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 in the range [1-], which is a straight forward method. The value of K 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 for every value of The second E(W) K. method is the hierarchical method, in which the cost function is computed for different E(W) values of in the range [1-]; is considered as the value of corresponds to the K K_opt K minimum residual variance. However, this method produces a set S of potential candidates for as the residual variance has more than one minimum [84]. Finally, the residual variance K_opt for all the values of from the set must be computed and the K_opt is chosen as the value of K S K corresponding to the minimum residual variance. In this work, the first method is used to optimize the parameter K. Local Linear Embedding Regression (LLER): 3.2.1. 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 is A decomposed as the product of the LLE matrix and the reconstruction factors P. Y = (4.3) A Y.P Where is the number of variables in the raw spectra, is the dimensionality of the D d embedded vector and is the number of training spectra. N 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
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
Patchava_thesis_Krishna_200318.docx