Post on 22-Jul-2020
Curve fitting – Least squares
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Curve fitting
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Km = 100 µMvmax = 1 ATP s-1
Excurse error of multiple measurements
Starting point: Measure N times the same parameterObtained 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
Mean = (x1 + x2 + ... + xN)/N
Standard deviation of mean(called standard error of mean) N
N1
3
N
NN
N
Excurse error propagation
-> Individual variances add scaled by squared partial derivatives (if parameters are uncorrelated)
Examples :
What is error for f(x,y,z,…) if we know errors of x,y,z,… (σx, σy, σz, …) for purely statistical errors?
Addition/substraction:
Product:
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squared errors add up
squared relative errors add up
Excurse error propagation
Ratios:
Powers:
Logarithms:
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squared relative errors add up
error is relative error
relative errortimes power
Curve 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 a
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Mmax ][
][
KS
Svv
Curve fitting – Least squares
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Mmax ][
][
KS
Svv
Probability to get yi for given xi]2/)];([ 22
2
1);( iii axfy
i
i eayP
Curve fitting – Least squares
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Mmax ][
][
KS
Svv
Prob. to get whole set yi for set of xi
N
i
axfy
i
NiiieayyP
1
]2/)];([1
22
2
1);,..,(
For best fitting theory curve (red curve) P(y1,..yN;a) becomes maximum!
Curve fitting – Least squares
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Prob. to get whole set yi for set of xi
N
i
axfy
i
NiiieayyP
1
]2/)];([1
22
2
1);,..,(
For best fitting theory curve (red curve) P(y1,..yN;a) becomes maximum!
Use logarithm of product, get a sum and maximize sum:
2ln);(
2
1);,..,(ln
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2
1 i
NN
i
iiN
axfyayyP
OR minimize χ2 with:
Principle of least squares!!!
Curve fitting – Least squares
Principle of least squares!!!(Χ2 minimization)
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 fitErrors not available: σi’s are set as constant → conventional fit
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Solve:
Reduced χ2
For weighted fit the reduced χ2 should become 1, if errors are properly chosen
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Expectation value of χ2 for weighted fit:
N
axfy N
i i
iN
i i
ii
12
2
12
2
2);(
Define reduced χ2:
MNred
red
22
number of fit paramters
12 redwith
Expectation value of χ2 for unweighted fit:
2
1
2
1
22 1);(
1
N
i
N
i
ii MNaxfy
MN
Should approach the variance of a single data point
Simple proportion
Differentiation:
Solve:
Get:
For σi = const = σ
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Simple proportion - Errors
Rewrite:
-> Error of m given by errors of yi-> Use rules for error propagation
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-> Standard error of determination of m (single confidence interval) is then square-root of V(m)
N
ii axfyN 1
22 );(1
‐> σ2 is estimated as mean square deviation of fitted function from the y-values if now errors are available
Straight line fit (2 parameters)
Differentiation w.r.t c:
Get:(solve eqn. array)
or
Differentiation w.r.t m: or
Errors:
For all relations which are linear with respect to the fit parameters, analytical solutions possible! 14
(or σi = const = σ)
Straight line fit (2 parameters)
Additional quantities for multiple parameters: Covariances-> describes interdependency/correlation between the obtained parameters
Covariance matrix
Straight line fit:
Vii = Var(x(i))
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)(
)(
)(
,cov
33231
32221
31211
333231
322221
312111
pVarVV
VpVarV
VVpVar
VVV
VVV
VVV
pp
pppp
pppp
pppp
pppppp
pppppp
pppppp
ji
Squared errors of fit parameters!
More in depth
http://www-zeus.physik.uni-bonn.de/~brock/teaching/stat_ws0001/
Online lecture: Statistical Methods of Data Analysis by Ian C. Brock
Numerical recipes in C++, Cambridge university press
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