Figure 2.1 Schematic diagram of Michelson interferometer used in FTIR spectrometer [24]. For a monochromatic source, the interferogram can be denoted by a cosine wave [25]. Intensity of interferogram is a function of retardation δ and is defined as shown in Equation 2.1 below. I0 [1+cos(2 π δ υ)] (2.1) I0 is the intensity of interferogram at zero retardation and υ is the wavenumber (cm-1) and equal to 1/λ, where λ is the wavelength of the light. In most of the FTIR spectrometers, a sample is placed before the detector and a broadband source is used to collect the raw spectrum of the sample. For a polychromatic or a broadband continuous light source, equation 2.1 can be extended to I ( = υ) cos (2 π δ υ) dυ (2.2) The location where and all the light frequencies interfere constructively is known as the point of Zero Path Difference (ZPD). As a result, interferogram has maximum intensity at the point of ZPD and is also termed as centerburst. From Equation 2.2, it is apparent that the sample spectrum B(υ) is the Fourier transform of the intensity of interferogram. B (υ) = δ) cos (2 π δ υ) dδ (2.3) As δ) is symmetric at Zero Path Difference, Equation 2.3 can be rewritten as shown below. B (υ) = δ) cos (2 π δ υ) dδ (2.4) In Equation 2.4, integration is performed with respect to δ and the maximum of value of it is chosen by the moving range of movable mirror used in the FTIR spectrometer.
To overcome this problem, two methods are introduced in [84] to optimize the neighborhood size (K). In the first method, the residual variance of the embedded data is computed for all the possible values of K in the range [1-], which is a straight forward method. The value of K corresponding to the minimum residual variance is considered as the optimum value . The drawback of this method is that it is time consuming, as it needs to optimize both and the local reconstruction error and the reconstruction error E(W) for every value of K. The second method is the hierarchical method, in which the cost function E(W) is computed for different values of K in the range [1-]; K_opt is considered as the value of K corresponds to the minimum residual variance. However, this method produces a set S of potential candidates for K_opt as the residual variance has more than one minimum [84]. Finally, the residual variance for all the values of K from the set S must be computed and the K_opt is chosen as the value of K corresponding to the minimum residual variance. In this work, the first method is used to optimize the parameter K. 3.2.1. Local Linear Embedding Regression (LLER): The LLER is a non-linear calibration model, in which, the LLE dimensionality reduction technique is used to map the high dimensional absorbance spectra (A) to a lower dimensional embedded vector (Y). The basic idea in the proposed LLER regression model is that the absorbance matrix A is decomposed as the product of the LLE matrix Y and the reconstruction factors P. A = Y.P (4.3) Where D is the number of variables in the raw spectra, d is the dimensionality of the embedded vector and N is the number of training spectra. In the LLER method, the embedded vectors computed using the LLE analysis represent the scores and loading matrix is computed by multiplying the pseudo-inverse of the embedded
Model RMSEP* RMSECV* Optimum parameters PLSR PLSR PLSR PLSR PLSR None SNV 1st derivative RReliefF FSDR 40.30 35.48 34.10 31.30 16.39 41.85 39.16 35.93 35.47 31.14 6LVs 8LVs 8LVs 8LVs 8LVs, FWHH =7.5 cm-1; truncation frequency (tp) = 0.04 f; 7: NIR
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