Let’s cover a few sound processing exercises in MATLAB!
So in this matlab video we’re going to do some exercises related to sound processing. So the first exercise we’re going to do is try and reverse the audio file, please pause this video and see if you can figure out how to do it yourself first. Ok, so, if you search on Google how to reverse a signal you see two functions that pop up, there’s fliplr and flipud. So now if we look at the sound file that we opened earlier, hello world from the last lecture, we see that it’s about 85,000 x 2. So this tells us that the samples are along the rows and each column is the left and right channel. Therefore, we want to use flipud not fliplr, because if you fliplr it’s just going to reverse the two channels, but if you flipldd it’s going to reverse the signal. So, let’s do that. So you see the shape is the same. If I plot d2 it’s not apparently obvious but this is the reverse of the signal that we had before. So now I want you to play this signal. Ok, so, I want to play this signal for you guys, but I have to save it first so that I can put it into the video editor that I’m using. I’m going to skip ahead a little bit and talk about the function audiowrite. So this is analogous to audioread, you pass in the data, and then you pass in the sampling rate. Ok, so, matlab does not support mp3 in audiowrite for some reason even though you can read mp3’s, let’s try mp4. Ok, so, now I’m going to read this file, I’m going to get the sampling rate also, going to play this back. So, that was hello world backwards. So the next thing I want to do is I want to speed up the sound waves so it plays twice as fast. I’m going to give you guys a minute to think about that, and so please pause this video and then come back once you think you’ve figured it out. Ok, so, the process of speeding up a sound wave is called downsampling. So we’re going to use an integer number like 2 so that it’s a little bit easier. So conceptually what you want to do is you basically want to drop every other sample of the original signal. There is some other algorithms and functions you could apply that theoretically might give you a better sound, but just to take every other sample is the simplest way. And so if you search for this earlier there is a method called downsample in matlab, ok. And so notice that we check the size of d4 the signal is half the length that it was before because we took every other sample. Now I want you to play this back with the original sampling rate. And so now it sounds like a chipmunk because it’s going twice as fast, and the pitch has increased. So if you think about a sine wave that doubles it’s frequency that’s the exact same thing that’s happening here. And now the next exercise I want you to do is theoretically maybe a little bit more complicated, but I think you guys can get it. So now that I have this downsampled version of hello world that’s playing every other sample of the original, how do I get it to play at the original speed so that it’s not high-pitched? I’m gonna give you guys a minute to think about that, so please pause this video and then come back once you think you’ve figured it out. Ok, so, the solution to this what we want to do is we want to play at half the sampling rate. “Hello world” and so this sounds pretty much like the original, you probably can’t tell that there’s some information loss, this has to do with a thing called compression. So if you ever study multimedia and you look into this field further you will learn more about that.
How do you open and process sound files in MATLAB?
In this tutorial we are going to talk about sound processing in matlab. For this tutorial there are no toolboxes needed. So the first thing we’re going to talk about is what sound is. So sound, as you may have heard, is a wave and it’s a wave of air particles, and so when you think of a wave usually we think of something like this like a sine wave, we call that a transverse wave. When we think of sound though it’s a variation in pressure right that bounces off your eardrums, so a wave like this doesn’t really make a lot of sense. There’s another kind of wave called a longitudinal wave, and this is the kind of wave that includes sound. So if we look at this visualization of a spring this describes what is happening to air as sound travels through it. Now interestingly enough when you record sound on a microphone and convert it into an electrical signal, that signal can be viewed as a wave like this. So, we again get a transverse wave. And so when we look at sound in matlab we’re going to be looking at a transverse wave even though in reality it’s a longitudinal wave. So, one thing you have to remember is that an electrical signal is an analog wave, and when we represent things in matrices on computers those are digital signals. So, for example, you could have a sound recording of say two seconds long and then that translates into a matrix of length 1000, so that would mean you have 500 samples per second, and so this process of sampling is what we call discretization. And so sampling is the discretization of data points along the x-axis, or in this case time. The other kind of discretization we have is along the y-axis. So, this is because numbers in matlab, numbers in any programming language, can only take on certain values. So you can’t have an infinitely precise number. This is because numbers are stored in quantities of say 32-bit or 64-bit and so we have to not only discretize the signal in time along the x-axis, but we also have to quantize the signal on the y-axis. And so both of these lead to some error, but in general say you’re playing a WAV file or you’re playing mp3 you don’t really notice these differences, they are imperceptible to the human ear. So now that we’ve covered a little bit of the theory behind what sound is and how it’s translated into a digital signal that we can read in matlab, we’re going to go ahead and open a sound file in matlab. But, before I do that I’m going to show you how I created the sound file. So there’s a free program on all major platforms linux, windows, and mac called Audacity. It’s free and open-source, you can download it just by searching on Google, and so if you have a microphone connected to your computer you can record things. So I’ve recorded myself saying hello world, hello world, and I’ve saved it to a WAV file. I have this WAV file in my workspace, and so there used to be a function in matlab called the