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Analysis and Selection of Myriad Estimate Tuning Parameter For SαS Distributions Roenko A.A., Lukin V.V., Djurović I.

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Symmetric α-stable distributions and their properties Characteristic function of random variable Х with SαS distribution: where α denotes the characteristic exponent or index and γ is dispersion; α=2 corresponds to Gaussian distribution with zero mean and variance 2γ: α=1 describes Cauchy distribution with location parameter equal to 0: Fig.1. SαS distributions for various α values 1 (1) (2) (3) Fig.2. Realizations of processes with SαS distributions for α=0.5 and different γ values equal to 0.5, 1 and 4, correspondingly γ 1/α denotes parameter characterizing the scale of the data sample with SαS pdf.

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Sample myriad based location parameter estimator The sample myriad relates to the class of М-estimators and is the efficient (optimal) estimator for Cauchy distribution. For the data sample х 1, х 2,…,х N and parameter К>0 it is defined as (5) An infinite values of K converts the myriad to sample mean: (4) When K tends to 0 the sample myriad precisely defines the pdf mode: (6)(6) The behavior of myriad estimator in case of fixed values of tuning parameter depends upon the values of processing realization: (7)(7) where с>0. Fig.3. The dependence of myriad estimator properties upon the K values 2

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Analysis of known dependencies of optimal К values upon SαS distribution parameters 3 Fig.4. The dependence K opt Arce ( α ) for γ=1 proposed in the works of G. Arce and J. Gonzalez The number of experiments was fixed and equal to 5000. The α value was varied with step size 0.1 and the increment of the parameter K value in case of fixed α and γ was equal to 0.02. The approximation formula for calculation of optimal К values proposed in the works of G.Arce and J.Gonzalez: (8)(8) Fig.5. Dependence σ 2 (K) for N=128 and 256; α=1.4, γ=1 Since the minima of the dependencies of σ 2 min (K opt ) for fixed α and γ are commonly not very obvious (see Fig.5) and because of the fact that small variations of К values (accordingly to Fig.5) dont considerably influence the estimator accuracy, we determined not only the optimal value К opt (α,γ) but also the values К opt min (α, γ) and К opt max (α, γ). Then σ 2 (К min, α, γ)= σ 2 (К max, α, γ)=1,1σ 2 min (К opt, α, γ) (9)

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Analysis of obtained dependencies Irrespectively to γ, the functions К opt min (α) and К opt max (α) are monotonically increase if α grows. The following conditions are valid: for α0, К opt (α,γ) 0; for α1,К opt (α,γ) γ; for α2, К opt (α,γ) (note that the curve К opt (α) goes in between the curves К opt min (α) and К opt max (α) for any given γ). Arces approximations К opt Arce (α,γ) are not absolutely correct: -for 0<α<1 the values К opt Arce (α) are larger than К opt (α); -for 1α<1,8 the values К opt Arce (α) are slightly smaller than К opt (α); -for γ>1 and 0<α<1 the values К opt Arce (α) start to quickly increase if α reduces; this fact is explained by the behavior of the factor γ 1/α used in Arces approximation formula. Thus, one needs more accurate approximation К opt (α,γ), especially for γ>1. 4 Fig.6. The plots of obtained optimal К values and К opt Arce (α,γ) curves depending upon α for fixed values of γ equal to 0.5, 1 and 4

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The proposed approximation of obtained dependencies К opt (α,γ) and its analysis 5 We propose to use the following approximation formula: (10) Fig.7. The plots К opt min (α, γ), К opt max (α, γ) and К optAp (α, γ) obtained by (10) for fixed γ values equal to 0.5, 1 and 4

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Adaptation parameters to the distribution tail heaviness and data scale 6 The percentile coefficient of kurtosis (PCK): (11) where Q=(Q 3 -Q 1 )/2 is the half of interquantile range; P 90, P 10 denotes the 90th and 10th percentiles. Absolute median deviation from median (MAD): (12) where median denotes the sample median; x 1, x 2,…, x N are the order statistics data sample with size N. Fig.8. The dependence of average PCK values upon the parameter α The MSE estimate for the case of Gaussian noise (α=2) is equal to (13) For the SαS pdf with α=2 and γ we can define: (14) Then (15)

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Analysis of PCK values for the cases of Cauchy and Gaussian distributions 7 Cauchy distribution with zero location parameter and scale parameter σ 2 С =γ 2 can be defined as: (16) Then, for Р 90 -Р 10 : (17) Taking into account that the distribution is the symmetric one obtains: (18) Since Р 50 coincides with the distribution location equal to 0, we can write: (19) Thus,Similarly, Q = Q 3 - Q 1 = Р 75 - Р 25 = σ C. Consequently, Similarly for Gaussian distribution with zero-mean and variance σ 2 G one obtains PCK=0.2636.

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Proposed adaptive procedure for calculation of tuning parameter value 8 Proposed adaptive procedure for determination of the parameter K value in case of a priori unknown α and γ: (20) Fig.9. The plots К opt min (α, γ), К opt max (α, γ) and К ADAPT (α, γ) for fixed values of parameter γ equal to 0.5, 1 и 4

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