Modelling of the tactical path selection of bicyclists at signalized intersections

Transcription

1 0 0 0 Modellg of the tactical path selection of bicyclists at signalized tersections Sasan Ai, M.Sc., Graduate Student (Correspondg Author) Chair of Traffic Engeerg and Control Technische niversität München, Arcisstraße, 0 Munich Tel: +--, Eail: sasan.ai@tu.de Heather Twaddle, M.Sc., Graduate Research Associate Chair of Traffic Engeerg and Control Technische niversität München, Arcisstraße, 0 Munich Tel: +--, Eail: heather.twaddle@tu.de Axel Leonhardt, Ph.D., Professor Transportation Planng and Traffic Engeerg Erfurt niversity of Applied Science, Altonaer Straße, 0 Erfurt Tel: +--00, axel.leonhardt@fh-erfurt.de Word Count: Words (excludg the references) + Figures + Tables = Words Subitted for presentation and publication to the Transportation Research Board th Annual Meetg Subission Date: July th 0

2 Ai, Twaddle, Leonhardt 0 0 ABSTRACT Bicyclg is becog ore and ore prevalent due to its societal and personal benefits. Consequently, understandg bicyclists behavior and considerg bicyclists as relevant eleents transport and traffic odellg is essential. To assess operational aspects, bicyclists behavior at tersections is particularly iportant, as tersections have a large ipact on overall syste perforance and safety. In contrast to otorized vehicles, bicyclists typically have ultiple (legal and illegal) path options to travel through an tersection. This study presents a discrete choice odel to predict the path on which left-turng bicyclists travel through signalized tersections. To accoplish this objective, revealed preference data fro busy tersections Munich, Gerany, has been collected through video observations. The exhibited left-turng aneuvers are categorized three types: bicycle turn, pedestrian turn and vehicular turn. After a careful analysis of the itial set of explanatory variables, unnecessary variables are oitted fro the odel. For the data analysis, a ultoial logit odel is developed order to identify the fluence of the dividual factors. A field effect variable is exaed, which reflects the fluence of the choice of the peer decision-akers. The results of the study reveal that aong the selected variables, seconds passed sce the begng of the red phase of the signal is the ost fluential paraeter followed by the approachg speed of the bicyclist. ltiately, an external validation was perfored with an dependent dataset fro the sae tersection, and the result shows % accuracy the odel prediction. Keywords: Bicyclists Behavior, Tactical Path Selection, Multoial Logit Model, Revealed Preference

3 Ai, Twaddle, Leonhardt INTRODCTION The personal and societal benefits of bicyclg has captured the terest of professionals and policy akers to facilitate the bicycle coutg trips by iprovg the safety and convenience of bicyclg (, ). As a result, bicyclg is becog creasgly prevalent for daily coutg trips, which has led to a heterogeneous traffic strea coposition urban areas. Microscopic traffic siulation is a widely used struent evaluation of the transportation and traffic control easures before their ipleentation. However, the reliability of these evaluations strongly depend on the realistic odellg of the road users attributes and dynaics (). The existg literature on the siulation odels for bicycle traffic, contrast to otor vehicles, is scarce. Previous research the field of bicycle transportation has aly focused on the fluential factors on the bicyclists route choice. For stance, () have proposed a route choice odel with revealed preference GPS data. Many studies have conducted a stated preference survey, which asks bicyclists to rank their preferences for different facility types ( ). Bicyclists are difficult to capture conventional odels sce they often share the rights of way with otor vehicles but their behavior is quite different due to their different physical and dynaic characteristics. Bicycles are narrower and have a greater lateral flexibility lettg the to utilize the lateral space with a traffic lane and switch easily between different types of available frastructure (). Because of these challenges and the coplexity, the ajority of the available traffic siulation software on the arket still lack a realistic odellg of bicyclists behavior, particularly the teraction of the bicyclists with other road users. Even if there is the possibility to clude bicycles the siulation, they are odeled through a siplistic approach assug the bicycles are saller and low-power vehicles or fast ovg pedestrians (). Nevertheless, a nuber of recent studies have proposed odels, which take bicycles to consideration as a separate ode of travel. For stance, as one of the first attept, Faghri and Egyhaziova (0) developed a coputer siulation odel called BICSIM (BICycle SIMulator), which is applicable to car-bicycle, bicycle-car and bicycle-bicycle followg. The higher degree of lateral flexibility of bicycles was first studied by Oketch (), who vestigated the idea of siultaneous utilization of two lanes and gradual lane changg, as opposed to an stantaneous one. These efforts have tried to tegrate non-otorized vehicles to the siulation odels, especially the short ter decision-akg. However, these odels are not properly calibrated due to lack of epirical data. In ost transportation studies three levels of huan behavior are distguished: strategic, tactical and operational level ( ). In this hierarchy, expected utilities at a lower level affect choices at a higher level and choices at a higher level govern behavior at a lower level. The difference between the tactical level of behavior and the operational and strategic level is rooted the coplexity of the goal and duration of the activity. The tactical level of behavior odellg focuses on huan behavior when seekg short-ter goals a tie scale of seconds to utes, while operational behavior odels assue sgular goals that are achieved at a tie frae of one second; whereas the strategic behaviors have ore coplicated goals and volve decision akg at ore than one level (). For stance, case of a bicyclist, at the operational level, one would odel the obstacle avoidance ability; at the tactical level the decisions to stop or cross a red light would be odeled, and the strategic level of behavior would focus on route choices. The focus of this research is on bicyclists path choice behavior at the tactical level. Previous research on the tactical behavior of bicyclists has focused on topics such as red light copliance, frastructural preferences and gap acceptance. Developg logit odels based on revealed preference or stated preference surveys is the ost coon approach deployed by the researchers order to fd the ost fluential factors on these decisions. The key fdgs and

