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The study aims to determine a mathematical model that can be used to describe the growth of the Adana pigeon. Since pigeons have only one breeding season, just one or two pairs of baby pigeons are raised per year. Hatchlings sometimes die before reaching adulthood. For this reason, measurements can be taken for 10, 15 and 60 days periods. Related with this issue, only 43-days measurements of 68 pigeons are used over a 6-year period. The study is modelled by taking the day-to-day average of the data (43 days) of 68 pigeons. The study was conducted on 68 Adana pigeons in the interval between the age of 1 and 43 days. The growth of pigeon cub was measured by daily live weight until 1 to 43 days. The estimation is carried out by writing the specific Matlab codes. Classical growth functions used in animals are in nonlinear form. Various numerical methods have been developed to estimate parameters in nonlinear functions. Special program routines have been developed to implement these methods. In these nonlinear models, there are more than one parameter to be estimated. Therefore, the number of mathematical operations in estimating the parameters is large. The most used models in the literature are Brody, Bertalanffy, Logistic, Generalized Logistic, Gompertz, Richards, Negative Exponential, Stevens, and Tanaka. However, as far as is known, there is no published article for Adana pigeons that uses all of these models and compares which one is better. These models are Brody, Bertalanffy, Logistic, Generalized Logistic, Gompertz, Richards, Negative Exponential, Stevens, and Tanaka. The best analysis was done by the Richards model in terms of both the Mean Squared Error (MSE), mean absolute percentage error (MAPE) and (Coefficient of Determination) R2 .

In this paper, COVID-19 cumulative cases are estimated with AKF based on total COVID-19 cases between January-September 9, 2020 in USA, Germany, United Kingdom, Italy, France, Russia, Brazil, India, Turkey, Spain, Peru, Colombia, South Africa, Argentina, Iran, Pakistan. The cumulative covid-19 cases time-series was modeled with a stochastic dynamic linear model (DLM). The estimation performance of the models is measured by the calculation of mean square error (MSE) and coefficient of determination (R2). Ca lculated MSE an d R2 values showed that the model and AKF could be used to estimate the number of cases in these countries. In this study, firstly, the cumulative number of cases was estimated. Secondly, using these estimates number of daily cases was calculated. Thirdly, the reproduction number was obtained by using these number of daily cases. The model and estimation method used is suitable. The AKF algorithm uses only the number of cases in the last day. We propose that the model and estimation method under consideration is a convenient tool for calculating the reproduction number depending on time.

In this work, Adaptive Extended Kalman Filter (AEKF) is introduced and its use for oscillation of an object connected to the end of a spring is shown. As a new approach, an AEKF is used as a nonlinear estimation tool for online estimation of the states and parameters of an oscillating object attached to the end of a spring model. Parameter states that do not change with time were examined. The simulation results revealed that with proper selection of initial values of AEKF, AEKF is a very useful tool for this particular application.

In this study, sleep electroencephalography (EEG) which is frequently used in statistical modelling has been modelled with the autoregressive (AR) time-series model and what kind of a structure the variance of the term white noise included in the model represented in different sleep stages has been observed. Taking all of the stages scored in accordance with Rechtschaffen and Kales criterion into account separately, epoches in each stage have been modelled with the AR and the variance of the term white noise in this model has been monitored. The study has evaluated the sleep EEG variances of a subject. In accordance with the results, the heterogeneity at Stage 2 was thought to be the reason why the objective differences appeared in scoring. It is thought that this data pointed out a necessity that the period in Rechtschaffen and Kales scoring which is called Stage 2 must be revised.

In this study, theemergences of postsynaptic action potentials are investigated using simulationstudy which is depended on the statistical model of acetylcholine (Ach) quantalrelease theory at motor end-plate. It is supposed that the end-plate potentials(EPP) are sum of miniature end-plate potentials (MEPP) which are released asquantal and random in number (N). By the hypothesis that N number has a Poissonand MEPPs have a Gaussian distribution, EPPsare obtained by simulating distributions parameters. Our study suggests thatthe simulation of the EPPs are aconvenient tool in understanding bothphysiological process of synaptic transmission and pathological process of neuromuscular junction.