wavread that you can use to extract data from a WAV file. Now notice I get a warning when I use wavread. When you search on Google for how to open a sound file this is the thing that’s going to come up. So, you’re going to get this warning and wonder what’s going on. So wavread is going to be deprecated and there is a method called audio read that’s going to be used instead, but this is currently not showing up in Google results. So let’s look at wavread first. So it only reads WAV files, and so if you don’t know a WAV file is just raw digital sound data. So we sample the sound, and we quantize it, and then we save each of those data points as an array essentially into the file. So there are different method signatures that we can use with wavread, we can get just the data which I’ve done or we can get the data and the sampling rate. So commonly we use FS to refer to the sampling rate the, F stands for frequency, S stands for sampling, so the sampling frequency. Now there’s some interesting things we can do once we have the data. So I can plot the data, not sure where it went, there we go. So you can see here it pretty much looks the same as what I had in audacity, also check out the size of the matrix. So there are 84,480 samples just for that maybe one second of data, and notice that the other dimension here is two. If you think about why that is let’s go back to audacity, there are two channels. So I was recording in stereo right, so there’s the left side, and the right side, so that’s why there are two dimensions to this matrix. Now another interesting thing you can do is play sound. So I can actually play this file back but for that I need to so there’s a method called sound in matlab that you can use to play sound, but we need to use a different method signature for a wavread in particular. We need to get the sampling rate. So I’m going to use wavread hello world and return both the data and the sampling rate. So I do sound, pass in the data, and pass in the sampling rate, “hello world” and it plays back the sound. So now you might be wondering what will happen if I try to read a mp3 file. So I’m going to save this as a mp3, the exact same file, so keep in mind that wavread of course only works with wav files. If we want to read mp3’s or any other kind of compressed sound file you should use audioread, the new function for reading audio in matlab which can also read waves by the way. Okay, so now let’s say one play this “hello world” and so it sounds exactly the same.
Creating subplots, 3-D plots, and labels in MATLAB
So in this lecture we are going to talk about more plotting. We’re going to talk about subplots, so that’s putting more than one plot into the same figure, we’re going to talk about three-dimensional plots, and we’re going to talk about how to label plot, so you want some field for the x-axis, the y-axis, and the title. So, let’s go into subplots. So suppose I have some
function of X, so let’s just say Y = sin(x), then I have another function which is just Z = Y + randn(1,100); Ok, so I can plot these individually, I can plot them together, I might want to stack them up or something to see if there is a correlation between them, so the way we do that is with subplot. So we call a function called subplot and it takes in three values. Ok, so the three values are the number of rows in the subplot, the number of columns in the subplot, and then which subplot you’re about to plot. So since I have two subplots let’s say I want two rows in one column and I want to set the first plot. So I just called subplot(2,1,1); and I say plot(X,Y); Ok, so now my figure has X,Y in the first row and first column. So now I do subplot(2,1,2); because I still have two rows in one column, but now I want to plot something in the second slot. Here I’m going to plot X and Z. So now if I look at my plot you can see the two plots stacked up on top of each other instead of overlaid. So the next thing we’re going to talk about is 3D plots. So suppose we have some two dimensional data set I’m just going to make it random noise. Okay, so we look at the matrix Z, we see that it’s 100 by 100, so the rows you can think of as the x-axis, the columns is the y-axis, and then the value as the third dimension. So how would we plot this? So there are a couple ways you may have seen already from your math courses, so we can plot an actual 3D plot so sort of how you would draw a cube on a two-dimensional piece of paper, or you could use what’s called a contour plot which draws lines where the value of the function is equal. So we’re going to do both of those. The first one is a function called surf, so you pass in the matrix and so you can see the x-axis and the y-axis both go from zero to a hundred. If you look up the documentation you can set the x-axis and y-axis manually, and then the height is the value of the function. So what I did just now as I clicked on the this thing that sort of looks like a circular arrow and so what that allows me to do is it allows me to rotate the surface plot, so now I can look at it from different angles which could be very useful if you’re doing data analysis. The other type of plot that we talked about is the contour plot. So the simplest way to call that is with the one argument and that generates just a 2D visualization of your data, the red is the really high values, and the blues the really low values. So the third thing we’re going to talk about today is how to label plots. So far we’ve only seen plain visualizations, so let’s go ahead and plot the sine function again. Alright, so I want to give this plot a title, all I do is use the title function title(‘A plot of sin(x)’) Ok, so now if I look at my plot you can see it now has the title A plot of sin(x). Now I want to label the x axis I’m going to label it with the very unoriginal label. Ok, so now you can see the x-axis is labeled and you can do a similar thing for the y label as well. Ok, so that’s pretty much it for labeling, suppose I want to label a subplot. Ok, so now I’m going to create another function, going to subplot(2,1,1), plot(X,Y), title(‘sin(x)’) Ok, so once I do that the first plot, first subplot gets the title sin(x). Now I call subplot(2,1,2), I plot(x,z) and I call title again so it’s the same function but it’s stateful so it knows that I called subplot earlier and that it corresponds now to the second plot. Ok, so I call title again and now my second plot has the title cos(x), and my first plot has the title sin(x).
