evolution Programs[END_REF][START_REF] Mitchell | When will a genetic algorithm outperform hill-climbing ?[END_REF] is used to extract rules from the past errors and trials of evolution. Such rules enable to a priori estimate the opportunity of next trials; by means of this estimate, several strategies of control, termed ML-based controls, are allowed to command the next steps of evolution.

The paper is organized as follows. Section 2 describes the automatic extraction of rules about evolution, from examples obtained through experimenting on a population or spying evolution. The use of such knowledge in order to guide the next evolution steps is discussed, and a hybrid algorithm interleaving evolution and induction is proposed. Section 3 presents an experimental study of several ML-based controls of evolution. Besides two well-studied GA problems, the Royal Road 17,[START_REF] Mitchell | The royal road for genetic algorithms : Fitness landscapes and ga performance[END_REF], and a GA-deceptive problem 31], a combinatorial optimization problem is considered, the multiple knapsack problem [START_REF] Jones | Crossover, macromutation and population-based search[END_REF][START_REF] Mitchell | Generalization as search[END_REF]. The scope and limitations of ML-based control are discussed in section 4, with respect to related works devoted to the control of evolution 8, 24, 29] and cultural algorithms 22, 1].

2 Knowledge-Controlled Evolution ML-based control of evolution is grounded on the following remark: evolution is made of good and bad events (crossovers and mutations). This section rst shows how ML, more precisely inductive learning, can be used to characterize the classes (sets) of good and bad events, through rules induced from examples. These rules provide an a priori estimate of the class, good or bad, of new incoming events. How such rules can be used on-line by evolution, is then discussed.

Inductive Learning

Let us rst brie y introduce inductive learning (see [START_REF] Michalewicz | Genetic Algorithms+Data Structures=Evolution Programs[END_REF][START_REF] Petersen | Computational experience with variants of the balas algorithm applied to the selection of r & d projects[END_REF] for a thorough presentation).

Examples are points of the search space which have been classi ed (e.g. by an expert). The goal of induction is to extract rules from training examples; a rule can be viewed as a schema of the search space associated to a given class. A rule R covers an example i the example belongs to the schema of R. A rule generalizes an example, i it covers this example and they both belong to the same class.

E 1 1 1 1 0 0 1 good E 2 0 0 0 1 1 1 good E 3 1 1 0 0 1 1 bad E 4 1 0 0 0 1 1 bad E 5 0 0 0 0 1 1 good R ? ? ?

0 1 ? bad Induction proceeds by exploring the training examples either in a top-down or in a bottom-up way. In the top-down approach 21], one builds rules or decision trees by repeatedly selecting the most discriminant genes, i.e. the gene whose value gives a maximal information regarding the class of the examples. In the bottom-up approach 16], one starts from a given example and nds out the rules that generalize this example and maximize some user-supplied quality function.

Then the examples generalized by these rules are removed from the training set, and another example is considered. The learning algorithm used in this paper is a bottom-up algorithm that determines all rules maximally general with a given prescribed (user-supplied) accuracy; a constraint-based formalism allows to build such rules with a polynomial complexity 27].

Induction ultimately allows for classifying any point E in the search space: E is associated to the class of the rules covering E, (with majority vote in case of con icts). In case where E is not covered by any rule, it is classi ed unknown.

Examples about evolution

In order to apply inductive learning, we need examples relevant to evolution, and easy to gather. The possibility investigated in this paper is to take as examples the elementary events of evolution, namely the birth of new individuals through crossover or mutation.

Description of examples.

A crossover event is de ned by a pair of parents and the crossover mask applied on these parents. Following Syswerda 30], a crossover c can be represented by a binary mask (c 1 ; : : : c N ), c i 2 f0; 1g:

x 1 : : : x N y 1 : : : y N ! x 0 1 : : : x 0 N y 0 1 : : : y 0 N with x 0 i = x i y 0 i = y i if c i = 1 x 0 i = y i y 0 i = x i otherwise Likewise, a mutation event is de ned by a parent and the mutation applied on this parent. A mutation can also be represented through a binary mask m = (m 1 ; ::m N ), such that x 1 : : : x N ! x 0 1 : : : x 0 N with :

x 0 i =

( 1 ? x i if m i = 1

x i otherwise Both kinds of events can then be represented through the operator mask and the parent(s).

In this paper, the description of an example consists of the operator mask, and optionally the parent the operator applies on (the most t parent in the crossover case).

Then, examples must be classi ed in order to permit induction. It seems natural, as far as learning intends to serve control, to classify events as to whether they contribute to the current optimization task. The choice made in this paper is the following: the class of an event depends on the way the tness of o spring compares to the tness of parent(s). The class of an event is: good if the (best) o spring has higher tness than the (best) parent, bad if the (best) o spring has lower tness than the (best) parent inactive if the (best) o spring and the (best) parent have the same tness.

Acquisition of examples. At the moment, examples are gathered through experimenting on a given population, termed reference population:

1. An operator mask is randomly generated according to the parameters of the evolutionary algorithm (e.g. mutation rate, n-point crossover or uniform crossover,...); inactive operators are rejected (e.g. mask 00:::0); 2. One or two chromosomes (depending on whether the operator is mutation or crossover) are randomly selected in the reference population; 3. The operator is performed according to the mask and parent(s) selected. The tness of the o spring is computed and compared to that of the parent(s). This comparison determines the class of the event, good, bad or inactive; 4. The example composed of the operator mask, optionally the (most t) parent, and the associated class, is stored1 .

Rules about evolution

Rules are induced from the gathered examples. Only signi cant rules (covering more than one example) are retained.

Scope of the rules. Table 2 shows examples of 2-point crossovers together with a rule induced from these examples.

Chromosome

Mask Class Rule R states that : The crossover of an individual in schema H = 111 ? ?? according to a crossover mask in schema ?01 ? ??, gives a bad result, i.e. the o spring are less t than the parents. This can be interpreted as: don't set a crossing point between bits 2 and 3 if the parent belongs to schema H .

