Erwan Moreau
email: moreaue@cs.tcd.ie

Arun Jayapal
email: jayapala@cs.tcd.ie

Carl Vogel
email: vogel@cs.tcd.ie

Exploring a Large set of Parameters using a Genetic Algorithm -Notebook for PAN at CLEF 2014

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In this author verification task, a training set containing 6 datasets was provided; each dataset consists of a set of problems (between 96 and 200) which belong to the same language and genre; each problem consists of a small set (between 1 and 5) of "known" documents written by a single person and a "questioned" document: the task is to determine whether the questioned document was written by the same person. More precisely, the system must provide its prediction as a value in the interval [0, 1], which represents the probability that the answer is positive (same author). The intended interpretation is for 0 to mean "different author" with maximum certainty, and for 1 to mean "same author" with maximum certainty, and any intermediate value describes the likeliness of a positive answer, and with 0.5 equivalent to the system saying "I don't know". The predictions are evaluated using the product of the area under the ROC curve (AUC) and the modified accuracy measure c@1 [6], which treats 0.5 answers as a particular case.

We consider the task as a supervised learning problem, where, for each dataset, the goal is to find a function which, when applied to a set of unseen problems in this dataset, maximizes the performance (product of AUC and C@1). This function must be generic enough to capture the stylistic characteristics of every author. It is meant to represent how to capture any author's style within a particular dataset, that is, in the context of a particular language and genre. For example, the type of observations (e.g., word bigrams) to take into account depends on the language, but whether a particular observation (e.g., the bigram "it is") is relevant or not is specific to a given author.

We define the space of all possible functions in the following way: each function is defined by a set of parameters, each parameter being assigned a particular value among a predefined set of possible values. The process of selecting features from the texts, combining them in any predefined way, and learning how to interpret the differences between the known documents and the questioned document is entirely driven by the values taken by these parameters. For example, a parameter indicates which distance metric should be used to compare the unknown document to the set of known documents. We call a particular combination of parameters a configuration. We define the two following strategies which share only a subset of common parameters:

-The fine-grained strategy, described in §3, in which there are many possible parameters, is intended to try as many configurations (or functions) as possible, in order to maximize the performance. -The robust strategy, described in §4, is a more simple method which uses only a small subset of parameters. It is intended to be safer (in particular less prone to overfitting), but probably not to perform as well as the fined-grained strategy.

For the fine-grained strategy, the space of all possible configurations is too big to be explored exhaustively. This is why we implement a simple genetic algorithm, which is supposed to converge to a (possibly local) optimal configuration. This algorithm is described in §3.4.

General architecture

For every problem to solve, we extract observations3 from the set of known texts, and try to measure the four following abstract characteristics:

their consistency, i.e. how consistently these observations apply among the known documents written by the author; their divergence (from a reference corpus), i.e. how much the frequency of these observations differs from the reference corpus (see §3.2); the confidence of the system in the reliability of these observations; the distance between the known documents and the questioned document with respect to these observations.

We consider multiple ways to compute and use these four characteristic values. In particular, the configuration file defines:

the types of observations to take into account; the method to compute every characteristic at the observation level; the method to extract the most relevant subset of observations;

the method to obtain a global value for each of the four characteristics; which subset of these values will be used as features in the machine learning stage.

The final step consists in training (or applying) a ML model based on these features. There can actually be two models: the first and most important one predicts the scores for each case in the dataset; the second optional one is meant to detect the ambiguous cases, so that they can be assigned 0.5 instead of their predicted score, in order to maximize the c@1 score.

In the robust strategy the parameters are restricted to a small set of possible configurations, whereas with the fine-grained strategy, on the contrary, we explore a vast space of parameters (about 10 19 possible combinations in the predefined space that we use). This is why the learning stage for the latter consists in learning an optimal configuration using a genetic algorithm.

3 The fine-grained strategy

Observations and frequency statistics

We consider a large set of observations types, among which the configuration can use any subset. These are typically various kinds of n-grams, but not only:

words (actually tokens) unigrams to trigrams; -Characters trigrams to pentagrams.

-Part-Of-Speech (POS) tags unigrams to tetragrams; -Combinations of POS tags and tokens, including skip-grams, e.g.:

"<POS tag> <token> <POS tag>" or "<token> _ <POS tag>"; -"stop words" n-grams, i.e. tokens n-grams considering only a predefined list of the the most frequent tokens in the language,4 from trigrams to pentagrams; -Token length classes, where the tokens are classified depending on their length into 6 categories: lower than 2, 3 to 4, 5 to 6, 6 to 8, 8 to 10, more than 10. -Token-Type Ratio (number of distinct tokens divided by total number of words).

The POS tags are computed using TreeTagger5 [8]. The lists of most frequent words in the language are computed from the complete set of documents in the training data: we consider the 200 most frequent words, except for Dutch (100 most frequent words).

A few thresholds are applied when extracting these observations, in order to remove some noise in the data and/or improve efficiency:

-Minimum absolute frequency in a document (possible values: 2, 3, 5); -Minimum proportion of documents among the reference corpus which contain the observation (possible values: 10%, 25%, 50%); -Minmimum proportion of known documents containing the observation (only for known documents) (possible values: 30%, 51%);

The frequency of all the observations which fulfill these conditions relativized to the total possible number of such observations lpq is stored for very observation type specified in the configuration. For every observation, various statistics are computed based on the set of frequencies extracted from the known documents: mean, standard deviation, median, etc. Practically, in the training stage, the observations and the statistics are computed only once and then stored, so that the data can be used as many times as necessary with different combinations of parameters.

Features

The consistency, divergence, confidence and distance values are based on the frequency statistics extracted during the first step. At first they are computed for every distinct observation. Then they can be "synthetized" in different ways according to the configuration; the final features can be either specific to each observation type or global.

Consistency The consistency value is meant to represent how constant the use of a particular observation is, so that it can be assessed whether the observation is used in a similar way in the unknown document. For example, the standard deviation of the (relative) frequency of the observation among the known documents is a valid indicator (the lower it is, the higher the consistency is). Other statistics are available, e.g. range between minimum and maximum, ratio between first and third quartile, etc. However these statistics are more reliable with a high number of known documents, and require at the very least two distinct documents.

The goal of the consistency measure is to distinguish as far as possible between the observations which are specific to the author and those which are only specific to the document (for example the subject of an essay). This is why the more known documents there are the most accurate the consistency is. Consequently, cases which contain only one known document are irrelevant for consistency. [START_REF] Peñas | A simple measure to assess non-response[END_REF] Divergence The divergence measure is meant to represent to what extent a particular observation is specific to an author. This value is calculated against a reference corpus, which should ideally be an independent set of documents in the same language and genre as the dataset. But since we do not have access to such a corpus for every dataset, we simply consider the whole set of documents (known and unknown) in the training set as the reference corpus. Because it is meant only to measure divergence, the only important assumption that we make is that it contains documents written by a sufficient number of different authors, and that it is not massively imbalanced (for instance if most of the documents were by the same author). [START_REF] Quinlan | C4.5: programs for machine learning[END_REF] The system can use different methods to measure the divergence of an observation: several simple statistics like the absolute difference between the frequency means (or medians), but also more complex measures which try to estimate the difference between the two distribution (known documents and documents in the reference corpus). Some of these measures assume a normal distribution of the observation frequency accross the documents: [START_REF] Schmid | Improvements in part-of-speech tagging with an application to german[END_REF] we use several measures based on the Bhattacharrya distance [1]. These measures are also more reliable when the number of distinct documents is high.

Confidence The confidence measure is intended to find the most discriminative observations, based on their consistency and divergence values. Thus, it simply combines these two values in order to rank the observations by their discriminative power for the given author: for instance, an observation which is very consistent but not distinctive (or the opposite) might be less interesting than another one which is less consistent but has a better divergence value.

We consider various ways to combine the two numerical values: product, mean, geometric mean, weighted product, etc. There is also an option to ignore the consistency score (i.e. use the divergence as confidence), and another one for ranking the two values, so that the rank is used instead of the actual score.

Distance Finally the distance measure is meant to capture how different the unknown document is from the set of known documents. The distance value is usually not meaningful at the level of the observation, but becomes meaningful only once computed on a selected set of observations.

Various standard similarity or distance measures can be used, like the cosine or Jaccard similarity, but also some more specific measures, like the probability that the observed frequency in the unknown document belongs to the distribution observed in the known documents (assuming this is a normal distribution). The distance can be weighted in different ways with a coefficient based on the confidence score.

Scoring stage

The configuration defines what kind and how many features will be used, as well as how to obtain them. Multiple possibilities have been implemented, including:

-There can be a set of features for each observations type, or all observations types can be combined in a generic set of features; -The maximum number of observations to consider for the distance feature(s); -Whether consistency, divergence and confidence scores should be included in the features.

A regression model is trained/applied to the features which have been computed for all the input cases (instances for the model). 9 We use the Weka [3] (version 3.6.10) implementation of SVM regression [4] (with polynomial or RBF kernel), and decision trees regression [7], with variants depending on their parameters.

Optionally, a second model can be generated/applied in order to evaluate the confidence in each answer, and possibly replace it with 0.5 (unanswered case). This classification "confidence model" can use any of the available features, as well as the score computed with the first regression model. 10

Genetic learning

The complete "author verification model" which is returned at the end of the training stage consists of the scoring model (that is, the regression model and optionally the confidence model), but also the configuration which was found to be optimal on the training set by cross-validation. This is achieved using the generic genetic algorithm described next.

The individuals in a population are the configurations, in which every parameter is assigned a particular value among a predefined set (the "genotype"). Starting from a random population, the algorithm iterates through each generation by selecting a proportion of the population to "breed" the next generation.

