Introduction

Insured claims may be time dependent. For example, an insured who has claimed one year might be more careful the following year and thereby alter the probability that a claim occurs. By the same token, an insured whose health has been impaired one year may be more prone to claim subsequently. Such persistence recurs in many instances motivated and implied in insured attitudes and expectations as well as recurring due to events inherent to the claim process (see for example [START_REF] Michel | Actuarial Theory for Dependent Risks: Measures, Orders and Models[END_REF]. Questions such as, does car failure induce additional and subsequent car failures? Are terrorist attacks correlated [START_REF] Telesca | Are global terrorist attacks time correlated[END_REF], one attack defining the propensity for a subsequent attack or a respite ? Does the severity of hurricanes one year imply the severity of the following year? etc., are obvously questions of importance for insurers. Similarly, does a patient relapses following a treatment or not? Do stock prices have an inertia, tending to increase (or decrease) following increases or decreases with the same probabilities? Are fear regarding financial markets persistent? (The Financial Times, February 7, 2008, p.28). Most actuarial counting approaches assume a Poisson distribution, implying that events are independent. In other words, prior events do not alter the basic probabilities laws that determine the occurrence of subsequent events. To circumvent this lacuna, the credibility theory approach in actuarial science, evaluates the objectivity and the subjectivity of a sourcethe insured that may potentially claim, and devises a statistical "learning" meachanism that allows the updating of the underlying claim probability. Using Bayesian statistics for example, credibility theory divides insured into classes that have various propensities to claim, which are updated using subjective prior estimates of risk classes and an accrued experience-the claim history of insured which is observed. The goal of credibility theory is then to set up an experience rating system to determine next year's premium, accounting for the individual and the collective group experience. Unlike credibility theory, this paper presumes that there may be an inherent persistence in an underlying process that will dictate the probability laws with which subsequent events occur. The probability of a subsequent claim for example, will then be determined by the past memory (in our case, the single past event of a claim or no claim) rather than be determined by a statistical estimator based on the accrued evidence of past claims.

Explicitly, while credibility theory seeks to integrate "experience" in estimating the propensity to claim, a persistence to claim is an inherent property of the underlying claiming (stochastic process) that determines, conditionally on the "claim memory", the actual probabilities with which an insured will claim or not. For example, if an insured claiming (or not) one year will be more careful (or less careful) the following year, then the underlying claiming process will account for such a behavior in defining the effects of the "process memory" on the claiming stochastic process. The credibility approach however, will use the fact that a claim has been made to revise the probability that he will claim again in the following year with a credibly (larger) probability. In this sense these approaches differ fundamentally.

The purpose of this paper is to consider such a counting persistent process based on the Markov memory of the immediate and past event (and not only its state) and claculate the generating function of such a process (for example, see [START_REF] Patlak | Random walk with persistence and external bias[END_REF], [START_REF] Weiss | Random walks: theory and selected applications[END_REF], Balinth [1986], Claes and Van den Brock [1987], [START_REF] Weiss | Aspects and Applications of the Random Walk[END_REF][START_REF] Weiss | Some applications of persistent random walks and the telegrapher's equation[END_REF], [START_REF] Pottier | Analytic study of the effect of persistence on a one-dimensional biased random walk[END_REF], [START_REF] Vallois | Memory-Based Persistence in A Counting Random Walk Process[END_REF] for prior research on such processes). The resulting persistent counting process will be shown on the one hand to depart from the Poisson counting process and recognize the effects of the past memory on the claim process. Given such a process, we assess the effects of persistent claims on an insurance contract and use such observation to better assess the risk premium needed to compensate its risk exposure. The result obtain in this paper extend the results obtained in [START_REF] Vallois | Memory-Based Persistence in A Counting Random Walk Process[END_REF]. Explicitly, we extend our previous result by providing the probability generating function of the persistent claim process, its moments as well as an explicit expresison for the probabilities of such a process. In particular, the process kurtosis due to persistence is expressed explicitly. Finally, a general expression for Persistent Compound processes is provided and an explicit recursive equations for its probability moments is given explicitly, generalizing thereby the often used Compount Poisson process. Numerical analyses are then used to highlight the effects of persistence counting compared to traditional counting and claiming processes.

The Markov Memory Based Persistent Counting Process

Assume that a representative insured can in any one year claim or not, denoting these events by (0,1). The event « 0 » states that no claim is made within the year while a « 1 » states that the insured has filed a claim during the year.

Claim records indicate that when a claim is made in a given year, then the following year, the probability of no claim is 0 > β . When a claim is not made in a given year, then the probability of a claim being made in the following year is α . This defines a simple two-states Markov chain, given by:

(1) 1 , 0 , 1 1

α α α β β β - ⎡ ⎤ = < < ⎢ ⎥ - ⎣ ⎦ P
If we denote by t y the values the random event can assume at time t then:

(2)

( ) ( ) ( ) ( ) 1 1 1 1 1 0 , 0 0 1 1 1 1 , 0 1 t t
⎧ = = = = = = - ⎪ ⎨ = = = - = = = ⎪ ⎩ Or (3) ( ) ( ) ( ) ( ) 1 1 0 1 1 0 1 1 t t t t P y P y P y P y α α β β - - ⎡ = = ⎤ - ⎡ ⎤ ⎣ ⎦ ⎡ = = ⎤= ⎢ ⎥ ⎣ ⎦ - ⎣ ⎦
Over a period of time t, the total number of events (claims made, etc.) is therefore given by:

(4)

0 t t j j x y = = ∑
The number of claims are conditional on the initial event (the current memory of the event), denoted by: 0 0 y = or 0 1 y = . In this memory-based persistent counting process, we calculate first the probability generating function (PGF), summarized in Proposition 1 and subsequently use this PGF to obtain some of the characteristics of an underlying claim process. P x = denotes the probability that a claim was made initially and set for notational convenience. Then:

(5)

1 ( ,1) ( ,0) ( , ) , 1 t t G G G t t λ λ λ λ ρ δ δ - = Ψ - Ψ ≥ with (6) ( ) ( )( ) ( ) 0 0 0 0 ( ,0) ( 0) ( 1)
,1 (1

) ( 0) 1 1 ( 1) G Px Px G P x P x λ λ λ α λ α λ λ α ρ = = + = = -+ = + + - + = where ( ) ( ) 2 0 1 0 1 1 , ( ) 4 1 , 2 1 2 t t t a a a a α λα δ λρ λρ μ μ μ λ ρ δ μ λ ρ δ = -+ = + - Ψ = - = + + = + - Proof: See Appendix 1
Such a generating function provides the means to calculate higher order moments of the underlying persistent counting claim process as well as the probabilities of the number of claims. Further, it clearly points out to the effects of the persistence index on the counting process. When 0 ρ = then 1 β α =and whatever the previous outcome (whether a claim or no claim), the subsequent probability to claim is α while that of no claim is 1 α -.

When the persistence index is positive,

0 ρ > then 1 β α ρ = --. That is, if
in a given year a claim is made, then the probability that in a subsequent year a claim is made has a smaller probability. And vice versa, when the persistence index is negative 0 ρ < and the underlying stochastic process would point out to a "contagious" claim process (for example, with Hurricanes of a high category following Hurricanes of High category).

Inversely, for a positive persistence index 0 ρ > it will indicate that the underlying claim process has a built-in "incentive effect", reducing a claim probability in a given year following a claim made in the previous one.

Our results in proposition 1 allow an explicit calculation of the moments of the persistent counting process, defined recursively as shown below. Note that:

( ) 1 ( , ) t x t G t E x λ λ λ - ∂ = ∂ , ( ) ( ) 2 2 2 ( , ) 1 t x t t G t E x x λ λ λ - ∂ = - ∂ , ( )( ) ( ) 3 3 3 ( , ) 1 2 t x t t t G t E x x x λ λ λ - ∂ = - - ∂
and so on for higher order terms. Using these terms and setting 1 λ = , we obtain the necessary equations which allow the calculation of the mean, the variance, the kurtosis and other moments of the persisting counting process distribution.

It is also useful to derive a number of special and well known cases to confirm the validity of our results. First note that when 1

λ = then (1, ) 1 G t = as expected.
Further, when there is no persistence (i.e 0 ρ = ), we have

( ,1) ( , ) t G G t a λ λ = Ψ , 0 1 , 0 and t t a a μ μ = = Ψ = and therefore: (7) { } 0 0 ( , ) ( 0) ( 1) t G t P x P x a λ λ = = + = In particular, if initially, 0 ( 0) P x = =1, then ( , ) t G t a λ =
which corresponds as expected to the Probability Generating Function of a binomial distribution.

