Least squares filter

We can show that the filter is of the form of Equation (5.40). The development of the formula to estimate the coefficients Bi, B2,. .., Bjv is quite complex. However the end result is simple to apply. Indeed the reader could skip to Equation (5.56) if happy to just to accept the result. 

The equation of the line is developed to minimise the sum of the squares between the predicted value of Y and the actual value, i.e. 

Partially differentiating with respect to each of m and c, and setting the derivative to 0 will identify the best choice of these values, i.e. 

This is a linear function of previous values of Y and so can be written in form of Equation (5.40). To determine the coefficients B) we use the formula for the sum of a series of consecutive integers  

Substituting these into Equation (5.52) we can determine the value of each coefficient so 

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We have carried out simulations using polynomial least-squares filters of the type described by Savitzky and Golay (1964) to determine the impact of such smoothing on apparent resolution. For quadratic filters, a filter length of one-fourth of the linewidth (at FWHM) does not seriously degrade the apparent resolution of two Gaussian lines in very close proximity. 

Many types of data treatment software allow the composition of a mixture to be obtained from its spectra. Kalman s least squares filter is one of the most widely known of these methods. Using successive approximations, it automatically finds the spectra of the sample solution by addition of the spectra of each compound contained in the spectral library (i.e. by additivity of the absorbances) and use of weight coefficients. These are called deconvolution methods (Fig. 11.26). 

FIGURE 10.16 Polynomial least-squares filtering. A quadratic, five-point polynomial filter is shown. 

Numerous software data treatments authorize the elucidation of mixture composition from spectra. One of the best-known methods is the Kalman s least squares filter algorithm, which operates through successive approximations based upon calculations using weighted coefficients (additivity law of absorbances) of the individual spectra of each components contained in the spectral library. Other software for determining the concentration of two or more components within a mixture uses vector quantification mathematics. These are automated methods better known by their initials PLS (partial least square), PCR (principal component regression), or MLS (multiple least squares) (Figure 9.26). 

A more general process known as least-squares filtering or Wiener filtering can be used when noise is present, provided the statistical properties of the noise are known. In this approach, g is deblurred by convolving it with a filter m, chosen to minimize the expected squared difference between / and m g. It can be shown that the Fourier transform M of m is of the form (1///)[1/(1 - - j], where S is related to the spectral density of the noise note that in the absence of noise this reduces to the inverse filter M = /H. A. number of other restoration criteria lead to similar filter designs. 

We can choose different coefficients for B in Equation (5.40) provided they sum to 1. An approach which does this is known as the least squares filter. It gets its name from the least squares regression technique used to fit a line to a set of points plotted on an XY (scatter) chart. Its principle, as shown in Figure 5.19 is to fit a straight fine to the lastiVpoints. The end of this line is 7 . 

Figure 5.20 Performance of averaging and least squares filters...

The advantage of this filter is that, because it uses the trend of the unfiltered measurement to predict the filtered value, it has no lag. Indeed its predictive nature introduces lead which partially counteracts the process lag. This effect is shown in Figure 5.21. Both filters have been tuned to give the same level of noise reduction. The least squares filter not only outperforms the exponential filter in that it tracks the base signal more closely but actually overtakes the base signal. 

If the filter order m is not fixed but increases with time index k instead, a recursive least-squares filter can be derived from the batch type least-squares [4]. 

Fading Memory Least-Squares Filter (FMLS)... 

Eqs. (7)-(10) constitute the structures of the fading memory least-squares filter for state estimation, which are in the recursive form. 

The optimal Kalman filter is derived under the optimality criterion of least-mean-square error of the state, while the fading memory least-squares filter (FMLS) is under least-squares error of the measurement. However, it is interesting to see that they turn out to have the same form of structures. Examing Eqs. (7)-(10), it is found that and in FMLS are equivalent to the a priori state error covariance and the posterior error covariance Pjt, respectively, in the Kalman filter. The formula for Pj i is... 

Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

To summarize, Wiener inverse-filter is the linear filter which insures that the result is as close as possible, on average and in the least squares sense, to the true object brightness distribution. 

Leach, R. A., Carter, C. A., and Harris, J. M., Least-Square Polynomial Filters for Initial Point and Slope Estimation, Anal. Chem. 56, 1984, 2304-2307. 

Kahn, A., Procedure for Increasing the Accuracy of the Initial Data Point Slope Estimation by Least-Squares Polynomial Filters, Anal. Chem. 60, 1988,... 

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 ... 

This assumption can be relaxed when the experimental error in the independent variable is much smaller compared to the error present in the measurements of the dependent variable, In our case the assumption of simple linear least squares implies that Xvdt is known precisely. Although we do know that there are errors in the measurement of Xv, the polynomial fitting and the subsequent integration provides a certain amount of data filtering which could allows us to assume that experimental error in Jxvdt is negligible compared to that present in S(t,) or P(t,). 

The Fourier transforms were performed in the standard way. No smoothing nor filtering was employed. Subtraction of the data from the least squares fit removes the constant or linear term characterizing a Markovian process. Fourier transform of the differences from the linear fit suppresses the enhancement of both the power and amplitude spectra at low frequencies. 

Historically, treatment of measurement noise has been addressed through two distinct avenues. For steady-state data and processes, Kuehn and Davidson (1961) presented the seminal paper describing the data reconciliation problem based on least squares optimization. For dynamic data and processes, Kalman filtering (Gelb, 1974) has been successfully used to recursively smooth measurement data and estimate parameters. Both techniques were developed for linear systems and weighted least squares objective functions. 

Extended Kalman filtering has been a popular method used in the literature to solve the dynamic data reconciliation problem (Muske and Edgar, 1998). As an alternative, the nonlinear dynamic data reconciliation problem with a weighted least squares objective function can be expressed as a moving horizon problem (Liebman et al., 1992), similar to that used for model predictive control discussed earlier. 

Gratton et. al. in [34] describes a least squares regression method to model heartbeat artifacts and to filter it out adaptively. Since the heartbeat rate is approximately 1 Hz, it is necessary that data is sampled at a sufficiently high rate (e.g. above the Nyquist rate) so as to represent and filter the artifacts out correctly and have minimal impact on the signal of interest itself [98]. 

In the present study we have used the phase and amplitude functions of absorber-scatterer pairs in known model compounds to fit the EXAFS of the catalysts. By use of Fourier filtering, the contribution from a single coordination shell is isolated and the resulting filtered EXAFS is then non-linear least squares fitted as described in Ref. (19, 20). 

Figure 2. a) X-ray absorption spectrum near the Mo K-edge of the Co/Mo = 0.125 unsupported Co-Mo catalyst recorded in situ at room temperature b) normalized Mo EXAFS spectrum c) absolute magnitude of the Fourier transform d) fit of the first shell e) fit of the second shell. The solid line in d) and e) is the filtered EXAFS, and the dashed line is the least squares fit. 

In this chapter different aspects of data processing and reconciliation in a dynamic environment were briefly discussed. Application of the least square formulation in a recursive way was shown to lead to the classical Kalman filter formulation. A simpler situation, assuming quasi-steady-state behavior of the process, allows application of these ideas to practical problems, without the need of a complete dynamic model of the process. 

Problems like overlapping and interfering of fluorophores is overcome by the BioView sensor, which offers a comprehensive monitoring of the wide spectral range. Multivariate calibration models (e.g., partially least squares (PLS), principal component analysis (PCA), and neuronal networks) are used to filter information out of the huge data base, to combine different regions in the matrix, and to correlate different bioprocess variables with the courses of fluorescence intensities. 

Adopting Eu=ql and Ey=0, then Equation l6 reduces to Equation 5 With Eu=ql and Ey=rl, Equation l6 has a format which is identical to the solution derived in (2T) through a deterministic minimum least squares approach for time-invariant systems. This is to be expected, because the Wiener filtering technique may be in fact Included as part of the general theory of least squares. 

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