Görsel uyarılmış potansiyellerin Kalman süzgeci ile kestirimi

Abstract

Fiziksel bir dış uyarımla EEG'de meydana gelen potansiyel değişimlerine uyarılmış potansiyeller denir. Bu tezde, görsel uyarılmış potansiyellerin kestirimi için bir yöntem incelenerek uygulanmıştır. Bu yöntemde yapılan ölçüm uyarılmış ve kendinden varolan iki bileşenin toplamı olarak düşünül muş t Ur. Her iki bileşenin de gözlenebilir özellikleri göz önüne alınarak bileşik bir durum uzayı modeli oluşturulmuş tur. Bu modelde kendinden varolan etkinlik öz bağlanımlı bir süreç olarak tanımlanmış ve karakteristikleri ölçülen etkinliğin uyarı öncesi kısmından kestirilmiş tir. Uyarılmış potansiyel doğrusal bir sistemin dürtü yanıtı olarak modellenmiştir. UP'nin karekteristikleri ortalama dalga şeklinden çıkarılmıştır. Parametrik sistem teorisi ile bütün modelin bir durum uzayı gösterimi elde edilmiştir. Modelin durum uzayı gösterimi esas alınarak, sistem durumunu gözlemek amacıyla her iki etkinlik içinde optimum kestirimlerde bulunan bir Kalman süzgeci tasarlanmıştır. Gerçeklenen yöntemin özellikleri benzeşim verilerine uygulanarak denenmiştir. Son olarak yöntem ölçülmüş gerçek verilere uygulanmıştır. Giriş ve çıkış işaret gürültü oranları hesaplanarak yapılan kestir imin hassasiyeti belirlenmiştir.

The recording of the electrical activity of the brain is known as electroencephalogram . Brain potantials which are elicited external stimuli is called evoked potantials , EP's are responses in the EEG to specific activation of sensory pathways. Evoked brain potentials are measured using one or more scalp elctrodes and most generally referencing the measurements to a position such as linked ears. In this thesis, a system theoretic approach is described which is based on parametric models describing the data generating processes. The poststimulus EEG activity is assumed to be an additive superposition of an evoked and a spontaneous part. The characteristics of the spontaneous EEG are estimated from the activity measured immediately before each stimulus, while those of the EP are derived from the average waveform. A compound state space model trying to incorporate the observable properties of both parts is formulated on the basis of additivity of the two components. The spontaneous EEG is modelled as the output of a linear system driven by white noise. It is described as an autoregressive CAR) process. We thus write, e+s CD where e is the white input noise. For estimating the AR parameters and the variance of the input noise from measured data the Burg algorithm used. The fixed values in the range from 5 to 13 are proposed for the model order p. The EP is modeled by an impulse response of a parametrically described system. The system is characterized by the vector of parameters c s where, s, c is the vector of parameters, and s is the vector of past values of the output and the input signal. The conventional averaged data served as a template for the single EP's. M y t=d... d+N-1 <3> MJ=i J Where y. 's are the measured single trials, and M is the number of stimulus repetitions. Therefore, an estimate of the vector parameters c is obtained by fitting the impulse response of the system to the average measured data, using y instead of unknown s(t). The least squres method was chosen for fitting the parameters. When the N values of the template y The estimated parameter vector c_can be determined with using standart numerical methods like cholesky decomposition. For the signals at hand, the model orders between S -11 were appropriate. In order to take into account of the variabilities of the EP under repeated stimulation, a stochastic amplification factor A and a stochastic latency shift L are introduced to form the entire EP model: sJ J J According to the definition, the mean of the stochastic amplification factor A has to be 1. From the mean and variance of both stochastic elements, the mean and variance of the input signal to the determi nistic element are easily obtained for unit impulse input u<u ~2 var <au] =P > A (6) <7> Where P is the probablity of shift L=t, and S is the variance of the random amplification factor A. The described model assumes the single responses to be a scaled and shifted version of the template. Using additivity, a state-space represantation of the combined EEG-UP model can be set up with a compound state vector x comprising the two partial models. x=Fx+Bu<t-d+l)+dw <8> y=Hx<t)+v <9> Where F is the system matrix, B is the input matrix, D is the noise input matrix, H is the output matrix, w is the - 2 output measurement noise with variance S. V X e F= 0. c d d 0.. m. O. n. 0...1 0 0. 0.... 0 o. 0.... o o. 0 0 0. o o....0 0...00. 0 a 0 0. 0 0 i 0 0 o. a 1 0 The input matrix B, the noise input matrix D the output matrix H: and is the input noise with variance given by <7>, and w is the noise input to EEG model. The compo nents of the noise input vector w are independent and the cross-ponding covariance matrix is diagonal, a </i-k</y</t)+v</t-d+l)+dw</au