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In the previous lesson we talked about how the Pearson correlation should only really be calculated
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on continuous data and also how outliers can kind of distort the picture and mislead us.
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The examples that I've used to illustrate this point was Anscombe's quartet and these four charts illustrated
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how very different patterns in the data can actually give you very, very similar descriptive statistics
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if you're not careful. What I want to show you now is how this isn't really just a hypothetical example.
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We can also see something similar happening in our housing data set when we don't apply the right tools
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to the right kind of data and aren't careful with our interpretations.
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Remember how a correlation of 0.9 or 1 is meant to look like? You'd expect to see a chart
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that kind of looks like this, where the data points almost form a straight line. Looking at a correlation
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matrix
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we've actually got a number here, a very high number, 0.91 as the correlation between RAD,
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our access to radial highways and our TAX feature. So given our expectation of what a high correlation
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between two variables would look like, let's visualize this relationship and see what we get.
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I'm actually gonna copy this cell here with our jointplot from seaborn and then I'm going to paste it
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in here. Move the cell down and then what I'm gonna do is going to change the references from distance
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and nitrous oxide to TAX and our RAD, and for the color I'm going to go with, I don't know, maybe dark
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red and then hit Shift+Enter. Voila! This is what we get. Now this picture is probably not what we would've
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expected to see had we just looked at the correlation, but we've said before that this very high correlation
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that we have between these two variables is a result of the fact that this data is not continuous and
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the fact that there are these big outliers in the top right corner that are driving this correlation.
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One thing that's actually very interesting to do as an exercise is to plot the linear regression onto
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this chart and see how it's affected by the data here.
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So let's see what would happen if we ran a linear regression between these two features.
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Now previously we've run our regressions separately using the scikit learn module and we've plotted
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our best fit lines on a matplotlib chart.
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Let me show you a shortcut on how you can use seaborn to do this very, very quickly.
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So if I come down here and in this cell I'm going to make use of a function called "lmplot".
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So "sns.lmplot()" and an x is gonna be our TAX and y is gonna be RAD and for the data
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that we're gonna feed in, the data keyword argument is gonna be our data frame.
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So it's gonna be "data = data, size = 7".
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So what this line of code is gonna do is gonna look in our data frame which is this bit of Python
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code here and it's gonna look for a column called RAD and it's also going to look for a column called
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TAX. The thing that might be a little bit confusing is this "data = data" Python code.
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But this first part data equals is the keyword argument and data
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here is the name of our Python data frame.
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So I'm gonna add another line "plt.show()" and hit Shift+Enter. Voila. And
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just like that,
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we've plotted a linear regression between our two features on this chart.
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Seaborn has done a lot of work for us here and I hope you're seeing just how useful this little Python
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module really is.
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Now let's interpret and make sense of what we're looking at on this chart.
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The data points in our top right corner are affecting the slope of our regression line significantly.
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It's like they're pulling up that regression line to make it steeper. And looking at this,
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you can see how a linear regression between RAD and TAX might not be such a good idea because the regression
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line is kind of meant to represent the data a bit better.
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Right?
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And here we're forcing this linear regression model onto a data set that's not really suited for it.
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Our computers will happily do this calculation but we end up with a model that's perhaps not so useful
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to capture the true relationship between accessibility to highways and tax.
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But luckily we're in the business of estimating house prices, right?
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So what we should actually be looking at is how our features relate to our target, namely our Boston
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property values. Coming down here,
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I want to give you another challenge,
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I want you to pause the video and run a few more scatter plots for data visualization.
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The first one should be between the number of rooms, RM, and our target.
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This is the price series in our data frame.
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Now you can either use matplotlib or seaborn to accomplish this.
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I'll give you a few seconds to pause the video.
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Ready?
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Here's the solution.
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Now I'm going to be very, very lazy and I'm going to come up here and copy the code that I've previously
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written
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and paste that down here and what I'm going to do is I want to change all the references from NOX and
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DIS to RM and TGT, target.
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So "data['RM']" and "data['
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PRICE']" for the correlations,
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and of course I'm going to change the inputs to the scatter plot. For the color,
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I'm gonna go with "skyblue" to differentiate a little bit and then I'm going to say "RM vs PRICE" and then for
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the correlation, of course, I'm going to go "rm_tgt_corr" and this is going to
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be RM which is the median number of rooms. And on the y label,
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it's gonna be "PRICE - property price in 000s". Let's see what we get.
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Voila.
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Now let me do the same thing with seaborn and you know what, I'm going to use "lmplot" to pull up the
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regression line as well. So "sns.lmplot(x='RM', y='
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PRICE', data = data, size = 7)"; "plt.show()".
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Okay, so how do we interpret what we're seeing here?
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Well the first thing is that our positive correlation of 0.7 approximately ties out really, really
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nicely with the relationship that we're seeing on the scatter plot.
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So this means that room size alone has a very, very clear relationship with our house price.
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And I think that's kind of neat to see.
