A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions: Basic Results. New York: Springer-Verlag, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in a neighborhood of 0, then (1 O(s) u w(u; s) log s w(u; s) 0 : 12) The resolution error O(s) tends to zero at least as fast as s, and can therefore be neglected at ne scales. Condition (11) on the covariance of R is quite weak, and is satis ed by most correlation functions [25]. This proves that at small scales s 0, the deformation gradient is the solution of the Texture Gradient Equation. A scalogram w(u; s) is displayed on the bottom of Figure 3(b) the positions of the scalogram maxima are transported in the (u; log s) plane, with a velocity equal to the ....

A.M. Yaglom. Correlation Theory of Stationary and Related Random Functions, volume 1. Springer-Verlag, 1987.

....are measured using the following unbiased estimator of autocorrelation function: where is the total number of data points and is the number of time correlation lags. For a finite trace, the estimator becomes less accurate as increases. A general rule of thumb is to set within 10 20 of [26]. Obviously, is the maximum timescale of the measurement. The time measurement window for trace is then defined by (18) Using the power spectral representation, the equivalent measurement window in the frequency domain is identified by (19) In the discrete frequency domain as processed using DSP ....

A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions I: Basic Results. New York: Springer-Verlag, 1987.

....either 3D Euclidean space or 2D Riemannian manifolds. The generalization of a continuous stochastic process defined in 1 to a higher dimensional abstract space is called a random field. For the general overview of random fields, see the books by R. J. Adler [1] E.R. Dougherty [38] and A.M. Yaglom [126]. Given a probability space, a random field defined in 1 is a function such that for every fixed x 1 , X(x) is a random variable on the probability space. The more precise measure theoretic definition can be found in [1, pp. 13] It is also possible to extend the definition of a random field onto ....

....(xl, x TM) x . x TM ) then a random field X is said to be stationary [1] For a stationary random field X, we can show (x) 0) and R(x,y) f(x y) for some function f. Although the converse is not always true, such a case has never been encountered in practical applications [126]. Since we are interested in more practical applications, throughout the thesis we will equate the stationarity with the condition (x) 0) and R(x,y) f(x y) An important class of random fields is Gaussian fields. A Gaussian random field X(x) is a random field whose finite joint distribution ....

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A.M. Yaglom. Correlation theory of stationary and related random functions vol. i: Basic results. 1987.

....d ij f( ij ) for some monotonic function f . If the kernel function r is monotonically decreasing then clearly 1 r is monotonically increasing. However, there are valid isotropic kernel (covariance) functions which are non monotonic (e.g. the exponentially damped cosine r( e cos( see [11] for details) and thus we see that f need not be monotonic in kernel MDS. Remark 3 One advantage of PCA is that it de nes a mapping from the original space to the principal coordinates, and hence that if a new point x arrives, its projection onto the principal coordinates de ned by the original ....

A. M. Yaglom. Correlation Theory of Stationary and Related Random Functions Volume I:Basic Results. Springer Verlag, 1987.

....for some particular fixed Z that QG [E 2 is defined to yield a zero mean circular complex random variable if ZF] E and so that QG E 2 aN b More generally, the weights can be made functions of frequency. c Many texts like Wong and Hajek [6] Rosenblatt [7] Adler [8] and Yaglom [9] use a Stieltjes style integral with respect to an orthogonalincrement process, but that is more natural in a single dimension. The present development instead follows Rozanov [10] and uses a Lebesguestyle integral with respect to a random measure. if Zd( eE . In this simple case, the field ....

....output is f 9 gih 8 9 #: 2 8 9 [ 2 j k , k V V EG # 32 EG [lm2; H n oU pY [ # q o. pL )r gih)P QW 1 32 PRQG [lm2 j k , k V k EG # 32Hk P tu # 32 (3) using basic result gih)P QW 1 32 PRQG [lm2 jv xwY 1 y Xlm2YP lzPRtu 1 32 from stochastic integration theory [6, 9]. Since the intensity gihYkR k j of the random field given by (2) is similarly given by PRtu 1 32 tu S T 2 , positive Borel measure t on S T should be interpreted as the distribution versus look vector of complex field intensity. A normalization further clarifies the picture. The ....

