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Simple Linear Regression

https://www.youtube.com/watch?v=nk2CQITm_eo&t=267s

In general, there are 3 main stages in Linear regression:

1. Using Least-squares to fit a line to the data

2. Calculating $R^{2}$

3. Calculating a $p-value$ for $R^{2}$

Using Least-squares to fit a line to the data

• First, draw a line through the data.

• Second, calculate the Residual sum of squares: Measure the distance from the line to each data point (residual), square each distance, and then add them up.

The distance from a line to a data point is called a residual

• Then, we rotate the line a little bit and calculate the RSS. We do this many times.
• ...

• Then, the line that represents the linear regression is the one corresponding to the rotation that has the least RSS. The regression equation:

$y=a+bx$

The equation is composed of 2 parameters:

• Slope: $b=r{\frac {S_{y}}{S_{x}}}$

The slope is the amount of change in units of $y$ for each unitchange in $x$.

• The $y-axis$ intercept.

The regression equation: $y=a+bx$

• Dependent variable: $y$:
• Independent variable: $x$:
• Slope: $b=r{\frac {S_{y}}{S_{x}}}$

The slope is the amount of change in units of $y$ for each unitchange in $x$.

• $y$ intercept: $a={\bar {y}}-b{\bar {x}}$: