Curve Fitting - Least Squares

Curve fitting – Least squares1Curve fittingKm 100 µMvmax 1 ATP s-12

Excurse error of multiple measurementsStarting point: Measure N times the same parameterObtained values are Gaussian distributed around mean withstandard deviation σWhat is the error of the mean of all measurements?Sum x1 x2 . xNVariance of sum N * σ2Standard deviation of sum N Mean (x1 x2 . xN)/N Standard deviation of mean(called standard error of mean)(Central limit theorem)N NN N 1N3Excurse error propagationWhat is error for f(x,y,z, ) if we know errors of x,y,z, (σx, σy, σz, ) for purelystatistical errors?- Individual variances add scaled by squared partial derivatives (if parameters are uncorrelated)Examples :Addition/substraction:Product:squared errorsadd upsquared relativeerrors add up4

Excurse error propagationsquared relativeerrors add upRatios:Powers:relative errortimes powerLogarithms:error is relativeerror5Curve fitting – Least squaresStarting 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 av v max[S ][S ] K M6

Curve fitting – Least squaresProbability to get yi for given xiP ( y i ; a) v v max221e [ y i f ( xi ;a )] / 2 i ]2 i[S ][S ] K M7Curve fitting – Least squaresNProb. to get whole set yi for set of xiP ( y 1,., y N ; a ) i 1221e [ y i f ( xi ;a )] / 2 i ]2 iFor best fitting theory curve (red curve) P(y1,.yN;a) becomes maximum!v v max[S ][S ] K M8

Curve fitting – Least squaresNProb. to get whole set yi for set of xi221e [ y i f ( xi ;a )] / 2 i ]2 iP ( y 1,., y N ; a ) i 1For best fitting theory curve (red curve) P(y1,.yN;a) becomes maximum!Use logarithm of product, get a sum and maximize sum:2 N1 N y i f ( xi ; a) ln P ( y 1,., y N ; a ) ln i 2 2 1 i1 OR minimize χ2 with:Principle of least squares!!!9Curve fitting – Least squaresPrinciple of least squares!!!(Χ2 minimization)Solve:Solve equation(s) either analytically (only simple functions)ornumerically (specialized software, different algorithms)2χ value indicates goodness of fitErrors available:USE THEM! so called weighted fitErrors not available: σi’s are set as constant conventional fit10

Reduced χ2Expectation value of χ2 for weighted fit: y i f ( x i ; a) 2N 2 i2i 1 i2 N2 ii 1N Define reduced χ2: 2red 2 redN M2 red 1withnumber of fit paramtersFor weighted fit the reduced χ2 should become 1, if errors are properly chosenExpectation value of χ2 for unweighted fit: 21 N MN yi 1 f ( x i ; a ) 2i1 N MN 2 2i 1Should approach the variance of a single data point11Simple proportionDifferentiation:For σi const σSolve:Get:12

Simple proportion - ErrorsRewrite:- Error of m given by errors of yi- Use rules for error propagation- Standard error of determination of m (single confidence interval) is thensquare-root of V(m)‐ σ2 is estimated as mean square deviation of fitted function from the yvalues if now errors are available1 N2 y i f ( x i ; a ) N 1213Straight line fit (2 parameters)(or σi const σ)Differentiation w.r.t c:orDifferentiation w.r.t m:orGet:(solve eqn. array)Errors:For all relations which are linear with respect to the fit parameters, analyticalsolutions possible!14

Straight line fit (2 parameters)Additional quantities for multiple parameters: Covariances- describes interdependency/correlation between the obtained parametersVii Var(x(i))Covariance matrix Vp1p1 Vp1p 2 Vp1p 3 Var ( p1 ) Vp1p 2Vp1p 3 Var ( p2 ) Vp 2 p 3 cov pi , p j Vp1p 2 Vp 2 p 2 Vp 2 p 3 Vp1p 2 Vp 2 p 3Var ( p3 ) Vp1p 3 Vp 2 p 3 Vp 3 p 3 Vp1p 3Squared errors of fit parameters!Straight line fit:15More in depthOnline lecture: Statistical Methods of Data Analysis by Ian C. Brockhttp://www-zeus.physik.uni-bonn.de/ brock/teaching/stat ws0001/Numerical recipes in C , Cambridge university press16

For best fitting theory curve (red curve) P(y1,.yN;a) becomes maximum! Use logarithm of product, get a sum and maximize sum: ln 2 ( ; ) 2 1 ln ( ,., ; ) 1 1 2 1 i N N i i i N y f x a P y y a OR minimize χ2with: Principle of least squares!!! Curve fitting - Least squares Principle of least squares!!! (Χ2 minimization)