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İki boyutlu sistemlerin yüksek mertebeden istatistik ile modellenmesi

Modelling of two-dimensional systems using higher order statistics

  1. Tez No: 39126
  2. Yazar: A. MAHİR ÖZDOĞAN
  3. Danışmanlar: PROF.DR. AHMET H. KAYRAN
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1993
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 89

Özet

ÖZET Bu tezin konusu sistem modeilemede önemli bir yer tutan parametrik yöntemler içerisinde özbağlanımlı-yürüyen ortalamalı (ÖBYO) modellerin katsayılarının yüksek mertebeden istatistik ile kestirilmesi sorununun incelenmesidir. ikinci mertebeden momentler ya da özünti işlevine üstünlükleri nedeni ile üçüncü mertebeden kümülantların kullanımı yaygınlaşmıştır. Bu üstünlükler, evre bilgisi taşıması, Gauss dağılımlı süreçlerin kümülantlarının sıfır olması, toplamsal sabitlere duyarsız olması, ve toplamsal Gauss dağılımlı gürültü eklenmiş veri dizileri ile yapılan kestirimlerde gürültünün etkisinin yokolmasıdır. Üçüncü mertebeden bir kümülantın iki gecikme değişkenine sahip olması nedeni ile hesap karmaşıklığı artmaktadır. Hesap karmaşıklığını azaltmak için kümülantın gecikme değişkenlerinden birisine sabit değer atanarak elde edilen kümülant dilimleri kullanılmaktadır. İki boyutlu sistemlerde ise bir değişken serbest bırakılmakta ve diğer üç sabit değer atanmaktadır. ÖBYO katsayı kestiriminde genellikle karşılaşılan durum, eldeki sınırlı veri dizisinden yararlanarak katsayıların kestirilmesidir. Model mertebesinin bilinmemesi durumunda ise önce model mertebesi kestirilerek sonra ÖB ve YO katsayıları sıra ile kestirilmektedir. Bunun için güçlü bir matematiksel teknik olan tekil değer ayrışımından yararlanılmaktadır. Sonuçların tutarlılığının arttırılması açısından ise denklem sayısını bilinmeyen sayısının üzerine çıkaran yöntemler kullanılmaktadır. Tez çalışmasında incelenen yöntemlerin kullanıldığı benzetimler yapılmış, gürültüsüz ve gürültülü veri dizileri kullanılarak ÖBYO mertebeleri ve katsayıları kestirilmiştir. Sonuçlar incelendiğinde tekil değer ayrışımı, kümülant dilimi kullanan tekniklerin ve q-dilim algoritmasının tutarlı sonuçlar verdiği görülmüştür. Gürültülü ve gürültüsüz veri dizileri ile yapılan kestirimler karşılaştırıldığında aralarında fark olmadığı görülmekte ve bu da üçüncü mertebeden istatistiğin üstünlüğünü ortaya koymaktadır.

Özet (Çeviri)

