Least Squares — the Gory Details
How Do We Find the Line of Best Fit?
Copyright © 2002–2013 by Stan Brown, Oak Road Systems
Copyright © 2002–2013 by Stan Brown, Oak Road Systems
You have a set of observed points (x,y). You’ve plotted them and they seem to be pretty much linear. How do you find the line that best fits those points?
“Don’t be silly,” you say. “Put them into a TI-83 or Excel and look at the answer.”
Okay, you got me. I’ll ask a deeper question: How does the calculator find the answer? What are the underlying equations?
To answer that question, first we have to agree on what we mean by the “best fit” of a line to a set of points. Why do we say that the line on the left fits the points better than the line on the right? And can we say that some other line might fit them better still?
Intuitively, we think of a close fit as a good fit. We look for a line with little space between the line and the points it’s supposed to fit. We would say that the best fitting line is the one that has the least space between itself and the data points, which represent actual measurements.
Okay, what do we mean by “least space”? There are three ways to measure the space between a point and a line: vertically in the y direction, horizontally in the x direction, and on a perpendicular to the line. We choose to measure the space vertically. Why? because our whole purpose in making a regression line is to use it to predict the y value for a given x, and the vertical distances are how far off the predictions would be for the points we actually measured.
If we know we want the line that has the smallest vertical distance between itself and the points, how do we compute that vertical distance? If the line is y=3x+2 and we have a point (2,9), the the predicted value is 3×2+2=8 and subtract the actual measured value 9. We say that the deviation is –1, negative because the predicted value is less than the actual value. In general, the deviation (vertical gap) between any given point (x,y) and the line y=mx+b will be mx+b–y.
But each deviation could be positive or negative, depending on whether the line fall above or below that point. We can’t simply add up deviations, because then a line would be considered good if it fell way below some points as long as it fell way above others. To prevent that, we square each deviation, and add up the squares. (This also has the desirable effect that a few small deviations are more tolerable than one or two big ones.)
And at long last we can say exactly what we mean by the line of best fit. If we compute the deviations in the y direction, square each one, and add up the squares, we say the line of best fit is the line for which that sum is the least. Since it’s a sum of squares, the method is called the method of least squares.
It’s always a giant step in finding something to get clear on what it is you’re looking for, and we’ve done that. The best-fit line, as we have decided, is the line that minimizes the sum of squares of vertical deviations between itself and the measured points. We can write that sum as where ŷ is the predicted value on the line for a given x (namely mx+b), and the y is the actual value measured for that given x.
Do we just try a bunch of lines, compute their E values, and pick the line with the lowest E value? No, we could never be sure that there wasn’t some other line with still a lower E — and of course it would be a lot of work, too.
Instead, we use a powerful and common trick in mathematics: We assume we know the line, and use its properties to help us find its identity. Here’s how that works.
What is the line of best fit? It’s y=mx+b, because any line (except a vertical one) is y=mx+b. We happen not to know m and b just yet, but we can use the properties of the line to find them.
What is the chief property of the line? It is that E is less for this line than for any other line that might pass through the same set of points. In other words, E is minimized by varying m and b. Let’s look at how we can write an expression for E in terms of m and b, and of course using the measured data points (x,y).
The squared deviation for any one point follows from the definition we gave earlier:
E is found by summing over all points:
Once we find the m and b that minimize that quantity, we will know the exact equation of the line of best fit.
As soon as you hear “minimize”, you think “calculus”. And indeed calculus can find m and b. Surprisingly, we can also find m and b using plain algebra.
It’s not entirely clear who invented the method of least squares. Most authors attach it to the name of Karl Friedrich Gauss (1777–1855), who first published on the subject in 1809.
But the Frenchman Adrien Marie Legendre (1752–1833) “published a clear explanation of the method, with a worked example, in 1805” according to Stephen Stigler in Statistics on the Table (Cambridge, Massachusetts; Harvard University Press, 1999; see Chapter 17). In setting up the new metric system of measurement, the meter was to be fixed at a ten-millionth of the distance from the North Pole through Paris to the Equator. Surveyors had measured portions of that arc, and Legendre invented the method of least squares to get the best measurement for the whole arc.
Using calculus, a function has its minimum where the derivative is 0. Since we need to adjust both m and b, we take the derivative of E with respect to m, and separately with respect to b, and set both to 0:
Each equation then gets divided by the common factor 2, and the terms not involving m or b are moved to the other side. With a little thought you can recognize the result as two simultaneous equations in m and b, namely:
The summation expressions are all just numbers, the result of summing x and y in various combinations.
(By the way, how do we know that these will give us a minimum and not a maximum or inflection point? Because each second derivative is 2 for all values of m and b, and if the first derivative is 0 and the second derivative is positive you have a minimum.)
These simultaneous equations can be solved like any others: by substitution or by linear combination. Let’s try substitution. The second equation looks easy to solve for b:
Substitute that in the other equation and you eventually come up with
And that is very probably what your calculator (or Excel) does: Add up all the x’s, all the x², all the xy, and so on, and compute the coefficients. It’s tedious, but not hard. (Usually these equations are presented in the shortcut form shown below.)
But you don’t need calculus to solve every minimum or maximum problem. Look back again at the equation for E, which is the quantity we want to minimize:
Now that may look intimidating, but remember that all the sigmas are just constants, formed by adding up various combinations of the (x,y) of the original points. In fact, collecting like terms reveals that E is really just a parabola with respect to m or b:
Both these parabolas are open upward. (Why? because the coefficients of the m² and b² terms are positive. The sum of x² must be positive unless all x’s are 0; and of course n, the number of points, is positive.) Since the parabolas are open upward, each one has a minimum at its vertex.
Where is the vertex for each of these parabolas? Well, recall that a parabola y=px²+qx+r has its vertex at -q/2p. These are parabolas in m and b, not in x, but you can find the vertex of each one the same way:
Now there are two equations in m and b. Substitute one into the other one, perhaps the second into the first, and the solution is
These are exactly the equations obtained by the calculus method.
The formula for m is bad enough, but the formula for b is a monstrosity. Actually it’s easier to compute m first, then compute b using m:
Just to make things more concrete, here’s an example. Suppose that x is dial settings in your freezer, and y is the resulting temperature in °F. Here’s the full calculation:
Now compute m and b:
5(−157) − (16)(−24) m = ------------------- 5(74) − 16² −401 m = ---- 114 m = −3.5175 −24 − (−3.5175)(16) b = -------------------- 5 32.28 b = ----- = 6.456 5
“These values agree precisely with the regression equation calculated by a TI-83 for the same data,” he said smugly.
Some authors give a different form of the solutions for m and b, such as:
where x̅ and y̅ are the average of all x’s and average of all y’s.
These formulas are equivalent to the ones we derived earlier. (Can you prove that? Remember that nx̅ is ∑ x, and similarly for y.) While the m formula looks simpler, it requires you to compute mean x and mean y first. If you do that, here’s how the numbers work out:
Whew! Once you’ve got through that, m and b are only a little more work:
m = −80.2 / 22.8 = −3.5175
b = −4.8 − (−3.5175)(3.2) = 6.456
The simplicity of the alternative formulas is definitely deceptive.