So in this tutorial I’m going to talk about loading and saving data. So a lot of the time when you’re doing data analysis your data is going to be passed around and files that you might download off the internet are stored on your hard drive or something like that, so you’re not going to be manually typing in matrices into matlab all the time. The first thing I want to talk about since it’s probably going to be really popular with Windows users is excel. So Excel uses a format called .xls and there are two functions in matlab that are useful for reading and writing .xls files. Ok, so if you type in help xls read you can see the documentation for the xls read function. There’s also a function called xls write that you may want to look into. So the reason I’m not going to cover xls read and write in this tutorial is because first of all it’s not perfect. So, I saved I don’t I’m on a mac so I used Google sheets to export and .xls document and it’s a very simple document, it’s just the three by three matrix with the numbers 123456789 in matlab. So xls read matlab example gives me an error when I’m trying to read this file, so we’re not going to be working with .xls. In addition to that, so if you’ve ever participated in a Kegel contest, or you get a file to work with from school, I’ve never seen anyone ever use .xls. Usually it’s a .csv, sometimes it’s a .mat file which is a matlab file. So we’re going to be working with those. So .csv I would say is the most common, it stands for comma-separated values, so your file will look something like this 123456789, and I’m going to save this CSV example .csv. So we have a function called csv read csv example, ok and so that did exactly what I wanted it to do. So you can see how the csv file pretty much one to one corresponds with the matrix that you would get after you read the file. Now let’s save a new matrix, let’s create something a little different. Ok, so just fives along the diagonal. I’m going to call csv write the another.CSV, so I did it backwards because I didn’t know. So I’m going to write the matrix B to a file called another.CSV. So if I look at this file…alright so when I double-click it matlab gives me kind of a CSV editor. So now what I’ve done is I’ve opened the same file in a text editor on the command line console and so you can see again what we’d pretty much expect a CSV of the matrix that we made. So that’s the nice thing about CSVs is they don’t just work with matlab they work with many other types of programs, it’s just a plain text file. So now the other way that we can read and write files is the .mat file which is matlab specific. So I’m going to clear my workspace, you’ll see why. I’m going to create one matrix A = [1,2,3,4], oops. Going to create another matrix B = [5,6,7,8] Ok, so now i have two matrices A and B so now what matlab lets you do is it allows you to save your entire workspace all into one file, and so all I do is I call save my workspace.mat, okay. So now if I look at this file, so I open this .mat file in a text editor, notice how it just shows me a bunch of garbage. So this is binary data specifically for matlab, so that’s why you might prefer to CSV so you can be cross-platform or cross program. So I’m going to clear my workspace again and I’m going to call load my workspace.mat and so what load does is it just reloads your entire workspace that you saved into the .mat, and so that could be useful if you don’t want to save every variable individually into their own CSV. What you can do is you can if you don’t want to save all the variables you can pick specific ones so let’s say I create another matrix C = [9,10,11,12] Ok, so I have variables A, B, and C but I only want to save A and C, so I’m gonna give it a file name AC.mat, I’m going to pass in so you can’t pass in the variable you have to pass in the name of the variable, so I’m going to pass in A and C to save it, I’m going to clear my workspace, and so now when I load AC.mat it just gives me back A and C only but not B.
What are the best plots for data analysis in MATLAB?
So in this tutorial we’re going to be talking about different types of plots that could be useful for data analysis. So the first type of plot if you’ve ever used Excel or watched a PowerPoint presentation, you’ve probably seen this probably seen most of these actually, so the bar plot. So suppose we have some sequence of data let’s, just say x equals 1 to 10, and we want to plot this on a bar chart. We just call the method bar, pass in X, and we see the bar chart. So this does pretty much the exact same thing as a plot function except instead of connecting them all in a line it connects well it doesn’t connect the points at all it gives them each their own individual bar. So, just for reference let’s do let’s do our sine example again. Ok, so we’re going to do bar( X, Y), right, so this is a bar chart of the sine function. Ok, so the next type of plot that I want to talk about is the histogram. So if you’ve ever seen, if you’ve ever studied probability you know about probability distributions and given a data set we might want to figure out what kind of distribution does it follow, so let’s generate some data where we know the distribution. So I’m going to create a thousand points that are normally distributed. So if I plot this it doesn’t do much it just looks like noise, so I won’t be able to see much from that. What you want to do is you want to call a function called hist which will automatically create a histogram out of your data. So I’m going to call the simplest version which is hist of just the data right, and so you see that it’s pretty much normally distributed. There’s a little bit more weight on the right bar than the left bar in the center and so the resolution for this chart is not that great. So what you can do is you can set the number of bins right, so if you have a lot of data, in our case a thousand points, you can set the bins to maybe say 50, ok, and so this gives us a much more granular representation of the data. So now we can see it still looks normal, you know there’s a little less data than expected kind of right here at zero, and then a little more to the right of that. But, generally speaking this plot is pretty much what we expect. The next type of plot that I want to talk about, again if you use Excel you’ve probably seen this or PowerPoint, is a pie chart. So we’re going to work with some simple data again let’s say x is 1 to 5, and all I do is call a function called pie, pass in the data right, and so it automatically splits up the data. So the dark blue is the first element, light blue is the second element, green is the third element, and so on. And so there are ways that you can label this to to make your visualization better and so when you show it to people they know what everything stands for, so there’s a lot of documentation on the matlab website for that. The last type of plot that I want to talk about, probably the most useful for data analysis, is called the scatterplot. So we’re going to again return to our sine example, so x = linspace(0,2*pi,1000); So now I’m going to do something a little different, I’m going to set it y = 10*sin(X) + randn(0,1000) and then I’m going to add some random noise with variants 1 centered at 0, and so the reason why I want to scale it by 10 is so that the sine wave is big compared to the noise otherwise the noise will just take over the shape of the sine wave. I did that wrong. Ok, so now if I were to just plot x and y so you can see a noisy sine wave which is ok, if I do it in a scatterplot I can see that pattern pretty much just as well. So the nice thing about our sine X example was that the data was ordered. Now suppose that my data is not ordered so instead of having linspace my X is just a bunch of random points. So X is normally distributed with a standard deviation of 2, and so Y is going to be let’s say 5*sin(X) plus some noise. Ok, so if I plot x and y now I don’t really see anything it just looks like a bunch of craziness, but scattering so a scatterplot that will just plot all the dots so all the individual data points right, and so I can now very clearly see the sine wave that I’m supposed to see. So if you’re doing data analysis and your points are not ordered, you want to use a scatterplot.
