Mathematics/Statistics/Regression Lines

A regression line is effectively an attempt to fit some data to a straight line.

Basically, given some set of data points, you create a line that as closely as possible fits the data. This is accomplished by trying to minimize the total distance between all points and the line itself.

Since this is regression line is a standard straight line, it can be represented by a standard $y=mx+b$ equation.

Residuals

Given a single data point, a residual is the distance between that single point and the regression line.

A positive residual value indicates that the data point is somewhere above the regression line. A negative residual value indicates that the data point is somewhere below the regression line. Larger values indicate the point is farther away.

To calculate the residual for some point at $(p_{x},p_{y})$, we have the following equation for some $m$ and $b$ from our regression line:

$residual_{p}=p_{y}-(m(p_{x})+b)$

We can then take this a step further and combine all residuals in our dataset to get an overall view of "how closely the regression line matches our dataset".

$\sum _{i=1}^{n}(r_{i})^{2}$

Squaring means means that negative residual values don't cancel out positive values. It also means points that are farther away from the line end up with more weight than points closer to the line.

Least Squares Regression

Least Squares Regression is one of the more popular ways to minimize the sum of all residuals in our dataset. We can calculate this regression line with a few steps:

First, we note that the equation we want to calculate is ultimately some form of $y=mx+b$.

Our regression line will always go through the point $({\bar {x}},{\bar {y}})$, which denotes the means of our x and y. So it's safe to use that as the $x$ and $y$ in our equation. From there, we can move to calculating our $m$ and $b$.

Calculating our M and B

Next, we can calculate line slope with the equation

$m=r{\frac {S_{y}}{S_{x}}}$

Where

• $r$ is the correlation coefficient for our dataset.
• $S_{y}$ is the standard deviation of our x.
• $S_{x}$ is the standard deviation of our y.

We can then proceed to get our $b$ with simple algebra:

$b=y-mx$

where

• $x$ is the mean of our x.
• $y$ is the mean of our y.
• $m$ is the m we just calculated, above.

Putting all of this together, we can replace the m and b in our $y=mx+b$, which will give us a line equation for our regression.

For further explanation, see this Khan Academy video.