However, if 0 ( 1) 1 P x = = , then ( , ) t G t a λ λ = .
A convenient recursive expression for the generating function can be found by noting that 0 1

t t t μ μ Ψ = - where 0 μ and 1 μ solve ( ) 2 ( ) 1 0 μ μ λ ρ α α λρ - + + -+ = ,
(proved in Appendix 1) and verify the second order equation:

(8) 2 1 ( ) 0 t t t a λρ λ ρ + + Ψ -+ Ψ + Ψ =
As a result, (5) implies that the probability generating function ( , ) G t λ satisfies as well the second order recursive equation given by: (9) ( )

( , 2) 1 ( ) ( , 1) ( , ) G t G t G t λ α λ α ρ λ λρ λ + = -+ + + -
Deriving ( 9) with respect to λ with 1 λ = we obtain a recursive expression for the moments of the counting process. Concentrating our attention on the first moments only, derivatives of (9) yields the following recursive equation:

(10) ( ) ( ) 1 1 1 1 ( , 2) 1 ( ) ( , 1) ( , ) ( , 2) ( , 1) ( , ) 1 ( ) ( , 1) ( , ) ( ) , 1, 2,3,.... k k k k k k k k k k G t G t G t G t G t G t G t G t k k k λ α λα ρ λ λ ρ λ λ λ λ α λ α ρ λρ λ λ λ λ λ α ρ ρ λ λ - - - - + = -+ + + - ∂ + ∂ + ∂ = -+ + - ∂ ∂ ∂ ∂ + ∂ + + - = ∂ ∂ With initial conditions: (11) ( ) ( )( ) ( ) ( )( ) ( ) ( ) 0 0 0 0 0 0 0 2 2 0 2 2 ( ,0) ( 0) ( 1) ,1 (1 ) ( 0) 1 1 ( 1) ( ,0) ( ,1) ( 1), ( 0) 1 2 1 ( 1) ( ,0) ( ,1) 0, 2 ( 1)
( ,0) ( ,1) 0, 0, for 3

j j j j G Px Px G P x P x G G P x P x P x G G P x G G j λ λ λ α λ α λ λ α ρ λ λ α λ α ρ λ λ λ λ α ρ λ λ λ λ λ λ = = + = = -+ = + + - + = ∂ ∂ = = = = + + - + = ∂ ∂ ∂ ∂ = = + = ∂ ∂ ∂ ∂ = = ≥ ∂ ∂
These initial conditions are specified using equation ( 6) in Proposition1. At λ =1, we can write these expressions in the following manner (together with equation ( 9)) which simplifies their numerical solution:

(12) ( ) ( )

1 1 1 1 (1, 2) 1 (1, 1) (1, ) (1, 2) (1, 1) (1, ) (1, 1) (1, ) 1 ( ) , k k k k k k k k k k G t G t G t G t G t G t G t G t k k ρ ρ ρ ρ α ρ ρ λ λ λ λ λ - - - - + = + + - ∂ + ∂ + ∂ ∂ + ∂ = + - + + - ∂ ∂ ∂ ∂ ∂
While the initial conditions stated above in equation ( 11) and leading to:

(13) ( ) ( ) ( ) 0 0 2 2 0 2 2 (1, 0) 1, 1,1 1 (1, 0) (1,1) ( 1), 1 ( 1)
(1,0) (1,1) 0, 2

(1, 0) (1,1) 0, 0, for 3.

j j j j G G G G P x P x G G P x G G j α ρ λ λ α ρ λ λ λ λ = = ∂ ∂ = = = + + = ∂ ∂ ∂ ∂ = = + = ∂ ∂ ∂ ∂ = = ≥ ∂ ∂
Similarly, we can calculate the probabilities of persistent counting process by setting λ =0 in the derivatives of the generating functions. In this case, the probabilities are given by:

1 ( , ) ( ) , 0,1, 2,3,..., ! i i i G t p t i t i λ λ ∂ = = ∂ with: (14) ( ) ( ) 1 1 1 1 (0, 2) 1 (0, 1) (0, 2) (0, 1) 1 (0, 1) (0, ) ( ) , k k k k k k k k G t G t G t G t G t G t k k α α λ λ α ρ ρ λ λ - - - - + = - + ∂ + ∂ + = - ∂ ∂ ∂ + ∂ + + - ∂ ∂
With the initial conditions (specified by equation ( 6)):

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 8 (15) ( ) ( ) ( ) 0 0 0 0 0 2 2 0 2 2 (0,0) ( 0), 0,1 (1 ) ( 0) (0,0) (0,1) ( 1), ( 0) 1 ( 1) (0,0) (0,1) 0, 2 ( 1)
( ,0) ( ,1) 0, 0, for 3.

j j j j G P x G Px G G P x P x P x G G P x G G j α α α ρ λ λ α ρ λ λ λ λ λ λ = = = - = ∂ ∂ = = = = + -- = ∂ ∂ ∂ ∂ = = + = ∂ ∂ ∂ ∂ = = ≥ ∂ ∂
These equations define a numerical approach to calculating both the moments and the probabilities of a persistent process. A more direct approach will be outlined subsequently however.

Explicit results for the first two moments are provided below with proofs found directly from equations ( 12) and ( 13).

Proposition 2

Let 0 0 x = , then: (16) ( ) ( ) ( ) ( )
( ) 1 ( 1) ( 2) , 2 E x t E x t E x t t ρ ρ α = + -- - + ≥ (17) ( ) ( ) ( ) ( ) 2 2 2 1 ( ) 1 ( 1) ( 2) (1 ) 2 1 2 , 2 1 1 t t E x t E x t E x t t t ρ ρ α α α ρ ρ α ρ ρ ρ + = + - - - + ⎡ ⎤ ⎛ ⎞ + - - + + - ≥ ⎢ ⎥ ⎜ ⎟ - - ⎝ ⎠ ⎣ ⎦
In these equations, note that we have as expected [START_REF] Vallois | Memory-Based Persistence in A Counting Random Walk Process[END_REF]):

(18) ( )

1 1 ( ) 1 1 1 t E x t t α ρ ρ ρ + ⎡ ⎤ - = + - ⎢ ⎥ - - ⎣ ⎦
This clearly indicates the nonlinear time effects of persistence in such counting processes. A verification of (17) can also be reached. First note that when there is no persistence, then ρ =0 and

( ) ( ) [ ] 2 2 ( ) ( 1) 1 2 ( 1) E x t E x t t α α = - + + -.
Summing for 1 t n ≤ ≤ , we obtain ( )

( ) ( ) 2 ( ) 1 1 E x n n n α α = + -. Since ( ) x n has a binomial distribution ( , )
B n α , we have

( ( )) E x n nα = , ( ) ar ( ) (1 ) V x n nα α = - and thereby: ( ) ( ) ( ) 2 2 2 ( ) (1 ) 1 1 E x n n n n n α α α α α = -+ = + - as expected.
Explicit expressions for the persistent counting probabilities can be determined as well using the recursive probability generating functions. In this case, we calculate ( ) ( );0 k p t k t ≤ ≤ by recurrence. Initially these are specified by:

(19) 0 0 1 (0) 1, (1) 1 , (1) p p p α α = = - = Further, ( ) ( 2);0 2 k p t k t + ≤ ≤ +
is defined as a function of ( )

(
k p t = if 0 k < or k t > .
Subsequent calculations will indicate the underlying process probabilities. In particular, we have for the first 3 probabilities:

(21) ( ) ( ) 0 2 1 2 2 4 2 ( ) (1 ) ( ) (1 ) (1 ) , 1 ( )( 1 )
( ) , 2 1 ( 1 ) ( 1 ) ( 2 ) ( 1 ) 2 2 t t t t p t p t t t p t t t t α α α α ρ ρ α ρ α α α α ρ α α ρ α ρ α α ρ - - - = - = - -- + ≥ ⎧ ⎫ + - ⎪ ⎪ = ≥ ⎨ ⎬ + - -- - -- + + -+ ⎪ ⎪ ⎩ ⎭
Of course, when there is no persistence, this is reduced as expected to:

(22) 0 1 1 2 2 2 ( ) (1 ) ( ) (1 ) , 1 1 ( ) (1 ) (
= - = - ≥ = - - ≥

Application: The Compound Counting Persistent Process

In many applications, counting is used for summing events that are independent (for example, the Compound Poisson processes). Expectedly, when counting is persistent (and therefore dependent), such a statistical characteristic has to be accounted for. The analytical results regarding the counting persistent process can be used and applied to numerous problems. For example, say that an insurance firm seeks to calculate the sum of claims (independently distributed but dependent on the persistent counting process-and therefore the Poisson distribution cannot be used). Let the sum of claims be ( ) ( )

1 t x i t i i E Z E x E Z = ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∑ % %
while claims variance is given by: ( )