The method for making a configuration which performs better more likely to get selected is as follows: all the configurations in the population are ranked from 1 to N by their performance in ascending order. The probability of an individual being selected is defined as r/N ×p×2, where r is its rank and p is the proportion to retain as breeders. For example, if the population N = 200 and p = 10% (that is, 20 breeders are selected at each stage), the probability for the best performing configuration (with relative rank N/N = 1) to be selected is 1 × 10% × 2 = 20%, and the probability for the worst performing configuration (with relative rank 1/200 = 0.05) is 0.01. Since the average relative rank r/N is 0.5 by definition, the method selects, on average, N ×0.5×p×2 = N × p breeders, as expected (consequently p must not be higher than 0.5).

Every new individual is generated based on two "parents" picked randomly among the breeders; every of its parameter is defined as one of its parents value (each having a 0.5 probability to be picked), but can be "mutated" with a (small) predefined probability. We also use two variants: one consists in reusing a few of the previous best individuals in each new generation (elitism), and the other in including a small proportion of totally random individuals.

The robust strategy

In the robust strategy, consistency and divergence features were used to verify whether the document X has been authored by the author of the given documents Y = {y 1 , y 2 , ...y n }, but in a slightly different way as above: the consistency defines how well the words or n-grams or character-grams were used consistently used across all the documents Y and X. Whereas, divergence defines how well document X is distinct from documents Y and viceversa. The intuition behind using this feature is that these features could provide an insight into how the document X and documents Y co-vary linguistically.

Divergence Motivated from the Jaccard similarity, we use a slight variant to compute the divergence of documents Y to document X (J 1 ) and of document X to documents Y (J 2 ):

(1)

J 1 = (p + q) (p + q + r) J 2 = (p + r) (p + q + r)
where p is the number of words found in both X and Y documents, q is the number of words found in Y but not in X and r is the number of words found in X but not in Y documents. J 1 will provide a measure on how distinct Y is from X, whereas J 2 will provide a measure on how distinct X is from Y .

The above provided are the document level metrics, which are used to compute the word-level divergence for X and Y . One assumption considered here for word-level metric; to compute divergence of word x i in X to Y , when the word w i is identified in Y , we assign a boolean value 0 assuming no divergence and when w i is not identified in Y , we assign 1 to a temporary variable F assuming complete divergence of word w i . With, F , J 1 , J 2 and relative frequency values (rf 1 i and rf 2 i ) for each word, we compute the divergence for words in X to Y (d i,J1 ) and Y to X (d i,J2 ) as:

(2)

d i,J1 = F * J 1 * rf 1 i d i,J2 = F * J 2 * rf 2 i
Consistency Consistency is defined as the difference between the relative frequencies:

(3)

c i,J1 = rf 1 i -rf 2 i c i,J2 = rf 2 i -rf 1 i
These measures are based only on the characters tetragrams' frequencies (the other observations types are not taken into account). In order to train or apply the model, the scoring stage defined in the fine-grained strategy is used.

Observations and results

Genetic learning process

Since the system was being implemented specifically for the task and we had to deal with the time constraints for the competition, the process of tuning the genetic learning parameters was not carried out in optimal conditions. In particular, we could not afford to run many different cases, especially cases which require a long time; moreover, bugs were fixed and features were added along the process. This is why we are not able to provide here a very detailed analysis of the impact of the parameters on the evolution of the performance. Yet we did several preliminary tests in order to determine a couple Average performance for various combinations of genetic parameters (preliminary test stage). GeneticParams contains the following parameters: population, breeders proportion; mutation probability; elitism proportion; random proportion. Example: on the longest curve (in green with square symbols), it can be observed that the average performance at the 20th generation was close to 0.6. 12 of optimal combinations of genetic learning parameters. Figure 1 shows the results of one of these.

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In general, the system was able to converge relatively quickly (a few tens of generations at most) to a high level of performance with most tested combinations of parameters. In particular, the size of the population did not have a major impact on the convergence (even though a larger population makes the evolution smoother). This is why we opted for using a small population size, in order to minimize the time required for the system to find optimal configurations. Figure 2 shows how the two selected sets of parameters performed during the main learning stage. 13 In total, between 13,400 and 26,700 configurations (in a space of 10 19 possibilities) by dataset were evaluated in 12 Moreover, the legend shows that this value is the average over a population of 30 configurations obtained after 19 iterations, where, at each stage: 20% of the previous configurations are selected as breeders (i.e., 6 configurations); the probability of a mutation (of an individual parameter) is 0.02; 10% (i.e., 3 configurations) of the new generation is made of the 10% best previous configurations ("cloned" directly without any alteration); 5% (i.e., 1 or 2 configurations) are totally random configurations. 13 A recent server (24 Intel Xeon 3GHz cores) was used for the computation, but with only one core for each pair dataset/genetic configuration. It is difficult to evaluate exactly the total time spent due to various technical interruptions. Computing a single generation took in average between 20 and 48 minutes (depending mostly on the size of the dataset) for the "fast" configuration (30 individuals) and between 50 and 117 minutes for the "slow" configuration (75 individuals).

the genetic learning process. Each configuration was evaluated using only 3 fold crossvalidation during the main genetic process; after this stage, a subset of the best configurations found was evaluated again using 10 fold and then 20 fold cross-validation.

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Observations

Below we present some characteristics observed in the configurations which were selected by the genetic learning process (in the case of the fine-grained strategy):

Observation types In most cases only a few observations types are selected (from 3 to 11). Several POS n-grams (as well as POS/tokens combinations) are selected in the five cases where they are available; words n-grams are also selected in most cases, but characters n-grams never are. The word length and the type/token ratio are used in half of the cases. At least one knid of stop words n-grams is selected in 4 datasets.

Consistency, divergence, confidence and distance methods The Bhattacharrya coefficient is used the main criterion for divergence in most cases, and in particular in all datasets where the median number of known documents is higher than 1. The consistency value is actually not used at all in most cases (4): the system choses to use only divergence as confidence to select the relevant observations. The most simple distance metrics are selected in general (mean of the difference, euclidean or cosine distance), but half of the time the frequency is weighted with the confidence score. The number of observations taken into account seems to depend mostly on the dataset.

Learning stage

The decision tree regression (M5P) is selected most of the time for the scoring model. The confidence model stage is selected in only one case; this must be because, in general, the errors made by this classification model are more costly in performance than the benefit of assigning 0.5 scores.

Earlybird test set and final model selection

Thanks to the Tira system [2], we were able to evaluate both strategies on the "earlybird corpus". Figure 3 shows the performance of the two strategies on the training set (by cross-validation) and the earlybird test set; the results obtained on the training set by cross-validation were always better with the fine-grained strategy, but in two cases they were better with the robust strategy on the earlybird test set. This is of course an expected consequence of how the two strategies were defined: the robust strategy is usually not as good as the fine-grained one, but is less data-independent; conversely the fine-grained strategy is more prone to overfitting. But, more interestingly, we noticed that, with the fine-grained strategy, the performance drops much more (between the training set and the earlybird test set) when the dataset has only a small number of known documents by case in average, especially on the datasets in which most cases contain only one known document. In table 1 we report the performance and compute the drop in performance in all cases, and for each dataset the difference between this value and the average value; based on this difference, we can observe that the drop in performance correlates quite strongly with the (low) number of known documents by case for the fine-grained strategy, whereas it does not at all for the robust strategy.

As a consequence, we decided to use both strategies in our official submission: for the three datasets where there is only one known document (Dutch essays and reviews and English novels), the model corresponding to the robust strategy is used instead of the fine-grained model.

Results

Table 2 shows the performance obtained on each dataset by both strategies on the training set, the earlybird test set and the final test set, as well as our official ranking. 14 In particular, it shows that our decision to use the robust approach in three cases was good: it performed better than any of the two original strategies taken independently. However our hypothesis that this was linked with the low number of known documents might not hold, since our results on the English novels are quite low compared to the other participants' results, and this would not have happened with the fine-grained strategy. Overall, our system was among the best in this task, ranking third among 13 in average performance. Table 1. Comparison of the performance on the training set and the earlybird corpus, and impact of the number of known documents by case depending on the strategy. The penultimate column is the difference in performance between the two datasets, and the last column is the difference between the aforementioned value and the average value for the same strategy. Finally the Spearman correlation is calculated between the value in the last column and the mean number of known documents.

Figure 1 .

 1 Figure1. Average performance for various combinations of genetic parameters (preliminary test stage). GeneticParams contains the following parameters: population, breeders proportion; mutation probability; elitism proportion; random proportion. Example: on the longest curve (in green with square symbols), it can be observed that the average performance at the 20th generation was close to 0.6.12

Figure 3 .

 3 Figure3. Performance of the fine-grained and robust strategies on the training set and the earlybird test set (the datasets are identified by their initials, their full name can be found in table 1 below).

We use the term "observation" here to avoid any confusion with the features used in the machine learning stage.

Other tokens are replaced with a special symbol, e.g. "the _ _ is _".

http://www.cis.uni-muenchen.de/ schmid/tools/TreeTagger. POS tags are not used for the Greek dataset.

It is possible then to use different parts of the document, but this is not reliable in general since the distinction between document specific and author specific observations cannot be made.

Although it is quite unlikely given the size of the training sets, we do not have any guarantee that the second condition is satisfied in all the datasets provided. Assuming these conditions are fulfilled, the fact that the reference corpus contains documents by the author of the problem studied is not an issue, because it is frequent that a particular stylistic feature can be observed

with several authors: the relative ordering of the observations according to their specificity to an author should not be impacted.[START_REF] Schmid | Improvements in part-of-speech tagging with an application to german[END_REF] We had observed in[START_REF] Moreau | Style-based Distance Features for Author Verification -Notebook for PAN at CLEF[END_REF] that this assumption holds in most cases for frequent n-grams.

When training the model, the Y/N answers are converted to 1/0, so that the predictions of the system are values in [0, 1], which is the expected output format.

In the learning stage, the model which was trained is applied to the instances. Depending on the configuration, the instances can be split up so that the second model is based on unseen instances (but then less instances are used to train each model, of course).

The results provided at the time of writing have not been made official yet, therefore changes can still happen in the ranking.

