Finding Line of Best Fit with MatricesDate: 06/30/2004 at 14:24:33 From: Dominic Subject: Line of best fit using linear algebra Hello, Dr. Math - I've recently been trying to recollect all that I've learned in my Linear Algebra classs. I remember my teacher showing us a way to find a line of best fit using matrices. I believe you need at least two matrices, but I don't remember much else. Can you refresh my memory? Date: 07/02/2004 at 08:42:36 From: Doctor Fenton Subject: Re: Line of best fit using linear algebra Hi Dominic, Thanks for writing to Dr. Math. In trying to fit a line y = mx + b to a set of data points, (x(1),y(1)),(x(2),y(2)),...,(x(n),y(n)), you are trying to solve an overdetermined linear system y(1) = m x(1) + b y(2) = m x(2) + b : : y(n) = m x(n) + b which can be written as a matrix equation [ x(1) 1 ] [ y(1) ] [ x(2) 1 ] [ m ] [ y(2) ] [ : : ] [ b ] = [ : ] [ x(n) 1 ] [ y(n) ] , or A X = B , where A is the n x 2 matrix whose kth row is [ x(k) 1 ], X is the 2 x 1 matrix [ m ] , and B is the column vector of y's. [ b ] You can show that the minimum norm of ||AX - B|| occurs when X satisfies the "normal equations" A^tAX = A^tB , where A^t is the transpose of A. This gives the system of two equations in two unknowns [ n n ] [ n ] [ --- --- ] [ --- ] [ \ \ ] [ \ ] [ / x(k)^2 / x(k) ] [ / x(k)y(k) ] [ --- --- ] [ --- ] [ k=1 k=1 ] [ m ] [ k=1 ] [ ] [ b ] = [ ] [ n ] [ n ] [ --- ] [ --- ] [ \ ] [ \ ] [ / x(k) n ] [ / y(k) ] [ --- ] [ --- ] [ k=1 ] [ k=1 ] which can be easily solved with Cramer's Rule, for example, to give the formulas for m and b in the least squares fit of the line to the data points. If you have any questions, please write back and I will try to explain further. - Doctor Fenton, The Math Forum http://mathforum.org/dr.math/ |
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