So today we’re going to learn about how to make plots in matlab. So one of the very simplest plots you can make is just to plot a series of data, so one very popular series is the Fibonacci sequence and that’s just adding the two previous numbers to make the current number, right. So here are the first 8 values of Fibonacci, and so to plot in sequence all I need do is call a function called plot and pass in the variable right, and so that’s Fibonacci. Now notice how the X values have been set automatically, so starts at one and increases sequentially. So the next thing we can do to make it a little more complex is to have actual values for X. So suppose I want X to increase by point one every time, so I might do 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7. So now I have an x and y which is what you’re probably more used to when you’re creating a two-dimensional plot. To plot x and y you just pass in x and then y into the plot function, right and so now you can see the y-axis is the same as before but the x-axis is now no longer 1-8 its 0-0.7. Notice that the size of both x and y have to be equal, right so if I do this it is not going to work right. Alright, so the next thing we’re going to do is something a little bit more complex, we’re going to plot a function. So the first thing we need to learn about is a function called linspace. Linspace what that does is it creates the range of values of x for us so we just pass in where we want it to start, where we want it to end, and how many points in between. So I’m saying X is going to go from 0-100 and has 200 points in between. Now to calculate Y all I have to do is sin(x) we’re calling the sine function. In matlab functions generally apply element-wise right so every value of x you call sin(x) and then it assigns the value of that sine X into y. So I’m going to plot x and y again, and so now I see sine X going from zero to a hundred. So one thing you might notice about this is the lines are pretty jagged and it’s not a smooth graph but you know that sine X is actually a smooth graph. So the reason why that is if we look at X again let’s look at the values remember there are 200 points for about a range of 100, and so that’s a very low resolution. The last value here is 100, the second to last values about 99.5, so there’s only about 0.5 distance for every unit of X. So to give us a better granularity we might want to say you might want to reduce the range first of all and then put more points in between. So, let’s recreate X let’s say it’s going to go from 0 to 2 pi and have a hundred points in between, alright, so y is going to be exactly the same when a plot XY again. Alright, and so now you can see that it’s a lot more smooth. So adding to the complexity of this a little bit suppose I want to plot two functions on the same plot and that might be useful if you want to compare two different series, so let’s say the second function we multiplied is cosine of X say y2 = cos(x) and so we type in plot we can see some different function signatures for plot. If you look at the third one it shows us x1 y1…xn yn, so what that’s telling us is that for every pair x and y that we want to plot we just pass the min adjacent to each other. So in our case we’re going to have x and y, and since it’s the same x for both y and y2 we’re going to just pass an x again and say y2 over there. Alright, and so now you can see both sin(x) which is the blue line, and cos(x) which is the green line on the same plot and matlab automatically changes the colors for us so we can differentiate between. One thing you can do if you want to make it a little more fancy is you can specify a line style and so those are also passed in adjacent to the x and y’s, so now it’s even easier to differentiate between the two lines. So the second chart cos(x) is a bunch of dots, and I made a mistake the dashes the default that just makes a regular line. If you put in two dashes it makes them into dashes that you can differentiate between like that. Ok, so now let’s say I want to save this plot to my hard drive or maybe put it on the internet or a document, so you go to file save as. So you see the default file format here is .fig and that is a matlab file so only mad matlab can read those files. There are some other types of formats here that we can look at such as PDF. If you want to save it to a PDF and then print it out later. You can save a PNG or jpg so if you want to put in a word document or you’re writing a paper, or you want to put it on your blog post you might want to use JPEG or PNG, and so I’m going to use .fig since you probably already know what a PNG and JPG look like. Alright, so I already made a test.fig overwrote it so now that I have it, so you can see right here saved into my current working directory. If I want to open an existing figure I do that through a function called openfig, I pass in the filename and it opens back up the figure. And so the nice thing about matlab figures is that I can still zoom in, I wouldn’t be able to do that really with the jpeg would look very blurry, of course i can zoom out, I can use the hand to move around the chart. So, it opens back up the same figurehead before so I can play with it as if I had made it for the first time.
