mathematics//probability//stochastic process
A stochastic process is a sequence of random variables indexed by time, \(x_0, x_1, x_2,\dots\), together with a structure that says how each one depends on those before it, and it is the model of how noise, drift, disturbances and degradation are generated over time. In a system that runs in time, what matters is how the sample of now relates to the sample of a second ago, and that relation is what lets the noise be predicted on average and filtered.
A stochastic process is a sequence of random variables indexed by time, x0,x1,x2,…x_0, x_1, x_2,\dotsx0,x1,x2,…, together with a structure that says how each one depends on those before it, and it is the model of how noise, drift, disturbances and degradation are generated over time. In a system that runs in time, what matters is how the sample of now relates to the sample of a second ago, and that relation is what lets the noise be predicted on average and filtered.
Three processes cover almost everything an engineer filters, and all three are built from a sequence wkw_kwk of independent Gaussian draws of variance qqq. White noise is the sequence itself, xk=wkx_k=w_kxk=wk, with no memory at all. The random walk adds it up, xk+1=xk+wkx_{k+1}=x_k+w_kxk+1=xk+wk, forgets nothing, and its spread grows as the square root of time, which is what happens to the angle of a drone whose gyroscope is integrated. The Gauss-Markov process adds it up with a leak, xk+1=a xk+wkx_{k+1}=a,x_k+w_kxk+1=axk+wk with ∣a∣<1|a|<1∣a∣<1, and wanders slowly without running away, the standard model of a sensor bias.
A stochastic process is a model of how the noise is made, and that is why it can be filtered.
Noisy data by themselves can only be smoothed; a model that says the noise is white, or walks, or wanders on a leash tells a filter what to expect at the next step and how much to trust each reading.
Two properties decide what can be learned from a recording. Stationarity asks whether the statistics stay the same over time, so that a baseline learned in July still holds in January; ergodicity asks whether one long record says the same as many machines watched at once. Memory itself is measured by the autocorrelation, zero beyond lag zero for white noise and decaying as aℓa^\ellaℓ for Gauss-Markov.
When the state is discrete (healthy, degraded, failed) the process is a Markov chain, and when each event spawns a random number of new ones (a failure loading its neighbours) it is a branching process. Events scattered in continuous time are a point process.
Which process dominates a real sensor at each timescale is read from its Allan variance; the series a plant historian stores are realizations of processes that time series models try to identify from a single recording.