How Do You Optimize a and b in Linear Regression for Minimal Deviation?

[oleandora]

Homework Statement

I've been given a set of data
x 0 0.5 0.7 1.5 1.75
y 0.5 0.72 0.51 1.5 1.63

Given y=ax+b

for this data points of linear model, I have to
1. minimize the sum of the absolute values of deviations between experimental value of Y and value predicted by the linear relation
2. minimiza the maximum value of deviation between all experimental values of Y and value predicted by linear relation

The question is
What are the optimal values of a and b for both cases and values of objective function.

Homework Equations

The Attempt at a Solution

What I know is
y=ax+b+e where e is the deviation but that's it. I don't even have the slightest idea for the next steps.
I know seems simplebut I have very little background on linear regression.
Help! T_T

 

[MaxManus]

Hey, this is the first time I tried to help someone on this forum, so don't shoot me if I am wronge. If we are at the same level I guess you are supposed to use OLS Ordinary Least Square.

b = [tex]\sum[/tex] (X_i-[tex]\bar{X}[/tex])(Y_i-[tex]\bar{Y}[/tex])/
[tex]\sum[/tex] (X_i-[tex]\bar{X}[/tex])²

a = [tex]\bar{Y}[/tex] - b[tex]\bar{X}[/tex]

where you sum from 1 to n.
[tex]\bar{X}[/tex] means the average

 

[Mene]

I would approach this problem by first understanding the objective of the task. The goal is to find the optimal values of a and b that minimize the sum of absolute values of deviations and the maximum value of deviation between the experimental values of Y and the predicted values by the linear relation. This is known as the least absolute deviations (LAD) method.

To solve this problem, I would use the LAD regression method, which involves minimizing the objective function:

min Σ|y - (ax + b)| for the first case
min max|y - (ax + b)| for the second case

To find the optimal values of a and b, I would use an optimization algorithm such as gradient descent or the simplex method. These algorithms would help me find the values of a and b that minimize the objective function.

Alternatively, I could also use a statistical software package such as R or Python's scikit-learn to perform linear regression and find the optimal values of a and b. These tools have built-in functions that can perform LAD regression and provide the optimal values of a and b as well as the objective function values.

In summary, the optimal values of a and b can be found by using an optimization algorithm or a statistical software package to minimize the objective function. This will provide the best fit line for the given data points and help in predicting future values.