In this matlab tutorial we’re going to be talking about different kinds of data types, so things that are not what we’ve seen so far such as numbers, scalars, vectors, and matrices. So one thing that we’ve been implicitly looking at are strings. So whenever I display something like this and I put it into single quotes the thing that is inside it is a string. Later on when we look at opening files we will specify the file names as strings. So a string is basically any sequence of characters and so we’ve seen the workspace but there’s also a function called whos that lets us look at all the variables in the workspace and see some stuff about it. So you’ll notice a column called class. So far everything I’ve created is a double so we have no actual strings yet. If I use the class function and I look at an actual string, let’s just use the I found it string again, you’ll see that it returns char. So if I set a variable called my_string = ‘I found it!’ and I type whos you’re going to see now that there’s a variable with the name my string size 1×11. So strings are considered row vectors and it’s of type char. So the next topic we’re going to talk about is structs. We’ve seen so far in matlab numbers, matrices, vectors, arrays, strings, but that doesn’t leave a lot of flexibility compared to other programming languages. So if you’re familiar with Python maybe you’ve seen the dictionary data type, or if you use Java it would be the hashmap, or if you’ve coded in javascript you may have used json. If you’ve ever used any of those before you know that those data types are very flexible because a json can store other jsons, it can store arrays, whereas in matlab we have so far been constricted to only numbers. So in matlab when you want to declare struct it’s very easy so call our struct my_struct.name = ‘My new struct’ Ok, so this creates a struct with one field which is referred to as name and the value is my new struct. If I do class on my struct it tells me that it’s a struct. So the cool thing is you can add new values to the struct dynamically. So I can say my_struct.age = 25, so now my _struct contains two fields name and age and notice that they are different types. So if I do class(my_struct.name) it’s a char, but if i do class(my_struct.age) it’s a double. So there are also some useful functions that act on structs. So is field will tell you if a field exists within a struct, so I just checked if the field name was part of my struct and it returned one because it is. Let’s try gender. So that returns false because gender is not a field in my struct. There’s a method called rmfield. So if I want to remove a field from a struct, I spelled it wrong, so now my struct only contains the field name because I removed the aid field. There’s a function called set field so you might want to use this if for example the name where the value is set dynamically while the program is running, so you can’t do my_struct.something you have to set the field some other way and I spelled field wrong again, so now you can see I’ve added a field called gender with the value F. So the other thing I talked about is that a struct can actually contain a struct. So suppose I want to add a new field to my struct called contact and I want to give a phone number to my contact. So I’ve set the phone number on this struct and so you can see when the struct prints out now the contact field is just a struct but it doesn’t print what’s inside, and so now since contact is a struct I can set other fields as well. Ok, so my struct now contains the contact field but as a struct and the contact is also instruct. So the last thing about structs that I want to mention is an alternative way to initialize a struct. So far we’ve been manually entering each struct value one at a time but you could also do something like this, S = struct(‘name’, ‘Bob’, ’email’, ‘[email protected]’) right and so what that does is it takes in an arbitrarily long list of parameters where the first parameter is a field, the second parameter is a value, third parameter is a field, the fourth parameter is a value, and so on. So the last data type that I want to talk about today is the cell. So the cell is very similar to a struct in that it contains arbitrary fields, the difference is you you might see a cell as more of like an array that can hold indices other than just numbers versus you might see a struct as more of an object that contains information about some some type of thing, and then a cell might be you know like an array of those things. So the cell syntax is very similar to the matlab matrix and array syntax. Ok, so to initialize our cell we’re going to use curly braces we want to set the field to 1, doesn’t have to be an integer, going to set it to the value hello world, ok. So now if I print out my cell it simply contains hello world. Ok, so now i’m going to set another field, I’m just going to call it A and it’s going to be a matrix. Ok, so now I have two keys and two values for my cell. I have the key one in the key A, and then the values hello world and then this two-by-two matrix. So what I can usually do with matlab variables just type it into the command window and see what it prints out. You can see that it has printed out a lot of stuff. So you generally won’t want to do that but notice how now I can access key 1 and it returns hello world, and I can access the key A and it returns the two-by-two matrix that I previously stored. So this is good for when you want to store things by name for example instead of trying to put it into an array, and you can also store heterogeneous information so different types.
