Nathalie Rey
email: nathalie.rey@univ-paris13.fr

Credit derivatives: instruments of hedging and factors of instability. The example of "Credit Default Swaps" on French reference entities

Keywords: credit derivatives, credit risk, credit default swap, inter-temporal relations between markets, VAR models JEL classification: G10; C32

Through a long-period analysis of the inter-temporal relations between the French markets for credit default swaps (CDS), shares and bonds between 2001 and 2008, this article shows how a financial innovation like CDS could heighten financial instability. After describing the operating principles of credit derivatives in general and CDS in particular, we construct two difference VAR models on the series: the share return rates, the variation in bond spreads and the variation in CDS spreads for thirteen French companies, with the aim of bringing to light the relations between these three markets. According to these models, there is indeed an interdependence between the French share, CDS and bond markets, with a strong influence of the share market on the other two. This interdependence increases during periods of tension on the markets (2001-2002, and since the summer of 2007).

Introduction

Over the last ten years, with the increase in the number of business bankruptcies and the introduction of a regulatory framework, lending institutions have started actively to manage their credit risk. This has resulted in the rapid development of markets in instruments of risk transfer, including credit derivatives. These latter can be defined as financial contracts reflecting the value of the risk attached to a loan contract. The principle is quite simple. The lending institutions seek to protect themselves against the credit risk, i.e. against the risk of loss incurred in the event of default on the credit. This can be done either by transferring the asset bearing the risk, through what are known as "funded" credit derivatives, or by simply transferring the risk attached to the credit by means of "unfunded" derivatives. In the case of funded derivatives, the underlying risky asset disappears from the balance sheet of the institution buying the protection, while the institution selling the protection buys a bond or debt which then appears in its balance sheet. Where unfunded derivatives are concerned, the underlying asset is not transferred and only the instrument of risk transfer appears, in or outside the balance sheet depending on the instrument used. In practice, however, the use of these instruments raises certain problems and criticisms. These problems are related to the very nature of the contracts, the means of determining the prices of these new products, the lack of transparency of credit derivative markets and the interdependence of these negotiated markets with other markets, including share and bond markets. This article has a double aim.

Through an analysis of credit default swaps, the most commonly-used form of credit derivative † , it will seek to expose both the advantages and the limitations of these instruments of risk transfer. Based on an empiricial study of the inter-temporal relations between the French CDS, share and bond markets over the period 2001-2008, it will show how these derivatives can be the source of financial instability. In the second section, it will situate credit derivatives in relation to other instruments of credit risk transfer, before focusing more particularly on credit default swaps and the literature analysing the relations between the markets for CDS, shares and bonds. The third section presents the credit derivatives market and the empirical data. The econometric method used, the main hypotheses tested and the principal results are presented in the fourth section.

Credit derivatives: "simple" instruments of coverage?

Credit derivatives: one of several instruments of risk transfer

Before drawing up a typology of the different instruments of credit risk transfer available to lending institutions and the place of credit derivatives within this typology, it is worth recalling the different forms of underlying credits involved. These can be grouped into two categories: consumer credits (credits outstanding on credit cards, residential real estate loans, loans and financing for the purchase of cars), and securities on transferable and nontransferable loans and debts (customer debts, financing on plant and machinery, commercial mortgages, private sector and sovereign debt). The underlying credit constitutes the primary criterion for the classification of instruments of credit risk transfer. Credit derivatives only concern transferable and non-transferable private sector and sovereign debts, in the form of loans and bonds. To this criterion, we can add that of the modality of transfer of the underlying asset (the funding), depending on whether it is transferred from the buyer of the protection to the seller, or whether only the risk is transferred. Unlike other instruments of risk transfer, the use of credit derivatives does not result in the loaan being removed from the balance sheet of the buyer of the protection, and consequently requires no financing on the part of the seller of the protection.

If a lending institution has sovereign and/or private sector debts among its assets, and if it wishes to keep these assets in its balance sheet, maintaining their legal characteristics and the commercial relation with its customers, while at the same time protecting itself against the risk of default by these customers, then it has the choice between four different instruments, including credit derivatives. These latter correspond to four main types of derivatives, described as "single name" when they cover one sole reference entity (the issuer of the debt for which the lender is seeking coverage), or "portfolio" when there is a basket or portfolio of reference entities (Table 1). -the initial contract: the underlying loan or asset that the lending institution is seeking to cover, characterised essentially by an amount, a maturity and a rate;

-the coverage contract: the derivative enabling the institution to protect itself against the credit risk, characterised by an amount covered, a maturity and a premium called the spread;

-the buyer of the protection or seller of the risk: two terms defining the institution which can present the initial loaan in its assets and which can use the credit derivative as an instrument of credit risk coverage;

-the reference entity or borrower deemed by the lending institution to represent a credit risk;

-the seller of the protection or buyer of the risk: the institution that undertakes to pay a certain sum in the case of a "credit event" on the reference entity. In the case of the seller of protection, the credit derivative is used as an instrument of investment, as it is buying a risk from the counterparty without having to fund and establish a commercial relation with this latter;

-the credit event corresponding to the occurrence of the credit risk; this may take different forms such as the bankruptcy of the reference entity, default on payment, or restructuring of its debt.

Most of these parameters must be defined within the frame of a contract drawn up between the buyer and seller of the protection. Credit derivative contracts have the particularity that they can be negotiated separately from the underlying asset. Because of this, they can be used as trading instruments, in other words with the sole aim of producing short-term value-added through the hoped-for evolution of the credit risk.

CDS: "first" among credit derivatives ‡

A CDS is a contract by which the buyer of protection (A) pays a regular premium (usually every quarter) to the seller (C), who undertakes to compensate (A) for the loss incurred on the underlying asset if a credit event takes place in the reference entity (B) (Figure 1). The CDS has no effect on the conditions of the initial contract; the lending institution (A) receives the interest and repayments of the capital loaned from the reference entity (B) as long as no credit event occurs. The lender (A) is not obliged to inform the borrower (B) of the establishment, with the counterparty (C), of an operation of coverage of the credit risk attached to their commercial relation. The buyer of the protection (A) pays the seller protection (C) a regular premium corresponding to the value of the CDS contract, expressed in "basis points". This annual premium, or its equivalent, is inversely proportional to the credit rating of the borrower (B). By paying this premium, the buyer of the protection acquires the right, if a credit event occurs in the reference entity (B), to receive a payment from the seller of the protection (C), corresponding to the market value at the date of ‡ For a presentation of the different instruments of risk transfer and other credit derivatives, see notably BATTEN J., [START_REF] Batten | A perspective on credit derivaties[END_REF]; BIS (2003); KIFF J., MICHAUD F. L. and MITCHELL J. (2003).

establishment of the CDS contract on the debts issued by a borrower with the same credit rating.

Rather than swaps, CDS correspond to put options on the credit risk of an underlying asset. In return for the payment of a premium, the buyer of the CDS has the right to "exercise" this protection in the case of a credit event on the underlying asset, and to receive payment from the protection seller of a sum defined in the CDS. In the case of standard options, however, the buyer's profit (the seller's loss) can be illimited. With CDS, when a credit event occurs, the one's profit and the other's loss are fixed, corresponding to the amount of the indemnity minus the premium. If, on maturity of a CDS contract (which need not correspond to the maturity of the underlying asset covered), the initial borrower (B) has not defaulted, then the protection seller (C) makes a profit equal to the total premium received, and the protection buyer (A) makes a loss equal to the total premium paid minus the reduction in regulatory capital requirements. For the intial lender (A), CDS has a double interest: not only can the lender transfer the credit risk, it can also reduce its capital requirements, by declaring as counterparty of the underlying asset not the initial borrower (B), but the protection seller (C), less risky and with a better credit rating. CDS also has a double interest for the protection seller, in that it makes a profit and diversifies its portfolio.

Let us take a concrete example: that of CDS on the entity France Telecom. On 08/02/2007 the five year CDS spread was 23.5 basis points, meaning that the protection buyer who, for example, wished to cover 10 million euros, would have to pay 5.875 basis points, i.e. a premium of 5,875 euros every quarter over five years: a total of 117,500 euros. If a credit event was to occur in the reference entity France Télécom before 08/02/2012, the protection seller would pay the buyer the total covered (10 million euros) minus the amount recovered from liquidation of the entity, that is to say, with a recovery rate of 40%, six million euros. On the other hand, if the entity France Telecom has not defaulted before 08/02/2012, then the protection seller will have paid nothing during the whole lifespan of the CDS. The CDS spread is an indicator of the market valuation of the credit risk on an entity, and the evolution of this spread gives us information about the market's perception of this risk. Thus, for France Telecom, the five-year CDS spread was 730 basis points on 26 June 2002. The buyer of this contract therefore undertook to pay 182,500 euros every quarter during five years (i.e. a total of 3,650,000 euros!). Between 26 June 2002 and 08/02/2007, the market therefore strongly downgraded the credit risk for France Telecom.

CDS contracts can be negotiated independently of the underlying asset. In this case, the protection buyer (A) has no lending contract with the entity (B), but it establishes a CDS contract on (B) with the protection seller (C) with the simple aim of making a profit. This is the profit it can make by re-selling the CDS contract at a higher price, in the event of an increase in the credit risk on the entity (B). CDS can be used as trading instruments, with the sole objective of producing short-term added-value on the predicted evolution of the credit risk. The buyer of the CDS expects a rise in the credit risk, while the buyer expects a fall.

