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All right.
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So, if you haven't fired up to put a notebook in a while then open your Anaconda command line
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if you're in Windows, or go to terminal if you're on Mac and then type in the command "jupyter notebook"
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and hit Enter. Your browser should fire up and drop you off on local host with your folders. Here,
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you want to navigate to the MLProjects folder that you've set up earlier and there you want to create
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a new Python 3 notebook.
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And,
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we're gonna click up here what says Untitled.
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I'm gonna rename it to "0 3
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Gradient,
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(oh man I can't type), gradient descent."
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We're gonna hit "Rename" and now we're ready to go. For our first cell in our new notebook,
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we're not gonna be typing any Python code.
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Instead we're going to be making our notebook a little bit more pretty, a little bit more readable by
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inserting a heading.
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So what I'm going to do is I'm going to go to "Cell" and then change the "Cell Type" from "Code" to "Markdown".
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So select "Markdown" here and you'll notice that the "in" here has disappeared because this cell will
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now no longer be evaluated as code.
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So we said we're gonna make a title.
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Our first title is gonna be "Notebook Imports and Packages".
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And, we hit Shift+Enter. So now we can see here this is just plain text. And in the cell below
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we're gonna be adding our import statements.
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So the first thing that we're gonna import is our old friend matplotlib pyplot - we're going to be doing
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some plotting in this notebook.
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So I'm going to write "import matplotlib.pyplot as plt" - I'm going to stick with the same naming convention
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here.
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And we're also going to import another library, another package, called numpy.
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And this is commonly referred to as "np".
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So a lot of people use np and we'll use that as well.
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So I'm going to write "import numpy as np" and we're also going to add the good ole Jupyter notebook statement
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"%
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matplotlib
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inline".
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So I'm going to be doing some plotting, so we're gonna add this matplotlib inline statement so that we can
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export our plots very, very nicely. I'm going to hit Shift+Enter now and before we go on I want to revert back
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to this markdown up here.
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I'd really like to have a section heading, and let me show you how we can get that, because "Notebook Imports
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and Packages" is very, very small.
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If we put a hashtag in front of this and hit Space then immediately the font becomes a lot bigger, bolder
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and blue and this is because this hashtag is telling Jupyter that now "Notebook Imports and Packages" should
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be considered as a heading. It's not actually gonna look like this when I hit Shift+Enter, it's going
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to look like this - it's gonna be big and bold.
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So this is a very nice way to create section headings in your notebook.
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Now if you wanted this a little smaller and one level down in the size and boldness then you can put
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two hashtags and you can see how the markdown adjusts really nicely and will look like this in contrast.
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Right.
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And you can even try three.
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Right.
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So there's different levels of boldness and size that you can you can play with for your section headings
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to keep your notebook organized.
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So I go with one, and this way we can find our imports very quickly.
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Now we're gonna dive straight into our first example.
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Now the thing I'm going to do though is, I'm going to take this cell here and I'm going to modify it to
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be a markdown cell as well.
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So this is my chance to share a little bit about what Jupyter notebook can do. So I'm going to say "Example 1",
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"Example 1" and example 1 is going to be about a very, very, very simple cost function and the cost
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function is going to look like this.
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It's going to be f(x) = x^2 + x + 1.
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Let's see what this markdown would actually look like.
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Would look something like this but, um, the cool thing about markdown is that you can actually make the
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mathematical notation look a lot better.
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So you can add a dollar sign in the front and you can add a dollar sign in the back.
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And if I press Shift+Enter now, then you can see that indeed we have this formatting here in the markdown
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cell looking like an actual equation.
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So this is gonna be Example 1 is
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going to look like this.
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I can make this bigger as well so I can give it a section heading if you will.
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And now it's going to look like this so this is gonna be a simple cost function.
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All right.
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Here we go.
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Now,
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if this is the first time that you're seeing this then you might be confused about what are these dollar
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symbols, right?
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Well this is a markup notation from a system called LaTeX and LaTeX uses tags like these dollar
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signs to mark a particular section as a mathematical notation. If I double up on the dollar
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signs, then I'm giving it a different kind of tag.
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And you can see that you'll have a different formatting as well.
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So in this case it's centered, so single dollar sign is in line
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and the other tag the double dollar sign is for display.
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So we're gonna be using a LaTeX a little more in this module and
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you might actually see this in in many, many other places as well - it's super popular for writing
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mathematical equations or scientific papers especially in academia.
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The best analogy that I can think of of how LaTeX works is that it works really similar to XML
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or HTML.
