Hello everybody,
Let's copnsider the problem of linear regression, i.e. given a set of values (xk, yk), finding (a,b) that minimize:
sum_{k=1}^N (axk+b - yk)2
The solution to this problem leads to the well known formula for (a,b) in linear regression.
Now, on the paper linked below they introduce a measure of uncertainty on the slope a.
(http://www.chem.mtu.edu/~fmorriso/cm3215/UncertaintySlopeInterceptOfLeastSquaresFit.pdf)
My questions are:
1. What is the rational beyond such an uncertainty? uncertainty with respect to what?
I would expect an uncertainty in (a) if the the value (xk,yk) come with some given uncertainty (DeltaX, DeltaY), but it does not seem to be the case.
2. Is there a way to extend the uncertainty formula for other types of regression, e.g. weighted linear regression?
with the latter I mean the values of (a,b) that minimize the following quadratic error:
sum_{k=1}^\infty wk (axk+b - yk)2
with wk > 0, sum wk = 1.
Thank you in advance!!!!
Hi, the uncertaintty comes from the underlying statistical interpretation. You can refer for example to the attached tutorial. As (xk,yk) (k=1...n) are noisy measures of random variables X and Y, the question is what is the confidence interval we have on coefficents a and b given a certain number n of samples.
Hello Giuseppe Papari
Every sample of size n is derived from a population of size N because it is impossible to work with the whole population. consequently it is possible to derive NCn samples from a population. In reality a reseracher conducting an experiment can obtain only one sample. In a regression analysis the true equation is say y = β + αx whereas using one sample you obtain yest = b + ax. The values of b and a based on a sample of size n are uncertain with respect to the population values β and α based on a population of size N. The confidence interval for say α is an attempt at bracketing the true value of α using the estimated value a obtained using a sample.
I hope this answer will be useful to you.
Hello Giuseppe Papari
With respect to your 2nd question "Is there a way to extend the uncertainty formula for other types of regression, e.g. weighted linear regression?" my answer is that
(a) the previous formulas for specifying the confidence intervals for b and a will still be valid but
(b) the formulas for determining the regression coefficients b and a in the relation y = b + ax will now be different and are given by
a = ( ∑wkxkyk - ∑wkxk∑wkyk)/[∑wkxk2 - (∑wkxk)2]
b = ∑wkyk - a ∑wkxk where the constraint ∑wk = 1 has been applied
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