Curve Fitting – Least Squares

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

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

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

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) lnP(y1,..,yN ;a)       ln i 2 2 1   i  1

OR minimize χ2 with:

Principle of least squares!!!

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

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  covpi ,p j  Vp1p2 Vp2p2 Vp2p3    Vp1p2 Var(p2 ) Vp2p3      Vp1p3 Vp2p3 Vp3p3   Vp1p3 Vp2p3 Var(p3 )

Squared errors of fit parameters!

Straight line fit: