1. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin National Aerospace University, Dept. of Transmitters, Receivers and Signal Processing, 17 Chkalova St, 61070 Kharkov, Ukraine e-mails: daemonius@mail.ru, roenkomail@mail.ru, lukin@xai.kharkov.ua roenkomail@mail.ru lukin@xai.kharkov.ua Igor Djurović University of Montenegro, Electrical Engineering Department, Cetinsky Put bb, 81000, Podgorica, Montenegro e-mail igordj@ac.meigordj@ac.me Presentation outline 1.Considered practical situations 2.Definition of a sample meridian 3.Main properties of sample meridian 4.Considered distributions 5.Analysis of statistical characteristics of sample meridian estimator 6.Dependence upon sample size 7.Comparison of estimators and practical examples 8.Conclusions and future work

2. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 2 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic CONSIDERED PRACTICAL SITUATIONS Quite often people rely on Gaussian model of noise (interference, disturbing process). However, there are many practical situations for which this assumption is not valid:  Atmospheric noise (is well described by symmetric α-stable distribution with α about 1.5);  Acoustic noise in indoor environment;  Clutter from sea surface and vegetation in maritime and micro-Doppler radar surveillance systems;  Noise in bispectrum domain and in Wigner distributions domain for low input SNR;  Noise components in spectral components of DFT in robust DFT framework;  Mixed noise in images where impulse noise arises due to coding/decoding errors, etc. In all these cases, noise distributions have heavier tails than Gaussian. Even if input noise is Gaussian, it might become having heavier tails after non-linear transformations. K.Barner and T.Aysal have shown that a product of two random variables has heavier tail distribution than any distribution of multiplied random variables. Thus, one needs robust estimators to cope with non-Gaussian noise.

3. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 3 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic PRACTICAL EXAMPLES Fig.1 - Micro-Doppler reflections from vegetation (bushes) in windy weather (a), PCK characterizing Gaussianity (b) and MAD characterizing interference intensity (c) a)b)c) Fig.2 - Image formed by maritime polarization radar (VH) (a) and histograms in clutter regions for middle (b) and large (c) distances a) b) c)

4. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 4 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic DEFINITION OF A SAMPLE MERIDIAN Analytically a sample meridian estimator of location is defined as where N denotes a sample size; x i is an i-th element of a data sample; δ is called medianity parameter. Its cost function is The function φ(β) monotonically decreases for β<x MIN and increases for β<x MAX where x MIN and x MAX are the minimal and maximal elements of a data sample, respectively. Thus, minima of cost function are from x MIN and x MAX and the number of minima is limited, it increases if δ reduces. Moreover, minima can be observed only for β that coincides with one of elements of original sample. These properties show that a very simple algorithm for finding meridian estimate can be realized. One has to calculate φ(x i ), i=1,…,N and to find such i for which φ(x i ) is the smallest. No sorting is needed, only N calculations of (2) and N-1 comparisons.

5. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 5 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic MAIN PROPERTIES OF SAMPLE MERIDIAN ESTIMATOR 1) 1) the meridian estimator is invariant to translation of distribution center (mean if it exists). This allows restricting consideration of meridian estimator properties by the case of distributions with zero mean (center); 2) 2) we concentrate on studying only distributions symmetric with respect to their means; for such distributions the meridian estimator produces unbiased estimates for arbitrary δ; 3) 3) for a rather large δ, a sample meridian coincides with a sample median and, thus, statistical characteristics of the corresponding estimators are practically the same; however, it is not clear what is “rather large δ” in practice; 4) 4) for a rather small δ, sample meridian estimator can serve as distribution mode finder; but it is not clear what is “rather small δ” in practice; There are also other questions There are also other questions:  Can the meridian estimator accuracy be better than accuracy for other robust estimators of location and under what conditions?  Is there some dependence of the meridian estimator accuracy on sample size?  What are practical recommendations for setting δ?

6. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 6 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic CONSIDERED DISTRIBUTIONS Special attention in our analysis should be paid to non-Gaussian heavy tailed distributions. The first one is Gaussian PDF with variance σ 2 G and the second PDF is Cauchy one which is a particular case of symmetric α-stable distributions: f 1 (γ,x)= γ/(π(γ 2 +x 2 )), where γ is the parameter characterizing PDF scale. For both PDFs, their maximums are bell-shaped and continuous derivatives. Let us also analyze PDFs f 2 (x), f 3 (x), f 4 (x) of the following three random variables: 1) Y 1 =X 1 X 2 (denoted as dgauss); 2) Y 2 =X 1 X 2 X 3 (denoted as tgauss); 3) Y 3 =(X 1 ) 3 (denoted as gauss3). Here X 1, X 2, and X 3 are independent zero-mean Gaussian variables with standard deviations σ X1, σ X2, σ X3, respectively. All three PDFs f 2 (x), f 3 (x), f 4 (x) have heavier tails than Gaussian. If σ X1 =σ X2 =σ X3 =1 then for the PDFs f 2 (x) and f 3 (x) their variances are equal to 1 and for the PDF f 4 (x) it variance approximately equals to 15. One more peculiarity of f 2 (x), f 3 (x), f 4 (x) is that they all have peaky (sharp, not bell-shaped) maximums.

7. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 7 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic CONSIDERED DISTRIBUTIONS For heavy tail PDFs, it is reasonable to characterize their scale by a robust estimate of scale. Median of absolute deviations (MAD) can serve as such a characteristic. For PDF f 2 (x), MAD=0.545σ X1 σ X2 ; for f 3 (x) one has MAD=0.292σ X1 σ X2 σ X3 ; for f 4 (x) MAD=0.462σ 3 X3. The following expressions are valid for Gaussian and Cauchy PDFs: MAD=σ G /1.483 and MAD=1.5γ. A characterization of tail weight could be percentile coefficient of kurtosis (PCK). For Gaussian PDF it is equal to 0.265, for heavier tail distributions the PCK values are smaller: 0.16 for Cauchy PDF; 0.178, 0.132, and 0.076 for PDFs f 2 (x), f 3 (x) and f 4 (x), respectively. Fig. 3 – Gaussian distribution and PDFs dgauss, tgauss, gauss3, full appearance (upper plot) and for the range of argument values (bottom plot).

8. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 8 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic ANALYSIS OF STATISTICAL CHARACTERISTICS OF SAMPLE MERIDIAN ESTIMATOR Consider conditions under which a sample meridian differs from median for the same data sample. Let us determine probability P that meridian(x i ; i=1,…,N; δ)=median(x i ). For N=3 the probability is equal to unity for all five considered PDFs and for any δ. So, let us study larger N, namely N=5, 9, 15, 25 that correspond to typical situations in signal and image processing in a sliding window manner. Fig. 4 - Dependences of P on δ for N=5 (left), 9 (right) All curves have similar behavior and they have “saturation” to unity for rather large δ and a flat region with P≠1 for rather small values of δ. Transition zones between these two “saturations” are observed for δ within the limits from 10 -2 to 10 2 or, more generally speaking, for δ/MAD from 10 -2 to 10 2. For heavy tailed distributions it is more correct to analyze the obtained dependencies with respect to normalized values δ/MAD since standard deviation is a non-robust characteristic of scale and theoretically it can be infinite.

9. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 9 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic ANALYSIS OF STATISTICAL CHARACTERISTICS OF SAMPLE MERIDIAN ESTIMATOR Fig. 5 - Dependences of P on δ for N=15 (upper plot) and 25 (bottom plot) At the same time, there are some differences and other interesting observations. First, P becomes practically equal to 1 for δ exceeding 10 standard deviations of Gaussian noise (σ G ). For sharp peak PDFs, P reaches 1 for δ/MAD of the order 10…50. The second observation is that the probabilities P depend upon N for small δ. For larger N, P becomes smaller. The smallest P takes place for Gaussian PDF, for other distributions that are heavy tailed the values of P are approximately the same. There is no sense to study them for δ/MAD larger than 100 since then the sample meridian and median coincide and, thus, have the same statistical characteristics. The properties of the sample meridian and median differ in the area of δ/MAD<100 although this does not necessarily mean that in this case the meridian estimator accuracy is better than that of the median estimator.

10. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 10 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic ANALYSIS OF STATISTICAL CHARACTERISTICS OF SAMPLE MERIDIAN ESTIMATOR For analysis of meridian estimator accuracy, we have determined: - root mean square error (RMSE) σ β of obtained estimates; - we also used where is the meridian estimate for the j-th sample of data obeying a given PDF, N exp defines the number of experiments (analyzed data samples, N exp =1000). Expedience of analyzing MAD β stems from the fact that has been found in experiments – PDF of the obtained meridian estimates of location for PDFs f 2 (x), f 3 (x) and f 4 (x) occurred to be symmetric but non- Gaussian. For Gaussian and Cauchy distributions, the meridian estimate distribution is close to Gaussian, especially if N is large enough. Similar effects have been observed for sample median estimates in the case of Laplacian distribution. In the case of Gaussian PDF, σ β (δ) and MAD β (δ) for δ/MAD 100. There is no minimums of the curves σ β (δ) and MAD β (δ). Both curves are monotonically decreasing. Both σ β (δ) and MAD β (δ) are proportional to σ G and they are approximately inversely proportional to N 0.5.

11. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 11 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic ANALYSIS OF STATISTICAL CHARACTERISTICS OF SAMPLE MERIDIAN ESTIMATOR Let us analyze data obtained for Cauchy PDF, N=64. As it is seen, all dependencies are monotonously decreasing. For δ/MAD 10 where Parameters σ β (δ) and MAD β (δ) are proportional to data scale defined by γ. For providing the best accuracy of the meridian estimator it is enough to set δ=(10…20)MAD X. Fig. 6 - Dependences of σ β and MAD β on δ for Cauchy PDF with γ=1 and 2, N=64 Thus, a common conclusion for Gaussian and Cauchy PDFs is that accuracy of the meridian estimator cannot be better than for median estimator. for a data sample at hand.

12. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 12 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic ANALYSIS OF STATISTICAL CHARACTERISTICS OF SAMPLE MERIDIAN ESTIMATOR Let us now study statistical characteristics of meridian estimator for peaky PDFs starting from the PDF f 2 (x). Both curves σ β (δ) and MAD β (δ) have minimums observed for δ≈1 and δ≈0.1, respectively. The ratio MAD β (δ)/σ β (δ)<0.5, i.e., considerably smaller than if meridian estimator is applied to Gaussian data. This indirectly shows that the estimates do not obey Gaussian distribution and are heavy tailed. This has been confirmed by analysis of their histograms. Fig. 7 - Dependences σ β (δ) and MAD β (δ) for PDF f 2 (x), N=64 Therefore, if δ is set optimal, there is sense to apply the meridian estimator.

13. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 13 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic ANALYSIS OF STATISTICAL CHARACTERISTICS OF SAMPLE MERIDIAN ESTIMATOR For data samples with PDFs f 3 (x) and f 4 (x), the plots σ β (δ) and MAD β (δ) have obvious minimums of both σ β and MAD β. For f 3 (x), minimums are observed for δ≈0.1 and δ≈0.01, respectively. It is possible to recommend using δ/MAD Y2 ≈0.01 that clearly corresponds to modal mode of the meridian estimator. MAD β (δ)/σ β (δ) for δ/MAD β (δ)≈0.01 becomes about 0.3. This means that PDF of meridian estimates is non-Gaussian. For f 4 (x), minimum of σ β (δ) has place for δ≈0.001, minimum of MAD β (δ) is observed for δ≈10 -5, i.e. again the meridian estimator looks for distribution mode. Thus, δ/MAD Y3 should be about 10 -4. This means that the meridian estimator with properly adjusted δ/MAD Y3 is able to perform considerably better than the median estimator. MAD β /σ β occurs to be very small, about 0.02. This means that meridian estimates have very heavy tails. Fig. 8 - Dependences σ β (δ) and MAD β (δ) for PDFs f 3 (x) (upper plot) and f 4 (x) (bottom plot), N=64

14. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 14 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic DEPENDENCE UPON SAMPLE SIZE Table 1. Accuracy of meridian estimator depending upon N for quasi-optimal δ As it is seen, σ β and MAD β decrease faster than N -0.5, especially for data samples with the most heavy tailed PDF f 4 (x). It is also worth stressing that optimal δ q opt decreases if N becomes larger. This is a specific property that has not been observed for the myriad estimator. This property makes more complicated designing an adaptive algorithm for determination of δ q opt for limited a priori information on PDF a data sample obeys to. We have also analyzed dependence of δ q opt on data scale. As expected, experiments carried out for all considered PDFs have demonstrated that δ q opt should be directly proportional to MAD of a given distribution where MAD X can serve as its estimate.

15. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 15 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic COMPARISON OF ESTIMATORS Table 2. Comparison of accuracy for the mean, median and meridian estimators for data samples with different N and PDFs Thus, the meridian estimator can be useful for applications when one deals with very impulsive noise environments where noise PDF is not bell-shaped. We also expect that the properties of meridian and median estimators can differ a lot in cases of processing data samples with asymmetric distributions.

16. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 16 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic COMPARISON OF ESTIMATORS AND PRACTICAL EXAMPLES Edge of unity amplitude (49-th sample) corrupted by Gaussian noise with SD=0.1. Fig.9 - Median filter output bias, residual noise variance and aggregate error Fig.10 - Meridian filter (δ=0.03) output bias, residual noise variance and aggregate error Meridian filter with optimally set δ is able to produce smaller bias and aggregate error in the neighborhood of noisy step edge.

17. National Aerospace University of Ukraine ANALYSIS OF MERIDIAN ESTIMATOR PERFORMANCE FOR NON-GAUSSIAN PDF DATA SAMPLES 17 Dmitriy Kurkin, Alexey Roenko, Vladimir Lukin, Igor Djurovic CONCLUSIONS AND FUTURE WORK  The studies carried out have shown that there exist non-Gaussian distributions for which the sample meridian with δ q opt is able to produce more accurate estimates of location parameter than one of the most known robust estimators, sample median and myriad.  All three PDFs for which the benefits of the sample meridian have been observed are peaky (not bell-shaped) ones that is explained by the peculiarities of the used cost function (2).  For PDFs with heavier tails one has to set smaller δ q opt. At the same time, δ q opt should be proportional to data scale.  These observations let us hope that adaptive algorithm for determination δ of the meridian estimator can be designed.