About this video
Data that curve like an exponential are hard to fit with a straight line. By plotting log base 10 of y against x, the data become close to linear, so the tools of linear regression can be used. The true x and y relationship is still nonlinear, and the linear model can be unwound back to an exponential one.
You'll be able to
- explain why a straight line is a poor fit for exponential-looking data.
- transform data by taking the logarithm of y to make the relationship look linear.
- explain why we might prefer linear regression on transformed data.
Key words
- Transforming data
- Applying the same function (like a logarithm) to every value so a pattern becomes easier to model.
- Logarithm (log base 10)
- The power you raise 10 to in order to get a number; log(1000) = 3.
- Exponential relationship
- A relationship where y grows by a constant factor, so the graph curves upward more and more steeply.
- Linear regression
- Fitting a straight line that minimises the sum of squared distances from the points.
Transcript
- [Instructor] So we have some data here that we can plot on a scatter plot that looks something like that. And so, the next question given that we've been talking a lot about lines of regression or regression lines is can we fit a regression line to this? Well, if we try to, we might get something that looks like this, or maybe something that looks like this, I'm just eyeballing it, obviously you can input it into a computer to try to develop a linear regression model to try to minimize the sum of the square distances from the points to the line.
But you can see it's pretty difficult. And some of you might be saying, well, this looks more like some type of an exponential. So maybe we could fit an exponential to it. So it could look something like that. And you wouldn't be wrong. But there is a way that we can apply our tools of linear regression to this dataset.
And the way we can is instead of plotting x versus y, we can think about x versus the logarithm of y. So this is the exact same data set. You see the x values are the same. But for the y values, I just took the log base 10 of all of these. So 10 to the what power is equal to 2,307.23.
10 to the 3.36 power is equal to 2,307.23. And I did that for all of these data points, I did it on a spreadsheet. And if you were to plot all of these, something neat happens. All of a sudden, when we're plotting x versus the log of y, or the log of y versus x, all of a sudden, it looks linear.
Now, be clear. The true relationship between x and y is not linear. It looks like some type of an exponential relationship, but the value of transforming the data, and there's different ways you can do it. In this case, the value taking the log of y, and thinking about that way, is now we can use our tools of linear regression because this data set, you could actually fit a linear regression line to this quite well.
You can imagine a line that looks something like this, it would fit the data quite well. And the reason why you might wanna do this versus trying to fit an exponential is because we've already developed so many tools around linear regression and hypothesis testing around the slope and confidence intervals and so, this might be the direction you wanna go at.
And what's neat is once you fit a linear regression, it's not difficult to mathematically unwind from your linear model back to an exponential one. So the big takeaway here is that the tools of linear regression can be useful even when the underlying relationship between x and y are non-linear and the way that we do that is by transforming the data.
Here, we took a logarithm of the y's and that helped us see a more linear relationship of log y versus x.
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