By P. R. Krishnaiah

Hardbound. This quantity is dedicated to pattern Surveys, that is the main usual technique in statistical perform. It covers many theoretical and sensible elements of social and organic investigations, and is a worthwhile advisor for these interested in designing pattern surveys.

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**Example text**

A detailed comparison of fractional Gaussian noises and fractional Brownian motions using the Fourier ﬁltering technique and the method of successive random additions has been given by Gallant et al. Ž1994.. These authors also considered a third method of generating synthetic fractional noises using Weierstrass ᎐Mandelbrot functions. Other relevant studies include those carried out by Li and McLeod Ž1986. and Osborne and Provenzale Ž1989.. 4. Semivariograms Using the deﬁnition for the semivariogram, ␥ k , given in Eq.

The variance converges to a ﬁnite value for large N when  - 1, but diverges to inﬁnity as N ª ϱ when  ) 1. The value  s 1 is a natural crossover between weak and strong persistence in a self-afﬁne Žlong-range persistent . time series. ; 32 BRUCE D. MALAMUD AND DONALD L. TURCOTTE FIG. 8. The variance, VN , Eq. , of a time series, yn , is plotted as a function of the length of record, N, for values of  in the range y1 F  - 1 and 1 -  F 3. As N gets larger the variance quickly converges for values of  - 1 and diverges for  ) 1.

For 1 F  F 3. , and squares Ž3 F  F 5. are explained in Table I. Quite good agreement is found in the range 1 F  F 3, where the fractional Brownian motions are expected to be self-afﬁne fractals. From Fig. 17, it is seen that Ha f 0 for fractional Gaussian noises in the range y1 F  F 1. From Eq. , n ; n H a, one can conclude that, in this range of  , the standard deviation, n , and thus the variance, Vn , are independent of the length of the signal, n. Therefore, these fractional noises are Žweakly.