Least Square Method to fit a line to a set of datapoints

[MatinSAR]

Homework Statement

Given a set of data points, we aim to fit a line to these points by minimizing the total error and finding the coefficients $a_0$(intercept) and $a_1$(slope).

Each of these data points has an associated error. Derive the expressions that give the errors (uncertainties) of $a_0$ and $a_1$.

Relevant Equations

I'll mention the relevant equations later in my solution.

We are given a set of points $(x_i , y_i)$. If we want to fit a line to these points which has the form of $y=a_0+a_1x$, we need to do it in a way which minimizes the total error E:$$E = \sum_{i=1}^n (y_i - a_0 - a_1x_i)^2$$So we set $\frac{\partial E}{\partial a_0} = 0$ and $\frac{\partial E}{\partial a_1} = 0$ and solve the system of equations. Then we get:

My problem , I have no idea how to start with errors to find uncertainties of $a_0$ and $a_1$.

 

[kuruman]

This reference has what you are looking for plus more.
https://pages.mtu.edu/~fmorriso/cm3215/UncertaintySlopeInterceptOfLeastSquaresFit.pdf

 

[MatinSAR]

This reference has what you are looking for plus more.
https://pages.mtu.edu/~fmorriso/cm3215/UncertaintySlopeInterceptOfLeastSquaresFit.pdf

Thank you for all the times you’ve helped me.

 

[WWGD]

I believe you need to make some assumptions on the residuales; IIRC, normality and homostadicity, in order to find the distribution of the slope, intercept. Under " reasonable" conditions, they are both normal and converge to the true value.

 

[haruspex]

homostadicity

homoschedasticity ?

 

[WWGD]

homoschedasticity ?

Something like that.

 

[kuruman]

According to this Wikipedia article,

In statistics, a sequence of random variables is homoscedastic (/ˌhoʊmoʊskəˈdæstɪk/) if all its random variables have the same finite variance; this is also known as homogeneity of variance. The complementary notion is called heteroscedasticity, also known as heterogeneity of variance. The spellings homoskedasticity and heteroskedasticity are also frequently used. “Skedasticity” comes from the Ancient Greek word “skedánnymi”, meaning “to scatter”.​

 

[BvU]

Homework Statement: Given a set of data points, we aim to fit a line to these points by minimizing the total error and finding the coefficients $a_0$(intercept) and $a_1$(slope).

Each of these data points has an associated error. Derive the expressions that give the errors (uncertainties) of $a_0$ and $a_1$.
...
My problem , I have no idea how to start with errors to find uncertainties of $a_0$ and $a_1$.

Here are a few links in an old thread (with a link to an even older thread, etc. etc. -- sigh -- turtles all the way down).

I specially recommend Kirchner

$\$

 

[MatinSAR]

Here are a few links in an old thread (with a link to an even older thread, etc. etc. -- sigh -- turtles all the way down).

I specially recommend Kirchner

$\$

Thank you for providing the links. Of course, I'll eventually find a post that doesn't link to an older one.

 

[Gavran]

Why is this thread placed in the Introductory Physics Homework Help?

 

[kuruman]

Why is this thread placed in the Introductory Physics Homework Help?

Because students in introductory physics courses with a lab component are sometimes required to use linear regression to analyze their data?

 

[Gavran]

Thank you, @kuruman.