4 Ai, Twaddle, Leonhardt the ethodology eployed soe of these studies is discussed below and is used as the basis for selectg fluential factors for this research. Several studies have been conducted to vestigate the behavior characteristics and associated factors of red light violation. The results of these studies reveal that age group, gender (, ) and helet use () as well as the duration of the red phase and the geoetry of the tersection () have significant fluence on red light violation behavior. Regardg the frastructure selection of bicyclists, the purpose of travel, ridg skills and gender are found as the ost iportant factors (0, ). In another study, () vestigated the fluence of weather condition, tie of day and segent characteristics addition to socio-deographic variables on the probability of ridg bicycle lane. To the knowledge of authors, there is no previous research on the path selection at signalized tersections; however, soe studies on crossg behavior of bicyclists at unsignalized tersections provide valuable sight of the proble. For stance, Huang and Wu () developed a fuzzy logic odel to describe a bicyclist s path planng ixedtraffic flow at an unsignalaized tersection Cha. The results of this study shows that bicyclists first try to ga rough foration of tersection situation and then sketch their preferred path; cofort, directness and efficiency are ajor criteria for path sketch. METHODOLOGY The ethodology eployed this study is used to vestigate bicyclists decision akg process while approachg a signalized tersection and the translation of these paraeters to an algorith that is suitable for tegration a icroscopic siulation tool. Thus, a revealed preference dataset fro ore than hours of recorded videos fro three tersections Munich, Gerany, has been created. A portion of these videos are only used to set the analysis fraework and draw reasonable assuptions. This preliary analysis is perfored on five hours of video data fro all three tersections, which are different ters of geoetric design and traffic volues (FIGRE ). This is an essential step order to understand the choice situation. Then, the priary dataset is collected fro hours of video data fro one of the tersections (tersection (b) FIGRE ) to estiate the odel. The high nuber of bicyclists, clear overview of the area as well as availability of accurate traffic signal data are the decisive factors to select this tersection as the study site. In the second stage of data collection, six hours of video of another day fro the sae tersection is collected, which is called the secondary dataset, and is used to validate the odel. Analysis Fraework Path selection proble arises when ore than one path exists to reach the destation. In general, there are any physically possible paths to turn left at an tersection, soe of which are depicted FIGRE. Note that for the sake of siplicity, soe paths fro/to vehicle lane or sidewalk are not shown the figure. This choice set will grow even further if the type of the behavior, for exaple walkg on the pedestrian crosswalk stead of ridg or crossg agast the red light, is also considered.

5 Ai, Twaddle, Leonhardt FIGRE Observed tersections preliary analysis phase (a) Marsstrasse- Seidlstrasse (b) Arnulfstrasse-Seidlstrasse (c) Karlstrasse-Luisenstrasse. FIGRE Possible paths for turng left at tersections.