The buyer and seller of protection must be able to evaluate the credit risk, to attribute a price to it. To determine this price or spread, they use credit risk assessment models. Two types of model are generally opposed: structural models [START_REF] Merton R | On the pricing of corporate debt: The risk structure of interest rates[END_REF] and reduced-form models (JARROW and[START_REF] Jarrow R | Pricing derivatives on financial securities subject to credit risk[END_REF]2000;HULL and[START_REF] Hull | Valuing credit default swapsI: No counterparty default risk[END_REF][START_REF] Hull | Valuing credit default swapsII: Modeling default correlations[END_REF]. They two categories of models differ on a number of points. In structural models, the financial structure of the entity (the weight of the debt) is taken into consideration, the value of the entity's assets is modelled, and the default can be predicted. In reduced-form models, the fundamentals of the entity are ignored, only the default time is modelled, without any reference to the entity's assets, and the default is totally unpredictable. Structural models enable the spread to be assessed on the basis of fundamental variables, while reduced-form models consider the spread as a given. However, whatever the type of model, three essential parameters must be determined in order to measure the credit risk: the probability of default, the loss in the event of default and the correlation between the two. Now, the determination of these three parameters depends on the information available. Accessible, good-quality information allows the default to be predicted, while a lack of information may make it impossible to predict the default, justifying the use of reduced-form modelling.

The reduced-form approach enables us to relate CDS spreads to bond spreads, that is to say, the difference between the bond yield and the yield of a risk-free asset (DUFFIE and SINGLETON, 1999). If we consider a market without arbitrage and a risk-neutral default probability, then there is a relation of equivalence between the two spreads. This means that the risk assessment is the same on the CDS market and the bond market. In a market without arbitrage, we can construct a replication portfolio, which consists in selling short the fixedrate bond with the same maturity as the CDS and investing in a risk-free asset. The value of this portfolio is equal to the CDS spread on the entity that issued the bonds. If, for a given entity and a given maturity, the CDS spread and the bond spread are not equal, then arbitrage is possible on the markets. When the CDS spread is higher than the bond spread, investors make a profit by selling the CDS on the derivatives market and buying a risk-free asset and the entity's bond on the bond market.

Using an econometric study of the relation of equivalence between CDS and bond spreads, [START_REF] Zhu | An empirical comparison of credit spreads between the bond market and the credit default swap market[END_REF] showed that over the long term, this relation is verified. The two spreads evolve together over the long term, but there are shoret-term differences. They react differently to changes in the credit quality of the entity. The derivatives market adapts more quickly than the bond market, where the movements lag behind those of the CDS. Several factors might explain a gap between the two spreads. Bonds are influenced by liquidity, by their life-span, which decreases every day, while the CDS market quotes a set of credit events for a entity with a constant due date. Bonds can include particular clauses such as convertibility into securities, the use of collaterals, which directly affects their value. The eventual impossibility of selling short on the bond market justifies a positive gap between the two spreads.

The structural approach enables us to bring to light a relation between the CDS market and the share market. This approach, developed by [START_REF] Merton R | On the pricing of corporate debt: The risk structure of interest rates[END_REF], proposes an interpretation of the relations between shareholders and lenders drawn from the theory of options. It considers that the return profile of the shareholder is similar to that of a call option, and the return profile of the creditor is similar to that of a put option. The more the value of the company's assets increases, the greater the profit the shareholder makes from his shares, after repayment of lenders. The value of the company's assets is determined on the basis of an option model and the information contained in the asset price is assumed to contain implicit information on default. For the lender, the value of the assets must be higher than or equal to the debt, if he is to be reimbursed in the event of bankruptcy. The company defaults when the value of its assets falls below a certain threshold, generally situated somewhere near the value of the debt. Assessment of the probability of default therefore depends on future probability distribution of the share price, relative to the level of debt. A fall in the share market leads to a rise in credit spreads, implying a rise in the probability of default and an increase in the demand for protection (through CDS). To honour their commitments, protection sellers take position on the share market by selling short or to close. The initial shock spreads and the liquidity of the share market falls. This shock may spread to other markets if the protection sellers are forced to sell assets other than shares.

From the moment that structural models can be used to determine, for a given entity, the proportions in which the CDS spread should evolve for a given variation in the share price, it is possible to use the share market to cover credit risk. An investor in an entity's bonds can cover his portfolio either by buying protection on the CDS market or by selling short shares in the same company in the proportions indicated by the structural model. If the bonds market deteriorates, the loss on the bonds will be compensated for by the gain from the protection or from the sale of shares. If all bond investors adopt this same covering strategy, the fall in the bond market will accelerate and spread to the other markets.

On a theoretical level, these two approaches bring to light the relations between the three markets: shares, bonds and CDS. In what follows, this article will seek to measure the extent of these relations, through an econometric study.

Presentation of the credit derivatives market and the data used

A credit derivatives market dominated by CDS

The credit derivatives market is experiencing strong growth § . According to surveys by the credit ratings agency Fitch, the total nominal value of credit derivatives grew from 3 trillion dollars in 2003 to 12 trillion dollars in 2005, with CDS contracts representing nearly half of this total. This market is dominated by short-term contracts; more than 80% of contracts have a maturity date of five years or less. Now, an instrument of coverage loses its efficiency when its maturity is lower than that of the underlying loan.

The banks are the main actors in this market, both as buyers and sellers of protection, although they are, overall, net buyers of protection (for a total of 268 billion dollars in 2005).

However, some British and Swiss banks have become net sellers of protection. They use credit derivatives more as financial instruments for the diversification of income sources rather than as an instrument of coverage for their asset portfolios. The main net sellers of protection are insurance and reinsurance companies, pension funds and hedge funds, with 20 to 30% of market activity in 2005. The credit risk is thus transferred from the big international banks to insurance companies and funds, but also to regional banks, notably German ones.

These regional banks are turning to this market as a means of diversifying their portfolios by regions and sectors of activity. They use credit derivatives as instruments to generate additional income. Today, credit risks are ultimately transferred to financial institutions that have less ability to manage the risk and to deal with the losses incurred when credit events do § The data on this market are widely dispersed. Public data remain insufficient and private data (British Bankers' Association (BBA) and Fitch are the main suppliers of data on this market) are based on questionnaires and should be treated with caution. The figures quoted in this paragraph (3.1.) are drawn from Fitch surveys.

occur. The credit derivatives market is very concentrated. In 2005, fifteen institutions were responsible for 86% of total volumes ** . Now, high concentration tends to increase the counterparty risk.

The reference entities of credit derivatives are few in number. In 2005, 62% of them were companies (with General Motors, DaimlerChrysler, Ford Motors and France Telecom far ahead of the others), 18% were financial institutions (Deutsche Bank, Goldman Sachs, JP Morgan) and 4% were States (Brazil, Italy, and Russia). To an ever increasing extent, the credit derivatives market involves the entities with the worst credit ratings and therefore the highest levels of risk † † . Between 2003 and 2005, the share of entities with a rating of at least "A" fell from 60% to 33%, while the share of "speculative grade" entities (rated below BBB) grew from 20% to 37%.

The data base on the French CDS, share and bond markets

We have chosen thirteen French companies for our study (including 11 in the CAC40), from twelve different sectors of activity and representing a total market capitalization of nearly 600 billion euros on 23/04/07. For each company, and over the period 2001 -February 2008, we have noted the daily share prices and the daily five-year CDS spreads, and we have calculated the five-year bond spreads (from January 2001 to February 2007). This gave us a total of 60,000 items of data (Table 2). Out of the thirteen companies, two, Alcatel and Rhodia were rated as "speculative grade", and both of them saw their rating slip over the period in question. None of the companies were rated triple A and only three were rated double A.

** The French banks are active on this market, as there are four of them among the top twenty-five operators. In terms of volume, at the end of 2005, BNP Paribas was in 10 th position; the Société Générale and Calyon were in 17 th and 18 th position respectively.

† † The principle of credit-rating is that the less risky the entity, the higher the rating. For Standard and Poor's, for example, the best rating is triple A (AAA); the worst is D for issuers who have defaulted. It uses the following rating grid: AAA, AA, A, BBB, BB, B, CCC, CC, D. An entity is considered "investment grade" when it is attributed a rating AAA, AA, A or BBB. Entities with ratings under BBB are considered "speculative grade" and present a very high credit risk. The daily share return, denoted RS, is calculated without dividends, and corresponds to a simple variation in price: observed price share the : P with

log 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = - t t t P P RS
The daily CDS spread (denoted by CDS) is calculated as the average between the bid and the ask price for the five-year CDS (the most widely-exchanged maturity). This spread is expressed in 100 base index.

The bond spread (denoted by BOND), expressed in 100 base index is obtained by calculating the difference between the actuarial return rate of the five-year bond with the daily interest rate of the five-year swap curve. For each company, we took the actuarial return rate of a fixed-rate, non-convertible, non-puttable and non-callable five-year bond. When such a bond did not exist, either we calculated the actuarial return rate by interpolation between the rate of bonds of more than five years and those of less than five years maturity, or we took the return rate of bonds of at least three and a half years maturity.

From observation of the three variables, we note ( Figures 2, 3 and4 In early 2007, out of the thirteen companies, only Rhodia had a return rate above 5%. The thirteen bond spreads appear to be non-stationary.