MODELLING OF TWO-DIMENSIONAL SYSTEMS USING HIGHER ORDER STATISTICS SUMMARY Many methods and approaches are developed on the system modelling and identification which becames an interdisciplinary subject in the last decades with the motivation that stems from the extensive areas of application. In signal processing, the way prefferred to model stochastic systems is spectral estimation. Particularly, power spectrum estimation is used. Chief methods in power spectrum estimation are conventional (Fourier type), maximum likelihood, minimum power and parametric methods. After the development of Kalman filter, parametric methods became more popular and constitutes a good representation of speech and signals processing systems. Parametric methods are more appropriately called model-based methods, because each technique first assumes a pre-specified model set (e.g. all-pole, all-zero etc.) and then estimates the appropriate model parameters. In fact, the model based approach consists of three essential ingredients, 1) data, 2) model set 3) criterion. The parametric signal processing method can be summarized in the following steps, 1 Selecting a representative model set (e.g. autoregressive (AR), moving average (MA), autoregressive-moving average (ARMA), lattice, state space), 2) estimate the model parameters from the data {y(t) or u(t)}, 3) Construct the signal estimates from the parameters. ARMA model is recursive and needs simple hardware and software in realization. The final objective in signal processing is to process a finite set of data and extract important information which is“hidden”in the data. This is usually achieved by combining the development of mathematical formulations which reach a certain level of estimation performance with their algorithmic representation and their application to real data. Various conflicting figures of merit are associated with digital signal processing techniques, namely, quality of the estimates, computational complexity, or data throughput rate, cost of implementation and statistical properties. VIAutocorrelation methods has been dominant in the estimation until the last decade. But the use of higher order statistics became more important, because it has the following motivations, 1) to extract information due to deviations from Gaussianness (normality), 2) to estimate the phase of non-Gaussian parametric signals, 3) to detect and characterize the non-linear properties of mechanisms which generate time series via the phase relations of their harmonic components. There are many papers published in the last three decades dealing with the applications of higher order statistics. Areas of application are oceanography, geophysics, geoscience, passive sonar, biomedicine, telecommunications, speech processing, economic time series, fluid mechanics, plasma physics, sunspot data analysis, pulse shapes, radio image reconstruction, sound quality. Cumuiants are third order moments and bispectrum is their Fourier transforms. Following important properties can be derived for cumuiants: 1) If x(m) is a Gaussian process, its cumuiants of order greater than two are identically zero, 2) If x(m) is a non-Gaussian process, then its higher-order cumuiants cannot be all identically zero. 3) Cumuiants are shift invariant, that is, the cumuiants of y(m) and y(m-n), where n is non-random, 4) Cumuiants are invariant to additive constants, that is, the cumuiants of y(m) and y(m)+a (a non-random, fixed)are identical, thus, if the given process, y(m) is not zero-mean, its cumuiants may be computed as the cumuiants of the process, y(m)-E{y(m)}, 5) Cumuiants of order greater than two, are generally not fully symmetric functions, and as such, carry phase information, 6) If z(m)=y(m)+g(m), where y(m) is a non-Gaussian process and g(m) is a (colored) Gaussian process of independent of of y(m), then the cumulant of z(m) is identical to the cumulant of the signal y(m). The cumuiants of linear processes carry both amplitude and phase information of the linear system. In power spectrum estimation of an ARMA process is realized by estimating the ARMA coefficients and constructing this model using the estimated coefficients and estimating the power spectrum. For estimation of ARMA coefficients second order moments or autocorrelations have been widely used, but the advantages stated above make third order moments more powerful. However, existence VIIof two lag variables gives rise to computational complexity in the use of third order cumuiants. Slices of cumuiants has been used in order to reduce dimension and as its consequence, computational complexity. Cumulant slice is obtained by setting one of the lag variables to a constant. In two-dimensional systems four lag variables appear and the cumulant slice is obtained by setting three of them to a constant value. The main task is to estimate ARMA coefficients using the available finite set of data. In the case that the model orders are known the corresponding samples of cumuiants (cumulant sample estimates) are computed. The problem of estimating ARMA coefficients are generally handled in two successive steps: estimating AR coefficients and estimating the MA coefficients using these estimated values. For estimating AR coefficients a linear set of equations is constructed as in the same way of Yule-Walker equations. The only difference is replacement of samples of autocorrelations by samples of third order cumuiants. However, the matrix associated with these equations and formed with the cumulant slices must be of full rank. Once the AR cofficients are estimated, by substitution of the estimated coefficients the problem reduces to pure MA case. Then the MA coefficients can be estimated using cumulant sample estimates. In general, the problem of estimating ARMA coefficients appears with unknown model orders. To determine the model order, a powerful mathematical tool, singular value decomposition is employed. Since the model order is unknown, the linear set of equations used to estimate AR coefficients are constructed in a quantity associated with an order chosen as much higher than the likely real order. Singular value decomposition is applied to the cumulant sample estimates matrix and gives the eigenvalues ot the system. As the singular values are ordered, a break will be observed. This break gives the effective AR order. The small eigenvalues next to ts breakpoint are nearly zero. But after reconstruction of the cumulant samples matrix after setting the small eigenvalues to zero, its dimension is still the same and the linear system of equations is overdetermined.The solution needs a special method. In addition to this unavoidable overdeterminacy, the number of equations may further be increased by computing the cumulant slices for a number of the second lag variable instead of fixing to a constant. Prior to determine MA order both AR order and coefficeints VIIIshould be determined. The time series is passed through the finite impulse response filter whose coefficients are estimated AR coefficients. The output of this filter is called 'residual time series' and using these output series another set of cumulant sample estimates are generated. The lag for which cumulant sample becomes too close to zero is the effective MA order. The data sequence may be corrupted with additive (colored) Gaussian noise. But, this corruption does not effect the performance of estimates because by the property 6 stated above, third order cumulant of additive Gaussian noise vanishes and its contribution to estimate also. In this thesis, there is two application examples realized on two ARMA models by simulations. AR coefficients are estimated by three different techniques, all are based upon obtaining a linear set of equations like Yule-Walker equations. One uses Hankei structured matrix and has number of equations which is equal to number of equations, but the overdeterminacy originating from the use of the“anticipated order”involves. Other two technique use equations whose number is more than number of unknowns. The reason to generate overdetermined set of equations is the more the equations provide the more the information on the system. Because each individual equation carries information on the system. Simulations are. made for a 256x256 array of data which are generated for 10 statistically independent Monte carlo runs. Cumulants are computed and averaged over 10 runs. The data or time series are generated for three cases, noiseless, noisy with signal-to- noise ratios 10 and 20 dB. ARMA coefficients are estimatid via three techniques and for three distinct set of data. Before estimating AR and MA coefficients, model orders are to be estimated. Anticipated orders are chosen as 10 which is much higher than the highest actual order 3. Singular value decomposition is successful in estimating the actual orders for all simulation examples. The robust technique“visual inspection”can be regarded as successful. The simulation method is based on the algorithm which imposes the denominator of trensfer function as product seperable in its individual arguments. Thus estimation of AR coefficients are made seperately for each argument. Free and costant choice of variables in cumulant slices are made accordingly. For a proper estimation, the IXsystem is constrained as free of pole-zero cancellation and causal or anti-causal. In the first technique, the cumulant slices are computed as one of variiables set to change free and the other three set to zero. The technique is successful in estimating AR coefficients. In the second technique, equations appeared in a quantity more than unknowns. This technique yielded superior result among tne three techniques. Third technique is adapted from Cadzow's work on ARMA coefficients estimation by using an overdetermined set of equations formed with autocorrelations. All three techniques have the same performance of estimation also for noisy data. Choice of“anticipated order”over 15 make the estimates diverge. This tells that overdeterminacy over a certain amount makes the solutions unstable. The works in this field seems to be concentrated on reduction of computational complexity, rather than mathematical deductions on theory. New algorithms towards the reduction of computational complexity introduces restrictions on the descripton of processes, such as being causal and having a seperabie denominator. The expansion in the application areas and especially growing applications in multidimensional systems give rise to searches towards developing computationally simple algorithms to overcome the limitations in hardware and software.

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