Acknowledgments

We are grateful to the reviewers for their valuable feedback, and to the organizers of the task for their hard work and their availability.

This research is supported by the Science Foundation Ireland (Grant 12/CE/I2267) as part of the Centre for Global Intelligent Content (www.cngl.ie) funding at Trinity College, University of Dublin.


Jackie Krafft
email: jackie.krafft@gredeg.cnrs.fr

Sebastien Lechevalier
email: sebastien.lechevalier@ehess.fr

Francesco Quatraro
email: francesco.quatraro@unice.fr

Cornelia Storz
email: storz@wiwi.uni-frankfurt.de

Collegio Carlo

Alberto Cornelia Storz

Keywords: Innovation, Technological Change, Path Dependence, Economic Development, Firms, Evolutionary Economics, Regions

published or not. The documents may come

Emergence and evolution of new industries: The path-dependent dynamics of knowledge creation.

Emergence and evolution of new industries: The path-dependent dynamics of knowledge creation. An introduction to the Special Section.

The path-dependent dynamics of emergence and evolution of new industries: Introduction Highlights

 We provide the background framework to the special section  We present the special section illustrating the coherence of the contributions  We provide a short description of the papers included in the section

The path-dependent dynamics of emergence and evolution of new industries: Introduction 1

The dynamics by which new industries are created, and how these develop over time, represent a crucial issue for understanding the evolutionary patterns of capitalistic economies.

Change is indeed the main engine of growth [START_REF] Nelson | Capitalism as an engine of progress[END_REF][START_REF] Metcalfe | Adaptive Economic Growth[END_REF]. Such dynamics are grounded on the interplay between the phenomena of structural change, i.e. the change in the sectoral composition of an economic system, and technological change [START_REF] Malerba | Innovation and the dynamics and evolution of industries: Progresses and challenges[END_REF][START_REF] Quatraro | The economics of structural change in knowledge[END_REF]. Actually, the creation of new industries alters the sectoral landscape in the economy, engendering structural changes. Very often, the creation of new industrial activities is made possible by technological progress, which provides the inputs to the design of new products. Creative destruction, enabled by innovation efforts, is a key part of the process leading to the emergence of new sectors [START_REF] Schumpeter | Capitalism, socialism and democracy[END_REF]. Moreover, the change in the industrial structural structure is likely to stimulate the creative response of incumbents, which will likely undertake innovation efforts [START_REF] Schumpeter | Creative response in economic history[END_REF]. In this direction, structural change becomes both an incentive and an outcome of an endogenous process of technological change.

Considering [START_REF] Schumpeter | Creative response in economic history[END_REF][START_REF] Schumpeter | Capitalism, socialism and democracy[END_REF] but also [START_REF] Marshall | Principles of Economics[END_REF][START_REF] Marshall | Some aspects of competition[END_REF] as the founding fathers in the field, there are alternative reasons why an industry may emerge. First, the emergence of an industry may be essentially driven by technology. This calls to mind Schumpeter"s vision of creative destruction in industrial dynamics. An entrepreneur sets up a firm to introduce his invention. This firm grows and holds a monopoly position for some time, but this firm is imitated by new entrants that compete with and eventually outperform the initial firm. This situation can last until another entrepreneur develops a new project involving the exit of older and larger firms and the entry of new ones [START_REF] Saviotti | Technological evolution, variety and the economy[END_REF]. But we can also think about the process in a different manner. We can consider that knowledge and competencies drive the emergence the industry. In that case, closer to Marshall and the post-Marshallianvision, the growth of knowledge is linked to the ability of firms to ensure 1 The authors wish to thank the editor Ed Steinmuller for his guidance during the whole process leading to the publication of this special section. We thank Pablo d"Este, DietmarHarhoff, Attila Varga, Torben Schubert, Alex Coad, Brian Loasby, Martin Fransman, Guido Bünstorf, Mark Lehrer, Bill Lazonick, Stan Metcalfe, Steven Casper for their help in the process. Jackie Krafft and Francesco Quatraro acknowledge the financial support of the European Commission, within the context of the FP7 research project "Policy Incentives for the Creation of Knowledge: Methods and Evidence" (PICK-ME), Grant number 266959. Cornelia Storzacknowledges the financial support of the German Research Foundation (DFG) for the project "Human Resource Management in New Industries", Grant number STO 860/1-1. coherence between internal economies (organization and direction of the resources of the firm) and external economies (general development of the economy, including the role of firms in the neighbourhood). From this perspective, the creation of new activities may be related to the firmsopportunity and ability to accumulate specific knowledge and competencies and generate novelty [START_REF] Loasby | Knowledge, Institutions and Evolution in Economics[END_REF].

Despite the richness of these early contributions, and some notable advances in recent industrial dynamics (the work by Steven Klepper has been here central here, from [START_REF] Gort | Time Paths in the Diffusion of Product Innovations[END_REF]Klepper, 1983, to Anderson and[START_REF] Andersson | Characteristics and performance of new firms and spinoffs in Sweden[END_REF] ,the understanding of this process is still limited in the field of economics of innovation. This is all the more surprising since economic development all times has been essentially nurtured by the emergence of new industries (see over the last two decades software and biotechnology, and today nanotechnology or next generation robotics).This special section of Research Policy 2 aims to improving our understanding of processes of industry emergence and the role of path dependencies within this process, and hereby hopes to substantiate innovation research.

In this framework, bringing about some "novelty" in the market is not necessarily linked to the introduction of a radically new product or process. The degree of novelty can be a relative concept, according to which a technology or a product can be new to some contexts, even though they are not new in absolute terms. Studies focusing on "new industries", too often assume that sectors emerge from nothing. On the contrary, the analysis of the conditions underpinning the emergence of new industries needs therefore to pay attention also to the specific characteristics featuring the environmentsto which they are new, i.e. the combination of firms and institutions that make them new (Dietrich and Krafft, 2012).

The evolutionary process by which competences are accumulated over time plays such a role that the patterns of change in industrial structure are expected to be path dependent.In this special section, the concept of "path" is used in conjunction with the simple idea that "history matters", and path dependence is taken as the expression of the influence of structures inherited from history [START_REF] David | Clio and the Economics of QWERTY[END_REF][START_REF] Arthur | Competing technologies, increasing returns, and lock-in by historical events[END_REF][START_REF] Arthur | Increasing Returns and Path Dependence in the Economy[END_REF]. Initial conditions as well 2 Some of the initial drafts of papers in this special section of Research Policy have been presented at a workshop co-organized by BRICK (Bureau of Research on Innovation, Complexity and Knowledge) and CNRS-GREDEG, hosted by the Collegio Carlo Alberto in Torino (Italy), with the support of Fondation France-Japon de l"EHESS. We would like to thank the participants of the workshops for their valuable comments. We are also grateful to the members of the workshopscientific committee, in particular BjörnAsheim, Cristiano Antonelli, Michel Callon, Steven Casper, DietmarHarhoff, Franco Malerba , Simone Strambach , JörgSydow, Nick Von Tunzelmann and SeiichiroYonekura. as the choices of economic agents at different moments in time are likely to shape the likelihood for some industries to emerge and their probability to survive. As an example, empirical evidence suggests that industrial diversification in local contexts is more likely to be successful when new industrial activities are to some extent related to those already in place [START_REF] Frenken | Related Variety, Unrelated Variety and Regional Economic Growth[END_REF][START_REF] Boschma | The emergence of new industries at the regional level in Spain.A proximity approach based on product-relatedness[END_REF], and that new industry creation shows strong cross-country differences due to different initial institutional and technological regimes [START_REF] Casper | Managing competences in entrepreneurial technology firms: a comparative institutional analysis of Germany, Sweden and the UK[END_REF][START_REF] Storz | Dynamics in Innovation Systems: Evidence from Japan's Game Software Industry[END_REF]. The appreciation of the very dynamics leading to the introduction of new industrial activities cannot therefore disregard the evolution of the techno-economic environment in which they are supposed to take root.

In this special section we analysethe initial conditions of the industry and how they affect the pattern of industry emergence. Indeed, a number of important industry sectors emerged as a result of certain radical innovations, which substantially changed the path of former sectors and their rebirth into new ones, becoming eventually sub-industries of former sectoral categories. In order to understand the processes of emergence of new industries, it is thus important to acknowledge that this process is strongly shaped by the dynamics of knowledge creation in the geographic and the technological space, the technological regimes as well as the institutional setting. Several driving forces are at stake in this framework, involving the creation of new innovative firms, for example through spinoffs, the mobility of skilled labour force enabling knowledge transfer and exploitation in new contexts, firms" diversification patterns and the establishment and evolution of network linkages.

Such variety of mechanisms will be explored in the articles contributing this special section, which is ideally structured in three parts: i) spatial factors affecting the emergence of a new industry, ii) microeconomic determinants of industry emergence and search behavior and iii) the role of institutional and technological regimes.

As far as the first part on how spatial factors affect industry emergence is concerned, the articles tackle the issue of space from a multidimensional perspective. The first paper of Mathijsde Vaan on "Interfirm networks in periods of technological turbulence" shows the interplay between the evolutionary patterns of inter-firms linkages and those of technological lifecycles. The emergence of new trajectories, and as a consequence of new products, in the videogames industry is better faced by incumbents featured by high diversity of network partners. This makes this industry deviate from traditional evidence about industry lifecycles (de Vaan, 2014). In their article on "The emergence of new technology-based sectors in European regions: A proximity-based analysis of nanotechnology", Alessandra Colombelli, Jackie Krafft and Francesco Quatraroshow that besides relational space, technological and geographical dimensions also matter when the emergence of new industries is at stake. The introduction of "new" activities indeed does not occur in the air, but rather in local contexts which exhibit specialization patterns in already existing activities. These in turn rely on specific competences, which have been accumulated over time and constrain the directions that local industrial and technological diversification may undertake. Geographical and technological spaces show therefore some degree of overlap, and the emergence of new activities is more likely to occur and be successful when these are related to the competences already developed by local agents. This dynamics is all the more evident when new activities are based on brand new technologies, like nanotechnology [START_REF] Colombelli | The emergence of new technology-based sectors in European regions : A proximity-based analysis of nanotechnology[END_REF].

