Christian Merkle

Degradation model for erbium-doped fiber amplifiers to reduce network downtime

Keywords: EDFA, aging curve, OpEx

The outage of optical components like optical amplifiers and cross connects reduces the availability of a network and increases the operational expenditures of network operators. Responsible for this is the repair process of optical components which takes very long, because of the large distances in the field. The detection of performance degradation of erbium-doped amplifiers (EDFAs) can be used to reduce the repair time by changing an EDFA before it fails. With the knowledge of the degradation state the remaining life time of an EDFA can be calculated, enabling the operator to plan the replacement of EDFAs to avoid longer outage times. In this paper, an 1,000 km link and a German backbone network are used to simulate the aging of EDFAs with aging curves. Additionally the operational expenditures are considered to compare the cost of the repair process of EDFAs.

Introduction

In today's optical networks EDFAs are widely used because they can amplify many signals on different wavelength at the same time. The most commonly used amplifier in transmission systems is the erbium-doped fiber amplifier (EDFA) [START_REF] Ramaswami | Optical Networks -A Practical Perspective[END_REF]. Such amplifiers may fail during their operation, for example due to aging effects of the pump laser, and lead to transmission failure on a link. Due to the high bandwidth which are used on backbone network links today an outage of an amplifier leads to a high loss of data packets transmitted over the failed link. This also has an impact on the revenues of the provider, who has to pay a penalty to the customers affected by the link outage. Hence, network operators monitor their networks to get information of the current network status. If a failure occurs in a backbone network, an alarm is sent from the network management system to the network operator center (NOC). Based on the failure alarms, the operator tries to localize the failure first and solves it afterwards. Today optical amplifiers are not monitored by the management system. The outage of an EDFA takes very long because the technician has to travel to the amplifier to repair it.

In this paper the monitoring of the aging of an EDFA is addressed which enables the operator to plan the repair process and to replace the EDFA before it fails. Thus, the repair process can be reduced which increases the network availability. In section ?? the aging of the EDFA performance is discussed and in section 4 two different models, a 1,000 km link and a Germany reference network, are discussed to compare different change strategies for EDFAs. In section 5 the operational expenditures (OpEx) are analyzed for the repair process.

Related Work

A detailed description of the functionality of an EDFA can be found in [START_REF] Ramaswami | Optical Networks -A Practical Perspective[END_REF]. An EDFA amplifies the signal by using a pump signal from a laser, typically at a wavelength of 980nm or 1480nm. But there are several imperfections that a system designer need to worry about when using an amplifier in a system. One disadvantage is the degradation of the performance due to aging of the components. An algorithm how the aging effects of an EDFA can be detected is described in [START_REF] Rapp | Quality Surveillance Algorithm for Erbium-Doped Fiber Amplifiers. Design of Reliable Communication Networks[END_REF]. The basic concept of the algorithm is to calculate the pump diode current that would be needed to create the measured gain at begin of life (BOL) conditions and to compare the results with the actual pump current. A pump laser of an optical amplifier is used to get a constant amplifier gain. With the monitoring of the actual pump current and the data from the data sheet of an amplifier, a network provider has a rough indication of the remaining life time of an amplifier.

The planning of the repair process is important for a network operator to reduce the OpEx of the network. In [START_REF] Verbrugge | Methodology and input availability parameters for calculating OpEx and CapEx costs for realistic network scenarios[END_REF] a methodology to calculate CapEx and OpEx of a telekom operator is described. Different input parameters like staff cost, energy cost, and the description of the repair process and the provisioning process are identified to calculate the OpEx of a German reference network. The OpEx calculation for the German reference network indicates that the repair processes are the second largest cost factor of the OpEx of a network. Hence, with the optimization of the repair process, the OpEx of a telecom operator can be reduced.

In [START_REF] Machuca | Expenditures Study for Network Operators[END_REF] a total cost of the network ownership model is presented. The paper describes the different factors which contribute to the capital expenditures and operational expenditures. The OpEx are defined as the cost related with the operation of a network. Several types of cost are described which can be distinguished for the reparation sub-process of a network failure. These factors are used in section 5 when the cost of the EDFA repair process are described.

EDFA Degradation model

In this section the degradation of the pump current of an EDFA and the dimensioning of the pump current threshold are described. As mentioned previously, every amplifier requires a pump laser to bring the erbium ions into a higher energy band to amplify an incoming WDM signal [START_REF] Ramaswami | Optical Networks -A Practical Perspective[END_REF]. Due to aging effects of the amplifier, the pump power of the amplifier has to be increased to obtain a constant gain. In this paper the pump current is used for the description of the pump power. The aging of an EDFA can be described by aging curves as shown in Fig. 1. The curves show the necessary pump current of an EDFA over a certain time period to achieve the same amplifier gain. There are different aging curves because not every EDFA has the same aging characteristic. It is possible that some EDFAs degrade faster than others. The second characteristic of the aging curves is that they have no constant gradient over the life time of an EDFA. At the begin of life the pump current gradient is higher than at the end of life. The begin of life pump current shown in Fig. 1 was taken from a data sheet of an EDFA. The used pump current of the EDFA at the begin of life is 550 mA. Beside the BOL pump current an EDFA has also a maximum pump current up to which it can be operated without it fails. This parameter is also given by the data sheet of a manufacturer of an EDFA. Here, the maximum pump current which can be used is 670 mA. With the algorithm described in [START_REF] Rapp | Quality Surveillance Algorithm for Erbium-Doped Fiber Amplifiers. Design of Reliable Communication Networks[END_REF] the actual needed pump current can be calculated. With this knowledge the pump current threshold which indicates that the amplifier has to be replaced can be calculated as the difference between the BOL current and the actual pump current. The network operator has to define the pump current threshold for the EDFAs. The value of the threshold depends on the remaining life time an operator wants to have for the EDFAs. When an EDFA reaches the threshold an alarm message is send to the network operation center (NOC) to inform the operator that it has to be replaced. The advantage of having a remaining life time for an EDFA is that the network operator does not need to change the EDFA immediately and can plan the repairing process. If an EDFA has a remaining life time of one year, for example, the operator can wait if further EDFAs reach also their degradation threshold in a certain time period after the first EDFA signals its degradation state. Hence, the repair team in the field has the possibility to repair more than one EDFA in one shift. Also the network availability is increased because the traveling time of the repair team does not account to the outage time of an EDFA anymore. Considering only the link breaks is not sufficient to analyze the availability of a backbone network. Also the repair time of the EDFAs and, hence, the downtime of the link are important factors for the network availability. An analysis of the repair time and link breaks is presented in section 4.

Another goal for the operator is to reduce the OpEx of backbone networks. One part of the OpEx is the repair process of failed network components. If the repair time of an EDFA can be planned by an operator, the traveling time can be optimized which reduces the traveling overhead and also the cost of the repair process. Further cost savings can be achieved due to the reduction of spare parts and stock size. The knowledge about the remaining life time of an EDFA can be used to order a new unit not until an old unit reaches the degradation threshold. Hence, only a minimum of spare parts is needed to repair an unforeseen outage of an EDFA for example. The OpEx of the repair process are discussed in section 5.

The signaling of the degradation threshold of an EDFA can be done with a network management protocol like the Simple Network Management Protocol (SNMP). All EDFAs in a network periodically send their actual pump current to a NOC. If an EDFA reaches its degradation threshold, it sends an additional message to the NOC to inform the operator that the degradation threshold was reached. The time interval of the EDFA aging messages are not constant over the EDFA life time. At the begin of life the management messages are sent in a larger interval and when the actual pump current is closer to the degradation threshold the management messages are sent more frequently.

Simulation Setup and results

In this section two different simulation scenarios are analyzed. First, a link with a length of 1000km is considered to simulate two different strategies to replace degraded EDFAs. In the second simulation scenario, a German 50 node reference network is used to analyze the repair process in a larger network. For both scenarios an EDFA spacing of 80 km is used. To simulate the aging of the pump current of an EDFA the curves in Fig. 1 are used. In the beginning of the simulation, one of the aging curves is randomly assigned to every EDFA. The replacement of an EDFA corresponds to a random allocation of a new aging curve.

Consideration of a 1000km network link

For the analysis of the 1,000 km link 12 EDFAs are used. These 12 EDFAs do not have the same aging process and, hence, degrade in different rates. The fiber at the link also has a failure rate which is given in failure in time (FIT). For the simulations 380, 22 F IT km are used for the fiber. One FIT means one failure per 10 9 hours. It is assumed that two repair teams are at one NOC which is located in the middle of the link at 520 km. Hence, every repair team has to maintain one half of the 1,000 km. The spare parts are also located at the NOC, hence, there is no additional traveling to a stock required. In the following two different replace strategies are analyzed for the 1,000 km link.

In the first simulation scenario, only the EDFAs which reached the degradation threshold are changed either at the end of every year or if a fiber fails. If the link is down due to a fiber failure, it is expedient to also replace all ED-FAs which received the degradation threshold because further downtimes due to EDFA changes are avoided. For the simulations a time period of 30 year is considered. In Fig. 2 the total number of changed EDFAs and the number of link breaks in 30 years are shown. Three different pump current thresholds were taken for the simulations. In average 67 EDFAs have to be changed in 30 years when a threshold of 620 mA for the pump current is used. The replacement of these 67 EDFAs lead to 25 link breaks in 30 years which means that there was a link break almost yearly. If a threshold of 660 mA is used for the pump current 4 link breaks occur in 30 years and lead to 13 replaced EDFAs. As expected the higher threshold leads to less replaced EDFAs and, hence, to less link breaks. The disadvantage of a higher threshold is that the rest life time of the EDFAs is shorter compared to threshold of 620 mA. Because the operator does not know how the aging curve of the EDFA exactly looks like, he can only determine a time interval when the EDFA will fail with a certain probability.

