LECT: Simple Linear Regression

[mmcky]

This PR is the initial work on a lecture showing the basics of Simple Linear Regression and solves the system of equations for this simple illustrative case for early Undergraduates that are learning calculus and basic estimation

There are alternative ways to present this information -- the contained is a pretty classic approach but the team has other ideas as well that could form a more unified approach:

Team Comments and Ideas

1. Fixed and Repeated Samples approach (Epsilon is the randomness)
2. Frame the lecture as an Experimental Design Question
3. Use the Fisher Original Example (Fertiliser)
4. Extend with other following lectures (multivariate, Linear Algebra (rather than $\sum$) etc.)

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[jstac]

Great work @mmcky ! I really like the style.

Comments:

• After "The Ordinary Least Squares (OLS) method..." I suggest replacing $\min$ with $\min_{\alpha, \beta}$.
• After "and setting it equal to" replace $\beta * x_i$ with $\beta x_i$ --- similar issues appear in the equations below.
• " solving for B" -> "and solving for $\beta$"
• ": I don’t think this surface plot adds a whole lot" --- I agree.
• "How does error change with respect to..." --- I wonder if this section should come before differentiation. First set up the cost function, explore how it changes the parameters using your figures, and then minimize systematically using calculus.
• As a suggested exercise, simply ask people to run a simple regression on a new data set, perhaps from real data.

[mmcky]

thanks @jstac -- super helpful comments. Thank you.

[mmcky]

thanks @jstac the exercise example was a great idea.

I have explored the relationship between life expectancy and GDP per capita.
The Our World in Data objects integrate nicely into the page as well and the exercise kind of turned into a blend of "working with data" and "regression work"

[mmcky]

• @mmcky to proof read