y x x. y = x y y x x. y = - x y y x x. y = 0 Two vectors x and y are orthogonal when their scalar product is zero x. y = 0and xy = 1= Two vectors x and y are orthonormal

Correlation Coefficient = cos  ’ r = (x – x). (y- y) x - xy - y Degree of overlapping and correlation coefficient

Correlation m.file

? Investigate the correlation between spectral overlapping and angle between vectors under following conditions: a) Two spectra with same max and different intensities b) Two spectra with same max and different band widths

? Plot the variation of correlation coefficient with change in degree of spectral overlapping

Orthogonal Projection y x p p = b x y – b x x. (y – b x) = 0 x. y = b x. x x. y x. x b = x T y x T x = x T y x T x p = x p = b x

Least Squares y = b x ^ E = b x - y ^ ^ E = ( b x - y). ( b x - y) ^ ^ = b 2 x T x - 2 b x T y + y T y ^ ^ ^ 2 d E d b = 2 x T x b - 2 x T y ^ x T y x T x b = x T y x T x y = x ^

Least Squares with MATLAB

Gram-Schmidt Orthogonalization x2x2 x1x1 x 1,x 2,…,x k v 1,v 2,…,v k Orthogonalization

Gram-Schmidt Orthogonalization v 1 = x 1 v 2 = x x 3. v 1 v 3 = x 3 - v 1 - v 2 v1v1 2 x 3. v 2 v2v2 2 x k. v 1 v k = x k - v 1 - v 2 - … - v k-1 v1v1 2 x k. v 2 v2v2 2 x k. v k-1 V k-1 x2x2 p v2v2 v1v1 2 x 2. v 1 v1v1 v1v1

x y M1 2x3x M2M3 2 M 3. x M 3 - x x 2 M 3. x M 3 - x x y 2y 3y x M1 M2 M3

x y M1 M2 M3  = 44.64° Angle between comp1 and comp2    = 22.77° Angle between comp2 and mixture1 Angle between comp1 and mixture1    = 21.87°      = 44.64°

? Plot the variation of angle between mixture vectors and one of the component vectors as a function of its concentration

Equal lengths of orthogonal component of mixture vector

Non-equal lengths of orthogonal component of mixture vector