Identifying Risk Groups in Flanders: Time Series Approach

Transcription

1 Identifying Risk Groups in Flanders: Time Series Approach RA-MOW D. Karlis, E. Hermans Onderzoekslijn Risicobepaling DIEPENBEEK, STEUNPUNT MOBILITEIT & OPENBARE WERKEN SPOOR VERKEERSVEILIGHEID

2 Documentbeschrijving Rapportnummer: Titel: RA-MOW Identifying Risk Groups in Flanders: Time Series Approach Auteur(s): D. Karlis, E. Hermans Promotor: Prof. dr. Geert Wets Onderzoekslijn: Risicobepaling Partner: Universiteit Hasselt Aantal pagina s: 38 Projectnummer Steunpunt: 6.1 Projectinhoud: In dit project worden prognoses op vlak van verkeersveiligheid in Vlaanderen gemaakt. Uitgave: Steunpunt Mobiliteit & Openbare Werken, juni Steunpunt Mobiliteit & Openbare Werken Wetenschapspark 5 B 3590 Diepenbeek T F E info@steunpuntmowverkeersveiligheid.be I

3 Samenvatting Titel: Identificeren van risicogroepen in Vlaanderen: Tijdreeksbenadering Jaarlijkse ongevallen- en blootstellingsdata voor Vlaanderen uit de periode worden gebruikt om state-space modellen op te stellen en verkeersveiligheidsvoorspellingen voor de periode te maken. We maken gebruik van het zogenaamde Latente Risico Tijdreeksmodel, dat geschikt is voor het modelleren van ongevallendata om op die manier inzicht te verkrijgen in de verkeersveiligheidssituatie die kan verwacht worden in de komende jaren. In dit model worden twee componenten, blootstelling enerzijds en verkeersdoden (of een andere categorie van verkeersslachtoffers) anderzijds, gelijktijdig beschouwd. Bovendien focussen we in de analyse ook op kleinere subgroepen, bepaald op basis van de leeftijd van het verkeersslachtoffer, het wegtype en het type weggebruiker (of zijn transportmodus). De voorspellingen duiden op een verwachte daling in het aantal verkeersdoden, hoewel dit niet aan hetzelfde tempo zal gebeuren voor de verschillende subgroepen. Steunpunt Mobiliteit & Openbare Werken 3 RA-MOW

4 English summary Annual accident and exposure data from Flanders covering the period are used in order to create state space models and make road safety predictions for the period We make use of the Latent Risk time series model, suitably developed for accident data in order to forecast the road safety situation that can be expected in the forthcoming years. In this model two components, the exposure measurement and the fatalities measurement (or another category of road casualties), are fitted simultaneously. Moreover, in the analysis we also focus on smaller subgroups, depending on the age of the road traffic victim, the road type and the road user type (or transport mode). Forecasts clearly show that the number of fatalities is expected to decrease, however not at the same rate for different subgroups. Steunpunt Mobiliteit & Openbare Werken 4 RA-MOW

5 Inhoudsopgave 1. INTRODUCTION STATE SPACE MODELS THE LATENT RISK MODEL RESULTS FROM AGGREGATE MODELS RESULTS FROM DISAGGREGATE MODELS By Road user type By Age category By Road type SOME STATISTICAL CONSIDERATIONS CONCLUSIONS REFERENCES APPENDIX... 32

