Hãy xem xét một tập hợp các vector ngẫu nhiên M-chiều, {x}, phạm vi trong đó là một phần hoặc tất cả các không gian Euclide chiều P. Một eigenbasis hiệu quả để đại diện cho {x} đòi hỏi rằng các eigenvectors ít nhất được sử dụng để mô phỏng {x} đến một mức độ mong muốn của MSE dự kiến | A zv A r . r I I IZ I I x -T- r Appendix 9A Description of the Karhunen-Loeve Transform 265 Consider a set of M-dimensional random vectors x the range of which is part or all of P-dimensional Euclidean space. An efficient eigenbasis to represent x requires that the fewest eigenvectors be used to approximate x to a desired level of expected MSE. Suppose that any sample pattern vector x x1 x2 . xM T from this set belongs to L possible pattern classes mi l 1 2 . L where the a priori probability of the occurrence of the lth class is p mi . Further assume that each class is centralized by subtracting the mean Hi of the random pattern vectors xi in that class. Denoting the centralized observation from mi by Zi we write Zi xi - Hi The centralized pattern vector Zi can be represented by a special finite expansion of the following form M Zi 2 cim m m 1 where m are orthonormal deterministic vectors satisfying the condition m k Smk and 8mk is the Kronecker delta function Smk f 1 m k mk 0 m k while the coefficients cim satisfy E cim 0 E ci 0 and are mutually uncorrelated random coefficients for which L 52 p mi E cimCik pmsmk i 1 The deterministic vectors m in are termed the KLT basis functions. These vectors are the eigenvectors also known as principal components of the covariance matrix R of z L R p mi E ZizT i 1 and Km pm 266 Introduction to Feature Extraction are their associated eigenvalues where pm are the standard deviations of the coefficients. Since the basis vectors are the eigenvectors of a real symmetric matrix they are mutually orthonormal. The eigenvectors m of the covariance matrix R and their corresponding eigenvalues km are found by solving R m Àm m Denoting the KLT basis vectors 1 2 . M in matrix notation the KLT transformation pair for pattern vector Zi and the coefficients of the expansion Ci may be expressed as Zi Ci Cl T Zi It is important to arrange the KLT coordinate vectors m in descending .