So in this lecture we’re going to talk about for loops, while loops, and if-else statements in matlab. The first thing I’m going to talk about is the basic syntax of a for loop. So the basic syntax of a for loop is for i= some array of numbers, so it can be anything. So, for argument’s sake i’m going to use just a bunch of random numbers [1 3 5 4 10 7] and then I’m just going to display them to the output. Ok, so all that does is it sets i to the first number for the first iteration, it sets i to the second number for the second iteration, and so on. We’ve seen before that matlab can create ranges with the colon though which is probably a more common use of the for loop, so you want to count from one to a thousand or 1 to 10 and so the way you would do that is for i=1:10 which actually in MATLAB is generating the array 1 2 3 4 5 6 7 8 9 10. So I’m just going to display(i) again and you can see that it goes through every value of i. So now that we have the basic for loop syntax in matlab, what I want to do is go through a little more complicated example. You’ve heard of a concept called the mean squared error. If you haven’t let me show it to you on Google. So the mean squared error is if you’re trying to predict something you’re going to have some error for all the data points between the true values and the predicted values, and so the mean squared error basically takes the difference between your prediction and the actual value squares them all, sums them together, and divides it by the number of points. So you can see how we might be able to use for loops to implement that. So now since we don’t have any data or any error we’re just going to pretend that we do. So I’m going to use the rand and function to create an array of size 1000 by one, these error points are going to be normally distributed which is often an assumption that we make any way with statistical models. Ok, so now I have a 1000 length array of errors which represents the difference so i don’t have to calculate the difference in this equation, and so now we can go ahead and calculate the error. So the first thing we want is the sum of square error, right, so that’s just the sum of all the individual elements. So I can do that by first initializing the sum of squared error to zero, and I’m going to do a for loop from 1 to 1,000 to accumulate the sum of squared error being a square of each individual error. Ok, and so now that I have the sum of squared error I’m going to divide that by 1,000 to give me the mean squared error. So one thing to note about MATLAB is that for loops, while loops and if statements are actually very inefficient, so you don’t want to use them unless you absolutely have to, and so I want to demonstrate how that is the case. So we have this mean squared error example, I’ve pre-written the code, and I use a statement called tick and talk to time how long it takes. So you basically you start your code with tic, you put all your actual code after that, and then you put a talk at the end and then it will
tell you the elapsed time it took to run that code. So I’ve already pre written it so i’m going to paste it out here, tic t-i-c, toc t-o-c, ok. So that took .005668 seconds. Now if we look
again carefully at the mean squared error we notice that we want to square each element and then sum them all together, and so that sounds a lot like the dot product, right. So if you take the dot product of a vector with itself you’re just squaring each element and then summing them altogether. So if I wanted to do the mean squared error equation in a more compact way, what I
might want to do is the dot product between e and itself which is e transpose times e as a matrix and then divided by a thousand, right, and so that gives us the same answer as before, and so if I put a tic and a toc around that I get .002896 seconds. So it’s significantly faster than using a for loop. Ok, so that’s for loops, now we’re going to talk about if statements. So what I want to do is I want to visit I want to revisit the is even problem that we talked about in an earlier tutorial. So if you recall what I did was I created a function that returns one for all the entries of X that are even and return 0 otherwise. So you could do this with if statements let’s say X = 1:10, alright and we want to say Y is initialized to a an array of zeros, and we will do for i=1:10 if mod(x(i),2) == 0 then x is even, right. So, that means we’re going to set Y(i) = 1; else Y(i) = 0. So we don’t really need that else since Y(i) is already 0, but I’m including it so you know what an else looks like. Okay, so now if we look at Y has the ones where we expect it which is where x is even. Ok, so now let’s do a similar example also using for loops and if statements. So the example is the problem is let’s say I want to sum all the elements of an array that are divisible by 3, so let’s just use the same X as before, let’s reinitialize Y to all zeros, sorry we don’t even need to do that, let’s set S = 0; and so now I’m going to loop through all X’s again for i=1:10. So if mod(X(i),3 == 0 so if it’s divisible by 3 we want to sum that element. S = S + X(i); and now this time we’re not going to include the else. So is 18 which is equal to 3 + 6 + 9 which we expect so that’s the correct answer. Now of course there is another way to do this, so if you only want to look at the elements in X so notice how whenever we use i we’re actually saying x(i) we can just do this for x=X right because the little x is going to just go through every element in the big X. We were going to start again so we’re going to reinitialize S = 0, say for x=X,if mod(x,3) ==0, S = S + x; end. So if we look at sum again we still get 18. So now let’s look at another problem. So suppose we want to find an element within a matrix. So suppose again we have the same X and I want to find the number 8, so I can use a while loop for that. So I can say I can set a variable called found = 0 meaning false, and then I can say while not found, so I’m going to initialize an index i = 0, gonna say while ~found, i = i + 1 if X(i) == 8 which is the thing I’m looking for. disp(‘I found it\n’); end Ok, so I made a little mistake there we went out of bounds because i went to 11 and the size of X is only 10, so what we did was we forgot to set found to true after we found x(i) == 8. So I’m going to reset this code, i = i + 1 if X(i) == 8, disp(‘I found it!’); set found to true found = 1; now end the if statement, right. So now we go up to 8, printout i found it, and then we quit the while loop. So while loops only go until the condition after the while is false, so it will continue to go while it’s true, and then when it’s false it will stop. Another way we could do this though is we could again use a for loop. So generally speaking whenever you can use a while loop you can also use a for loop, it’s just the syntax that will change and a little bit of the structure of the code. Suppose I want to look through all of X again, so I want to say if X(i) == 8 disp(‘I found it!’) So now this works but there’s a problem right because in the original while loop we would only go up to 8 and then we would quit, versus this for loop goes from 1 to 10 so that’s inefficient because it’s doing extra steps. So what we can do to avoid that is use the break statement. So just to prove to you that it works i’m going to print out for i=1:10 so again if X(i) == 8 disp(‘I found it!’); break; and so now you see I only goes up to 8. It finds 8, it breaks, and then it doesn’t go to 9 and 10.
