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So now that we understand the intuition behind our regression algorithm, we can run and evaluate our
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regression.
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We will write the Python code to actually crunch the numbers.
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So I hope that at this point you still have your Jupyter notebook open and your session is still connected.
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While the big advantage of using the online version of Jupyter notebook is that you can get started
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right away and you don't have to install anything, but if you're inactive for a while and you haven't
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been using it then it's possible that you can get disconnected and lose your work.
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So in that case you might see something like this.
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And it's important to remember that you can always save your work by saying "Download as" and then "Notebook"
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and you can always restore your work by going back to try Jupyter with Python and then simply uploading
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the Jupyter notebook that you downloaded previously and your data file.
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So if you upload those, then you can continue where you left off. In the next module,
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I will walk you through how to install Jupyter locally on your machine.
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You might only encounter this situation if you are trying it out using binder through the web portal.
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Now without further ado, let's give our notebook the capability to run a regression. This capability, just
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like the others, it's going to come from a module. In this case this module is going to be called scikit-learn.
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Scikit-learn is one of the most popular machine learning modules in Python and we can get hold
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of it in our Jupyter notebook simply by typing "import sklearn", but we're only looking for something
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very specific out of scikit-learn, so instead of importing all of scikit-learn, what we're gonna do instead
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is we're going to say from as sklearn.linear_model and I hit Tab on my keyboard here to bring
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up this option, we're gonna import linear regression.
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Once again, I typed the first few characters there and then I hit Enter to insert the rest of the code
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and this avoids any sort of typos like, you know, having a lowercase r here for example. After we've
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added this line of code, let's hit Shift+Enter on our keyboard or click "Run" to run the cell.
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Alternatively, if you've opened this notebook from fresh, you might have to go to "Cell" and then "Run All".
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Let me add a few more cells here at the bottom, and then we can add the code to run our linear regression.
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The task of running a linear regression for calculating the slope of our line and the intercepts is
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once more going to be done by an object; so we're gonna create this object and we're gonna give it a name.
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So what I'm going to call it - regression, and I'm going to set it equal to LinearRegression with some
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parentheses at the end.
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This bit of code here will create our object and we're gonna be storing it inside here so we can always
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refer to our linear regression by the name "regression".
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So now that we've created our object we can actually tell it to do something; we can tell it to run our
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regression. And the way we're gonna do that is simply by using the regression and then putting a dot
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after it and then writing "fit". Inside the parentheses we have to tell it two things: namely the features -
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our X, and our labels - our lowercase y. When we hit Shift+Enter on the cell it will crunch the numbers
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and, just like that, we've actually run our regression.
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Now, I know we don't see any output here but trust me,
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the numbers have been crunched and to prove this we can pull up the slope coefficient and the intercept
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that were calculated by our regression. We can get hold of the slope through our regression object.
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So regression.coef and here I'm hitting Tab on my keyboard to bring up that menu there and then hit
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Enter to insert the code.
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Now you can, of course, type this out and you'll get the same result but just remember that there's an
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underscore here at the end.
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That's part of the name.
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So let me run the cell and let's take a look at what the slope coefficient is.
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So here we see that it's 3.11.
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Now one of the great things about Jupyter notebook is that we can add some markdown cells so I can add
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a cell here and I can change it to "Markdown" to add a little bit of explanation to what my code is doing.
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That way if I come back to this notebook in the future and I'm wondering what regression.coef_ does,
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I can look at the markdown cell above and I can leave myself a little note; for example, I can say slope
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coefficient and Shift+Enter and then I'll get this text inserted here.
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Now, if I wanted to leave myself little notes inside the actual cells where I've got my Python code,
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then I have to use something called a comment and I can add a comment here with a hashtag or a pound
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symbol and I can add this here and right, for example, theta _1 and you can see that the text
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here is green and that means that this text is considered to be a comment and will be ignored.
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It's not going to be treated as code and I can actually execute the cell and I'll run just fine, but
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if I were delete this little symbol here - this pound symbol, and try to run it then I would get an error
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and it would tell me "invalid syntax".
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So that little pound symbol is very, very important.
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That's what's going to differentiate code from comments.
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OK, so we've got our slope coefficient.
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What about our intercept? Our intercept
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we can pull up similarly through this regression object.
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So "regression.", and then put "intercept" with the underscore at the end.
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Once again I hit Tab on my keyboard there - I'm not such a fast typer, and it inserted the code for
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me.
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So let's take a look at what the value of this intercept is.
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Here we can see that it's about negative 7.2 million.
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So, now we know the regression slope and the intercept.
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And while it's not bad to have these two numbers to hand, we can actually do better than this.
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Wouldn't it be nice if we could just plot a line on our chart?
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Because we've painstakingly visualized our data.
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Why not just plot the regression line on this chart as well?
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So let's do just that.
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What I'm going to do is I'm gonna take this entire cell and I'm going to copy it and then I'm going
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to come down here and I'm going to paste it.
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And the reason I'm doing this is simply to have a reference.
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So here I'm going to have the chart without the line and here I'm going to modify my code here so I
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get the chart with the line. And you can move these cells around as well.
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So if I use this little up arrow here, then I can move it above this cell here. Wonderful!
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So, how do we plot a line on here?
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Well, we can use matplotlib once again.
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So, matplotlib
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has a functionality called plot.
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So plt.plot will allow us to plot a line on this chart, but we have to supply some information; we
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have to tell
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matplotlib what exactly to plot.
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And for that it will actually need two things.
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It will need the Xs and the y's so I'll need some information for where it should plot things on this axis,
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and where should plot things on this axis. Now,
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production budgets are our feature.
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So for this axis we can use our Xs, right?
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So I'm going to put an X here, and then a comma and I have to supply the y value. And what should this
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be?