TheKalman filter is most popular estimation technique for solving state estimation problems of dynamical systems and it has been the most frequently used algorithm in applications from different areas such as science, military and economics etc. The Kalman filter works best with predictive performance as long as system characteristics are known correctly.However, the performance of the Kalman filter will dramatically decrease when system characteristics are either unknown or partially known. Numerous studies have been published so far to get over the problem of performance loss in the Kalman filter.Someresearchers introduceda fading factor to improve the performance of the Kalman filterunder unknown or partially known initial information. “Adaptive estimation of multiple fading factors in Kalman filter for navigation applications” (AEMFFKF) is one of these studies. In this paper, adaptive fading Kalman filter with the multiple forgetting factors is consideredandan adaptive estimation algorithm is proposed to determineforgetting factors which can not be determinedintheAEMFFKF. In addition, A Monte Carlo simulation is performed to compare the estimation performances of the Kalman filterwith the adaptive filters., Kalman filtresi dinamik sistemlerde durum tahminprobleminin çözümü için kullanılan popüler bir tahmin yöntemidir. Fen, mühendislik, ekonomi, askeri vb. olmak üzere birçok alandan probleme kolayca uygulanabilir. Sistem karakteristikleri doğru olarak bilindiği sürece Kalmanfiltresi en iyi tahmin performansı ile çalışır. Ancak sistem karakteristiklerinin kısmen bilindiği durumlarda veya yanlış bilindiği durumlarda filtrenin tahmin performansında ciddi kayıplar olması kaçınılmazdır. Kalman filtresindeki performans kaybı probleminin üstesinden gelebilmek için şu ana kadar çok sayıda çalışma yayınlanmıştır. Bir kısım araştırmacı tarafından Sistem karakteristiklerinin kısmen veya tamamen hatalı bilinmesi durumunda, filtrelemede bazı güçlendirmelerin yapılmasını sağlayacak unutma faktörü ile uyarlanmış Kalman Filtresi tanıtılmıştır. “Adaptive estimation of multiple fading factors in Kalman filter for navigation applications” (AEMFFKF) bu çalışmalardan bir tanesidir.Bu çalışmada, çoklu unutma faktörüyle uyarlı Kalman filtresi incelenmiş ve AEMFFKF yönteminde belirlenemeyen unutma faktörlerini belirleyebilmek için adaptif bir tahmin algoritması önerilmiştir. Ayrıca yapılan simülasyon çalışmasıyla Kalman filtresinin performansı ile uyarlı filtrenin tahmin performansı karşılaştırılmıştır.

In this paper, we employ a non-linear state space model and the extended Kalman filter to simultaneously estimate the time-varying parameters in an optimal control problem, where the objective (loss) function is quadratic. Our methodology also allows us to derive the difference between the optimal control and the observed control variable. A simulation exercise based on a simple intertemporal model shows that the estimated parameter values are very close to their population values, which provide further support for the estimation methodology introduced in this paper

Kalman filtresi dinamik sistemlerde durum tahmin probleminin çözümü içinkullanılan popüler bir tahmin yöntemidir. Fen, mühendislik, ekonomi, askeri vb.olmak üzere birçok alandan probleme kolayca uygulanabilir. Sistemkarakteristikleri doğru olarak bilindiği sürece Kalman filtresi en iyi tahminperformansı ile çalışır. Ancak sistem karakteristiklerinin kısmen bilindiğidurumlarda veya yanlış bilindiği durumlarda filtrenin tahmin performansındaciddi kayıplar olması kaçınılmazdır. Kalman filtresindeki performans kaybıprobleminin üstesinden gelebilmek için şu ana kadar çok sayıda çalışmayayınlanmıştır. Bir kısım araştırmacı tarafından Sistem karakteristiklerinin kısmenveya tamamen hatalı bilinmesi durumunda, filtrelemede bazı güçlendirmelerinyapılmasını sağlayacak unutma faktörü ile uyarlanmış Kalman Filtresi tanıtılmıştır.“Adaptive estimation of multiple fading factors in Kalman filter for navigationapplications” (AEMFFKF) bu çalışmalardan bir tanesidir.Bu çalışmada, çoklu unutma faktörüyle uyarlı Kalman filtresi incelenmiş veAEMFFKF yönteminde belirlenemeyen unutma faktörlerini belirleyebilmek içinadaptif bir tahmin algoritması önerilmiştir. Ayrıca yapılan simülasyon çalışmasıylaKalman filtresinin performansı ile uyarlı filtrenin tahmin performansıkarşılaştırılmıştır. TheKalman filter is most popular estimation technique for solving stateestimation problems of dynamical systems and it has been the most frequentlyused algorithm in applications from different areas such as science, military andeconomics etc. The Kalman filter works best with predictive performance as longas system characteristics are known correctly. However, the performance of theKalman filter will dramatically decrease when system characteristics are eitherunknown or partially known. Numerous studies have been published so far to getover the problem of performance loss in the Kalman filter. Some researchersintroduced a fading factor to improve the performance of the Kalman filter underunknown or partially known initial information. “Adaptive estimation of multiplefading factors in Kalman filter for navigation applications” (AEMFFKF) is one ofthese studies.In this paper, adaptive fading Kalman filter with the multiple forgetting factors isconsidered and an adaptive estimation algorithm is proposed to determineforgetting factors which can not be determined in the AEMFFKF. In addition, AMonte Carlo simulation is performed to compare the estimation performances ofthe Kalman filter with the adaptive filters.