So in this matlab tutorial we’re going to talk about some basic linear algebra in matlab. The first thing I want to talk about is how to solve linear equations. So in general these are in the form of A(X) + B where A is a matrix, X is a vector of the variables you want to solve for, and B is a vector of constants. So if you remember from linear algebra a system of linear equations is when you have more than one variable and the same number of linear equations. So, in this example I’m looking at right here the variables are X, Y, and Z and because there are three of them I have three different equations. So one way of writing the problem is just writing the equations out explicitly like this, or you can write it out in matrix form. So, in this example A = [3 2 -1; 2 -2 4; -1 0.5 -1]; OK, so that’s A, b = [1 -2 -2]; ok. So, when you’re working with equations like this you might be tempted to do just simple algebra, so you could say x = inv(a)*b because mathematically that works. So, I can do that oh and right I have to make b a column matrix, so a*b. Sorry, I used the wrong b so inv(A)*b so I get 1 -2 -2 which is the answer given on Wikipedia. So sometimes when you try to do something like this matlab will give you a warning this is a very inefficient way of calculating this answer. It’s matlab already contains algorithms to solve this problem. So if you remember from before when we were talking about operations we were talking about the division operations and how there’s a forward slash division and a backslash division. So, this is the use case where you want to use the backslash division. I mean if you kind of think about it if these were scalars you know x would be b/a, but since they’re matrices we write the answer like this so A\b. So, it’s kind of like A is underneath the b, right, and so if I do that you can see that I get the same answer as when I did inv(a)*b. So that’s how you solve systems of linear equations in matlab, So now we’re going to talk about some other linear algebra concepts that you can do in matlab. One of them is taking the determinant of a matrix,, and so that is just a simple function called det, ok. So remember A = [3 2 -1; 2 -2 4; -1 0.5 -1]; from the last example we can also compute the eigenvalues of a matrix. So, iaeig and now you know when we compute the eigenvalues we usually want the corresponding eigenvectors. So one interesting thing about matlab that differs from other languages is that what is returned by the function is not set in stone, its dynamic. So if we look at the documentation for the eig function we see that if there is one output parameter it returns this little e, and the matlab documentation says e contains the eigenvalues of the matrix a. But, say I want the eigenvectors also so you can see these different function signatures. So if I assign the returned values of eig to two different variables I can get back let’s read here a diagonal matrix D of eigenvalues in a matrix V whose columns are the corresponding right eigenvectors. So the eigenvalues are going to be returned in a matrix called d, and the eigenvectors are going to be a columns of the matrix V. So let’s try that. [V,D] = eig(A) so we get a diagonal matrix D which has the same eigenvalues as before, and then the eigenvectors are all in V. So let’s just confirm that the definition of eigenvalues and eigenvectors holds, so A*V should equal V*D right, and so here’s another thing interesting about MATLAB is that if we want to check if two things are equal we can generally use the == as we do in regular programming. So 1 == 1 is 1 which means true, 1 == 0 is false. So if we do A*V == B*D we get false which is surprising because they should be equal given that that’s the definition of eigenvalues and eigenvectors, but we have to remember that there is always in programming some round off error. So if we subtract A*V – V*D we should get back a very small number, alright. So, we do that we see 1×10^14 which is indeed very small, so for all intents and purposes they are still equal even though equals equals returns false. Now so that brings us to the next thing I want to talk about which is so we have this matrix A times V minus V times D which kind of represents all the differences in those two matrices, and so we want to ensure that all the values are small. Now there are nine different values in this matrix so it’s kind of hard to tell how big or small it is collectively. So one thing we want to do sometimes when we have vectors or matrices is we want to calculate the distance from the origin, and so we can do that with vectors with matrices we call that the norm, and so typically all the elements so in a euclidean space for example it’s all the elements squared, added together, and then you take the square root. So in matlab there’s a function called norm that does this. So let’s just see what that gives us. So it gives us one very small numbers, so 5×10^-15 which we again can use to confirm that it’s very close to zero, so A*V is very close to equal to V*D, and if you remember from your math courses there are different kinds of norms. So I talked about the Euclidean norm which is a norm 2, we can actually do a p norm which is so instead of taking the square and summing them all together and taking the square root we do every element to the power of p, sum them all together and take the p of root, and that’s just the second argument of norm. So if I put a 2 there I should get the same answer because p equals 2 by default. So you can’t use P =3 you can only use P = 1 to infinity or fro, and so one would be the Manhattan distance.
QUESTION: What are the popular builtin functions/constants in MATLAB? How do you define your own functions?