In the second part on microeconomic determinants in industry emergence,Marco Guerzoni, T. Taylor Aldridge, David B. Audretsch and Sameeksha Desai examine in their paper on "A New Industry Creation and Originality: Insight from the Funding Sources of University Patents" the determinants of patents" originality, as a condition to the emergence of new technology-based industries, with particular focus on the role of university research [START_REF] Guerzoni | A New Industry Creation and Originality: Insight From The Funding Sources of University Patents[END_REF].Using data on patented cancer research, they analyse how the research context, in particular the funding source for each scientist, is associated with patent originality. By looking at "The emergence of the education tool industry: Opportunities and challenges for innovation in education", D. [START_REF] Foray | The emergence of an educational tool industry: Opportunities and challenges for innovation in education[END_REF] provideevidence of the microeconomic conditions favouring the emergence of new industries. They investigate the population of small firms that begun to invent and commercialize (mainly ICT-based) technologies related to education practices, and provide evidence of the new accumulation of competences in this new branch as well as of the difficulties in finding a viable market for these technologies, above all as far as the public education is concerned.

The third part on the role of institutional and technological regimes tackles the issue of the emergence of new industries by focusing on the historically evolving structure in which new industries emerge. SébastienLechevalier, Junichi Nishimura and Cornelia Storz investigate the role of institutional regimes and key actors in processes of industry emergence.

In their paper on "Diversity in patterns of industry evolution: how an intrapreneurial regime contributed to the emergence of the service robot industry", they analyse, by using data of Japanese patent applications, the role of new firms and of collaborations in the process and industry emergence and find that the emergence of the industry is mainly triggered by established large firms, rather than by new firms [START_REF] Lechevalier | Diversity in patterns of industry evolution: how an intrapreneurial regime contributed to the emergence of the service robot industry in Japan[END_REF]. The following article of Kenneth L. Simons and Susan Sandersonon "Light Emitting Diodes and the Lighting Revolution: The Emergence of a Solid-State Lighting Industry" analyses how a solid-state lighting industrygrew out of light emitting diode (LED) technologies that evolved for half a century. This paper investigates how a diversity of key actors like tens of thousands of researchers in universities, national laboratories, and firms affected the process of industry emergence (Sanderson and Simon, 2014). This special section is intended present and discuss some of the key issues that are rising today in the research field of the path-dependent dynamics of emergence and evolution of new industries. While the contributions contained in the special section do not exhaust the possible investigations, we think they contribute to structuring the future research agenda in the field. Avenues of research over the next year will lie in the in-depth exploration of the relations existing between spatial aspects, micro-foundations and technological/institutional regimes in the emergence of new industries.


Iulian Voicu

Jean-Marc Girault

SIMILARITY QUANTIFICATION ANALYSIS TO DETECT SUFFERING FETUS

Keywords: dynamical system, recurrence plot, similarity plot, RQA parameters, fetal heart rate

To characterize a dynamical system, a number of descriptors or invariants can be used. Here we proposed to review and generalize the analysis of recurrence by introducing the concept of similarity. In this study, we showed that it was possible to distinguish the fetal heart rate of healthy from suffering fetus using the similarities of a time series, while the analysis of recurrence does not allowed it.

INTRODUCTION

Most of the nonlinear dynamical systems are characterized by differential equations. These equations usually are not known and the only available information is often the output signal of the system. The lack of knowledge of differential equations leads to two strategies to characterize the system.

The first strategy is to calculate parameters using directly the output signal of the system. In this case, several parameters such as Lyapunov exponents, dimensions and entropies [START_REF] Pincus | Approximate Entropy (ApEn) as a Measure of System Complexity[END_REF][START_REF] Lake | Sample Entropy Analysis of Neonatal Heart Rate Variability[END_REF][START_REF] Costa | Multiscale Entropy to Distinguish Physiologic and Synthetic RR Time Series[END_REF][START_REF] Bandt | Permutation Entropy: A Natural Complexity Measure for Time Series[END_REF] have been calculated to describe the different operating modes (periodic, random or chaotic) of the nonlinear system. Among them, the entropies count identical patterns of length m in the signal. The identical patterns are the patterns identical within a tolerance r.

The second strategy is to use the output signal to obtain the phase space of the system and then to calculate parameters. Although the real phase space can not be reconstructed without the knowledge of the differential equations, it is possible to construct an equivalent phase space (m-dimensional) [START_REF] Takens | Detecting Strange Attractors in Turbulence[END_REF]. In this equivalent phase space the topological properties of the attractor are preserved.

To characterize the phase space Eckman [START_REF] Eckman | Recurrence Plots of Dynamical Systems[END_REF] introduced the recurrence plot (RP) method. The RP has enabled a two dimensional (matrix) representation of the recurrent information in the D-dimensional equivalent phase space. An element of the recurrence matrix is 1 if two points belonging to the trajectories in this space are recurrent (are confined within a sphere of radius r).

Different structures such as isolated points or lines, characterize the recurrence matrix [START_REF] Marwan | Encounters with Neighbours: Current Developments of Concepts Based on Recurrence Plots and Their Applications[END_REF]. To quantify these struc-tures Zbilut and Webber [START_REF] Zbilut | Embeddings and Delays as Derived from Quantification of Recurrence Plots[END_REF][START_REF] Webber | Dynamical Assessment of Physiological Systems and States Using Recurrence Plot Strategies[END_REF] proposed different parameters known as recurrence quantification analysis (RQA) parameters.

The analysis of the RQA parameters is important because the parameters change according to the periodic, random or chaotic regime of the system.

Because a pattern of m consecutive non-constant samples reflects a local variation of the signal, looking for recurrences involves to search in the signal only those variations with almost the same amplitudes. The disadvantage of this definition is that it removes identical variations of the signal located at different levels.

To quantify all similar variations regardless the signal level, we proposed a new approach based on similarity patterns. This approach that generalized the notion of recurrence to that of similarity was novel and simple to implement, since it applied directly to the output of a dynamical system.

The paper is structured as follows: section 2 presents the recurrence and modified similarity matrices together with the RQA parameter. The results obtained applying the RQA parameter in the two cases were presented in section 3, while in section 4 we discuss them. Finally, we conclude our study in section 5.

METHODS

In this section we presented the steps to obtain the recurrence and the similarity matrices. The steps to find the recurrence matrix, noted in this paper RP D m , are described in section 2.1, while the steps corresponding to similarity matrix, noted SP D m , are presented in section 2.2. At the end of this section we present the parameters used to distinguish between healthy and suffering fetus.

Recurrence plot matrix

The recurrence matrix was constructed by using patterns (noted X i ) composed from m consecutive samples of the time series of length N . Thus we considered a phase space of dimension D = 1. Mathematically, the recurrence matrix computed in case of patterns of size m can be expressed as:

RP 1 m (r, i, j) = Θ(r -X i -X j ), X i , X j ∈ R m , (1)
where Θ (•) is the Heaviside function, • is the L ∞ norm, r is a fixed tolerance and i, j = 1, . . . , Nm + 1. This matrix is a binary matrix. To fill this matrix we made:

1. for each i = 1, . . . , Nm + 1 we formed the vectors;

X i = [x(i), x(i + 1), . . . , x(i + m -1)], (2)
where x(i) was the i th sample of the signal, and X i was the pattern composed of m samples.

2. we computed the maximum value of the m-dimensional absolute difference vector;

d(X i , X j ) = max(|X i -X j |), (3)
3. using a tolerance r, we assigned for each pair (i, j), i, j = 1, . . . , N ;

RP 1 m (r, i, j) = 1, d(X i , X j ) ≤ r, 0, d(X i , X j ) > r, (4)

Similarity plot matrix

The key idea to obtain the similarity matrix was to remove the distance between the means of the compared patterns.

1. for each i = 1, . . . , Nm + 1 we formed the vectors;

X i = [x(i), x(i + 1), . . . , x(i + m -1)], (5)
where x(i) was the i th sample of the signal, and X i was the pattern composed of m samples.

we computed

X ′ i = X i -M , (6)
where M was the mean of the vector X i 3. we computed the maximum value of the m-dimensional absolute difference vector;

d(X ′ i , X ′ j ) = max(|X ′ i -X ′ j |), (7)
4. using a tolerance r, we assigned for each pair (i, j), i, j = 1, . . . , N ;

SP 1 m (r, i, j) = 1, d(X ′ i , X ′ j ) ≤ r, 0, d(X ′ i , X ′ j ) > r, (8)
To quantify the similarities present in the similarity matrix we introduced the notion of similarity quantification analysis (SQA).

SQA and RQA parameters

One of the parameter proposed by Zbilut [START_REF] Zbilut | Embeddings and Delays as Derived from Quantification of Recurrence Plots[END_REF] was the recurrence rate. This parameter was nothing else that the number of recurrent/similar points in the recurrence/similarity matrix. We computed this parameter according with the relation

RR = 1 N 2 N i,j=1 RP 1 m (r, i, j), (9)
for the RP 1 m matrix, and with the relation

SR = 1 N 2 N i,j=1 SP 1 m (r, i, j) (10)
for the SP 1 m matrix. We called this parameter the similarity rate.

RESULTS

In this section we exemplified in section 3.1 the difference between the recurrence and similarity matrix using a sine wave, while in section 3.2 we present the results obtained on fetal heart rate signals.

Recurrence and similarity matrices applied to the sinusoidal signal

Firstly we illustrated the difference between the two matrices RP 1 m et SP 1 m , calculated in the case of a sine wave. Towards a better understanding of our approach we used for the sine wave, shown in figure 1, a frequency of 10 Hz sampled at 250 Hz.