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One of the other things that I find quite interesting when looking at this chart is that there almost
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seems to be like a ceiling on the property prices at the 50000 dollar mark, because there's a lot of
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data points that are on this line here.
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And I think that's quite, quite curious because it could be coincidence, but it might also have something
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to do with how the data was collected in the 1970s. But for the purposes of this tutorial I'm not really
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going to dig into these kind of details.
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There's so much other stuff that I want to focus on.
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One of the things that you might already be thinking at this point is: you've got like 13 variables and
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there are just too many combinations to graph individually. We can't possibly copy and paste like our cells
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in our code to graph all these scatter plots individually.
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Right? And
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yeah,
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we're not going to do that because remember, we have to think like lazy programmers and what would
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lazy programmers do?
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They would graph every single combination at once and this is where seaborn is going to come to the
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rescue once again, because seaborn has a function called "pairplot" which will do just that.
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So "sns.pairplot()" and then as an argument it needs our entire data frame,
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so "sns.pairplot(data)", "plot.show()" will create scatter plots between all our
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features and our target.
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But I hope you didn't type Shift+Enter just now and run this thing already because there's something
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I want to show you before I execute this cell.
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I want to show you a little bit of Jupyter a notebook specific code, a little bit of Jupyter notebook
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magic if you will, cause the thing is pairplot actually has to do a lot of calculations and it takes
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some time to run.
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And what I want to show you is how you can benchmark or time specific bits of code.
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People actually call this microbenchmarking, because we're benchmarking a few lines of code,
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we're benchmarking an individual cell and there's a magic command for Jupyter notebook and this command is
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"%%time".
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And now I'm going to hit Shift+Enter to run the cell. Ready?
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Here we go. Now my computer has produced all these plots.
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Check it out. And because I've added that Jupyter notebook magic code, I've got some additional information
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printed out here as well. This is showing me how long it took to run my code.
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Now when you're running this at home, I don't know how long it'll take you.
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I don't know if it'll take you like 19 seconds like it took me
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or like 40 seconds, or maybe five seconds depending on the kind of machine that you've got.
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But I think this is a really neat feature from Jupyter notebook, because suppose you want to make a decision
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between two different types of algorithms or two different ways of running your code, say if you've got
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something that takes a little bit longer and you're looking to like optimize it a little bit or do a
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horse race between two different algorithms, this timing capability of Jupyter notebook can come in
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quite handy.
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So we've done this a little bit out of curiosity and so that you can see that, you know, your computer
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didn't freeze up or whatever because it took me some time as well to run this bit of code.
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But that said, this has some other practical applications as well.
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Another thing to know about Jupyter notebook actually,
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on this note, is that up here in the right hand corner there is a little dot here that for me is currently
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not filled in.
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And when I hover over it I can see it says "Kernel idle".
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This means that Jupyter notebook is currently not doing anything.
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If I was to execute this cell again, during that execution you could see that this dot is filled in
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and it says "Kernel busy".
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So this gives you an indication that at the moment our Jupyter notebook is doing a lot of work and that
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even though it might look like there is nothing happening,
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our computer is actually working.
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This is another handy little thing to know about Jupyter notebook.
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Now let's take a look at our output here.
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So what you can actually see here is kind of a cropped version or a zoomed out version of the actual
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size of this image.
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This image in fact here is a lot larger than what's being displayed.
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So we're kind of zoomed out. If you want to see the full size image which I encourage you to try and
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do, what you can do as you can right click on this thing click "Save As",
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stick it in your downloads folder as jointplot or what have you.
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And then in your Downloads folder open it up and pull it up in your graphics program of choice.
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So I've got a Mac and I'm using Preview here so I can go here and go to "Actual Size".
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And here you see a zoomed in, actual size, 100 percent version of this image.
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Now the other thing that you can actually do is you can open this image up in a new tab.
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So if I right click on this image and, I'm using Firefox here, this will be different on Chrome or Safari
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or Internet Explorer of course but you can easily do is you can do something like this and then you
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don't have to download this image and dig it out of your Downloads folder.
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So I think this is quite handy.
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Now I've got two tabs - 
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one with the full sized version of the image and one with my notebook.
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So what we can see here is that we've got scatter plots for every single column in our data frame.
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So we've got Pice vs Crime, Price vs Zone, Price vs Industry, Price vs CHAS, Price vs
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NOX or RM,
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And then we've also got LSTAT vs crime,
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LSTAT vs Zone, LSTAT vs industry, but the really neat thing is that, in the diagonal down
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the middle,
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yeah,
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so this would be Crime vs Crime,
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we get a histogram.
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So this is the histogram for the ZN feature.
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This is the histogram for INDUS.
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This is the histogram for, this is the histogram for CHAS and so on.
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And down here we've got the histogram for Price.
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So the diagonal shows is the histograms and everything else shows us a scatter plot. Scrolling down to
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the last row, namely the Price row,
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we can check out for which features we can very, very clearly see a relationship between property prices
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and the feature.
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So here this is Crime and Price. Going a bit to the right we've got Price and Industry and we can definitely
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picture some sort of relationship here,
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right?