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We don't want to use the actual value that we have for the gross revenue.
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Instead, what we would like to do is we'd want to use the predicted value from our regression and we
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can get hold of those values by calculating a predicted value for each of the X values.
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To do that we'll need the regression object from scikit-learn that we used earlier.
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All we need to do is type regression.predict(x),
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then it will calculate a prediction for each of the budget values in our
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data and plot it on the graph.
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So let me hit Shift+Enter on the cell and let's take a look at what we've got scrolling down.
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I can see we've got a regression line right here.
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Fantastic!
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But, it'd be a bit nicer if it stood out a little more, right?
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Let's give it a color.
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Let's give it a width.
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Let's make it a bit thicker. And we can do that by going into this line of code here.
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And just before this ending parenthesis we can add a comma and then we're gonna add a color here.
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So I'm gonna see color is equal to red in single quotes and that will make our line red,
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as you can guess. And to make it thicker, we can specify the line width.
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So linewidth, all in lowercase in one word is equal to, let's try 4; if I hit Shift+Enter to refresh
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my cell, then I can see that my regression line now is a lot thicker and it's changed in color.
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What we see now is, we see the relationship between our production budgets and our movie revenue as predicted
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by our linear regression model.
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And that means we can move on to the final part of our data science workflow, namely evaluating and analyzing
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our algorithms performance.
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How do we do? Did we do a good job or a bad job?
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What can the movie's budgets tell us about the movie revenue?
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This is the point where we have to think very, very hard about our model.
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The question that we should ask ourselves at this point is: are these parameters actually plausible?
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Let's take a look at that slope coefficient.
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It's 3.11.
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It means that there is a positive relationship between budget and revenue and not only that, it means
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that for each dollar that we spend on producing the movie, we should get around 3.1 dollars
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in revenue in return.
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And I actually think this seems to make sense.
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Bigger budget films tend to do better.
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So that's good news for us, right? Because we've put our life savings on the line for this zombie movie,
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right?
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Now, what about the other parameter? The intercept.
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This one is at -7.2 million.
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How do we interpret that?
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What this intercept is literally telling us is that a movie with a budget of zero would actually lose
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over 7 million dollars.
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So that's a bit problematic.
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Right?
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That seems quite unrealistic because if you and I went out to make a movie with a thousand dollars, it's
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pretty unlikely that seven million would just disappear from our bank accounts.
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So, this is a lot less realistic.
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And this begs the question - what should we conclude about our model?
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Well it means that we should actually take it with a grain of salt.
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We just have to accept that our model is a dramatic simplification of the real world.
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And as such we should be a little bit careful on how much we believe the predictions of our model, especially
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at the extreme ends.
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Just look at the distance here look at the gap between this data point and our line - the predictions
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of our model seem to fit the data a lot worse at the extreme.
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So how would we use this model to make a prediction, anyhow? Say we need to predict the revenue for a film with
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a 50 million dollar budget.
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We know what our intercepts are.
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We know what the theta zero and the theta one is equal to.
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And if we want to know what the revenue would be for a film with a budget of 50 million, all we would
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have to do is to substitute the values of our parameters that we estimated into this equation.
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So our X is going to be 50 million and if we do the math then we get our prediction, at least according
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to our model.
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Right?
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On a chart it would look something like this.
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We can draw a line up from 50 million and then predict how much,
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what it is that we're actually going to make of the movie?
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And it's gonna be a little bit less than three times the amount that we invested so about 148 million
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dollars - which is not bad, right?
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But, how do we know if it's accurate?
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How can we measure how good our model is?
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So even though it is very very simplistic we can still ask the question of how much of the real world
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data it actually explains.
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And for that we need some kind of measure.
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We need some kind of statistic and the measure that we're going to look at is called R squared.
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Also called the goodness of fit. To look at our R squared we'll simply take our regression and we write
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regression.score.
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And within the parentheses we supply our capital X and our lowercase y - our feature and our labels and
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we hit Shift+Enter and here what we see is that the R squared is approximately 0.55.
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This number here is the amount of variation in film revenue that is explained by the film's budget.
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And I got to say that 55 percent is actually pretty good because think about it this way are very simplistic
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model with a single feature, namely the production budget, can explain around 55 percent of the variation
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that we see in worldwide movie earnings.
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I'd say that's pretty good news for a first try but, of course, we should be a little bit cautious
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reading into this model too much, because we've still got a lot to learn.
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For example how would our model do if we added more features, like how long it took to make or if it's
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a sequel?
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Would we get more realism?
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Would it make our model perform better and make our predictions more accurate?
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And perhaps we should evaluate our model, not just on the data that we used for training it, but on new
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data, data that it hasn't seen yet and also, what if the relationship that we have here is actually non-linear.
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What if we somehow need to transform the data to get a better fit?
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So in a way our analysis has left us with a lot more questions that we should investigate and we will
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do just that in the upcoming modules.
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Well, you got started on the first project and you went through the whole data science workflow; you've
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gathered the data, you've cleaned the data, you've visualized it and then you ran a machine learning algorithm
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and then you've evaluated the results and you've even made a prediction, but this is only the start.
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We've got a whole lot more ground to cover and we'll dive deep into a lot of these concepts and the
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techniques that we've introduced here. In the next module we're going to install Jupyter and we're also
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going to learn a little bit more about regression.
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But the real focus will be on learning more Python programming.
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From there, we're going to learn about gradient descent and how optimization works for many machine learning
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algorithms that we're going to encounter.
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And after that we're going to use multivariable regression and predict some real estate prices in Boston.
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From there we're going to build an actual spam filter from scratch using a Naive Bayes classifier and
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then we're gonna take it up a notch,
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and we're gonna dive into deep learning with neural networks and TensorFlow.
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So for all of that and more, I'll see you on the next lessons.