A. J . Yaglom, Correlation Theory of Stationary and Related Random Functions, Volume 1: Basic Results, SpringerVerlag, New York, 1987.

....positive de niteness is equivalent to the process having a spectral distribution function (Mat ern 1986) p. 12] Assuming further the covariance function to be isotropic in the full spatio temporal domain, all valid functions can be represented as (possible in nite) mixtures of Bessel functions (Yaglom 1987)[p. 106] Such representations have been used for de ning nonparametric covariance functions (see Ecker Gelfand (1997) and the references therein) Because the time dimension has a di erent interpretation compared to the spatial ones, an isotropic assumption is often unrealistic. Other ....

Yaglom, A. (1987), Correlation Theory of Stationary and Related Random Functions I, Springer-Verlag, New York.

....simple choice for Phi(d ) is a step function that assigns positive weights a k at points b k , k = 1; 2; Delta Delta Delta ; K. Following S G we will use here a similar representation for the covariance function where Y p (t) is replaced by the Gaussian type kernel expf Gammat 2 g (Yaglom, 1986), which is valid in any dimension. Therefore the general form proposed for g is a mixture of K Gaussian correlation functions, specifically, g(h) K X k=1 a k exp( Gammab k h 2 ) 7) where the number of components K is, for reasons of parsimony, as small as possible consistent with the ....

Yaglom, A.M. (1986) Correlation Theory of Stationary and Related Random Functions I Basic Results. Springer Verlag.

....time series models seems to have been by hydrologists Thomas Fiering (1962) Since that time there have been very extensive developments in the theory and applications of periodically correlated time series. For a review of the probabilistic literature on periodically correlated processes, see Yaglom (1986, x26.5; 1987). Miamee (1990) and Sakai (1991) have derived new theoretical results and conditions on the spectral density function of periodically correlated time series. On the statistical methodology side, contributions to periodically correlated time series modelling have been made by Jones Brelsford ....

Yaglom, A.M. (1987). Correlation Theory of Stationary and Related Random Functions II. Supplementary Notes and References. New York: Springer-Verlag.

....of a stationary covariance kernel are exp ik:x. Many commonly used covariance functions are also isotropic, so that C(h) C(h) where h = x Gamma x 0 and h = jhj. For example C(h) exp( Gamma(h=oe) is a valid covariance function for all input dimensionalities d 1 and for 0 2 (Yaglom, 1987, pg. 137) Note that in this case oe sets the correlation length scale of the random field, although other covariance functions (e.g. those corresponding to power law spectral densities) may have no preferred length scale. A simple example of a non stationary covariance function is obtained by ....

....some random component such as a magnitude. In this case we can write Y (x) X n Vnh(x; xn ) 15) 3 Although many commonly used covariance functions are non negative, there are examples such as the exponentially damped cosine C(x; x ) exp( Gammaff ) cos which obtain negative values; see Yaglom (1987), pp. 366 367. 5 where the xn s are the Poisson distributed, h( Delta; Delta) is the kernel (which will often be a function of x Gamma x i ) and the Vn s are iid random variables. It can be shown that the mean and covariance functions of this shot noise process are given by (see Parzen ....

Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions Volume I:Basic Results. Springer Verlag.

....that we need (and that are satisfied by these families) are that the covariance function be non negative and decrease monotonically with distance d = ks Gamma uk, with limiting values of 1 at d = 0 and of 0 at d = 1. This is the most common kind of association found in spatial data; see Yaglom (1987) and Jones and Vecchia (1993) for other covariance models. Spherical, l = 1; 2; 3. Wackernagel, 1995) K S # (d) ae 1 Gamma 3 2 d 1 1 2 ( d 1 ) 3 if 0 d 1 0 if d 1 ; 1 0: This family is a popular model with the distinctive feature that any two observations taken ....

....are uncorrelated. Power Exponential, l 1. De Oliveira et al. 1997) K PE # (d) e Gamma(d= 1 ) 2 ; 1 0; 2 2 (0; 2] 2 This family contains the exponential ( 2 = 1) and squared exponential ( 2 = 2) models, which are often used in applications. Rational Quadratic, l 1. (Yaglom, 1987). K RQ # (d) 1 d 1 2 Gamma 2 ; 1 0; 2 0: Mat ern, l 1. Mat ern, 1986; Handcock and Stein, 1993) K M # (d) 1 2 2 Gamma1 Gamma( 2 ) d 1 2 K 2 d 1 ; 1 0; 2 0; where K 2 ( is the modified Bessel function of second kind and ....

