- Understand how regression lines can help us make predictions about data
- Understand how the components of slope and y-intercept determine the output of a regression line
- Understand how to represent a line as a function
Now we know a bit about plotting data and we have written some functions, like trace_values, layout, and plot, to help us do so. You can view them here.
Imagine we are hired as a consultant for a movie executive. The movie executive receives a budget proposal, and wants to know how much money the movie might make. We can help him by building a model of the relationship between the money spent on a movie and money made.
To predict movie revenue based on a budget, let's draw a single straight line that represents the relationship between how much a movie costs and how much it makes.
Eventually, we will want to train this model to match up against an actual data, but for now let's just draw a line to see how it can make estimates.
from lib.graph import trace_values, plot, layout
regression_trace = trace_values([0, 150], [0, 450], mode = 'lines', name = 'estimated revenue')
movie_layout = layout(options = {'title': 'Movie Spending and Revenue (in millions)'})
plot([regression_trace], movie_layout)By using a line, we can see how much money is earned for any point on this line. All we need to do is look at a given
- Spend 60 million, and expect to bring in about 180 million.
- Spend 20 million, and expect to bring in 60 million.
This approach of modeling a linear relationship (that is a drawing a straight line) between an input and an output is called linear regression. We call the input our explanatory variable, and the output the dependent variable. So here, we are saying budget explains our dependent variable, revenue.
Instead of only representing this line visually, we also would like to represent this line with a function. That way, instead of having to see how an
Let's take an initial (wrong) guess at turning this line into a function.
First, we represent the line as a mathematical formula.
Then, we turn this formula into a function:
def y(x):
return x
y(0)y(10000000)This is pretty nice. We just wrote a function that automatically calculates the expected revenue given a certain movie budget. This function says that for every value of
Take a look at the line that we drew. Our line says something different. The line says that spending 30 million brings predicted earnings of 90 million. We need to change our function so that it matches our line. In fact, we need a consistent way to turn lines into functions, and vice versa. Let's get to it.
We start by turning our line into a chart below. It shows how our line relates x-values and y-values, or our budgets and revenues.
| X (budget) | Y (revenue) |
|---|---|
| 0 | 0 |
| 30 million | 90 million |
| 60 million | 180 million |
Next, we need an equation that allows us to match this data.
- input 0 and get back 0
- input 30 million and get back 90 million
- and input 60 million and get back 180 million?
What equation is that? Well it's
- 0 * 3 = 0
- 30 million * 3 = 90 million
- 60 million * 3 = 180 million
Let's see it in the code. This is what it looks like:
def y(x):
return 3*xy(30000000)y(0)Progress! We multiplied each
By multiplying
Let's make sure we understand what all of our variables stand for. Here they are:
-
$y$ : the output value returned by the function, also called the response variable, as it responds to values of$x$ -
$x$ : the input variable, also called the explanatory variable, as it explains the value of$y$ -
$m$ : the slope variable, determines how vertical or horizontal the line will appear
Let's adapt these terms to our movie example. The
A higher value of
$m$ means a steeper line. It also means that we expect more money earned per dollar spent on our movies. Imagine the line pivoting to a steeper tilt as we guess a higher amount of money earned per dollar spent.
There is one more thing that we need to learn in order to describe every straight line in a two-dimensional world. That is the y-intercept.
- The y-intercept is the
$y$ value of the line where it intersects the y-axis.- Or, put another way, the y-intercept is the value of
$y$ when$x$ equals zero.
Let's add a trace with a higher y-intercept than our initial line to the movie plot.
regression_trace_increased = trace_values([0, 150], [50, 500], mode = 'lines', name = 'increased est. revenue')
plot([regression_trace_increased, regression_trace], movie_layout)What is the y-intercept of the original estimated revenue line? Well, it's the value of
- Our formula is no longer
$y = 3x$ . - It is
$y = 3 x + 50,000,000$ .
In addition to determining the y-intercept from a line on a graph, we can also see the y-intercept by looking at a chart of points.
In the chart below, we know that the y-intercept is 50 million because its corresponding
$x$ value is zero.
| X | Y |
|---|---|
| 0 | 50 million |
| 40 million | 170 million |
| 60 million | 230 million |
The y-intercept of a line usually is represented by b. Now we have all of the information needed to describe any straight line using the formula below:
Once more, in this formula:
-
$m$ is our slope of the line, and -
$b$ is the value of$y$ when$x$ equals zero.
So thinking about it visually, increasing
In the context of our movies, we said that the the line with values of
Let's translate this into a function. For any input of
def y(x):
return 3*x + 50000000y(30000000)y(60000000)In this section, we saw how to estimate the relationship between an input variable and an output value. We did so by drawing a straight line representing the relationship between a movie's budget and it's revenue. We saw the output for a given input simply by looking at the y-value of the line at that input point of
We then learned how to represent a line as a mathematical formula, and ultimately a function. We describe lines through the formula