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notes-least-squares - Notes on the least squares method...

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Notes on the least squares method Patrick De Leenheer * January 14, 2009 You have probably heard about a ”least squares fit”, and when you read papers in mathematical biology this phrase often pops up. These notes provide justification to the existence and uniqueness of the least squares fit to a given data set. The problem is as follows: Given a data set ( x i , y i ) with i = 1 , 2 , . . . , n , find the equation of a line y = ax + b that is the best fit (in some sense to be specified later) to these data points. Geometrically, you are trying to draw a straight line in the ( x, y )-plane that approximates the n given data points in the best possible way. The best fit is by definition the one that minimizes the sum of squares of the errors between the predicted y -values on the line, and the data y -values. This explains the terminology of ”least squares fit”. The precise mathematical problem is to Minimize n X i =1 ( ax i + b - y i ) 2 over all a, b R . Closer inspection of the problem tells us that we are trying to minimize a quadratic function in two unknowns a and b . Let us call this function 1 f ( a, b ) = n X i =1 ( ax i + b - y i ) 2 = X x 2 i a 2 +2 ab X x i - 2 a X x
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