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Now I hope that you can't wait to calculate the mean squared error. From the previous cell up here,
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we already know what the lowest cost theta zero and theta one parameters are for our regression.
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And this is thanks to the scikit learn package.
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But let's work out what the mean squared error actually is for these two values.
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And for that we're going to need our estimated y values.
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Looking at our equation up here we need to calculate our y hat.
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What's this y hat equal to? Well, our y hat is gonna be equal to theta_0 + theta_1*x.
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This is the equation that underlines our linear regression in this case - our y hat is based on the intercept
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plus the slope,
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times our X values.
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Let's translate that to Python code. So I'm going to create a variable called y_hat
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and I'm gonna set it equal to our theta_0 value which was this one here.
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I'm going to copy this.
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Come down here.
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Paste it in, add a plus sign.
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Take the theta_1 value,
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copy and paste it in and then write
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"* x_5".
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This after all are all the X's in our dataset.
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And that way we can calculate all the predicted values in our dataset.
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Let's print these out for good measure.
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So I'm going to say "print" and then the string "'Est values y_hat are: ', 
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y_5".
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Let me hit Shift+Enter and let's see what these look like.
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So here they are.
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These are all the predicted values. Now,
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one thing that you can do is if you don't want this first value here on the same line as your string
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what you can do is you can come in here with a string is and insert a special character -
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the new line character.
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So that's backslash and then
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"n". When I hit shift enter now that 0.969 will move to a new line, "\n" is
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like a special character for hitting return on your keyboard.
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So how do these predicted values compared to our actual values?
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Let's print those out as well.
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So I'm going to say "print" and then write "In comparison, the actual y values are \n, y_
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5".
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These are actual values.
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There we go.
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Ideally we want these estimated values to be as close to these actual values as possible.
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And I'm looking at this.
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They're actually not too far off.
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This is not too bad.
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Given this theta_0 and this theta_1.
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So now that we know how to calculate our y hat values, let's work out the mean square error of the regression.
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And I think this actually makes a really good challenge.
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So can you write a Python function that takes two inputs y and y hat and returns the mean squared error?
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And after you've done that, after you've written this MSE function can you call this function and print
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out the mean squared error with the y hat calculated above, so feed in this y hat value into your MSE
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function and print out what the mean squared error is.
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Now remember the mean squared error equation looks like this.
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We've got that formula in LaTeX notation, so all you need to do is translate it into Python code and
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look up what code to use for that pesky summation symbol right here.
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I'll give you a few seconds to pause the video and try this on your own
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All right.
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Ready?
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So I hope you figured this out.
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Now in terms of the solution.
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I'm not just going to show you one solution.
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I'm going to show you three different ways that you can implement this, because there's actually many
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ways you can skin a cat as they say.
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So while you have the video paused and were solving the challenge,
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I was busy looking up possible uses for all that cat skin.
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And the best one I came across was a Japanese instrument called shamisen.
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Apparently the drum on this banjo was actually traditionally made from from cat skin.
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So uh there's your random fact of the day. If you've come across any other uses do let me know in the
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comments section.
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The Python code that you will have written will probably look something like this.
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You're going to have have "def mse():" and then for the parameters you'll have y and say y_hat, colon. And the
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first possible approach would look something like this.
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You might have a variable say call it "mse_calc" and that would have been equal to 1 divided by 7 because
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we've got 7 data points and you multiply that by the sum,
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this is an inbuilt Python function;
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"sum()" and then inside the sum you'll have another set of parentheses and there you'll have
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"y-y_hat" to the power of 2.
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So **2.
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And because this function needs an output you would return the results of your calculation.
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So mse_calc and this is one possible way to do it.
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Let me call this function and print out the mean squared error of y hat that we calculated.
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So it'll be "mse(y_5,", because these are the actual y values and then I'll supply y_hat that we
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calculated a few cells above. I'm going to hit Shift+Enter and see what we get. We get 0.95 approximately.
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Now there is one improvement that we can make to this function.
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So this 1/7 will probably be bothering you a little bit because it's hardcoded and
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it would only work or calculate the mean squared error correctly
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if we had 7 data points - if we had 8 or 9 then the mean square error formula wouldn't be apt
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anymore, wouldn't be correct anymore with this Python code.
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So I'm going to comment this out.
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But I'm going to copy, paste it below,
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and what I want to do is I'm going to replace this 1/7 with some different code.
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I'm going to say "1/y.size" - y.size will return the length or the number of
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samples that are fed in to the place holder for our y values.
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So this will also work.
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Let me hit Shift+Enter on this cell and the one below to prove just that. Writing the code this way makes
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the MSE function much more general because it figures out the number of samples inside of the function.
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It doesn't have to be supplied as an extra parameter and it's not hardcoded in the instructions themselves.
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So this is already quite nice but there is another way you could have done this as well, maybe you did
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a bit of googling and you figured out that numpy actually has a function called "average" and this does
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both the summation and the dividing by the number of samples for us.
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So let's see what this would look like.
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This would be "mse_calc = np.average()" and then it needs two things -
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it needs the function itself that it's averaging and if it should be averaging the rows, the columns
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or the whole thing.
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So first things first, the function its averaging would be (y-y_hat)**2
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and the way to tell it whether it should average the rows or the columns would be by supplying the "axis"
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argument, so "axis = 0" sums up the rows instead of both the rows and columns which is not what
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we want.
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So when I hit Shift+Enter on this and then run the cell below again, we can see that this works as well.
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But just because we've printed out a mean square error in this cell here doesn't mean it's correct.
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So let's check it against an inbuilt function from scikit learn.
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So we wrap this in a print statement. Write
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print and then the string
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"Manually calculated MSE is:" and then we have our mse function in here. To use the inbuilt function
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that inbuilt mean square error function from scikit learn,
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we're gonna have to import it.
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I'll have to scroll to the very top and then here we're gonna add another import statement, we're going to say
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"from sklearn.metrics import mean_squared_error .
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Sorry sorry.
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I hope you're not too upset and will forgive me that I made you do all this work even though there is
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an inbuilt function for this already. Let's call this mean_squared_error function with both our
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manually calculated y_hat as well as from the output of the regression directly.
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So where we were adding our print statement let's add another one let's add "print('MSE
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regression using manual calc is:', )", and then "mean_squared_error()" and it will
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supply two inputs: y_5, our actual y values and y_hat, which we've calculated
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above just here.
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And now let me copy this line and paste it below.
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And this one will read "'MSE regression is', mean_squared_error(y_5,)"  and instead of our manually calculated
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one we can use "regr.predict(x_5)" and when we hit Shift+Enter,
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we see that indeed all the outputs agree with one another. Our manually calculated mean squared error is
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the same as what we get from the machine learning package.
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So we've dissected and implemented this cost function correctly.
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Brilliant.
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Now it's time to plot our cost function and visualize it. We'll do that in the next lesson.
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I'll see you there.