On the Relationship Between Mean Absolute Error and Age of Incorrect Information in the Estimation of a Piecewise Linear Signal Over Noisy Channels

We consider the remote estimation of a stochastic piecewise linear signal, observed by a sensor, at a monitor. The sensor transmits a packet whenever the observed signal's slope changes. The packets are transmitted from the sensor to the monitor through an unreliable channel which randomly loses packets. The monitor sequentially estimates the signal using the information obtained from successfully received packets. The sensor does not have any feedback from the monitor. We derive an analytical expression for the average age of incorrect information, a recently proposed information freshness metric. The average age of incorrect information is shown to be a function of success probability of transmission and signal parameters representing the rate and clustering of slope changes. We obtain an upper bound on the mean absolute error of the remote estimate using the slope-weighted age of incorrect information. The age of incorrect information is also studied for a homogeneous multisensor scenario, where sensors use slotted ALOHA and the links between the sensors and the monitor are unreliable due to contention.