Least-squares regression equations
Introduction to residuals and least squares regression
- [Narrator] Though I'm interested in finding the relationship between people's height in inches and their weight in pounds. And so I'm randomly sampling a bunch of people measuring their heights, measuring their weight and then for each person I'm plotting a point that represents their height and weight combination. So for example let's say I measure someone who is 60 inches tall, that'll be about five feet tall and they weigh 100 pounds. And so I'd go to 60 inches and then 100 pounds. Right over there so that point right over there is the point 60 comma, 60 comma, 100. One way to think about it, height we could say is being measured on our X axis or plotted along our X axis and then weight along our Y axis. And though this point from this person is the 0.60, 100 representing 60 inches, 100 pounds. And so so I've done it for one, two, three, four five, six, seven, eight, nine people and I could keep going but even with this I could say, well look, it looks like there's a roughly linear relationship here. It looks like it's positive, that generally speaking as height increases so does weight. Maybe I could try to put a line that can approximate this trend. Let me try to do that so this is my line tool. I could think about a bunch of lines. Something like this seems like it would be, you'd be, most of the data is below the line so that seems like it's not right. I could do something like, I could do something like this but that doesn't seem like a good fit. Most the data seems to be above the line. And so once again I'm just eyeballing it here, in the future you will learn better methods of finding a better fit. But that's something like this and I'm just eyeballing it looks about right. So that line, you could view this as a regression line. We could view this as y equals mx plus b. Where we would have to figure out the slope and the Y intercept and we could figure it out based on what I just drew or we could even think of this as weight. Weight is equal to our slope times height. Times height plus whatever our Y intercept is, if you think of the vertical axis as the weight axis you could think of it as your weight intercept. But either way this is the model that I'm just through eyeballing, this is my regression line. Something that I'm trying to fit to these points. But clearly it can't go through, one line won't be able to go to all of these points. There's going to be for each point some difference or not for all of them but for many of them, some difference between the actual and what would have been predicted by the line. And that idea, the difference between the actual four point and what would have been predicted, given say the height that is called a residual. Gonna write that down. The A residual for each of these data points. And so for example if I call this right here, if I call that point one, the residual for point one. Is going to be well, for our variable, for our height variable 60 inches. The actual here is 100 pounds. From that we would subtract what would be predicted. So what would be predicted is right over here. I could just substitute 60 into this equation so it would be M times 60 plus b. So I could write it as M, maybe let me write it this way, 60 M plus B. Once again I would just take the 60 pounds and put it into my model here and say, well what weight would that have predicted. And I can even, just for the sake of having a number here. I can look, I can, let me get my line tool out. And try to get a straight line from that point. So from this point let me get a straight line. So that doesn't look quite straight, okay a little bit, okay. Okay so looks like it's about 150 pounds. So my model would have predicted 150 pounds. So the residual here is going to be equal to negative 50. So a negative residual is when your actual is below your predicted. So this right over here. This is our one, it is a negative residual. If you had, if you tried to find, let's say this residual right over here, for this point. This r2, this would be a positive residual because the actual is larger than what would have actually been predicted. And so a residual is good for saying, well how good does your line, does your regression, does your model fit a given data point or how does a given data point compare to that. What you probably want to do is think about some combination of all the residuals and try to minimize it. Now you might say, well why don't I just add up all the residuals and try to minimize that. But that gets tricky because some are positive and some are negative and so a big negative residual could counterbalance the big positive residual and it would look, it would add up to zero and then it look like there's no residual so you could just add up the absolute values. So you could say, well let me just take the sum of all of the residual, of the absolute value of all the residuals. And then let me change M and B for my line to minimize this and that would be a technique of trying to create a regression line. But another way to do it and this is actually the most typical way that you will see in statistics is that people take the sum of the squares of the residuals. The sum of the squares and when you square something whether it's negative or positive, it's going to be a positive so it takes care of that issue of negatives and positives canceling out with each other. And when you square a number, things with large residuals are gonna become even larger, relatively speaking. You know, if you square a large, you know one is, if you think about this way, let me put regular numbers, one, two, three, four. These are all one apart from each other but if I were to square them, one, four, nine, 16, they get further and further apart and so something, the larger the residual is when you square it, when the sum of squares is going to represent a bigger proportion of the sum. And so what we'll see in future videos is that there is a technique called least squares regression. Least squares regression. Where you can find an M and a B for a given set of data so it minimizes the sum of the squares of the residual. And that's valuable and the reason why this is used most is it really tries to take in account things that are significant outliers. Things that sit from pretty far away from the model, something like this is going to really, with a least squares regression. It's going to try to be minimized or it's going to be weighted a little bit heavier because when you square it becomes even a bigger factor in this. But this is just a conceptual introduction. In future videos we'll do things like calculate residuals. And we'll actually derive the formula for how do you figure out an M and a B for a line that actually minimizes the sum of the squares of the residuals.