Analysis and Selection of Myriad Estimate Tuning Parameter For S α S Distributions

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

2. 1 Symmetricα-stabledistributionsand their properties Characteristic function of random variable ХwithSαSdistribution: (1) whereαdenotesthe characteristicexponentor index andγ isdispersion; γ1/αdenotesparameter characterizing the scale of the data sample with SαSpdf. α=2 corresponds to Gaussian distribution with zero mean and variance 2γ: (2) α=1 describesCauchy distribution with location parameterequal to 0: Fig.1. SαSdistributions for various α values (3) Fig.2. Realizations of processes with SαSdistributions for α=0.5 and differentγvaluesequal to 0.5, 1 and 4, correspondingly

3. 2 Sample myriad based location parameter estimator • The sample myriad relates to the class of М-estimatorsand is the efficient (optimal) estimator for Cauchy distribution. For the data sample х1, х2,…,хN and parameter К>0 it is defined as (4) An infinite values of K converts the myriad to sample mean: (5) When K tends to 0 the sample myriad precisely defines the pdf mode: (6) The behavior of myriad estimator in case of fixed values of tuning parameter depends upon the values of processing realization: (7) where с>0. Fig.3. The dependence of myriad estimator properties upon the K values

4. 3 Analysis of knowndependenciesof optimalКvalues uponSαSdistribution parameters The approximation formula for calculation of optimal К valuesproposed in the works of G.Arce and J.Gonzalez: (8) The number of experiments was fixedand equal to 5000. The α value was varied with step size 0.1 andthe increment of the parameter K value in case of fixed α andγwas equal to 0.02. Fig.4. The dependence Kopt Arce(α) for γ=1 proposed in the works ofG. Arce and J. Gonzalez Since the minima of the dependencies of σ2min(Kopt) for fixed α and γ are commonly not very obvious (see Fig.5) and because of the fact that small variations of Кvalues (accordingly to Fig.5) don’tconsiderably influence the estimator accuracy, we determined not only the optimal value Кopt(α,γ)but alsothe valuesКopt min(α, γ)andКopt max(α, γ). Then (9) σ2(Кmin, α, γ)= σ2(Кmax, α, γ)=1,1σ2min(Кopt, α, γ) Fig.5. Dependence σ2(K) for N=128 and 256; α=1.4, γ=1

5. 4 Analysis of obtained dependencies • Irrespectively toγ,the functions Кopt min(α)andКopt max(α)are monotonically increase if α grows. • The following conditions are valid: forα→0, Кopt(α,γ) →0; forα→1,Кopt(α,γ) → γ;forα→2, Кopt(α,γ) → ∞ (note that the curveКopt(α)goes in betweenthe curvesКopt min(α)andКopt max(α)for any given γ). • Arce’s approximationsКoptArce(α,γ) are not absolutely correct: -for 0<α<1 the valuesКoptArce(α)are largerthan Кopt(α); -for 1≤α<1,8 the valuesКoptArce(α)are slightly smaller than Кopt(α); -forγ>1 and 0<α<1 the valuesКoptArce(α)start to quickly increaseifα reduces; this fact is explained by the behavior of the factor γ1/αused in Arce’s approximation formula. • Thus, one needsmore accurate approximationКopt(α,γ), especiallyforγ>1. Fig.6. The plots of obtained optimalКvalues and КoptArce(α,γ) curves depending uponαfor fixed values of γequal to 0.5, 1 and 4

6. 5 The proposed approximation of obtained dependencies Кopt(α,γ)and its analysis We propose to use the following approximation formula: (10) Fig.7. The plotsКopt min(α, γ),Кopt max(α, γ)andКoptAp(α, γ)obtained by (10) for fixedγ values equal to 0.5, 1 and 4

7. 6 Adaptation parameters to the distribution tail heaviness anddata scale (11) The percentile coefficient of kurtosis (PCK): whereQ=(Q3-Q1)/2 is the half of interquantile range; P90, P10 denotes the 90thand 10thpercentiles. Absolute median deviation from median (MAD): (12) wheremedian denotes the sample median; x1, x2,…, xN are the order statistics data sample with sizeN. The MSE estimate for the case of Gaussian noise (α=2) is equal to (13) For the SαSpdfwithα=2 andγwe can define: (14) Then Fig.8. The dependence of average PCK values upon the parameter α (15)

8. 7 Analysis of PCK values for the cases of Cauchy and Gaussian distributions Cauchy distribution with zero location parameter and scale parameter σ2С=γ2 can be defined as: (16) Then, for Р90-Р10: (17) Taking into account that the distribution is the symmetric one obtains: (18) SinceР50coincides with the distribution location equal to 0, we can write: (19) Thus, Similarly, Q = Q3 - Q1 = Р75 -Р25 = σC . Consequently, Similarlyfor Gaussian distribution with zero-mean and variance σ2Gone obtainsPCK=0.2636.

9. 8 Proposed adaptive procedure for calculation of tuning parameter value Proposed adaptive procedure for determination of the parameter K value in case of a priori unknown αandγ: (20) Fig.9. The plotsКopt min(α, γ),Кopt max(α, γ)andКADAPT(α, γ)for fixed values of parameterγ equal to 0.5, 1 и 4