A further change strategy of the EDFAs was simulated to compare it with the results of the first scenario. In the second scenario all EDFAs at the link are changed if one EDFA reaches the degradation threshold or the fiber fails. In Fig. 3 again the average number of replaced EDFAs and the number of link breaks are shown. For a threshold of 620 mA, the average number of changed EDFAs is 84. Compared with the values of the first scenario it can be seen that, as expected, more EDFAs were changed in 30 years. The advantage of the second change strategy appears if the number of link breaks is considered. The 84 replaced EDFAs lead to 7 link breaks. Hence, in the second scenario the number of link breaks is reduced about 72% in comparison to the first scenario. The number of link breaks are further reduced if the the threshold is increased. For a threshold of 660 mA the number of link breaks is 2.

These two cases illustrate that the replacement of more than one EDFA on one link at the same time can reduce the number of link breaks. Hence, if one EDFA has reached its degradation threshold it can be an advantage also to replace other EDFAs on the same link. Therefore, the current value of the threshold of the other EDFAs on the same link can be an indication which of the EDFAs the operator should replace additionally.

To further evaluate the different replace strategies the repair time to replace degraded EDFAs is calculated. For both scenarios, a degradation threshold of 620 mA is assumed. In the first scenario only the EDFAs are replaced which have reached their degradation threshold. The repair time t Repair for both scenario is given by the maximum of the repair times of both teams and the travel time t EDF A X -EDF A Y between the EDFAs.

t Repair = max(t RepairT eam1 , t RepairT eam2 ) + t EDF A X -EDF A Y (1)
The repair time of each EDFA is 2h, as described previously. The traveling time to the first EDFAs can be neglected for both scenarios because it is assumed that the link is disabled not until the repair team has arrived the first EDFA. The travel time t EDF A X -EDF A Y depends on the distances between the EDFAs that have to be replaced. For the first scenario, an average distance of 200km was obtained. Hence, the repair time t Repair is 171.42 h in 30 years. When all EDFAs are changed if one or more EDFA have reached their degradation threshold, the total repair time is 126 h. In this model the travel distance is constant and has a length of 420km. The result shows that the repair time and also the downtime of the link is reduced by 26.5% in comparison to the first scenario. Hence, the availabilty of the network can be increased if always all EDFAs are changed on one link. Besides the downtime also the OpEx of network has to be considered. The OpEx of the two scenarios is discussed in the next section.

Germany 50 node reference network

In this section the Germany 50 node reference network [START_REF]SNDlib: Germany 50 node reference network[END_REF] is used to compare the EDFA degradation model with a FIT rate model. The reference network consists of 50 nodes and 87 links and has an average link length of 131 km. The total link length of the network is 8,874.13 km and the number of used EDFAs is 241.

There are two NOCs in the network, each responsible for half of the 241 EDFAs.

For the simulations a time period of 100 years was used to analyze the total number of replaced EDFAs and the number of EDFAs which can be repaired at the same shift. The difference between the two models is the time when an EDFA is replaced. By the FIT rate model the EDFAs are replaced after they failed as it is done in networks today. The following assumptions are used for both models: the number of required technician to repair an EDFA is 1 and it takes 2 hours to repair and test an EDFA [START_REF] Verbrugge | Methodology and input availability parameters for calculating OpEx and CapEx costs for realistic network scenarios[END_REF].

Simulation with FIT rate In this model the number of EDFA outages for a EDFA FIT rate of 2,850 [START_REF] Mello | Availability Maps for Connections in WDM Optical Networks[END_REF] are simulated. The simulation results of this model serve as reference values for the EDFA degradation model which is analyzed in the next section. The failure probability of an EDFA in the time interval t is given by:

P (T out ≤ t) = 1 -exp(-λ F IT × t) (2)
Hence, the probability that exact one EDFA fails within one year is 2.47% and the probability that at least one EDFA fails within one year is 99.76%. This failure probabilty leads to 5.77 failed EDFAs per year in average. The average repair time is calculated with equation 3

tRepair = N k=1 (t N OC-EDF A k + t repair ) N (3)
and is 6.69 h for each EDFA. N is the number of failed EDFAs, t N OC-EDF A k is the traveling time from the NOC to the EDF A k , and t repair is the repair and test time of an EDFA. With this values a whole repair time of 3,857.97 h in 100 years is obtained.

Simulation with degradation threshold By using the pump current threshold described in section 3, an EDFA can be changed before it fails. This allows the planning of the repair process and the reparation of more than one EDFA in one working shift. This means that a repair team can plan a route between the NOC and the EDFAs they have to repair. The goal of planning the repair process is to repair as much EDFAs as possible without traveling back to the NOC. An example for a repair process with two degraded EDFAs is illustrated in Fig. 4 and in Fig. 5. In Fig. 4 the two EDFAs are repaired independently from each other. The repair team must always travel from the NOC to the EDFA and back. This is the case in the FIT rate model described above. In Fig. 5 the repair process was planned in advance and the two EDFAs can be repaired by one repair team in one cycle. In order that it is worth to repair the two EDFAs in on cycle the travel time t travel must be shorter than the sum of the single traveling times to the single EDFAs as shown in Fig. 4. The repair time for an EDFA is the same for both cases and can be neglected. The traveling time for the repair process in Fig. 5 is given by following equation

t travel = t N OC-EDF Ax + x × t EDF Ax-EDF Ay + t EDF Ay-N OC (4)
The whole travel time t travel of the technician is the sum of the traveling time from the NOC to the first EDFA t N OC-EDF Ax , the traveling time between the EDFAs t EDF Ax-EDF Ay , and the traveling time from the last EDFA back to the NOC t EDF Ay-N OC . When the degradation curves in Fig. 1 1. Major results for the German reference network is higher then for the FIT rate model. This is caused because the EDFAs are replaced before they fail. A further result is that it was only possible to repair two EDFAs in one shift because of the large distances between a NOC and the EDFA. If more than two NOCs will be used for the simulation it should be also possible to repair three EDFAs in one shift. The small number of NOCs is also a reason for the small number of years in which two EDFAs can be repaired in one shift. The advantage of the degradation model can be seen by considering the repair times of the EDFAs. Because the EDFAs are changed before they fail the traveling time does not account to the total repair time. Hence, the total repair time for the EDFAs is only the repair time of the replaced EDFAs and is 1544 h per 100years. This means that the repair time and hence, the downtime of the network is reduced by 60% in comparison to the FIT rate model. But in the degradation model, the traveling time is higher because more EDFAs had to be replaced than for the FIT rate model. As described above it was possible to repair two EDFAs in one shift in 42% of the 100 years and this lead to a reduced travel distance of 26,520 km. With an average speed of 70 km h the saved traveling time is 378.86 h. Hence, the whole travel time is 3,036.43 h per 100years for the degradation model. Thus, the travel time is increased by 10.95%.

OpEx analysis of the repair process

In this section the operational expenditures (OpEx) of the repair process of the EDFAs are discussed. For the OpEx calculations, only the Germany reference network is considered. Important for the cost calculation are the basic salary of a technician, the repair time for an EDFA, and the traveling time in the field.

For the calculation, it is assumed that the stock is located at the NOC and the repair team must not drive there to collect the spare parts. In the following, the FIT rate model and the degradation models are analyzed.

OpEx analysis for EDFA outages given by the FIT rate

To calculate the cost of the repair process following equations are used:

c Repair = (c Salary × t repair ) + c EDF A + c stock (5)
C Stock = 0.05 × P EDF A × N EDF A (6)
The cost of the repair process is the sum of the cost for the technician, the cost of the EDFA, and the cost of the stock. Further cost like gas are neglected for OpEx calculations. The basic salary of a technician is assumed to be US $36,19 per hour. This value is based on a network technician salary per year of US $37.000 [START_REF]Salary Survey for Job: Computer/Network Support Technician[END_REF]. The cost of an EDFA is US $11.850 [START_REF] Verbrugge | Methodology and input availability parameters for calculating OpEx and CapEx costs for realistic network scenarios[END_REF] and the cost of maintaining the stock is assumed to be 5% of the equipment cost and is given in eqn. 6. How much the operator has to pay to maintain the stock depends on the number of EDFAs in the stock and the price of an EDFA. The analysis of the FIT model has shown that the repair time is 3,857.97 h and that 557 EDFAs had to be replaced per 100 years. Because in average 6 EDFAs are replaced per year it is assumed that 6 backup EDFAs are in the stock. With these values and the equations above the total cost of the repair process are US $70,955.69 per year.

OpEx analysis for the EDFA degradation concept

For the degradation model the same equations are used to calculate the cost of the repair process. It is assumed that only one EDFA is in the stock because the operator has enough time to order a new EDFA if one needs to be replaced. With the values obtained by the analysis of repair time, the cost of the repair process are US $87,570.15 per year. As it can be seen from the results, the OpEx of the FIT rate model are minor then those of the degradation model. But it has to be mentioned that the penalties that a operator has to pay if he cannot fulfill the service level agreements are not included in the OpEx calculations. Furthermore, the selected pump current threshold for the simulations was 620 mA which means that the EDFAs have in average a rest life time of 6 years. If a higher threshold is used the OpEx for the degradation model should be minor.

