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Chapter 5 : Creating a dynamic bayesian network


All the Bayesian networks presented up to now represent a static knowledge, i.e. a knowledge that is time independent. To take into account the temporal dimension, one has to use dynamic bayesian networks.

The following diagram presents a system with three valves (V1, V2 and V3) that control the flow of a fluid[1].

images 1

In that dynamic system, a valve has two failure mode, and then three possible states: 

  1. OK : the valve functions normally
  2. RO : the valve always remains open (failure) 
  3. RC : the valve remains closed always (failure) 
The problem is being able to predict whether or not the system remains controllable (this means, whether or not we can always control the passage of the fluid in the system). 

 

Example :

If V2 and V3 remain closed, then the system won't let the fluid flow.  

If V2 and V3 remain open, then the passage of the fluid can be controlled with V1 (if it is functioning correctly). 

Here is the bayesian representation of this system :

images 2

Valve1, Valve2 and Valve3 represent the state of the valves at time t. Valve1 t+1, Valve2 t+1 and Valve3 t+1 represent the state of valves at time t+1. Remains Open and Remains Closed can be used to classify the system's failure (for a system that distributes gas, these two types of failure certainly don't have the same consequences). In short, Fluid Distribution determines whether or not the system is controllable.

Rates of failure for each valve are as follows:  

images 3

These rates are represented in the parameters associated to Valve1 t+1, Valve2 t+1 and Valve3 t+1. The table of conditional probability for a given node Valve i t+1 gives the probability for this valve to be in a given state (OK, RO or RC) at time t+1, knowing the state at time t of Valve i.  

images 4

In order to let BayesiaLab know that there is a temporal relation between the nodes Valve1 and Valve1 t+1, simply right click on the arc that links these two nodes and select "Temporal Relation" from the pop-up menu.

images 5

A temporal relation is represented by a red arc.


Analysis

The deterioration of the system over time can be determined by selecting Fluid Distribution as the target node and then monitoring this variable as well as Valve1, Valve2 and Valve3.

images 6

At the beginning of the simulation, the initial state of the input variables must be specified (the blue monitors). This state can be fully observed or it can correspond to a probability distribution when there is some uncertainty about these variables (the probabilities can be dragged to the desired level and then entered by using the green button). 

When the graph is temporal, a new toolbar appears in Validation mode:

images 7

The simulation can be done step-by-step or by pre-determining the number of time steps to simulate. For the step-by-step, you have just to press the arrow images 8. That will execute one step at each time it will be pressed. If you want to simulate a specified number of steps, simply enter the value in the fieldimages 9  and press Enter. 

During the simulations, you can see the deterioration of the system on the monitors.  

In the simulation the user must select the state for the variables to follow temporally (right click on the node and select the option "follow the temporal evolution"; the perimeter of the node will become red).

As the simulation can be stopped at any moment thanks to the red light of the state bar images 9, this number of step can be set to values extremely high.

We chose 2000 time steps which gave the following diagram after pressing the button: images 11

images 12

The red curve represents the probability that the system is operational at time t; the other two curves represent the evolution of the probabilities of the two types of failure.

Also appear in the legend, for each followed modality, its average probability on the simulation period.



[1] We illustrate how BayesiaLab's dynamic Bayesian networks function through a reliability problem taken from « Weber P., Jouffe L., Reliability modelling with Dynamic Bayesian Networks, SafeProcess 2003, 5th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Washington D.C. ».