Keywords: Mortality statistics, cause of death, coding, reliability, intercoder agreement, intracoder agreement

Background: Cause-of-death statistics are a major source of information for epidemiological research or policy decisions. Information on the reliability of these statistics is important for interpreting trends in time or differences between populations. Variations in coding the underlying cause of death could hinder the attribution of observed differences to determinants of health. Therefore we studied the reliability of cause-of-death statistics in the Netherlands.

Methods:

We performed a double coding study. Death certificates from the month of May 2005 were coded again in 2007. Each death certificate was coded manually by four coders.

Reliability was measured by calculating agreement between coders (intercoder agreement)

and by calculating the consistency of each individual coder in time (intracoder agreement).

Our analysis covered an amount of 10 833 death certificates.

Results:

The intercoder agreement of four coders on the underlying cause of death was 78%. In 2.2% of the cases coders agreed on a change of the code assigned in 2005. The (mean) intracoder agreement of four coders was 89%. Agreement was associated with the specificity of the ICD-10 code (chapter, three digits, four digits), the age of the deceased, the number of coders and the number of diseases reported on the death certificate. The reliability of cause-of-death statistics turned out to be high (> 90%) for major causes of death such as cancers and acute myocardial infarction. For chronic diseases, such as diabetes and renal insufficiency, reliability was low (< 70%).

Conclusions:

The reliability of cause-of-death statistics varies by ICD-10 code/chapter. A statistical office should provide coders with (additional) rules for coding diseases with a low reliability and evaluate these rules regularly. Users of cause-of-death statistics should exercise caution when interpreting causes of death with a low reliability. Studies of reliability should take into account the number of coders involved and the number of codes on a death certificate.

Introduction

Statistics Netherlands has been reporting on mortality since 1901 by publishing underlying causes of death for deceased persons. 1 2 An underlying cause of death is defined as "the disease or injury that initiated the train of morbid events leading directly to death or the circumstances of the accident or violence which produced the fatal injury". [START_REF]International Statistical Classification of Diseases and Related Health Problems[END_REF] The underlying cause of death is selected from a death certificate and coded according to the ICD-10. The coding of an underlying cause of death depends on (i) the information available on the death certificate, (ii) the interpretation of explicit coding or selection rules and (iii) individual deliberations of coders. 4 5 6 These factors vary in time and place, so the coding is apt to variations which could hinder the interpretation of trends in time or differences between populations. 7 8 9 Therefore we studied the reliability of coding. In general, reliability is measured by repeating an observation and calculating agreement between observers (this study: intercoder agreement) or calculating the consistency of the same observer in time (this study: intracoder agreement). Reliability differs from validity, e.g. the correspondence between the code for the cause of death and the clinical opinion, the medical record or the autopsy report. 10 11 We studied the reliability of coding an underlying cause of death in the Netherlands. In this article, we report on the intercoder and intracoder agreement; we discuss some explanations of our findings and we compare our results with the results of other studies of reliability.

Methods

In the Netherlands, death certificates are issued by the attending physician or medical examiner. They are sent to Statistics Netherlands by the municipality where the death occurred. At Statistics Netherlands the death certificates are coded manually. Four coders are employed. Each coder handles about 30 000-40 000 death certificates a year. The 10th revision of the International Classification of Diseases (ICD-10) has been used for coding the underlying cause of death since 1996. In the Netherlands, a maximum of four diseases can be coded per deceased. We studied death certificates from the month of May 2005 (n = 11 266). This month was chosen, because (i) all four coders worked on it (no vacations), (ii) it is not too big as a burden (work load) for coding it a second time, and (iii) it can be considered representative for deceased persons in the Netherlands with regard to age, sex and frequency of the most important causes of death. Stillbirths and death certificates which (originally) needed additional information from the certifying physician in order to be coded were excluded (n = 393). After additional information is obtained, the code of the underlying cause of death is written on this kind of certificates; so they are not suitable for coding them a second time. For reasons of efficiency, cases with just one code per deceased (n = 5236) were sampled at random every one in two. These (easy) cases require no selection of an underlying cause of death and are expected to exhibit less variation. The four coders of Statistics Netherlands coded all the remaining death certificates (n= 8 215), independently of each other during the second half of 2007, two years after the certificates had passed in the routine coding process. This time period is long enough to prevent direct recall of cases by coders and short enough to prevent a noticeable influence by a change in coding rules.

Analysis was performed on weighted data in order to compensate for the sampling of cases with (originally) just one code per deceased (n= 2618). These cases were weighted by counting them for 2, resulting in a weighted number of 10 833 death certificates. For analysis we used SPSS 14.0. First, we calculated the intercoder agreement by comparing the four coders on underlying cause of death per deceased. Second, we calculated the intracoder agreement for each coder by comparing the code for the selected underlying cause of death during the routine coding process in 2005 with the code assigned when coding the same death certificate again in 2007. The number of death certificates differs per coder now from 1500 to 2700, according to the initial number of certificates coded by each coder in 2005. For identifying significant differences between intracoder agreements, we calculated 95% confidence intervals.

In our study, the strongest measure of agreement seems to be four coders agreeing on a code for the underlying cause of death while this code also corresponds with the code assigned in 2005. This measurement is shown in the tables 4 and 5 according to the European-65 short list for causes of death (Eurostat, August 1998). This list contains 65 causes of death agreed on as being important for health policy purposes in Europe. Twenty causes of death were omitted, because of low frequencies (n < 20). For the remaining causes of death, the agreement on underlying cause of death by ICD-10 chapter can be found in table 4. Table 4 presents agreement on different levels of specification of the ICD code in order to enable a comparison between coding on chapter level (one digit) and the specification of a code within a chapter (three of four digit codes). The agreement on ICD-10 codes representing specific diseases as a cause of death can be found in table 5 on a four digit code level (maximum level of specification). We added some causes of death not on the European-65 short list in order to make our results more specific.

For comparing our results with other studies of reliability, we calculated the intercoder agreement for each combination of three or two coders. We did not use a (Fleisch) kappa measure for inter-rater reliability, as the chance on expected agreement among the four coders in our study is very small due to the large number of ICD-10 categories (about 8000 code entries). [START_REF] Brennan | Satistical methods for assessing observer variability in clinical measures[END_REF] We present our results by specificity of an ICD-10 code. The ICD-10 has twenty-two chapters, representing disease entities and external causes of death. Letters from the alphabetusually called digits -are used to identify the chapters. The chapters or disease entities are divided into groups, representing pathological entities or manifestations of diseases. Two digits are used for coding these entities. Pathological entities or manifestations are further divided into specific subgroups by different principles. Sometimes location is used (neoplasms), sometimes causal mechanisms (infections), and external causes of death are further divided by place of injury. The fourth or fifth position of an ICD-10 code is used for this. In this way a disease can be coded by a 4-digit code (for example: acute transmural myocardial infarction of anterior wall = I21.1), a 3-digit code (for example: acute myocardial infarction = I21) or a one-digit or chapter code (for example: disease of the circulatory system = I). In our study, all cases were coded at a 4-digit level. For diseases with just a 3-digit code (for example stroke: I64) a zero is added by default.

Results

Our study included 10 833 death certificates. Eighty percent of the deceased was 65 or older. There was a slightly higher proportion of women (51.1%) than men (48.9%). There is an increase in the number of diseases reported on the death certificate by age of the deceased. The relatively high proportion of people aged 0-49 years with more than one code is explained by the high proportion of external causes of death in this age category. For external causes of death the ICD-10 always requires two codes. The first code represents the circumstances of the accident or violence which produced the fatal injury and the second code represents the injury itself -see table 1.

(Table 1 here) Table 2 shows the intercoder agreement on underlying cause of death for sex and age of the deceased and for the number of codes reported on the death certificate per deceased. There is a relationship between intercoder agreement and the number of coders participating in the study: in 78.0% of the cases four, in 81.5% three and in 86.9% two coders agree on the underlying cause of death. Intercoder agreement increases with a decrease in specification level of ICD-10 code. It is 78.0% at a 4-digit level, 81.8% at a 3-digit level and 90.5% at chapter (one digit) level. There is no significant difference in agreement for coding the underlying cause of death of men or women (p > 0.05). Intercoder agreement first increases and then decreases with an increase in age of the deceased. Intercoder agreement increases with a decrease in the number of diseases (codes) reported on the death certificate -see table 2.

(Table 2 here) Table 3 shows the intracoder agreement. The (weighted) mean intracoder agreement of four coders was 89.0% at a 4-digit, 89.4% at a 3-digit and 95.3% at ICD-10 chapter level. Coder D appeared to be more consistent than coders A or B at a three or 4-digit ICD code level.

There was no significant difference of intracoder agreements at ICD-10 chapter level -see table 3.

(Table 3 here) Table 4 presents the agreement on the underlying cause of death by ICD-10 chapter as the percentage of cases on which all four coders agree while the code is also in agreement with the code assigned in 2005. The agreement varies by chapter and by specification level of a code. At chapter level, agreement is high for cancers, diseases of the circulatory system and for external causes of death such as transport accidents and suicide; it is low for diseases of the bloodforming organs or skin. These findings indicate how easy or difficult it is to assign a case to an ICD-10 chapter. The difference in agreement between a code at chapter level and a code at 3-digit level is relatively large ( > 15%) for diseases of the bloodforming organs, the perinatal period, congenital malformations and transport accidents, indicating difficulties in specifying a code within a chapter. The difference in agreement on a 3-digit and a 4-digit code is relatively large ( > 15%) for endocrine diseases, diseases of the musculoskeletal system and for external causes of death such as suicide, indicating difficulties in specifying complications of diabetes, the site of musculoskeletal involvement, or the circumstances of a suicide respectively -see table 4.