Conclusion

A degradation concept for EDFAs has been presented that allows the change of EDFAs before they fail. Every EDFA has a specific aging curve of its pump current which is used to determine the remaining life time of an EDFA. The remaining life time depends on the pump current threshold which is set by the network operator. The evaluation of the link model shows that the replacement of all EDFAs on one link reduces the downtime of the link by 26.5%. Furthermore, the degradation model was compared with a FIT rate model of EDFAs where the EDFAs are only replaced when they fail. This was done by using a Germany reference network. The results show that for the degradation model the downtime of network links can be reduced by 60% because the traveling time to the EDFAs can be neclected. But the former change of the EDFAs leads to a higher number of changed EDFAs in comparison to the FIT rate model. Finally, the OpEx for the repair process of the Germany reference network are analyzed. The results show that the repair cost of the degradation model is higher, but service level agreements penalties were not considered in the calculation which increase the OpEx of the FIT model.

Fig. 1 .

 1 Fig. 1. Different aging curves of EDFAs

Fig. 2 .Fig. 3 .

 23 Fig. 2. Number of EDFA outages and link breaks for scenario 1

  are used, the average Two EDFAs repaired in one shift total number of replaced EDFAs is 7.2 per year and in 42% of the 100 years it was possible to repair two EDFAs in one shift. In the other years, the distance between the replaced EDFAs is to large to repair it in the same shift. The results are shown in table1. For the degradation model, the number of replaced EDFAs

	NOC		NOC
	EDFA	EDFA
	Failed EDFA	Signaling EDFA
	Fig. 4. EDFAs repaired independently Fig. 5. Parameters FIT rate model Degradation model
	Outage probability	2850	aging curves	-
	Replaced EDFAs per year	5.77	7.2	-
	Total travel time	2703.97h/100years 3036.43h/100years +10.95%
	Total repair time	3857.97h/100years 1546h/100years	-60.03%
	Table

Acknowledgement The author is grateful to Lutz Rapp (Siemens AG) for insightful discussions about the aging of an EDFA.


MSc Sarrazin Samuel

PhD (e Poupon Cyril

PhD Julia Linke

PhD, Phillips Wessa Michèle

PhD Delavest Marine

MD, PhD (g Versace Amelia

MD, PhD (g Almeida Jorge

PhD (e,i), Duclap Guevara Pamela

PhD (e Mangin Jean-François

Le Dudal

PhD Daban Claire

MD Hamdani Nora

MD Albis Marc-Antoine

MD Leboyer Marion

MD Houenou Josselin
email: josselin.houenou@inserm.fr

MD Josselin Houenou

A multicenter tractography study of deep white matter tracts in bipolar disorder: psychotic features and interhemispheric disconnectivity

Importance. Tractography studies investigating white matter abnormalities in patients with bipolar disorder have yielded heterogeneous results due to small sample sizes. This limits their generalizability, a critical issue for neuroimaging studies of biomarkers of bipolar disorder.

Objective. To study white matter abnormalities using whole-brain tractography in a large international multicenter sample of patients with bipolar disorder type 1 and to compare these alterations between patients with or without psychotic features during mood episodes.

Design.

A cross-sectional multicenter international q-ball imaging tractography study comparing patients with bipolar disorder 1 to healthy controls, and patients with bipolar disorder with a past history of psychotic features to patients without such a history. Setting. University hospitals. Participants. 118 patients with bipolar disorder and 86 healthy controls. Intervention. Participants were assessed using the Diagnosis Interview for Genetic Study at the French site or the Structured Clinical Interview for DSM-IV at the German and US sites. Diffusion-weighted magnetic resonance images were acquired using the same acquisition parameters and scanning hardware. Q-ball imaging tractography and an automatized segmentation technique were used to reconstruct 22 known deep white matter tracts. Main Outcome Measures. Generalized fractional anisotropy along each reconstructed white matter tract.

Results. Compared to controls, patients with bipolar disorder had significant reductions of mean generalized fractional anisotropy along the body and the splenium of the corpus callosum, the left cingulum and the anterior part of the left arcuate fasciculus, when controlling for age, gender and acquisition site (corrected for multiple testing). Patients with a past history of psychotic features had a lower mean GFA value than those without, along the body of the corpus callosum (corrected for multiple testing).

Conclusions.

In this multicenter sample, patients with bipolar disorder 1 had reduced white matter integrity in interhemispheric, limbic and arcuate white matter tracts. Interhemispheric pathways were more disrupted in patients with psychotic symptoms than in patients without psychotic symptoms. Taken all together, these results highlight the existence of an anatomical disconnectivity in bipolar disorder and further underscore a role for interhemispheric connectivity in the pathophysiology of psychosis in BD.

I-Introduction.

White matter (WM) abnormalities have been widely involved in the pathophysiology of bipolar disorder (BD) [START_REF] Mahon | A role for white matter abnormalities in the pathophysiology of bipolar disorder[END_REF] , mainly using diffusion tensor imaging (DTI) studies [START_REF] Vederine | A meta-analysis of whole-brain diffusion tensor imaging studies in bipolar disorder[END_REF] . DTI uses the properties of water molecule motion to provide insights into the microscopic structure of brain tissues, i.e. with FA, a quantitative index that reflects the integrity and coherence of WM [START_REF] Beaulieu | The basis of anisotropic water diffusion in the nervous system -a technical review[END_REF] Few studies have applied tractography methods to BD despite several advantages. Indeed, the virtual reconstruction of entire WM tracts provides comprehensive anatomical information and allows for anatomically driven hypothesis testing on changes of FA. Despite discrepancies [START_REF] Houenou | Increased white matter connectivity in euthymic bipolar patients: diffusion tensor tractography between the subgenual cingulate and the amygdalo-hippocampal complex[END_REF][START_REF] Torgerson | DTI tractography and white matter fiber tract characteristics in euthymic bipolar I patients and healthy control subjects[END_REF] , most studies in BD reported decreased FA values along tracts linking regions involved in emotion processing: the uncinate fasciculus [START_REF] Versace | Elevated serum measures of lipid peroxidation and abnormal prefrontal white matter in euthymic bipolar adults: toward peripheral biomarkers of bipolar disorder[END_REF][START_REF] Benedetti | Tract-specific white matter structural disruption in patients with bipolar disorder[END_REF][START_REF] Lin | Abnormal frontal cortex white matter connections in bipolar disorder: a DTI tractography study[END_REF][START_REF] Mcintosh | White matter tractography in bipolar disorder and schizophrenia[END_REF] , the anterior thalamic radiations [START_REF] Lin | Abnormal frontal cortex white matter connections in bipolar disorder: a DTI tractography study[END_REF][START_REF] Mcintosh | White matter tractography in bipolar disorder and schizophrenia[END_REF] and the cingulum [START_REF] Versace | Elevated serum measures of lipid peroxidation and abnormal prefrontal white matter in euthymic bipolar adults: toward peripheral biomarkers of bipolar disorder[END_REF][START_REF] Emsell | Limbic and Callosal White Matter Changes in Euthymic Bipolar I Disorder: An Advanced Diffusion Magnetic Resonance Imaging Tractography Study[END_REF] . Such impairments are thought to underpin the emotional dysregulation observed in the patients [START_REF] Phillips | A neural model of voluntary and automatic emotion regulation: implications for understanding the pathophysiology and neurodevelopment of bipolar disorder[END_REF] and are further considered to be a relevant candidate biomarker for bipolar disorder [START_REF] Mahon | A role for white matter abnormalities in the pathophysiology of bipolar disorder[END_REF] .

Tractography studies in BD suffer from relatively small sample sizes (at most 40 patients with BD [START_REF] Mcintosh | White matter tractography in bipolar disorder and schizophrenia[END_REF] ), thus limiting the generalizability of their conclusions. Multicenter imaging studies including larger samples can help to address these issues [START_REF] Hallahan | Structural magnetic resonance imaging in bipolar disorder: an international collaborative mega-analysis of individual adult patient data[END_REF] . Furthermore, a relatively larger sample allows for the disentangling of a heterogeneous sample of patients by focusing on a clinical feature of interest. This possibility is worth being exploited in a complex clinical condition like BD. On this note, at least 50% of patients with BD experience psychotic features during acute illness phases [START_REF] Dunavevich | Prevalence and description of psychotic features in bipolar mania[END_REF] . Despite substantial overlap in psychotic features and genetic susceptibility between patients with BD and schizophrenia , growing evidence from both clinical [START_REF] Marneros | Bipolar I disorder with moodincongruent psychotic symptoms: a comparative longitudinal study[END_REF] and genetic [START_REF] Goes | The genetics of psychotic bipolar disorder[END_REF] studies support the view that BD with psychotic features can be seen as a homogeneous subtype of BD. However, despite the high prevalence of BD with psychotic features, whether this subtype relating to different WM alterations when compared to non-psychotic BD remains poorly explored. Thus, we designed a large multicenter whole-brain tractography study in patients with BD I. Our main objective was to probe microstructural properties of 22 major deep WM tracts, covering the whole brain, in a large sample of patients with BD and controls using Qball imaging (QBI) tractography. Our secondary objective was to investigate whether these disruptions were different in subsamples of patients with or without a past history of psychotic features.

II-Methods.