6 Ai, Twaddle, Leonhardt The target group of this study is left-turng bicycles regardless of their lateral position on the road (bicycle lane, vehicle lane or sidewalk) and cludg those that exhibit law-breakg behaviors. The analysis fraework is set based on the results of the preliary observation on left-turng bicyclists. The a conclusions ade at this step are: Bicyclists who arrive durg the green phase of the straight-through signal follow the sae path. If they are bicycle lane, first they go straight and then wait for the green light of the second signal, and if they are the left ost vehicle lane, they turn diagonally. This iplies that two groups of bicyclist are confronted with two different sets of choices and have to be studied separately. Bicyclists ridg on the sidewalk as well as the red light violators are rarely observed and can therefore be neglected. Bicyclists change their approachg lane and speed accordance to their preferred path around 0 eters upstrea the stop le. Those who want to stop at the straight-through signal soothly reduce their speed, but other riders that want to cross the pedestrian crosswalk are less patient and ride faster to avoid two stops at the tersection. In order to address this issue this research, left-turng path selection is defed as a sequence of actions; as the bicyclist gets closer to the stop le, starts to analyze the tersection, particular, the traffic signals status, speed and the position of other road users. Bicyclists assess all the available paths contuously for a short period of tie and ake their decision a few eters (three eters this case) prior to the stop le. This is where they are able to observe the pedestrian signal status and have enough space to adjust their speed and lateral position le with their decision. An observation zone on each segent of the tersection has been defed that is siilar to the concept of dilea zone which has been used for analyzg drivers reaction to signal change at tersections (). Sce there is not a clear view of 0 eters on all segents of the tersections, the begng of the observation zone is considered to be 0 eters upstrea of the stop le; bicyclists speed would be recorded onset of enterg the observation zone and they choose their desired path as they reach the decision akg pot. In case this distance is not covered by the caera, the farthest observable distance will be used. Data Collection To study bicyclists behavior, a collection of video data is gathered by ountg a high-defition caera fps forat at the top of a buildg that has a full view of the tersection. As the autoated extraction of variables describg the tactical behavior is very difficult, the video fraes are anually analyzed. The tersection of two iportant arterial roads, Arnulfstrasse and Seidlstrasse, the Central Busess District (CBD) of Munich close to the Central Tra Station (Hauptbahnhof) create a four leg-tersection as shown FIGRE b. It is a large, high-volue tersection with a fully actuated traffic signal control. Three approaches of the tersection have two vehicle lanes and a dedicated bicycle lane. The east approach does not have a bicycle lane. A tra le runng through the iddle of Arnulfstrasse is prioritized over the private car traffic. The duration of the cycle tie varies between 0 to 0 seconds. The south approach leads to the tersection through an underpass that liits the caera view range. This causes liitations on speed easureents on this approach. For this reason a bary variable that dicates the relative speed coparison with other bicyclists is used place of a easures speed value.

7 Ai, Twaddle, Leonhardt 0 Modellg Approach The selection of a path fro a set of possible paths is a special case of discrete choice proble. The atheatical ethod which is by far the ost coonly used approach to represent the characteristics of discrete choice proble is the logit odel. Therefore, Multoial Logit (MNL) odel is deployed, which has shown to be effective traffic siulation tools as well (). The first and foreost step the estiation process of the logit odel is to identify the choice set. As shown FIGRE, there are an extreely large nuber of fite paths that can be selected by a bicyclist. In order to siplify the choice situation, left-turng aneuvers are categorized three types: 0 0 FIGRE Categorization of possible alternatives for turng left at an tersection; blue paths are for bicycle lane group and red ones for the vehicle lane group of riders. The fundaental difference between two groups of bicyclists, bicycle lane and vehicle lane group, iplies that a standard MNL odel is not applicable for this proble, and two dependent MNL odels are needed. This is siilar to the approach followed by Bierlaire et al. () to cobe the results of two revealed preference and stated preference surveys. The idea is to forulate two dependent MNL odels with the sae explanatory variables. After identifyg the choice set, the utility functions that are assigned to the alternatives should be forulated. In general, a lear paraeter function is used (): V X k k k Where V is the deteristic portion of the utility, which is defed by β, a vector of k paraeters fluencg the utility and ε is the error ter. The crucial liitation of this odel is that only the observable attributes of the alternatives and characteristics of the decision aker are considered. Many psychologists and econoists, however, have specifically ephasized the fluence and constrats iposed by the peer group to which the decision aker belongs (). One solution is to tegrate a field effect variable to the utility function that captures the average effect of peer group choice on the probability of an alternative (). Thus, the utility function is forulated as: V ( X, S ; ) F n () ()