(Table 4 here) Table 5 shows the reliability of codes for underlying causes-of-death at the 4-digit level. All four coders agree on a code and this code is in agreement with the original one assigned to a case in 2005. For major causes of death, such as cancers and acute myocardial infarction, the reliability appeared high (>90%). The reliability was low (< 70%) for chronic diseases such as diabetes, atrial fibrillation and flutter, cardiac arrhythmias, gastrointestinal ulcers, chronic liver diseases, renal failure, and for congenital malformations of the nervous systemsee table 5.

(Table 5 here)

Discussion

Information on reliability can be used for interpreting cause-of-death statistics. 13 14 First of all, we identified causes of death with a low reliability, notably chronic diseases (table 5). This low reliability is caused by difficulties in coding (for example specifying diabetes mellitus on a 4-digit level) on the one hand and difficulties in selecting a chronic disease (such as renal failure) as underlying cause of death on the other. There are several reasons for this. First of all, for many chronic diseases it is hard to assess their contribution in the process of dying, even with a full medical record at hand. For example, diabetes mellitus may or may not be the cause of an acute myocardial infarction. Moreover, all chronic diseases that severely restrict mobility may or may not cause infections of bladder or lung. As chronic diseases are, by definition, not very lethal and may well start 20 years or more before death, it is hard to draw the line where they are considered the starting point of a causal chain of events leading to death. The ICD provides rules for this, but as Lu et al. discovered, these rules leave room for individual interpretations. [START_REF] Lu | Factors associated with errors in death certificate completion: a national study in Taiwan[END_REF] For example, according to the ICD-10 selection rule 3 a coder can select a disease as a cause of death disregarding any causal connection reported on the death certificate when death is obviously a direct consequence of this disease. [START_REF]International Statistical Classification of Diseases and Related Health Problems[END_REF] Interpretation of "obviously" is left to the discretion of the coders, forming a source of variations in coding. A second reason lies in the coding process itself. Table 2 shows the agreement decreases with age and with the number of diseases coded on a death certificate. Chronic diseases tend to become more common with age, so the number of codes on a death certificate increases with age (see: table 1). The number of diseases on a death certificate has a direct influence on agreement. If there is only one coded disease present, only coding variation and no variation in selecting the underlying cause of death can occur.

With two, three or four codes present, there are increasingly more ways to interpret their causal ranking. Therefore the number of coded diseases per death certificate determines for a large part the differences in reliability.

Our results fit in with several other studies of reliability. Studies of reliability were published in the USA, [START_REF]A methodological study of quality control procedures for mortality medical coding[END_REF] Austria, [START_REF] Leitner | Control of coding in Austria[END_REF] Germany, 19 20 Sweden, 21 Switzerland, [START_REF] Minder | Zur Qualität der Schlüsselung der Todesursachen in der Schweiz[END_REF] and Taiwan [START_REF] Lu | Accuracy of cause-of-death coding in Taiwan: types of miscoding and effect on motality statistics[END_REF] -see table 6.

(Table 6 here)

In comparison to other countries we seem to perform well. However, comparing different studies of reliability is difficult as a variety of study protocols and measures have been applied. Some countries, like Sweden and Taiwan, report an intercoder agreement much lower than found in our study. The aim of these studies differs from ours. They compare a coder with an expert, e.g. one coder is a standard for the other based on his or her knowledge and experience. This looks more like a study of the validity of coding. We found a difference between the agreement of four coders as such (table 2) and the agreement of four coders on a code for the underlying cause of death while this code corresponds with the code of 2005 (table 4): 78.0 versus 75.8% at a 4-digit ICD-10 code level, e.g. in looking back, coders seem to agree on a change of code in 2.2% of the cases. This seems to be a general measure for the validity of coding in the Netherlands. In Germany and Switzerland, samples of death certificates were exchanged among coders of different countries and recoded manually. These studies look more like studies of reliability, e.g. the consistency of coding.

However, the second coding took place apart from the routine coding process and could thus be influenced by context (coders correcting other coders from different countries). The intercoder agreement in these studies is also (much) lower than in our study. A recent German study of reliability showed low agreement when comparing two coding offices. [START_REF] Winkler | Reliability of coding causes of death with ICD-10 in Germany[END_REF] This study is restricted to the mortality of ethnic German migrants from the former Soviet Union and includes 372 death certificates. Studies differ in study population and the number of death certificates. This might explain the difference in findings. However, the number of coders involved seems to be the most important explanation for the different findings of the different studies of reliability. Agreement decreases with an increasing number of coders on all levels of specificity of ICD codes. Moreover, in our study, the intracoder agreement (table 3: 88-90% at a 4-or 3-digit level and 95-96% at chapter level) was more or less the same in magnitude as the intercoder agreement for pairs of coders (table 2: 87% at a 4-digit, 89% at a 3-digit and 94% at chapter level). This is an important finding. If intracoder and intercoder agreement are about the same, this means that the coding process in itself has limited reproducibility and is not bound by individual preferences of coders. Therefore, reporting on reliability should include the number of coders.

In contrast to most other studies of reliability (see: table 6), we found that both intercoder and intracoder agreement did not differ significantly between the third or fourth digit of codes: 78% versus 81.5% and 89.0% versus 89.4% respectively. This finding could be explained by the structure of the ICD-10. The ICD-10 has about 8 000 code entries. Some of them do not have a fourth digit at all, such as stroke (I64), unspecified dementia (F03) or Parkinson"s disease (G20). This kind of codes -in our study 20% -has no variation at a 4-digit level (table 4). Furthermore, we noticed that in 53% of the cases an 8 ("other") or 9 ("unspecified") is assigned as a fourth digit. The death certificate did not provide the information for a more accurate classification in these cases. Approximately 73% of the cases have no informative fourth digit and a limited variation at a 4-digit level of the ICD-10 code. This is an important finding as it indicates the upper level of detail that can be expected from a cause-of-death statistics.

From a user perspective, the results of our study can be used as an estimate of possible bias when studying trends in time or comparing causes of death between places or populations.

For major causes of death, such as cancers or acute myocardial infarction, this bias is low.

Observed differences can be attributed to underlying causal factors such as unhealthy behaviour, smoking, food or environment. For chronic diseases, such as diabetes or renal failure, the bias is high. Observed differences could be caused by the coding practice as such. Bias can be reduced by using software for coding and selecting the cause of death, like ACME or IRIS, as the reliability study of Sweden shows (table 6). 24 25 The impact of these automatic coding systems on reliability is large. [START_REF] Th | The automated classification of medical entities (ACME) system objectively assessed the approriateness of underlying cause-of-death certification and assignment[END_REF] However, automatic coding systems reject 15-20% of the death certificates, because the system does not recognise medical terms on the death certificate, encounters ambiguous causal relationships or cannot handle external causes of death or complications of surgery. 27 These rejected death certificates have to be coded manually. They are still exposed to intercoder or intracoder variations. Therefore, information on the reliability of manual coding is important for understanding cause-of-death statistics now and in the near future.

  system used ** Agreement of three coders

Table 1

 1 The study population, according to sex and age of the deceased by the number of diseases reported on the death certificate* (percentages, n = 10 833)

			number of diseases per deceased *	Total
			1	2	3	4
	Sex of deceased
		Male	47.5	33.5	13.5	5.6	48.9
		Female	49.1	31.8	13.7	5.4	51.1
	Age of deceased
	(years)	0-49	46.9	37.1	13.3	2.7	6.1
		50-64	61.2	27.0	7.2	4.5	13.0
		65-74	51.4	30.6	13.4	4.7	17.8
		75+	45.0	33.8	15.0	6.2	63.1
	* In the Netherlands, a maximum of four diseases can be coded per deceased

Table 2

 2 Intercoder agreement on underlying cause of death by specification of the ICD-10 code for age of the deceased and the number of codes (diseases) per deceased (percentages, n = 10 833)

	4-digits	3-digits	IDC-10 chapter
			(one digit)

*Every possible combination of three or two out of four coders

Table 3

 3 Intracoder agreement on underlying cause of death (percentages with 95% CI between brackets, n = 10 833)

		Coder A	Coder B	Coder C	Coder D	Mean*
	4-digit level	87.6	87.9	90.0	91.1	89.0
		(86.2-89.0 )	(86.8-89.0 )	(88.9-91.1 )	(89.8-92.4 )	(88.4-89.6)
	3-digit level	88.1	88.3	90.3	91.3	89.4
		(86.7-89.5 )	(87.2-89.4 )	(89.3-91.3 )	(90.0-92.6 )	(88.8-90.0)
	ICD-10 chapter	95.7	94.8	95.4	95.7	95.3
		(94.1-95.9 )	(94.1-95.5 )	(94.7-96.1 )	(94.8-96.6 )	(94.9-95.7)
	* weighted mean, CI: mean ± 1,96 x (1/∑wi)

Table 4

 4 Agreement on underlying cause of death by ICD-10 chapter * (percentages, n = number of death certificates with corresponding code in 2005, e.g. 100% in row) Agreement of four coders on ICD-10 code in 2007 and correspondence of this code with the code in 2005 ** For external causes of death, reliability is also influenced by adjustments made after May 2005 based on information concerning the circumstances of death from other sources (police, courts).