Participants

Adult inpatients and outpatients with BD type I (DSM-IV-R) were recruited from university-affiliated participating centers: APHP Henri Mondor hospitals, Créteil and Fernand-Widal hospital, Paris, France; Western Psychiatric Institute and Clinic, UPMC, Pittsburgh, USA and the Central Institute for Mental Health, Mannheim, Germany. Controls, recruited from media announcements and registry offices, had no personal or family history of axis I mood disorder, schizophrenia or schizo-affective disorder. All subjects were clinically assessed by trained raters using the Diagnosis Interview for Genetic Study at the French sites [START_REF] Nurnberger | Diagnostic interview for genetic studies. Rationale, unique features, and training. NIMH Genetics Initiative[END_REF] and the Structured Clinical Interview for DSM-IV at the German and US sites [START_REF] First | Structured Clinical Interview for DSM-IV-TR Axis I Disorders, Research Version[END_REF] and the Montgomery-Asberg Depression Rating Scale [START_REF] Montgomery | A new depression scale designed to be sensitive to change[END_REF] or the Hamilton Depression Rating Scale-17 [START_REF] Hamilton | A rating scale for depression[END_REF] , the Young Mania Rating Scale [START_REF] Young | A rating scale for mania: reliability, validity and sensitivity[END_REF] and the National Adult Reading Test 21 at all of the sites. Symptomatic patients were defined as having a score > 7 on the mood scales. Past history of psychotic features was defined as at least one manic or depressive episode with delusions or hallucinations (DSM-IV-R). Exclusion criteria for all subjects comprised of: the history of neurological disease or head trauma with loss of consciousness and MRI contraindications.

Twenty-four patients (20.34%) had participated in a previous single center tractography study [START_REF] Versace | Elevated serum measures of lipid peroxidation and abnormal prefrontal white matter in euthymic bipolar adults: toward peripheral biomarkers of bipolar disorder[END_REF] with different aims, DTI sequence and processing pipeline. Local ethics committee of each center approved the study and after complete description of the study to the subjects, written informed consent was obtained.

Data acquisition

To minimize between-sites bias, diffusion-weighted (DW) images and T1-weighted images were obtained for all subjects using the same hardware in the 3 MRI acquisition sites (MRI Siemens Magnetom TrioTim 3T Syngo MR B17 (Siemens Medical Solutions), 12channel head-coil). The MRI protocol included a high-resolution T1-weighted acquisition (echo time = 2.98 ms, repetition time = 2300 ms, 160 slices, 1.0x1.0x1.1 mm) and a shared DW sequence along 41 directions (2.0x2.0x2.0 mm, b=1000 s/mm² plus one b=0 image, echo time 87 or 84 ms, repetition time 14000 ms, 60 or 64 axial slices). Data was assessed for movement, susceptibility and noise artifacts blindly to the diagnosis by two operators (JH and SS) and 12 subjects with significant artifacts or movements were consensually dropped out from the initial sample leading to an analyzed sample of 204 subjects.

Data processing

DW MRI data was processed using Connectomist 2.0 and T1 using BrainVISA 4.2 (www.brainvisa.info). The DW slices were corrected for eddy currents, susceptibility induced geometrical distortions and spikes using fieldmap based and Q-space interpolation corrections respectively. Then, an orientation distribution function (ODF) was computed at each voxel included in this mask using an analytical Q-ball model (spherical harmonic order 6, regularization factor λ=0.006) [START_REF] Descoteaux | Regularized, fast, and robust analytical Q-ball imaging[END_REF] . The ODF embeds information about the angular profile of diffusion within each voxel, thus allowing to detect the principal directions of diffusivity similarly to the main eigenvector of the DTI models. However, QBI models render complex fibers configurations such as crossing or bending fibers better by allowing mixture of populations of fibers characterized by different directions within each voxel [START_REF] Farquharson | White matter fiber tractography: why we need to move beyond DTI[END_REF][START_REF] Tuch | Diffusion MRI of complex neural architecture[END_REF] . Equivalently to DTI FA, the generalized fractional anisotropy (GFA) was evaluated from all the computed ODF [START_REF] Tuch | Diffusion MRI of complex neural architecture[END_REF] . A decreased GFA value is thought to indicate loss of integrity or coherence of WM 3 .

Whole-brain tractography

The definition of the 3-dimensional space within which the fibers are tracked is necessary for tractography algorithms. In most studies, this relies on thresholding the FA map using an arbitrary threshold yielding a mask supposed to mostly contain white matter. But in practice, many regions of interest included in WM are missed especially when partial voluming effects occur due to the limited resolution of FA maps, or due to the inadequacy of the FA threshold in crossings for instance. In order to compute a more robust mask, we used a T1-based propagation tractography mask computed using a published method [START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF][START_REF] Guevara | Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas[END_REF][START_REF] Guevara | Accurate tractography propagation mask using T1-weighted data rather than FA[END_REF] . In this approach, such a mask is driven by the T1 anatomical data. The propagation mask thus includes the entire brain tissue (rather than regions with high FA) and excludes specific areas such as the sulci skeleton to prevent the creation of implausible fibers (figure 1A). The mask in T1 space is then registered to the DW data by a linear rigid transformation.Each registration between T1 and DW data and the quality of the propagation mask were visually checked.

We performed whole-brain tractography in each subject native space using a regularized streamline deterministic algorithm (one seed per voxel, forward step 0.5mm, bilateral propagation) [START_REF] Perrin | Connectivity-based parcellation of the cortical mantle using q-ball diffusion imaging[END_REF] (figure 1B), which allows for the reconstruction of WM tracts using a step-bystep approach following the multidirectional diffusion orientation [START_REF] Descoteaux | Regularized, fast, and robust analytical Q-ball imaging[END_REF] . Algorithm propagation was interrupted if the tract length exceeded 300mm, if the tract streamline propagated outside the mask or if the curvature between two steps exceeded 30°. No between-subject registration was performed.

Clustering-based segmentation

To reconstruct WM tracts, whole-brain tractography volumes were then segmented using an automatic segmentation pipeline. This method, based on a clustering technique relying on the definition of a pairwise distance between fibers and described in [START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF] and [START_REF] Guevara | Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas[END_REF] , allows for an automatized reconstruction and segmentation of anatomical WM tracts in a large sample of subjects. Briefly this method follows five main steps, all performed in the native space of individuals. For a more in depth description of this method, the reader is invited to refer to [START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF] and [START_REF] Guevara | Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas[END_REF] . Firstly, the tracts dataset is divided to three subsets depending on their anatomical localization: left/right hemisphere or interhemispheric region (eFigure 1).

Secondly, each of these subsets is split into groups of similar length (figure 1C). Thirdly, the WM is fragmented into multiple clusters with dense connectivity using average-link hierarchical clustering and fiber clusters are extracted from this parcellation. Fourthly, an extremity density analysis regroups these clusters according to their extremity into homogeneous fascicles. Fifthly, a centroid tract representing each fascicle is generated and an average-link hierarchical clustering is performed to merge similar fascicles (figure 1D).

Finally, each volume containing each centroid is registered with a mean centroid atlas in Talairach space using a linear affine transformation and each centroid is matched to its corresponding deep WM bundle of the atlas [START_REF] Guevara | Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas[END_REF] (figure 1E). This leads to a segmentation of the tractography datasets into 22 known deep WM bundles, allowing a whole brain exploration of WM connectivity.

These WM bundles are comprised of the corpus callosum (CC) (rostrum, genu, body, splenium), and bilaterally of the cingulum with two populations of fibers depending on their length (cingulate part, short and long fibers), the uncinate fasciculi, the arcuate fasciculi (direct, anterior and posterior segments), the inferior longitudinal fasciculi, the inferior frontooccipital fasciculi and the frontal thalamic radiations. Each reconstructed bundle was visually assessed. All bundles were reconstructed for the whole sample, except the direct segment of the right arcuate fasciculus which was missing for 5 patients and 4 healthy subjects (4.4% of the total sample). These subjects were not included for the comparison of mean GFA along the direct segment of the right arcuate fasciculus. Examples of WM segmentations are provided in eFigure2.

For each bundle, we extracted the mean GFA using BrainVisa 4.2 (figure 1F). In this paper the terms "bundles" or "tracts" synonymously refer to reconstructed deep WM tracts.

INSERT FIGURE 1

Statistical analyses

We used linear mixed models to compare the mean GFA along each bundle between patients with BD and HC. The model included diagnosis (BD/HC) as a factor of interest, gender as a confounding factor, age as a confounding covariate and the site of inclusion as a random-effect factor to control for potential site-specific effects [START_REF] Hallahan | Structural magnetic resonance imaging in bipolar disorder: an international collaborative mega-analysis of individual adult patient data[END_REF] .

When we identified a significant difference of GFA value along a tract between patients with BD and HC, we compared in a post-hoc analysis the mean GFA along this tract between patients with a positive history of psychotic features (BD-PF group) and those with no such a history (BD-noPF group).

To check the robustness of the results, we repeated the primary analysis on homogeneous subsamples of patients. We removed patients taking lithium medication [START_REF] Hafeman | Effects of medication on neuroimaging findings in bipolar disorder: an updated review[END_REF] and patients with past history of alcohol dependence or abuse from the dataset. To explore the effect of mood symptoms, we repeated our analyses on a subsample of patients with mixed or elevated mood symptoms. We also compared mean GFA values between the 31 depressed patients of our sample and 31 HC matched for age, gender and IQ. A medication load score was computed following a method described in [START_REF] Versace | Elevated left and reduced right orbitomedial prefrontal fractional anisotropy in adults with bipolar disorder revealed by tract-based spatial statistics[END_REF] . We explored the relationship between GFA and medication load as well as with illness duration by partial correlation analyses controlling for age and gender. Lastly, we checked that adding IQ to the model did not change the results.

