Gram polynomials

Separating measured data vectors or matrices into independent lower order approximations and residual terms is useful both in process performance evaluation, as variance contributions can be clearly separated, and in feedback process control, as the number of decision variables can be significantly reduced while the adverse effects of autocorrelation are eliminated. In the following two sections orthogonal decomposition approaches using Gram polynomials and principal components analysis (PCA) will be introduced. 

Gram polynomials are orthogonal and defined uniquely for discrete data at equidistant positions much like the spatial data collected in sheet forming processes. For N data positions, discrete-point scalar components of the mth-order polynomial vector pm = [Pm,i---Pm,n---Pm,jv] are defined as 

Gram polynomial approximation has essentially removed all low-frequency variations from the profile. 

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Example Consider the CD profile examined in Figure 10.6. The measurement vector ycD has = 54 data positions with the corresponding Gram polynomials pj through P53 that form = [P1P2...P53] and as defined earlier. Computationally S can be easily... 

Figure 10.10. Gram polynomials as basis functions, first-order through fifth-order.

Figure 10.11. Fourth-order Gram polynomial approximation of mean CD profile and comparison with the residual measurement signal, ycD R) = ycD - ycD(M)-...

Figure 10.12. Accounting for total CD profile variability through Gram polynomial approximations and the cumulative power spectra for ycd and yCD R)-...

A reasonable observation after these results may be that an effective CD controller should be able to reduce CD variability by at least a factor of 2. Another way of stating the same observation from a performance monitoring point of view would be that as long as a CD controller is functioning properly both plots in Figure 10.12 should indicate insignificant contributions from Gram polynomials up to order 4 or 5, which would also mean essentially similar power spectra for ycd and ycd r) ... 

Principal components analysis (PC A) (see Section 3.1) provides a technique to define orthogonal basis functions that are directly constructed from process data, unlike Gram polynomials which are dependent on the data length only. PCA is also uniquely suitable for extracting the dominant features of two-dimensional data like the residual profile obtained after MD/CD decomposition, Yr. 

It also calls the subroutine ConvolutionFactors, which calculates the Gram polynomials and the corresponding convolution weights. 

The form of the polynomials is similar to the standard, discrete Gram polynomials. 

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