8 Ai, Twaddle, Leonhardt Where F is the proportion of the people the peer group of agent n who chose alternative i and γ is the coefficient that reflects the strength of the fluence. This choice odel cludes the social fluence and can be estiated by conventional ethods. However, it ay suffer fro endogeneity resultg consistent estiate of the odel paraeters. The endogeneity arises because the effect of other decision akers is captured by both ε and F, and therefore, these two variables ight be correlated. A correction algorith known as BLP ethod has been proposed by Berry, Levsohn and Pakes (0). This approach solves the endogeneity at a arket level, which cludes a group of siilar decision akers that fluence each other. The error ter is decoposed to two portions: the endogenous-causg part and a rando ter. Sce the endogeneity occurs at arket level, the utility function is odified by considerg arkets. The equation then becoes: V ( X, S n ; ) [ F i i ] The ter [γf + ε ] this equation reflects both the observable and unobservable part of the utility relevant to the decision aker s peer group. This ter will be replaced by a arket constant αi which will be estiated by usg a two-stage lear regression odel (readers are referred to () for ore foration). The ultiate for of utility function is expressed as: V ( X, S n ; ) i It should be noted that if the estiated coefficient for this field effect variable is close to zero or statistically significant, then the distribution of the probability of that particular alternative is not relation with other decision akers choices. In other words, it is not beneficial to tegrate the social fluence the odel and consequently, the traditional approach should be used. Once the odel is defed, selectg the fluential paraeters which ust be recorded durg data collection phase plays an iportant role. A list of selected factors is given the followg table. The video data does not provide adequate foration for the selection of the personal explanatory variables (i.e. age group, gender, helet use) due to height of the caera: TABLE Coded Paraeters Paraeter Sybol Xred Xrs Xs Xph Paraeter Defition Seconds passed sce the begng of the red phase of the signal Contuous variable Pedestrian crosswalk signal status Dichotoous variable: = green and 0 = red Bicyclists approachg speed at the begng of the observation zone Contuous variable Peak-hour Dichotoous variable: = peak-hour and 0 = non-peak () ()

9 Ai, Twaddle, Leonhardt 0 0 Xst Xcc Xq Xped Xw DATA ANSLYSIS Stop le direct accessibility (bicyclists that are already waitg at the stop le have blocked the direct access to the pedestrian crosswalk for other bicyclists) Dichotoous variable: = blocked and 0 = direct access Nuber of the conflictg cars fro the opposg traffic strea Contuous variable Nuber bicyclists waitg at the stop le Contuous variable Nuber of conflictg pedestrians Contuous variable Effective width of pedestrian crosswalk (su of the body width of present pedestrians is deducted fro actual width of the pedestrian crosswalk and converted to a ratio value between 0 and ) Contuous variable Choice Distribution In total, left-turng bicyclists arrivg durg the red phase of the straight-through signal were observed; ridg bicycle lane and ridg the left ost vehicle lane. No other paths apart fro the ones cluded the itial choice set, which were illustrated FIGRE, have been observed; therefore, the universal choice set reas the sae. However, aong the riders the bicycle lane group no one selected the vehicular turn, and as a result this path will be oitted fro the choice set. The choice distribution for bicyclists ridg bicycle lane is alost the sae on all segents of the tersection. It is likely due to the frastructural siilarities on the segents. Alost 0% of bicyclists have chosen bicycle turn and 0% have selected the pedestrian turn. However, the vehicle lane group, the vehicular turn is ore attractive than the pedestrian turn. The descriptive statistics reveals that for the bicycle lane group, % of those who ride faster than the average speed of the approach and % of peak-hour riders have selected the bicycle turn versus the bicycle turn. These values are % and % for the vehicle lane group respectively. Below, the choice distribution per segent for both group of riders is suarized: TABLE Choice Distributions Bicycle Lane Vehicle Lane Approach Bicycle Pedestrian Pedestrian Vehicular Total Total turn turn turn turn North East South West 0 Total 0