Differences between continuous demographic variables were assessed using Student's t-tests and differences between categorical variables using Pearson's chi-squared tests when suitable assumptions were met. Data was analyzed using PASW 18 (Chicago, IL, USA) and R 2.15.1 with Multtest package (www.r-project.org). Statistical tests were considered significant if corrected p values were < 0.05 when applying the False Discovery Rate (FDR) [START_REF] Benjamini | Controling the false discovery rate: a practical and powerful approach to multiple testing[END_REF] method to correct p values for comparisons of GFA along the tracts (26 comparisons) and of sociodemographic data. Model check was performed using residual QQ plots and outliers detection using leaf plots. If detected, analyses were repeated after removing outliers.

III-Results.

Clinical sample.

118 patients and 86 controls were actually included in the analysis (Table 1). Detailed demographic information for each center is provided in eTable 1. Information about history of psychotic features was missing for one patient (0.85%).Concerning mood state, 38 patients (32.2%) were symptomatic at the time of scanning: 31 patients had HRS-17 >7 (range: 0-32) or MADRS >7 (range: 0-25), 2 had YMRS>7 (range 0-24) and 5 had both.

INSERT TABLE 1

Comparison between patients with BD and HC.

Compared to HC, patients with BD had significantly reduced mean GFA along the anterior segment of the left arcuate fasciculus, the body and the splenium of the CC and the left cingulum (Table 2). These results remained significant after removing identified outliers.

INSERT TABLE 2

Comparison between BD-PF and BD-noPF.

Among the four bundles included for this comparison, BD-PF patients had a significantly lower mean GFA value than BD-noPF patients along the body of the CC (Table 3, Figure 2). This result remains significant after removing identified outliers.

INSERT FIGURE 2 INSERT TABLE 3

Effect of confounding factors on the BD vs HC comparison.

Results were unchanged when patients with elevated or mixed symptoms (n=7) (eTable 2), patients currently taking lithium (n=39) (eTable3), or with past alcohol abuse/ dependence (n=25) (eTable 4) were removed from the sample. Similarly, we found significant differences of GFA between the 31 depressed patients and the 31 matched HC except along the splenium (eTable 5). We found no correlation nor between the mean GFA and the medication load (eTable 6) nor between mean GFA and the duration of illness (eTable 7).

Adding IQ as a covariate did not change the results (eTable 8). When adding bundle volume as a covariate, all results remained significant.

IV-Discussion.

Difference of mean GFA between patients and controls.

This multicenter MRI study compared microstructural properties of 22 WM tracts between BD I patients and HC, using QBI tractography and automatic WM segmentation.

Patients with BD had WM alterations along the left cingulum (long fibers), the CC (body and splenium), and the anterior segment of the left arcuate fasciculus. BD-PF patients had lower GFA mean value than BD-noPF patients along the body of CC. To our knowledge this study has the largest recruitment to date of patients with BD in a tractography study. This relatively large sample 32 also allows for the comparison of white matter connectivity with tractography between patients with and without psychotic features.

To date interhemispheric disconnectivity has not been incorporated in neural models of BD [START_REF] Phillips | A neural model of voluntary and automatic emotion regulation: implications for understanding the pathophysiology and neurodevelopment of bipolar disorder[END_REF][START_REF] Strakowski | The functional neuroanatomy of bipolar disorder: a consensus model[END_REF] . However, some WM abnormalities of the CC have repeatedly been reported in patients with BD [START_REF] Versace | Elevated serum measures of lipid peroxidation and abnormal prefrontal white matter in euthymic bipolar adults: toward peripheral biomarkers of bipolar disorder[END_REF][START_REF] Emsell | Limbic and Callosal White Matter Changes in Euthymic Bipolar I Disorder: An Advanced Diffusion Magnetic Resonance Imaging Tractography Study[END_REF][START_REF] Bellani | DTI studies of corpus callosum in bipolar disorder[END_REF][START_REF] Leow | Impaired Inter-Hemispheric Integration in Bipolar Disorder Revealed with Brain Network Analyses[END_REF] , suggesting a relationship between interhemispheric connectivity abnormalities and BD [START_REF] Pettigrew | A 'sticky' interhemispheric switch in bipolar disorder?[END_REF] . In line, we identified a reduced anisotropy along fibers belonging to the body and the splenium of the CC. Interestingly, the patients who experienced psychotic symptoms during mania or depression had lower GFA in the CC. In parallel, similar decreased callosal WM anisotropy was reported in schizophrenia [START_REF] Knochel | Interhemispheric hypoconnectivity in schizophrenia: fiber integrity and volume differences of the corpus callosum in patients and unaffected relatives[END_REF][START_REF] Patel | A meta-analysis of diffusion tensor imaging studies of the corpus callosum in schizophrenia[END_REF] and neuropsychological results suggest that interhemispheric transfer may be impaired in patients with psychosis [START_REF] Gorynia | Intermanual coordination in relation to different clinical subgroups in right-handed patients with schizophrenic and other psychotic disorders[END_REF] . Thus, our results support that bipolar disorder with psychosis could be a biologically relevant subtype of BD. However, the substantial overlaps in white matter abnormalities between BD and schizophrenia prevent us to conclude on the specificity of such difference. Further neuroimaging studies with a dimensional rather than categorical approach of psychosis are needed to understand more-in-depth the role of callosal alterations in the pathogenesis of pychotic features.

Emotional dysregulation is a core feature of BD 11 33 . Functional disturbances have been described in the anterior cingulate cortex of patients with BD [START_REF] Wessa | Emotional processing in bipolar disorder: behavioural and neuroimaging findings[END_REF] , a region involved in emotional processing. Interestingly we found an altered microstructure of the cingulum bundle, formerly reported [START_REF] Mahon | A role for white matter abnormalities in the pathophysiology of bipolar disorder[END_REF][START_REF] Lin | Abnormal frontal cortex white matter connections in bipolar disorder: a DTI tractography study[END_REF][START_REF] Emsell | Limbic and Callosal White Matter Changes in Euthymic Bipolar I Disorder: An Advanced Diffusion Magnetic Resonance Imaging Tractography Study[END_REF][START_REF] Wang | Abnormal anterior cingulum integrity in bipolar disorder determined through diffusion tensor imaging[END_REF][START_REF] Pavuluri | Diffusion tensor imaging study of white matter fiber tracts in pediatric bipolar disorder and attention-deficit/hyperactivity disorder[END_REF] . We were able to specify the localization of these WM alterations along a subsample of cingulum fibers linking the anterior cingulate cortex to posterior parts of the cingulate cortex. This adds evidence of an anatomical disconnectivity within the cingulum in BD. Former studies reported a decrease of FA along the arcuate fasciculi in patients with BD 6,8 that we specified on the anterior segment of the left arcuate fasciculus. Interestingly, this segment links Broca's area to Geschwind's area and is thought to be involved in memory processing and speech comprehension [START_REF] Martin-Loeches | Brain activation in discourse comprehension: a 3t fMRI study[END_REF][START_REF] Catani | Atlas of human brain connections[END_REF] . We found no difference of GFA along this tract between BD-PF and BD-noPF patients in our sample. This does not support the hypothesis of an involvement of the arcuate in hallucinatory or delusional symptoms of patients with BD [START_REF] Lin | Abnormal frontal cortex white matter connections in bipolar disorder: a DTI tractography study[END_REF] . Notably, left-sided results have been reported in tractography studies [START_REF] Houenou | Increased white matter connectivity in euthymic bipolar patients: diffusion tensor tractography between the subgenual cingulate and the amygdalo-hippocampal complex[END_REF][START_REF] Lin | Abnormal frontal cortex white matter connections in bipolar disorder: a DTI tractography study[END_REF] , challenging those from a meta-analysis of voxel wise DTI studies 2 that found right-sided clusters of reduced anisotropy. Methodological heterogeneity may explain this apparent discrepancy. Indeed, we averaged GFA along each tract and very focal reductions of GFA may have remained undetected.

Reductions of GFA have been related to microstructural modifications including decreased axonal diameter or density, reduced myelination or decreased WM coherence [START_REF] Beaulieu | The basis of anisotropic water diffusion in the nervous system -a technical review[END_REF] .

Involvements of pro-inflammatory cytokines [START_REF] Stertz | Is bipolar disorder an inflammatory condition? The relevance of microglial activation[END_REF] , a dysfunction of myelination-related genes [START_REF] Mcintosh | The effects of a neuregulin 1 variant on white matter density and integrity[END_REF] and oligodendrocytes dysfunction [START_REF] Konradi | Mitochondria, oligodendrocytes and inflammation in bipolar disorder: evidence from transcriptome studies points to intriguing parallels with multiple sclerosis[END_REF] have been suspected. However, the precise mechanisms underpinning these alterations in BD remain unknown.

It has been questioned whether mood stabilizers could have an influence in neuroimaging measures. In line with the majority of existing DTI studies [START_REF] Hafeman | Effects of medication on neuroimaging findings in bipolar disorder: an updated review[END_REF] , we found no relationship between current medication and GFA measures. In order to draw a definitive conclusion, a replication in medication free patients would be worthwhile. Evidence suggests that mood symptoms may influence FA measures in BD [START_REF] Benedetti | Disruption of white matter integrity in bipolar depression as a possible structural marker of illness[END_REF] . We did not replicate such findings. Indeed, we found similar significant differences of GFA in a subsample of 31 depressed patients compared to 31 matched HC, except along the splenium where there was no significant difference. However, interpretation is limited by the possible underpowerment of this comparison.

Concerning our results, we found medium effect sizes for GFA. Larger samples with adequate statistical power may help to decrease the risk of overestimating "true" effect sizes [START_REF] Button | Power failure: why small sample size undermines the reliability of neuroscience[END_REF] . On the other hand, we used a multiple scanner design which may have lead to a systematic error. We reduced such potential site effect by sharing hardware, scanning protocol and analysis pipeline across sites.