We remember that finding the recurrence patterns means to find all patterns located at the same level of the signal. For example, the recurrence patterns of the plus sign-pattern (m = 2) in figure 1 3 ); ; (c) the similarity plot matrix using m = 3 (SP 1 3 ). The matrices were constructed for a value of r equal to 0.1 from the standard deviation of the sinusoidal signal. plus-pattern) had a similar pattern (circle-pattern), which was located at a different level in the signal.

We applied the equations 2-4 for two cases: m = 1, m = 3. The tolerance r was chosen equal to 0.1 from the value of the standard deviation of the sinusoidal signal. To obtain a better resolution of the figures that illustrated the recurrence and similarity matrices we increased the sample frequency at 1000 Hz for the sinusoidal signal of 10 Hz.

As we can see in figure 2a the recurrence matrix for m = 1 is formed by the diagonal and anti-diagonal lines structures. The main diagonal of the matrix indicates that a pattern was identical (recurrent) with itself. The distance between the lines that are parallel with the main diagonal line indicates the frequency of the signal, which in our case was 100 samples. Each column i of the matrix identified recurrences of the pat- tern that started at i th sample.

We increased the size of the pattern and we used now m = 3 in the equations 2-4. We observed that when we computed the recurrence matrix for the pattern with m = 3, the antidiagonal vanished.

Next step was to compute the similarity matrix using equations 5-8 for m = 3 and the same values r. We illustrated the similarity matrix in figure 2c. The similarity matrix has more points coded in black, which means that the number of similar patterns was larger (we found vertical lines for each pattern) than the recurrent patterns from a sinusoidal signal.

Recurrence and similarity matrices applied to fetal heart rate

We monitorized the cardiac rhythm of the fetuses over 30 minutes using the ultrasound Actifoetus device [START_REF] Voicu | Robust Estimation of Fetal Heart Rate from US Doppler Signals[END_REF]. The Doppler signals were acquired at CHRU "Bretonneau" Tours, France. The consent of each patient was obtained and the study was approved by the ethics committee of the Clinical Investigation Centre for Innovative Technology of Tours (CIC-IT 806 CHRU of Tours). Patients were older than eighteen years and pregnancy was singular. The recordings were made during the twenty-fifth and fortieth gestational weeks. The fetuses with congenitally malformations were not considered by the physicians in the group of suffering fetuses. In the group of suffering fetuses were included all fetuses for which a reversed end diastolic flow velocity in the fetal umbilical artery was present.

The figure 3 shows the heart rates of a healthy and suffering fetus, respectively. The rhythm was evaluated every 250 ms, obtaining 7200 values for a period of 30 minutes, let 240 values per minute.

We computed the recurrence and the similarity matrices for the fetal heart rates series. However, before computing the matrices a normalization of the heart rates was made, thus both of them were converted into zero mean variables with unity variances. Because the signal is non-stationary we applied a moving window technique by fixing a three minutes value for the window size. The RP 1 3 and SR 1 3 were evaluated for each window (720 values per window) together with the recurrence rate (RR) and similarity rate (SR) parameters (the equations 9 et 10).

The evolution in time of the RR parameter for both healthy and suffering fetus was illustrated in figure 4a, while We can observe in figure 4a that the parameter RR ranged in the interval 0.006-0.118 when it was evaluated for the healthy and suffering fetus. The two curves could not be separated in this case.

In contrast, when we evaluated the parameter SR from the similarity matrix, the two curves shown in figure 4b increased, but the increase was more pronounced in the case of the healthy fetus and a clear separation between the two curves was possible. We found that for the healthy fetus the SR curve varied in the interval [0.37, 0.92], whereas in the case of suffering fetus the curve varied in the interval [0.11, 0.35]. In this case we were able to separate the two fetus.

DISCUSSIONS

The RP 1 3 matrix quantified the percentage of "similar" patterns in the signal that had the same variation. In comparison, the SP 1 3 matrix quantified the percentage of patterns with the same variation and modulated in amplitude according to Groth [START_REF] Groth | Visualization of Coupling in Time Series by Order Recurrence Plots[END_REF]. In contrast to symbolic patterns considered by Groth, the amplitude modulation of the similar patterns used to compute the similar matrix is more restrictive. If the approach of Groth took into account only the sign of the variation between consecutive elements of the pattern, we also considered the value of this variation (the value of the derivative).

The more pronounced increasing of the parameter SR in the case of the healthy fetus means that the same kind of variation (similar patterns) is more present comparing with the case of the suffering fetus. The increased number of the similar patterns in the case of healthy fetus was determined by the increased number of different patterns and by the increased probabilities of the similar patterns. A higher diversity and probability of similar patterns is the characteristics of a chaotic system.

On the other hand we observed, see in figure 5a-5b, that the decreasing of the parameter SR in the case of suffering fetus was the consequence of a loss or fragmentation of the vertical lines in the matrix SP 1 3 . According to Marwan [START_REF] Marwan | Encounters with Neighbours: Current Developments of Concepts Based on Recurrence Plots and Their Applications[END_REF], the vertical lines of the matrix characterize chaos-chaos transitions. The disappearance of the vertical lines can signify the loss of the chaotic nature of the heart.

If the diversity of similar patterns decreases or there is a too much diversity of very unlikely patterns, then we can imagine that the operating mode of the fetal heart tends to a random one. On the other hand, if diversity decreases but the occurrence of the patterns is important, the operating mode of the heart tends to a periodic regime.

In the end of this section we mention that the similarity matrix could be applied only for patterns of length m ≥2. For the pattern m=1 we could not applied this method because all the patterns became similar after the subtraction of the mean.

CONCLUSIONS

We modified the definition of the recurrence of the patterns. Using the new definition we calculated the similarity matrix. From this matrix we were able to discriminate the heart rate of a healthy from a suffering fetus while it was not possible using the recurrence matrix. A loss of the chaotic nature of the fetal heart, measurable by the similarity matrix, could explain the discrimination.

Fig. 1 :

 1 Fig. 1: The sinusoidal signal. Example of a pattern m=2 placed at different levels in the signal.

Fig. 2 :

 2 Fig. 2: (a) The recurrence plot matrix using m = 1 (RP 1 1 ); (b) The recurrence plot matrix using m = 3 (RP 13 ); ; (c) the similarity plot matrix using m = 3 (SP1 3 ). The matrices were constructed for a value of r equal to 0.1 from the standard deviation of the sinusoidal signal.

Fig. 3 :

 3 Fig. 3: The fetal heart rate: (a) of a healthy fetus; (b) of a suffering fetus.

Fig. 4 :

 4 Fig. 4: (a) The recurrence rate RR and (b) the similarity rate SR parameters computed for a pattern m = 3 in the case of a healthy and suffering fetus.

Fig. 5 :

 5 Fig. 5: (a) The RR and (b) the SR parameters computed for a pattern m = 3 in case of a healthy and suffering fetus.


Jean-Marc Girault
email: jean-marc.girault@univ-tours.fr.

Denis Kouamé

Abdeldjalil Ouahabi

Analytical formulation of the Fractal dimension of filtered stochastic signals

Keywords: Length, fractal dimension, stochastic signals, Lorentzian filters

The aim of this study was to investigate the effects of a linear filter on the regularity of a given stochastic process in terms of the fractal dimension. This general approach, described in a continuous time domain, is new and is characterized by its simplicity. The framework of this problem is general since it emerges when a fractal process undertakes a transformation, as is the case in denoising or measurement processes.

INTRODUCTION AND DEFINITION

Introduction

Fractal processes are processes whose ubiquity spans innumerable fields. In contrast to the well-known family of ARMA processes, these intrinsically scale-invariant processes, also known as 1/f processes, are characterized by long-term correlation structure. Moreover, these fractal signals are continuous and underivable signals presenting deterministic or statistical self affinities [START_REF] Mandelbrot | The fractal geometry of nature[END_REF], [START_REF] Mandelbrot | Self-affine fractals and fractal dimension[END_REF]. Invariants such as the fractal dimension D, the Hurst exponent H or the Hölder exponent h are often evaluated to characterize such irregular signals. Stochastic signals can be continuous and not differentiable, presenting strong irregularities.

Our study was performed in the context of analysis and synthesis of fractal signals where fractal and stochastic signals can be related through fractional differentiation [START_REF] Hosking | Fractional differencing[END_REF] or filtering [START_REF] Mandelbrot | A Fast Fractional Gaussian Noise Generator[END_REF]. Others have studied 1/f noise in electronic devices [START_REF] Barnes | A statistical model of flicker noise[END_REF][START_REF] Barnes | Efficient numerical and analog modeling of flicker noise processes[END_REF][START_REF] Keshner | 1/f noise[END_REF], roughness and texture of surfaces [START_REF] Gagnepain | Fractal approach to twodimensional and three-dimensional surface roughness[END_REF][START_REF] Majumdar | Fractal model of elastic-plastic contact between rough surfaces[END_REF][START_REF] Wu | Characterization of fractal surfaces[END_REF][START_REF] James | Fracture surface examination of dental ceramics using fractal analysis[END_REF] and biomedical signals and images [START_REF] Sarkar | An efficient differential box-counting approach to compute fractal dimension of image[END_REF][START_REF] Loussot | An oriented fractal analysis for the characterization of texture: application to bone radiographs[END_REF][START_REF] Qingrui | Mammogram analysis based on wavelets and fractals[END_REF][START_REF] Lee | A robust algorithm for the fractal dimension of images and its applications to the classification of natural images and ultrasonic liver images[END_REF].