E 1 1 1 1 0 0 0 1 1 1 0 0 1 good E 2 1 1 1 0 0 0 0 0 0 1 1 1 good E 3 1 1 1 0 0 0 1 0 1 1 1 1 bad E 4 1 1 1 0 0 0 0 0 1 1 1 1 bad R
Rules re ect the reference population: note that R cannot be learnt before schema H is discovered, and will hardly be learnt if many individuals in the population belong to H . ML-based control. Such rules enable to a priori estimate whether a next event (crossovers or mutations) is bad, good, or inactive. This estimate can accommodate several strategies of control:

Favoring desirable events, by actuating only good events. However, this strategy would likely break the balance between exploration and exploitation in favor of the latter. Limiting the disruptiveness of operators, by rejecting bad events. This strategy of control is termed classical.

Increasing the diversity of the population, by rejecting inactive events. This strategy of control is termed modern.

Neither classical nor modern control actually breaks the balance between exploration and exploitation: rather, the rules delineate regions where exploration or exploitation have led to bad or null results. This allows to both biase exploration and exploitation toward other regions. ML-based control involves two kinds of cost:

The Limitations. The presented approach can fail in two ways: control may be disabled, or, even worse, misleading.

Control is disabled when induction fails to deliver signi cant or usable rules. This may be the case if the reference population (2.2) does not contain relevant schemas; then no trends about disruptive or inactive operators can be learnt. It may also happen that all acquired examples fall in the same class; discriminant induction then does not apply.

A much worse case is that of a misleading control, discouraging the discovery of optimal regions: control would then be properly deceptive. The deceptivity of control is to blame on the rules: Rules may become globally erroneous, for instance if the reference population is too di erent from the populations undergoing control. (Similarly, the estimations made from random individuals may be not reliable as evolution goes toward regions of better and better tness 9]). And rules may be locally erroneous, since they generalize rather than compact the available examples. However, would the rules only compact examples, they would also allow for very few classi cations, leading again to a disabled control.

Some of these limitations are addressed by the following coupling of evolution and induction.

Interleaving evolution and induction

We propose to distinguish three phases in the \game" of evolution. The beginning of the game is characterized by a (relatively) high probability of getting o spring more t than parents. During this phase, evolution obviously needs not be controlled. Practically, the rst generations do not undergo any control. ML-control then waits until relevant schemas appear, so that signi cant rules can be learnt. This prevents the rst risk of disabled control. The middle of the game is characterized by a high probability of getting o spring less t than parents. During this phase, relevant schemas likely have emerged, but not yet crowded the population. The main concern here is to limit the disruptiveness of operators, which can be done through a classical ML-control (discarding disruptive operations).

The end of the game is characterized by a (relatively) high probability of getting o spring as t as the parents. During this phase, the population is getting homogeneous. A main concern would then be to preserve the diversity of the population, which can be done through a modern ML-control (discarding inactive operations).

The deceptivity of the control is partially prevented through periodically updating the rules.

Every M generations, the reference population being set to the current population, new examples are gathered. If these new examples do not enable induction (characterized as, the fraction of examples in the majority class exceeds some user-supplied threshold D, with D < 100%) then control is disabled. The next M generations undergo darwinian evolution.

Otherwise, if the age of evolution is quali ed as \middle of the game" (characterized as, the fraction of inactive examples is less than some user-supplied threshold I, with I D), then a classical ML-control is performed in order to limit disruptiveness during the next M generations, termed classical period. Otherwise, the age of evolution is quali ed as \end of the game" and a modern control is performed in order to preserve diversity during the next M generations, termed modern period.

The number M of successive generations controlled through the same rules (in case of classical or modern periods) is experimentally set to 3: a large value of M may lead to a deceptive control in the last generations of the period; and small values of M increase the overall cost of controlled evolution, without de nite bene ts.

ML-controlled evolution can then be viewed as a mixture of darwinian, classical and modern periods. The occurrences of darwinian periods are governed by parameter D: as D decreases, the majority class tends to be represented by more than D% of the examples. Similarly, the occurrences of modern periods are governed by parameter I.

Experimental Validation

The aim of the presented experimentations is twofold. The behavior of an ML-controlled evolution is studied through varying values of D and I, which allows to compare di erent mixtures of darwinian, classical and modern periods. Besides, this approach gives a unique opportunity to study the roles respectively devoted to mutation and crossover, by comparing what happens when mutations only, then crossovers only, are controlled. Three problems are considered: the Royal Road problem 17], a GA-deceptive problem 31], and a combinatorial optimization problem 11].

Experimental settings

The evolutionary algorithm is a standard GA 7] with bit-string encoding, roulette wheel selection with tness scaling, two-points crossover at a rate of 0.6 with both o spring replacing the parents. Mutation is performed at a rate of 0.005. The evolution stops after 15000 tness evaluations. Fitness scaling is used with a selective pressure 1.2 or 2. The size of the population is 25. The ML algorithm used, called CBI for Constraint-Based Induction, is described in detail in 27].

Acquisition of examples and induction are performed every 3 generations, the rst three generations being darwinian.

The results are given in terms of percentage of success over 15 independent runs (success is intended as hitting the maximum, known for all considered problems). The dynamics of evolution is visualized by plotting the average best tness (over 15 runs) obtained for a given number of tness calculations. These include of course the extra calculations required by MLcontrol.

Several evolution schemes are compared: A classical GA rst (legend GA) that serves as reference. Then two GAs with a GA-based control of crossover are experimented: the crossover control described by Spears 29] (legend Sp) where an additional bit commands the kind of crossover, uniform or 2-point, to be applied on the individual ; and the crossover control described by Scha er and Morishima 24] (legend S-M) where individuals are augmented by the crossover mask to be applied on them. Last, four schemes of ML-based control are experimented: Control applies on crossovers only, and the underlying rules are induced from only examples of crossovers (legend X-X).

Control applies on crossovers and mutations, and the underlying rules are induced from only examples of crossovers (legend X-XM).

Control applies on mutations only, and the underlying rules are induced from only examples of mutations (legend M-M).

Control applies on crossovers and mutations, and the underlying rules are induced from only examples of mutations (legend M-XM).

The fact that a given kind of operations can be controlled through rules learnt from operations of another kind, can be justi ed as follows. Mutating an individual x through a mutation mask m can be viewed as crossing-over x with its complementary :x through crossover mask c = m. (See also 10]). This implies that rules learnt from mutations enable a too severe control of crossovers (x is usually crossed with an individual nearer to x than :x ; and crossover gives two o spring), and reciprocally, rules learnt from crossovers enable a loose control of mutations. In both cases, the control is still worth trying.