Yaglom, A.M. (1987) Correlation Theory of Stationary and Related Random Functions I. Basic Results. New York: Springer-Verlag.

....properties that we need (and that are satis ed by these families) are that the covariance function be non negative and decrease monotonically with distance d = ks uk, with limiting values of 1 at d = 0 and of 0 at d = 1. This is the most common kind of 2 association found in spatial data; see Yaglom (1987) and Jones and Vecchia (1993) for other covariance models. Spherical, l = 1; 2; 3. Wackernagel, 1995) K S # (d) 1 3 2 d 1 1 2 ( d 1 ) 3 if 0 d 1 0 if d 1 ; 1 0: This family is a popular model with the distinctive feature that any two observations taken more ....

....units apart are uncorrelated. Power Exponential, l 1. De Oliveira et al. 1997) K PE # (d) e (d= 1 ) 2 ; 1 0; 2 2 (0; 2] This family contains the exponential ( 2 = 1) and squared exponential ( 2 = 2) models, which are often used in applications. Rational Quadratic, l 1. (Yaglom, 1987). K RQ # (d) 1 d 1 2 2 ; 1 0; 2 0: Mat ern, l 1. Mat ern, 1986; Handcock and Stein, 1993) K M # (d) 1 2 2 1 ( 2 ) d 1 2 K 2 d 1 ; 1 0; 2 0; where K 2 ( is the modi ed Bessel function of second kind and order 2 ; see ....

Yaglom, A.M. (1987) Correlation Theory of Stationary and Related Random Functions I.

....is the second order stationarity, defined through the covariance function. In general, for every random field Z we introduce the mean function m(s) E(Z(s) and the covariance function r(s, t) Cov(Z(s) Z(t) 1 General theory on stochastic processes appears in Adler [1] Matern [43] and Yaglom [65]. 24 If m(s) # and if r is a function of s t only, Z is called second order stationary. Similarly, if m(s) # and if r depends only on #s t#, Z is called second order isotropic. In this chapter we consider Gaussian fields, defined by the property that all finite collections (Z(s 1 ....

Yaglom, A. M., (1987), Correlation Theory of Stationary and Related Random Functions I. Basic Results, New York: Springer-Verlag.

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A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions: Basic Results. New York: Springer-Verlag, 1987.

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A. M. Yaglom. Correlation Theory of Stationary and Related Random Functions I: Basic Results. Springer Series in Statistics. Springer-Verlag, New york, 1986.

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A.M. Yaglom. Correlation Theory of Stationary and Related Random Functions, volume 1. Springer-Verlag, 1987.

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A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions, Vol. I, Springer Verlag, 1987.

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A. Yaglom. Correlation Theory of Stationary and Related Random Functions, volume I. Springer, 1987.

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A. Yaglom. Correlation Theory of Stationary and Related Random Functions, volume I. Springer, 1987.

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M. YAGLOM, Correlation Theory of Stationary and Related Random Functions I, Springer-Verlag, New York, 1987.

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A. M. Yaglom. Correlation Theory of Stationary and Related Random Functions, volume 1. Springer-Verlag, 1987. CERMICS, Ecole Nationale des Ponts et Chauss ees INRIA, 06902 Sophia-Antipolis, France. Maureen.Clerc@cermics.enpc.fr CMAP, Ecole Polytechnique, Palaiseau, France. Courant Institute of Mathematical Sciences, New York University, NY 10012. Stephane.Mallat@polytechnique.fr

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A. M. Yaglom. Correlation Theory of Stationary and Related Random Functions I. SpringerVerlag, New York, 1987. 19

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A. M. Yaglom. Correlation Theory of Stationary and Related Random Functions. Volume 1 and 2, Springer-Verlag, New York,

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Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. New York: Springer-Verlag.

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Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions. Volume I: Basic Results. Springer-Verlag.

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J. R. Statist. Soc. A 146, 150--157. Yaglom, A.M. (1986). Correlation Theory of Stationary and Related Random Functions I. Basic Results. New York: Springer-Verlag.

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