10 Ai, Twaddle, Leonhardt Explanatory Variables Analysis In this section the itial set of explanatory variables is analyzed to discover which variables strongly fluence the bicyclists choice (dependent variable) and to vestigate potential correlation between the dependent variables. First, the field effect variable is discussed; the value of Fn is the portion of bicyclists selectg path (pedestrian turn) and is 0., 0., 0. and 0. for north, east, south and west approach of the tersection respectively. The value of this variable is ore or less 0. all segents for the bicycle-lane group, which is a first dication that the field effect variable could be significant. For the vehicle-lane group, due to sall saple size of each segent, it is not possible to clude the field effect variable to the utility functions. Pearson s correlation coefficient is coputed for each pair of dependent variables order to vestigate their lear correlation. High values of correlation coefficient (e.g. 0. or 0.) dicates a strong lear correlation, and thus, only one of the two tercorrelated variables should be kept the odel. Developg the correlation atrix, reveals that the queue length (X q ) is strongly correlated to a nuber of dependent variables like X red, X cc, X st. Thus, X q is reoved fro the odel and X st (the stop le accessibility) is used to reflect the fluence of the waitg bicyclists the odel. Furtherore, cludg X cc the bicycle-lane odel is not necessary as none of the bicyclist selected the vehicular turn. In addition to the statistical analysis, observational conclusions have been ade to add or drop soe variables fro the utility functions. Conflicts with pedestrians ostly occur on the north crossg, and additionally, ost cases even if there is a huge crowd walkg across the crosswalk, bicyclists do not change their path, but try to slightly adjust their trajectory to avoid collision with the crowd. Therefore, X ped and X w will be discarded fro the odel. To conclude the fdgs of the descriptive statistics, the odels which will be estiated the next section are forulated below: Bicycle lane odel (cludg the social fluence): Choice set = {bicycle turn (path ), pedestrian turn (path )} red X rs X red rs s X s X st st ph X If the field effect variable is statistically significant or close to zero, the standard forat of the odel will be used, which cludes the Alternative Specific Constants (ASC) as well: ASC ASC red X rs X red rs X s s X st st ph ph X ph Vehicle lane odel: Choice set = {pedestrian turn (path ), vehicular turn (path )} ASC ASC X rs red X rs X red s X cc s X cc st st It is worth to note that these are not the fal for of the functions; the ultiate odel will be obtaed after conductg the necessary statistical tests, external validation and refg the explanatory variables to reach the best fit odel. (-) (-) (-) (-) (-) (-)

11 Ai, Twaddle, Leonhardt Logit Model Estiation In this section, first a logit odel would be estiated usg BIOGEME freeware () and then, the backward approach is eployed to reove the significant variables fro the odel. More specifically, variables with lower p-value will be oitted stepwise until all p-values are saller than the defed threshold. Even though odern odels criticize the usefulness of hypothesis testg approaches (), for this specific case study due to liited nuber of explanatory variables these ethods are siple and useful. The outcoe of the estiation are assessed by three criteria:. Model perforance: it is evaluated based on rho-squared and adjusted rho-squared values which is a easure for goodness of fit.. Model significance: this easure analyzes the significance of the coefficients. This is done via t-test at % confidence terval, which eans the value of t-test ust be equal or higher than. to be able to reject the null hypothesis.. Model correctness: correctness of the coefficients is the last easure to check if the sign of the coefficients are the sae as the expected sign. The bicycle-lane odel was estiated first with consideration of the field effect variable. However, the t-test value for all the arket specific constants shows they are significant at the 0% terval. Consequently, the field effect variable is excluded and the odel is estiated without considerg social fluence. At this step, the odel is estiated by considerg the utility functions as forulated Equations (-) and (-). Then, the odel is refed by eployg the backward eliation procedure, which the less significant coefficients are oitted one by one to see if the perforance of the odel decreases considerably. For the bicycle lane odel, β ph and β st have the greatest p-value and despite the fact that they both have the expected sign, they should be discarded stepwise (odel significance criterion is violated). ltiately, a likelihood ratio test, which is a useful test to copare a full odel with a restricted odel, was conducted to select the best odel. The followg table suarizes the results of the paraeter estiation: TABLE Estiation Results for the Bicycle Lane Model Coefficient Value Standard Error T-Test P-Value ASC 0 fixed - - ASC βred βrs βs Nuber of observations Null log-likelihood -0. Fal log-likelihood -0. Likelihood ratio test. Rho-squared 0. Adjusted rho-squared 0.0