( ) ( ) 2 1 var ( ) t x i i t i t i Z E Z Var x Var Z E x = ⎛ ⎞ ⎡ ⎤ = + ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ∑ % % % .
However, when claims are persistence-dependent a more general expression can be found. Explicitly, consider the following random claims:

(23)

{ } { } 0 1 0 1 0 0 1 1 i i t t t i i y y i i S Z Z = = = = = + ∑ ∑ % % where 0 t t i i x y = = ∑
where 0 i Z % is a "normal claim" occurring in any regular period (defined by the fact that no specific event has occurred) while

1 i Z % is a "large claim" (of course, if 0 1 i i Z Z = % % then t t S ξ = as stated above). We assume that { } 0 1 , i i Z Z % %
are random variables independent of each other and independent of the Markov (persistent) claims. In this case, the compound claim mean and variance and the claim probability generating function are given by the following (with proofs provided in the appendix):

(24) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 0 1 1 1 ( 1) t t E S t E Z E Z E Z E x = + + - % % % (25) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 0 0 2 1 0 1 1 1 1 1 0 0 1 0 1 1 1 1 1
( 1) var ( 1) var v ar 2( 1)

t t t E S t Z t E Z E x E Z E Z E x Z Z t E Z E Z E Z ⎡ ⎤ ⎡ ⎤ = + + + + - ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ + - + + - ⎣ ⎦ % % % % % % % % %
While their Laplace Transform is:

(26) ( ) ( ) ( ) ( ) ( ) 1 1 0 0 1 1 , , t i Z t S Z Z E e E e E e G z t z E e λ λ λ λ - + - - - ⎡ ⎤ = = ⎢ ⎥ ⎣ ⎦ % % % , 0 λ ≥
where ( ) , G z t is a probability generating function:

( ) ( ) 1 0 ( , ) t t x i t i G z t E z z P x i + = = = = ∑ .
Note that when 0 1 Z % is null, then, from equation ( 26) we find that:

(27) ( ) ( ) ( )

1 1 t t E S E Z E x = % and ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 1 1 t t t Var S E x Var Z Var x E Z = + % %
These results can then be used to obtain approximate prices for the premium to be charged (based on mean-variance rules) when the counting process is persistent. Note however that the variance of this process has increased over time due to the process persistence. Further, based on these moments, the Value at Risk (VaR) risk exposure can be determined which would use these two moments as a first approximation (although higher order moments can be calculated as well using ( 26), the generating function of the persistent Compound Process).

Some Numerical Results

A numerical analysis of our equations will reveal some of the characteristics of a persistent process. As expected, the mean evolution of the persistent process has an almost linear growth as indicated in our equation. In the long run, the variance turns out to be also almost linear, as it is the case for random walks. However, persistence ( ρ >0) has the effect of increasing the variance as shown in Figure-1 below. In the short term however, the variance evolution is nonlinear as our equations have indicated.

Interestingly, the rate of change in variance is not constant and growing over time which indicates a "persistent volatility". Of particular interest is the evolution of the third moment of the persistent claim distribution. Initially, it was increasing (over 4 periods) and subsequently declining (although remaining positive for ρ positive). When ρ is negative we note that for the first few periods the evolution of the mean and the variance are indeed nonlinear. This is particularly the case for the variance as shown in Figure 2 below. In this Figure, the variance initially declines, then increases and again decreases.

Finally, it converged to a linear growth. This behavior is indicative of the short term effects of memory on the stochastic process as indicated earlier. In Figure 3, we note the divergence in the growth of volatility when the persistence parameter is negative. Finally, the third moment is positive which demonstrate that the distribution is skewed reflecting the process memory-

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

12 persistence. This latter observation is particularly important for it may be used to explain partly the skew of certain time series, presuming that this skew is due to the short term memory effects prevalent in such series (for example, in financial time series). Similarly, numerical analysis of some of the first persistent probabilities reveal a certain cyclicality, a function of the peristent indexr. As shown in Figures 4 and5, the evolution of the claim probability is complex when there is persistence (memory) compared to that without persistence, oscillating initially and subsequently converging.

Conclusion

The persistent counting process in insurance is both of theoretical and practical importance. Such a process, appropriately structured and based on the relevant data that insurance companies possess may be used as an underlying process with respect to which premiums, risk exposure and risk management procedures may be determined when persistence is inherent to the insurance environment. While applications to insurance were emphasized, the results obtain here are equally applicable to numerous problems where counting processes are persistent. Further research on the implications of this approach to multiple periods memories as well as the effects of memory on skewness and process cyclicality are needed however. The numerical results and the moments calculations in this paper have indicated however that persistence matters and leads to numerous stochastic phenomena that are observed in practice and hardly explained.

( , ) 1 = 1 1 1 (1 ) t t t t t t t t x x x x y y y y F t E E E E λ λ λ λ λ α λ λ β - - - - - + = = = = = = + -.
As a result, we can write:

0 0 1 1 ( , ) ( , 1) 1 ( , ) ( , 1) (1 ) F t F t F t F t λ λ α β λ λ λα λ β - - ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ - - ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Or in a matrix notation:

0 1 ( , ) 1 , , ; ; , ( , ) (1 )
F t F t λ α β λ λ λ λ λα λ β - ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ - ⎝ ⎠ ⎝ ⎠ t F( t) = A F( 0) A = F( t)
Note that the matrix eigenvalues μ are a solution to: ( ) ,0 1 ( 1).

2 1 4 , 0,1 2 2 i a a i λρ μ λ ρ λ ρ + = ± + - = Or ( ) ( ) 2 0 2 1 1 4 , 2 2 1 4 2 2
x x x x F E Px F E Px λ λ λ λ λ = = = = = = = = While ( ) { } ( ) ( ) ( ) 1 1 0 0 1 0 ,1 1 (1 ) ,0 ,0 . x x F E F F λ λ α λ β λ = = = - +
And therefore ( )

0 0 0 ,1 (1 ) ( 0) ( 1). F Px Px λ α λ β = - = + =
Which we rewrite conveniently by replacing β by 1 α ρ --, or:

( ) ( )

0 0 0 ,1 (1 ) ( 0) 1 ( 1). F Px Px λ α λ α ρ = - = + -- = Similarly, ( ) ( ) ( ) 1 0 1 ,1 ,0 (1 ) ,0 . F F F λ λα λ λ β λ = + -
And therefore ( ) ( )

2 1 0 0 ,1 ( 0) ( 1)
F v v F v v λ λ μ μ = + = +
These lead to the following solutions:

0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 ( ,1) ( ,0) ( ,1) ( ,0) = .
( ,0) ( ,1) ( ,0) ( ,1) .

F F F F u F F F F u λ λ μ λ λ μ μ μ δ λ μ λ λ μ λ μ μ δ - - = - - - = = -
By the same token:

1 0 1 1 1 1 0 1 ( ,0) ( ,1) ( ,1) ( ,0) , F F F F v v λ μ λ λ λ μ δ δ - - = = 0 0 1 0 0 0 0 0 1 1 0 1 1 1 1 1 0 1 ( ,1) ( ,0) ( ,0) ( ,1) ( , ) ( ,0) ( ,1) ( ,1) ( ,0) ( , ) t t t t F F F F F t F F F F F t λ λ μ λ μ λ λ μ μ δ δ λ μ λ λ λ μ λ μ μ δ δ - - = + - - = + Since, 0 1 ( , ) ( , ) ( , ) t x G t E F t F t λ λ λ λ ⎡ ⎤ = = + ⎣ ⎦
, we obtain the following probability generating function:

( )( ) ( ) ( ) 0 1 0 1 1 0 0 1 0 0 1 1 ( ,1) ( ,1) ( ,0) ( ,0) ( , ) ( ,0) ( ,0) ( ,1) ( ,1) . t t F F F F G t F F F F λ λ λ λ μ λ μ δ λ λ μ λ λ μ δ ⎛ ⎞ + - + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + - + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
Or expressed in terms of the known initial conditions

0 1 0 1 ( ,0) ( ,1) ( ,1) ( ,0) ( , ) . t t G G G G G t λ μ λ λ λ μ λ μ μ δ δ ⎛ ⎞ ⎛ ⎞ - - = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠
Which can be rewritten by: ( ) ( )

1 1 0 1 0 1 0 1 ( ,0) ( ,1) ( , ) . t t t t G G G t λ μ μ λ λ μ μ μ μ δ δ - - = - - -
Note that 0 1 μ μ λρ = and thereby, in a recursive form, we have:

1 0 1 ( ,1) ( ,0) ( , ) , t t t t t G G G t λ λ ρ λ λ μ μ δ δ - = Ψ - Ψ Ψ = -
As stated in the proposition. In addition:

( ) ( ) ( )

0 1 0 0 ,0 ,0 ,0 ( 0) ( 1) G F F Px Px λ λ λ λ = + = = + = and ( ) ( )( ) 0 0 0 ,1 ( 0) ( 1) 1 ( 1) G aP x P x P x λ λ λλ α ρ = = + = + - + = As stated in Proposition 1. ∴ ( ) ( ) ( ) ( ) ( ) 0 1 0 0 0 1 t t t i i i i i i E S E Z P y E Z P y = = = = + = ∑ ∑ % % However, ( ) ( ) ( ) ( ) 0 0 1 1 1 1 and , 1 i i E Z E Z E Z E Z i = = ∀ ≥ % % % % , consequently, ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 1 0 0 1 1 1 t t t i i i i E S E Z P y E Z P y = = = - = + = ∑ ∑ % % And ( ) ( ) ( ) ( ) ( ) ( ) 0 1 0 1 1 1 0 ( 1) 1 t t i i E S t E Z E Z E Z P y = = + + - = ∑ % % % But ( ) ( ) 0 1 t t i i E x P y = = = ∑ which proves (2).