	ICD-10 chapter

* ++ not on European-65 short list of causes of death

(August 1998)

Table 6

 6 Intercoder agreement in studies of reliability

	ICD code	Germany	Netherlands	Austria	Sweden	Taiwan	Germany	Switzerland	USA
		2009	2007 **	2006 **	2002 *	2000	1989	1981 **	1980
	4-digit level	46.0%	81.9%	70%	94.6%		61.5%		90.3%
	3-digit level	56.0%	84.4%	74%	96.7%	80.9%	67.7%	75.4%	90.2%
	1-digit level	78.2%	91.8%	87%	98.7%	83.9%	86.9%
	(chapter)
		n = 372
		(ICD-10)

Pavillon G, Coleman M, Johansson LA, Jougla E, Kardaun J. Coding of causes of death in European community. Luxembourg: Eurostat 1998 (report 96/S 99-57617/EN).

Acknowledgements

We would like to thank the four coders of Statistics Netherlands for their participation in this study. We thank Onno van Hilten and the staff members of Health Statistics for their critical comments on earlier drafts of this paper.


Elemér E Rosinger
email: eerosinger@hotmail.com

Marie-Louise Nykamp

Reducing the Quantumness of Composite Quantum Systems to Two Classical Compositions

A family of quantumness spaces is identified and precisely defined. They are spaces which characterize the difference between states given by classical compositions of systems, and on the other hand, states corresponding to their quantum compositions. Consequently, the quantum composition of systems is reduced to two classical compositions. A family of rankings is also defined for the respective family of quantumness.

"History is written with the feet ...

Introduction

As seen in [START_REF] Rosinger | Non-Cartesian Systems : an Open Problem[END_REF][START_REF] Rosinger | Further on Non-Cartesian Systems[END_REF][START_REF] Rosinger | On Systems[END_REF], there is a natural division of physical systems into Cartesian ones, and on the other hand, non-Cartesian ones, according to the way their state spaces compose. Namely, given two Cartesian systems X and Y with the respective state spaces E and F , then the composite system "X and Y " has the state space given by the Cartesian product E × F . Classical physical systems are in this sense Cartesian. On the other hand, quantum systems, for instance, have considerably larger state spaces for their composites. Namely, if X and Y are two such systems and their state spaces are the complex Hilbert spaces E and F , respectively, then the state space of the composite quantum system "X and Y " is the tensor product E F . And indeed, this is a considerably larger space than the Cartesian product E × F , since we have the injective mapping, which for convenience we shall consider to be an embedding

(1.1) E × F (x, y) -→ x ⊗ y ∈ E F
thus is seen according to the inclusion

(1.2) E × F ⊂ E F
and the difference between the two sets, in this case both complex Hilbert spaces is clearly illustrated already in the finite dimensional case when, if m, n are the dimensions of E and F , respectively, then m + n is the dimension of E × F , while E F will have the dimension mn. Thus in general, the set of entangled elements

(1.3) (E F ) \ (E × F )
is considerably larger than the set E × F of non-entangled elements.

Here it is important to note the following related to (1.2). Let E and F be abelian groups. Then the sets E × F and E F obtain cor-responding structures of abelian group. However, E × F it is not a subgroup of the abelian group E F . Instead, E × F is a generator of E F . Consequently, the mapping in (1.1) is not a group homomorphism. Clearly, similar situations happen when E and F are vector or Hilbert spaces.

An essential difference, therefore, between Cartesian and non-Cartesian physical systems is that in the state spaces of the composites of two of the latter kind there are states which can be seen as entangled in some appropriate sense, namely, those states which cannot be expressed simply in terms of a pair of states, with each state in the pair taken from one of the two component systems. So far, physics happens to know only about two ways of composing systems, namely, the classical Cartesian one, and the quantum one. However, there is no known reason to expect that these two ways would be the only ones physical systems are indeed possible to be composed. In this sense, being an entangled state of the composite of two physical system may eventually have a more general meaning than being an element in the set (1.3).

As far as quantum systems are concerned, it is well known that entangled composite states are most important, for instance, in quantum computation, and in general, quantum information technology.

As mentioned, so far, it appears that the only known non-Cartesian physical systems are the quantum ones. In this regard, in [START_REF] Rosinger | Non-Cartesian Systems : an Open Problem[END_REF][START_REF] Rosinger | Further on Non-Cartesian Systems[END_REF][START_REF] Rosinger | On Systems[END_REF], the problem was formulated to find physical systems other than the quantum ones and which are non-Cartesian. Needless to say, there may be various applicative advantages in such systems. Among others, they may be used to build computers which -due to the presence of entangled states -could have advantages over usual electronic digital computers.

The main message, see pct. 2) in Remark 1 in section 4, is that for two quantum systems X and Y , one has where c denotes the classical composition of systems, that is, the Cartesian product of their state spaces, while q denotes the quantum composition of systems, thus the tensor product of their state spaces, see [START_REF] Rosinger | On Systems[END_REF].

It follows that the quantum composition X q Y is reduced to two classical compositions, namely, first

X c Y
the result of which is then further composed classically with quantumness of X and Y Needless to say, by the above reduction the quantum aspects are not eliminated completely, since they remain in the quantumness of X and Y However, the interest in such reduction is in the consequent relegation of whatever the quantum aspects may ever be as such to a specific well circumscribed place, namely, as a mere component in two successive classical compositions.

Constructing Tensor Products

Let us start with a more general, and thus simpler setup in order to better highlight what is going on. Let E and F be two abelian groups.

Then their tensor product E F is constructed in the following five steps, [START_REF] Rosinger | Non-Cartesian Systems : an Open Problem[END_REF][START_REF] Rosinger | Further on Non-Cartesian Systems[END_REF][START_REF] Rosinger | On Systems[END_REF].

Step 1 :

Let G be the free monoid generated by the elements of the usual Cartesian product E ×F . In other words, the elements of G are all the finite sequences

(2.1) (a 1 , b 1 ), (a 2 , b 2 ), (a 3 , b 3 ), . . . , (a n , b n ) where n ≥ 1 and a 1 , a 2 , a 3 , . . . , a n ∈ E, b 1 , b 2 , b 3 , . . . , b n ∈ F .
We also include the empty sequence, which thus corresponds to n = 0.

Step 2 :

We recall that the monoidal composition of these sequences is done simply by their concatenation. Furthermore, in order to simplify the notation, the comas between pairs of brackets in (2.1) will be omitted.

It will be convenient to denote the resulting monoid by

(2.2) (G, )
Clearly, (G, ) is a noncommutative monoid, whenever at least one of the groups E or F has more than one single element. Further, we have the injective mapping

(2.3) E × F (a, b) -→ (a, b) ∈ G which in fact is the embedding (2.4) E × F ⊆ G
and this embedding is strict, whenever m + n < mn, where E has at least m elements, while F has at least n elements.

Step 3 :

We define an equivalence relation ≈ on G as follows. Given two elements

g = (a 1 , b 1 )(a 2 , b 2 )(a 3 , b 3 ) . . . (a n , b n ), h = (c 1 , d 1 )(c 2 , d 2 )(c 3 , d 3 ) . . . (c m , d m ) ∈ G
they are equivalent, if and only if any of the following conditions holds :

(2.5) g = h or one of the elements g or h can be obtained from the other by a finite number of applications of any of the following operations : It follows easily that ≈ is an equivalence relation which is compatible with the monoid (G, ).

Step 4 :

Finally, one defines the tensor product as the quotient space

(2.12) E F = G/ ≈
and in view of (2.3), (2.4), obtains the injective mapping

(2.13) E × F (a, b) -→ a ⊗ b ∈ E F
where a ⊗ b denotes the coset, or in other words, the equivalence class of (a, b) ∈ G, see (2.4), with respect to the equivalence relation ≈ on G.

Step 5 :

Since the equivalence ≈ is compatible with the monoid structure of (G, ), and in view of (2.8), (2-9), it follows that the tensor product E F obtains an abelian group structure.

It is useful to note the fact that in the above steps 1 and 2, there is absolutely no need for any structure on the sets E and F , and thus they can be arbitrary nonvoid sets.

Furthermore, in step 3 above, the only place the structure on the sets E and F appears is in (2.8), (2.9). And the way this structure is involved allows for wide ranging generalizations, far beyond any algebra, [2-6].

Universal Property of Tensor Products

For convenience, we recall here certain main features of the usual tensor product of vector spaces, and relate them to certain properties of Cartesian products.

Let K be a field and E, F, G vector spaces over K.

Cartesian Product of Vector Spaces

Then E × F is the vector space over K where the operations are given by λ(x, y) + µ(u, v) = (λx + µu, λy + µv)

for any x, y ∈ E, u, v ∈ F, λ, µ ∈ K.

Linear Mappings

Let L(E, F ) be the set of all mappings

f : E -→ F such that f (λx + µu) = λf (x) + µf (u)
for u, v ∈ E, λ, µ ∈ K.