12 Ai, Twaddle, Leonhardt 0 0 Fro the table it can concluded that β red is the ost fluential paraeter. The positive sign of this factor eans that as the end of red phase of the straight-through signal nears, bicyclists are ore likely to select the bicycle turn. Moreover, faster bicyclists are expected to choose the pedestrian turn, possibly because they are less patient. The value of β rs iplies that if the pedestrian crosswalk signal is green upon arrival, the probability of choosg the pedestrian turn is higher than the bicycle turn. Regardg the vehicle lane odel, as discussed earlier, the saple size is too sall to provide enough variation aong the selected variables. This issue can be observed better when the paraeters are estiated: TABLE Estiation Results for the Vehicle Lane Model Coefficient Value Standard Error T-Test P-Value ASC ASC 0 fixed - - βcc βph βred βrs βs βst Nuber of observations Null log-likelihood -. Fal log-likelihood -. Likelihood ratio test. Rho-squared 0. Adjusted rho-squared 0. All paraeters except β red and β cc are statistically significant. Therefore, the estiation procedure of the vehicle lane odel is terated and this odel will not be discussed further. Model Validation Two types of validation are perfored: first a face validity by coputg the adjusted percentage of right value as presented by (): 00 PR Pn ( i) N n i y Where P n (i) is the probability of choosg alternative i for person n, and y is if the highest predicted probability by the odel corresponds to the chosen alternative, and 0 otherwise. This adjusted statistic reflects the value of log likelihood function better and is ore sensitive to ()

13 Ai, Twaddle, Leonhardt low values of probabilities for the chosen alternative. The bicycle lane odel predicts out of correctly which eans a PR value of.. Second type of validation is analyzg the accuracy of the odel on an dependent dataset at both aggregate and disaggregate levels, which is also known as external validation. Six hours of video data fro the sae tersection but fro another day with siilar conditions (weather, weekday, traffic, etc.) is collected and analyzed with the sae approach. In total 00 bicyclists are recorded, of the rode bicycle lane. The external validation is perfored at both the aggregate and disaggregate level. At aggregate level, the choice distribution of the odel is copared with the actual choice distribution of the epirical data. At disaggregate level, the choice distribution is copared for each path separately. More specifically, the accuracy of prediction for each path is checked with dividuals who have chosen that specific path. The followg table suarizes the results of the external validation: TABLE External Validation of the Bicycle Lane Model Epirical Data Aggregate Disaggregate Path Total Percentage Nuber Percentage Correct Percentage Bicycle turn.%.%.% Pedestrian 0.%.%.% turn Total 00% - -.% CONCLSION This study presented an MNL odel to predict the path selection of left-turng bicyclists at signalized tersections. As a case study, video data was collected at a busy tersection Munich, Gerany and data extraction was perfored anually to create a revealed preference dataset. Discrete choice analysis was eployed to develop the MNL odel, which is successful accurately predictg the choice behavior of the bicyclists ridg bicycle lane. The predicted path selection ratio is only.% different fro the epirical data. However, due to the sall portion of bicyclists that ride the vehicle lane, it was not possible to estiate a consistent odel. Nevertheless, fro the bicycle lane odel it can be concluded that X red is the ost fluential variable. Another key fdg of this research dicates that bicyclists who are ridg faster than the average of a segent, are ore likely to select the pedestrian turn. Moreover, external validation denotes that the odel correctly predicts the selected path alost % of the ties. With further studies, the ipact of gender and helet use as well as frastructural factors such as signal cycle tie and crossg distances can be vestigated. The nature of the observation zone that has been considered this study easily suits the route decision algoriths on the existg traffic siulation tools. The tegration of the presented choice odel to an existg icroscopic traffic flow siulation tool and its deployent will be presented a future work. REFERENCES. Noland, R. B., and H. Kunreuther. Short-run and long-run policies for creasg bicycle transportation for daily couter trips. Transport Policy, Vol., No.,, pp... Dill, J. Bicyclg for transportation and health: the role of frastructure. Journal of public health policy, Vol. 0, 00, pp. S S0.

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