Q

In order to calculate the variance of the Compound persistent claim process we first state the following Lemma:

Lemma 1:

Conditionally on , t x k = the Compound Claim t S is distributed as: 1 1 0 1 1 k t k i i i i Z Z + - = = + ∑ ∑ % % Proof:
Let 0 1 , ,..., t i i i be a series assuming values 0 and 1 and let k be the number the index j assumes a value 1,

1 j i = .
Conditionally on 0 0 1 1 , ,..., . The first sum consists of k terms while the second consists of t+1-k terms.

As a result, their sum is distributed as defined in the Lemma.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 0 0 1 1 1 0 0 1 0 1 1 1 1 1 2 2 1 0 1 1 ( 1)
2 1 2 1 0 1 1 1 1 2 1 1 1 0 1 0 1 1 1 1 1 1 1 1 2 1 0 1 0 1 1 1 1 0 1 0 0 1 1 1 1 ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( 1) ( 1) The proof is based on Lemma 1 and follows a procedure similar to Lemma 2. Namely, (

)

1 1 0 0 1 0 1 1 1 1 1 1 1 0 1
+ - - + + + - - - - + - - ⎛ ⎞ ⎡ ⎤ ⎡ ⎤ = = = ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ ⎡ ⎤ ⎢ ⎥ ⎡ ⎤ = ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ % % % % % % % % %
Further, it is evident that:

( ) ( ) ( ) Q

  Then, for independent (Poisson) counting processes, the mean claims up to time t are:

Figure 1 :Figure

 1 Figure 1: Persistence and Variance of the Counting process

Figure 4 :.Figure 5 :

 45 Figure 4: The Probability of a Claim, ρ =-0.3

  v are parameters calculated by the initial conditions we have defined previously. As a result, we have:

  desired result stated in the Lemma above.

( , ) 1 and ( , ) 1

And therefore,

% % are independent of t y , we have:

2( 1)

As stated in the Lemma.

Q


Fabian M Suchanek
email: fabian@xsuchanek.name

Aparna S Varde
email: vardea@xmontclair.edu

Richi Nayak
email: r.nayak@xqut.edu.au

Pierre Senellart
email: pierre@xsenellart.com

The Hidden Web, XML and the Semantic Web: Scientific Data Management Perspectives

Keywords: H.2.8 [Database Management]: Database Applications -Scientific databases, H.3.3. [Information Search and Retrieval] -Search process Design, Human Factors, Performance Deep Web, Domain-Specific Markup Languages, Hidden Web, Multidisciplinary work, Scientific data, Semantic Web, XML

The World Wide Web no longer consists just of HTML pages. Our work sheds light on a number of trends on the Internet that go beyond simple Web pages. The hidden Web provides a wealth of data in semi-structured form, accessible through Web forms and Web services. These services, as well as numerous other applications on the Web, commonly use XML, the eXtensible Markup Language. XML has become the lingua franca of the Internet that allows customized markups to be defined for specific domains. On top of XML, the Semantic Web grows as a common structured data source. In this work, we first explain each of these developments in detail. Using real-world examples from scientific domains of great interest today, we then demonstrate how these new developments can assist the managing, harvesting, and organization of data on the Web. On the way, we also illustrate the current research avenues in these domains. We believe that this effort would help bridge multiple database tracks, thereby attracting researchers with a view to extend database technology.

INTRODUCTION 1.1 Background and Motivation

The Web has revolutionized the way we seek information. It covers domains as diverse as education, entertainment, business, and even health. It has evolved into a publishing medium, a global information repository and a global electronic market. From a user's perspective, the Internet is mainly a source of Web pages. Yet, there is much more to the Web than HTML pages. Much of its content lies in databases behind the Web. This content, called the hidden Web, is accessible only through Web forms or through specific Web services. It is estimated that the hidden Web contains orders of magnitude more data than the visible Web. Yet, despite its providing structured data, it is not obvious how to access this data by a machine. In order to make use of this wealth of information, Web forms have to be found and filled in, and the requested answers have to be analyzed and understood. Through the medium of this tutorial, we show the newest research trends in the area of the hidden Web and tell the story of various early successes. We also focus on the Web developments in two other exciting areas, namely XML and the Semantic Web.

The increasing use of XML is an interesting trend on the Web. XML, the Extensible Markup Language, is a W3C standard for the exchange of semi-structured data. Through XML, it has become possible to transfer data between systems as diverse as databases, Web services, semantic knowledge bases or end user applications in one common file format. Yet, XML by itself defines just the abstract syntax, the hull of the information. It does not standardize the semantics or the structure of the data. This is the role of domain-specific markup languages (DSMLs). In this tutorial, we explain the design principles of DSMLs and how they blend into XML. We first give an overview of the most established DSMLs, thereby diving into the realm of real-world XML as it is used on the Web. We then also show how various mining techniques can be applied to XML data for better knowledge discovery.

The latest development on the Web is the Semantic Webthe newly growing structure of computer-processable meaningful information. With its W3C standards RDF and OWL, the Semantic Web aims to unify the way semantic information is stored and exchanged, making it possible for one system to not just read, but also -understand‖ the data from another source. Our tutorial provides basic literacy with these standards. We also show where the Semantic Web has already taken offwith example knowledge bases from the science domain, the music domain or indeed general common sense knowledge. Still, the Semantic Web is in its infancy. Numerous challenging research questions, e.g., in areas of information reconciliation and knowledge representation, still long to be answered.

Goals

The overall goal of this tutorial is to show how all these diverse trends can be used for data management. As an example, we choose the scientific domain. Interdisciplinary research has attracted tremendous interest in recent years with the emerging need for computer scientists to work across domains. This has led to significant development of techniques and standards that use computational principles to solve challenging real-world problems in specific domains, with new theories being proposed as needed. In the last decade, a large part of this research has occurred in the field of scientific domains. Management of scientific data has accordingly become an exciting interdisciplinary field of study. Moreover, in this age of globalization, it is imperative to share the information with communities of scientists and related professionals from all over the world through the Web. We therefore shed light on new methods of scientific data management on the Web. On the way, we illustrate the various research questions in these domains that are still unanswered.

We aim to use various examples from scientific data throughout including markup languages specifically developed for scientific domains. For instance, we show how data about Cédric Villani (recipient of the Fields Medal 2010) and his works can be stored and retrieved using each of these technologies. Data on his research, not so obvious from the surface Web, can be found using intentional and extensional approaches in the hidden Web. Meaningful data extraction about him, including personal information that requires some human reasoning, can be carried out using Semantic Web technologies. XML and its domainspecific markup languages such as MathML, can store additional information, such as mathematical data, in an expressive Webbased format.

These technologies are by no means finalized. This tutorial thus endeavors to highlight the open issues in these domains and point out the hot spots of today's research. It would thus encourage future work across these areas, opening new avenues for research and development, especially propelling further research in areas that blend these technologies.

Outline

This tutorial is organized in three parts, each one focusing on one aspect of scientific data management. The first part delves into the hidden Web (also known as the deep Web or invisible Web), explaining how it can be used for storing and retrieving scientific information. The second part focuses on XML data management along with domain-specific markup languages (DSMLs), particularly with respect to modeling and mining scientific data. The third part presents the details of the Semantic Web with emphasis on its usefulness for managing scientific data.

SCIENTIFIC DATA MANAGEMENT 2.1 The Hidden Web

It is impossible to reach the whole content of the World Wide Web by just following hyperlinks. Rather, some Web pages create their content on demand if the user fills out and submits a Web form. These forms are typically interfaces to databases stored on Web servers, which are usually not directly accessible. A reason behind this may be that the Web site owner wants to keep control on how the information is accessed. Some hidden Web pages are -Yellow pages‖ services and other kinds of directories; others are library catalogs, weather forecast services, geo-localization services, commercial product catalogs or administrative sources such as the database of the United States census bureau.