Bilinear Mappings

Let L(E, F ; G) be the set of all mappings

g : E × F -→ G
such that for x ∈ E fixed, the mapping F y -→ g(x, y) ∈ G is linear in y, and similarly, for y ∈ F fixed, the mapping

E x -→ g(x, y) ∈ G is linear in x ∈ E.
It is easy to see that

L(E, F ; G) = L(E, L(F, G))

Tensor Products

The aim of the tensor product E F is to establish a close connection between the bilinear mappings in L(E, F ; G) and the linear mappings in L(E F, G).

Namely, the tensor product E F is :

(3.4.1) a vector space over K, together with (3.4.2) a bilinear mapping t : E × F -→ E F , such that we have the following : UNIVERSALITY PROPERTY ∀ V vector space over K, g ∈ L(E, F ; V ) bilinear mapping :

∃ ! h ∈ L(E F, V ) linear mapping : h • t = g
or in other words :

(3.4.3) the diagram commutes

E × F E E F t d d d d d d d d d d © V g ∃ ! h
and (3.4.4) the tensor product E F is unique up to vector space isomorphism.

Therefore we have the injective mapping

L(E, F ; V ) g -→ h ∈ L(E F, V ) with h • t = g
The converse mapping

L(E F, V ) h -→ g = h • t ∈ L(E, F ; V )
obviously exists. Thus we have the bijective mapping

(3.4.5) L(E F, V ) h -→ g = h • t ∈ L(E, F ; V )

An Application of the Universal Property of Tensor Products

We shall particularize (3.4.3) as follows. Let V = E × F and g : E × F -→ E × F any bilinear mapping. Then we obtain a unique linear mapping (3.5.1)

h g : E F -→ E × F such that (3.5.2) g = h g • t and clearly (3.5.3) g surjective =⇒ h g surjective moreover, in general (3.5.4) g(E × F ) ⊆ h g (E F )
Now, if we consider the image of h g , namely

(3.5.5) (E × F ) g = h g (E F ) ⊆ E × F
which in view of the linearity of h g is a vector subspace in E × F , then obviously we have the vector space isomorphism

(3.5.6) (E × F ) g ∼ = (E F )/ker h g
In this way, we obtain the vector space isomorphism

(3.5.7) E F ∼ = (E × F ) g × ker h g ⊆ (E × F ) × ker h g in view of

Lemma

Let A be a vector space over the field K, while B is a vector subspace in A. Then we have the vector space isomorphism (3.5.8)

A ∼ = B × (A/B) Proof.
Let C be a vector subspace in A, such that A = B ⊕ C. Then we have the vector space isomorphism C ∼ = A/B. However, we also have the vector space isomorphism A ∼ = B × C.

In view of (3.5.7), we introduce Definition 1.

Given two vector spaces E and F on a field K, and a bilinear mapping

g : E × F -→ E × F .
Then the vector subspace ker h g in E F is called the g-quantumness of the tensor product E F . Now in view of (3.5.3), (3.5.5) and (3.5.7), we obtain (3.5.9)

g surjective =⇒ E F ∼ = (E × F ) × ker h g
As for ker h g , (3.4.3) gives for 1≤i≤n x i ⊗ y i ∈ E F the relation

h g ( 1≤i≤n x i ⊗ y i ) = (0, 0) ∈ E × F if and only if 1≤i≤n h g (x i ⊗ y i ) = (0, 0) ∈ E × F .
Thus we obtain (3.5.10)

ker h g = { 1≤i≤n x i ⊗ y i ∈ E F | 1≤i≤n g(x i , y i ) = (0, 0) ∈ E × F }
We further note that every bilinear mapping g :

E × F -→ E × F is of the form (3.5.11) E × F (x, y) -→ g(x, y) = (g E (x, y), g F (x, y)) ∈ E × F where (3.5.12) g E : E × F -→ E, g F : E × F -→ F
are bilinear mappings. Consequently, (3.5.10) takes the form

(3.5.13) ker h g = { 1≤i≤n x i ⊗ y i ∈ E F | 1≤i≤n g E (x i , y i ) = 0 ∈ E, 1≤i≤n g F (x i , y i ) = 0 ∈ F } Remark 1.
1) The decomposition (3.5.7), and its particular instance in (3.5.9), of the tensor product E F of two arbitrary vector spaces has, in terms of (1.1) -(1.3), the interest of bringing a certain clarity about the difference between the simple Cartesian product E × F , and on the other hand, the more involved tensor product E F . And that clarification is made in terms of a further simple Cartesian product, as well as of a special vector subspace ker h g in the tensor product E F , a vector subspace called g-quantumness of E and F .

Here g is an arbitrary bilinear operator on E × F . Therefore, there is in fact a whole family of decompositions (3.5.7), (3.5.9), corresponding to the various mentioned bilinear mappings g.

The respective quantum mechanical interpretation of the family of decompositions (3.5.7), (3.5.9) was mentioned in section 1.

2) In case the dimensions of E and F are finite, then (3.5.5) obviously gives

(3.5.14) dim (E × F ) g ≤ dim E + dim F
hence in view of (3.5.7), one has

(3.5.15) dim ker h g ≥ dim E dim F -dim E -dim F
while in the particular case of (3.5.9), one obtains

(3.5.16) dim ker h g = dim E dim F -dim E -dim F
3) All the above results extend trivially to any finite number of quantum compositions.

Universality Property of Cartesian Products

Let A, B be two nonvoid sets. Their cartesian product is :

(3.6.1) a set A × B, together with (3.6.2) two projection mappings p A : A × A -→ A, p B : A × B -→ B, such that we have the following :

UNIVERSALITY PROPERTY ∀ Z nonvoid set, f : Z -→ A, g : Z -→ B : ∃ ! h : Z -→ A × B : f = p A • h, g = p B • h
or in other words :

∀ G ∀ f E d d d d d d ∀ g F E × F d d d d d d s pr E © pr F E ∃ ! h t d d d d d d ∀ k E F ∀ V c ∃ ! l

The Quantumness of Composite Quantum Systems

Based on the above, and specifically (3.5.7) and its particular instance (3.5.9), let us see in some detail what is the difference between the Cartesian product E × F and the tensor product E F in (1.1) -(1.3), since it is precisely this difference which stands for the assumed quantumness that distinguishes between the classical composition of the respective systems leading to the Cartesian product E × F , and on the other hand, the quantum composition which gives the tensor product E F . For convenience, we shall assume here that both E and F are vector spaces on a given field K, a case which obviously contains in particular the situation of quantum interest, when E and F are complex Hilbert spaces.

Clearly, E F is in general considerably larger than E × F . An important fact here, however, is that we have (3.5.7), and in particular, (3.5.9). Consequently, we obtain Theorem 1.

Given two vector spaces E and F on a field K, and a bilinear mapping g : E × F -→ E × F . Then for every 1≤i≤n x i ⊗ y i ∈ E F , there exists a unique representation (4.1)

1≤i≤n x i ⊗ y i = x ⊗ y + 1≤j≤m u j ⊗ v j
where x i , x, u j ∈ E, y i , y, v j ∈ F , and

(4.2) 1≤j≤m g(u j , v j ) = (0, 0) ∈ E × F
which condition, in view of (3.5.11), (3.5.12), is equivalent with

(4.3) 1≤j≤m g E (u j , v j ) = 0 ∈ E, 1≤j≤m g F (u j , v j ) = 0 ∈ F Consequently, x ⊗ y ∈ (E × F ) g
is the classical term in (4.1), while 1≤j≤m u j ⊗ v j ∈ ker h g corresponds to the g-quantumness term.

Further, we have the following implicit conditions on the classical term x ⊗ y ∈ (E × F ) g , namely (4.4) 1≤i≤n g(x i , y i ) = g(x, y) ∈ E × F or equivalently (4.5)

1≤i≤n g E (x i , y i ) = g E (x, y) ∈ E, 1≤i≤n g F (x i , y i ) = g F (x, y) ∈ F Corollary 1.
Given two vector spaces E and F on a field K, and a bilinear mapping

g : E × F -→ E × F .
Then the quantum composition E F is the Cartesian composition of the vector subspace (E × F ) g of the Cartesian composition E × F , with the g-quantumness space ker h g .