3.2

The Royal Road

The Royal Road problem was conceived by Holland and Mitchell 17] to study into details the combination of features most adapted to GA search (laying a Royal Road). An analysis of the unexpected di culties of this problem can be found in [START_REF] Mitchell | The royal road for genetic algorithms : Fitness landscapes and ga performance[END_REF][START_REF] Fogel | On the e ectiveness of crossover in simulated evolutionary optimization[END_REF]. Table 3 shows the results obtained on the Royal Road problem, modi ed as in 18], for selective pressure 1.2 and 2. Results indicated for ML-controlled evolutions correspond to the average of the results obtained for D = 95% and I in f50%; 67%; 95%g (see Table 4

A GA-deceptive problem

An elementary deceptive tness is de ned on = f0; 1g 3 , by F(x) = 3 if x = 111; F(x) = 2 for x in 0 ? ?, and F(x) = 0 otherwise. The deceptive problem we considered is composed of 10 concatenated elementary deceptive problems 31].

The percentages of success are indicated in Table 5. Results of ML-controlled schemes are averaged over 45 runs, corresponding to D = 95% and I in f50%; 67%; 95%g (detailed results

for selective pressure 1.2 are given in Table 6).

The multiple knapsack problem

The multiple knapsack problem 11] is a combinatorial optimization problem de ned as follows:

? Let P knapsacks have respective capacities c 1 ::c P , ? Let O denote a set of N objects, whose cost is respectively p 1 ::; p N , ? Let w i;j be the overall dimension of object i regarding knapsack j; Determine a subset of O, noted X = x 1 ; ::x N , with x i boolean, that is feasible, i.e. satis es the constraints relative to the maximal capacities of all knapsacks, and maximizes the overall pro t: Max f N X i=1 p i :x i ; 8j = 1::P; N X i=1 w i;j x i < c j :g Much attention has been paid to evolutionary constrained optimization [START_REF] Levenick | Inserting introns improves genetic algorithm success rate : Taking a cue from biology[END_REF][START_REF] Scha Er | An adaptive crossover distribution mechanism for genetic algorithms[END_REF]. A usual heuristics consists in reducing the tness of non feasible individuals by a penalty term. We considered a multiplicative penalization:

F (X) = 8 > < > : P N i=1 p i x i if X is feasible r 2 P N i=1 p i x i
if r is the percentage of satis ed constraints Table 7 reports the results obtained on the fourth problem de ned by Petersen 20], with N = 20 and P = 10. Similar results are obtained on the other data sets. Again, results indicated for ML-based controls are averaged on several values of D and I , which are detailed in Table 8. The dynamics of evolution (Figure 1) shows that ML-based control of mutations reaches sooner better solutions.

sel. press. GA Controlled GA GA-based Ctrl ML-based Ctrl Sp S-M M-M M-MX X-X X

Number of function evaluations

Remarks

On these three arti cial problems, the ML-based control built from examples of mutations signi cantly and consistently improves on classical GA and other GA-based controls. In the meanwhile, the ML-based control built from examples of crossovers shows disastrous.

In the Royal Road and the GA-deceptive problems, the best option is that of a classical permanent control (D = I = 95%), preventing disruptive mutations only. In the combinatorial optimization problem, the best control is also classical, but prevents disruptive crossovers as well as disruptive mutations.

Discussion

From the above results, it appears that controlling the disruptiveness of mutation can be more e ective than that of crossover. After an attempt to explain this fact, we focus on the ML aspects of the presented control, with respect to some related works.

Controlling mutation

The disruptiveness of crossovers seems at rst to deserve more attention than that of mutations, since the crossover rate is one or several orders of magnitude greater than the mutation rate 14, 29, 24, 3, 28]. However, the homogenization of population can e ciently counteract the disruptiveness of crossovers, and does so in the end of evolution. In the meanwhile, nothing can ever counteract the disruptiveness of mutations, but control. A controlled mutation then appears a powerful means to prevent the loss of near-optimal schemas in the end of evolution. This way, it improves the \memory" of evolution.

Such e ect was so far expected from selection only: the loss of good individuals can also be prevented through elitist replacement or strong selection.

If the memory of evolution is too e cient, due to either controlled mutation, elitism or strong selection, this favors premature convergence. But controlled mutation leaves less room than selection to premature convergence: First, mutation tends to increase the diversity of a homogeneous population; in opposition, selection and elitism always decrease this diversity. Second, controlled mutation tends to increase the number of active bits in a mutation mask2 , thereby increasing the mutation rate.

A ML approach

The presented approach involves three points.

First, we formalize the goal of control in terms of what should be avoided (disruptiveness or loss of diversity); in opposition, previous approaches of control rather attempt to determine what should be done [START_REF] Sebag | Controlling crossover through inductive learning[END_REF][START_REF] Scha Er | A study of control parameters a ecting online performance of genetic algorithms for function optimization[END_REF][START_REF] Schoenauer | Constrained ga optimization[END_REF]. We claim that a negative control (made of inhibitions), is safer than a positive one (made of recommendations). On one hand, suitable recommendations are outnumbered by suitable inhibitions, especially in the end of evolution. On the other hand, we know part of the suitable inhibitions (e.g., the past errors of evolution) while we know nothing like a priori suitable recommendations for non-trivial problems.

Second, we express control within the formalism of logical rules. Previous approaches aim at controlling evolution either at a global level (e.g., operator rates 8, 9, 13]) or at the level of each individual (e.g., suited type or mask of operators [START_REF] Sebag | Controlling crossover through inductive learning[END_REF][START_REF] Scha Er | A study of control parameters a ecting online performance of genetic algorithms for function optimization[END_REF][START_REF] Schoenauer | Constrained ga optimization[END_REF]). Rules o er a tractable and compact way to handle schemas of operators: the rule-based control applies on the whole population, and can still take into account the topology of the search space (e.g. don't mutate a given bit; mutate simultaneously a set of bits,: : :).

Last, we propose a procedure to extract the rules underlying control: inductive learning from examples. A further perspective of research deals with setting the rules through evolution itself: according to the fans of Nature Only, evolution can handle all choices pertaining to the representation space, and does so in an optimal way. Experimentations will tell whether rules of control are better adjusted by evolution, or faster extracted by an ad hoc external algorithm.