The well known conventional Kalman Filter requires an accurate system model and exact stochastic information. However, in a number of practical situations, the system information is partly known or incorrect. In this situation, the filter may diverge or give based estimate. To overcome the problem of divergence, it is necesary to made some changes and robustness in filtering process. Here arises the notion of adaptive filter. In this study, we propose a new adaptive Unscented Kalman Filter to compensate the effects of incomplete information and the estimation results are evaluated with a simulation study, Bilinen Geleneksel Kalman Filtresi doğru bir sistem modeli ve tam stokastik bilgi gerektirir. Fakat, pek çok gerçek uygulamada sistem bilgisi tam olarak bilinmez veya yanlış bilinir. Bu nedenle filtre ıraksayabilir veya yanlı tahminler elde edilebilir. Bu ıraksama probleminin giderilebilmesi için filtreleme işleminde bazı değişikliklerin ve güçlendirmelerin yapılması gerekir. Uyarlı filtre kavramı burada ortaya çıkar. Bu çalışmada, lineer olmayan durum-uzay modellerinde, eksik sistem bilgisinden meydana gelebilecek problemlerin üstesinden gelebilmek için yeni bir Uyarlı Kokusuz Kalman Filtresi önerilmiş ve bir simülasyon çalışmasıyla tahmin sonuçları değerlendirilmiştir

Bu çalışmada lineer regülatör probleminde bulunan kayıp fonksiyonunda bilinmeyen parametre olması durumunda bu parametrelerin en çok olabilirlik yöntemiyle tahmini ele alınmıştır. Bu amaçla, ilk önce durum-uzay modellerinde model parametrelerinin tahmini için en çok olabilirlik yöntemi, daha sonra lineer regülatör problemi ve çözümü ile ilgili bilgi, son olarak kayıp fonksiyonunda bulunan bilinmeyen parametrelerin en çok olabilirlik yöntemiyle tahmini verilmiştir. Simülasyon çalışması ile yöntemin işleyişi gözlemlenmiştir., In this study, maximum likelihood estimation of loss function parameters in linear regulator problem is investigated. For this reason, in state- space models, maximum likelihood estimation of model parameters, linear regulator problem and this solution and maximum likelihood estimation of unknown parameters in loss function are explained. With simulation study working this giving method is observed.

Bu çalışmada İlerletilmiş Kalman Filtresi ve onun bazı düzenlemeleri tanıtılmaktadır. Bir yayın ucuna bağlı bir cismin salınımını etkileyen çevrenin sürtünme katsayısının tahmin edilmesi problemi üzerine bir simülasyon çalışması yapılmaktadır. İlerletilmiş Kalman Filtresi ve onun bir düzenlemesi ile elde edilen tahmin sonuçları grafiksel olarak karşılaştırılmaktadır., In this study the Extended Kalman Filter and some of it’s modifications are introduced. A simulation study is carried out on the estimation of the coefficient of friction of the medium, which affects the oscillation of a body tied to the end of a spring. The estimation results obtained by the Extended Kalman Filter and the Modified Extended Kalman Filter are compared grafically.