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So if you were to go to a Web site like say example.com and you right click on it and you go to
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"View Page Source", then you'll see the HTML documents.
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So while the website will look like this - "Example
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Domain" is actually surrounded by two HTML tags - title and title,
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so a beginning tag and an end tag. And LaTeX sort of works a bit like this as well.
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You've got a markup, meaning these tags, that kind of give structure to your document.
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So it's through these tags that the Jupyter notebook knows how to format a particular section of text
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in the in the markup cells. Now that we've added the markup and our section heading,
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we can now actually, you know, write the Python code for this function and it would look like this "def
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f(x):",  new Line, "return x", double multiplication sign for the power,
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so "**2+x+1" .
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And that's our function.
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Right?
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This is our function in Python code.
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Now what we're going to do is we're going to generate the data.
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I'm just going to add a little comment here "Make Data" and the way we're going to do this is by using num-
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py.
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So our data is going to sit in a variable, I'm going to call it x_1 - "_1"
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because this is our first example, so x_1 is gonna be equal to something and it's
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gonna get its value from a numpy function - numpy was np and the function we're gonna call is "lin-
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space".
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Yeah.
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"np.linspace(start = - 3, stop = 3)"
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and then "num" is gonna equal, say, 100. I'm going to hit Shift+Enter and explain what I did just there.
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So the linspace function is something that comes from the numpy library - I'm going to pull up the
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documentation for you guys and I've got a couple of arguments that I gave it  - I gave it a start value,
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a stop value and a value for this third one called num. Now linspace creates a sequence, a sequence
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of numbers that creates our data for us and it creates a sequence of numbers between the start value
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and the stop value and the number of samples is set by that third parameter so that third value, the num
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value. Back in our Python notebook we can actually take a look at what this would look like
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so if I put x_1 here and hit Shift+Enter then we can see what it is that we've actually got
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back; we've got back an array that starts at -3 and goes to 3 and has a hundred individual
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data points -
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one hundred individual values that are spaced out equally between -3 and 3.
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If this, you know, was I don't know the number 10 instead of 100 then I'd get an array with much fewer
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values, right?
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Just 1,2,3,4,5,6,7,8,9,10.
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Right?
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So you can see this linspace function here is very, very handy for generating data.
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And one of the things I quite like doing is actually adding the names for the arguments
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when I make a call to my function because I just find it's so much more readable than having it like
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-3, 3, 10.
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Right?
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That's much harder to read, especially when you're coming back to it and you might
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not remember it so well.
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So x_1 is an array with approximately 100 values.
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We can even give it more, we can give it like say 500.
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Right?
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I'm going to hit Shift+Enter
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And that's us having made our data. Now that we've got our data,
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let's plot it, right?
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Let's graph it using our function, so we can add the plot by using our matplotlib - plt, so we can actually
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do this in one line.
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Right?
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We can say "plt.plot", and then "(x_1, )",
and then what,
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f(x_1)" - we're going to feed the actual data that we have in x_1 into our
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function here and we are going to plot f(x_1)  - this is our y right.
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And x_1 - this is our x.
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So let's call plt.show().
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See what happens.
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And here it is.
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This is what our function actually looks like - it's a parabola. I'm going to actually center this a little bit as well,
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so I'm going to set the axes by writing "plt.xlim()", 
and I'm going to give it a range of
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-3
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uh, to, I don't know, 3.
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Right.
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This is what we generated our data from, right - from -3 to 3 and the y axis,
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we can set with "plt.ylim(0,8)"
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See what we get.
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This is starting to look a little bit better.
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I'm going add two labels as well with "plt.xlabel()" and then the string 'X', I'm going to give it a font size of
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16. And for the y axis.
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we're gonna just write f(x), so "plt.ylabel()" and then I'm going to give it the string
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"f(x)", font size is also equal to 16.
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There we go.
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All right so that's the plot of our function.
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That's pretty good, right?
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So we've got a cost function -
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sn example of one: x^2+x+1.
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And the way we've kind of broken this down is we've created a function, a Python function, right?
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Also called it f(x) - less confusing.
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We've created some data, and the reason we had to do this was so that we could generate a nice graph,
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right?
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So x_1 used numpy's linspace and then we've just used matplotlib again to graph our
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data here. In the next lesson,
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we're gonna set the stage for minimizing our cost. If f(x) is our cost,
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then the lowest cost will be at the bottom of this graph.
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Right?
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So it'll be somewhere around here.
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All we have to do now is find out what's the lowest cost and for what value of x is our cost
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the lowest.