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Data assimilation does not only take the observation into simulation models and process it as extra information, but also, via data assimilation algorithm, generate better system state estimation based on the observation. Data assimilation algorithms require the system model to be a dynamic state space model, while in most cases simulation models are not. Dynamic state space model defines the relation between system states and their observation by system transition and measurement function.

st=f(st−1, µt) (3.12a)

mt=M(st, νt) (3.12b)

In equation (3.12a), f is the system transition which forwards the system by one time step. We

note that the time step t−1 and t are used to indicate the stepwise nature of the process. The

actual time interval between two consecutive steps is usually defined by how often observation can

be obtained, for example, in every 30 minutes. The random component, denoted as µ, is included

in system transition function since this dynamic state space model has to be stochastic. There- fore, the next state then becomes a stochastic variable which follows the probability distribution

which generates observation from system state. st is the system state at time step t and νt is the

measurement noise. Similarly, mt, the measurement at time step t is conditionally dependent on

st, of which the probability distribution is p(mt |st).

We assume an arbitrary simulation model has a system transition function st0 = sim(s_t0₋₁)

which advances the simulation from time t0−1 to t0 . To become a dynamic simulation model, it

needs to be modified in several aspects.

First, system transition function of the simulation model should make sure the time is consistent

with the dynamic state space model, that is t0 ≡t. In (3.12a), system state changes over time and

the simulation time advances by the given amount of time. For example, if system state st is

advanced by 4t, then the result state will be st+4t and the simulation time becomes t +4t.

However, some simulation models do not inherently simulates in this way. For example, in DEVS model, the simulation time will not be guaranteed when given a mount of time since the system transition function of DEVS model advances the time according to scheduled events. A DEVS model won’t be able to stop at a certain time unless a event is scheduled at the same time.

In addition, a measurement function needs to be implemented for simulation models, which maps the system state to observation in every time step. This observation is one of the essential component of data assimilation algorithms, which will be used for better estimation. However, simulation models do not include this ability originally. Even dynamic data driven simulation [67], which can take the real time sensor data as input, does not necessarily retrieve the data by itself (through deployed sensor or history data). Therefore, we have to build up the measurement function based on the system states to be estimated, and implement a component to define how the observation is retrieved and what data can be observed. For example, in [56], we develop a sensor component to deploy sensors along each road segment and define the sensor data as a composition of average vehicle velocity and the number of vehicles in the detection area. The measurement function maps the average behaviors on all road segments into a list of sensor data.

Furthermore, the simulation model is required to be a stochastic model instead of a determinis- tic model. As is shown in equation (3.12a), system transition function requires a random component so that the posterior system states follows system transition distribution. If the simulation model

is a deterministic model, we need to add a random component to build a stochastic model. For example, in our previous work [50], DEVS-FIRE model is originally a deterministic model. The next fire state is determined by the fuel type of the burning area, as well as the wind speed and direction. In order to perform data assimilation algorithm on DEVS-FIRE model, the wind speed and direction have been changed to random variables that follows normal distribution with an aver-

age of a given (µ, σ). Besides, the design of the random component is related to the model error or

the measurement error. A good random noise would increase the possibility that a sampled system state becomes closer to the real system state, or a sampled measurement is less biased than the sensor data (sensors are likely to be incorrect). For example, in wildfire simulation, the real data of wind speed and direction is observed in a sparse time interval. However, the wind direction and speed is actually changing constantly in a real wildfire. There are chances that a random noise that added to this parameters can be more accurate than the observed data, which results in higher weight and more likely to be selected in re-sampling step.