Cultural algorithms

The presented approach can be viewed as a particular type of cultural algorithm 22, 1]. Cultural algorithms involve both the space of individuals, and the space of schemas of individuals, termed beliefs. A belief is built by generalization of individual experience in the population; a lattice of shared beliefs is then built, which allows via a communication protocol, to biase the next operations to be applied on individuals.

Both beliefs and control rules encapsulate some knowledge about evolution, that is automatically extracted; this knowledge is similarly used to guide the next evolution steps.

The main di erence lies in the way this knowledge is updated. The update of the beliefs is mainly a cumulative mechanism, as it proceeds by generalization of individual experiences. Furthermore, these experiences themselves are biased by the control. The risk is then to gradually validate some erroneous generalizations; simply put, this mechanism is apt to build prejudices as well as beliefs.

In opposition, we propose a quite rough update mechanism: every M generations, rules are learnt anew. Furthermore, they are learnt from experiences which are not biased by the control.

In other words, the \memory" of the control is erased. This gives opportunity to get rid of old prejudices (erroneous rules). Other prejudices may be introduced, but long lasting prejudices are less likely to distort the control and the course of evolution.

Conclusion and Perspectives

This work is oriented toward building and using an explicit memory of evolution, expressed through rules. The rule formalism allows for handling knowledge that is both general (related to the whole population) and speci c (related to particular genes or sets of genes).

Rules are used to express the signi cant trends regarding disruptive and inactive operations; these are periodically built by induction from experimentations conducted on the current population. These rules enable to a priori estimate the e ects of further operations. Two modes of control are then possible: Classical control aims at preventing disruptiveness, through rejecting disruptive operations. Modern control aims at increasing the diversity of the population through rejecting inactive operations.

An hybrid evolution, interleaving darwinian periods and periods undergoing a classical or modern control, is described. The strategy of control is inspired from the analogy between games and evolution. Evolution is darwinian during the beginning phase, then it undergoes a classical control during the middle of the game, and it undergoes a modern control during the end of the game. Indicators of transition are suggested.

This approach addresses the control of both crossovers and mutations. Quite unexpectedly, experimentations demonstrate the control of mutations to be much more e cient than that of crossovers, in spite of the fact that the crossover rate is much greater than the mutation rate. A tentative explanation is given (4.1).

These results suggest several avenues for further research. First, the strategy of control could be de ned in a more exible way. For instance, the description of an individual could include the mode of control, classical, modern or darwinian, to be applied on this individual. Evolution would thereby optimize for free the strategy of control, a la Spears 29].

Second, ML-control will be experimented in the evolutionary programming frame. Further experimentations will be conducted to understand the potentialities of controlled mutation, and see to what extent it constitutes an alternative to crossover 5, 10].

Third, this approach will be extended to handle real-valued search spaces. The feasibility of this extension is straightforward: Mutation and crossover can be given a mask representation with masks in ?1; 1] N

  1g 6 representing classes good and bad, together with a rule. Induction attempts to optimize a quality function involving several features: (a) Generality, i.e. order of the schema in the rule; (b) Signi cance, i.e. number of examples the rule generalizes. (R generalizes Ex 3 and Ex 4 ), and (c) Accuracy, i.e. ratio between the number of examples the rule generalizes and the number of examples it covers (R covers Ex 3 ; Ex 4 and Ex 5 ; its accuracy is 2/3).

  acquisition of K examples implies at most 2 K tness computations. The number of examples considered by induction is experimentally set to the number P of individuals in the population. This extra cost could be avoided if examples were gathered through spying evolution instead of experimenting on the reference population. The cost of induction from examples (in O(P 2 N), where N denotes the dimension of the search space, for the learner used in our experiments).

Fitness 8 :

 8 With and without control (b) ML-controls Figure 1 : The knapsack problem. Dynamics of evolution. Sel. pressure 1.2. ; D = I = 95% ML-control D = I = The knapsack problem. Detailed results of ML-based control. Selective pressure = 1.2

  [START_REF] Levenick | Inserting introns improves genetic algorithm success rate : Taking a cue from biology[END_REF]. Many learners allow to extract rules (hyper rectangles) from examples in IR N . But unexpected problems are likely to appear with experimentations.

Table 1 :

 1 Induction from examples

Table 1

 1 shows some examples in f0;

Table 2 :

 2 Induction from examples of crossover

1 1 1 ? ? ? ? 0 1 ? ? ? bad

  for detailed results).

	sel. press. GA 1.2 80 83 GA-based Ctrl Sp S-M 73 2 93 100 100	Controlled GA ML-based Ctrl M-M M-MX X-X X-MX 93 95 55 44 100 100 95 84

Table 3 :

 3 The Royal Road. Percentage of success of GA with and without control Obviously, there is few room for control when the classical GA is e cient, i.e. for selective pressure 2. But globally, the ML-control built from examples of crossovers (X ?X and X ?XM) is harmful, and in any case much less e cient than other GA-based controls of crossover.In opposition, the ML-control built from examples of mutations (M ?M and M ?MX) reaches the same results than GA-based control for selective pressure 2., and signi cantly supersedes other evolution schemes for selective pressure 1.2.The in uence of parameters D (commanding the occurrences of darwinian periods) and I (commanding the occurrences of modern periods), is shown in Table4, and discussed in 3.5.

	ML-control D = I = 50% M ? M 73 M ? M X 47 X ? X 33 X ? X M 40	D = 67% I = 50% I = 67% I = 50% I = 67% I = 95% D = 95% 60 93 93 100 87 80 73 100 87 100 47 27 60 53 53 73 80 33 47 53

Table 4 :

 4 The Royal Road. Detailed results of ML-based control. Selective pressure = 1.2

Table 5 :

 5 A GA-deceptive problem. Percentage of success of GA with and without control

	sel. press. GA 1.2 80 83 GA-based Ctrl Sp S-M 87 2 93 90 87	Controlled GA ML-based Ctrl M-M M-MX X-X X-MX 93 93 61 42 100 100 58 49
	ML-control D = I = 50% M ? M 87 M ? M X 93 X ? X 40 X ? X M 27	D = 67% I = 50% I = 67% I = 50% I = 67% I = 95% D = 95% 73 93 100 80 100 93 80 93 87 100 53 40 73 57 53 7 47 27 13 87

Table 6 :

 6 A GA-deceptive problem. Detailed results of ML-based control. Sel. pressure = 1.2

Table 7 :

 7 The knapsack problem. Percentage of success of GA with and without control

	-MX

Note that crossing over parent1 with parent2 according to a given crossover mask may happen to be good, while crossing over parent1 with parent3 according to the same crossover mask is bad. Then, if only one parent (say parent1) is considered in the example description, one gets two examples with same description belonging to distinct classes, i.e. examples are inconsistent. Inconsistencies are still more likely, if the parent is omitted in the example description. Fortunately many learning algorithms can deal with a limited amount of inconsistencies, so this is not a real limitation.