Examples

1) Clearly, for u ∈ E, v ∈ F , we have

(5.1) u ⊗ v ∈ ker h g ⇐⇒ g E (u, v) = 0 ∈ E, g F (u, v) = 0 ∈ F 2) For u 1 , u 2 ∈ E, v 1 , v 2 ∈ F , we have (5.2) u 1 ⊗ v 1 + u 2 ⊗ v 2 ∈ ker h g ⇐⇒ ⇐⇒   g E (u 1 , v 1 ) + g E (u 2 , v 2 ) = 0 ∈ E, g F (u 1 , v 1 ) + g F (u 2 , v 2 ) = 0 ∈ F   3) Let now E = F = R 2 , with K = R. Further, let any bilinear map- ping g : E × F -→ E × F .
We consider the non-normalized Bell state-like element

(5.3) z = | a > ⊗ | a > + | b > ⊗ | b > ∈ E F
where | a >, | b > ∈ R 2 form an orthonormal basis in R 2 , and consider its unique representation (4.1), namely

(5.4) z = x ⊗ y + 1≤j≤m u j ⊗ v j

In order to find the terms in the right hand above, we recall (4.4) and obtain

(5.5) g(x, y) = g(| a >, | a >) + g(| b >, | b >) while (4.2) gives (5.6) 1≤j≤m g(u j , v j ) = (0, 0) ∈ E × F
Let us now choose a specific bilinear mapping g :

E × F -→ E × F . For instance, given (x, y) = ((x 1 , x 2 ), (y 1 , y 2 )) ∈ R 2 × R 2 , we can take (5.7) g(x, y) = ((x 1 y 1 , x 2 y 2 ), (x 1 y 2 , x 2 y 1 )) ∈ R 2 × R 2 Then, assuming that | a > = (a 1 , a 2 ), | b > = (b 1 , b 2 ) ∈ R 2 × R 2
, we obtain from (5.5) the relations (5.8)

x 1 y 1 = |a 1 | 2 + |b 1 | 2 , x 2 y 2 = |a 2 | 2 + |b 2 | 2 , x 1 y 2 = x 2 y 1 = a 1 a 2 + b 1 b 2
As for the relation (5.6), it becomes (5.9) 1≤j≤m u j,1 v j,1 = 1≤j≤m u j,2 v j,2 = = 1≤j≤m u j,1 v j,2 = 1≤j≤m u j,2 v j,1 = 0 if we assume that u j = (u j,1 , u j,2 ),

v j = (v j,1 , v j,2 ) ∈ R 2 × R 2 .
Now, in the decomposition (5.4), the unknowns are x, y, u j , v j , each of which is a pair of complex numbers. Thus (5.4) contains 4m + 4 unknown complex numbers, namely x 1 , x 2 , y 1 , y 2 , u j,1 , u j,2 , v j,1 , v j,2 .

On the other hand, (5.8), (5.9) are 8 equations in those 4m + 4 complex numbers. Thus it may appear that, in general, one should have 4m + 4 ≤ 8, which means that m ≤ 1.

However, the g-quantumness term 1≤j≤m u j ⊗ v j in (5.4), although unique in its value in ker h g , need not have a unique form as such, due to a well known property of sums of tensor products. And then, it may happen that m > 1.

This issue is addressed in principle next.

Rank of Quantumness

In view of the above, we are led to Definition 2.

Given two vector spaces E and F on a field K, and a bilinear mapping g : E × F -→ E × F .

Then for z ∈ ker h g , we define (6.1) rank g z to be smallest m ∈ N for which we have (6.2) z = 1≤j≤m u j ⊗ v j with (6.3) 1≤j≤m g(u j , v j ) = (0, 0) ∈ E × F

In view of the unique representation (4.1), one can extend Definition 2, as follows Definition 3.

Given z ∈ (E F ), we define (6.4) rank g z = rank g w where one has the unique representation (4.1) (6.5) z = x ⊗ y + w with w ∈ ker h g .

X

  quanutm composed with Y = = ( X classical composed with Y ) classical composed with ( quantumness of X and Y ) or more compactly written (1.4) X q Y = ( X c Y ) c ( quantumness of X and Y )

  (2.6) a permutation of pairs (a, b) in g (2.7) a permutation of pairs (c, d) in h (2.8) replacement of a pair ((a + a ), b) in g with the pair of pairs (a , b)(a , b), or vice-versa (2.9) replacement of a pair (a, (b + b )) in g with the pair of pairs (a, b )(a, b ), or vice-versa (2.10) replacement of a pair ((c + c ), d) in h with the pair of pairs (c , d)(c , d), or vice-versa (2.11) replacement of a pair (c, (d + d )) in h with the pair of pairs (c, d )(c, d ), or vice-versawhere + is the group operation in the respective abelian groups E and F .

  Cartesian and Tensor Products seen together


Anne Bonneau

Vincent Colotte

Automatic Feedback for L2 Prosody Learning

Introduction

The emergence of speech signal processing functions has allowed speech scientists to analyse and modify speech with the aim of improving the perception and/or the production of speech in adverse conditions and language learning. In most cases, these tools are all the more efficient that they take into account the phonological system of each language. The applications of these aids are numerous and include the improvement of speech intelligibility in noise [START_REF] Hazan | The effect of cue-enhancement on the intelligibility of nonsense word and sentence materials presented in noise[END_REF], and for hearing aids [START_REF] Loizou | Mimicking the human ear[END_REF], computer assisted-aids for language learning devoted to speech therapists or learners of a foreign language. They are concerned either with speech intelligibility (comprehension), or with the perception and production of speech sounds and prosody. One of the best known works in this domain, the study by Tallal [START_REF] Tallal | Language comprehension in language-learning impaired children improved with acoustically modified speech[END_REF], published in Science, proposed speech modifications for hearing impaired children. Among signal modifications, the enhancement and slowing down of specific regions of speech signals has been the object of numerous studies and applications. [START_REF] Ortega | Enhancing acoustic cues to aid L2 speech perception[END_REF] applied these techniques to the improvement of speech intelligibility in second language learning. [START_REF] Colotte | Percpetual experiments on enhanced and slowed down speech sentences for second language acquistion[END_REF] also tackle speech intelligibility in second language learning. By means of an entirely automatic method, they enhanced unvoiced stops and fricatives and slowed down transitions. This paper deals with the elaboration of advanced feedback devoted to aid learners in the acquisition of the prosody of a foreign language, and presents a pilot experiment investigating the immediate impact of such feedback on learners. Note that, from now on, we will use L2 for second (non-native) language and L1 for the learners' first (native) language. A computer-assisted aid in language learning (a CALL system) and more precisely in prosody can offer at least three kinds of feedback: (1) visual feedback, such as visual displays of the learners' melodic curves, often associated with those of reference speakers (2) automatic diagnoses, based upon acoustical analyses of learners' realisations and (3) "advanced" perceptual feedback, through speech manipulations. Visualisation of melodic curves has been proposed since the early 60s, not only in second language learning but also in other domains such as hearing deficiencies. [START_REF] Vardanian | Teaching English through oscilloscope displays[END_REF] was one of the first scientists to use melodic curve visualisation in second language learning and test its impact on learners (Brasilian students learning English). Her results, may be due to the poor quality of visualisation (oscilloscope displays) were far from convincing. Nevertheless, after a first period of scepticism, speech specialists, like [START_REF] James | The Acquisition of a Second-Language intonation Using a Visualizer[END_REF], who used Philippe Martin's melodic curve detection [START_REF] Germain-Rutherford | Utilisation d'un logiciel de visualisation pour l'apprentissage de l'oral en langue seconde[END_REF], or De [START_REF] De Bot | Visual feedback on intonation I: Effectiveness and induced practice behaviour[END_REF], agreed on the efficacy of visual patterns. Current commercial computer-assisted aid in language learning systems, such as "Tell me more" of Auralog, LangMaster or Better Accent [START_REF] Kommissarchik | BetterAccent Tutor-Analysis and Visualization of Speech Prosody[END_REF] propose visual display of learners' melodic curves. The simple visualisation of melodic curves, although interesting, is not sufficient if one wants to provide learners with clear feedback about their production. Indeed, as [START_REF] Chun | Signal analysis software for teaching discourse intonation[END_REF] noted, if no further feedback is provided, the learners have to "extrapolate" their deviations themselves. A more efficient aid would provide learners with indications about their deviations and the way to improve their realisations. But the elaboration of automatic diagnoses encounters at least two major problems. Firstly, an automatic diagnosis of a learner's intonation presupposes the segmentation and labelling of the signal in syllables and in speech sounds. Yet this automatic segmentation, made by alignment if the text is known, is risky for non-native speech. (See 2.1.). A second major problem stems from the notion of "error" or "deviation". Indeed, we face issues such as "what are the links between acoustic cues and categories?" and "what are the accepted deviations?" For the moment the way to deal with this problem consists in proposing an evaluation of the degree of difference between prosodic patterns produced by a reference (or references) and those of the learner. Impact of feedback (diagnosis) on English intonation for French advanced learners (students in English at university) has been elaborated and tested by [START_REF] Herry | Subjective and Objective Evaluation of the Prosody of English Spoken by French Speakers: the Contribution of Computer Assisted Learning, Speech Prosody[END_REF]. Learners did not improve their realisations due to diagnosis. The authors underlined that students participating in the test were volunteers, which might have influence results. Another interesting way of helping learners in language learning relies upon speech modifications. Winpitch LPL [START_REF] Martin | WinPitch LTL II, a Multimodal Pronunciation Software[END_REF], a speech signal editor, proposes functions especially designed for L2 teachers and learners, which enable the user to modify by hand fundamental frequency1 and duration and annotate prosodic displays. Manipulations of prosodic cues are intended to make learners aware of prosodic patterns that do not exist in their first language; they are in general realized through PSOLA (Pitch Synchronous Over-Lap and Add) resynthesis. An interesting exploitation of speech manipulation consists in replacing the learner's prosodic cues by those of a reference without modifying the learner's timbre. WinSnoori [START_REF] Laprie | Snoori, a software for speech sciences[END_REF], software for speech analysis, enables users to realize this substitution by hand. An automatic version of this substitution has been realized [START_REF] Henry | Tools devoted to the acquisition of the prosody of a foreign language[END_REF] and implemented in WinSnoori. Other speech transformations based on an accent morphing technique have been recently proposed in the domain of foreign language intelligibility [START_REF] Ingram | Voice morphing and the manipulation of intraspeaker and cross-speaker phonetic variation to create foreign accent continua: A perceptual study[END_REF][START_REF] Yanagisawa | Accent morphing as a technique to improve the intelligibility of foreign-accented speech[END_REF]. In particular, the last authors modified the accent of the speaker whilst maintaining his identity to improve the intelligibility of foreign-accented speech. In this paper, we will present the various kinds of feedback (visual displays, diagnosis, and perceptual feedback based upon prosodic cues substitution)2 we elaborate for L2 prosody learning (section 2), and a pilot experiment devoted to test their immediate impact (section 3). Results are given in section 4.

