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In a previous lesson we've already written this line of code "from sklearn.linear_model import
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LinearRegression".
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In other words we've already imported our linear regression functionality into this Python intro notebook.
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For consistency,
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let's follow the same pattern that we employed when we were estimating our movie revenue.
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We're going to create a variable called "regr" and this variable is going to store our linear regression
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object.
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To run our regression,
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all we have to do is call the good old fit method.
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So we're gonna see regr.fit,
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open the parentheses and then for our explanatory or independent variable we're going to use the amount
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of drugs in the tissue.
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So we're gonna type LSD and then put a comma after it,
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and now we can add our dependent variable or the values that we're going to try to predict.
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In this case this is the score. And that's it.
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Now if you remember previously our explanatory variable was called capital X and our dependent variable
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was called lowercase y.
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Now remember our scikit-learn's fit method essentially computes the parameters of this equation.
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We've got our theta zero which is our intercept and we've got our theta one which is our coefficient
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in front of our explanatory variable.
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Let's see what happens now when we try to run our regression.
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We in fact get an error.
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Looking down, we see that we get a value error: "Found input variables with inconsistent numbers of samples".
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Now this is odd, right?
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Why would there be an inconsistent number of samples?
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We've got seven rows in each of our two columns.
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We've got seven rows in LSD and we've got seven rows in the math score.
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So why would there be an inconsistent number of samples?
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Now, even though this error message seems a little bit like a red herring,
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the reason that we're getting this problem actually has to do with the fact that we are working with
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series and not data frames. We're getting this error because this notation here data[],
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and then the column name actually extracts an object of type series instead of an object of type data
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frame.
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Now, if you recall, the way to get a data frame from another data frame is to double up on the square
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brackets.
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So if we add an additional pair of square brackets, this notation now will extract an object of type
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data frame and store it in LSD and in score.
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So if I press Shift+Enter now, we are now no longer working with series we are working with data frames.
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Going back down to where we're fitting our regression I can press Shift+Enter again to rerun this line
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of code and we can see that our regression runs without a problem.
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Now you don't have to take my word for it
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regarding the change in types - you can double check this yourself. So you can always add a cell below
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and check the types, so if I say type(LSD), then I see it is a data frame and if I take away the square brackets,
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press Shift+Enter and rerun this again, I can see that it is a series in this case and this is what we've
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talked about when working with data frames in the previous lesson.
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So now that we've successfully fitted our regression, let's take a look at the values of this theta one
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and this theta zero parameter.
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If you remember the way we did this in the past it was looking at the attribute of our regression object -
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the attribute in question was called "coef" with an underscore.
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So let's add that here "regr.coef_" and let's hit Shift+Enter, and see what happens.
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Our Jupyter notebook will print out array, then parentheses,
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then one set of square brackets, two set of square brackets and then the value of our theta one parameter.
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So we can see that our coefficient is stored inside an array.
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It's an array of one element.
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Let's pick that element out.
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So I'm going to say [0] to access the first element in the array.
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Let's see what happens when I press Shift+Enter now.
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Huh.
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In this case one of the square brackets has disappeared, but we're still left with an array.
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We're not yet able to access the number inside directly.
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We're still getting a collection containing just one element.
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So you might ask, did this just work?
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We access the first element of our array and we still got an array.
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It seems like a bug, right?
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Well the answer is is that we have to go one level deeper to get the raw value.
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You've probably noticed that there was one square bracket less, right?
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So if I don't have this at the end, I have two square brackets but if I do access an element inside my
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array I have one square bracket.
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And the reason we get this is that we have to go two levels deep.
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We actually have an array of arrays.
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Mind blown, right? Our coefficient is buried inside an array that's inside another array.
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So how do we access an array inside an array?
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Well, we can access the first element which is the array and then we can access the first element of
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that array again to get the raw value.
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And this is how you would access a particular value of a nested array.
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Let me add this to a print statement.
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So I'm going to say "print", and just say "Theta 1 : "comma and then close our brackets here.
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So I'm going to add this to a print statement like this.
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Now let's take a look at our intercept - our intercept was the intercept_ attribute from
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our regression object. So I can say "regr.intercept_" and  press Shift+Enter and we see that
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our intercept is also inside a collection that's also inside of an array.
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But there's only one set of square brackets here, so we can access the raw value inside just having that
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[0] following the name of the attribute.
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I hit Shift+Enter we see the raw value printed out. And again I can wrap this inside a print statement
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'Intercept: ', regr.intercept_intercept[0], and close the parentheses at the end.
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There we go.
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So here's our intercept.
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Now what about the goodness of fit or our R squared.
