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### lsq

Course: ASTR 415, Fall 2008
School: Maryland
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Word Count: 391

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of Modeling Data NRiC Chapter 15. Model depends on adjustable parameters. Can be used for &quot;constrained interpolation&quot;. Basic approach: 1. 2. 3. 4. Choose figure-of-merit function (e.g. 2). Adjust best-fit parameters: minimize merit function. Compute error estimates for parameters. Compute goodness-of-fit. Least Squares Fitting Suppose we want to fit N data points (xi,yi) with a function...

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of Modeling Data NRiC Chapter 15. Model depends on adjustable parameters. Can be used for "constrained interpolation". Basic approach: 1. 2. 3. 4. Choose figure-of-merit function (e.g. 2). Adjust best-fit parameters: minimize merit function. Compute error estimates for parameters. Compute goodness-of-fit. Least Squares Fitting Suppose we want to fit N data points (xi,yi) with a function that depends on M parameters aj and that each data point has a standard deviation i. The maximum likelihood estimate of the model parameters is obtained by minimizing: = = Assuming the errors are normally distributed, a "good fit" has 2 ~ , where = N -M. Fitting Data to a Straight Line For this case the model is simply: y(x) = y(x; a, b) = a + bx Derive formula for best-fit parameters by setting 2/a = 0 = 2/b. Derive uncertainties in a and b using: = = 2 Want Q = gammq((N -2)/2, /2) > 0.001. General Linear Least Squares Can generalize to any linear combination: = = e.g. y(x) = a1 + a2x + a3x2 + ... + aMxM-1. Define N M design matrix Aij = Xj(xi)/i. Also define vector b of length N: bi = yi/i and vector a of length M: ai = a1,...,aM. Then we wish to find that a minimizes: 2 = Aa -b2 This is what SVD solves! General Linear Least Squares, Cont'd Recall for SVD we had A = UWVT. Rewriting the SVD solution we get: = = where U(j) and V(j) denote columns of U and V. As before, if wj is small (or zero), can omit. Useful because least-squares problems are both overdetermined (N > M) and underdetermined (ambiguous combinations of parameters exist)! Nonlinear Models Suppose model depends nonlinearly on the aj's... e.g. y(x) = a1 exp(-a2 x2). Still define 2, but must proceed iteratively: Use anext = acur - 2(acur) far from minimum (steepest descent), where is a constant. Use anext = acur - D-1[2(acur)] close to minimum, where D is the Hessian matrix. The Levenbe...

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Maryland - ASTR - 415
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