We have calculated the mean and the volatility for each variable for each year. Three subperiods stand out:

-2001-2002: share return rates were mainly negative and very volatile, the CDS and BOND credit spreads were very high and volatile. The companies with the worst credit ratings had the highest and most volatile spreads.

-2003-2004: the credit spreads showed a downward trend and became less volatile. Overall, share return rates were low, with lower volatility. -2005-2007: overall, share return rates rose, with relatively low volatility, and the credit spreads continued their downward trend. Companies whose ratings had been downgraded saw their spreads increase, while those whose ratings had been upgraded saw their spreads fall.

Since the subprime crisis, there has been negative and volatile average share return rates for the thirteen companies, together with a strong rise in CDS spreads. For six of the thirteen companies, CDS spreads reached their highest levels since 2001. According to the CDS market, the credit risk on these companies from different sectors of activity had all risen sharply. To give one example: from 01/06/2007 to 21/02/2008 the CDS spread on the entities Société Générale and Danone rose from 6.8 to 101.54 and from 8.42 to 100.18 respectively!

The losses incurred by the Société Générale may partly explain the strong rise in the spread for this entity, but how can we explain the massive increase in the credit risk assessment for Danone and other companies outside the banking sector? It would appear that this is the consequence of behaviour by investors using the CDS market more to diversify theit portfolios than to cover credit risks.

We have calculated the coefficients of correlation between the three variables for the period 2001-2007 (Table 3). From this matrix of average correlations, we can observe negative correlations between share return rates and bond spreads, and between share return rates and CDS spreads. The correlation between the two credit spreads is positive but weak. This result shows the existence of diversification gains between the three markets and partly explains the behaviour of protection sellers, who see the derivatives market as a source of high returns.

The models and their main results

The VAR models used

The aim of our study is to analyse the inter-temporal relations between three French markets:

shares, bonds and CDS, represented by thirteen French companies. We therefore felt that multivariate modelling would be appropriate, allowing to explain the evolution of one variable on the basis of an extended set of information (the history of the explained variable and the history of the explanatory variables) ‡ ‡ .

We therefore built two difference VAR models, to bring to light the inter-temporal relations between the markets and to verify certan hypotheses:

-Hypothesis 1: positive share return rates are accompanied by negative variations in CDS and bond spreads.

-Hypothesis 2: the share market and the CDS market influence the bond market.

-Hypothesis 3: the relation between the share market and the CDS market is stronger than the relation between the bond market and the share market.

-Hypothesis 4: the relation between the share return rate and the variations in credit spreads depends positively on the company's credit risk. The higher the credit risk, the stronger the relation.

-Hypothesis 5: the relation between the share return rate and the variations in credit spreads depends negatively on the size of the company. The smaller the company (in terms of capitalization), the stronger the relation.

Observation of our series shows that the share-price, CDS-spreads and bond-spreads variables are non-stationary. We therefore conducted stationarity tests, which confirmed the nonstationarity of the series (Table 4). We transformed our variables into share return rates, firstdifference or variation of bond spreads and (DBOND) first-difference or variation of CDS spreads (DCDS). The three stationarity tests carried out on the three variables for each company show that the series RS, DBOND and DCDS are indeed stationary. This result is supported by the autocorrelation test for a lag of p = 5 days (Table 5). Number of firms for which the null hypothesis of non-stationary data can be rejected at 5% (*rejected at 1%). ‡ ‡ Our work is in the line of those carried out by BLANCO, BRENNAN and MARSH (2004), NORDEN andWEBER (2004), and[START_REF] Zhu | An empirical comparison of credit spreads between the bond market and the credit default swap market[END_REF].

+ + + + = + + + + = + + + + = - = - = - = - = - = - = - = - = - = ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

The results

For each company, we estimated the two VAR models for the whole period and for two subperiods (from 01/01/01 to 31/12/03 and from 01/01/04 to 08/02/07 for the VAR1 model; from 01/01/01 to 31/12/03 and from 01/01/04 to 21/02/08 for the VAR2 model).

Over the whole period, and on average for the thirteen companies, the VAR1 model explains and predicts the credit spreads better than the share return rates. The average 2 R are higher for the credit-spread variables and the significant coefficients are numerous for these two variables. The coefficients obtained have the expected signs. Overall, share return rates are not very sensitive to variations in bond and CDS spreads; they react negatively to their past changes, and increases in DBOND and DCDS spreads lead to a very weak fall in RS return rates. The DBOND spread variable reacts essentially to its past evolutions, negatively; it falls with a lag when the share return rate rises and it increases weakly with the CDS spread. The DCDS spread variable reacts very negatively to an increase in share return rates, positively to an increase in bond spreads and positively to its past evolutions (Table 6). Columns (1), (3), ( 5) represent mean coefficients. In brackets, we report the mean t-student. Columns (2), ( 4), ( 6) report the number of firms for which the coefficient of the explanatory variable is significantly different from zero at the 5% level.

Over the sub-period January 2001 -December 2003, the results of the VAR1 model are better. The average 2 R of the model are higher than those for the whole period. The model explains better a period of declining share market and high, volatile credit spreads. The relations between the three markets are stronger during a period of market tension. There may, therefore, be a risk of contagion between the three markets. During a period when the share market is rising, on the other hand, the relation between the three markets tends to weaken (Table 7). Columns (1), (3), (5) represent mean coefficients. In brackets, we report the mean t-student. Columns (2), ( 4), ( 6) report the number of firms for which the coefficient of the explanatory variable is significantly different from zero at the 5% level.

For the sub-period 2004 -February 2007, the average 2 R obtained are lower and the coefficients are less significant. For this sub-period, however, we observe that the credit spread variables explain the share return rate variable better than they do over the whole period (with the expected negative sign) and the bond spreads appear to react less to the share return rate and more to CDS spreads (with the expected positive sign). Finally, the CDS spreads react less to increases in share return rates (with the expected negative sign) than to their own past evolutions (Table 8). Columns (1), (3), (5) represent mean coefficients. In brackets, we report the mean t-student. Columns (2), ( 4), ( 6) report the number of firms for which the coefficient of the explanatory variable is significantly different from zero at the 5% level.

The relations that appear to dominate are those of the share market towards the CDS market, the share market towards the bond market and the CDS market towards the bond market. As our results highlight the existence of a relation of causality between share returns and credit spreads, we thought it appropriate to study the effet of a share market shock of the CDS and bond markets (the three markets being represented by our thirteen companies). We expected an instant rise in credit spreads following a fall in share return rates, and conversely a rise in share return rates should cause an instant fall in spreads. Figures 5, 6 and 7 display the impulse response functions: how do credit spreads react to a share return rate shock? § § .

We consider that the amplitude of the shock is equal to the standard deviation, and we observe the effects of the shock over fifteen periods, i.e. fifteen days *** . The share return rate shock has an immediate impact on CDS and bond spreads and the consequences of the shock disappear after seven days. The reaction of the CDS market to a share return rate shock is stronger and lasts longer than the reaction of the bond market. According to the impulse response functions obtained using the VAR2 model applied to the sub-period 2004-2008, the CDS spreads react immediately to a share return rate shock, with a fall in the share market causing an instant rise in spreads on the CDS market. This rise is absorbed in almost linear fashion in less than six days.

Conclusion

The credit derivatives market has developed rapidly over the last ten years. The credit default swaps that dominate this market are used as instruments of credit risk coverage, but they are also used by investors for portfolio diversification. Investors adopt strategies of coverage, behaviours of arbitrage between the markets that are conducive to the propagation of instability between markets. The theoretical models of credit risk assessment bring to light relations between the share, bond and CDS markets. In this article, we have sought to conduct a long-period analysis of the relations between the French markets between 2001 and 2008.

Analysis of the survey data and the results of the VAR models used both suggest that the three markets are interconnected, and that this interconnection is all the stronger when the markets experience tensions. The influence of the French share market on the CDS and bond markets is proven, as is the influence of the CDS market on the bond market. During a period of tension on the share market, this interconnection between the markets can favour financial instability.

  ): -Sharp falls in share prices and negative, very volatile share return rates in 2001 and 2002 for all the companies. Rising prices and positive return rates from 2005-2006 up until summer 2007 (the start of the subprime crisis). The thirteen series of share returns appear to be stationary. -The CDS spreads reached their highest level in 2002. They experienced a period of decline from 2004 before starting to rise again in summer 2007. The gaps between bids and asks fell strongly over the last few years, expressing greater market maturity. The thirteen CDS spreads appear to be non-stationary. -The bond spreads were very high in 2001, 2002 and 2003 and have been falling since then.

Figure 2 :

 2 Figure 2 : Share returns (2001-2008)

  VAR model (denoted VAR2) for the period 2001 -February 2008.