As an example in the scientific data management domain, let us consider the following task: -Retrieve all articles published by Cédric Villani.‖ This information is most likely available from the surface Web, but a classical Web crawler would have to explore a large number of documents, and then use some heuristics or classification techniques to determine the relevant ones. This is a time-consuming and imprecise process. On the other hand, using services of the hidden Web, this becomes a very simple task: simply use the advanced querying capabilities of a publication database, such as Google Scholar.

The question is how users can benefit from this vast source of information. How can they discover it, index it, and be able to use it automatically for a given task? We review some ways to tackle these challenges. We focus here especially on unsupervised and semi-supervised approaches for knowledge discovery over the hidden Web such as [START_REF] Senellart | Automatic Wrapper Induction from Hidden-Web Sources with Domain Knowledge[END_REF], more adapted to the scale and diversity of the Web than supervised methods. Most current approaches can be roughly classified [START_REF] Ru | Indexing the invisible Web: A Survey[END_REF] into extensional or surfacing strategies (retrieving information from the hidden Web and storing it locally to process it) and intentional or integration strategies (analyzing services to understand their structure, store this description, and use it to forward users' queries to the services). By covering these two avenues, we give an overview of the most important current approaches of data management on the hidden Web.

XML and DSMLs

With the continuous growth in XML-based data sources, especially in scientific applications, the ability to manage collections of XML documents becomes increasingly important [START_REF] Li | Online mining of frequent query trees over XML data streams[END_REF]. In such applications, it is useful to capture the reasoning process of domain experts, especially in modeling and mining the data. Modeling of XML documents, which are semistructured, requires finer details than unstructured (text) documents and fully structured documents. XML allows the representation of semi-structured and hierarchal data containing not only the values of individual items but also the relationships between data items. Due to the inherent flexibility of XML, in both structure and semantics, mining of XML documents significantly differs from structured data mining and text mining [START_REF] Kutty | Frequent Pattern Mining on XML documents[END_REF]. Consequently, management of XML data is faced with new challenges as well as benefits [START_REF] Rusu | Mining Association Rules from XML Documents[END_REF][START_REF] Wan | Mining Association rules from XML data mining query[END_REF].

A considerable advancement in the general area of XML is the advent of domain-specific markup languages (DSMLs). These languages are designed such that they follow the syntax of XML and encapsulate the semantics of the concerned domain in order to cater to the needs of targeted user bodies. Accordingly, they serve as communication standards in the respective domains. The storage of data in this format has several advantages such as more meaningful information retrieval and knowledge discovery. This is particularly useful in scientific applications where it is crucial to incorporate the domain experts' perspective in various realworld situations.

We discuss the issues in managing XML-based scientific data. We address XML modeling and mining, focusing on both structure as well as content of XML documents. We give details such as vector, tensor, tree and graph structure representation for XML data, and their usefulness in scientific data management. We cover DSMLs with emphasis on scientific domains, giving illustrative examples from MML (Medical Markup Language) [START_REF] Guo | The Latest MML (Medical Markup Language) -XML based Standard for Medical Data Exchange / Storage[END_REF] and MathML (Mathematical Markup Language) [START_REF] Carlisle | Mathematical Markup Language (MathML)[END_REF]. We explain the steps in DSML development [START_REF] Varde | XML Based Markup Languages for Specific Domains[END_REF], the desired properties of DSMLs, and the application of XML constraints to preserve semantic restrictions [START_REF] Davidson | Propagating XML Constraints to Relations[END_REF]. Furthermore, we describe how the XML model used for data storage facilitates information retrieval with languages in the XML family such as XQuery [START_REF] Boag | XQuery 1.0: An XML Query Language[END_REF]. We also explain the advantages of using classical data mining techniques such as association rules and clustering over XML documents in conjunction with DSMLs to enhance knowledge discovery, especially from scientific data. Suitable examples from domains such as medicine are provided here [START_REF] Nayak | Fast and Effective Clustering of XML Data Utilizing their Structural Information[END_REF][START_REF] Varde | XML Based Markup Languages for Specific Domains[END_REF]. Finally, open research questions are addressed pertaining to structure and content mining and a greater synergy between XML and DSMLs.

Semantic Web Technologies

The Semantic Web project envisions that people will publish semantic information in a computer-processable formalism that allows the information to be globally interlinked. For this purpose, the World Wide Web Consortium (W3C) has developed the knowledge representation formalisms RDF [START_REF]RDF Primer, W3C Recommendation[END_REF] and OWL [START_REF]OWL Web Ontology Language (W3C Recommendation 2004-02-10[END_REF]. These formalisms are based on XML, but go beyond it by specifying semantic relationships between entities and even logical constraints on them. A collection of world knowledge in these formalisms is commonly called an ontology [START_REF]Handbook on Ontologies, 2nd edn[END_REF]. In this tutorial, first give an introduction to semantic knowledge representations and ontologies in general. We also explain the knowledge representation formalisms RDFS and OWL, their syntax and semantics. We explain the vision and the applications of the Semantic Web project.

We then show how the Semantic Web can be used for scientific data management. We introduce different existing semantic resources for scientific data, from publication databases such as DBLP, which have been published in RDF, to general purpose ontologies such as YAGO [START_REF] Suchanek | YAGO: A Core of Semantic Knowledge[END_REF] or DBpedia [START_REF] Auer | DBpedia: A nucleus for a web of open data[END_REF], each containing thousands of scientists and scientific concepts and theories. Several large-scale ontologies are available online and are interlinked in the spirit of the Semantic Web.

We also explain how this information was gathered from different sources and how it can be queried using the SPARQL query language [START_REF]SPARQL Query Language for RDF[END_REF].

CHALLENGES

We highlight some of the challenging issues leading to current research questions in each of these Web developments addressed in this work.

In the area of the hidden Web, although there are techniques to derive the type of input parameters and output records of a Web form, it is still challenging to get the precise semantics of a service. If a service, given a person, returns a year, how do we know whether this is the birth date, death date, or graduation date? Moreover, dealing with complex forms (such as those used to access specialized scientific databases), especially when there are dependencies between form fields, required and optional fields, etc., is a fully open problem by itself. Finally, a knowledge discovery system for the hidden Web that would not cover specific Web sites or a specific domain of interest but would work at the scale of the whole Web is still to be constructed.

In the area of XML and DSMLs, some of the current issues and challenges in managing scientific data are: 4. The Semantic Web can also be mined, for example to find sub-and super-class relationships or to find schema information.

With the presentation of these challenges, we aim at encouraging even more research in the areas of the hidden Web, XML, and the Semantic Web, to further enhance scientific data management.

CONCLUSIONS

This tutorial would show that developments in the areas of the hidden Web, Semantic Web, XML and DSMLs have influenced the current state of the Web by improving the way different computer applications and services communicate, and relevant information is identified, especially in scientific domains.

Most of the data on the Web is so unstructured that only humans can understand it. However, the volume of the data is so huge that only machines can process it. This paradox highlights the importance of techniques -in between‖ the extremes: semistructured languages such as XML, the Semantic Web and the hidden Web. This tutorial sees itself as an invitation to researchers to make use of these technologies on one handand to contribute to these developments on the other hand. We believe that there is exciting research potential in bridging the different new developments on the Web.

  One of the main open research questions is the reconciliation of different semantic conceptualizations in different ontologies. Techniques such as record linking and matching carry over from the database world, but present themselves in the new light of the Semantic Web, with more information and constraints available to support the mapping process. 2. Another challenge is the growing of the Semantic Webbe it through community work, by converting existing databases into RDF, or by Information Extraction. 3. Reasoning on Web scale is likewise still an open issue. How can we apply automated reasoners on huge, potentially noisy data sets?

	1. Effectively modeling both structure and content features for
	XML documents to adequately represent scientific data and
	investigating how DSMLs can be useful here
	2. Combining structure and content features in different types of
	data models which do not affect the scalability of the mining
	process
	3. Integrating background knowledge of scientific processes in
	XML mining algorithms and harnessing DSMLs here
	4. Developing new standards as needed especially to foster
	knowledge discovery by synergizing XML and DSMLs
	The Semantic Web is still relatively young. Numerous challenges
	still wait to be solved:
	1.

* The work was partially funded by the ERC grant Webdam.

