mathematics//probability//stochastic process//stationarity

Stationarity is the property of a stochastic process whose statistics do not change with time, and it is the assumption that lets anything learned from past data be applied to future data: a noise level, a baseline, a forecasting model, an alarm threshold. In the form engineers use (**wide-sense** or weak stationarity) it asks three things: a constant mean, a constant variance, and an autocorrelation that depends only on the lag between two samples and never on when they were taken.

Stationarity is the property of a stochastic process whose statistics do not change with time, and it is the assumption that lets anything learned from past data be applied to future data: a noise level, a baseline, a forecasting model, an alarm threshold. In the form engineers use (wide-sense or weak stationarity) it asks three things: a constant mean, a constant variance, and an autocorrelation that depends only on the lag between two samples and never on when they were taken.

A stationary process can move a great deal. White noise jumps at every sample and is perfectly stationary, because the rules that generate it never change, while a random walk moves no faster and fails the test because its variance keeps growing. Stationarity fixes the law that generates the values and leaves the values themselves free.

Every model learned from history assumes stationarity, and plants rarely provide it.

A vibration baseline learned on a pump in July fails in January: the ambient temperature, the fluid viscosity and the load have all moved, so the detector reads the season as a fault. Seasons, wear, operating regimes, a replaced sensor and a new product all break the assumption quietly.

A linear model says exactly when its output is stationary. An autoregressive process is stationary when every eigenvalue of its companion matrix has modulus below one, the usual discrete stability test (autoregressive model); a random walk has an eigenvalue on the unit circle and is differenced first.

In practice stationarity is arranged: a plant is treated as stationary within each operating regime, a trend or a daily cycle is removed before the rest is modelled, and a model trained on a fixed window is retrained when monitoring shows its inputs have moved (data drift). A baseline that adapts by itself (EWMA) follows slow changes and, for the same reason, slowly absorbs a slow fault.

Ergodicity is the stronger and quieter assumption that the average over one long record equals the average over many copies of the process watched at the same moment. It is made every time a noise level or a failure rate is estimated from a single machine. It fails when each unit has a personality: a year of data from one pump with its own offset gives that pump's offset precisely and says nothing about the spread across the fleet. (In a Markov chain the word names a related property, convergence to one stationary distribution.)