M. Subbotin, \On the law of frequency of errors," Mathematicheskii Sbornik 31, pp 296-301, 1923  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Consider the class of densities f(o; jdet j 1=2 exp( q) 2 ) 22) q = o ) o ) 23) 3= 1= and = 24) This class was recently suggested and studied in [5] The one dimensional case appears to have rst been suggested by Subbotin, [14]. The class (22) will be referred to as the the power exponential distribution. It is also known as the error function, p Gaussians or as Gaussians. Following (14) a model is considered where each state in the system is modelled by a mixture of power exponential distribution, i.e. f(o; ....

M. Subbotin, \On the law of frequency of errors," Mathematicheskii Sbornik 31, pp 296-301, 1923

....Gaussian distribution where p(v) ffi 1(v) ffi vr (v) is the Kronecker delta function. This class also includes the Cauchy distribution as another standard case. By appropriately modifying the distribution of v it is possible to alter both the tails and the peakiness of the distribution. In [8] an EM scheme is described for ML estimates of and Sigma that does not require explicitly obtaining the distribution p(v) which remains unaltered during training. The discrete version of (1) was described in [9] Here p(v) P r wr ffi vr (v) with wr 0, P r wr = 1. Then f(o; Sigma; ....

....described in (1) Power exponential distributions can not in general be modelled with Richter distributions. This fact can be verified by noticing that functions in the class (1) are all log concave, whereas the power exponentials are not log concave for 0 ff 1. This makes the framework of [8] unsuitable for parameter update for 0 ff 1. 6 4 2 0 2 4 6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 The power exponential distribution for 4 different powersa likelihood a=0.7 a=1 a=2 a=5 Figure 2: The power exponential function for various values of ff. 3.1. Parameter Estimation The ....

M Subbotin. On the law of frequency of errors. Mathematicheskii Sbornik, 31:296--301, 1923.

.... for suitable additional constraints on d(x) to be defined below, as Hyper Generalized Gaussian Distributions (HGDs) As discussed below, they are a superset of the well known generalized gaussian distributions (GGDs) also known as exponential power distributions or Box Tiao distributions) [37, 9, 4, 5, 6, 16, 24, 38, 17, 40, 39]. Here, we treat A, p, and q as known parameters, and thus x and y are jointly distributed as P (x; y) P (x; y; p; q; A) Bayes rule yields, P (xjy; p; A) 1 fi P (yjx; p; A) Delta P (x; p; A) 1 fi P (y Gamma Ax) Delta P p (x) 5) where fi = P (y) P (y; p; q; A) Z P ....

....community) We are concerned in this paper only with the undetermined case for which A is onto and many to one. Independent Component Analysis (ICA) An important class of densities is given by the generalized gaussians for which d p (x) kxk p p = n X k=1 jx[k]j p ; 9) for p 0 [37, 9, 4, 5, 6, 16, 24, 38, 17, 40, 39]. This is a special case of the p class (the p class ) of functions which allow p to be negative which is discussed in [34, 18] Note that this function has the special property of separability, d p (x) n X k=1 d p (x[k] which corresponds to factorizability of the density P p (x) P ....

M.Th. Subbotin, "On the Law of Frequency of Error," Matematicheskii Sbornik, 1923, Volume 31, pp. 296-301.

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