Dört bölümden oluşan bu çalışmanın birinci bölümü kesikli-zaman durum- uzay modellerine ve Kalman Filtresinin değişik ölçütlere göre elde edilişine ayrılmıştır. İkinci bölümünde Kalman Filtresi için Merkezi Limit Teoreminde yakınsama hızları verilmiştir. Üçüncü bölümde, modelin hatalı kurulması durumunda Kalman Filtresinde ortaya çıkan ıraksama problemleri ve bu problemlerin giderilmesine yönelik yapılan çalışmalar hakkında bilgi verilmiştir. Son bölümde, otomatik kontrol kavramları hakkında bilgi verilmiş ve geri- beslemeli kendini uyarlayabilen en küçük varyanslı kontrol yöntemi ile bilgisayar destekli sıcaklık kontrolü yapılmıştır. The first part of this work consisting four parts is devoted to discrete-time state-space models that asserts the Kalman Filter according to several criteria. In the second part, the convergence rates in the Central Limit Theorem for the Kalman Filter is given. In the third part, the divergence problem occurring in the Kalman Filter when the model is misspecified and some information related to work to solve such a problem are given. In the last part, some information about automatic control concept is given and computer supported heat control application is presented for which the method of feed-back adaptive minimum variance control has been utilized. 88

II ÖZET Yüksek Lisans Tezi KESÎKLİ-ZAMAN DURUM-UZAY MODELLERİ VE İNDİRGEMELİ TAHMİN Levent ÖZBEK Ankara Üniversitesi Fen Bilimleri Enstitüsü istatistik Anabilirin Dalı Danışman : Doç. Dr. Fikri ÖZTÜRK 1993, Sayfa : 72 Jüri : Doç. Dr. Fikri ÖZTÜRK Y.Doç.Dr. Ayşen APAYDIN Doç.Dr. Müslüm EKNÎ Dört bölümden oluşan bu çalışmanın birinci bölümü kesikli-zaman durum-uzay modeli ve modele ilişkin baza temel kavramlara ayrılmıştır. İkinci bölümde indirgemeli tahmin incelenmiş, indirgemeli tahmin ve durum-uzay modeli birlikte ele alınarak Kalman Filtresi elde edilmiş ve bazı özellikleri verilmiştir. Üçüncü bölümde durum-uzay modelindeki hata terinüerinin normal dağılıma sahip olması durumunda Bayes tahmin edicisi olarak Kalman Filtresi elde edilmiştir. Son bölümde Kalman Filtresinin işlerliğini görmek amacıyla iki simülasyon çalışması yapılmıştır. Ayrıca 1968-1991 yılları gayri safi milli hasıla zaman serisi kullanılarak 1992-1993 yılları gayri safi milli hasıla değeri tahmin edilmiştir. Anahtar kelimeler : Durum-uzay modeli, Dik izdüşüm, İndirgemeli tahmin, Kalman Filtresi, Bayes tahmini ABSTRACT Masters Thesis DISCRETE-TIME STATE-SPACE MODELS AND RECURSIVE ESTIMATION Levent ÖZBEK Ankara University Graduate School of Natural and Applied Sciences Department of Statistics Supervisor ; Assoc. Prof. Dr. Firki ÖZTÜRK 1993, Page: 72 Jury: Assoc.Prof.Dr. Fikri ÖZTÜRK Asst.Prof.Dr. Ayşen APAYDIN Asgoc.Prof.Dr. Müslûrn EKNl In this work which consists of four parts, the first part is devoted to discrete-time state-space models and some fundamental concepts of the model. In the second part:, recursive estimation is considered. Kalman Filter is obtained as a recursive estimation for state-space models, and some properties of the Kalman Filter are given. In the third part, the Kalman Filter is obtained as a Bayesian estimator in state- space models with normally distributed errors. In the last part, two simulation studies have been done in order to see the performance of the Kalman Filter. The gross national product for the years 1992-1993 is estimated by using the gross national product time series observations from 1968 to 1991. Key words : State-space models, Orthogonal projection, Recursive estimation, Kalman Filter, Bayesian estimation. 72

The Extended Kalman Filter (EKF) is the often used filtering algorithm for nonlinear systems. But it does not usually produce desirable results. Recently a new nonlinear filtering algorithm named as Unscented Kalman Filter (UKF) is introduced. In this paper, we propose a new modified Unscented Kalman Filter (MUKF) algorithm for nonlinear stochastic systems that are linear in some components. These nonlinear systems can be considered as having linear subsystems with parameters and aim is to estimate the system parameters. In simulation study, performance of the EKF, its known variant Modified Extended Kalman Filter (MEKF), UKF and the proposed MUKF is demonstrated for a nonlinear system that is linear in some components. The results show that MUKF gives the best solution for parameter identification problem