So in this tutorial we’re going to talk about matlab functions and constants, so built-in functions and constants to be specific, and then how you can define your own functions later on. So by functions what I mean is mathematical functions like sine, cosine, tangent, exponential log, and square root and so those are pretty much what you’d expect if you’ve ever coded in any other language before. So I can do sine of 0 is going to be 0, I can do cosine of 0 gives me 1, tangent of 0 of course would be also 0. So let’s look at these trigonometric functions for a little bit. So one thing is that because matlab works on matrices I can actually pass in more than just one number into sine. So suppose I use the matrix grid before 1,2,3,4, if I pass that to sine what that does is it calculates the sine of every element of A and returns a matrix the same size as A. So I can do all that separately of course just to show that it is indeed doing that individually for every element and so all matlab functions generally work like that. Some other functions you might want to use is the exponential function that’s exp, there’s the log which is the opposite of the exponential. If I do log of exp of one I should get back one. I can do square root is sqrt ok, so those are some very simple elementary functions in matlab. There are also some built-in constants into matlab that you should know about. So, for example, pi that’s just pi. So if you do something like have a variable called pi and you assign it to something else, now I cannot use the original variable pi anymore. So what I would do if I want pi back is I would go into my workspace and delete pi alright, and so now if I type pi again it’s now back to original pi. Generally, I would avoid overriding built-in constants. So, one weird thing about MATLAB is it doesn’t have a variable e, so if you want the variable e you actually have to do exp of one which is e to the power of 1 which is e. Another important constant is i. So if I do the square root of minus one I get 0 plus 1 times i. I can also just type in I and it would give me the same answer. So you can also do things like i*i which should give you -1, or you can do i^2. So you do exponents using the hat symbol so that also gives me -1. So there are also some important functions for initializing matrices that you should know about. So the first one is the I function. So you’ve seen I in the sense that if I have a matrix A then A*I should equal A, and so I is the identity matrix and it’s a square matrix with ones on the diagonal and zeros everywhere else. So suppose I want to create a 3 by 3 identity matrix. I could do something like this which would be a little bit inefficient right, or I could do eye(3) which gives me the same thing. Another thing you might want to do is create a matrix of all zeros, so that’s just the function zeros. So that gives me a three-by-three matrix of all zeros. If I want to specify different dimensions, so I don’t want it to be a square, what I can do is I can pass in multiple dimensions. There’s also a function called ones which does a similar thing it creates a matrix of all ones. You can pass in the size the same way. So the next thing I want to teach you guys to do is how to define your own functions. So generally you want to for every function you create you wanna put it into a new file, so I’m going to go new function. So it gives me some boilerplate code that I could use and this is good because it will help me explain the syntax of a function to you. So when we create a function it starts with the word function. If your function returns things you can specify the output arguments over here. If there are more than one output argument you can specify it as an array. If there aren’t any output arguments you can get rid of this part completely, and if there’s only one output argument you can get rid of the brackets. So let’s suppose I want to define the hyperbolic sine function which matlab probably already has but we’re going to do it anyway. So it’s going to take one input argument called X and take an output of the Y. So matlab has some useful syntax highlighting it will give us a warning you know stating input argument X is unused since we haven’t written any code yet, and then it’s returning a value Y but we haven’t set Y to anything yet. So let’s save what we have so far, you can do command s or go to file save. So we’ll save it. So one thing to notice you have to save the function name has to be the same as the filename, and so like I mentioned before matlab already has its own sine H so we don’t want to overwrite matlab sine H. So we’re going to call our sine H something else, I’m going to call it my sine H. I’m going to rename this file to my sine H. Alright, so now I’m not getting that warning anymore. Ok, so we’re going to define our own hyperbolic sine which is defined as the exponential of the input, minus the exponential of the negative of the input, and then taking one half of that, so let’s do that. So inside the function I have to set Y remember because I was getting a warning that Y was not set. Remember, I want to put a semicolon here so that I don’t see the output of this expression. I’m going to get rid of this comment because this function is pretty trivial but in general if you have a complicated function you should probably comment it so you remember what you were doing later on or whoever else is reading your code will understand what you’re doing, and so you can write comments using the % sign to comment code. So I’m going to save that, I’m going to use this function so my sine H let’s say one, zero so we can see from this image that hyperbolic sine crosses zero and then it increases as x increases, so that’s what you should be seeing. Remember that matlab works with matrices though so if I wanted to I can say X goes from 0 to 10 and I want to calculate the hyperbolic sine for all of those values I can just pass in X and I’ll get back hyperbolic sine for all of those values. So that’s how you define your own function, probably yours in the future will be more complicated than that but hopefully that was a useful simple example. So we’re going to do one more function example here it will illustrate some new concepts. So I’m going to instead of going clicking the new button and choosing function I’m actually going to right-click on the workspace and I can see an option to create a new file and I can create a new function that way. So I’m going to call this new function is even and so you can see matlab has already replaced so before when we created a function from a new file it called it untitled, but now that I’ve given it a name to start matlab already knows to change the function name to that name. Ok, and I’m only going to have one output argument again so I can get rid of these brackets. The input argument is going to be X, and I’m going to add a comment returns 1 if X is even. Ok, so if you’ve done programming before you’ve probably heard of the modulo operator although I haven’t mentioned it in this tutorial series yet it does exist in matlab, so in matlab because the % sign is used for comments we do not use the % for modulo, if you’ve done that in c or java before. So instead we use modulo by calling a function called mod. So, basically it returns us the remainder. Ok, so this should be very simple. We want to return 1 if X is even so we know we want to mod 2, so y = mod(X, 2). Ok, but we have a problem here because when you take the modulo of an even number with two you will get 0, but if you take an odd number modulo 2 you get one so that’s kind of the opposite of what we want, but what we can do is we can use the utility operator which does a logical negation. So, until the 0 gives us one, until the one gives us 0, so what we want is to put the tilde in front of the mod, ok. And so now if I call is_even(1,2,3,4,5) we should get ones in the spots where the input is even which we do. So for 2 and 4 we get 1, for 1, 3 and 5 we get 0.
