Squared error of regression line

In the next few videos I'm going to embark on something that will just result in a formula that's pretty straightforward to apply. And in most statistics classes, you'll just see that end product. But I actually want to show how to get there. But I just want to warn you right now. It's going to be a lot of hairy math, most of it hairy algebra. And then we're actually going have to do a little bit of calculus near the end. We're going to have to do a few partial derivatives. So if any of that sounds daunting, or sounds like something that will discourage you in some way, you don't have to watch it. You could skip to the end and just get the formula that we're going to derive. But I, at least, find it pretty satisfying to actually derive it. So what we're going to think about here is, let's say we have n points on a coordinate plane. And they all don't have to be in the first quadrant. But just for simplicity of visualization, I'll draw them all in the first quadarant. So let's say I have this point right over here. Let me do them in different colors. And that coordinate is x1, y1. And then let's say I have another point over here. The coordinates there are x2, y2. And then I can keep adding points. And I could keep drawing them. We'd just have a ton of points. There and there and there. And we go all the way to the nth point. Maybe it's over here. And we're just going to call that xn, yn. So we have n points here. I haven't drawn all of the actual points. But what I want to do is find a line that minimizes the squared distances to these different points. So let's think about it. Let's visualize that line for a second. So there's going to be some line. And I'm going to try to draw a line that kind of approximates what these points are doing. So let me draw this line here. So maybe the line might look something like this. I'm going to try my best to approximate it. Actually, let me draw it little bit different. Maybe it looks something like that. I don't even know what it looks like right now. And what we want to do is minimize this squared error from each of these points to the line. So let's think about what that means. So if the equation of this line right here is y is equal to mx plus b. And this just comes straight out of Algebra 1. This is the slope on the line, and this is the y-intercept. This is actually the point 0, b. What I want to do, and that's what the the topic of the next few videos are going to be, I want to find an m and a b. So I want to find these two things that define this line. So that it minimizes the squared error. So let me define what the error even is. So for each of these points, the error between it and the line is the vertical distance. So this right here we can call error one. And then this right here would be error two. It would be the vertical distance between that point and the line. Or you can think of it as the y value of this point and the y value of the line. And you just keep going all the way to the endpoint between the y value of this point and the y value of the line. So this error right here, error one, if you think about it, it is this value right here, this y value. It's equal to y1 minus this y value. Well what's this y value going to be? Well over here we have x is equal to x1. And this point is the point m x1 plus b. You take x1 into this equation of the line and you're going to get this point right over here. So that's literally going to be equal to m x1 plus b. That's that first error. And we can keep doing it with all the points. This error right over here is going to be y2 minus m x2 plus b. And then this point right here is m x2 plus b. The value when you take x2 into this line. And we keep going all the way to our nth point. This error right here is going to be yn minus m xn plus b. Now, so if we wanted to just take the straight up sum of the errors, we could just some these things up. But what we want to do is a minimize the square of the error between each of these points, each of these n points on the line. So let me define the squared error against this line as being equal to the sum of these squared errors. So this error right here, or error one we could call it, is y1 minus m x1 plus b. And we're going to square it. So this is the error one squared. And we're going to go to error two squared. Error two squared is y2 minus m x2 plus b. And then we're going to square that error. And then we keep going, we're going to go n spaces, or n points I should say. We keep going all the way to this nth error. The nth error is going to be yn minus m xn plus b. And then we're going to square it. So this is the squared error of the line. And over the next few videos, is I want to find the m and b that minimizes the squared error of this line right here. So if you viewed this as the best metric for how good a fit a line is, we're going to try to find the best fitting line for these points. And I'll continue in the next video. Because I find that with these very hairy math problems, it's good to kind of just deliver one concept at a time. And it also minimizes my probability of making a mistake.