Figure

  Figure 5: Reponse of DCDS TO RS (VAR1, 2001-2007)

Figure

  Figure 6: Response of DBOND to RS (VAR1, 2001-2007)

Table 1 : Credit derivatives

 1

	« Funded »	« Unfunded »

Whatever the type of credit derivative, it is characterised by at least six parameters:

Table 2 : Firm characteristics

 2

		Period	Sector	Market Capitalization (€)	Number off
				the 23/04/07		observations
						(CDS,
						bonds,
						stocks)
	Alcatel	26/06/2001 -21/02/2008	Communications Equipment	21 480 016 011,30	3,61	4 941
	Sanofi-Aventis	04/09/2001 -21/02/2008	Pharmaceutical Industry	90 959 774 639,53	15,28	4 849
	BNP Paribas	03/09/2001 -21/02/2008	Banks	79 761 110 895,00	13,40	4 825
	Bouygues	19/06/2002 -21/02/2008	Construction and Related Machinery	20 043 133 894,21	3,37	4 428
	Carrefour	04/09/2001 -21/02/2008	Grocery Stores	40 574 200 332,96	6,82	4 850
	Danone	04/09/2001 -21/02/2008	Packaged Foods	32 396 792 805,74	5,44	4 607
	France Télécom	29/08/2001 -21/02/2008	Integrated Services Telecommunication	54 713 063 223,00	9,19	4 811
	Pinault PR	15/03/2002 -21/02/2008	Department Stores	16 314 170 907,18	2,74	4 404
	Rhodia	25/03/2002 -21/02/2008	Specialty Chemicals	3 588 474 798,52	0,60	4 516
	Renault	29/08/2001 -21/02/2008	Automobile	26 539 043 170,52	4,46	4 835
	Société Générale	03/09/2001 -21/02/2008	Banks	69 836 607 458,70	11,73	4 664
	Sodexho	22/07/2002 -21/02/2008	Restaurants	8 981 811 806,24	1,51	4 254
	Total	04/09/2001 -21/02/2008	Integrated Oil & Gas	129 927 028 347,90	21,83	4 643
	Totaux		12	595 115

228 290,80 100,00 60 627

Table 3 : Average correlations 2001-2007

 3

		Share returns	Bond spread changes	CDS spread changes
	Share returns	1	-0,0765	-0,1124
	Bond spread changes	-0,0765	1	0,0641
	CDS spread changes	-0,1124	0,0641	1

Table 4 : Stationarity tests

 4

		Augmented DF Test Philipps-Perron Test	KPSS test
	Share returns	13	13	13
	Bond spread changes	13	13	13*
	CDS spreads changes	13	13	13

Table 5 : Autocorrelation tests, average period 2001-2007

 5

									Lag 1							Lag 2	Lag 3	Lag 4	Lag 5
	Stock returns			0,0052						-0,0188	-0,0292	0,0096	-0,0240
	Bond spread changes			-0,1608				-0,0570	-0,0175	-0,0058	0,0165
	CDS spreads changes			0,0025						0,0320	0,0344	0,0075	0,0335
	We used the following two VAR models:
	-3 dimensional VAR model (denoted VAR1) for the period 2001 -February 2007;
	RS	t	a	S		b	Sp	RS	t	p					c	Sp	t DBOND	p	d	Sp	DCDS	t	p	e	St
						p							p						p
	t DBOND	a	B		b	Bp	RS	t	p			c	Bp	t DBOND	p	d	Bp	DCDS	t	p	e	Bt
									p									p			p
	DCDS	t	a	C			Cp b	RS	t	p			c	Cp	t DBOND	p	d	Cp	DCDS	t	p	e	Ct
								p									p				p

Table 6 : VAR1 estimation results over the period 2001-2007

 6

		RS (1)	(2)	DBOND (3)	(4)	DCDS (5)	(6)
	RS(-1)	0,0047 (0,1718)	4	-17,0458 (-1,3743)	4	-34,1031 (-4,6753)	10
	RS(-2)	-0,0207 (-0,7219)	2	-9,9410 (-0,7181)	2	-29,5817 (-3,5251)
	RS(-3)	-0,0264 (-0,9164)	3	-2,6326 (-0,0011)	2	-9,7021 (-2,4597)
	RS(-4)	0,0159 (0,5242)	4	-3,0859 (0,0320)	1	-24,9162 (-2,5052)
	RS(-5)	-0,0201 (-0,6998)	1	12,8331 (1,2435)	6	2,5084 (0,2555)
	DBOND(-1)	0,0000 (-0,5144)	2	-0,4262 (-15,0158)	13	0,0196 (0,4837)
	DBOND(-2)	0,0000 (-0,1031)	2	-0,2745 (-8,6074)	13	0,0073 (0,2737)
	DBOND(-3)	0,0000 (0,3662)	2	-0,1837 (-5,6549)	12	0,0181 (0,5625)
	DBOND(-4)	0,0000 (0,3139)	1	-0,1129 (-3,5586)	9	0,0096 (0,4231)
	DBOND(-5)	0,0000 (-0,5842)	2	-0,0415 (-1,4929)	3	0,0016 (0,1497)
	DCDS(-1)	0,0006 (0,9458)	3	0,0980 (0,8936)	5	-0,0538 (-1,8801)
	DCDS(-2)	0,0003 (0,0765)	3	0,0332 (0,7522)	4	-0,0202 (-0,6175)
	DCDS(-3)	0,0003 (0,2520)	2	0,0244 (1,3224)	6	0,0036 (0,2024)
	DCDS(-4)	0,0004 (1,2430)	2	0,0134 (0,5118)	3	-0,0174 (-0,5510)
	DCDS(-5)	0,0004 (0,9333)	3	0,1451 (0,8609)	4	0,0460 (1,6601)
	Const. (a)	0,0002 (0,4691)		-0,0505 (-0,3136)		-0,0827 (-0,4476)
	R-squared	0,0265		0,1915		0,1044
	Adj, R-squared	0,0146		0,1816		0,0935
	F-statistic	2,2278		21,1609		9,7340
	Log likelihood	3080,9115		-4282,0556		-2826,7628
	Akaike AIC	-4,9032		6,8487		4,4798
	Schwarz SC	-4,8374		6,9145		4,5456
	Mean dependent	0,0001		-0,0222		-0,0478
	S,D, dependent	0,0218		9,3616		6,3449

Table 7 : VAR1 estimation results over the period 2001-2003

 7

		RS (1)	(2)	DBOND (3)	(4)	DCDS (5)	(6)
	RS(-1)	0,0143 (0,3289)	2	-20,5233 (-1,1429)	4	-40,7411 (-3,3373)	9
	RS(-2)	-0,0218 (-0,4596)	0	-16,6965 (-0,8500)	1	-37,3088 (-2,7262)	8
	RS(-3)	-0,0269 (-0,5765)	1	-2,0749 (-0,1405)	1	-13,5768 (-1,9093)	6
	RS(-4)	0,0244 (0,4909)	3	-3,3614 (-0,1255)	1	-28,5470 (-1,7941)	5
	RS(-5)	-0,0278 (-0,6021)	1	17,6284 (1,0188)	1	-1,0238 (0,0116)	1
	DBOND(-1)	0,0000 (-0,4130)	0	-0,4453 (-9,6624)	13	0,0138 (0,0331)	1
	DBOND(-2)	0,0000 (-0,2228)	0	-0,2914 (-5,5713)	11	0,0103 (0,2914)	2
	DBOND(-3)	0,0000 (0,1874)	2	-0,2015 (-3,7864)	10	0,0232 (0,4995)	1
	DBOND(-4)	0,0000 (0,2167)	1	-0,1314 (-2,5183)	7	0,0150 (0,3589)	2
	DBOND(-5)	-0,0001 (-0,3997)	1	-0,0420 (-0,9499)	2	0,0037 (0,0930)	1
	DCDS(-1)	0,0007 (0,7611)	3	0,0906 (0,4365)	2	-0,0884 (-1,8004)	6
	DCDS(-2)	0,0004 (0,3474)	1	0,0042 (0,5249)	2	-0,0263 (-0,5090)	4
	DCDS(-3)	0,0004 (0,1924)	0	0,0211 (1,0207)	3	-0,0031 (0,0599)	7
	DCDS(-4)	0,0004 (0,9100)	2	0,0290 (0,5476)	2	-0,0137 (-0,2518)	3
	DCDS(-5)	0,0005 (0,7192)	1	0,1714 (0,5777)	3	0,0480 (1,0930)	6
	Const. (a)	-0,0003 (-0,1190)		-0,0756 (-0,3287)		-0,1451 (-0,4031)
	R-squared	0,0428		0,2190		0,1399
	Adj, R-squared	0,0109		0,1931		0,1114
	F-statistic	1,3422		9,5659		5,0390
	Log likelihood	1065,1197		-1788,3300		-1281,4857
	Akaike AIC	-4,3044		7,3390		5,2688
	Schwarz SC	-4,1631		7,4803		5,4101
	Mean dependent	-0,0004		0,0188		0,0087
	S,D, dependent	0,0293		11,5342		9,1311