Strengths and limits of the study.

Several strengths of our report should be emphasized. We have constituted a multicenter and relatively large sample of patients with BD in a tractography study thus reducing the biases associated with a single center recruitment. We have harmonized the MRI acquisition protocol and hardware to reduce site-specific effects. Finally, we have performed a multiple testing correction and performed secondary analyses to check for influences of confounding factors.

Such advantages are to be balanced against several limitations. We have not explored the inter-rater and inter-sites reliability of the scales used in this study. Despite our effort at harmonization, we have not included a phantom procedure to check the inter-center quality of the acquisitions. We cannot exclude that there was an effect of past medications on our results. Considering the large number of tracts to assess we have not exploited other metrics derived from the ODF. We averaged GFA values along each tract to perform comparisons; very localized decreases of GFA may have been undetected. A common limitation of crosssectional studies is the inability to determine whether observed alterations precede the onset or develop during the course of the disease. However, our results do not support the hypothesis of a relationship between the duration of the illness and GFA.

V-Conclusion.

Using whole-brain Q-ball imaging tractography in the first and relatively large international sample of patients with BD and HC, we reported decreased GFA values along interhemispheric tracts, limbic and frontotemporal WM tracts in patients with BD. Patients with psychotic features had lower GFA values than patients without such features along the CC.

These results highlight the role of interhemispheric disconnectivity in BD and further suggest that BD with psychotic features could be a relevant subtype of BD with specific pathophysiological features. They also provide additional evidence of the involvement of disturbed connectivity of the cingulum and the arcuate in BD. Future large multisite studies comparing BD patients to other psychotic disorders are warranted to clarify the involvement of these tracts in the pathophysiology of BD and to study the clinical relevance of such neuroimaging biomarkers. Abbreviations: GFA, generalized fractional anisotropy; FDR p, false discovery rate p value; SD, standard deviation; 95% CI, 95% confidence interval; DF, degrees of freedom; F, F value. Bold: significant after FDR correction. Effect sizes for diagnosis effect are reported as Cohen's f.

Figure 1 :

 1 Figure 1: Illustration of the processing pipeline. (A) Tractography mask. (B) Whole brain tractography. (C) Segmentation based on fiber lengths. Colors represent different fiber length. (D) Centroid tracts calculated from each fascicle. (E) Final segmented bundles of the left hemisphere and of the interhemispheric region. (F) Extraction of the mean GFA along the left inferior longitudinal fasciculus.

Figure 2 :

 2 Figure 2 : Mean GFA along the body of the corpus callosum for patients and controls.Individual values of generalized fractional anisotropy (GFA) along the body of the corpus callosum for patients with BD (BD), with (BD-PF) or without (BD-noPF) positive history of psychotic features and healthy controls (HC). * p<0.05 corrected for multiple testing.

Figure 1 :

 1 Figure 1: Illustration of the processing pipeline. (A) Tractography mask. (B) Whole brain tractography. (C) Segmentation based on fibers lengths. Colors represent different fibers length. (D) Centroids tracts calculated from each fascicle. (E) Final segmented bundles of the left hemisphere and of the interhemispheric region. (F) Extraction of the mean GFA along the left inferior longitudinal fasciculus.

Figure 2 :

 2 Figure 2 : Mean GFA along the body of the corpus callosum for patients and controls. Individual values of generalized fractional anisotropy (GFA) along the body of the corpus callosum for patients with BD (BD), with (BD-PF) or without (BD-noPF) positive history of psychotic features and healthy controls (HC). * p<0.05 corrected for multiple testing.

Table 1 :

 1 Demographic and clinical characteristics of patients with bipolar disorder (BD) and healthy controls (HC).

	BD	HC

Table 2 :

 2 Comparison of mean GFA between patients with bipolar disorder (BD) and healthy controls (HC).

		BD (n=118)	HC (n=86)
		Mean GFA (SD) Mean GFA (SD)	Difference	95% CI	F (DF)	Cohen's f for	FDR p	Cohen's f for
				of			diagnosis effect		covariates
				estimated					(scanner, sex, age)
				means
	Interhemispheric bundles
	Corpus callosum
	Body	0.101 (0.004)	0.102 (0.004)	-0.002	[-0.003; -0.001]	10.145 (1,198)	0.23	0.03	0.21, 0.03, 0.38
	Genu	0.094 (0.004)	0.094 (0.004)	-0.001	[-0.002; 3.18 10 -4 ]	1.803 (1,198)	0.09	0.37	0.36, 0.08, 0.66
	Splenium	0.113 (0.005)	0.115 (0.005)	-0.002	[-0.003; -0.001]	8.391 (1,198)	0.21	0.03	0.31, 0.003, 0.23
	Rostrum	0.096 (0.006)	0.096 (0.006)	-0.001	[-0.002; 0.001]	0.825 (1,198)	0.06	0.53	0.68, 0.002, 0.63
	Left hemisphere
	Left arcuate
	Direct segment	0.086 (0.003)	0.086 (0.003)	<10 -3	[-0.001; 0.001]	0.832 (1,198)	0.06	0.53	0.11, 0.04, 0.38
	Anterior segment	0.079 (0.004)	0.081 (0.003)	-0.001	[-0.002; -0.001]	8.403 (1,198)	0.21	0.03	0.09, 0.08, 0.39
	Posterior segment	0.075 (0.004)	0.075 (0.004)	<10 -3	[-0.001; 0.001]	0.538 (1,198)	0.057	0.60	0.11, 0.16, 0.35
	Left uncinate	0.078 (0.004)	0.078 (0.004)	<10 -3	[-0.001; 0.001]	0.122 (1,198)	0.03	0.81	0.28, 0.03, 0.33
	Left cingulum
	Long fibers	0.090 (0.005)	0.092 (0.005)	-0.002	[-0.003; -0.001]	7.645 (1,198)	0.20	0.04	0.05, 0.16, 0.14
	Short fibers	0.075 (0.004)	0.077 (0.005)	-0.002	[-0.003; -2.91 10 -4 ]	5.969 (1,198)	0.17	0.08	0.09, 0.11, 0.21

Table 2 :

 2 Comparison of mean GFA between patients with bipolar disorder (BD) and healthy controls (HC) (continued).

		BD (n=118)	HC (n=86)
		Mean GFA (SD) Mean GFA (SD)	Difference	95% CI	F (DF)	Cohen's f for	FDR p	Cohen's f for
				of			diagnosis effect		covariates
				estimated					(scanner, sex, age)
	Left inferior longitudinal	0.087 (0.004)	0.087 (0.003)	means <10 -3	[-0.001; 0.001]	0.041 (1,198)	0.01	0.87	0.19, 0.03, 0.21
	Left inferior fronto-	0.095 (0.004)	0.096 (0.004)	-0.001	[-0.002; 2.82 10 -4 ]	2.107 (1,198)	0.10	0.33	0.13, 0.08, 0.24
	occipital
	Left frontal thalamic	0.074 (0.004)	0.073 (0.004)	0.001	[-2.53 10 -4 ; 0.002]	2.207 (1,198)	0.10	0.33	0.36, 0.28, 0.22
	radiations
	Right hemisphere
	Right arcuate
	Direct segment	0.083 (0.004)	0.084 (0.004)	-0.001	[-0.002; 0.001]	0.631 (1,189)	0.05	0.58	0.20, 0.03, 0.19
	Anterior segment	0.082 (0.003)	0.082 (0.004)	<10 -3	[-0.001; 0.001]	0.003 (1,198)	0.004	0.96	0.05, 0.21, 0.35
	Posterior segment	0.073 (0.004)	0.074 (0.004)	-0.001	[-0.002; -3.19 10 -5 ]	4.145 (1,198)	0.15	0.16	0.03, 0.04, 0.32
	Right uncinate	0.079 (0.003)	0.080 (0.004)	-0.001	[-0.002; 1.67 10 -4 ]	2.584 (1,198)	0.11	0.30	0.20, 0.06, 0.36
	Right cingulum
	Long fibers	0.085 (0.006)	0.086 (0.005)	-0.001	[-0.003; -9.85 10 -5 ]	3.945 (1,198)	0.14	0.16	0.15, 0.11, 0.12
	Short fibers	0.074 (0.004)	0.076 (0.004)	-0.001	[-0.002; -9.92 10 -4 ]	4.588 (1,198)	0.15	0.15	0.08, 0.10, 0.11
	Right inferior longitudinal	0.087 (0.004)	0.087 (0.003)	<10 -3	[-0.001; 0.001]	0.789 (1,198)	0.06	0.53	0.22, 0.06, 0.11
	Right inferior fronto-	0.094 (0.004)	0.095 (0.004)	-0.001	[-0.002; 4.72 10 -4 ]	1.138 (1,198)	0.08	0.49	0.11, 0.03, 0.32
	occipital
	Right frontal thalamic	0.074 (0.004)	0.074 (0.004)	<10 -3	[-0.001; 0.001]	0.039 (1,198)	0.01	0.87	0.24, 0.19, 0.17
	radiations
	Abbreviations: GFA, generalized fractional anisotropy; FDR p, false discovery rate p value; SD, standard deviation; 95% CI, 95% confidence interval; DF, degrees of
	freedom. Bold : significant after FDR correction. Effect sizes for diagnosis effect are reported as Cohen's f.

Table 3 :

 3 Comparison of mean GFA between patients with bipolar disorder and at least one episode with psychotic features (BD-PF) and patients with bipolar disorder and no history of psychotic features (BD-noPF).

