Least-square fitting to data (sine function) : what is the error of the derived fit parameters

analysisdata analysisleast squaresmathematical physicsmean square error

I have a set of data. I want to fit it to a sine function of the form :
\begin{equation}
f(x)=A sin(\omega x+B)+C
\end{equation}
I use the least-square method to find the appropriate fit-parameters which are $A$, $B$ and $C$. In this method, each term of the cost-function has a weight calculated from the error-bar of each point in my dataset.

Now I want to calculate the visibility $V$ for the fitting curve. The visibility is defined by :
\begin{equation}
V=\frac{A}{C}
\end{equation}

I obtain a good value of $V=0.95$, but now I want to know how to calculate $\Delta V$, the error of the visibility. To get it, I need to know $\Delta A$ and $\Delta C$.

Do you know how to do it ?

EDIT : Some people suggested to post the data, so here is the figure on the link below.
Figure representing the data

Basically, each point has a poissonnian error bar $\Delta Y= \sqrt{Y}$. I did the weighted least-square method to obtain my fit-function which is the solid line you can see on this plot (there is two data-set actually, red and blue). The area in red/blue represent standard deviation of the distance in errorbar unit from the datapoints to the fit, multiplied by the poissonian error $\sqrt{Y}$ [I don't know if this is okay, maybe it's false to do like that].

Best Answer

This fitting problem can be equivalently rewritten as fitting function of form:

$$ f(x) = K \sin(\omega x) + L \cos(\omega x) + C $$

And your original $A$ is just $A =\sqrt{K^2+L^2}$

This reduces it to just ordinary least squares problem. We get least squares estimators for $K,L$ from the equation

$$\begin{bmatrix} K \\ L \\ C \end{bmatrix} = (X^TX)^{-1}X^{T}y$$

Where $X$ is matrix formed by values of $\sin(\omega x),\cos(\omega x), 1$ evaluated at consecutive values of $x$ coordinate of your observations and $y$ are values of said observations.

This way you can see that $K,L$ are some linear functions of $y_i$. I'm not sure what Poissonian error bar is but in general finding variance of sum of variables can be done if we know variance of individual variables.

Assuming $y_i$ uncorrelated we get:

$$ {Var}(K) = (1,0,0) . (X^TX)^{-1}X^{T} . Var(y_i)$$

and analog for $L$.

And $C$ is even simpler as it is just the mean value of the observations.

This way we have found $Var(K),Var(L),Var(C)$ assuming these are small enough you can just propagate error in the formula

$$ V = \frac{\sqrt{K^2+L^2}}{C} $$

either by direct computation or using this helpful lookup.

Related Solutions

Total Least Square fitting

In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models.

Least Square Fit of Ellipse and Border Thickness

You can make the least-squares method work, but you have to be careful about which least-squares system you solve.

Clearly, the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ for the ellipse isn’t unique: multiplying $A, B, C, D, E, F$ by the same constant gives you another equation for the same ellipse. So you can’t simply minimize

$$\sum (Ax^2 + Bxy + Cy^2 + Dx + Ey + F)^2$$

over $A, B, C, D, E, F$, because you’ll get $A = B = C = D = E = F = 0$; you need to add a normalizing constraint to exclude this solution. But if you add the wrong constraint, for example $F = 1$, you’ll bias the solution towards ellipses where $A, B, C, D, E$ are smaller relative to $F$.

The right constraint is $A + C = 1$, because $A + C$ is invariant over isometries of $(x, y)$. Minimize

$$\sum (Bxy + C(y^2 - x^2) + Dx + Ey + F + x^2)^2$$

over $B, C, D, E, F$, and then let $A = 1 - C$.

An advantage of this method over one using barycenters and inertial moments is that it still works well with a non-uniform distribution of points.

Python code for these figures:

import numpy as np
from matplotlib.patches import Ellipse
import matplotlib.pyplot as plt

with open("ellipse_data.txt") as file:
    points = np.array([[float(s) for s in line.split()] for line in file])
xs, ys = points.T

# Parameters for Ax² + Bxy + Cy² + Dx + Ey + F = 0
B, C, D, E, F = np.linalg.lstsq(
    np.c_[xs * ys, ys ** 2 - xs ** 2, xs, ys, np.ones_like(xs)], -(xs ** 2)
)[0]
A = 1 - C

# Parameters for ((x-x0)cos θ + (y-y0)sin θ)²/a² + (-(x-x0)sin θ + (y-y0)cos θ)²/b² = 1
M = np.array([[A, B / 2, D / 2], [B / 2, C, E / 2], [D / 2, E / 2, F]])
λ, v = np.linalg.eigh(M[:2, :2])
x0, y0 = np.linalg.solve(M[:2, :2], -M[:2, 2])
a, b = np.sqrt(-np.linalg.det(M) / (λ[0] * λ[1] * λ))
θ = np.arctan2(v[1, 0], v[0, 0])

ax = plt.axes(aspect="equal")
ax.add_patch(
    Ellipse((x0, y0), 2 * a, 2 * b, θ * 180 / np.pi, facecolor="none", edgecolor="red")
)
ax.scatter(xs, ys, s=0.5)
plt.show()

Best Answer

Related Solutions

Total Least Square fitting

Least Square Fit of Ellipse and Border Thickness

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