Table 8 : VAR1 estimation results over the period 2004-2007

 8

		RS (1)	(2)	DBOND (3)	(4)	DCDS (5)	(6)
	RS(-1)	-0,0225 (-0,6074)	1	-12,2426 (-0,6805)	2	-8,9482 (-1,5527)	5
	RS(-2)	-0,0192 (-0,5137)	1	1,0875 (0,2959)	0	-4,6657 (-0,4390)	1
	RS(-3)	-0,0239 (-0,6657)	2	-1,6128 (0,2015)	1	-0,6615 (0,1389)	1
	RS(-4)	-0,0033 (-0,0933)	1	3,2225 (0,5618)	0	-3,8490 (-0,4020)	1
	RS(-5)	-0,0090 (-0,2347)	0	5,2822 (0,4607)	0	1,9936 (-0,1578)	2
	DBOND(-1)	-0,0001 (-0,5506)	3	-0,4814 (-13,0205)	13	0,0157 (0,8276)	2
	DBOND(-2)	0,0000 (-0,1882)	3	-0,3283 (-7,8160)	13	0,0080 (0,3509)	2
	DBOND(-3)	0,0000 (-0,0989)	1	-0,2103 (-4,8521)	12	0,0075 (0,2236)	2
	DBOND(-4)	-0,0001 (-0,5279)	1	-0,1192 (-2,8616)	10	0,0044 (0,3554)	0
	DBOND(-5)	-0,0001 (-0,7223)	2	-0,0520 (-1,4108)	2	0,0084 (0,6590)	1
	DCDS(-1)	0,0004 (0,4340)	0	0,2337 (2,0010)	5	0,0353 (0,9499)	13
	DCDS(-2)	-0,0006 (-0,8405)	2	0,2593 (0,6450)	2	-0,0334 (-0,8514)	5
	DCDS(-3)	-0,0001 (0,1561)	0	0,1991 (1,1296)	2	0,0229 (0,6557)	4
	DCDS(-4)	0,0005 (0,6229)	0	0,1121 (0,7927)	1	-0,0009 (-0,0075)	2
	DCDS(-5)	0,0002 (0,4894)	2	0,0211 (0,5253)	2	0,0339 (0,9372)	3
	Const. (a)	0,0005 (0,9891)		-0,0224 (-0,1653)		-0,0691 (-0,8092)
	R-squared	0,0254		0,2140		0,0953
	Adj, R-squared	0,0058		0,1982		0,0771
	F-statistic	1,3157		14,0721		5,3888
	Log likelihood	2192,8224		-2309,4307		-936,5150
	Akaike AIC	-5,7117		6,0967		2,4292
	Schwarz SC	-5,6143		6,1942		2,5267
	Mean dependent	0,0004		-0,0455		-0,0843
	S,D, dependent	0,0145		6,9592		2,1017

Table 10 : Granger causality Tests (VAR1 model)

 10

		DBOND	RS cause	DCDS cause	RS cause	DCDS cause	DBOND
		cause RS	DBOND	RS	DCDS	DBOND	cause DCDS
	Alcatel	yes	yes	yes	yes	yes	yes

Table 11 : Granger causality Tests (comparison of the VAR models results)

 11

		Global period*	Sub period 1**	Sub period 2***
		RS cause	DCDS cause	RS cause	DCDS cause	RS cause	DCDS cause
		DCDS	RS	DCDS	RS	DCDS	RS
	VAR1 model	11	8	10	2	4	1
	VAR2 model	11	6	11	3	9	4

*: 2001-2007 for VAR1 and 2001-2008 for VAR2; **: 2001-2003 for VAR1 and VAR2; ***: 2004-2007 for VAR1 and 2004-2008 for VAR2.

† According to the 2006 Fitch survey, the total nominal value of credit derivatives at the end of 2005 stood at nearly 12 trillion dollars, nearly half of which was constituted by CDS.

To test the consequences of the current subprime crisis on the relations of interdependence between the share market and the CDS market, we estimated the VAR2 model. Two main conclusions can be drawn from our results. Firstly, the bond market appears to be relevant for explaining the evolutions of the other two markets, especially the CDS market (for the three periods, the average 2 R obtained for the DCDS equation of the VAR2 model are lower than those of the VAR1 model). Secondly, the relations of interdependence between the markets are stronger during periods of financial instability. The VAR2 model explains better the unstable sub-period 2001-2003 (average 2 R higher than for the whole period and average 2 R of the DCDS equation close to those obtained with the VAR1 model) and show a strengthening of the relation between the share returns and CDS spreads of the companies most affected by the crisis (the 2 R of the RS and DCDS equations are higher over the reccent period 2004-2008 that for the whole period for the Société Générale, BNP Paribas, Danone, Renault, and Total). For the period 2004-2008, we obtain higher and more significant average coefficients and the share return rates appear to react more strongly to variations in CDS spreads than to their own past evolutions (Table 9).

We have verified the relations of causality between these three markets over the whole period using the Granger test. For eleven of the thirteen companies, there is a relation of causality from share return rates to CDS spreads, and this causality is bidirectional for eight of the eleven companies. For eight out of thirteen companies, there is a relation of causality from CDS spreads to bond spreads, and this causality is bidirectional for six of the eight companies. For the companies with high credit risk, and therefore with low ratings, there is a relation of causality between their share return, their bond spread and their CDS spread. On the contrary, there is no relation of causality between these markets for the company with the best credit rating, which is also a company with large market capitalization (Table 10). We have sought to verify whether the current crisis has weakened the relations of causality between the share market and the CDS market. Our results show that this crisis has led to an increase in the number of causalities. According to the VAR1 model and for the sub-period 2004-2007, there is a relation of causality from share returns to CDS spreads for four companies; when we introduce the crisis (VAR2 model) this relation is verified for nine companies: the previous ones plus those in the financial sector and the big groups. Before the crisis, the relation of causality from CDS spreads to share returns existed for one sole company; since the crisis, this relation concerns four companies (Table 11).


Mohamed Laaraiedh
email: mohamed.laaraiedh@univ-rennes1.fr

Implementation of Kalman Filter with Python Language

In this paper, we investigate the implementation of a Python code for a Kalman Filter using the Numpy package. A Kalman Filtering is carried out in two steps: Prediction and Update. Each step is investigated and coded as a function with matrix input and output. These different functions are explained and an example of a Kalman Filter application for the localization of mobile in wireless networks is given.

I. Introduction

Within the significant toolbox of mathematical tools that can be used for stochastic estimation from noisy sensor measurements, one of the most well-known and oftenused tools is what is known as the Kalman filter. The Kalman filter is named after Rudolph E. Kalman, who in 1960 published his famous paper describing a recursive solution to the discrete-data linear filtering problem [START_REF] Kalman | A New Approach to Linear Filtering and Prediction Problems[END_REF]. The Kalman filter is essentially a set of mathematical equations that implement a predictor-corrector type estimator that is optimal in the sense that it minimizes the estimated error covariance when some presumed conditions are met. Since the time of its introduction, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. This is likely due in large part to advances in digital computing that made the use of the filter practical, but also to the relative simplicity and robust nature of the filter itself.

A complete tutorial about Kalman filtering is given in [START_REF] Welch | An Introduction to the Kalman Filter[END_REF].

I.1. Mathematical Formulation of Kalman Filter

The Kalman filter addresses the general problem of trying to estimate the state n x ℜ ∈ of a discrete-time controlled process that is governed by the linear stochastic difference equation

1 1 - - + + = k k k k w Bu Ax x ( 1
)
with a measurement

m y ∈ ℜ that is k k k y Hx v = + (2)
The random variables k w and k v represent the process and measurement noise respectively. They are assumed to be independent of each other, white, and with normal probability distributions

) , 0 ( ) ( Q N w p ≈ (3) ) , 0 ( ) ( R N v p ≈ (4)
In practice, the process noise covariance Q and measurement noise R covariance matrices might change with each time step or measurement, however here we assume they are constant. The n n × matrix A relates the state at the previous time step to the state at the current step, in the absence of either a driving function or process noise. The 1 n × matrix B relates the optional control input

I.2. How does Kalman Filter work?

The Kalman filter process has two steps: the prediction step, where the next state of the system is predicted given the previous measurements, and the update step, where the current state of the system is estimated given the measurement at that time step. The steps translate to equations as follows [START_REF] Welch | An Introduction to the Kalman Filter[END_REF]:

• Prediction k k k k k U B X A X + = - - - 1 1
(5)

1 1 1 1 - - - - - + = k T k k k k Q A P A P (6) • Update - - = k k k k X H Y V (7) k T k k k k R H P H S + = - (8) 1 - - = k T k k k S H P K (9) k k k k V K X X + = - ( 10
) T k k k k k K S K P P - = - (11)
where • - k X and - k P are the predicted mean and covariance of the state, respectively, on the time step k before seeing the measurement.

• k X and k P are the estimated mean and covariance of the state, respectively, on time step k after seeing the measurement.

• k Y is mean of the measurement on time step k .

• k V is the innovation or the measurement residual on time step k .

• k S is the measurement prediction covariance on the time step k .

• k K is the filter gain, which tells how much the predictions should be corrected on time step k .

I.3. Applications of the Kalman Filter

The applications of the Kalman Filtering in real world are diverse. An example application would be providing accurate, continuously updated information about the position and velocity of an object given only a sequence of observations about its position, each of which includes some error. For a similar, more concrete example, in a radar application, where one is interested in tracking a target, information about the location, speed, and acceleration of the target is measured at each time instant, with a great deal of degradation by noise. The Kalman filter exploits the dynamics of the target, which govern its time evolution, to remove the effects of the noise and get a good estimate of the location of the target at the present time (filtering), at a future time (prediction), or at a time in the past (interpolation or smoothing). Other applications are weather forecasting, speech enhancement, economics, autopilot... etc.

II. Python Code of the Kalman Filter

We have chosen to divide the Kalman Filtering Code in two parts similarly to its mathematical theory. The code is simple and divided in three functions with matrix input and output.

II.1. Prediction Step

This step has to predict the mean X and the covariance P of the system state at the time step k . The Python function kf_predict performs the prediction of these output ( X and P ) when giving six input:

X : The mean state estimate of the previous step ( 1 k -). P : The state covariance of previous step ( 1 k -). A : The transition n n × matrix.

Q : The process noise covariance matrix. B : The input effect matrix. U : The control input.

