mathematics//probability//stochastic process//Gauss-Markov process
A Gauss-Markov process (first order) is a stochastic process in which each value is a fixed fraction of the previous one plus a fresh Gaussian step, and it is the standard model of a sensor error that wanders slowly without running away, such as the bias of a gyroscope drifting with temperature or the slow error of a GPS position. It behaves like a random walk on a leash.
A Gauss-Markov process (first order) is a stochastic process in which each value is a fixed fraction of the previous one plus a fresh Gaussian step, and it is the standard model of a sensor error that wanders slowly without running away, such as the bias of a gyroscope drifting with temperature or the slow error of a GPS position. It behaves like a random walk on a leash.
xk+1=a xk+wk,∣a∣<1x_{k+1}=a\,x_k+w_k,\qquad |a|<1xk+1=axk+wk,∣a∣<1
With aaa close to one the process remembers for a long time; with aaa small it forgets almost at once and looks like white noise. Its memory is the correlation time τ=−Δt/lna\tau=-\Delta t/\ln aτ=−Δt/lna, with Δt\Delta tΔt the sampling period, and its autocorrelation decays as aℓa^\ellaℓ. The leak bounds it: while each step adds variance qqq, the pull back towards zero removes more the further out it is, and the variance settles at
Var(x∞)=q1−a2.\operatorname{Var}(x_\infty)=\frac{q}{1-a^2}.Var(x∞)=1−a2q.
A gyroscope bias with a correlation time of 100 s sampled at 100 Hz has a=e−0.0001≈0.9999a=e^{-0.0001}\approx0.9999a=e−0.0001≈0.9999, and if it wanders by 0.010.010.01°/s in steady state the step that drives it has a standard deviation of only about 1.4×10−41.4\times10^{-4}1.4×10−4°/s.
An error with predictable structure stops being noise and becomes a state.
Put a Gauss-Markov bias into a filter's state (state augmentation), with aaa in the model and qqq in the process noise, and the filter learns the bias as it goes instead of mistaking it for the signal.
In continuous time it is white noise passed through a first-order low-pass filter with time constant τ\tauτ (the Ornstein-Uhlenbeck process), which is often how it arises physically: a fast random disturbance smoothed by a slow thermal or mechanical response.
Its two parameters, τ\tauτ and the steady-state spread, are read from a long static recording, from the decay of the autocorrelation or from the shape of its Allan variance plot around the bias-instability floor. When the mission is short compared with τ\tauτ, the leak never acts and a random walk with the same step does as well.
It is the first-order case of an autoregressive model (autoregressive model); errors with oscillating memory, like vibration at a fixed frequency, need a second-order model.