		BD-PF (n=57)	BD-noPF (n=60)
		Mean GFA (SD) Mean GFA (SD)	Difference	95% CI	F (DF)	Cohen's f for	FDR p	Cohen's f for
				of			diagnosis effect		covariates
				estimated					(scanner, sex, age)
				means
	Interhemispheric bundles
	Corpus callosum, body	0.100 (0.004)	0.101 (0.003)	-0.002	[-0.003;-0.001]	8.902 (1,111)	0.27	0.03	0.27, 0.003, 0.46
	Corpus callosum, splenium	0.113 (0.006)	0.113 (0.004)	-0.001	[-3.68; 4.825 10 -4 ]	2.485 (1,111)	0.15	0.30	0.34, 0.04, 0.27
	Left side
	Left arcuate, anterior	0.080 (0.003)	0.079 (0.003)	-0.001	[-0.002;0.001]	1.158 (1,111)	0.12	0.49	0.14, 0.005, 0.35
	segment
	Left cingulum, long fibers	0.089 (0.005)	0.089 (0.004)	<10 -3	[-0.002; 0.002]	0.046 (1,111)	0.02	0.81	0.11, 0.04, 0.17

Acknowledgements.

We thank all subjects for their participation in this study. We also thank Soufiane Carde, MD François-Eric Vederine, MD and the personel of participating centers for their help regarding collection of data and David Kupfer, MD, for his support to this project. This work was supported by public funding from the Alliance pour les Sciences de la Vie et de la Santé (ITMO Neurosciences), the French Agence Nationale pour la Recherche (ANR MNP 2008), the German Deutsche Forschungsgemeinschaft (SFB636/C6 and We3638/3-1), and the NIMH R01 MH076971. S.S. work was supported by a grant from the Agence Régionale de Santé Ilede-France.

The funders did not participate in the design and conduct of the study, in the collection, analysis and interpretation of the data; or in the preparation, review or approval of the manuscript.

Samuel Sarrazin and Josselin Houenou had full access to all of the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis.

Conflict of interest disclosures.

All authors report no competing financial interest to declare.


Younes Guessous

Maurice Aron
email: maurice.aron@ifsttar.fr

Neila Bhouri

Simon Cohen

Euro Working Group on Transportation 2014 Estimating Travel Time Distribution under different Traffic conditions

Keywords: congestion, traffic flow, travel time, motorway, modeling, statistics, reliability, distribution, Bureau of Public Roads

published or not. The documents may come

Introduction

Traffic congestion impacts speed, thus travel time. When traffic increases and approaches the full capacity of the network, the flow becomes unstable and much more vulnerable to incidents, road works or bad weather. This increases the variability of travel time, to which users are very sensitive. Therefore, travel time reliability has become an important performance criterion for transportation facilities, complementing the traditional measures such as delay and average travel time. In recent research, a variety of performance measures have been proposed to quantify reliability and monetize it. This includes planning time, buffer time, standard deviation, coefficient of variation, skewness,... -an overview is given in [START_REF] Lomax | Report of Selecting Travel Reliability Measures[END_REF]. These indicators are based on percentiles of travel time. The knowledge of the travel time distribution is then needed.

Different distributions are presented in the literature as the best way to model the travel time distribution. [START_REF] Richardson | Travel time variability on commuter journeys[END_REF], [START_REF] Rakha | Estimating Path Travel-Time Reliability[END_REF], [START_REF] Pu | Analytic relationships between travel time reliability measures[END_REF] and [START_REF] Arezoumandi | Estimation of Travel Time Reliability for Freeways Using Mean and Standard Deviation of Travel Time[END_REF] concluded for a Lognormal distribution. [START_REF] Polus | A study of travel time and reliability on arterial routes[END_REF] concluded for a Gamma distribution; however Al-Deek and Eman (2006) proposed a Weibull one. In (Taylor & Susilawati 2012) and [START_REF] Susilawati | Distributions of travel time variability on urban roads[END_REF] the Burr XII distribution is adopted, the advantage of this latter method is that its tails often fit the empirical ones. Aron, Bhouri and Guessous (2012) presented a comparison of six statistical distributions used to model travel time. Tests were conducted to identify the parameters of these different statistical distributions on the basis of real time data collected on a weaving section of the A4-A86 French urban motorway.

Based on the same data, this paper uses the Burr XII distribution, completed by a scale parameter introduced by [START_REF] Singh | A function for the size distribution of income[END_REF] to model travel time over five levels of service. The next section is dedicated to data collection. A method for levels-of-service extraction using the fundamental diagram is given in section 3. Calculation of the travel time and modeling of its distribution over five levels-of-service are presented in section 4. In section 5, the travel time distribution calibration is improved, using relations linking travel time to flow.

Data collection

The data used in this paper was collected on a weaving section of the A4-A86 French urban motorway. A twolane urban motorway ring (A86) round Paris and a three-lane West-East urban motorway (A4) meet in the east of Paris and share a four-lane 2.3 km-long section. Traffic is particularly dense at some hours, and causes the greatest traffic bottleneck in Europe. Data used in this paper were collected in the year 2002, on a 3-km long stretch (2.3 on the weaving section, 0.7 km downstream), in the Eastbound direction. Four inductive loops (three on the weaving section, one downstream) provide every six minutes flow, occupancy and average speed by lane.

Although the data are generally very good, some are missing, inaccurate or irrelevant. A mean speed for one lane lower than 2 km/h or higher than 150 km/h, is considered as an outlier. Other anomalies in traffic data are identifiedoccupancy greater than 100% or 6-minute flow (by lane) greater than 400 vehicles. In these cases the data for the corresponding period and lane are cancelled and considered as missing. When this occurs in 2002, the missing data for a given period and lane is substituted, when possible, by data from a corresponding period from the year 2001 or 2000, the same day of the week, the same exact time and approximately the same date.

Level-Of-Service

Fundamental Diagram

The three macroscopic traffic variables -traffic flow q, traffic density k and average speed v-are linked by the equation q=k.v. Furthermore, when traffic density increases, speed decreases. This phenomenon is modeled by a relation between speed and density (or flow), called "Fundamental Diagram". An example of a fundamental diagram is provided in Figure 1. The first part of the graph represents the free flow, where the interaction between vehicles is light; then traffic flow increases along with traffic density until the critical density value k c . Corresponding flow is the maximal flow q max sustained by the infrastructure. Above this density, vehicles are bunched and flow decreases. Flow is again equal to zero when density reaches its maximum.

Fundamental diagram is plotted on the basis of experimental data, and allows a better understanding of traffic.

Construction of a fundamental diagram for consecutive sections

In order to extract a simple information, the scatter plot is usually fit by a solid line. A division of this line in a few more homogeneous traffic states or "levels-of-service" is particularly useful.

So as to extract levels-of-service, we need first to construct a global fundamental diagram, since the data is presented by lanes (i=1..n) and section (j=1..m). Aggregation of speed, flow and density variables must be performed carefully to reflect reality, and the homogeneity of units must be thoroughly checked. We assert here that speed will be expressed in km/h, traffic flow in vehicles/hour and traffic density in vehicles/km.

Aggregation of data over lanes in one section is obtained by applying simple operations to the three traffic variables. We add traffic flows to obtain the flow for an entire section and we similarly add traffic densities. Speed for the entire section is the result of dividing the flow by the density. Formulas are the following:

1 n j ij i q q    ; ij ij ij q k v  or 1 ij ij ij k vq  ; 1 n j ij i k k    B ; 1 1 1 n ij ij i ij n j ij i k q q v q           = j j k q (1)
The last equation giving the harmonic average speed, weighted by traffic flows. Let L j be the length and the travel time of section j, thus L j /v j is and its travel time. Travel times on consecutive sections being additive, the equation of the global average speed, on consecutive sections, is obtained by:

1 1 1 1 1 / ;
As vehicles are present on the sections, thus the density is:

/ (2) m m m m m j j j j j j j j j j j j j L v L N L k k L k L v              1 1 1 1 1 1 1 1
The global flow is the product of speed and density:

(3)

m j j m j j j m m m j j j j jj L j j j j j m L m m j j v j j j j j j L q q L Lk v v q L L L v v                        
The flow is then the average of sections' flows weighted by travel times. It is also the total travel time (for all users) divided by the average travel time.

The global fundamental diagrams in 2002 and 2006 are shown in Figure 2(a) and 2(b). In 2006 points become less dense around 160 vehicles/km, whereas in 2002 they do so around 180 vehicles/km. That can be interpreted as the positive effect of opening hard shoulders to vehicles when congestion exceeds a certain level.

Fitting the fundamental diagram

The levels of services are defined with respect to road capacity and critical density. Road capacity is not assumed to be the maximum observed traffic flow, which could be an outlier. Here road capacity is determined after fitting, in a first step an analytic curve to the scatter plot; the maximum of this curve gives the capacity.

Numerous models are used to fit the fundamental diagram. Some are summarized in Table 1 : For each model we apply a nonlinear regression analysis, based on the least-square method -which provides acceptable results. All computations in this article are performed using R, an open-source statistical software. We compared only the generalized models. Figure 3 The Generalized Exponential, giving the best representation of the scatter plot's tail, is selected here.

Level-of-service computation

The road capacity and the critical density appear on the fitted fundamental diagram; separating the diagram in levels of service (LOS) is straightforward, using the LOS thresholds in terms of capacity percentage and density.