The Python code of this step is given by: from numpy import dot def kf_predict(X, P, A, Q, B, U): X = dot(A, X) + dot(B, U) P = dot(A, dot(P, A.T)) + Q return(X,P)

II.2. Update Step

At the time step k , this update step computes the posterior mean X and covariance P of the system state given a new measurement Y . The Python function kf_update performs the update of X and P giving the predicted X and P matrices, the measurement vector Y , the measurement matrix H and the measurement covariance matrix R . The additional input will be: The Python code of these two functions is given by:

IV. Conclusions and future work

In this paper, we presented the Python code for the Kalman Filter implementation.

  The m n × matrix H in the measurement equation relates the state to the measurement k y . For more deep investigation of the Kalman Filter you may see the reference[START_REF] Welch | An Introduction to the Kalman Filter[END_REF].

K

  : the Kalman Gain matrix IM : the Mean of predictive distribution of Y IS : the Covariance or predictive mean of Y LH : the Predictive probability (likelihood) of measurement which is computed using the Python function gauss_pdf.

Fig- 1 :

 1 Fig-1: Kalman Filter Applied to ToA Based Localization

We presented a two step based implementation and we give an example of using this kind of filters for localization in wireless networks. The next steps will be the implementation of others Bayesian filters like Extended Kalman Filter, Unscented Kalman Filter and Particle Filter. A third step of smoothing of estimations may be introduced later.

V.

III. Example of application: Tracking of mobile in wireless network

The most interesting field of application of Kalman Filter, in telecommunications, is the tracking of a mobile user connected to a wireless network. In this section, we will present a simple tracking algorithm of a mobile user who is moving in a room and connected to at least three wireless antennas [START_REF] Shen | Performance Comparison of ToA and TDoA Based Location Estimation Algorithms in LOS Environment[END_REF].

The matrix of measurement Y describes the estimated position of the mobile using a trilateration algorithm based on a least square estimation and the knowledge of at least three values of Time of Arrival (ToA) at time step k . These values are computed using ranging procedures between the mobile and the three antennas [START_REF] Shen | Performance Comparison of ToA and TDoA Based Location Estimation Algorithms in LOS Environment[END_REF].

Starting by an initialization of different matrices and using the updated matrices for each step and iteration, we plot in Fig- 1 the estimated, the real trajectory of the mobile user, and the measurements performed by the least square based trilateration. We show here that the Kalman Filter enhances the accuracy of tracking compared to the static least square based estimation. The Python code describing the tracking process is given as below. In order to simplify the understanding of this code, we draw the matrix Y randomly centered on the true value of mobile position.


Brigitte Chebel-Morello

Karim Haouchine

Noureddine Zerhouni1

A METHODOLOGY TO CONCEIVE A CASE BASED SYSTEM OF INDUSTRIAL DIAGNOSIS

Keywords: Diagnostics, Case-based reasoning, Knowledge management, Functional safety, Hierarchical and contextual models, Adaptation-guided retrieval 1

The objective of this paper is to address the diagnosis knowledge-oriented system in terms of artificial intelligence, particular by the Case-Based Reasoning (CBR) approach. Indeed, the use of CBR, which is an approach to problem solving and learning, in diagnosis goes back to a long time with the appearance of diagnostic support systems based on CBR. A diagnostic system by CBR implements an expertise-base composed of past experiences through which the origins of failure and the maintenance strategy are given according to a description of a specific situation of diagnostic. A study is made on the different diagnostic systems based on CBR. This study showed that there was no common methodology for building a CBR system. This design depends primarily on the case representation and knowledge models of the domain application. Consequently, this paper proposes a general design approach of a diagnostic system based on the CBR approach.

INTRODUCTION

In the field of diagnosis the different studies can be classified in three approaches [START_REF] Chiang | Fault Detection and Diagnosis in Industrial Systems[END_REF]:

The first one is the analytical approach needing mathematical models, the second one is the data driven approaches like neural networks, systems-based rules, Bayesian networks, neuro-fuzzy networks, etc. And the third one is the knowledge based on causal analysis or expert knowledge.

The choice of the approach depends on the type of available models or information like data or knowledge.

Our choice is directed towards the knowledge-oriented methods and particularly the case based reasoning (CBR). In effect most of the data driven techniques require getting a complete set of learning, while the CBR that allows reasoning on a minimum cases. In effect the reasoning can be done in a few cases, and is less costly for a company than other methods which require a complete learning set to be effective. [START_REF] Althoff | Case-Based Reasoning[END_REF] thinks the CBR is the technology of choice to implement a knowledge-based system.

Domain resources

Richter [START_REF] Richter | The Knowledge Contained in Similarity Measures[END_REF] defines four knowledge containers involved in a CBR process:

• The vocabulary containers which describing a vocabulary of the domain. This vocabulary is represented by ontology, rules, etc.

• The case base which present a practical solving knowledge coming from experience.

• The similarity containers identifying indexes similarity metrics,

• The adaptation knowledge developing the solution transformation rules, adaptation operators, adaptation cases.

Knowledge is involved in the CBR cycle. The first "classical" CBR cycle was proposed by Aamodt and Plaza in 1994 [START_REF] Aamodt | Case-Based Reasoning: Foundational Issues, Methodological Variations, and System Approaches[END_REF]. The reasoning is composed into four steps; know as the four" Re": Retrieve, Reuse, Revise and Retain.

The retrieve phase find the most similar case or cases according to the similarity between the request and previously experienced cases stored in the case base. The reuse phase uses solutions of the similar cases in order to solve the new problem. The differences between the reminded case and the new case are taken into account and the old solution is adapted to the new situation. The phase "revise" of the proposed solution evaluates the proposed solution in the real world. And the retain phase store a new case in the case base.

The figure1 presents an evolution of this cycle, by integration of the elaboration step. This step permits to formulate a request problem on a new problem case. In the core of the cycle we can find the four knowledge containers. However, Cordier in [START_REF] Cordier | Interactive and Opportunistic Knowledge Acquisition in Case-Based Reasoning[END_REF] summarise in the table1 the main form that the knowledge containers take and the roles played by the knowledge unit in the different steps of the CBR process.

The containers allow the organisation of the knowledge units but are considered together as a single knowledge base. The typology of CBR knowledge is as follows (see Table 1) To present expert knowledge as past and concrete experiences makes its comprehension by human users easy. Lamontagne & Lapalme [START_REF] Lamontagne | Raisonnement à base de cas textuels -état de l'art et perspectives[END_REF] represent by two complementary processes a generic model of case based reasoning (see figure 1). A process off line generates the knowledge containers, and a phase of case authoring, and an on line process composed of the different step of the cycle of CBR. To conceive a system oriented knowledge, we must establish the two processes of this model, the off line process with the different containers knowledge, by authoring case and acquisition knowledge, and the process on line by the different algorithm corresponding at the five step of the CBR process.

Case based reasoning on the diagnosis system

Diagnosis field are recurrent and the previous documented solutions can be used again. In the retrieval of similar cases, algorithms can be used even if problems are not completely understood. The case based reasoning is initially a cycle of four main activities. The retrieval phase determines most similar cases. The reuse phase solves the new problem by using information and knowledge in the retrieved cases. And the revise phase evaluates the applicability of the proposed solution in the real-world. The retain phase updates the case base with the new case for future problem solving.

In [START_REF] Althoff | Case-Based Reasoning[END_REF] two different types of diagnostic systems based on CBR exist:

• In a "help desk" application, CBR is exploited as a decision support, and is used by non-specialists in domains with lots of technical equipment.

• In a "general diagnosis and repair" application, CBR is used as a decision help system for the diagnosis of complex equipment or for medical diagnosis and points out failure cause research and problem exploitation.

The object of the CBR application can vary from a car studied in diagnostic technique system named CREEK [START_REF] Aamodt | Knowledge-Intensive Case-Based Reasoning and Sustained Learning[END_REF] to a locomotive defined for remote diagnostics [20] and to a Boeing 747 aircraft developed in CaseLine [21] used as a demonstrator by British Airways and to the Boeing 737 in CASSIOPEE [START_REF] Bergmann | Developing Industrial Case-Based Reasoning Applications: The INRECA Methodology[END_REF] owned by CFM International. There are also an industrial printer studied by Domino UK Ltd in CHEKMATE presented in [START_REF] Grant | Fault Diagnosis for Industrial Printers Using Case-Based Reasoning[END_REF] and gas turbines studied in a fault diagnosis system of General Electric Energy in Atlanta [START_REF] Devaney | Case-Based Reasoning for Gas Turbine Diagnostics[END_REF].

The current research in case-based reasoning focuses on detailed knowledge representation and adaptation phase. Indeed, some systems such as Nodal CBR [START_REF] Cunningham | A Comparison of Model-Based and Incremental Case-Based Approaches to Electronic Fault Diagnosis[END_REF] and Gas Turbine Diagnostics [START_REF] Devaney | Case-Based Reasoning for Gas Turbine Diagnostics[END_REF] do not develop an adaptation phase. While others, such as FormTool [START_REF] Cheetham | Case-Based Reasoning in Color Matching[END_REF], apply a transformational adaptation but have no knowledge model. Creek system [START_REF] Aamodt | Knowledge-Intensive Case-Based Reasoning and Sustained Learning[END_REF] composed of a network of semantic knowledge is handled by three sub tasks namely activation, explanation and focus.

There is a wide variety of method ranging from classification problem when there is just a weak domain theory, to knowledge based systems. In most systems, a case characterizes a diagnostic experience without reference to any kind of model.

