Gain clarity on regression analysis, a crucial step toward machine learning, by understanding how variables relate and how predictions can be made. Learn how linear regression efficiently finds a best fit line to minimize variance and enhance predictive accuracy.
Key Insights
- Regression analysis is utilized to predict relationships between variables, allowing one to estimate variable y based on known values of variable x.
- Linear regression creates a "best fit line" by minimizing the variance, mathematically finding the line that yields the lowest cumulative squared distance (sum of squared areas) from all data points.
- An illustrative metaphor compares linear regression to strategically placing a street among houses to ensure all "driveways" (distances from data points to the line) remain minimal, avoiding overly long deviations for any individual points.
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Let's move on to regression. We're going to do some pretty neat stuff now with data and with predictions, starting on our road to machine learning. So regression is used to predict the relationship between variables.
In this case, we'll do two variables, x and y. How do we, given these x and y points, given just x, we want to be able to predict y. Let's take a look at the image actually here so that we can visualize this. Let's run this cell block to output this image. So here is some x and y points, right? Y is vertical, x is horizontal.
So this point is at 5 on the x-axis, 2.5 on the y-axis, right? We have these points and we can kind of see, we can kind of eyeball that they go up and to the right a little bit, right? The more x increases, the more y increases, but they're not, it's not a perfect relationship. This data point is kind of far outside the line. This one's a little high, this one's a little low, right? So what a linear regression will do, it will make a best fit line and that line will minimize the distance from the line to all points, right? It'll plot out this line and the way it's calculating that line is by decreasing, by finding the line that has the least amount of difference from all dots, not just one dot, right? We could just draw a line like that goes basically right through these four, but we'd be maximizing the distance to this one.
So a metaphor somebody gave me on this is that this is like driveways and if you wanted to draw, you want to make a street with short driveways to these little red houses here. The street we want is the one that where nobody's too upset, right? Even this one with a long driveway isn't so upset, whereas if we did it over here, right through the lines, it would make this person over here very upset if they had a super long driveway, right? So it finds the line that minimizes overall the distances between the line and the points, or in other words, minimizes the variance. And in fact, it is a square mathematical value, it's a variance.
So this is actually, it's actually the sum of the square areas is what it's minimizing here. All right, we'll take a look at some real data and see what we can do with it.