The rst mutation examples have very few bits active. By rejecting the schemas containing some of them, mutation masks are gradually biases toward regions with more and more active bits.


E Cataldo
email: ecataldo@id.uff.br

L Monteiro

C Soize

A novel source-filter stochastic model for voice production

Keywords: Voice production, jitter, stochastic models, voice pathologies

The novel stochastic model to produce voiced sounds proposed in this paper uses the source-filter Fant theory to generate voice signals and, consequently, it does not consider the coupling between the vocal tract and the vocal folds. Two novelties are proposed in the paper. The first one is the new model obtained from the unification of two other deterministic one mass-spring-damper models obtained from the literature and the second one is to build a stochastic model which can generate and control the level of jitter resulting even in hoarse voice signals or with pathological characteristics but using a simpler model than those ones discussed in the literature. An inverse stochastic problem is then solved for two cases, considering a normal voice and other obtained from a case of paralysis on the vocal folds. The parameters of the model are identified in the two cases allowing the validation of the model.

Introduction

Voice has a fundamental importance in the transmission of knowledge, feelings and emotions. It has also an important role in the culture of a people, such as singing and acting. Over the years, researchers have had interests in some aspects of the human voice due to its peculiarities and actions in society, considering that it is the main work tool for many agents, such as singers, speakers, teachers, and others. To study the production of the human voice, a brief introduction on how its generation takes place is needed.

It should be noted that the framework of this paper is not that of voice recognition and speech synthesis but is that of the development of a simple stochastic model that allows generating voice signals with jitter, even hoarse voice and/or with pathological characteristics.

The phonation occurs when the glottis closes (adduction) and a column of air, expelled from the lungs, forces passing through the vocal folds. Then, pulses of air are produced forming a (quasi-)periodic acoustic pressure signal called the glottal signal, which goes into the vocal tract (portion that goes from the glottis up to the mouth), where it is filtered, amplified, and finally, radiated by the mouth generating the sound we hear.

The glottal signal is not exactly periodic due to small random deviations in relation to a mean value of the glottal time interval called jitter. This phenomenon has practical applications as to help in the identification of pathologies from the vocal folds, identification of voice aging, voice recognition, speaker recognition, and other [START_REF] Wilcox | Age and changes in vocal jitter[END_REF]Li et al., 2010;[START_REF] Mendonza | Classification of Vocal Aging Using Parameters Extracted From the Glottal Signal[END_REF]. There are different objective ways to measure jitter as, for example, the absolute jitter and the relative jitter. In general, values of relative jitter between 0.1% and 1.04% indicate normal voice, that is, a voice that is not symptomatic of a pathology (Wong et all, 1991).

Models of jitter could suggest or confirm the mathematical form of markers that would characterize perturbed cycle lengths statistically rather than heuristically. It has been verified empirically that vocal jitter increases for some dysphonic voices [START_REF] Pinto | Unification of perturbation measures in speech signals[END_REF][START_REF] Schoentgen | Time series analysis of jitter[END_REF]. In addition, models of jitter can be used to improve naturalness or simulate hoarse voices [START_REF] Bangayan | Analysis by synthesis of pathological voices using the Klatt synthesizer[END_REF]. Another interesting application of models of jitter is the generation of voice signals to calibrate signal processing algorithms and to help in detecting glottal cycles [START_REF] Muta | A pitch-synchronous analysis of hoarseness in running speech[END_REF]. Some stochastic models of jitter have already been proposed taking into account only mathematical expressions of the glottal signal without considering a mechanical model for the vocal folds (Schoentgen et al., 1997(Schoentgen et al., , 2001)). Other authors have recently described a mechanical model for the vocal folds considering the generation of jitter [START_REF] Cataldo | Using Bayesian method for updating the probability density function related to the tension parameter in a voice production model[END_REF]Cataldo andSoize, 2016, 2018). However these models consider the coupling between the vocal folds and the vocal tract causing a relative model complexity that induces a significant computational cost for carrying out its identification by solving a statistical inverse problem.

In this work, a simplified model is proposed for the generation of jitter, considering the unification of two deterministic models proposed by [START_REF] Qureshi | A one-mass physical model of the vocal folds with seesaw-like oscillations[END_REF] and also by [START_REF] Titze | Parametrization of the glottal area, glottal flow, and vocal fold contact area[END_REF][START_REF] Titze | The physics of small-amplitude oscillation of the vocal folds[END_REF] with posterior modifications [START_REF] Lucero | A theoretical study of the hysteresis phenomenon at vocal fold oscillation onset-offset[END_REF]Lucero et al., 2001), disregarding a coupling equation between the vocal tract and the vocal folds, and considering the stiffness as a stochastic process following the ideas proposed by [START_REF] Cataldo | Stochastic mechanical model of vocal folds for producing jitter and for identifying pathologies through real voices[END_REF].

The fact that the model is relatively primitive is exactly one of the main ideas of this work. With this model that is experimentally validated, it is possible to generate the random phenomenon that is present in all voice signals. By numerical simulation, a big dataset can be generated for different voice signals, with different levels of jitter, and for different kinds of pathologies. Such a big dataset can be used for training an artificial neural network. Finally, it should be noted that simple models have been used for the vocal folds, even nowadays, to better understand their movement as, for example, in the recently published paper by [START_REF] Lucero | Phonation threshold pressure at large asymmetries of the vocal folds[END_REF]. In addition, with this simplified model it is possible to obtain intelligible voiced sounds, characterizing normal voices but also hoarse voices or voices indicating pathologies due to the high level of jitter. It is important to note that [START_REF] Cataldo | Stochastic mechanical model of vocal folds for producing jitter and for identifying pathologies through real voices[END_REF] have used concatenated tubes to simulated the vocal tract and here only a frequency response function with bandwidths associated with the resonance frequencies is considered contributing to reduce the computational cost of the model.

