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Suppose there is a system where certain objects move through a network. The objects are detected only when they pass through a sparse set of points in the network. For example, the objects could be vehicles moving along a road network, and observed by a radar or other sensor as they pass through (or originate or terminate at) certain key points in the network, but which cannot be observed continuously and tracked as they travel from one point to another. Alternatively they could be data packets in a computer network. The detections only record the time at which an object passes by, and contain no information about identity that would trivially allow the movement of an individual object from one point to another to be deduced. It is desired to determine the statistics of the movement of the objects through the network. I.e. if an object passes through point A at a certain time it is desired to determine the probability density that the same object will pass through a point B at a certain later time.
The system might perhaps be represented by a graph, with a node at each point where detections are made. The detections at each node can be represented by a signal as a function of time, where the signal is a superposition of delta functions (one per detection). The statistics of the movement of objects between nodes must be deduced from the correlations between the signals at each node. The problem is complicated by the possibility that a given object might move between two nodes along several alternative routes (perhaps via other nodes or perhaps not), or might travel along the same route but with several alternative speeds.
What prior knowledge about the network, or constraints on the signals, are needed to make this problem solvable? Is it necessary to know the connections between the nodes or the pdfs for the transition time between nodes a priori, or can this be deduced? What conditions are needed on the information content of the signals? (I.e. if detections are very sparse on the time scale for passage through the network then the transition probabilities can be built up by considering each cascade of detections independently, while if detections are dense then it will presumably be necessary to assume that objects do not move through the network independently, but instead tend to form convoys that are apparent as a pattern of detections that persist for some distance on average). What limits are there on the noise in the signal or amount of unwanted signal, i.e. false detections, or objects which randomly fail to be detected at a particular node, or objects which are detected at one node but which do not pass through any other nodes? Is any special action needed to enforce causality, i.e. positive time delays for transitions between nodes?
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