Evaluation of the innovation

Before we can apply an adaptive filter, we should define a criterion to judge the validity of the model to describe the measurements. Such a criterion can be based on the innovation defined in Section 41.2. The concept of innovation, /, has been introduced as a measure of how well the filter predicts new observations  

Thus /(/ ) is a measure of the predictive ability of the model. For the calibration example discussed in Section 41.2, x(/ - 1) contains the slope and intercept of the straight line, and h (/) is equal to [1 c(/)] with c(j) the concentration of the calibration standard for the yth calibration measurement. For the multicomponent analysis (MCA), x(/ -1) contains the estimated concentrations of the analytes after y - 1 observations, and h (/) contains the absorptivities of the analytes at wave-lengthy. 

It can be shown [4] that the innovations of a correct filter model applied on data with Gaussian noise follows a Gaussian distribution with a mean value equal to zero and a standard deviation equal to the experimental error. A model error means that the design vector h in the measurement equation is not adequate. If, for instance, in the calibration example the model was quadratic, should be [1 c(j) c(j) ] instead of [1 c(j)]. In the MCA example h (/) is wrong if the absorptivities of some absorbing species are not included. Any error in the design vector appears by a non-zero mean for the innovation [4]. One also expects the sequence of the innovation to be random and uncorrelated. This can be checked by an investigation of the autocorrelation function (see Section 20.3) of the innovation. 

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