6 1. I N T R O D U C T I O N Road traffic crashes are one of the world s largest public health and injury prevention problems. The problem is all the more important because the victims are overwhelmingly healthy prior to their crashes. A report published by the World Health Organization (WHO, 2009) estimated that approximately 1.3 million people die each year on the world's roads, between 20 and 50 million sustain non-fatal injuries and traffic accidents were the leading cause of death among children of years of age. Undoubtedly there is awareness in most societies about this issue and reducing the fatalities from road accidents is always on every political agenda. Also, the issue of traffic safety is high in the academic agenda and a lot of research is undertaken in order to examine and improve traffic safety issues. Lately, there was a downward trend in the number of fatalities in most countries in Western Europe, North America and Oceania (see Elvik, 2010, see also Lassarre, 2001), reflecting the awareness of the problem as well as all the measures undertaken to decrease it. However, apart from fatalities there is also a great concern for the public with respect to other types of non-fatal accidents as they also produce significant losses and thereby contribute to the economic costs. In Flanders, almost casualties were registered in 2009 (FOD Economie, 2011). The purpose of the current report is to forecast the (disaggregate) level of road safety in Flanders up to The data used in the analyses are yearly data up to 2007 covering different subgroups like different age categories, different road user types as well as different road types. A close look in such subgroups is also of primary importance in safety research as it can reveal vulnerable subgroups for which particular measures are most urgently needed. Our forecasts are based on a state space model developed by Bijleveld et al (2008) and is suitable for road safety data as it captures the basic ideas in road safety research. The model assumes that road casualties (of a particular severity level, e.g. fatalities) are the result of the road risk and the exposure of individuals to that risk. While exposure can be approximated using real data the risk is a latent factor not directly observable. So we make use of the Latent Risk time series model that on the one hand treats the risk as latent and on the other hand models the exposure and the casualties at the same time. The model is applied to fatalities, casualties, serious and slight injuries using data from the entire population. As exposure we use the total number of kilometres travelled (in millions). Then we focus on subgroups, namely road type, road user and age. As exposure now we use relevant detailed data if available or proxies if not. Moreover, we primarily focus on fatalities in the disaggregate analyses. The report proceeds as follows: Section 2 briefly introduces the state space models, while section 3 describes the Latent risk model used in the report. Section 4 contains the main results for the entire population. In section 5 we provide some disaggregate analysis focusing on particular subgroups of the population. Working with subgroups offers interesting challenges as for example, most measures affect only a part Steunpunt Mobiliteit & Openbare Werken 6 RA-MOW

7 of the population. Section 6 deals with some statistical considerations about the model fitting. Concluding remarks can be found in section 7. Additional results have been put in the appendix. Steunpunt Mobiliteit & Openbare Werken 7 RA-MOW

8 2. S T A T E S P A C E M O D E L S Road safety data are typically data observed in subsequent time points, creating, hence, a time series. The density of the observations depends on the way they are collected and can have very different time spans. In this report we consider annual data 1, which implies that seasonality has been cancelled out (and consequently, seasonality issues will not be described). A powerful class of time series models are the dynamic models, i.e. models where the parameters may change over time. There are two main classes of univariate dynamic models: ARIMA models studied by Box and Jenkins and unobserved component models which are also called structural models, by Harvey and Sheppard (1993). In a structural model each component or equation is intended to represent a specific feature or relationship in the system under study. State space methods described in this section, belong to the latter group of models. A typical time series may be decomposed in a trend, a seasonal and an irregular part. An important characteristic is that the components are stochastic. Models without stochastic component are called static. Moreover, explanatory variables can be added and intervention analysis carried out. The principal structural time series models are therefore nothing more than regression models in which the explanatory variables are functions of time and the parameters are time-varying. The key to handle structural time series models is the state space form, with the state of the system representing the various unobserved components. State space time series analysis began with the path breaking paper of Kalman (1960) and early developments in the subject took place in the field of engineering. Once in state space form, the Kalman filter may be applied and this in turn leads to estimation, analysis and forecasting. The state space model in its simple form can be expressed as y Z a, a t t t T a R, t ~ N(0, H ), ~ N(0, Q ), t 1 t t t t t t with initial value a1 ~ N( 1, P1 ) t t where matrices be relaxed). Z, t, Ht, Tt Rt and t Q are assumed known (however this assumption can Note matrices equations and H t and Z T, R t t t Q t are covariance matrices associated with the errors of the, are matrices used to appropriately define a multitude of models and they may contain coefficients to be estimated as well. 1 Disaggregate models, providing detailed insights, require more detailed data. At subgroup level, exposure data are difficult to find and often non-existing on a e.g. monthly basis. Steunpunt Mobiliteit & Openbare Werken 8 RA-MOW