Unified deterministic model for the vocal folds

The complete model presented here is based on the source-filter Fant theory (Fant, 1981), illustrated in Fig. 1. The voice signal generated, s(t), is given by the convolution of the glottal signal u g (t), the corresponding impulse response function h f (t) of the filter that models the vocal tract, and the radiation by the mouth for which the impulse response function is r(t). We then have,

s(t) = (r * (h f * u g ))(t) , (1)
or, in the frequency domain,

s(ω) = r(ω) h f (ω) u g (ω) , (2)
where . means the Fourier Transform. This equation is well discussed in the literature about voice synthesis [START_REF] Rabiner | Theory and Applications of Digital Speech Processing[END_REF][START_REF] Prasad | Backend Tools for Speech Synthesis in Speech Processing[END_REF]. With this formulation, there is no coupling between the vocal folds and the vocal tract, simplifying the model. In this paper, u g (t) is constructed using the proposed model, h f (t) is one coming from the literature and detailed later, and r(t) is a first order high-pass FIR (finite impulse response) filter, such as suggested in [START_REF] Rabiner | Theory and Applications of Digital Speech Processing[END_REF]. Before discussing the stochastic model, the also original deterministic model is constructed based on two other models from the literature. Both models generate the glottal signal and the source. The final idea is to consider the best characteristics of each model to construct the proposed unified deterministic model. The sketch considered is the one proposed by [START_REF] Qureshi | A one-mass physical model of the vocal folds with seesaw-like oscillations[END_REF] and reproduced in Fig. 2. The movement of each vocal fold is given by a rotary motion about its support point P 0 . A single mass-spring-damper system is attached to the glottis at its entrance. The model is assumed to be symmetric about its central line so that the left side of the vocal fold is the same as its right one. The glottal entry displacement is represented by x 1 (t), while x 2 (t) corresponds to the glottal exit displacement and P 0 divides the glottis in two parts that are assumed here to have the same lengths.

The characteristic of a nonlinear damping is introduced in the model, according to [START_REF] Laje | Continuous model for vocal fold oscillations to study the effect of feedback[END_REF], and the equation of the motion for a single vocal fold is described by [START_REF] Lucero | A lumped mucosal wave model of the vocal folds revisited: recent extensions and oscillation hysteresis[END_REF]:

M ẍ1 (t) + B (1 + η x 2 1 (t)) ẋ1 (t) + K x 1 (t) = P g (t) . (3)
where M, B, and K are the mass, the damping, and the stiffness, per area unit, and where η is the nonlinear coefficient damping. The mean glottal pressure P g is obtained using the Bernoulli law, and after some simplifications given by Eq. ( 4) [START_REF] Titze | The physics of small-amplitude oscillation of the vocal folds[END_REF][START_REF] Lucero | A theoretical study of the hysteresis phenomenon at vocal fold oscillation onset-offset[END_REF], is written as,

P g (t) = 2τ P sub (t) ẋ1 (t) k t (x 01 + x 1 (t)) , (4)
where x 01 is the initial glottal displacement, pre-phonation, in relation to the mean point of oscillation, in the entry, τ is a short glottal delay time, k t is the coefficient of transglottal pressure, and P sub (t) is the subglottal pressure. The input glottal area is written (Qureshi,2011) as,

A 1 (t) = 2L(x 01 + x 1 (t)) , x 1 (t) > -x 01 2Lx 01 , otherwise , (5)
and the output glottal area is given by Eq. ( 6):

A 2 (t) = 2L(x 02 + x 2 (t)) , x 2 (t) > -x 02 2Lx 02 , otherwise , (6)
in which x 02 is the initial glottal displacement, pre-phonation, in relation to the mean point of oscillation, L is the width of the vocal folds, and

x 2 (t) = -(x 1 (t) -x 0 )
, where the mean position of the oscillation is x 0 , which is specified by the horizontal position of the fulcrum point. In this text, the same values are considered for x 01 and x 02 . Finally, according to [START_REF] Qureshi | A one-mass physical model of the vocal folds with seesaw-like oscillations[END_REF] the glottal flow u g (t) is given by Eq. ( 7):

u g (t) = 2P sub (t) k t ρ A 2 (t) , ( 7
)
where ρ is the air density. After generating u g (t), the convolution with the filter (vocal tract and the mouth) should be performed to produce the sound, which will be done further in this paper. The next step is the construction of the stochastic model of the vocal folds based upon the unified deterministic model proposed here.

Proposed stochastic model

The objective is to vary the frequency of the voice signal. As the mass is fixed, the stiffness is considered as a stochastic process following some ideas proposed by [START_REF] Cataldo | Stochastic mechanical model of vocal folds for producing jitter and for identifying pathologies through real voices[END_REF] with the corresponding changes, because the model created here does not consider the coupling between the vocal tract and the vocal folds. The consideration is that jitter is generated due to the variation of the stiffness of the vocal folds, giving possible additional information to the biomechanics of the vocal folds.

In [START_REF] Cataldo | Stochastic mechanical model of vocal folds for producing jitter and for identifying pathologies through real voices[END_REF]) a more complex model was considered, taking into account a coupling equation between the vocal folds and the vocal tract. The idea is to reproduce similar results using the same consideration of the stochastic model associated with the the stiffness and to show that it is possible to generate jitter and very good intelligible voice sounds. The details about the construction of the stochastic model will not be explained here, because they can be obtained in the reference [START_REF] Cataldo | Stochastic mechanical model of vocal folds for producing jitter and for identifying pathologies through real voices[END_REF]. Only, the most important characteristics will be listed.

Let E be the mathematical expectation. The stochastic process {K(t), t ∈ R} is constructed according to the properties defined as follows.

(i) For all t in R, it is assumed that 0 < K 0 ≤ K(t) almost surely, where K 0 is a positive constant independent of t.

(ii) {K(t), t ∈ R} is a non-Gaussian stationary stochastic process such that E{K(t) 2 } < +∞ for all t (second-order stochastic process), for which its mean function (that is independent of t) is written as E{K(t)} = K > k 0 > 0, and which is assumed to be mean-square continuous in order to guaranty the existence of a power spectral measure.