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To find out how much of the variation in our data is explained by the amount of drugs in the volunteers
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tissue,
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we call the score method on our regression.
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So we type "regr.score" and then we have to provide two values.
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One is our explanatory variable and the other one is our dependent variable - which was score. So 
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regr.score(LSD, score)
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We're going to print this out, and we see that our R squared is approximately 0.88. Let's wrap this inside a print
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statement as well - " 'R-Square: ', " - there we go.
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So in this cell we fitted our regression, so we've run our machine learning model and we're printing
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out a couple of stats about our regression. A couple of the statistics that describe
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what went on with the calculation. One of them is the coefficient, another one is the intercept of our
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line and another one is the R-squared or the goodness of fit.
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So we've got some basic information about our regression and we see that the amount of drugs in the
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contestants tissue explains close to 88% of the math test performance and we also see
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that for every increase in LSD parts per million, our volunteers math performance was approximately 9
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percent worse than the control - this is what the theta one is telling us.
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Now even though this is all very well and good, it'd be really nice to represent this graphically because
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we like pictures - pictures are very very important for making sense of data so let's create another plot.
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I'm going to do this in the cell below.
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Now one thing that you've already seen a little bit in the Python code is that when creating nice looking
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graphs it's a two part process. In the first part,
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we do all the styling and in the second part we plot the data and show it off.
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So what I'm going to do is I'm going to do the second part first.
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I'm going to plot the data and then I'm going to add my styling code later on.
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So plotting the data as it is I can write plt.scatter and then provide the inputs to our scatter
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plot and that's going to be the LSD parts per million, comma, and then the score.
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These are the math scores.
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So let's see what this plot looks like
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by adding by adding plt.show() beneath.
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Here we go.
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This is what our plot looks like before we've done any styling on it.
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Now I think this chart looks,
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I think this looks super ugly actually so we're going to have to do something about this. For starters
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let's add some arguments by keyword to this plot.
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So in your scatter method you're going to put a comma at the end after score and then write
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color = 'blue'
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Let's hit Shift+Enter to see what it looks like. Now we've got our data points in blue.
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So this is a slight improvement to the black and white version.
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Now we don't have many dots on here.
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We don't have many, many data points.
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So let's increase the size of these individual dots on our chart.
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And I want to leave this to you as a challenge.
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So I've got the documentation of the scatter method up in front of you right now.
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And what I would like you to do is I'd like you to look at this documentation and see if you can figure
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out how to increase the size of these data points
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and also maybe add some transparency - in other words, instead of having it a solid blue color make those
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blue dots slightly transparent.
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I'll give you a few seconds to pause the video.
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The hint I'll give you is that it's going to be in the keyword arguments of the scatter function. And,
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here's the solution.
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So we wanted to increase the size of our dots.
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So the way to do this is to look at these keyword arguments.
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So, for example, "s" is the size in points of our dots and the transparency is this alpha value here.
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The alpha value will be between zero which is transparent and one which is opaque. Coming back to our
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Python code,
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we can add these key word arguments to our scatter method.
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So after 'blue', I'm going to add a comma and then I'm going to see "s=" and let's experiment here
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a little bit.
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So what happens if I say "s = 500" and hit Shift+Enter?
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I get enormous blue dots. This actually doesn't look half bad but I think I'm going to go with something
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like, maybe 100 is the right value here. So that's the size of our data points covered.
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Let's change our transparency now.
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This was in the alpha parameter. So I'm going to say
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"alpha = ", I don't know,
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0.7 - it's going to be a value between 0 and 1, remember?
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So hitting Shift+Enter, I get a nice little bit of transparency here on my data points which are now
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little bit larger so we can actually tell what's going on.
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OK, so I'm going to leave it at that.
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And now I'm going to add some labels to our chart and make it look a little nicer. Since we've done this
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before,
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I'm going to leave this to you as a challenge so you can return and remember the Python code that you
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wrote.
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Can you set the title of the plot as a whole as "Arithmetic vs LSD-25" and then add some labels on
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the side - one for the x axis that reads "Tissue LSD ppm" and one for the y axis that reads
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"Performance Score"?
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And here's the solution.
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So we take our plotting object, put a dot after it and write "title()", and then provide the string
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"Arithmetic vs LSD 25" and then we do the same for the labels on the x axis and y axis, so plt.xlabel(
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"Tissue LSD ppm") and plt.ylabel(
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"Performance Score").
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There we go.
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Let's  hit Shift+Enter and take a look at what this looks like and we see that may be the thing to do is to increase
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the font size a little bit on these three labels.