AUTHOR BIOGRAPHIES


Arnaud Hamon
email: arnaud.hamon]@irccyn.ec-nantes.fr

Yannick

Y Aoustin

Compliance for a cross four-bar knee joint

Keywords: Humanoids, Knee joints with springs, Cross four-bar mechanism, Parametric optimization

Compliance for a cross four-bar knee joint

Introduction

The biomechanics researchers have done important progress in the comprehension of the human lower limb and especially on the knee joint 1 and the ankle joint. 2 Indeed, these two joints have a complex structure formed by non symmetric surfaces. They can produce more complex movements than a simple revolute joint. In addition to the flexion in the sagittal plane, there are an internal rotation with displacement of the Instantaneous Center of rotation(ICR) of the knee joint and a posterior translation of the femur on the tibia. These motions cannot be represented by one or two revolute joints. Different studies have confirmed these results by an observation of the movements of the human knee in the 3D space. 3 From these studies, a new kind of prosthetic knee is appeared, called polycentric knee. A classical polycentric knee is the four-bar linkage, 4 which used in most prosthetic knees. This structure forms a closed mechanism, which allows a combined rotation and translation of the knee joint in the sagittal plain without using of artificial ligaments to keep the rigidity of the mechanism. The dimensions of the four-bar structure can be chosen by measurement, on a real subject, of the length of the anterior and posterior ligaments, and the position of the cross ligament attachments on the tibia and the femur, projected in the sagittal plain in the maximum extension position. 5 This choice of the dimensions produces similar motions in the sagittal plain of the knee than those obtained with the human knee. 6 Our objective is to improve the bipedal robot performance by a new conception of the knee joint. Several papers devoted to bipedal robots equipped with complex knees, like G. Gini et al. [START_REF] Gini | Human-oriented biped robot design : insights into the development of a truly antropomophic leg[END_REF] which used knee joints based on the human knee surfaces. F. Wang et al. [START_REF] Wang | Design and implementation of coordinated control strategy for biped robot with heterogeneous legs[END_REF] developed a bipedal robot with two different joints, a rotoide joint and a four-bar joint. However, a four-bar structure presents a configuration of singularity which limits the flexion of the knee contrary to a cross four-bar solution for which the singular position is out of the range of use of the knee joint. In, [START_REF] Hamon | Cross four-bar linkage for the knees of a planar bipedal robot[END_REF] we have proved a cross four-bar joint is a better solution for the knee than a rotoide joint in term of energy consumption. In this paper, we improved the study by adding a compliance in the knee with different solutions of springs.

q 5 q 3 q 2 q 1 q 4 q p1 q p2 Γ p1 Γ p2 Γ 2 Γ 3
Fig. 1. Diagram of a planar bipedal robot. Absolute angular variables and torques.

2. Effect of the springs on the knee joints on the energy consumption.

In this paper, we compare the effect of a cross four-bar knees with spring units on the energy consumption of the biped during a cyclic walking gait. We use different springs : a torsion spring in parallel of the actuator of the knee or two extension springs on the two side of the cross four-bar knee. Indeed, several papers pay interest to the effect of springs equipping the bipedal robot joints. [START_REF] Yang | [END_REF] We produce a set of optimal trajectories for the bipedal robot where the stiffness coefficient is an optimization parameter. The stiffness coefficient of the springs depends on the walking velocity and can be different for the support leg and for the swing leg. This choice implies the spring units used, have a variable stiffness. We can note, the work of S. Wolf, 11 which designs a rotoide joint with a control of the stiffness coefficient. Different types of springs can be used for the knee joint. We can see on Fig. 3, the energy consumption according to the walking velocity of the bipedal robot equipped of cross four-bar knees without springs and with the two solutions of springs. We observed the biped equipped of cross fourbar knees with extension springs is more efficient than the biped equipped of simple cross four-bar knees for the walking velocity lower than 2.5 Km/H. Moreover, the biped equipped of cross four-bar knee with torsion spring have a similar energy consumption than the biped with cross four-bar knees with extension springs for the walking rates lower than 2 Km/H. For the higher walking velocities the cross four-bar knees with torsion springs is the better solution of knee.

Conclusions

We have proposed a mechanical structure for the knee of a planar bipedal robot called cross four-bar knee and different solutions of actuation with spring units. We have produced with a parametric optimization method a set of optimal reference walking trajectories for a bipedal robot for different solutions of knee structures. We have compared the energy consumption of the bipedal robot for the different structure of the knees. This study proved that the cross four-bar knee with spring units can reduce the energy consumption of the biped. The perspective of this study is to extend this work for a 3D bipedal robots. Moreover a measure of the evolution of center of rotation of the human knee during a walking gait will be used to choose more precisely the dimension of the cross four-bar mechanism.

Fig. 2 .

 2 Fig. 2. Details of the cross four-bar joint and position of the Instantaneous Center of Rotation (ICR)

Fig. 3 .

 3 Fig.3. Evolution of the energy consumption according to the walking rate for the cross four bar knee without springs and with extension springs and torsion springs.


Anne Bilgot
email: anne.bilgot@ujf-grenoble.fr

Laurent Desbat
email: laurent.desbat@imag.fr

Valérie Perrier
email: valerie.perrier@imag.fr

Filtered backprojection method and the interior problem in 2D tomography

Keywords: Tomography, Filtered backprojection (FBP), local data, interior problem

We address here the interior problem in local tomography, by means of filtered backprojection (FBP). This algorithm, traditionally used in the context of complete data, is usually not considered as valuable for the interior problem. However, in this article, we prove that as well as more sophisticated methods, the FBP algorithm ensures that the difference between the original and the reconstructed functions is continuous. We verify numerically that the FBP method can supply satisfactory images of discontinuities (on Shepp and Logan phantom and real data). Nevertheless, we also show limits of FBP, by pointing up examples on which the dependence on exterior structures damages the reconstruction quality in the Region Of Interest.

Introduction and Notations

Filtered Backprojection Method

Let first introduce some notations and classical formulas: let S 1 denotes the unit circle in R 2 ; the Radon Transform of f ∈ L 1 (R 2 ) is defined by

∀(Θ, s) ∈ S 1 × R, R θ f (s) def = Rf (Θ, s) def = {x∈R 2 |x•Θ=s} f (x)dx
where "•" is the euclidean inner product, θ ∈ [0, 2π[, Θ = (cos θ, sin θ). The projectionslice theorem states that, for ω ∈ R,

R θ f (ω) = √ 2π f (ωΘ),
where for all n ∈ N * , the Fourier Transform of f ∈ L 1 (R n ) is normalized in the following way:

for k ∈ R n , f (k) = 1 √ 2π n R n f (x)e -ix•k dk
This leads to the well-known filtered backprojection inversion formula [START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF]:

f (x) = 1 √ 2π 3 π 0 R R θ f (ω)|ω|e iωx•Θ dωdθ
which can be reformulated as [START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF]:

f (x) = 1 4π 2 R # (g) (x)
where

R # (g) (x) = 2π 0 g(Θ, x • Θ)dθ and g(Θ, s) = vp 1 s * ∂ s R θ f (s) = lim ǫ→0 + u∈R;|s-u|>ǫ ∂ s R θ f (u) s -u du 1.

The interior problem

Let a (0 < a < 1) be the radius of the Region Of Interest (ROI), which, here, is assumed to be also the region of exposure. It is well known [START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF][START_REF] Maass | The interior radon transform[END_REF][START_REF] Louis | Incomplete data problems in x-ray computerized tomography[END_REF], that the interior problem, i.e. the reconstruction of f from

(Rf ) int = Rf (Θ, s); Θ ∈ S 1 , s ∈ [-a, a]
is not possible. More precisely, the interior problem is not uniquely solvable. Indeed, let us suppose that we apply the inverse of the Radon transform R -1 to the interior data only, in order to estimate f in the ROI. The set of non-considered data in the interior problem will be denoted by:

(Rf ) ext def = Rf (Θ, s); Θ ∈ S 1 , |s| > a .
As illustrated in Fig 1:

• left: when a structure with support included in the ROI is considered, the Radon transform has only interior components (Rf = (Rf ) int );

• right: when a structure with support outside the ROI is considered, the Radon transform has exterior components ((Rf ) ext ), but also interior components ((Rf ) int = 0).

Therefore, when a function f = f int 2D +f ext 2D with structures both in the ROI (f int 2D ) and outside the ROI (f ext 2D ) is dealt with, the available data in the interior problem are made of two components:

(Rf ) int = Rf int 2D + (Rf ext 2D ) int
and the inversion of these data using global inversion operator, denoted here by R -1 , leads to the reconstruction of the following function:

f loc = R -1 ((Rf ) int ) = R -1 (Rf int 2D )+R -1 ((Rf ext 2D ) int ) = f int 2D +R -1 ((Rf ext 2D ) int )
Thus, in the ROI, the initial function f int 2D is corrupted by R -1 ((Rf ext 2D ) int ) which generally does not cancel in the ROI in 2D. This bias is present in all even dimensions and observable in reconstructions.