Issues dealing with filtering and fractal signals have previously been addressed in three ways: the denoising of fractal signals corrupted by an additive noise [START_REF] Lundahl | Fractional Brownian Motion: A maximum likelihood estimator and its application to image texture[END_REF][START_REF] Wornell | Estimation of fractal signals from noisy measurements using Wavelets[END_REF][START_REF] Deriche | Signal Modeling with filtered discrete fractional noise processes[END_REF][START_REF] Chen | Deconvolution filter desing for fractal signal transmission systems: a multiscale Kalman filter bank approach[END_REF][START_REF] Chen | Maximum Likelihood Parameter Estimation of F-ARIMA Processes Using the Genetic algorithm in the Frequency Domain[END_REF], the filtering of fractal signals by a measurement system [START_REF] Mandelbrot | Fractal character of fracture surfaces of metals[END_REF][START_REF] Gagnepain | Fractal approach to twodimensional and three-dimensional surface roughness[END_REF][START_REF] Qingrui | Mammogram analysis based on wavelets and fractals[END_REF][START_REF] Loussot | An oriented fractal analysis for the characterization of texture: application to bone radiographs[END_REF][START_REF] Wu | Characterization of fractal surfaces[END_REF][START_REF] James | Fracture surface examination of dental ceramics using fractal analysis[END_REF][START_REF] Lee | A robust algorithm for the fractal dimension of images and its applications to the classification of natural images and ultrasonic liver images[END_REF] and the synthesis of fractal signals from stochastic processes [START_REF] Saletti | A comparison between two methods to generate 1/f noise[END_REF][START_REF] Mandelbrot | A Fast Fractional Gaussian Noise Generator[END_REF][START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF][START_REF] Kasdin | Discrete simulation of colored noise and stochastic process and 1/f power law noise generation[END_REF][START_REF] Guglielmi | 1/f α signal synthesis with precision control[END_REF].

In the synthesis of 1/f β noise, the framework which predominated for many years was naturally the spectral domain [START_REF] Barnes | A statistical model of flicker noise[END_REF][START_REF] Keshner | 1/f noise[END_REF][START_REF] Mandelbrot | A Fast Fractional Gaussian Noise Generator[END_REF][START_REF] Saletti | A comparison between two methods to generate 1/f noise[END_REF]. The relationships between the spectral and irregularity properties of the process appeared later in reports by Berry [1] who demonstrated the link between the slope β of the spectrum and the fractal dimension D = (5 -β)/2 for a fBm. The spectral estimator consisted of evaluating the slope -β in a log-log plot (Power/frequency), and then applying the previously introduced Berry equation. Although much used in the past, due to its simple formulation (for a continuous or a discrete framework), this spectral estimator is less often used today. The main reasons for its obsolescence are its poor performance and to the need for setting the spectral bandwidth of interest when the power spectral density presents several slopes.

The approaches based on maximum likelihood [START_REF] Lundahl | Fractional Brownian Motion: A maximum likelihood estimator and its application to image texture[END_REF][START_REF] Wornell | Estimation of fractal signals from noisy measurements using Wavelets[END_REF][START_REF] Chen | Maximum Likelihood Parameter Estimation of F-ARIMA Processes Using the Genetic algorithm in the Frequency Domain[END_REF] and the estimators based on multi-scale analysis [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF][START_REF] Wornell | Estimation of fractal signals from noisy measurements using Wavelets[END_REF][START_REF] Chen | Multiscale Wiener filter for the restoration of fractal signals: Wavelet filter bank approach[END_REF][START_REF] Hirchoren | On the Optimal number of scales in estimation of fractal signals using wavelets and filter banks[END_REF] were mainly used in denoising of fractal signals (or images). Because the likelihood function was a non-linear function according to D, it was not possible to obtain an explicit formula and only a numerical computation was possible.

For the approaches based on wavelet analysis, they were mainly focused on estimation of the slope of variance β = 2H + 1 of the wavelet coefficients d j in the log-log plot. This estimator was very interesting since it could easily be extended to the study of multi-fractal processes.

When the fractal signals (mono or multidimensional) are accessible only by measurement, as is the case for example when measuring surface roughness [START_REF] Gagnepain | Fractal approach to twodimensional and three-dimensional surface roughness[END_REF][START_REF] Wu | Characterization of fractal surfaces[END_REF][START_REF] James | Fracture surface examination of dental ceramics using fractal analysis[END_REF] or image texture [START_REF] Qingrui | Mammogram analysis based on wavelets and fractals[END_REF][START_REF] Loussot | An oriented fractal analysis for the characterization of texture: application to bone radiographs[END_REF][START_REF] Lee | A robust algorithm for the fractal dimension of images and its applications to the classification of natural images and ultrasonic liver images[END_REF], the measurement or acquisition systems act like a linear filter. Quantification of the impact of such linear filtering on the regularity of the processes studied is thus necessary. For these applications, the most commonly used fractal dimension estimators have been based on the box counting method (for example see [START_REF] Sarkar | An efficient differential box-counting approach to compute fractal dimension of image[END_REF]), the spectral slope (for example see [START_REF] Gache | Fractal Dimension Estimators for Fractional Brownian Motions[END_REF]) and length measurement [START_REF] Burlaga | Fractal structure of the interplanetary magnetic field[END_REF].

There is not, to our knowledge, a simple framework which directly shows the link between the filter characteristics and the regularity property of such filtered signals through explicit relationships. We therefore investigated the impact of linear filtering on irregularity for a given stochastic process in terms of the fractal dimension.

The novelty of our approach is based on an explicit formulation, in continuous time, linking the spectral properties of a given filter and the fractal properties of a given process. Our approach shows the relationship between the fractal dimensions at the input and the output of a filter. Due to its simple formulation and the good performance of its estimator, we focused on the fractal dimension formulation through the length measurement. The only a priori knowledge required to use our approach is the probability density of the input process and the correlation coefficient of the filter. Our approach is original since the issue has never been previously addressed in denoising, synthesis or measuring processes of fractal signals. Our study is complementary to other studies based mostly on a discrete formulation. The general framework and the simplicity of our formulation are its main assets.

Before explaining the details of our approach, we summarize the definitions used related to the length measurement of fractal stochastic processes.

Definition

The Euclidian length of a deterministic signal x(t) for a limited duration T can be defined as follows:

L 0 = T λ, (1)
where

λ = 1 T T /2 -T /2 λ(t)dt, (2)
is the arithmetic mean of the elementary length and where

λ(t) = 1 + dx(t) dt 2 . ( 3
)
On the hypothesis that x(t) is homogeneous to Volt, then λ is homogeneous to Volt per second. The elementary length λ can be rewritten according to the Pythagorean theorem as:

λ(t) = lim τ →0 1 τ τ 2 + ∆(t, τ ) 2 , (4)
where the increment ∆(t, τ ) is written in a symmetric form as:

∆(t, τ ) = x(t + τ /2) -x(t -τ /2). (5)
Finally, the signal length is written:

L x = lim τ →0 T /2 -T /2 1 τ τ 2 + ∆(t, τ ) 2 dt. (6)
Note that we had already used such a definition for deterministic signals (see [START_REF] Girault | Length and frequency of band-limited signals[END_REF]). Unlike deterministic signals, which are almost always derivable and have a finite length, stochastic signals do not follow the laws of Euclidian geometry. Stochastic signals cannot be described by continuous mathematical curves and their irregularities are still present even when examined with higher and higher resolutions. This leads to the paradoxical conclusion that the lengths of such signals are not finite. To overcome the fact that (as fractal signals) stochastic signals do not have a finite length, Richardson [START_REF] Richardson | The problem of contiguity: an appendix of statistic of deadly quarrels[END_REF] assumed that the length L x of a fractal signal depends on the resolution at which it is measured, i.e. the gauge λ of the measurement. To overcome this indetermination for λ → 0, Richardson proposed measuring the curve length for each value λ, ∀ λ > 0. Mandelbrot demonstrated that measuring L x (λ) as a function of λ was one way to determine the fractal dimension:

L x (λ) = K x λ 1-Dx , (7)
where D x is the fractal dimension and K x is a constant. Length definition becomes a function of λ and is written like equation (1):

L x (λ) = T λ(τ ), (8)
where

λ(τ ) = 1 T T /2 -T /2 λ(t, τ )dt, (9)
and where the elementary length is

λ(t, τ ) = 1 τ τ 2 + ∆(t, τ ) 2 .
The corresponding length estimator of equation ( 8) illustrated in Fig. 2 is:

L x (kT s ) = 1 kT s N -k n=1 (kT s ) 2 + ∆(n, kT s ) 2 , (10)
where T s is the sampling period and N is the number of elementary lengths between [-T /2, T /2].

The definition of equation ( 8) extends easily to stationary and ergodic stochastic signals by replacing the arithmetic mean by the mathematical expectation E[.]. The definition then becomes:

L x (λ) = T E[λ], (11)
where

λ = 1 τ √ τ 2 + ∆ 2 .
Finally, the definition becomes:

L x (τ ) = T τ E √ τ 2 + ∆ 2 . ( 12
)
The calculation of the mathematical expectation of equation [START_REF] Gagnepain | Fractal approach to twodimensional and three-dimensional surface roughness[END_REF] does not lead to an analytical formulation, and only a numerical resolution is possible. In order to present an analytical formulation, we propose using another definition based on that proposed by Burlaga and Klein [START_REF] Burlaga | Fractal structure of the interplanetary magnetic field[END_REF]:

L x (τ ) = T τ E |∆| . (13)
This definition supposes that τ << ∆, which is often true, particularly for a high sampling frequency. However, if this is not the case, an amplitude gain can be applied. A discussion of the sense of such a definition is reported in [START_REF] Tavernier | La méthode de Higuchi pour la dimension fractal[END_REF].

The length estimator that we finally used which corresponded to a discrete version of equation ( 13) was:

L x (k) = N -k n=1 |∆(n, k)|.
The performance of such estimator in its average version, known as the Burlaga and Klein [START_REF] Burlaga | Fractal structure of the interplanetary magnetic field[END_REF] estimator L BK (k), was excellent and outperformed the spectral estimators and the maximum likelihood estimators (see [START_REF] Gache | Fractal Dimension Estimators for Fractional Brownian Motions[END_REF]). However, the performance was similar to that obtained with estimators based on wavelet analysis (see [START_REF] Gache | Fractal Dimension Estimators for Fractional Brownian Motions[END_REF]). Note that Flandrin [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF] demonstrated the link between length measurement L BK (k) and variance of the wavelet coeffi-

cient d j (k): E[L BK (2 j-1 )] = 2 3-5j/2 2 π var(d j (k)
). Though this estimator, which is based on wavelet analysis, gave an excellent performance, it did not lead to a simple formulation in a continuous framework. The strong point of this estimator based on wavelet analysis was its adaptability to study multi-fractal processes.

