least squares equation for a vertical line

Question

Given the following 2d points:

213 106.8

214 189

214 293.4

213 324

223 414

I want to find an equation for the least squares vertical axis line that runs through them. My plan is to get a line equation so I can test subsequent points for their distances to that least squares line.

Thanks

Answer 1

Strictly speaking, a least squares fit is not defined for a vertical line (since the error for each point is measured parallel to the Y axis).

However, if you swap X and Y, you can find the horizontal line with the best least squares fit. It works out to simply the mean of the Y coordinate values:

The equation for a horizontal line is simply y = b.

The error at each point (x_i, y_i) is (y_i - b).

The sum of the squares of the errors is SSE = sum( (y_i - b)²). We wish to find the value of b that minimizes SSE. Take the partial derivative of SSE with respect to b and set it to zero:

sum(-2(y_i - b)) = 0

Simplifying,

sum(y_i) - Nb = 0

and

b = sum(y_i)/N

So in your case, averaging the X coordinates gives you the X coordinate of the vertical line that best fits your points.

Answer 2

The most generic solution would be to apply Total Least Squares

This finds (a, b, d) to minimize the sum of squared perpendicular distances (ax+by=d (a^2+b^2=1): |ax + by – d|). This can handle vertical lines, such as 0x+1y=0.

However, this is a bit more difficult to implement, so the solution offered by @Jim Lewis might be fine and more practical..

Answer 3

If you want the line of best fit to be vertical (i.e. x = constant), the y-values are irrelevant. Simply take the square root of the mean of the squares of the x-values.