pp 1-7 | Cite as

Non-Recursive Representation of an Autoregressive Process Within the Magic Square

  • Ina LothEmail author
  • Boris Kargoll
  • Wolf-Dieter Schuh
Part of the International Association of Geodesy Symposia book series


A stochastic process can be represented and analysed by four different quantities in the time and frequency domain: (1) the process itself, (2) its autocovariance function, (3) the spectral representation of the stochastic process and (4) its spectral distribution or the spectral density function, if it exits. These quantities and their relationships can be clearly represented by the “Magic Square”, where the quantities build the corners of this square and the connecting lines indicate the transformations into each other.

The real-valued, time-discrete, one-dimensional and covariance-stationary autoregressive process of order p (AR(p) process) is a frequently used stochastic process for instance to model highly correlated measurement series with constant sampling rate given by satellite missions. In this contribution, a reformulation of an AR(p) to a moving average process with infinite order is presented. The Magic Square of this reformulated process can be seen as an alternative representation of the four quantities in time and frequency, which are usually given in the literature. The results will be evaluated by discussing an AR(1) process as example.


Autoregressive process Moving average process Spectral analysis Stochastic process Time series analysis 


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Geodesy and GeoinformationUniversity of BonnBonnGermany
  2. 2.Geodetic InstituteLeibniz University HannoverHannoverGermany

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