A representation of the stochastic process K is chosen as described by

K(t) = K 0 + (K -K 0 )(z + Z(t)) 2 . ( 8
)
The random generator of stochastic process {Z(t), t ∈ R} can be constructed using the linear Itô stochastic differential equation,

dZ(t) = -b Z(t) dt + a dW(t) , t > 0 , (9)
with the initial condition Z(0) = 0, where {W(t), t ≥ 0} is the real-valued normalized Wiener stochastic process [START_REF] Krée | Mathematics of Random Phenomena[END_REF][START_REF] Soize | The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions[END_REF]. The power spectral density function of stochastic process Z considered here is given by Eq. ( 10):

S Z (ω) = a 2 2π(ω 2 + b 2 ) . ( 10
)
The level of jitter will mainly be controlled by a and b. Following Eq. (3), the displacement x 1 (t) of the vocal folds becomes the stochastic process X 1 (t). The dynamics of the vocal folds is then given by the following stochastic differential equation,

M Ẍ1 (t) + B (1 + η X 2 1 (t)) Ẋ1 (t) + K(t) X 1 (t) = P g (t) . (11)
All the other equation related to the unified deterministic model should be rewritten substituting x 1 (t) and x 2 (t) by random variables X 1 (t) and X 2 (t), respectively. The realizations of stochastic process K are obtained using Eqs. ( 8) and ( 9). Then the realizations of stochastic process X 1 are computed by solving Eq. ( 11) and the realizations of stochastic process X 2 are deduced. Finally, realizations of stochastic process U g are computed. The voice signal is generated through the convolution given by Eq. ( 1). The voice signals generated will indicate the presence of jitter and to quantify its level some measures can be used.

Jitter measures

Let T g be the random variable associated with the duration of the glottal cycle, which is defined as the duration between two successive times, the first one corresponding to the instant the vocal folds (glottis) opens and the second one the instant when it closes completely. To calculate T g from U g (t), it is used an algorithm based on an implementation of the RAPT pitch tracker [START_REF] Talkin | A robust algorithm for pitch tracking (rapt)[END_REF].

For each glottal cycle j, and each realization θ j of the random variable T g , a duration denoted by T g (θ j ) can be associated with. Considering that the set {T g (θ j ), j = 1, . . . , N} constitutes N realizations of random variable T g (corresponding to all the glottal cycles of the voice signal), jitter can be measured by the following equations.

(i) The absolute jitter, denoted by Jit abs , is defined by

Jit abs = 1 N -1 N-1 j=1 |T g (θ j+1 ) -T g (θ j )| . ( 12
)
(ii) The relative jitter, denoted by Jit rel , is defined by

Jit rel = 1 N -1 N-1 j=1 |T g (θ j ) -T g (θ j+1 )| 1 N -1 N-1 j=1 T g (θ j ) . ( 13
)
In general, values of Jit rel from 0.1% up to 1.04% are considered non pathological characteristics for the voice.

Simulations

In this section, simulations are performed using the proposed stochastic model. The variation of the parameter a (see Eq. ( 10)) is considered. The subglottal pressure pattern [START_REF] Lucero | A lumped mucosal wave model of the vocal folds revisited: recent extensions and oscillation hysteresis[END_REF] is given by characterized by having only poles associated with the resonance frequencies (the formants) related to voiced sounds. The transfer function considered here takes into account the speech-forming frequencies and the effects of soft-wall, friction, thermal conduction losses and radiation on lips, which will be given by the bandwidths associated with the formant frequencies. Table 2 shows the values (in Hertz) for the first four resonance frequencies of the vocal tract, the formants, for five vowels, and also losses in the vocal tract represented by the bandwidths (Bw) of the formant frequencies based on the findings of [START_REF] Titze | Benchmarks for time-domain simulation of sound propagation in softwalled airways: steady configurations[END_REF]. For the simulations, it is important to perform a convergence Parameters analysis, mainly for the solution of the Itô stochastic differential equation used to generate realizations of K(t). Let K = E{K(t)} be the mean value and K 2 = E{K(t) 2 } be the second-order moment of K(t), which are classically estimated using the asymptotically stationary and ergodic solution. The convergence for K(t) is warranty from 3 × 10 5 time steps. Figure 4 show three cases of glottal signals considering different levels of jitter, taking into account different values of a, showing that it is possible to generate the phenomenon with the model proposed.

P sub (t) =                0, i f 0 ≤ t ≤ t 0 or T f -t 0 ≤ t ≤ T f P m sin( π(t-τ) 0.280 ), i f t 0 ≤ t ≤ t 0 + 0.140 P m , i f t 0 + 0.140 ≤ t ≤ T f -(t 0 + 0.140) P m sin( -π(t+t 0 -T f ) 0.280 ), i f T f -0.140 -t 0 ≤ t ≤ T f -t 0 , (
Values M 0.476×10 -3 kg/m 2 B 1 000 N × s/m K 4.2 × 10 6 N/m 3 K 0 2 × 10 5 N/m 3 η 5 000 × 10 4 /m 2 P m 800 Pa k t 1.1 τ 0.001 s x 0 4 × 10 -5 m x 01 10 -5 m x 02 10 -5 m ρ 1.15 kg/m 3 L 0.014 m b 10 6
Table 3 shows different values of jitter, and two corresponding measures, considering different values for the parameter a, in the case of normal voices, without pathological characteristics (the first two values of a) and also three cases of voices with pathological characteristics (the last three values of a). The values presented here correspond to the vowel /a/ generated. Let F 0 = 1/T g be the random variable called the fundamental frequency. Considering a Jit abs Jit rel 160 2.27 × 10 -5 0.34% 200 3.25 × 10 -5 0.50% 500 1.72 × 10 -4 2.70% 700 1.88 × 10 -4 3.00% 1000 3.78 × 10 -4 6.13% the values of the parameter a from Tab. 3, it is possible to construct the probability density function of the random variable F 0 . The probability density functions are estimated by using the Gaussian kernel estimation method from the nonparametric statistics [START_REF] Bowman | Applied smoothing techniques for data analysis: The kernel approach with S-Plus illustrations[END_REF] and shown in Fig. 5, for all cases of the Tab. 3. Some synthesized sounds corresponding to the five vowels can be heard following the link:

www.dropbox.com/sh/fw0qmqfn3amo5yv/AABFHfe6Dt6-MpEuw7Ha9wcPa?dl=0 and are described in Tab. 4: Although the levels of jitter in Tab. 4 correspond to the vowel /a/, all of the other voice signals corresponding to the other synthesized vowels have similar levels of jitter.