9 The key idea of state space models is that a certain parameter a t relates to the parameter at the previous time point, inducing a dynamic linear model. The first equation is called the observation (or measurement) equation and the second equation is called the state equation. The state space formulation for time series models is quite general and encompasses most of the classical time series models like MA and ARIMA models for example. Also since the state equation(s) can capture in a very flexible way the behaviour of the underlying (and unobservable) variables it offers great flexibility with real data. The advantages of state space modelling can be summarized (see, e.g. Durbin and Koopman, 2001) as: They are based on a structural analysis of the problem at hand. The different components that may comprise a time series model, can themselves be modelled separately. They offer greater generality. In fact, several other models can be seen as special case of the state space models. They satisfy the Markovian property and hence the necessary calculations can be put in a typical recursive manner. Forecasting with state space models is relatively easy and simple. State space models in fact apply some smoothing in the data and hence forecasts are also smooth. In addition, diagnostic checking is simple as the Kalman filter employed provide such a framework. State space models are adaptive and the benefits of this are usually realised by implementing them in real time since only minor calculations are needed. Finally, they offer great flexibility as they can be used in certain circumstances, allowing for refined modelling in several problems. At the same time, some disadvantages should be mentioned. The models are usually more complicated and less interpretable than standard time series models, especially for non-treated researchers making their acceptance in some problems not easy. In addition, some added computational effort is needed with respect to much simpler models. Finally, note that while for certain models state space modelling is well established and easy to use, there are models where it is not so easy, like for example discrete valued time series models. The model developed by Zeger (1988) is in fact a state-space model for modelling discrete time series. However, assuming a Poisson distribution leads to a rather complicated recursion for the state equation and makes estimation difficult. State space models are currently popular models for accident prediction mainly due to their generality and flexibility (see e.g. Gould et al, 2004, Hermans et al, 2006a, 2006b, Bijleveld, 2008). Several software packages (like R, EVIEWS, MATLAB just to name a few) are available for fitting such models (see the special issue of Journal of Statistical Software, Commandeur et al, 2011). State space models provide a convenient Steunpunt Mobiliteit & Openbare Werken 9 RA-MOW

10 and powerful framework for analyzing time series data. More details can be found in several textbooks devoted to these models, see e.g. Durbin and Koopman (2001) and Commandeur and Koopman (2007). Steunpunt Mobiliteit & Openbare Werken 10 RA-MOW

11 3. T H E L A T E N T R I S K M O D E L The Latent Risk Time series Model (LRT) was introduced by Bijleveld et al. (2008). The LRT model is a particular case of state-space models. It has been developed in order to capture the idea of risk in road safety, an unobservable quantity which in fact plays a very important role in accident analysis. Road safety is usually affected by two factors: the risk and the exposure of the individuals to that risk. This approach was first developed by Oppe (1989, 1991). This decomposition implies that in order to analyze issues related to road safety one must be able to measure both quantities. While exposure can be measured using several different indicators, measurement of the risk is not easy. The cornerstone assumption is that traffic safety is the product of the respective developments of exposure and risk (Bijleveld, 2008); typically, exposure can be measured by traffic volume while number of fatalities (or casualties in general) is the product of exposure and (fatal) risk (which is unobservable). The stochastic model considered implies also some errors added to the above relationships, i.e. traffic volume is a proxy of exposure and not a full observation of it while the product of exposure and risk does not fully determines the fatalities. Typically one works with logarithms. A plain explanation for this is that firstly road safety quantities are positive numbers so logarithmic transformation guarantees consistent estimation. Secondly, taking logarithms implies a linear relationship in the logarithmic scale which is a more realistic assumption and thirdly, this makes the developed models easier to be fitted with real data. The LRT model developed in Bijleveld (2008) contains two measurement equations: one for traffic volume, and one for fatalities. In fact the model simultaneously fits two dependent variables (traffic volume and fatalities). In addition to each of these measurement equations two state equations correspond: For traffic volume the measurement equation is (3.1) while the state equations are (3.2) For the fatalities, the measurement equation is: while the state equations are: R (3.3) R (3.4) where is the traffic volume at time t, is the exposure variable at time t, is the number of fatalities at time t, and is the risk at time t, which is not observed, i.e. it is latent. Several extensions of this basic model can be considered, by allowing additional explanatory variables to be present, including the case of dummy variables, usually with Steunpunt Mobiliteit & Openbare Werken 11 RA-MOW