The paper is organized as follows. Section 2 presents the approach in detail, setting out the notion of length ratio of colored processes, then stating the link between the length ratio and the fractal dimension of colored processes. Section 3 presents several examples of filters and the simulation results are discussed. Finally we propose a conclusion and some future applications.

LENGTH RATIO AND FRACTAL DIMENSION OF FILTERED STOCHASTIC SIGNALS

The method that we propose is new and was investigated in a general stochastic framework with an explicit formulation in continuous time. This approach requires some a priori information such as the probability density of the input process and the correlation function of the filter.

In a linear system where the impulse response is h(t), the input signal x(t) and the output signal y(t), and the length L x (τ ) of the input signal, can be evaluated by:

L x (τ ) = T τ E |x 1 -x 2 | , (14)
where

x 1 = x(t 1 + τ /2), x 2 = x(t 2 -τ /2) ∀t 1 , t 2
and where

E |x 1 -x 2 | = |x 1 -x 2 |p x 1 ,x 2 (x 1 , x 2 )dx 1 dx 2 . ( 15
)
p x 1 ,x 2 (x 1 , x 2 )
is the joint probability density.

Filtered Gaussian noise

In the special case where x 1 and x 2 are two random Gaussian variables, the joint probability density p x 1 ,x 2 (x 1 , x 2 ) is written:

p x 1 ,x 2 (x 1 , x 2 ) = (1 -ρ 2 x ) -1/2 2πσ 1 σ 2 e -1 2(1-ρ 2 x ) x 2 1 σ 2 1 + x 2 2 σ 2 2 - 2ρxx 1 x 2 σ 1 σ 2 , ( 16
)
where σ 2 1 and σ 2 2 are the variances of the two random variables x 1 and x 2 , respectively. ρ x is the correlation coefficient defined by

ρ x = E[x 1 x 2 ]/E[x 2 1 ]E[x 2 2 ]
. The probability density of the increment ∆ = x 2 -x 1 is written:

p ∆ (∆) = +∞ -∞ p x 1 x 2 (x 1 , x 1 + ∆)dx 1 . ( 17
)
After calculation, the probability density of the increment is reduced to a normal law N(0, σ ′2 ):

p ∆ (∆) = 1 √ 2πσ ′ e -∆ 2 2σ ′2 , (18)
with

σ ′ = σ 2 1 + σ 2 2 -2ρ x σ 1 σ 2 .
x 1 and x 2 belong to the same signal x(t), then σ 1 = σ 2 = σ x and the standard deviation

σ ′ is simplified, σ ′ = σ x 2(1 -ρ x ).
The mathematical expectation of the increment becomes:

E |∆| = 2 +∞ 0 ∆p ∆ (∆)d∆ = 2 π σ ′ , (19)
and finally:

L x (τ ) = 2 √ π T σ x τ 1 -ρ x (τ ). ( 20
)
Similarly the length of the output signal y(t) is written:

L y (τ ) = 2 √ π T σ y τ 1 -ρ y (τ ). ( 21
)
Note that for a white noise (ρ x = 0), the length of the input signal x(t) is written:

L x (τ ) = T σ x τ γ, (22)
where the factor γ is equal to 2/ √ π for a Gaussian noise and the factor γ is equal to 2 √ 3 for a noise uniformly distributed between [-a/2, a/2] with a standard deviation σ x = a/ √ 12. The factor γ (and hence the length) seems to reveal the statistic nature of the white noise.

In order to establish a link between the impulse response or the correlation function of the filter ρ y (τ ) and the fractal dimension D y of the filtered signal y(t), it should be remembered that in the case of fractal signals, the length L y of the output signal is related to the fractal dimension D y as follows (see also equation [START_REF] Chen | Maximum Likelihood Parameter Estimation of F-ARIMA Processes Using the Genetic algorithm in the Frequency Domain[END_REF] for the input):

L y (τ ) = K y τ 1-Dy , (23)
where K y is a constant. Consequently, the fractal dimension D y of the signal y(t) can be defined as follows:

D y = 1 -lim τ →0 log L y (τ ) log(τ ) . (24)
Similarly:

D x = 1 -lim τ →0 log L x (τ ) log(τ ) . ( 25
)
By using equations ( 20), ( 25) and ( 21), [START_REF] Majumdar | Fractal model of elastic-plastic contact between rough surfaces[END_REF] the equations become:

D y = 2 -lim τ →0 1 2 log 1 -ρ y (τ ) log(τ ) , (26)
D x = 2 -lim τ →0 1 2 log 1 -ρ x (τ ) log(τ ) . ( 27
)
For unidimensional signal, the fractal dimension of signals x(t) and y(t) was the topological dimension of the space minus a term related to the inner correlation of x(t) and y(t).

In order to simply qualify the influence of linear filters, we propose evaluating a new parameter, i.e. the length ratio Ω between the input and the output of the linear system:

Ω(τ ) = L y (τ ) L x (τ ) . ( 28
)
By using equations ( 20) and ( 21), equation ( 28) becomes:

Ω(τ ) = σ y σ x 1 -ρ y (τ ) 1 -ρ x (τ ) . (29)
Furthermore, the following function Λ(τ ) is defined:

Λ(τ ) = lim τ →0 log Ω(τ ) log(τ ) . (30)
Λ(τ ) is an important parameter since it quantifies the filtering impact in terms of reduction irregularity. By using equations ( 7), ( 23) and [START_REF] Mandelbrot | Fractal character of fracture surfaces of metals[END_REF], it is easy to show that:

Λ = D x -D y . (31)
In this case D y is defined by:

D y = D x -lim τ →0 log Ω(τ ) log(τ ) . ( 32
)
When the input signal is a Gaussian noise and when the system is linear, by using equations ( 29) and [START_REF] Richardson | The problem of contiguity: an appendix of statistic of deadly quarrels[END_REF], this becomes:

D y = D x -lim τ →0 1 2 log(1 -ρ y (τ )) log(τ ) + lim τ →0 1 2 log 1 -ρ x (τ ) log(τ ) . ( 33
)
The fractal dimension of the filtered signal is then equal to the fractal dimension of the signal before filtering minus a term related to the filter plus a term explaining the inner correlation of the input signal x(t).

Before testing different types of filter, we present the filtered fractional Brownian motion.

Filtered fractional Brownian motion

Let's consider the fractional Brownian motion (fBm) [START_REF] Mandelbrot | Fractional Brownian Motions, Fractional Noises and applications[END_REF] as being in the input of the filter. This self similar process which generalizes the Brownian motion or the Wiener process is composed of stationary and independent increments. These increments follow a Gaussian law1 : x 1 -x 2 ≈ N(0, 2τ 2Hx ) where H x is the Hurst exponent. Note that an average power spectrum density of fBm can be defined in the frequency domain [START_REF] Flandrin | On the spectrum of Fractional Brownian Motions[END_REF] and it turns out that it is proportional to 1/f α with α = 2H x + 1. By using equation ( 14) to equation ( 20), we show that:

L x (τ ) = 2 √ π T τ Hx-1 . ( 34
)
Note that we recognize the factor 2/ √ π which underlines the Gaussian nature of the process. By using equations [START_REF] Chen | Maximum Likelihood Parameter Estimation of F-ARIMA Processes Using the Genetic algorithm in the Frequency Domain[END_REF][START_REF] Deriche | Signal Modeling with filtered discrete fractional noise processes[END_REF][START_REF] Flandrin | On the spectrum of Fractional Brownian Motions[END_REF][START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF][START_REF] Gache | Fractal Dimension Estimators for Fractional Brownian Motions[END_REF][START_REF] Gagnepain | Fractal approach to twodimensional and three-dimensional surface roughness[END_REF][START_REF] Girault | Length and frequency of band-limited signals[END_REF][START_REF] Guglielmi | 1/f α signal synthesis with precision control[END_REF][START_REF] Hirchoren | On the Optimal number of scales in estimation of fractal signals using wavelets and filter banks[END_REF][START_REF] Hosking | Fractional differencing[END_REF][START_REF] James | Fracture surface examination of dental ceramics using fractal analysis[END_REF][START_REF] Keshner | 1/f noise[END_REF][START_REF] Kasdin | Discrete simulation of colored noise and stochastic process and 1/f power law noise generation[END_REF][START_REF] Lee | A robust algorithm for the fractal dimension of images and its applications to the classification of natural images and ultrasonic liver images[END_REF][START_REF] Tavernier | La méthode de Higuchi pour la dimension fractal[END_REF][START_REF] Loussot | An oriented fractal analysis for the characterization of texture: application to bone radiographs[END_REF][START_REF] Lundahl | Fractional Brownian Motion: A maximum likelihood estimator and its application to image texture[END_REF][START_REF] Majumdar | Fractal model of elastic-plastic contact between rough surfaces[END_REF][START_REF] Mandelbrot | Fractional Brownian Motions, Fractional Noises and applications[END_REF], it becomes:

D x = 2 -H x , (35)
where

H x = 1 + lim τ →0 log L x (τ ) log(τ ) . ( 36
)
Note that H y has the same definition as H x . As for equation [START_REF] Saletti | A comparison between two methods to generate 1/f noise[END_REF], it becomes:

H y = H x -lim τ →0    1 2 log 1 -ρ y (τ ) log(τ )    . (37)
Similarly, the Hurst exponent H y of the filter output is the Hurst exponent H x before filtering minus a term related to the inner filter properties.