The L2 prosody learning platform

A platform designed for manual and automatic feedback of learners' prosody has been implemented in a research version of WinSnoori software. The platform contains classical functions such as spectrographic displays, speech recording, real-time F0 (fundamental frequency) display, and integrates three particularly interesting functions with respect to language learning: an automatic text-to-speech alignment for native and non-native speech, a module for modifying the speech signal manually (the fundamental frequency and duration can be modified at the same time), and another module displaying automatic feedback on learners' (non-native) productions and exploiting most of the above-mentioned functions. Besides classical F0 displays, two kinds of feedback are provided to learners, each of them based upon a comparison between a reference and the learner's production. The first feedback, a diagnosis, provided both in the form of a short text and visual displays such as arrows, comes from an acoustic evaluation of the learner's realisation; it deals with two prosodic cues: the melodic curve, and phoneme duration. The second feedback is perceptual and consists in a replacement of the learner's prosodic cues (duration and F0) by those of the reference. We will first describe the phonetic alignment, necessary for automatic diagnoses and speech modifications, then the method developed to modify speech signals, and, finally, perceptual feedback, diagnosis and the modus operandi.

Automatic text-to-speech alignment

Prosodic cues generally appear on well determined linguistic and phonetic entities. So a preliminary segmentation into words, phones and syllables is necessary to localize the prosodic events and to compare the learner's realization with that of a reference. After users produce a linguistic entity (a word, a group of words, or a sentence) from the corpus, a segmentation of their realization is performed. First, a phonetization of the text is carried out using the CMU dictionary. Then, the segmentation is computed with a text-to-speech alignment, which establishes the correspondence between phonetic units and parts of the speech signal. Text-to-speech alignment is achieved using Hidden Markov Models [START_REF] Fohr | Utilisation de modèles de Markov pour l'étiquetage automatique et la reconnaissance de BREF80[END_REF]. Two different kinds of model have to be used: one for native speakers and another one for non-native speakers (learners). Indeed, since learners of a foreign language tend to replace the sounds they do not know by sounds of their first language, models used for native speakers (learned on the TIMIT database and developed for automatic speech recognition -ASR-purposes) should be adapted to non-native speakers. Although we have already at our disposal an ASR system designed for non-native speech recognition [START_REF] Bouselmi | Fully Automated Non-Native Speech Recognition Using Confusion-Based Acoustic Model Integration[END_REF] we are still working on non-native alignment. Indeed, if ASR systems do not need precise segment boundaries for typical applications such as automatic speech transcription or translation, this is not the case for language learning applications where the production of a subject is analysed and corrected. The need for precise boundaries is obvious for diagnosis about segment durations, but is also important for speech modification functions such as the ones we design. Since the detection of very precise segment boundaries is not always possible, we need to associate confidence levels with detections. Then feedback can be avoided in the cases where detections are not sufficiently reliable. We are presently working on these two aspects of ASR for non-native speech: the improvement of segment boundary precision and the elaboration of confidence levels. The syllabification program of NIST was applied to the CMU dictionary in order to obtain a database of syllabified words.

Speech signal modification algorithm

Signal modification functions have been included in the platform. These functions are based on an improved version of well known TD-PSOLA method [START_REF] Colotte | Higher pitch marking precision for TD-PSOLA[END_REF][START_REF] Moulines | Pitch synchronous wave-form processing techniques for a text-to-speech synthesis using diphones[END_REF] and allow users to manually modify F0 contours, speech rates as well as phoneme durations. It means that we can apply a global or local modification factor. For instance, we can slow down a particular part of the signal and speed up another one of the same signal. The modification method is based on the decomposition of the speech signal into overlapping pitch synchronous frames and the modification of pitch or duration is obtained by duplication/decimation of some frames. Firstly, this method supposes the detection of the pitch marks: the signal periods are marked at the maximum or minimum relevant peaks (for the voiced parts 3 ). These marks are spaced every pitch period and indicate the centre of the (analysis) frames (see fig. 1 and2). Secondly, we need to compute the new position of these marked frames in the modified signal. The main requirement is to maintain the consistency of mark location between frames in order to preserve the original temporal structure of the signal under analysis. This marking directly influences the quality of the resulting signal. In [START_REF] Colotte | Higher pitch marking precision for TD-PSOLA[END_REF], we have proposed a high precision algorithm for pitch marking at two levels: analysis and synthesis marks. In one hand, dynamic programming selects peaks in the signal for marking periods. Through correlation and pruning strategies, the algorithm overcomes errors which may appear with other algorithms. In addition, the algorithm is very fast in computation, which is very suitable for TD-PSOLA method. In the other hand, the combination of our pitch marking with a fast re-sampling method (to obtain the true synthesis frame) during the synthesis step increases the signal quality. This gain in accuracy avoids the reduction of quality between original and synthetic signal observed with the classical TD-PSOLA method: the level of noise between harmonics is reduced with our method. Thirdly, each mark in the future signal is associated with a mark of the original signal. If we want to slow down (resp. speed up) the signal, the space between frames needs to be preserved (to keep the same pitch) and an original frame could be duplicated (resp. removed) to obtain the good length of the signal (see fig. 1). If we want to modify the pitch, preserved, as for the slowing down, an original frame could be duplicated (or removed) to 3 For unvoiced part, by definition without period, an arbitrary spacing is used (for instance 5ms).

Analysis pitch marks

Synthetic pitch marks P' P Synthetic short time signals

Original waveform

Synthesis waveform obtain the good length with the good spacing (see fig. 2). The pitch and duration modifications can be merged together. Finally, the signal is rebuilt from these frames (removed and/or duplicated) thanks to the principle of overlapping. The advantages of this method are that we work in the temporal domain without any computation of transformation into the frequency space and the principle of using frames synchronized with the pitch does not destroy the coherence of the signal for each frame (notion of shape-invariance). We obtain a high quality of resynthesis without modification of the timbre of the voice. The method allows us to lengthen the pitch period until 3 times the initial period (i.e the F0 can be lowered by 3). For the duration, the slow down is only limited by the fact that a numerous duplication of the same part can create a new period (for unvoiced sound) and a noise of voicing can appear where there was no voicing (the factor > 4-5 ). But the slowdown is sufficiently local and generally small enough to avoid this drawback. These modification functions can be use independently of the diagnosis process. The learner or teacher can (manually) modify the signal as he wants, to become aware of the link between prosodic cue variations and perception.

Automatic perceptual feedback

Such functions can be exploited to imitate the prosodic cues of a model, so that learners can appreciate the differences between their realization and what they are expected to realize. The signal is resynthesized with the required modifications, and the users can listen to the modified signals as well as visualize their spectrographic representations and the new melodic curves. We have developed a module which realizes this imitation automatically. In a first step, the relative durations of the learner's phones are aligned with that of the reference. In a second step, a new F0 contour for the learner's utterance is computed using a linear interpolation of the model's normalized F0 contour. Then the learner's realization is resynthesized and then he/she can appreciate the resulting speech signal. Note that the used resynthesis algorithm keeps the timbre of the learner's voice, since only F0 and phoneme durations are modified. The copy of the F0 takes into account the mean F0 (pitch) of the speaker.

Diagnosis

The diagnosis is based upon a comparison between the realisation of the learner and that of the reference speaker, as well as phonetic knowledge about prosodic patterns of L1 and L2. Let us take the example of the realisation of English lexical stress in isolated words by French speakers (the object of the experiment presented in section 3). The syllable which should be stressed is assumed to be the one exhibiting the higher F0 in the reference's production. Thus the system evaluates, in semi-tones, the peak height of this syllable with respect to other syllables, in both realisations (the native and non-native ones) and returns a comment indicating whether the prominent F0 peak appears on the expected syllable of the non-native realisation and if it is sufficiently marked. At the same time, visual displays are shown on the spectrogram of the learners' realizations: arrows indicate whether the pitch of the target syllables should be raised or lowered (the colour of each arrow provides an indication about the degree of difference between native and non-native realisations with respect to F0 height); a red curve represents the F0 contour; the syllable and vowel durations of the reference and those of the learner appear in the form of bars on the top of the learner's realization; the length of bars varying with the duration of these segments (see Fig. 3). The reduction phenomenon requires a specific treatment. If learners do not reduce syllables which have been strongly reduced by the English reference, they are invited to repeat their realization and to reduce the appropriate syllable.

Modus operandi

Let us present the procedure proposed for providing automatic feedback to learners of a foreign language. Before clicking on the function in the menu, the subject should record a word or a small sentence, belonging to our database. He can, depending upon his wish or the task he is involved in, listen to the English reference before producing the item. When the subject selects the "correction" function, he is asked to choose a reference (one male and one female English speakers have recorded the corpus). Then the software realizes a text-tospeech automatic alignment of the native and non-native versions, and substitutes the prosodic cues of the native speakers to those of the non-native speakers.