In [START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF] the variations of R -1 ((Rf ext 2D ) int ) are shown to be low. Moreover, the singular value decomposition [START_REF] Maass | The interior radon transform[END_REF] and the wavelet analysis [START_REF] Olson | Wavelet localization of the Radon transform[END_REF][START_REF] Berenstein | Wavelets and local tomography[END_REF] show that R -1 ((Rf ext 2D ) int ) has low frequency in the ROI. Nevertheless, the existence of this bias usually motivates the search for other reconstruction approaches, such as singularity or detail local reconstruction methods.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 exposition Figure 1.
Interior problem in 2D parallel geometry: on the left, the object is entirely included in the ROI (dashed circle): the Radon transform reduces to interior data. On the right, the object lies in the exterior of the ROI: the Radon transform data have not only exterior contributions, but also interior contributions.

Local methods for the interior problem

Local methods aim at "reconstructing discontinuities" in the ROI (in the sense: to reconstruct a function with discontinuities exactly located as in the initial function, and sometimes, even with the same jumps). The legitimacy of such a purpose relies on a theorem of micro-local analysis, established by Quinto [START_REF] Quinto | Singularities of the X-ray transform and limited data tomography in R 2 and R 3[END_REF]: this theorem links the singular supports (which correspond to discontinuity curves) and more precisely the wavefronts (which additionally give the normal directions) of f and Rf (see [START_REF] Quinto | Singularities of the X-ray transform and limited data tomography in R 2 and R 3[END_REF] for details).

The literature provides several approaches to the interior problem:

• Geometrical approaches (see for instance [START_REF] Ramm | The Radon Transform and Local Tomography[END_REF][START_REF] Thirion | Segmentation of tomographic data without image reconstruction[END_REF]), but singularities are highly difficult to detect in the data.

• Wavelet based-methods where separable [START_REF] Farrokhi | Wavelet-based multiresolution local tomography[END_REF][START_REF] Berenstein | Wavelets and local tomography[END_REF] or non separable [START_REF] Bonnet | Tomographic reconstruction using nonseparable wavelets[END_REF] multiresolution analysis are recovered using local data: the 2D wavelet transform of a function f can be computed from a 1D wavelet transform of its Radon transform Rf ; it can be seen as a generalization of the FBP algorithm. The locality of the wavelet filtering enables to reconstruct wavelet coefficients of the function f up to a given scale, according to the range of availability of data [START_REF] Olson | Wavelet localization of the Radon transform[END_REF][START_REF] Delaney | Multiresolution tomographic reconstruction using wavelets[END_REF], (fine details, and therefore discontinuities, are thus visible). See also [START_REF] Holschneider | Inverse Radon transforms through inverse wavelet transforms[END_REF][START_REF] Bilgot | Méthodes locales d'identification de surfaces de discontinuité à partir de projections tronquées pour l'imagerie interventionnelle[END_REF] for an alternative wavelet approach.

• Λ-Tomography (see for instance [START_REF] Faridani | Introduction to local tomography[END_REF]) where one reconstructs Λf instead of f (the pseudo-differential operator Λ, defined by Λf (k) = |k| f (k), is locally recoverable, and known to preserve the discontinuities);

• Pseudolocal Tomography [START_REF] Katsevich | Pseudolocal tomography[END_REF], where one copes with the non-locality of the ramp filter by truncating it in the direct domain. We detail this method below.

In the following, we show that the classical FBP method is also relevant in this context.

Truncated formulas for truncated data

In the following the filtered backprojection applied to interior data is called Truncated Filtered Backprojection or TFBP. From now, we suppose that f is compactly supported in the open unit disk Ω of R 2 .

Pseudo-local tomography

This method has been designed by A. Katsevitch and A. Ramm [START_REF] Katsevich | Pseudolocal tomography[END_REF]. The ramp filter is truncated in the direct domain so as to filter local data with a filter restricted to an interval [-d, d] (where d > 0 and for treatment of local data d << a); thus, instead of reconstructing f with

f (x) = 1 4π 2 2π 0 lim ǫ→0 + s∈R;ǫ<|x•Θ-s| ∂ s R θ f (s) x • Θ -s ds dθ
one reconstructs the function f d , defined by:

f d (x) = 1 4π 2 2π 0 lim ǫ→0 + ǫ<|x•Θ-s|<d ∂ s R θ f (s) x • Θ -s ds dθ
f d can be reconstructed locally, in the sense that the reconstruction of f at point x only requires the knowledge of Rf across the disk of center x and radius d. To identify the information on f present in f d , the authors introduce the function f C d , difference between f d and the reference function f , defined by: f

C d (x) = f (x) -f d (x)
and they show the following theorem.

Theorem 2.1 (Discontinuities and pseudo-local tomography [START_REF] Katsevich | Pseudolocal tomography[END_REF]) For all d > 0, the function f C d is continuous in the direct domain. It follows that for all d > 0, the discontinuities of f d are exactly the same as those of f (location as well as amplitude). The proof of this result is based on continuity properties of integrals with parameters, and can be found in [START_REF] Katsevich | Pseudolocal tomography[END_REF]. In this approach the filtering steps do not use all the range of data that are available in (Rf ) int (the filter is the same wherever it is applied).

Truncated FBP

In the context of TFBP, we introduce the following reconstructed function:

f a TFBP (x) = 1 4π 2 2π 0 Λ a TFBP R θ f (x • Θ)dθ where Λ a TFBP R θ f (s) = lim ǫ→0 + s-ǫ -a ∂ s R θ f (u) s -u du + a s+ǫ ∂ s R θ f (u) s -u du
In contrast with pseudo-local tomography, all available data, a priori, contribute to the value of f a TFBP (x), ∀x ∈ Ω. Whereas in pseudo-local tomography the same filter of constant width is applied to all projections, in Truncated Filtered Backprojection the filtering operator is only a filter when applied to the truncated data completed by extrapolated data (applied on truncated projections only, it is shift-variant).

The question that should be addressed now is: what can be learnt about f according to f a TFBP ? In [START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF], Natterer proposes to complete the interior data Rf by consistency into the whole domain. Applying the FBP to these completed data, he proves that the reconstructed function only differs from the original one up to an essentially constant function (in the framework of C ∞ -functions).

We introduce the difference between the original function f and f a TFBP :

(f a TFBP ) C (x) = 1 4π 2 2π 0 -a -1 ∂ s R θ f (s) x • Θ -s ds + 1 a ∂ s R θ f (s) x • Θ -s ds dθ (1)
and we prove in the following that (f a TFBP ) C is a continuous function in the ROI. Therefore the discontinuities of f a TFBP are exactly the same as the discontinuities of f within the ROI, just as in pseudo-local tomography. A close result is obtained independently in [START_REF] Yu | Compressed sensing based interior tomography[END_REF] with different and stronger assumptions (the exterior data are supposed to be smooth), using different arguments.

Theorem 2.2 (Discontinuities and Truncated Filtered Backprojection) For all a > 0, the function (f a TFBP ) C is continuous in the disk {x ∈ R 2 ; |x| ≤ r}, for all r such that 0 < r < a.

Proof : First, remark that for all f ∈ L 2 (Ω), ( 1) is well-defined; indeed, if f ∈ L 2 (Ω), then, for all θ ∈ [0

; 2π], R θ f ∈ H 1 2 ([-1; 1]), and ∂ s R θ f ∈ H -1 2 ([-1; 1]). (1) rewrites (f a TFBP ) C (x) = 1 4π 2 R # vp 1 s * χ [-a,a] ∂ s R θ f (x) Since χ [-a,a] ∂ s R θ f belongs also to H -1 2 ([-1; 1]) then vp 1 s * χ [-a,a] ∂ s R θ f remains in H -1 2 ([-1; 1]), which leads finally to R # vp 1 s * χ [-a,a] ∂ s R θ f ∈ L 2 (Ω).
Integration by parts on (1) (we recall that R θ f (-1) = R θ f (1) = 0 for all θ) yields: 2 ds dθ Let us now consider the closed disk D r , of radius r, strictly included in the region of exposure (ie r < a), and a point x within D r . In the following, we check that the conditions that are required to apply the results about continuity of integrals with parameters are fulfilled, so as to prove that (f a TFBP ) C is continuous at point x. For the first integral in (2), we have:

(f a TFBP ) C (x) = 1 4π 2 2π 0 R θ f (s) x • Θ -s -a -1 + -a -1 R θ f (s) (x • Θ -s) 2 ds + R θ f (s) x • Θ -s 1 a + 1 a R θ f (s) (x • Θ -s) 2 ds dθ = 1 4π 2 2π 0 R θ f (-a) x • Θ + a - R θ f (a) x • Θ -a dθ (2) + 1 4π 2 2π 0 -a -1 R θ f (s) (x • Θ -s) 2 ds + 1 a R θ f (s) (x • Θ -s)
• for all θ ∈ [0, 2π[, the functions x → R θ f (-a) x • Θ + a and x → R θ f (a) x • Θ -a are continuous in D r (because for each θ, x • Θ ± a never vanishes, as |x • Θ| ≤ |x| ≤ r < a);
• moreover for all x ∈ D r , -r < x • Θ < r, we have 0 < ar < x • Θ + a, and then

R θ f (-a) x • Θ + a ≤ |R θ f (-a)| a -r
which is a function (of θ), integrable on [0, 2π]. A similar argument can be used for the function x → R θ f (a)

x • Θa ; it follows that the first integral of ( 2) is a continuous function of x in D r .

