Recursive regression of a straight line

Before we introduce the Kalman filter, we reformulate the least-squares algorithm discussed in Chapter 8 in a recursive way. By way of illustration, we consider a simple straight line model which is estimated by recursive regression. Firstly, the measurement model has to be specified, which describes the relationship between the independent variable x, e.g., the concentrations of a series of standard solutions, and the dependent variable, y, the measured response. If we assume a straight line model, any response is described by  

A recursive algorithm which estimates Pq and P, has the following general form New estimate = Previous estimate + Correction 

After each new observation, the estimates of the model parameters are updated (= new estimate of the parameters). In all equations below we treat the general case of a measurement model with p parameters. For the straight line model p = 2. An estimate of the parameters b based ony - 1 measurements is indicated by b(/ - 1). Let us assume that the parameters are recursively estimated and that an estimate h(j - 1) of the model parameters is available from y - 1 measurements. The next measurement y(j) is then performed at x(j), followed by the updating of the model parameters to b(/). 

The first step of the algorithm is to calculate the innovation (/), which is the difference between measured y(j) and predicted response y(j) at x(j). Therefore, the last estimate b(/ - 1) of the parameters is substituted in the measurement model in order to forecast the response y(j), which is measured at x(j)  

The innovation I(j) (not to be confused with the identity matrix I) is the difference between measured and predicted response at x(j). Thus /(/ ) = y(j) - y(j). 

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