Then the subject is provided with three versions of the utterance analysed, displayed in three windows, one per each version (see fig. 3). The first version consists in the native speaker's production, the second version is the learner's production, and the third version is the learner's modified production. Each window contains the representation of the melodic curve onto the spectrographic display, with the automatic alignment just under the spectrogram. The automatic diagnoses appear onto the spectrogram of the second window, that of the learner's original production. The subject can listen to the three versions and is invited to give a special attention to his own modified voice. The subject is free to produce the word again or select another one.

Experiment

We conducted a pilot experiment to analyse the immediate impact of diagnosis and advanced perceptual feedback on French subjects speaking English as a second language. In France, teaching of English prosody for nonspecialists focuses on the main intonation patterns as well as the place and strength of English lexical stress accent. We have chosen to test the production of English lexical accent in isolated words by French speakers. French and English are very different from a prosodic point of view since French is considered as a syllable-timed language and English as stress-timed language [START_REF] Dauer | Stress-timing and syllable-timing reanalysed[END_REF]. The place of the English lexical accent is free, whereas the French one is fixed, and the English accent is very well marked on an acoustical point of view whereas the French one is relatively weakly marked. Indeed, in English, the stressed syllable of isolated words is more intense, higher in pitch and longer than the unstressed ones. Furthermore, post-stressed syllables are often reduced: their vocalic nucleus tends to be very short, weak, and its vocalic timbre can become similar to that of a schwa (the schwa is, as an example, the first and last vowels in the word "America"). The French accent is essentially characterized by a lengthening of the last syllable of a word or a group of words; non accentuated vowels are always produced with their "full" timbre (there is no reduction).

It is commonly said that learners of a foreign language are "deaf" to its prosodic system. This explains why learners tend to use the prosodic features of their mother language instead of the prosodic features of the target language. For example, when speaking English, French learners generally lengthen the last syllable of a word, even when they know that this syllable is unstressed, and produce post-stressed vowels with their full timbre (without any reduction).

Experimental protocol 3.1.1 Corpus and subjects

The corpus, made up of 40 English transparent words, has been recorded by two English speakers, the "reference" speakers, one male and one female, born and educated in England, and currently living in France. Transparent words have similar spelling in French and English but do not have the same pronunciation (e.g. "important", "favourite"). They constitute a good way of making learners aware of the differences between accentuation in both languages. The corpus was made up of two and three-syllable words of the type: `S1S2 (the accent is on the first syllable), S1`S2S3 (accent on the second syllable) and `S1(S2)S3.

Words with this last syllabic structure (e.g. "governor", "family"), are generally pronounced by English speakers with a very strong degree of reduction on the second vowel whereas French speakers pronounce this vowel with its full timbre. We kept ten words for a "familiarization" phase, which precedes all the recording sessions; these words were pronounced by speakers but not analysed. Thus thirty words -ten per each type-remained for the analysis. Note that, in this corpus, the lexical accent never appears on the last syllable of a word. Ten French subjects, five male and five female speakers from 15 to 50 years old, participated in the experiment. Taking into account learners' pronunciation, we chose four (relatively) advanced speakers and six low (production) level speakers. The advanced speakers have lived for a short period in USA or made frequent trips in this country, whereas the six speakers with a low oral production level (let us call them beginners, for short) have studied English at school for at least five years but do not master English pronunciation. Since ten subjects produced thirty words in each of the two recording sessions (presented below), the corpus contains 600 words uttered by non-native speakers. English reference speakers and French subjects recorded the corpus in a quiet office with a Sennheiser headset microphone connected to a laptop. The mono signal was digitized at 16 bits with a sampling frequency at 22500 Hz. The software Audacity was used for all the recording sessions except for the one requiring the use of the platform (second session of the test condition).

Experimental conditions

There were two experimental conditions: the test and control ones, and two sessions of recording for each condition. We selected five speakers for each condition, two advanced speakers and three "beginners", trying to balance both groups. Note that we do not let learners chose between the test or the control conditions. The first session was the same for both conditions, subjects from both groups (the test and control groups) recorded the corpus without listening previously to English references. This gives us an indication of the learners' mastering of English lexical accent at the beginning of the experiment. Each word was written on a sheet of paper in orthographic transcription.

To avoid the production of a list intonation, a short pause was enforced between each word.

In the second session, subjects participating in the test condition were submitted to the platform for prosody learning. The procedure was the following: subjects were invited to select a word, written in orthographic transcription, from a list corresponding to the corpus.

For the purpose of this experiment we asked subjects to follow the list order, and select a male or a female reference according to their gender. Then subjects uttered the selected word without listening previously to its English reference; the system analysed the realisation, and visual, perceptual and textual (diagnosis) feedback was displayed on three windows as explained in section 2.5. After subjects took knowledge of the various kinds of feedback, they recorded the word one more time, and selected the following word to repeat the procedure until the end of the list. Only word repetitions (the pronunciation after feedback) have been analysed and compared to words pronounced during the first session.

In order to avoid false corrections due to erroneous segment boundary detection, the system stops just after the automatic alignment of the word under analysis to ask the experimenter whether he/she desires to continue the process or modify speech boundaries before. This is a necessary step if we want to test the impact of feedback without incurring the risk of disconcerting the learner by inappropriate corrections. Let us recall that we are currently working on the elaboration of confidence levels to limit the risk of erroneous corrections.

In the second session, subjects participating in the control condition read each word on a sheet of paper, listened to its reference, and uttered it.

Both conditions enable us to compare the effect of simple auditory feedback, received by learners in control condition, with that of advanced feedback, received by learners in test condition.

Acoustic cues

We estimated differences in height (F0) and duration ratios for `S1S2 and S1`S2S3 words with the lexical accent falling on the penultimate vowel, and we analysed the presence or absence of reduction in the second syllable of `S1S2S3 words.

To analyse all productions, including those not analysed by the platform during the test condition (second session), we segmented words into speech sounds, and estimated the values of the acoustic cues taken into account, i.e. fundamental frequency and segment duration. F0 was evaluated in semi-tones.

For words with `S1S2 and S1`S2S3 syllabic structures, we estimated the following criteria.

1. The differences between F0 values of the (to be) stressed vowel (VS) and the following unstressed vowel (VUF, for unstressed final vowel) : F0(VS) -F0(VUF) . F0 was averaged across all the frames of each vowel except the frames close to the vowel boundaries. 2. The number of times the F0 maximum fell on the right syllable (the theoretically stressed one), given in percentage.

3.

The ratio of the duration of the stressed vowel to that of the following unstressed vowel:

D(VS)/D(VUF).

Note that VUF is the last vowel of the word, and could be lengthened by French subjects. Taking into consideration ratio and not difference in the above formula allows us to remove the effect of temporal variations due to speech tempo. For `S1(S2)S3 words, we noted whether French learners pronounced the second vowel (V2) with its full timbre. To estimate the presence/absence of reduction, we calculated the duration of V2, when it was audible, and we compared it to the averaged duration of V1 and V3. If the duration of V2 was inferior to 40 ms and at least two times shorter than the averaged duration of other vowels in the word, we considered that the vowel had been reduced. With these criteria, all the second vowels in `S1(S2)S3 words uttered by the English speakers used as references in this study are considered as reduced. We had also compared F0 height and durations between the stressed and the initial unstressed vowels (for S1`S2S3 words) but these parameters did not contribute a lot to global results and we do not give them here in order to simplify the discussion.

We used Student's t-tests, paired samples, associating the data of each speaker and each word to compare the pronunciation of the words in the first session (without feedback) to their pronunciation in the second session (with feedback). Each condition (auditory feedback and advanced feedback) and each group of speakers (i.e. advanced speakers and beginners) were tested separately. We also used Student's t-tests to compare results obtained in both conditions (only words uttered during the second session were taken into account). Once more advanced speakers and beginners were considered separately. We accepted a level of 0.05 for significant effect and considered that results were highly significant when this level was inferior to 0.001. We submit the following parameters to statistical tests: the first parameter (F0 height), the third parameter (relative duration ratio) as well as, for vowel reduction in 'S1S2S3 words, the ratio of the duration of the second vowel to that of other vowels in the words.

Results

Table 1 provides, for each session, the results obtained for advanced learners and beginners as well as the results for both reference speakers. Parameter values are averaged across all words and all speakers of a given level (advanced or beginner). For estimating the averaged differences in height (in semi-tones) between stressed and unstressed vowels (column 2), we only took into account the cases where the F0 maximum is on the correct syllable. The third column displays the number of times the F0 peak is located on the right syllable, and the fifth column the number of times the second vowel in `S1S2S3 words is reduced. The duration ratios are given in the fourth column. We also calculated the averaged values obtained for each speaker in each parameter, and we discuss them below, when interesting. 2.9 43 0.83 37 Beg.

Speakers

3.9 75 1.1 63

Table 1. Results for the first session, without feedback (blue line) and for control and test conditions with feedback (white and green lines, respectively), for advanced learners (Adv), and beginners (Beg). Values for reference speakers, the male (M) and female (F) are given on the top of the table. See text for more explanations.

First session

In this session, all learners (from test and control groups) uttered the words of the corpus without listening previously to their English references.