12 respect to interventions. Also, note that in the models above we assume normal distributions for the errors considered. This allows to create models inside the normal family. Estimation is not straightforward due to the recursive way in which the model is defined. Kalman filters are of special importance for such models. The LRT allows to consider together all the important aspects of road safety. Risk is latent and quantified via this model. The errors are considered to be normally distributed, which implies that the two dependent variables are normally distributed in the logarithmic scale. For details about estimation, prediction and other statistical properties we refer the interested reader to Bijleveld et al. (2008). Steunpunt Mobiliteit & Openbare Werken 12 RA-MOW

13 4. R E S U L T S F R O M A G G R E G A T E M O D E L S In this section we present the results on the aggregate forecasts for Flanders. Annual observed data from 1991 to 2007 were used. The road safety indicators considered are the number of fatalities, the total number of casualties, the number of severely injured persons and the number of slightly injured persons. The official Flemish casualty data was obtained from the FOD Economie. With respect to exposure we used the number of total kilometres travelled for that period in millions (Federaal Planbureau). The LRT model described in section 3 was fitted, thereby jointly modelling exposure on the one hand and a road safety outcome indicator (e.g. fatalities) on the other. Note that we have used the same model for casualties and injuries since the idea of risk is the same for these kinds of measurements of traffic safety. Figure 1 presents the real data and the forecasted values. The vertical dotted line implies the period where the forecasting started (2008 in this study). On the left, the observed values are shown while on the right we can see the forecasted values (dots) based on the model, together with a 95% forecasting interval to present the uncertainty around the forecast. The fitted model implied a linear forecasting, but this applies to the logarithmic scale as described in section 3, so we can see some curvature in the predictions. Figure 1 Forecasted values for the years together with the observed values for the number of fatalities in Flanders (yearly data available for ) Steunpunt Mobiliteit & Openbare Werken 13 RA-MOW

14 From Figure 1 a downward trend in the number of fatalities can be deduced resulting in a forecasted number of 360 fatalities by As expected a longer forecasting horizon implies a larger uncertainty. Note that the interpretation of the figure is the same for all figures presented in this section. Figure 2 deals with the number of casualties (i.e. the sum of fatalities, severe injuries and slight injuries). The uncertainty is much larger now. There is again a downward trend yet it is less than the one for fatalities. An explanation for this is that (see the right panel in Figure 3) the slight injuries are not expected to decrease a lot and they make up a larger part of the casualties. Figure 2 Forecasted values for the years together with the observed values for the total number of casualties in Flanders (yearly data available for ). Figure 3 presents the forecasts for the severe injuries (left) respectively slight injuries (right). Severe injuries are forecasted to decrease to 3690 by For slight injuries one notices that the variability is very large and that the overall trend in is not decreasing but rather stable (keeping in mind the large fluctuations that were present). Thus forecasted values are quite close to the 2007 level and not expected to decrease a lot. As already mentioned this has an effect on the overall number of casualties as slight injuries are the largest contributor to this number. Steunpunt Mobiliteit & Openbare Werken 14 RA-MOW

