Kalman innovation

The above example illustrates the self adaptive capacity of the Kalman filter. The large interferences introduced at the wavelengths 26 and 28 10 cm have not really influenced the end result. At wavelengths 26 and 28 10 cm , the innovation is large due to the interfered. At 30 10 cm the innovation is high because the concentration estimates obtained in the foregoing step are poor. However, the observation at 30 10 cm is unaffected by which the concentration estimates are restored within the true value. In contrast, the OLS estimates obtained for the above example are inaccurate (j , = 0.148 and JCj = 0.217) demonstrating the sensitivity of OLS for model errors. 

During the standard Kalman filter calculations, the matrix M has been evaluated from Eq. (8.38). When the test gives an alarm of malfunction, one or more elements of the innovation sequence vector is supposed to be at fault. In order to satisfy the abnormal situation, if one element is assumed to be at fault, the corresponding term in the matrix M would be greater than under normal circumstances. Thus,... 

The proposed technique is based on an extension to time-varying systems of Wiener s optimal filtering method (l-3). The estimation of the corrected chromato gram is optimal in the sense of minimizing the estimation error variance. A test for verifying the results is proposed, which is based on a comparison between the "innovations" sequence and its corresponding expected standard deviation. The technique is tested on both synthetic and experimental examples, and compared with an available recursive algorithm based on the Kalman filter ( ). 

Using the basic Kalman filter, the prediction error (or innovation), can be defined as... 

Using the basic Kalman filter as given in Theorem 5.5 and the initial parameter estimate, 6q, obtain the innovations and error covariances for t=l,..m. 

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