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I think in our previous chart 17 for the title and 14 for the labels worked really well. So I'm going to say 
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"fontsize=17" for the title, and then I'm going to add another keyword argument to our X labels and Y labels
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"fontsize = 14" and "fontsize = 14" again.
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So let's take a look.
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That's starting to look pretty good.
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Now to round things off a little bit,
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we can try again setting a limit on the range.
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So ylim
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is gonna be between 25 and 85, 25 and xlim is gonna be between maybe 1 and 6.5
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Yeah.
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Doesn't need to go all the way to seven,
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I reckon. And for the style maybe "plt.style.use"
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then we can choose our good old friend
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'fivethirtyeight'. Let's hit Shift+Enter and to apply the changes.
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I don't think that worked.
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Let's try again.
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Okay, so here we go.
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This is how it would look like with our styling as it is currently.
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We've got our range set.
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We've got our colors set and we've got the font size set as well.
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At the very top of the cell I'm going to add again this little percentage sign and write 'matplotlib
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inline'.
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And what this does is it tells Jupyter notebook to export this graph as it is when we say File > Download
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as > Notebook.
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So there's really only one thing left to do which is plotting our regression line on here.
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Because at the moment we've got our data points we've got our chart nicely formatted and looking good,
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all we have to do now is plot our predictions from our machine learning model on here. So our machine
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learning model will have a prediction for every level of LSD tissue concentration in the data set. To
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get hold of these predictions,
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we use a method called predict so we would write "regr.
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predict",
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And as a parameter here, as an argument here,
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we would supply the LSD tissue concentration.
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So this predicts a math score based on the amount of drugs in the tissue.
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Now we'll want to store that information somewhere.
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So I'm going to create a variable called "predicted_score" and set it equal to the output
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from this method right here.
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Now remember, you've got a press Shift+Enter to actually run this cell.
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Otherwise the cells below won't know about this code that we've just written.
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So I'm going to hit Shift+Enter now.
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So looking down at our chart, we see the actual scores indicated by the blue dots,
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and now we just have to plot the predicted scores alongside these actual ones. And all these predicted
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scores are gonna be connected by a line.
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This is the line that we want to superimpose on our graph, so we can write "plt.plot" and then provide
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the line that we want to draw.
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It's gonna be the LSD tissue concentration on the x axis and then on the y axis it's gonna be
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our predicted scores, right?
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"predicted_score"
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Now let's hit Shift+Enter.
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And here we go.
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We've got our predicted values connected by a line superimposed upon our scatter plot.
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Of course we can style this line any way we want to.
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So I'm going to say " color = 'red' " and " linewidth = 3 ".
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Now we've got a chart with even more contrast between the blue data points and our red fitted regression
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line.
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So I think that concludes all the analysis that we're gonna do for our Python intro.
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And what have we learned from all this?
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Well, drugs are bad for you,
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boys and girls, especially if you're studying math tests.
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But the other thing that you'll notice is that if you look at the original paper and you look at the
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equation that the researchers have estimated we can actually see what their estimate was for the intercept
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and the coefficient that we've estimated as well.
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So they've got 89.7 for the intercept and -9.44 for
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the coefficient in their equation. In contrast, our coefficient is -9.0 and our intercept
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is 89.1, not 89.7.
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So we can see that we're not able to reproduce the researchers' output exactly.
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Now I suspect that's because the researchers and we are not working off exactly the same numbers. You
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see, they actually don't provide the information on parts per million
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and the math scores in the PDF that we're looking at.
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I actually had to hunt around the web to get these numbers and they might be slightly different from
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what's in the original paper.
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But, that said, I think that our results are so close that we can say that we've successfully reproduced
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the research that's in the paper there.
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Oh and, by the way, this is in no way relevant to the study at all.
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But many of the calculations that we just did the original authors ran on something called an IBM 360
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computer.
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The reason you've probably never heard of the IBM 360 is because you don't have one of these monstrosities
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sitting in your living room.
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Now,
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I find this so funny that the researchers actually mentioned this particular computer model in their
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actual paper and I can't figure out if it's maybe some 1968 humble brag about how high tech they are
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or if IBM actually paid them for this shout out. In any case, plugging the computer model that you've
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done your research on in your scientific paper has probably gone a little bit out of fashion these
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days.
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But yeah, I I did find this interesting.
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Now, if your reaction of me telling you about this just now was "Wait a minute,
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IBM made computers?". Then,
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I highly, highly recommend watching this documentary called Silicon Cowboys. Silicon Cowboys is a really,
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really fascinating film about a little startup called Compaq that battled it out with big blue in days
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gone by.
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So yeah watch it and I'll see you in the next lessons.
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Take care.