Impact of the correlation coefficient

From definitions ( 33) and (37), it becomes obvious that the asymptotic behavior of the correlation coefficient when τ → 0 is a key point in the estimation of fractal dimension. Indeed, by exciting filters by a white Gaussian noise, all correlation coefficients ρ y (τ ) whose Taylor expansion satisfies the equation:

lim τ →0 ρ y (τ ) ≈ 1 -kτ n , ( 38
)
where k is a known constant and where n ∈ R and 0 ≤ n ≤ 2, involve a fractal dimension and a Hurst exponent described as follows:

D y = D x -n 2 ; H y = H x + n 2 ;
(39

)
where D x = 2 and H x = 0 for ρ x = 0. Note that for 2 ≥ D y ≥ 1, n satisfies 0 ≤ n ≤ 2. Moreover, by using definition [START_REF] Saletti | A comparison between two methods to generate 1/f noise[END_REF] and equation ( 38) for ρ x = 0, it becomes:

D y = 2 -lim τ →0 1 2 log(kτ n ) log(τ ) = 2 - n 2 , (40)
and n = 4 -2D y . Without making out an exhaustive list, we propose the following correlation coefficients (or a combination) satisfying equation ( 38) for τ > 0:

• ρ y (τ ) = α + (1 -α)cos(kτ n/2 ). For α = 0.5 and for n = 2 we recognize the Hanning function;

• ρ y (τ ) = (1 -kτ n ). For n = 1 we recognize the triangle function;

• ρ y (τ ) = e -kτ n . For n = 2 we recognize the Gaussian function and for n = 1 we recognize the Lorentzian function.

EXAMPLES

Lorentzian filters

The impulse response h(t) of a Lorentzian filter is defined by:

h(t) = ω c e -ωct cos(ω 0 t)u(t), (41)
where u(t) is the Heaviside function, ω c the cut-off pulsation and ω 0 the central pulsation of the filter. This linear system of impulse response h(t) is excited by a Gaussian white noise. For linear filters, the crosscorrelation function of the output signal is written:

R y (τ ) = R h (τ ) * R x (τ ), ( 42
)
where * is the convolution operator. Note that the process coming from this type of filter is also known as being the "first order Markov process". For the Lorentzian filter the correlation function is R h (τ ) = ωc 4 e -ωc|τ | cos(ω 0 τ ), the correlation function of the input signal is R x (τ ) = σ 2

x δ(τ ), and the correlation function of the output signal reduces to R y (τ ) = σ 2 y e -ωc|τ | cos(ω 0 τ ), where

σ y = σ x √ ω c /2. The correlation coefficient defined by ρ y (τ ) = R y (τ )/R y (0), is reported in Table I for τ > 0.
As already mentioned, the length of the white Gaussian noise is ∀ τ > 0:

L x (τ ) = 2T σ x √ πτ , ( 43
)
since the correlation coefficient is ρ x = 0, ∀τ > 0. The length of the output signal is ∀ τ > 0:

L y (τ ) = 2T σ y √ πτ 1 -e -ωcτ cos(ω 0 τ ). ( 44
)
The length ratio ∀ τ > 0 is written as follows:

Ω(τ ) = G 1 -e -ωcτ cos(ω 0 τ ), ( 45
)
where the gain G is written G = σ y /σ x .

In order to test different kinds of Lorentzian filter, four filters were considered: a wide band filter, a narrow band filter, a base band filter and an all pass filter. The correlation coefficients and length ratios for these different kinds of filter are summarized in Table I. We verified that for these different kinds of filter, the parameter Ω/G was related to the intrinsic filter parameters, ω c and ω 0 .

When the input is a white Gaussian noise, the fractal dimension D y of the output is:

D y = 2 -lim τ →0 1 2 log(1 -e -ωcτ cos(ω 0 τ )) log(τ ) . (46)
• For a base band filter ω 0 → 0 and for τ → 0, ω c τ << 1:

D y = 2 - 1 2 = 3 2 , (47) with lim τ →0
(1 -e -ωcτ ) ≈ ω c τ,

• For a narrow band filter ω c → 0 and for τ → 0, ω 0 τ << 1:

D y = 2 -1 = 1, (48)
with lim

τ →0 1 -cos(ω 0 τ ) ≈ ω 2 0 τ 2 /2.
The output signal is a sinusoid, i.e. a continuous signal whose fractal dimension is the same as a line. Note that for a Gaussian correlation coefficient ρ y (τ ) = e -ωcτ 2 , the fractal dimension is also

D y = 1 since lim τ →0 (1 -e -ωcτ 2 ) ≈ ω c τ 2 .
• For an all pass filter:

D y = 2 -lim τ →0 1 2 log(1) log(τ ) = 2 -0 = 2.
The signal is a white noise, i.e. an irregular signal which fills all the space.

When the input is a fractional Brownian motion, the Hurst exponent of the output signal is:

• for a base band filter ω 0 → 0 and for τ → 0, ω c τ << 1:

H y = H x + lim τ →0 1 2 log(ω c τ ) log(τ ) .
By imposing H x = 0.3 (high irregularity), the Hurst exponent of the output is: H y = 0.3 + 0.5 = 0.8. This confirms the fact that the greater the filtering the greater the regularity.;

• for a narrow band filter ω c → 0 and for τ → 0, ω 0 τ << 1:

H y = H x + lim τ →0 1 2 log(ω 2 0 τ 2 /2) log(τ )
.

By imposing H x = 0.3 (high irregularity), the Hurst exponent of the output is: H y = 0.3 + 1 = 1.3. As 0 ≤ H y ≤ 1, this means that H y = 1.

Simulations

As an illustration, Figs. 3 and4 show the length ratios for the four types of Lorentzian filter. Time series of 10000 samples are simulated with a sampling frequency of 10000 Hz for different values of the cut-off frequency f c = {1, 10, 100, 1000, 5000} Hz and for different values of the central frequency of the filter f 0 = {1, 10} Hz. The minimum and maximum values of the gauge are τ min = 10 -4 s and τ max = 0.5s. The frequency is thus an important setting parameter since it fixes the minimum resolution.

Fig. 3 shows that for a cut-off frequency f c = 5000 Hz and a central frequency f 0 = 1 Hz (similar to an all pass filter), the length ratio Ω of such a signal has a slope equal to zero, involving a fractal dimension D y = 2. The output signal is a white Gaussian noise which entirely fills the space. Note that for a white noise, we verified that Ω(τ min )/G ≈ 1. When the cut-off frequency was reduced to f c = 1 Hz and f 0 = 1 Hz (low pass filter), the slope for the length ratio was 1/2 for a gauge lower than the cut-off gauge τ < τ c = 1/ω c , involving a fractal dimension D y = 3/2.The output signal was a standard Brownian motion characterized by its Hurst exponent H = 1/2. Beyond the cut-off gauge, the slope was equal to zero.

Fig. 4 shows that for a central frequency f 0 = 100 Hz and for a cut-off frequency f c = 10 Hz, oscillations where the position of its first minimum corresponds to τ = 1/f 0 appear in Ω. For a gauge less than the cut-off gauge τ c = 1/ω c , the fractal dimension is between 1.5 and 1 for a narrow band filter and between 2 and 1.5 for a wide band filter. When the cut-off frequency tends towards zero, the fractal dimension tends to 1, thus involving the presence of a sinusoidal signal. When the cut-off frequency tends towards infinity (whatever the central frequency) then the fractal dimension tends to 2, involving the presence of a white noise.

By using an impulse response of the form h(t) = e -ωcτ n with 0 ≤ n ≤ 1, time series of 5000 samples are simulated with a sampling frequency of 10000 Hz and a cut-off frequency f c = 1 Hz. The input signal is a white Gaussian noise N(0, 1). The minimum and maximum values of the gauge are τ min = 10 -4 s and τ max = 0.5s. Simulation results are depicted in figure 5. Figure 5a shows synthesized fractal signals: the higher the order n, the greater the regularity of the time series. Figure 5b shows the length ratio Ω/G with respect to the gauge τ : the higher the order n, the higher the slope and the lower the fractal dimension. For n = 0, the corresponding filter is an all pass filter, the output signal is a white Gaussian noise and the corresponding fractal dimension is D y = 2. For n = 1, the corresponding filter is a base band filter, the output signal is a Brownian motion and the corresponding fractal dimension is D y = 1.5 (H y = 0.5). For 0 ≤ n ≤ 1 the corresponding fractal dimensions of the output filter are between 2 ≥ D y ≥ 1.5.

DISCUSSION & CONCLUSION

We propose here a simple explicit formulation of the filtering effect on the regularity of a given stochastic process in a general framework and in a continuous time context. The simulations studied validated the theoretical calculus but also permitted extraction of the filter characteristics such as cut-off frequency and central frequency.

Though limited to mono-fractal processes, this approach is interesting since it requires only commonly used a priori knowledge such as the probability law of the process and the spectral characteristic of the filter. The need for a priori information such as the probability density and the correlation coefficient of the filter constitute both an advantage and a drawback. It is an advantage since they provide a simple formulation, but it is also a disadvantage since the study presented here was limited to this information. Another drawback is that our approach is currently limited to mono-fractal processes. The next step will be to extend our framework to the study of multi-fractal processes.

Figure 1 :

 1 Figure 1: Low pass filtering increases the regularity of the fractal signal. a) Fractal signal (H = 0.5). b) Filtered fractal signal.

Figure 2 :

 2 Figure 2: Principle of length measurement of a stochastic signal.

Figure 3 :

 3 Figure 3: Length ratio of colored stochastic signals. First order Lorentzian low pass filter for different cut-off pulsations and for a fixed central pulsation.

Figure 4 :Figure 5 :

 45 Figure 4: Length ratio of colored stochastic signals. First order Lorentzian low pass filter for different cut-off pulsations and for a fixed central pulsation.

Note that the increments follow a Gaussian law written as follows: x 1 -x

≈ N (0, τ 2Hx ). However, in order to make a direct link between equations in this subsection and equations in subsection 2.1, we voluntary modified this formula by doubling the variance and this modification does not modify the definition of fBm.

List of Tables