The analysis of these different systems pointed out the lack of a proven methodology to register a typical diagnostic case, and methodology of building a case based diagnostic system. This paper has the objective to propose a methodology to conceive a diagnosis and repair help system based on case based reasoning for human operators.

Three special challenges are solved in this project:

The first one is the proposal of a general representation of the case, based on the diagnosis definition.

The second one is to define knowledge models of the case based reasoning systems. Models adapted to the retrieval and reusing phase of the technical diagnosis.

The third one is to propose three different phase of the on line process the elaboration, retrieval and reusing models.

After presenting a previous works in the elaboration of a methodology of conceives a diagnosis system, we propose a specific formalization of the diagnosis case, based on two descriptor types. Our case formalisation is linked to the dysfunctional mode and to the analysis decomposition of the equipment [START_REF] Rasovska | A Case Elaboration Methodology for a Diagnostic and Repair Help[END_REF].

After, we propose the different knowledge containers of the system, and we develop a retrieval phase based on the knearest algorithm within two measures (similarity and adaptation measures) depending on the dysfunctional mode and information relevant to the case.

In the last we applied our methodology in industrial equipment namely a diesel motor of the Renault Company.

THE DESIGN APPROACH

Our design methodology of diagnostic system by CBR takes as a starting point the generic model of Lamontagne and Lapalme [START_REF] Lamontagne | Raisonnement à base de cas textuels -état de l'art et perspectives[END_REF]. We develop in each process the various containers and steps of CBR cycle.

Previous works and methodology following

Among the work on the diagnosis, the systems are describing without defining the methodology for obtaining the components of the system. We are interested in previous work on this issue. We are proposed a mix method of knowledge capitalization and case based reasoning in order to develop a diagnosis system. The knowledge capitalization cycle was adopted as the underlying principle Rasovska et al [START_REF] Rasovska | A Case Elaboration Methodology for a Diagnostic and Repair Help[END_REF]. This method integrates a representation and a reasoning model both completing each other and suitable to represent and manipulate the domain knowledge.

Our system, like Creek for example, is based on the operating safety tools [START_REF] Rasovska | A Case Elaboration Methodology for a Diagnostic and Repair Help[END_REF] and proposes functional, dysfunctional or causality models. These models can be aggregated in a network of semantic knowledge as a Creek or be proposed in the case of our study, in the form of two models: hierarchical and contextual.

Off-line process

For the diagnostic system design we first define the off-line process (see Figure 2).

Documents

We propose to use the documents produced for a given maintenance process (alarm handling, diagnosis, etc.) as our basic building blocks for the case model. And, because all this information may be retrieved from a given intervention report, we choose the intervention reports database as our case base model. These two models will be completed and improved by the experience feedback

Authoring

In [START_REF] Rasovska | A Case Elaboration Methodology for a Diagnostic and Repair Help[END_REF] we were interested in case formalisation with the proposal of ontology. In this paper we go further and we define the case as a set of descriptors based on the diagnostic reference definition: "these are the actions carried out for the fault detection, its localization and the identification of the cause" [3]. Consequently, the diagnostic is composed of three parts: the fault detection part, the part concerning the zone localization in which the failure occurred and the identification part of the failing component. We develop these three phases in the case design; moreover, will base on the equipment knowledge models to be diagnosed. We summarize the three parts as follows:

Detection: we determine on the equipment the failure class based on the FMEA (Failure Mode and Effects Analysis). This class belongs to the solution of our case.

Localization: this part is described in the first problem part of the case. It allows determining thanks to a context model the equipment failure zone.

Identification: this identification concerns on the one hand, the second part of the case problem which one names "functional part" and on the other hand, the solution part while allowing to identify the failing component. The functional part is based on a components hierarchal model in which the classes of this model are found in the functional descriptors in order to obtain a generic case.

We represent the industrial diagnostic problem in the most general possible way.

A case has object formalization and allows defining a hierarchy of descriptors containing as well the problems descriptors as the solution descriptors.

The descriptors are represented by three attributes. We associate to each attribute the modal values forming a partition contiguous to the considered attribute. One of specificities of our study case is to determine a state and a operating mode, with each component to be diagnosed [START_REF] Haouchine | Adaptation-Guided Retrieval for a Diagnostic and Repair Help System Dedicated to a Pallets Transfer[END_REF]. Therefore, we assign to a given descriptor:

• An attribute on its value itself (the descriptor is an electrical or mechanical component for example);

• An attribute on the its status;

• And an attribute relating to the operating mode which will reflect the normal and/or abnormal state of the equipment components.

The problem descriptors of the functional part will have consequently three attributes relating to the value of the component, its state and its operating mode: ds i = (

value si d , state si d , FM si d ).
By taking account of all these specificities, we can schematize the case structure as shown in the Figure 3.

Problem part Solution part

Localization Functional Class We identify the relevant descriptors for the case representation according to the studied field and a suitable structure of the associated cases with the general knowledge models. This identification allows the installation of the two retrieval and adaptation phases and the exploitation of the comparison between these two phases through the creation of the appropriate measures.

The knowledge containers

The case-base container contains all the previously definite cases. Thereafter, we will develop the three remaining containers.

The vocabulary container

The vocabulary container contains the knowledge models of the CBR system.

The representation of knowledge of our CBR system dedicated to diagnostic is shown on Figure 1.

It consists of two models supplementing the dysfunctional part of our system which is the case-base, and by decision rules determining the component operating mode within the operating context of this one.

The equipment model is resulting from a equipment decomposition of the studied system which determines the functions performed by the equipment and its components. The components taxonomy model is determined from their functional analysis. Each components set is gathered by functional class. The knowledge representation is based on two models namely the context model and the components taxonomy model. The context model is based on flows of specific magnitudes of the system and on the equipment decomposition analysis of the studied system. This decomposition determines the functions provided by the equipment and its components. As for the components taxonomy, it determines the equipment sets starting from their functional analysis. Each components set is grouped according to the operating similarity or ensured by the same functions. We associate with these two models and the case-base another module concerning the decision rules for determining the system operating mode to be diagnosed.

The similarity measures container

Before the Nineties, the two retrieval and adaptation phases were exploited in a completely independent way. Until Smyth and Keane [START_REF] Smyth | Retrieving Adaptable Cases. The role of adaptation knowledge in case retrieval[END_REF] bring a new breathe and suggest the unification of these two phases. This unification goes in the direction where the chosen case in the retrieval phase is that most easily adaptable in order to optimize the adaptation phase results. Our recent work goes in this direction and we have proposed in [START_REF] Haouchine | Adaptation-Guided Retrieval for a Diagnostic and Repair Help System Dedicated to a Pallets Transfer[END_REF] two measures, one of similarity and one of adaptation.

In our study, the retrieval phase is based on the k nearest neighbours algorithm, on the components taxonomy and on the measures being in the similarity measures container. In order to select, among the retrieved cases, the most favourable case to the adaptation, we must initially evaluate the similarity of the source cases to the case to solve by applying a global similarity measure which is composed by a set of local similarity measures. Then, we will take account of the descriptors operating mode of the functional part via a Ȝi weight in the second measure that called: Adaptation Measure (A M ).

• Retrieval Measure

We initiated the Retrieval Measure (R M ) in [START_REF] Haouchine | Adaptation-Guided Retrieval for a Diagnostic and Repair Help System Dedicated to a Pallets Transfer[END_REF]. We recall that this measure is composed of four local similarities: The global similarity measure (1) is obtained by aggregation of these functions on the whole set of descriptors. From this measure, a set of cases can be selected.

¦ ¦ = = × × × = m i ésence i m i F M i ésence i Etat i Valeur i M R 1 Pr 1 . Pr ϕ ϕ ϕ ϕ ϕ (1)
Where n: represent the number of problem descriptors.

Thus, this measure will provide a set of the most similar cases to the case to solve. In order to select the most adaptable retrieved case, we have introduced a measure called: Adaptation Measure (A M ).

• Adaptation Measure

The Adaptation Measure (A M ) takes account of the components operating mode by attaching importance to the abnormal operating mode in the case source descriptors (dsi) and the case target descriptors (dti), by providing them with greater weight. This importance will be characterized by the Ȝi weight.

The adaptation measure (A M ) (2) takes into account the source cases descriptors which are different from case target and will be linked to the class and to the functional mode compared to the solution descriptors.

¦ ¦

= = + ⋅ = n i value i state i M F i n i i Class i M A 1 . 1 ). ( ϕ ϕ ϕ λ ϕ (2)
Where Ȝi is the associated weight according to the functional mode If FM=normal AE Ȝi= 2 0 ; if FM= abnormal AE Ȝi= 2 2 ; if FM= nor/ab AE Ȝi= 2 1 .

A weight is associated to the functional mode because this last is considered as being important in the determination of the failing component. The number of different descriptors is determined by the denominator in the equation [START_REF] Aamodt | Case-Based Reasoning: Foundational Issues, Methodological Variations, and System Approaches[END_REF]. The retrieved source case having the greatest adaptation measure value among the retrieval source cases will be the candidate chosen for the adaptation step.

The adaptation container

We set up in an adaptation algorithm the dependency relations between the various problem and solution descriptors. These relations allow to express the influence of a problem descriptor "ds" on the solution descriptors "Ds" in the various sources cases of the case-base. These relations will help to define relevant descriptors set which could be used in the adaptation phase. Their implementation is explained in [START_REF] Haouchine | Adaptation-Guided Retrieval for a Diagnostic and Repair Help System Dedicated to a Pallets Transfer[END_REF]. Three values are assigned to the various cases source descriptors according to the link between problem and solution spaces, DR = {high, low, none}.

