Theory of Intrinsic Random Functions

2019 ◽  
pp. 45-100
Author(s):  
Georges Matheron
Keyword(s):  
Space Variable ◽  
Nugget Effect ◽  
Formal Expression ◽  
Random Functions ◽  
Intrinsic Random Functions ◽  
Mean And Variance

Chapter 3 discusses intrinsic random functions Y(x) of the space variable x, i.e. functions whose mean and variance of the increments Y(x + h) − Y(x) depend on h only. Half of this variance defines the variogram γ‎(h). The behaviour of the variogram near the origin, such as continuity, the nugget effect, etc., expresses the regularity of these functions. Regularisations of them, by grading and convolutions, produce new variograms, via the same rules as for transitive methods. When Z(v) and Z(v′) are the averages of these functions in v and v′, respectively, then the variance of Z(v) − Z(v′) by attributing to v′ the grade in v is called extension variance. Its formal expression is given and calculated for various patterns of sampling v, in dimensions 1, 2, or 3, via the de Wijsian scheme and the spherical scheme, and for various models of variograms, such as the semi-variogram.

scholarly journals Intrinsic random functions for mitigation of atmospheric effects in terrestrial radar interferometry

2017 ◽  
Vol 11 (2) ◽  
Author(s):  
Jemil Butt ◽  
Andreas Wieser ◽  
Stefan Conzett
Keyword(s):  
Ordinary Least Squares ◽  
Deformation Monitoring ◽  
Radar Interferometry ◽  
Atmospheric Effects ◽  
Random Functions ◽  
Mountainous Regions ◽  
Temporal Averaging ◽  
Intrinsic Random Functions ◽  
Mean And Variance ◽  
Trend Functions

AbstractThe benefits of terrestrial radar interferometry (TRI) for deformation monitoring are restricted by the influence of changing meteorological conditions contaminating the potentially highly precise measurements with spurious deformations. This is especially the case when the measurement setup includes long distances between instrument and objects of interest and the topography affecting atmospheric refraction is complex. These situations are typically encountered with geo-monitoring in mountainous regions, e.g. with glaciers, landslides or volcanoes.We propose and explain an approach for the mitigation of atmospheric influences based on the theory of intrinsic random functions of order k (IRF-k) generalizing existing approaches based on ordinary least squares estimation of trend functions. This class of random functions retains convenient computational properties allowing for rigorous statistical inference while still permitting to model stochastic spatial phenomena which are non-stationary in mean and variance. We explore the correspondence between the properties of the IRF-k and the properties of the measurement process. In an exemplary case study, we find that our method reduces the time needed to obtain reliable estimates of glacial movements from 12 h down to 0.5 h compared to simple temporal averaging procedures.

scholarly journals Intrinsic random functions and universal kriging on the circle

2016 ◽  
Vol 108 ◽  
pp. 33-39 ◽  
Author(s):  
Chunfeng Huang ◽  
Haimeng Zhang ◽  
Scott M. Robeson
Keyword(s):  
Universal Kriging ◽  
Random Functions ◽  
Intrinsic Random Functions

The intrinsic random functions and their applications

1973 ◽  
Vol 5 (03) ◽  
pp. 439-468 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Statistical Inference ◽  
Realization Theory ◽  
Practical Applications ◽  
Random Functions ◽  
Stationary Increments ◽  
Turning Bands ◽  
Generalized Covariance ◽  
Intrinsic Random Functions ◽  
Generalized Processes ◽  
Unique Realization

The intrinsic random functions (IRF) are a particular case of the Guelfand generalized processes with stationary increments. They constitute a much wider class than the stationary RF, and are used in practical applications for representing non-stationary phenomena. The most important topics are: existence of a generalized covariance (GC) for which statistical inference is possible from a unique realization; theory of the best linear intrinsic estimator (BLIE) used for contouring and estimating problems; the turning bands method for simulating IRF; and the models with polynomial GC, for which statistical inference may be performed by automatic procedures.

scholarly journals Bootstrapped models for intrinsic random functions

1988 ◽  
Vol 20 (6) ◽  
pp. 699-715 ◽  
Author(s):  
Katherine Campbell
Keyword(s):  
Random Functions ◽  
Intrinsic Random Functions

scholarly journals Intrinsic random functions on the sphere

2019 ◽  
Vol 146 ◽  
pp. 7-14 ◽  
Author(s):  
Chunfeng Huang ◽  
Haimeng Zhang ◽  
Scott M. Robeson ◽  
Jacob Shields
Keyword(s):  
Random Functions ◽  
Intrinsic Random Functions

scholarly journals The intrinsic random functions and their applications

1973 ◽  
Vol 5 (3) ◽  
pp. 439-468 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Statistical Inference ◽  
Realization Theory ◽  
Practical Applications ◽  
Random Functions ◽  
Stationary Increments ◽  
Turning Bands ◽  
Generalized Covariance ◽  
Intrinsic Random Functions ◽  
Generalized Processes ◽  
Unique Realization

The intrinsic random functions (IRF) are a particular case of the Guelfand generalized processes with stationary increments. They constitute a much wider class than the stationary RF, and are used in practical applications for representing non-stationary phenomena. The most important topics are: existence of a generalized covariance (GC) for which statistical inference is possible from a unique realization; theory of the best linear intrinsic estimator (BLIE) used for contouring and estimating problems; the turning bands method for simulating IRF; and the models with polynomial GC, for which statistical inference may be performed by automatic procedures.

scholarly journals On a class of space–time intrinsic random functions

Bernoulli ◽  
2013 ◽  
Vol 19 (2) ◽  
pp. 387-408 ◽  
Author(s):  
Michael L. Stein
Keyword(s):  
Space Time ◽  
Random Functions ◽  
Intrinsic Random Functions
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