For the second integral in (2), we have:

• for each couple (θ, s) ∈ [0, 2π[×[-1, -a] (resp. each couple (θ, s) ∈ [0, 2π[×[a, 1]), the function x → R θ f (s) (x • Θ -s) 2 is continuous in D r (because x•Θ-s never vanishes); • moreover for all x ∈ D r , for all (θ, s) ∈ [0, 2π[×[-1, -a], 0 < -r + a ≤ -r -s ≤ x • Θ -s therefore R θ f (s) (x • Θ -s) 2 ≤ R θ f (s) (a -r) 2
the r.h.s. being a function of (θ, s), integrable on [0, 2π[×[-1, -a].

A similar proof can be made for the third integral of (2): the sum of the second and the third integral is thus a continuous function of x in D r .

Discussion and tests

First experiments

Example 1: Shepp and Logan phantom We first show reconstruction results obtained by Truncated Filtered Backprojection applied to Shepp and Logan phantom (displayed on the first line of figure 2, with a zoom-in on the ROI displayed on the right). We fix a > 0 and simulate the truncated Radon transform of the phantom:

∀(Θ, s) ∈ S 1 × [-1, 1], g(Θ, s) = Rf (Θ, s) if |s| ≤ a 0 else (3)
The results are displayed on the left of figure 2 (second and third line). The initial image is the square [-1, 1] 2 sampled on 256 2 pixels. The radius of the ROI is a = 0.25, and the data are acquired on 450 equiangular projections on [0, π[, each of them sampled with a step 2 256 . There is only a bias, roughly constant in the ROI, between the phantom and the reconstruction, and as Theorem 2.2 asserts, the discontinuities of the phantom are clearly visible in the reconstruction. Although this reconstruction is satisfactory if one aims at localizing discontinuities, it is common in local tomography literature to use a procedure to reduce artefacts at the ROI boundary [START_REF] Farrokhi | Wavelet-based multiresolution local tomography[END_REF][START_REF] Bonnet | Tomographic reconstruction using nonseparable wavelets[END_REF]: it consists in erasing discontinuities at the truncated sinogram boundary, by extending by continuity interior data in a constant way: ∀Θ ∈ S 1 , if s < -a then g(Θ, s) = Rf (Θ, -a), if s > a then g(Θ, s) = Rf (Θ, a), else g(Θ, s) = Rf (Θ, s). Theorem 2.2 is still valid: discontinuities of the reconstruction are the same as the ones of the phantom. The results are displayed on the right of figure 2 (second and third line). The artefacts in the reconstruction are clearly reduced at the borders at the ROI boundary, as well as the amplitude of the bias in the ROI. Discontinuities are still clearly visible, but the visual similarity (as regards gray levels) with the phantom is much improved: local reconstruction really "looks like" the phantom.

Example 2: Real data We have then applied TFBP technique on real data (human trabecular bone sample), kindly provided by F. Peyrin [START_REF] Apostol | Relevance of 2d radiographic texture analysis for the assessment of 3d bone micro-architecture[END_REF], and acquired on the medical line of ESRF Grenoble (European Synchrotron Radiation Facility). The projections were sampled on 1024 pixels of 15 µm, for 900 directions. The length of the medical line is 145 m, thus we consider that parallel beam conditions are fulfilled. Results are displayed on figure 3. They are fully satisfactory: even small details in the ROI are well localized in the reconstruction.

Some limits

Both previous examples tend to show that TFBP is a very satisfactory method for the interior problem. In order to assess the range of efficiency of TFBP, we have tried to build phantoms for which TBFP behaviour is less convincing. As explained in paragraph 1.2, the bias created in the ROI in case of local data comes from exterior structures. We thus compare influences of two kind exterior structures, by designing two phantoms where only exterior structures are present. Results are displayed in figure 4. It thus appears that in case of Shepp and Logan phantom, the different contributions of external structures tend to compensate in the ROI, thus creating a bias which is roughly constant, and thus leading to visually very satisfactory local reconstruction results. On the contrary, for the second phantom, we have put structures with high density, non-symmetrically distributed around the ROI. The bias implied in the ROI has no discontinuity (as theory proves), but has significant decay in the ROI (with even an inflection point). Therefore, the identification of discontinuities in the ROI is not compromised, but dissymmetry is forced in the ROI. Consequently, in case of such exterior structures, two similar structures in the ROI can be reconstructed in two different ways: such an example is given in figure 5 where disks are not reconstructed with the same gray levels whether they are close to the white exterior rectangles or not. In such a case, the reconstruction image tends to show that the two structures have different densities, whereas they actually have the same.

Discussion and conclusion

In this paper we have studied the behavior of "Truncated Filtered Backprojection" (that is to say FBP applied on interior problem data). As for pseudo-local tomography, we have proved that the difference between the reference function and the reconstructed function is continuous on the ROI. This supplies a justification to an experimental intuition: this simple method provides with very satisfactory results if one is only interested in the location of discontinuities in the reconstruction. Furthermore, we have shown that on some examples, such as the well-known Shepp and Logan phantom, this method goes further: the visual similarity between the two images can be very high. Nevertheless, we have also shown that this is not true for all images: TFBP method remains strongly dependent on exterior structures. As shown on an example, this similarity can be significantly damaged.

The main advantage of TFBP in comparison with other local methods is its simplicity: conventional FBP algorithm, easily available, can be used without any modification. Thanks to the theoretical result stated in this paper, TFBP, like Λtomography and pseudo-local tomography, provides a reconstructed function which has the same discontinuities as the reference function. TFBP has exactly the same complexity as FBP; it is well-known that the most expensive step is the backprojection, present in all considered local methods. Therefore, the complexity argument is not very significative. The main drawback of TFBP is its still unpredictable "qualitative performance", due to its might-be dependence on exterior structures. For a given function f , other local algorithms, which implement truly local filters, reconstruct in the ROI the same function whether the sinogram is complete or not; it is not the case for TFBP.

In conclusion, if not only discontinuities but also symmetry preservation is crucial in the application, other local methods should be considered. If only discontinuity localization is sufficient, the TFBP algorithm is a very simple and efficient method.

Figure 2 .

 2 Figure 2. Results of TFBP applied on interior data on Shepp and Logan phantom.First line, the phantom, with, on the right, a zoom-in on the ROI. Second and third line, on the left: reconstructions from raw local data (exterior data are zero), and on the right, reconstructions from data extended by continuity. Third line, in both cases, a zoom-in on the ROI of the reconstruction and comparisons between the phantom and the reconstruction horizontal cross-sections are displayed (the darker line stands for the reconstruction). Note that the values of the jumps are well estimated.

Figure 3 .

 3 Figure3. FBP-reconstruction results on real data (human trabecular bone sample data, kindly provided by F. Peyrin, ESRF Grenoble[START_REF] Apostol | Relevance of 2d radiographic texture analysis for the assessment of 3d bone micro-architecture[END_REF]). These data were acquired at ESRF (European Synchrotron Radiation Facility) on the medical line. The projections were sampled on 1024 pixels of 15 µm, for 900 directions. The similarity between the reconstruction from global data (left) and local data (right) is fully satisfactory in the ROI, in particular for small details.

Figure 4 .

 4 Figure 4. Comparisons of the influence of two kinds of exterior structures on the TFBP-reconstruction within the ROI. Phantoms are displayed on the first line, and the reconstructions obtained by TFBP applied on interior data, extended by continuity, are displayed on the second line. For the Shepp and Logan phantom (left), only a constant bias is created. For the second phantom with high density exterior structures localized only on some sides of the ROI, the bias has significant decay across the ROI: a dissymmetry is thus created in the ROI.

Figure 5 .

 5 Figure 5. Example where exterior data have influence on the TFBP-reconstruction in the ROI. The phantom is displayed on the first line, with a zoom-in on the ROI, the reconstruction obtained by TFBP on the second line, and the comparison of crosssections on the third line. A significant dissymmetry is created in the ROI by TFBP.

Acknowledgments

The authors would like to acknowledge Françoise Peyrin for fruitful discussions and for providing the ESRF data presented in figure 3.