The six LOS defined by the High Capacity Manual are replaced in France by a simpler categorization of 4 LOS. In this article we are using customized five LOS which are defined as following:

 LOS 1 : density under critical density, and flow under 75% of capacity  LOS 2 : density under critical density, and flow between 75% and 90% of capacity  LOS 3 : density under critical density, and flow above 90 % of capacity  LOS 4: density above critical density, and flow above 90% of capacity  LOS 5: density above critical density, and flow under 90% of capacity Four points separate the five LOS. Point 1 (between LOS 5 and 4); point 2 (between LOS 4 and 3); point 3 (between LOS 3 and 2); and point 4 (between LOS 2 and 1). Their flow, density and speed are given in Table 2.

Modeling travel time distribution by Level_Of-Service

Figure 5(a) and 5(b) show the placing of the five LOS and of the four separating points. Furthermore, in Figure 6 are displayed the travel time histograms by LOS; these are useful to select the distributions required for modeling. For >0:

F( )=1-[1+( / ) ] f( ) ( ) [1 ( / ) ] ( ) (1 ) 1 (4) a q a- q / q a x x x b ; x a q / b x b ; P b  
The 1 st and 3 rd parameters, a & q are shape parameters, whereas the second parameter b is a scale parameter.

For the first t0hree LOS, a normal mixture model is also fitted. We evaluate the quality of the models by using the Akaike inform0ation criterion (AIC), and select the model that minimizes this criterion (Table 3). LOS 3, 4 and 5, are the most interesting ones, because they occur near capacity or in congestion. Their adjustments to the models are presented here. Adjustments with a normal mixture have not been performed for LOS 4 and 5 because the empirical histograms have a single mode. For LOS 3, three modes appear, and normal mixture outperforms Singh-Maddala. For LOS 4, the AIC for the lognormal distribution is better than the one for the Singh-Maddala. However, the difference in this criterion between lognormal and Singh-Maddala is not significant enough compared to the same difference in LOS 3. Besides, we do not notice any significant difference graphically.

For LOS 5, Singh-Maddala outperforms the lognormal distribution from the AIC and graphics points of view.

With the AIC, the superiority of the Singh-Maddala distribution is confirmed from 2006 data for LOS 1, 4, 5, whereas it is lightly outperformed by normal mixture in LOS 2 and 3 (Table 4). The lognormal distribution lags far behind Singh-Maddala and normal mixture, except for LOS 4 where it is close to Singh-Maddala. Let us recall that Singh-Maddala distribution has 3 parameters, while a normal mixture with 2 components has 2×2=4 parameters. We recommend using Singh-Maddala distribution: it is the most stable distribution and adapts to various levels-of-service. It provides a good trade-off between fitting quality and model simplicity.

However the numerical values obtained for the Singh-Maddala parameters are not validated: 2006 values (given in Table 5 for the first LOS) differ from 2002 values -. This is probably due to changes in infrastructure.

Improving travel time models

One way to improve travel time prediction is to find a relationship between travel time and flow. The Bureau of Public Roads (BPR) function (here applied by LOS) is the most used formula in this case:

0 max 1 l l q TT q                
Where (l=1..5) represents the LOS number.  and  are dimensionless parameters; their values and the quality of the regressions (in terms of p-value and residual standard error) are presented in Table 7. As p-values are below 0.05 (except for LOS 2), the BPR functions, relating travel time and traffic flow are significant for LOS 1, 3, 4 and 5. Residual standard errors are generally acceptable (see Table 7), given the facts that in free-flow (LOS 1) the speed is not constrained by the flow, and that by high congestion (LOS 5) the relation between travel time and flow vanishes. Non-linear regressions are illustrated for LOS 3 & 5 in Figure 8. Then, applying the BPR functions on every travel time, we obtain a new "adapted" travel times series, more homogeneous, making possible a better adjustment. This happened. The improvement is very impressive for AIC values (Table 8). Due to the travel time queue, the slightest improvement is for LOS 5.The improvement remains slightly visible graphically for this LOS -see Figure 9, providing the original and adapted Singh-Maddala fits.

Conclusion

Basing reliability on aggregated travel times (here on 6-minute periods) and not on individual travel times is justified because it bases the information which is presented to users and which is taken into account in economics studies. The passage by Levels-Of-Service is now widespread in traffic studies because the homogeneity of a LOS induces more accurate treatments -this is confirmed here. The Singh-Maddala distribution is both appropriate (given the quality of the fit) and practical (for deriving percentiles, which are used in the reliability indicators). The use of BPR functions relating travel time to traffic flow (by LOS) improves the adjustments. However the numerical values of the parameters were not stable from one year to another, due to changes in the infrastructure. All of this contributes to a better understanding of travel time and of its reliability.

Fig. 1 .

 1 Fig. 1. Fundamental diagram in 2002 lane 1 section 1

Figure 2

 2 Figure 2 (a) Global fundamental diagram in 2002 (b) Global fundamental diagram in 2006 direction Paris to the East

  *exp(-b*k) q=a*k*exp(-b*k^alpha)

Figure 3 .

 3 Figure 3. Global fundamental diagram in 2002 fitted with (a) a generalized power model (b) a generalized exponential model Number of data: 53677; Residual standard error for (a) : 707.4 ; for (b) : 696.5

  above and Figure 4 below show the adjustments of the generalized power and exponential models on the global fundamental diagram in 2002 and 2006.

Figure 4 .

 4 Figure 4. Global fundamental diagram in 2006 fitted with (a) a generalized power model (b) a generalized exponential model; Number of data: 53677; Residual standard error for (a) : 655; for (b) : 588.2

Fig. 5 .

 5 Fig. 5. (a) Levels-of-service selection (b) Computed points

Fig. 7 .

 7 Fig. 7. Histogram of LOS 3, 4 and 5 travel times fit by different distributions in 2002

Figure 8 .

 8 Figure 8. Non-linear regression between travel times (variable y) and flows (x) in 2002 for (a) LOS 3 (b) LOS 5

Figure 9 .

 9 Figure 9. Histogram of original and adapted LOS 5 travel times fit by Singh-Maddala distribution in 2002

Table 1 .

 1 Statistical models for fundamental diagrams

	Model	Equation
	Greenshields	q=a*k+b*k^2

Table 2 .

 2 Flow, density and speed at four points separating the fiveLOS for 2002 and 2006

			2002			2006
	Points	Flow	Density	Speed	Flow	Density	Speed
	1	6753.8	184.0	36.7	6497.2	161.6	40.2
	2	7504.2	139.2	53.9	7219.1	122.8	58.8
	3	6753.8	98.0	68.9	6497.2	86.9	74.7
	4	5628.2	74.1	75.9	5414.3	65.9	82.1

Table 3 .

 3 AIC for the five LOS and three distribution models in 2002 (best distribution highlighted)

	LOS	Number of data	Singh-	Lognormal	Normal mixture
			Maddala
	1	24036	221876.36	230326.73	227150.70 (with 2 components)
	2	7878	73971.82	74746.84	73530.72 (with 2 components)
	3	6599	60821.46	60512.36	59872.96 (with 3 components)
	4	10989	101480.51	101325.77	Not performed
	5	4144	46622.16	49089.62	Not performed

Table 4 .

 4 Akaike information criterion for the five LOS and three distribution models in 2006

	LOS	Number of data	Singh-Maddala	Lognormal	Normal mixture
	1	23816	165571.60	178098.76	166346.62 (with 2 components)
	2	9508	73814.98	78407.83	73757.59 (with 2 components)
	3	10306	79115.66	82729.76	78913.22 (with 2 components)
	4	6248	56111.66	56284.97	Not performed
	5	3760	39100.80	40434.25	Not performed

Table 5

 5 provides estimates for Singh-Maddala parameters in 2002 (and in parenthesis for 2006, LOS 1 only).

Table 5 .

 5 Estimates and confidence intervals forSingh-Maddala parameters (five LOS in 2002 and LOS 1 in 2006)

	LOS	Parameter	Lower bound	Estimate	Upper bound
		shape1.a	31.98 (47.87)	32.01 (47.90)	32.05(47.92)
	1	Scale (in seconds)	115.65 (108.63)	115.65 (108.63)	115.65 (108.63)
		Shape 3 q	0.11 (0.27)	0.15 (0.31)	0.19 (0.34)
		shape1.a	16.49	16.53	16.57
	2	Scale (in seconds)	136.31	136.32	136.33
		Shape 3 q	0.31	0.38	0.44
		shape1.a	13.61	13.65	13.69
	3	Scale (in seconds)	150.67	150.68	150.69
		Shape 3 q	0.58	0.66	0.74
		shape1.a	18.42	18.45	18.48
	4	Scale (in seconds)	228.28	228.29	228.29
		Shape 3 q	0.73	0.79	0.85
		shape1.a	29.35	29.44	29.52
	5	Scale (in seconds)	260.40	260.41	260.42
		Shape 3 q	0.03	0.13	24

Table 7 .

 7 Statistical information for the 5 LOS in 2002

	LOS	Number	Residual standard				
		of data	error (seconds)	estimate	p-value	Estimate	p-value
	1	24036	76.61	1.419	<2e-16	0.454	<2e-16
	2	7878	30.46	0.871	<2e-16	0.005	0.87
	3	6599	24.32	0.926	<2e-16	0.135	7.06e-05
	4	10989	24.21	1.750	<2e-16	-0.255	<2e-16
	5	4144	139.4	2.606	<2e-16	-0.832	<2e-16

Table 8 .

 8 AIC values for the five LOS travel times (originals and adapted) fit bySingh-Maddala in 2002

	LOS	Number of data	Original	Adapted
	1	24036	221876.36	134976
	2	7878	73971.82	24641.99
	3	6599	60821.46	20289.9
	4	10989	101480.51	66013.44
	5	4144	46622.16	40341.57

Acknowledgements

work has been done with the support of Ecole des Ponts-ParisTech.