Reduction phenomon in `S1S2S3 words

Subjects utter these words the French way: they realize post-stressed vowels with their full timbre. We note only few exceptions: one advanced speaker reduces the vowels in approximately a third of the cases (probably when he knows well the word pronunciation) and one beginner realizes one reduced vowel. For all other realizations the duration of the second vowel varies between 45 and 130 ms and its averaged duration with respect to other vowels of the word is approximately 0.8, for both groups. The averaged percentages of reduction were 7.5% and 2% for advanced learners and beginners, respectively.

Vowel duration ratio in `S1S2 S1`S2S3 words

Most speakers do not master L2 vowel durations (the averaged ratios are 1.05 and 0.85 for advanced speakers and beginners). Indeed, for all learners but one, the duration ratio (d(VS)/d(VUF)) is in between 0.7 and 1.1 (vs. 1.7 for English speakers). This confirms the tendency, well known for French learners, to lengthen the last syllable of the word and make relatively weak differences between syllable durations. The duration ratio is relatively high (1.5) and close to that of English speakers for only one speaker (an advanced one).

F0 pattern in `S1S2 and S1`S2S3 words

The realisation of this pattern seems to provide a good indication of the degree of proficiency of the subjects involved in this experiment. Let us first present the case of speakers with a low production level. We observe two kinds of pattern for these learners.

Most speakers exhibit relatively flat patterns, with sometimes a tendency to rise F0 at the end of word, a behaviour typical of beginners and people apprehensive about speaking in a foreign language. One speaker exhibits systematically falling patterns. Since in the corpus, the accent never falls on the last syllable of the word, the percentage indicating the number of times the F0 maximum is on the right syllable is relatively low (33%). The difference in height, only taken into account when the F0 maximum is on the correct syllable, is weak (2.5 semi-tones). The advanced learners tend to exhibit correct F0 patterns, with the location of the F0 peak most of the times (90%) on the expected syllable and a substantial difference between the height of the stressed vowel and that of the post-stressed vowel (6.9 semitones).

Summary

To summarize the results of this first session, we remark strong differences in the mastering of F0 pattern and duration cues. Concerning duration, all speakers but one apply L1 duration rules when speaking English. Most of them are not aware of the reduction of poststressed vowels, and others seem to encounter problems in predicting its occurence. On the contrary, some speakers, the more advanced ones, realize correct F0 patterns.

Second session. Effect of feedback for advanced learners

In this subsection, we examine the effet of feedback for advanced learners: auditory feedback (control condition) and diagnosis, F0 displays, as well as speech modification (test condition). There is no statistical difference between results for advanced speakers in test and control conditions so results for both groups are discussed at the same time.

Reduction phenomon in `S1S2S3 words

Feedback has a very high impact on advanced speakers, both in amplitude and degree of significance, whatever the condition. Indeed, in both condition, all speakers change their pronunciation to realize strongly reduced vowels for all words. We thus obtain a score of 100% for all speakers. It then appears here that auditory feedback was sufficient to enable advanced learners to improve their realization.

Vowel duration ratio in `S1S2 and S1`S2S3 words

All speakers who exhibit relatively low averaged ratios -i.e. all but one-improve themselves, whatever the condition, showing once more the high impact of auditory feedback for advanced learners. The averaged ratios are 1.2 and 1.25 in control and test conditions, respectively. This increase is mainly due to the lengthening of the stressed vowel, rather than a shortening of the last one.

F0 pattern in `S1S2 and S1`S2S3 words

We observed slight but significant increases (of about 0.6 semi-tones) in F0 relative height in both conditions. This might not be important on a perceptual point of view, but it seems that learners are aware of the important difference in height exhibited by the reference speaker between the stressed and the following unstressed vowel and try to imitate it. There was no more error in the location of F0 maximum.

Summary

Due to effect of feedback in both conditions, advanced speakers change the relative duration of the vowels, and reduce vowels that have been strongly reduced by reference speakers.

For learners involved in this study, auditory feedback appears to be as efficient as more complex feedback such as the one proposed in this study. This result appears to be in agreement with that obtained by [START_REF] Herry | Subjective and Objective Evaluation of the Prosody of English Spoken by French Speakers: the Contribution of Computer Assisted Learning, Speech Prosody[END_REF] who tested advanced learners.

Second session. Effect of feedback for low production level speakers 4.3.1 Reduction phenomon in `S1S2S3 words

Auditory feedback does not have a high impact on low production level speakers.

According to the comments they made after the experiment, they seem to have been disturbed by the way English speakers utter seemingly familiar words (transparent words) and slightly (but significantly) improved their realizations. On the whole, all speakers diminish the averaged duration of the second vowel but the averaged percentage of reduction is relatively low (37%).

Of course the effect of advanced feedback is far better. Indeed, the system detects the number of syllables and when this number is different from that of the reference, subjects were informed of this deviation and asked to change their pronunciation. Then, aware of what is expected, subjects make strong efforts to reduce vowels. The results varies with speakers and words: only one speaker reduces all vowels, but all speakers drastically reduce V2 averaged duration (this reduction was highly significant). The average percentage of reduction was 63%. The difference between both conditions is highly significant.

Vowel duration ratio in `S1S2 and S1`S2S3 words

In control condition, there is no significant improvement with respect to vowel duration. The averaged duration ratio (d(VS)/D(UF)) estimated for all speakers stay in the same range as that observed in the first session.

In test condition, subjects significantly improved their realization, making generally longer stressed vowels (the averaged ratio rises from 0.84 up to 1.1). The difference between test and control condition is highly significant.

F0 pattern in `S1S2 and S1`S2S3 words

In control condition, the observed improvement is very small in amplitude (0.4 semi-tones) and significance and varies with speakers and words. Two speakers exhibit clear tendencies to raise the pitch of the stressed syllables with respect to the unstressed one, but this rise not systematic, i.e. not observed for all words. The overall percentage of words with the maximum of F0 on the right syllable increases from about 10%. In test condition, there is a very significant improvement. All speakers were clearly informed of what was expected from them (thanks to small texts and arrows) and improved their realization most of the times (the percentage concerning the location of the F0 maximum raises from 30% up to 75%). The amplitude of the modification (1.4 semi-tones in the average) varies with speakers and words. Once more, the difference between test and control conditions is highly significant.

Summary

Due to effect of "advanced" feedback, subjects with a low oral production level significantly improve their realizations. Advanced feedback appears to more interesting than simple auditory feedback for these subjects.

Conclusion

Feedback on L2 prosody based upon visual displays, speech modifications and automatic diagnosis has been elaborated and a pilot experiment undertaken to test its immediate impact on listeners. Results show that the various kinds of feedback provided by the system enable French learners with a low production level to improve their realisations of English lexical accents more than (simple) auditory feedback. These results shoud be reinforced with a large number of speakers but based upon the important differences between results obtained for speakers in test and control conditions, we are confident in the interest of the system presented here. In particular, the system analyses learners' realisations and provide indications on what they should correct, a guidance which is considered as necessary by specialists in the oral aspects of language learning, such as [START_REF] Chun | Signal analysis software for teaching discourse intonation[END_REF] or [START_REF] Germain-Rutherford | Utilisation d'un logiciel de visualisation pour l'apprentissage de l'oral en langue seconde[END_REF]. The perspectives of this work are twofold, technological and experimental. On a technological point of view, results from the experiment encourage us to pursue the work undertaken with the speech team ("Parole" team) at LORIA on speech alignment for nonnative speakers. This work concerns both the detection of precise segment boundaries when possible and the elaboration of confidence levels on boundary detections. The design of such confidence levels would allow the system to propose feedback only when it can rely upon detections with high confidence levels and thus avoid erroneous corrections. We also plan to refine our method for modifying learners' voice. We use (an improved version) of TD-PSOLA method to modify segment durations and melodic curves. This algorithm does not allow us to modifiy vowel quality. Yet automatic modifications of vowel quality would be interesting to take into account vowel (timbre) reduction in post-stressed vowels. This modification could be obtained by using techniques such as voice morphing or techniques working in the frequency space. Since these techniques could generate degradations in speech quality, compromises should be found.

On an experimental point of view, it would be interesting to separate the impact of each kind of feedback (visual displays, diagnosis and perceptual feedback based upon speech modification). We also plan to investigate the long-term effect of automatic feedback should in collaboration with teachers in foreign language at the University of Lorraine.

Fig. 1 .

 1 Fig. 1. Example of slowing down: the duration of the original signal is lengthened by a factor 2 (P'=P).

Fig. 2 .

 2 Fig. 2. Example of pitch modification (here a lengthening of the period/a lowering of the F0): the original pitch is lengthened by a factor 1.2 (P'=1.2P).

Fig. 3 .

 3 Fig. 3. Example of automatic diagnosis and speech modifications. Each panel shows the spectrogram, the melodic curve (in red) and the waveform (at the bottom, also in red) of a realization of the word "important". The first panel show the realization of this word by an English speaker; the second panel its realization by a French learner, with the diagnosis provided by the software (see text), and the third panel the modification of the learner's voice.

The variations of the fundamental frequency (F0) generate the melodic curve.

Parts of these tools have been presented in[START_REF] Henry | Tools devoted to the acquisition of the prosody of a foreign language[END_REF]

Acknowledgment

This work has been partially funded by the INTERREG (european project) ALLEGRO. We would like to thank Odile Mella for her help concerning the manuscript, the reference speakers, and the subjects participating in the experiment, as well as Yves Laprie, Guillaume Henry and Christian Gillot for their contribution to the software.