The method has been validated in our works of [START_REF] Haouchine | Adaptation-Guided Retrieval for a Diagnostic and Repair Help System Dedicated to a Pallets Transfer[END_REF].

On-line process

The on-line process contains various cycle phases. We develop three phases namely: elaboration of the target case, retrieval and adaptation.

Elaboration phase

At the time of failure, the user requests the diagnostic help system which proceeds in three steps: a failure localization step which is done thanks to the equipment model. This model provides a certain components set that could potentially failing by their geographical proximity of the concerned zone. Thus, the elaboration phase will be initialized by the filling of the associated descriptors with the "failure localization". By a consultation of the supervisor SCADA (Supervisory Control And Data Acquisition) step, giving us the actual state of the various components. Then, by a decision phase, which is starting from the actual position of these components, implements decisions rules giving the operating mode of the implied components. This allows the part of the problem descriptors of the target case to knowing its localization part and its operating mode part (normal/abnormal) to be informed (Figure 5). It is to be noted that the cause and effect relation is taken into account during the case-base creation. The various components are linked by their state, which results in the case-base descriptors values and therefore the definition of their operating mode. Indeed, a component can be in an abnormal operating mode state but it is not failing. This can happen because of the influence of another failing component.

Figure 1 .

 1 Figure 1. The CBR cycle (Mille, 1999).

Figure 2 .

 2 Figure 2. Generic model of a CBR system.

Figure 3 .

 3 Figure 3. Structure générique du cas de diagnostic.

Figure 4 .

 4 Figure 4. Représentation des connaissances d'un système de RàPC dédié au diagnostic de pannes.

  account the information in descriptors, a local similarity presence ϕ is developed. presence ϕ =1, when the descriptor is indicated in the source case and presence ϕ =0, if not.

Figure 5 .

 5 Figure 5. Les différentes étapes d'élaboration du cas cible.

Table

Varma A & Roddy N. (1999) ICARUS: A case_Based System for Locomotive Diagnostics. Applications of Artificial Intelligence Journal.

Watson I & Marir F. (1994) Case-Based Reasoning : A Review, The Knowledge Engineering Review, 1994.

Adaptation guided retrieval

The retrieval phase is based on the two installation measures (A M ) and (R M ) in order to guide the adaptation. Indeed, a first retrieval phase allows thanks to the k nearest neighbours algorithm to define a set of most similar cases to the case to be solved. The (A M ) measure is used to select among all retrieved cases the most easily adaptable case.

Adaptation

The adaptation phase is carried out thanks to an algorithm relies on the context model, the descriptors hierarchical model and the dependency relations. The dependency relations are inspired by Fuchs' work in [START_REF] Fuchs | An Algorithm for Adaptation in Case-Based Reasoning[END_REF].

The adaptation algorithm is presented in [START_REF] Haouchine | Conception d'un Système de Diagnostic Industriel par aisonnement à Partir de Cas[END_REF]. This algorithm adapts descriptor by descriptor. Three standard adaptation cases have been identified, namely:

• DR high involving at least a solution descriptor with a problem descriptor belonging to the same operating class;

• DR high involving at least a solution descriptor with a problem descriptor belonging to different operating classes;

• DR low.

The adaptation guided retrieval was validated on supervised industrial system of pallets transfer (SISTRE) [START_REF] Haouchine | Algorithme d'Adaptation pour le Diagnostic Technique[END_REF]. Thus, we have proved the feasibility of our diagnostic help system. This allowed to set up an design approach of diagnostic help system applied to another industrial equipment, in this case a diesel motor 1.5 dCi K9K 105ch of Renault company available to the following link: http://v3.renault.com/cfm/module-K9K/fr/index.html.

APPLICATION

A functional and dysfunctional study of a motor diesel was introduced for the construction of case-base and knowledge models of the diagnostic help system. We develop in what follows the installation of the used documents, the case base and the knowledge models.

Les documents

The used documents result from the diesel motor technical documentation which is available at: http://v3.renault.com/cfm/module-K9K/fr/index.html. This documentation enabled to determine various existing flows in the motor and its components. Moreover, we called upon an expert, in particular a mechanic, who helped us to define the components operating modes and to analyze the causal relations.

The motor case-base

Through the functional and dysfunctional study of the motor, we built a case-base containing 130 cases. The case problem part is composed of 12 descriptors. The first four descriptors determine the failure zone field of motor equipment. This zone is determined by "ds1: the motor state", "ds2: the glow plugs state", "ds3: temperature of the motor" and "ds4: failure sub zone".

Figure 6 shows an overview of the parts problem and solutions of seven cases of the diesel motor case-base. Figure 6. A part of the case-base of Renault gas motor. Let us take the example of case 1. This last reflects a failure on the injection pump. The problem part determines in its "localization" part the failing component place which is defined by the context model. According to the first four descriptors, the motor is running, the glow plugs are in a bad condition and the two remaining descriptors are not indicated. When referring to the context model, we find a number of components which are potentially failing in the specified zone. Then, the case functional part indicates the components state in this zone. The "ds5" descriptor which reflects the "pump injection" component belonging to the "injection" class is in a "no trained" state generating an "abnormal" operating mode. The "ds7" descriptor reflects the candle state which is in a "normal" operating mode because it produces sparks. The "ds9" descriptor describes the filter component that normally circulates gas. The "ds12" descriptor indicates that the crankshaft has a continuous motion and is in "normal" operating mode. The other descriptors are not indicated.

As for the solution part, the "Ds1" descriptor indicates that the concerned component failure class is "coupling mobile". "Ds2" specifies that the failing component is the "injection pump" with a remark that this pump does not deliver fuel. Thus, the associated repair action which is described by "Ds3" descriptor recommends a change of the injection pump. Finally, the "Ds4" descriptor indicates that the failure occurred on the fuel passage flow.

The knowledge models

The knowledge representation is based on two models associated with the case-base, namely: the context model and the components taxonomy model.

The equipment analysis determines sets of these components. Each set includes components which provide the same functions. This functional analysis allows having a hierarchical modelling of the equipment. Figure 1 shows an example of the components hierarchical model of a diesel motor. As the motor has a multiplicity of components, we show that part of the components hierarchical model. This model includes four classes: Ancillary, Coupling mobile, Basic elements and Timing. Each class is composed of subclasses until we arrive at the level of components.

The context model is based on flows of the system entities and on the equipment decomposition analysis of the studied system. This decomposition determines the functions provided by the equipment and its components. These components constitute the context in which the failing component is identified.

The context reflects the cause-and-effect relations permitting to locate components with problems and to select the good descriptors compared to the set. Figure 2 shows an overview of the diesel motor context model. The retrieval phase will take account of three attributes types of the case functional part. The adaptation phase will exploit the dependency relations by traversing a context model which defines the cause and effect relations between descriptors.

To evaluate our method of adaptation guided retrieval and the adaptation algorithm, we set up a specific protocol. This protocol relates to the division of the diesel motor case-base in two subsets. The first subset contains 40% of cases drawn randomly from the case-base forming the training set. The 60% of the remaining cases will constitute the test set. The goal of this protocol is to calculate the training case-base accuracy. We indicate that the accuracy relates to the determination of the good failure class. We found that our retrieval and adaptation method ensured a very high accuracy. In fact, we can conclude that CBR can reason from a limited number of cases, based on knowledge models and on adaptation phase adjusted to the considered application. Moreover, the retrieval phase, which uses two retrieval and adaptation measures, takes account of the adaptation effort and selects the most easily adaptable cases and amongst other things the cases which have the same classes.

CONCLUSION

Within the study framework made on the industrial diagnostic help and repair systems, we proposed a system of diagnostic based on the case-based reasoning approach (CBR) with the deployment of the different phases. The designed system includes an object formalization of case associated with a hierarchical model common to the problem and solution descriptors of the case-base cases and a model in relation to the application context. All CBR phases depend on the cases formalization associated with the knowledge models. Recently in [START_REF] Haouchine | Adaptation-Guided Retrieval for a Diagnostic and Repair Help System Dedicated to a Pallets Transfer[END_REF], we formalized a case for SISTRE equipment. This formalization is adapted to our method. The selected modelling influences proposed retrieval and adaptation measures. This latter is directly related to the operating mode of the monitored components (a specific attribute of a descriptor). The developed retrieval phase is guided by the adaptation in exploiting the relations of the two installation measures namely: Retrieval Measure (R M ) and Adaptation Measure (A M ). These relations enable to select among the retrieved cases those which are most easily adaptable. The adaptation phase operates the dependency relations between the problem and solution descriptors. These dependency relations are determined through relevant descriptors or using the context model reflecting the cause and effect relations between the equipment components. We have detected three standard cases, in the adaptation algorithm, associated with the different values of the dependency relations.

In order to prove the feasibility of our approach, we have established in this paper our CBR diagnostic system on a diesel motor 1.5 dCi K9K 105ch of the "Renault" company.

We plan in a near future to apply a maintenance method of the installation CBR system. Indeed, the system's evolution through time is done by the introduction of new cases following the appearance of new failures. This evolution may deteriorate the case-base quality and that two retrieval and adaptation measures can become obsolete. This maintenance method would allow the evolution of these two measures (as example while inserting weights) and of the case-base according to the system evolution.