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Curve Fitting – Least Squares

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Curve Fitting – Least Squares Curve fitting – Least squares 1 Curve fitting Km = 100 µM -1 vmax = 1 ATP s 2 Excurse error of multiple measurements Starting point: Measure N times the same parameter Obtained values are Gaussian distributed around mean with standard deviation σ What is the error of the mean of all measurements? Sum = x1 +x2 + ... + xN Variance of sum = N * σ2 (Central limit theorem) Standard deviation of sum N Mean = (x1 +x2 + ... + xN)/N N 1 Standard deviation of mean N (called standard error of mean) N N N 3 Excurse error propagation What is error for f(x,y,z,…) if we know errors of x,y,z,… (σx, σy, σz, …) for purely statistical errors? -> Individual variances add scaled by squared partial derivatives (if parameters are uncorrelated) Examples : Addition/substraction: squared errors add up squared relative Product: errors add up 4 Excurse error propagation squared relative Ratios: errors add up Powers: relative error times power Logarithms: error is relative error 5 Curve fitting – Least squares Starting point: - data set with N pairs of (xi,yi) - xi known exactly, -yi Gaussian distributed around true value with error σi - errors uncorrelated - function f(x) which shall describe the values y (y = f(x)) - f(x) depends on one or more parameters a [S] v vmax [S] KM 6 Curve fitting – Least squares 1 2 2 Probability to get y for given x [y i f (xi ;a)] / 2 i ] i i P(y i ;a) e 2 i [S] v vmax [S] KM 7 Curve fitting – Least squares N 1 2 2 Prob. to get whole set y for set of x [yi f (xi ;a)] / 2 i ] i i P(y1,..,yN ;a) e i 1 2 i For best fitting theory curve (red curve) P(y1,..yN;a) becomes maximum! [S] v vmax [S] KM 8 Curve fitting – Least squares N 1 2 2 Prob. to get whole set y for set of x [yi f (xi ;a)] / 2 i ] i i P(y1,..,yN ;a) e i 1 2 i For best fitting theory curve (red curve) P(y1,..yN;a) becomes maximum! Use logarithm of product, get a sum and maximize sum: 2 N N 1 y i f (xi ;a) lnP(y1,..,yN ;a) ln i 2 2 1 i 1 OR minimize χ2 with: Principle of least squares!!! 9 Curve fitting – Least squares Principle of least squares!!! (Χ2 minimization) Solve: Solve equation(s) either analytically (only simple functions) or numerically (specialized software, different algorithms) χ2 value indicates goodness of fit Errors available: USE THEM! → so called weighted fit Errors not available: σi’s are set as constant → conventional fit 10 Reduced χ2 Expectation value of χ2 for weighted fit: 2 N N 2 y i f (xi ;a) 2 i N 2 2 i 1 i i 1 i Define reduced χ2: 2 2 2 red with 1 red N M red number of fit paramters For weighted fit the reduced χ2 should become 1, if errors are properly chosen Expectation value of χ2 for unweighted fit: 1 N 1 N 2 y f (x ;a) 2 2 2 i i N M i 1 N M i 1 Should approach the variance of a single data point 11 Simple proportion Differentiation: For σi = const = σ Solve: Get: 12 Simple proportion - Errors Rewrite: -> Error of m given by errors of yi -> Use rules for error propagation -> Standard error of determination of m (single confidence interval) is then square-root of V(m) ‐> σ2 is estimated as mean square deviation of fitted function from the y- values if now errors are available N 2 1 2 y i f (xi ;a) N 1 13 Straight line fit (2 parameters) (or σi = const = σ) Differentiation w.r.t c: or Differentiation w.r.t m: or Get: (solve eqn. array) Errors: For all relations which are linear with respect to the fit parameters, analytical solutions possible! 14 Straight line fit (2 parameters) Additional quantities for multiple parameters: Covariances -> describes interdependency/correlation between the obtained parameters Covariance matrix Vii = Var(x(i)) V V V Var(p ) V V p1p1 p1p2 p1p3 1 p1p2 p1p3 covpi ,p j Vp1p2 Vp2p2 Vp2p3 Vp1p2 Var(p2 ) Vp2p3 Vp1p3 Vp2p3 Vp3p3 Vp1p3 Vp2p3 Var(p3 ) Squared errors of fit parameters! Straight line fit: 15 More in depth Online lecture: Statistical Methods of Data Analysis by Ian C. Brock http://www-zeus.physik.uni-bonn.de/~brock/teaching/stat_ws0001/ Numerical recipes in C++, Cambridge university press 16.
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