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Linear discriminant analysis (LDA), normal discriminant analysis (NDA), canonical variates analysis (CVA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or ...
Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions. The algorithm is named after Joseph Fourier [ 1 ] who proposed the method in 1826 and Theodore Motzkin who re-discovered it in 1936.
Gaussian elimination has O(n 3) complexity, but introduces division, which results in round-off errors when implemented using floating point numbers. Round-off errors can be avoided if all the numbers are kept as integer fractions instead of floating point. But then the size of each element grows in size exponentially with the number of rows. [1]
After that, in each step the expressions R k ij are computed from the previous ones by R k ij = R k-1 ik (R k-1 kk) * R k-1 kj | R k-1 ij. Another way to understand the operation of the algorithm is as an "elimination method", where the states from 0 to n are successively removed: when state k is removed, the regular expression R k-1
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
where is the number of examples of class . The goal of linear discriminant analysis is to give a large separation of the class means while also keeping the in-class variance small. [ 4 ] This is formulated as maximizing, with respect to w {\displaystyle \mathbf {w} } , the following ratio:
The method of reduction of order is used to obtain a second linearly independent solution to this differential equation using our one known solution. To find a second solution we take as a guess y 2 ( x ) = v ( x ) y 1 ( x ) {\displaystyle y_{2}(x)=v(x)y_{1}(x)} where v ( x ) {\displaystyle v(x)} is an unknown function to be determined.
This is the case of the theory of polynomials over an algebraically closed field, where elimination theory may be viewed as the theory of the methods to make quantifier elimination algorithmically effective. Quantifier elimination over the reals is another example, which is fundamental in computational algebraic geometry.