Gaussian process
Topic of Dependence along a series, part of Time series
Gaussian process
| Type | Topic |
| Taught in | Dependence along a series |
| Part | Time series |
A process whose every finite collection of values is multivariate normal, so that it is fixed by a mean function and a covariance function. A covariance that favours smooth paths acts like the roughness penalty of a smoothing spline.
In Dependence along a series
A series we record is a finite piece of a process that could have been running for a long time, and often in continuous time. A city’s midday temperature on 365 days is a sample from a continuous temperature path, taken at one hour of each day. The whole process can be studied first, and finite collections of times examined afterwards, which is the probabilist’s direction. Or a model can be written for the finite collection observed, which is the statistical direction. If every finite subcollection is multivariate normal, the continuous-time process is a Gaussian process, and the converse holds too. That equivalence is special to the normal. It is also why a Gaussian-process prior has become a fashionable smoother: one draw of the process is a curve, and a penalty on how irregular the draw is plays the role of \lambda in the smoothing chapters.
Used in
See also
Roughness penalty · Cubic smoothing spline · Multivariate normal distribution · Covariance · Wold decomposition · Autoregression and moving averages · Stationarity of a time series · Random walk with drift