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Going a bit further,
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we've got Price and Room size,
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this is a scatter plot that we've created earlier
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and here there was definitely also a relationship. Going further all the way to the right,
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we see this one here, Price and LSTAT.
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Now again this scatter plot is actually showing us a very, very clear relationship between these two
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things.
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It almost reminds me of Pollution and Distance in terms of its shape,
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right?
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But what is LSTAT? We haven't really talked about this feature before,
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so let's take a look at the description.
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So coming back to the Jupyter notebook, to the top where we printed out the description, we see that for
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LSTAT it's "% lower status of the population".
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Now this description might not be sort of the politically correct wording that people might use these
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days, but in a nutshell this feature measures the socioeconomic background of people in the neighborhood.
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I actually wanted to get a little bit more detail on LSTAT,
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so I checked back with what it said in the original research paper to get a little bit more of a complete
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description and what it says here is that by "lower status" what the data's actually capturing are things
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like what percentage of people don't have a high school education and what proportion of male workers
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are classified as laborers.
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So when I read this description, I immediately looked back at my correlation matrix because what I wanted
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to check was the correlation between LSTAT and Industry,
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so the proportion of industry in a neighborhood and the proportion of people who are classified as laborers
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or who don't have a high school education.
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And that correlation, it turns out, is 0.6.
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So it's both positive and fairly high.
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So reading this description now all of sudden this correlation made a whole lot more sense and coming
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back to this scatter plot between LSTAT and house prices is that what we can also clearly see is
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that the property prices tend to be lower in areas that have a high proportion of people who have no
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high school education or who work in factories.
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In other words, the property prices appear to be affected by the kind of people who live in the neighborhood.
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But we shouldn't jump to conclusions just yet, our multivariable regression will tell us a lot more.
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But
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speaking of regression and looking at these charts and the pairplot, wouldn't it be nice to be able
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to plot a regression line onto all of these plots as well?
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I think that would make it a lot more clear, don't you think?
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Let's go back to our Jupyter notebook and go down into the next cell, because "sns.pairplot(data,
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kind='reg')" will show us that seaborn will happily oblige if we add
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an extra argument to the powerful pairplot function.
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All we have to do is change the kind argument from "scatter", which is the default, so pressing Shift+Enter,
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we can see that "kind = 'scatter'" is the default for this function and change that to regression, which
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has the code "reg". Now I know what this regression will look like because I've already run this code
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before, but the thing is, it will be colored in exactly the same way as our data points.
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So it'll be blue on blue, which won't be very, very helpful.
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We want to color our regression line in a little bit differently.
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And I want to do that now,
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I don't want to execute this code,
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wait 20 seconds and then add more code,
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wait another 20 seconds.
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So I'm going to add another keyword argument here and that's gonna be the "plot_kws"
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keyword argument, "plot_kws". I'm gonna set that equal to a dictionary.
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This is how we're gonna add a splash of color to our regression line. A dictionary always has the curly
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braces and then it has a key and a value. The key is gonna be "line_kws" and then the value
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comes after the colon.
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So what's the value that we're gonna put here?
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It turns out it's actually gonna be another dictionary, so we're gonna have curly braces and then another
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key which is gonna be "color" and then a colon,
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and here's the value, "cyan", which is gonna be the color I'm going to use.
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So what we're looking at here now is a dictionary "{'color': 'cyan'}" which is the value for another dictionary,
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which is the "line_kws" key.
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In other words, we've got a nested dictionary, a dictionary inside a dictionary,
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it's like the most boring version of the movie Inception, but this is how we're going to get a color
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for our regression line.
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All that's left to add is "plt.show()".
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And because I'm a very curious person, I'm also going to add the "%%time" Jupyter notebook
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magic. Now I'm going hit Shift+Enter. *drum roll*, what is happening? I should really get a coffee while this
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is running. There we go,
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now it finished.
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How long did it take to run? 47 seconds.
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It's absolutely brutal. But let's take a look at what this looks like now. Again,
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we can see just how big this image is.
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It's about 2500 pixels
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times 2500 pixels.
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This is at 25 percent. 100 percent
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looks more like this. With the help of our regression line on our chart,
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we can now also visually see the positive relationship between our Industry variable and LSTAT. So
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we know the correlation was around 0.6
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and now we can also see that borne out in the positively sloped regression line right here and in the
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last row we can see a regression line between each feature and our target variable individually.
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This is really neat because we can see if the slope is positive or negative and it's basically doing
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a univariate regression 13 times.
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But the power of our analysis isn't gonna come from looking at our regressions against that target in
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isolation.
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The power comes from combining all our explanatory variables and running a multivariable regression.
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Now I have a feeling this air pollution themed cartoon reference is betraying my age a little bit.
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The point I'm trying to make is that we're going to combine all the explanatory power in our features
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to estimate our property prices in Boston and our model of choice is called multivariable regression.
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After we've done all that, we're going to evaluate our results and see if we can improve our model further.
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I'll see you in the next lessons.