Perspectives of the inverse problem

In order to validate the model proposed, parameters a, b, and K are identified using experimental voice signals. The role of K is to fit the fundamental frequency and a and b are to fit the values of jitter. This identification is carried out by introducing a cost are obtained and they will serve as start for the grid variation of the values of the parameters a, b, and K and, consequently, three loops are constructed.

• Step 3: For each value of the triplet (a, b, K), the Monte Carlo Method is used for the computation of cost function V cost (a, b, K).

•

Step 4: The minimum value of the cost function estimated in Step 3 is the objective that has to be reached.

5.2. Cases 5.2.1. First case: sustained vowel /a/ -voice signal from a woman without pathological characteristics.

After solving the inverse stochastic problem, the obtained optimal values are a opt = 100, b opt = 1 100 000, and K opt = 5 700 000 N/m 3 . Table 6 shows the values of the expected value of the fundamental frequency and the values of jitter calculated for the experimental voice and for the simulated voice, after solving the inverse problem. sound is synthesized considering these optimal values of the parameters, a voice signal has been simulated with the optimal values of the parameters. The sounds of the experimental signal ( f irstexperimentalcase.wav) and the corresponding optimal simulated one ( f irstsimulatedcase.wav) can be heard following www.dropbox.com/sh/fw0qmqfn3amo5yv/AABFHfe6Dt6-MpEuw7Ha9wcPa?dl=0 .

5.2.2. Second case: sustained vowel /e/ -voice signal from a woman with unilateral paralysis.

Similarly, after solving the inverse stochastic problem, the obtained optimal values are a opt = 200, b opt = 1 200 000, and K = 900 000 N/m 3 . Table 6 shows the values of the expected value of the fundamental frequency and the values of jitter calculated for the experimental voice and for the simulated voice, after solving the inverse problem. As a way to verify what happens when a sound is synthesized considering these optimal values of the parameters, a voice signal has been simulated with the optimal values of the parameters. The sounds of the experimental signal (secondexperimentalcase.wav) and the corresponding optimal simulated one (secondsimulatedcase.wav) can be heard following www.dropbox.com/sh/fw0qmqfn3amo5yv/AABFHfe6Dt6-MpEuw7Ha9wcPa?dl=0

Conclusions

A stochastic model to generate voiced sounds with jitter has been proposed based on the unification of two deterministic models from the literature and considering the parameter corresponding to the stiffness as a stochastic process. Two control parameters have been considered and it is shown that a model based on the source-filter theory, without taking into account the coupling between the vocal tract and the vocal folds, can generate synthesized sounds near real voices, even voiced sounds with pathological characteristics. The level of jitter is measured for each sound synthesized and also the probability density functions related to random variable associated with the fundamental frequency are constructed showing that jitter is created. Two cases are considered for the corresponding inverse problem helping to validate the model. In addition, the sounds produced with the stochastic model proposed are intelligible, which shows that the model created is very reasonable. Consequently, such a stochastic model is useful for generating big datasets involving different voice signals, with different levels of jitter, and for different kinds of pathologies. Such big datasets can be used for training artificial neural networks.

Acknowledgments

This work was supported by CNPq.

Conflict of interest statement

The authors disclose any financial and personal relationships with other people or organizations that could inappropriately influence their work.

Figure 1 :

 1 Figure 1: Sketch of source-filter.

Figure 2 :

 2 Figure 2: Sketch of the Qureshi model.

  14)in which P m is the maximum glottal pressure, where t ∈ [0, T f ]. In this work, the values considered are P m = 800 Pa, t 0 = 0.01 s, and T f = 1 s. All the other parameters are fixed and their values are given in Tab. 1. Figure3shows the graph of function t → P sub (t). In general, the transfer function corresponding to the digital filter (the vocal tract) is

Figure 3 :

 3 Figure 3: Graph of the subglottal-pressure function t → P sub (t).

Figure 4 :

 4 Figure 4: (a) Glottal signal without jitter a = 0, (b) Glottal signal with Jit rel = 0.5% and (c) Glottal signal with Jit rel = 6.13%.

Table 1 :

 1 List of parameters and their values (M , B, K and K 0 are given per area unit).

	F1	F2	F3	F4
	/a/ 860 1513 2489 3600
	/e/ 423 1899 2017 3546
	/i/ 283 2113 2800 3566
	/o/ 504 905 2624 3439
	/u/ 352 809 2394 3450
	Bw 20	25	200	50

Table 2 :

 2 Formants, in Hertz, of the vocal tract, the digital filter, for the case of vowels production and the corresponding bandwidths.

Table 3 :

 3 Absolute and local jitter for voices without pathological characteristics and also with pathological characteristics.

  As a way to verify what happens when a

	Parameters Experimental	Simulated
	E{F 0 }	272.14 Hz	273.76 Hz
	Jit abs	8.57×10 -6 s 7.72×10 -6 s
	Jit rel	0.23%	0.21%

Table 5 :

 5 Parameters for a female production of a vowel /a/, without pathological characteristics.

Table 6 :

 6 Parameters for a female production of a vowel /a/, without pathological characteristics

For each given value of vector (a, b, K), an utterance of the voice signal is obtained, which allows for computing the expected value of the random variable F f und and the values of jitter (Jit abs and Jit rel ) given by E{F 0 (a, b, K}, Jit abs (a, b, K), and Jit rel (a, b, K), respectively. For each experimental signal, it is possible to calculate the expected value of the fundamental frequency E{F exp 0 } and the values of jitter Jit exp abs and Jit exp rel . The optimal values a opt , b opt , and K opt are then computed by solving the following optimization problem, in which C is the admissible set.

(16)

Methodology used

• Step 1: From the experimental voice signal obtained with the vowel produced all the values corresponding to the random variable F 0 = 1/T g are obtained. The mean value of random variable F 0 (denoted by F exp 0 ) is calculated and it is used in the other steps. From this signal, the two measures of jitter are obtained: Jit abs and Jit rel , denoted by Jit exp abs and Jit exp rel , respectively.

• Step 2: Values for a, b, and K have been calculated so that values of the expected value of the fundamental frequency and values of jitter are not so far from those calculated for the experimental signal. This step takes time due to the essays. Values