15 Figure 3 Forecasted values for the years together with the observed values for the number of severely injured persons on the left panel and slightly injured persons on the right panel, in Flanders (yearly data available for ). Table 1 summarizes the forecasts for all four measurements of traffic safety. Regarding casualties we present two forecasts: one when using the aggregated data (column (1)) and one when each component is forecasted separately and then summed to obtain a forecast for casualties (column (5)). The differences are rather small, as the maximum proportion is around 1% for 2015, which implies a rather good correspondence between both forecasts. The forecasts in column (1) are shown in Figure 2 as they allow for a better estimation of the standard erros. To conclude, table 1 clearly shows that all road safety outcomes are expected to decrease in the following years. Year Casualties (1) Fatalities (2) Severe Injuries (3) Slight Injuries (4) Prediction of casualties from separate components (5)= (2)+(3)+(4) (5)-(1) Relative difference % % % % % % % % Table 1. Forecasts for the different road safety outcomes. Column (5) predicts the casualties as the sum of the forecasted number of fatalities, severe and slight injuries. The difference from the direct forecast is negligible. Steunpunt Mobiliteit & Openbare Werken 15 RA-MOW

16 As far as exposure is concerned, all 4 models provided forecasts for the total kilometres travelled (in millions). They are presented in Figure 4. In the Appendix (A2) we also depict the uncertainty around the 4 forecasts which clearly shows that the forecasts mostly agree and the observed small differences are due to the model and the uncertainty of the different variables used in the LRT model. Similar analyses are reported for other variables in Appendices A3 and A4. The values of the forecasts can be read from Table 2 for all the models. Figure 4. Forecasted values about exposure (in million of total kms travelled) for the years together with the observed values in Flanders (yearly data available for ). The four predictions are based on the traffic safety variable used in the LRT model. Forecast based on Severe Injuries Slight Injuries Year Casualties Fatalities Table 2. Forecasts for the exposure variable, i.e. the total kilometres travelled in millions. We obtained 4 forecasts, one from each model depending on the traffic safety variable used. Steunpunt Mobiliteit & Openbare Werken 16 RA-MOW

17 Summarizing so far, the fatalities are forecasted to be reduced to 360 by Also the severe injuries are expected to decrease but for slight injuries the decrease is expected to be very small. Since we use data up to to predict the period from 2008 up to 2015, the data for (which became available in the meantime) can be used to comment on the prediction accuracy of the developed model. The comparison is shown in Table 3. Slight injuries Severe injuries Fatalities Observed Forecasted Observed Forecasted Observed Forecasted Table 3. Forecasts based on the developed model and the real observed values for slight injuries, severe injuries and fatalities ( ). One can see that while for 2008 the forecasted values are close to the real figures, as time passes the forecasts are less accurate. Recall that in almost all the cases the values are within the 95% forecast intervals, i.e. taking into account the uncertainty the model does not fail to forecast. However, as time passes, the observed values are closer to the lower limit of the forecasted intervals. Slight injuries are forecasted worse by the model, while fatalities are forecasted more reasonably. About the target of maximum 250 fatalities and maximum 2000 severely injured persons by the year 2015, this seem not to be validated by the model. Concerning the number of fatalities, 250 is still inside the forecasting 95% interval but very close to the boundary, while for the number of severe injuries, the value of 2000 is outside. Hence, the model shows that the targets are hard to be met by In Appendix A1 a small comparison with other simpler models is shown. The findings are similar was the most recent year for which detailed data was available at the start of the analyses. Steunpunt Mobiliteit & Openbare Werken 17 RA-MOW

18 5. R E S U L T S F R O M D I S A G G R E G A T E M O D E L S Disaggregate models are tools for assessing different policy options, setting goals for safety programmes and predicting future safety developments at the disaggregated level. This makes their development of particular importance for better understanding the problem but also for policy and decision making purposes. While disaggregate models can suffer from lack of data, in our case quite accurate and detailed data for certain subcategories exist and thus we present such an analysis in this section. We primarily focus on the fatalities as for this variable the data are more accurate and detailed. However, in Appendix A5 the results from the disaggregated analyses using the (larger) number of casualties are presented. Note that there are two issues that tend to limit the scope for disaggregation. The first one refers to the fact that the numbers (of e.g. fatalities) in each group are typically much less than the overall number (of fatalities), which leads to increased variability. Consequently, it is more difficult to identify trends and hence the uncertainty on predictions is larger. This implies limitations on the level of disaggregation that can be used. The second issue relates to the availability of exposure measures which may be available for the whole population but not for each group separately. In this section, we present the results of applying the LRT model focusing on the following subgroups: Age classes split in 4 categories (ages 0-18, 19-45, 46-64, 65+). Type of road user (cars, trucks, small vans and motorcycles). Type of road (motorways and non-motorways). We have fitted a separate LRT model to each subgroup. Details follow when describing each subgroup. 5.1 By Road user type We worked with 4 categories of road user namely cars, small vans, motorcycles, and trucks. There were data available for other categories like buses but the number of fatalities were too small to build any interesting model. Recall that in the disaggregate analyses we primarily focus on the fatalities as we aim to identify the subgroups with a large share in the forecasted number of fatalities or with a high (or even increased) fatal risk in the future. As exposure variable for the 4 categories we used data on the number of kilometres travelled by this mode (Federaal Planbureau). Road user types for which no (good) exposure data was available (such as pedestrians) were not considered for analysis. Results from the model, with respect to fatalities are reported in Figure 5. One can notice the wide confidence intervals, implying that the uncertainty around the forecasts is large, perhaps invalidating the forecasts themselves. The overall trend is decreasing. There is a clear downwards trend for cars, motorcycles, trucks and a smaller one for small vans. However, the large uncertainty Steunpunt Mobiliteit & Openbare Werken 18 RA-MOW

19 prohibits deriving clear conclusions for all the road user types and thus any result should be interpreted with care. Note also that the data availability covered a smaller time period than the aggregate data, namely only from 1997 to Figure 5: Forecasted fatalities and corresponding 95% intervals for different road user types. The available data cover the period Table 4 contains the forecasted values. The last column is the sum of the values for the 4 user types which is smaller than the number of fatalities forecasted in section 4 since we miss data for some accidents (covering an inhomogeneous class named other which is not used in the forecasting) but also some road users were excluded due to nonavailability of reliable exposure data. Steunpunt Mobiliteit & Openbare Werken 19 RA-MOW

20 Year car Road user small van motorcycle truck Total 2007 (observed) Change from % % % % % % % % Proportion of each user type to the total % 7.74% 16.33% 3.44% % 6.52% 25.76% 1.85% Table 4. Forecasts for the number of fatalities for different types of road users. Forecasts are derived from the LRT model covering the period The last column presents the percentage of decrease from The models forecast a large decrease up to 50% for the year Also at the bottom of the table we have calculated the share of each of the four considered road user types to the total. Interestingly while the car fatalities will decrease, a large increase on the fatalities in motorcycles is expected (from 16.3% in 2007 up to 25.8% in 2015). Also note that the overall decrease concerning motorcycle fatalities is the smallest. Finally, forecasts for the traffic volumes can be read from Table 5 (in millions of vehicle kilometres). The general trend is increasing for all modes. It is interesting however to note that after a small decrease, the model forecasts an increase which is up to 4.8% for 2015 (compared to 2007). The corresponding graphs can be found in Appendix A3. Year Car Road user small van motorcycle truck Total Change since % % % % % % % % Table 5 Forecasts for the traffic volumes for 4 categories of road user type. Steunpunt Mobiliteit & Openbare Werken 20 RA-MOW