(1) Generate 1000 samples of length 1000 drawn from the uniform distribution on the interval [0, 1].

Transcription

1 PRACTICAL EXAMPLES: SET 1 (1) Generate 1000 samples of length 1000 drawn from the uniform distribution on the interval [0, 1]. >> T=rand(1000,1000); The command above generates a matrix, whose columns (and rows) are filled with random numbers drawn from a uniform distribution on the interval [0, 1]. The same matrix could have been generated with the command "T=rand(1000);" the two indices in the example above allow one to generate non-square random matrices. A single uniform random deviate can be generated by just typing "rand" in your MATLAB window: repeated calls to "rand" will produce a set of random numbers. See "help rand" for more info.! Note that the command entry is followed by a semicolon; if you skip the semicolon, your matrix entries will be displayed in your MATLAB window (in general, you don't want this to happen, especially if the data set is large; however, the result of a single computation or the contents of small arrays can in principle be output in this way for example, you can type "rand" in your MATLAB window and see the resulting number).

2 ! Your workspace variables (a variable "T" and its dimensions in the example above) and command history ("T=rand(1000,1000);" in the example above) are displayed either in separate windows or on the left of the main MATLAB screen. You can also see your workspace variables in your main MATLAB window by typing "whos" (it will also take the commands of the type "whos T" and "whos T* a* b* c*" in the latter case, all variables that start with letters T, a, b, c will be displayed). To clear workspace, use command "clear"; you may also clear selected variables for example, "clear a*". MATLAB window is a UNIX window and you can use there UNIX commands such as "cd" to change a directory etc. To see the contents of the directory you are currently in, you should type "!ls". You can also use an upper-arrow key to get (within MATLAB) to a previously issued command (try it to issue the command "T=rand(1000,1000);" again). You can also explore other ways of displaying a MATLAB session and/or moving within your computer by playing with the Desktop layout options of the MATLAB "View" command on the top of your screen (at least if you are using Mac:). (2) Compute the average and dispersion of each sample as a function of sample size N. Use N=100, 200,..., 1000 and get the total of 1000 estimates of each of the two quantities for each of 10 sample sizes. >> for i=1:10 a(i,:)=mean(t(1:i*100,:));

3 end >> for i=1:10 s2(i,:)=var(t(1:i*100,:)); end The above example introduces several MATLAB operations setting up a for-loop cycle, working with matrix indices, as well as two intrinsic functions: mean and var. The for-loop cycle's text is placed in between the two commands: "for i=1:10" initiates the loop computations over a range of the loop index i; this command should be followed by pressing Enter and then you can type the cycle commands "a(i,:)=mean(t(1:i*100,:));" or "s2(i,:)=var(t(1:i*100,:));" (you could type several commands in one loop and separate them by pressing Enter). Finally, the cycle ends with the command "end" (then press Enter).! MATLAB is a vector-based software; the for-loops should only be used if the operations they perform cannot be written in a matrix form (as in the example above) the latter form produces a much faster computational speed. Two-dimensional matrices in MATLAB are stored in a column-wise order, so that a(i,j) refers to the i-th element of the j-th column. You can define a column vector as bc=a(:,j) or a row vector as br=a(i,:) (i/j are the indices of the column/row you would like to save as a vector); you can also form matrices of different size based on your original matrix, as is done in our example above: T(1:i*100,:) defines the matrix with 1000 columns and i*100 rows. The MATLAB intrinsic function mean(x) and var(x), if x is a vector

4 (one-dimensional sample of data), compute this data set's average and dispersion, respectively. In our case, the variable T is a matrix, so the result of applying mean(t(1:i*100),:) is a vector of dimension (1,1000) a row vector so that the averaging was also applied column-wise (you can also do row-wise averaging by using mean(t,2) and var(t,2); see "help mean" or "help var" for further details). Now you can see that the two for-loops exercised above produced exactly the data we wanted 1000 independent estimates of the average and dispersion based on random samples of size 100*i (i=1,2,..., 10) stored in vectors a(1:10,1:1000) and s2(1:10,1:1000).! Check out how your workspace variables list and your command history list have changed.! How to compute skewness, kurtosis or an arbitrary moment of a data sample in MATLAB? type "help skewness," "help kurtosis," or "help moment" to figure out! (3) Let's summarize: for our original samples of size N=100, 200,..., 1000, drawn from a uniform distribution on the interval [0, 1], we have 1000 independent estimates of the sample average a(n) and dispersion s2(n). We can now compute numerically the distribution of these derived quantities. The simplest way to do so is to plot the histogram (see "help hist"): >> hist(a(1,:),15)

5 The command above takes our vector data (in this case, 1000 estimates of sample means for N=100; "hist(a(2,:),15)" would plot the data that came from samples with N=200 etc.), computes its maximum and minimum values and divides the interval between these two values into 15 bins. It then counts, over the whole record, the number of data points with values belonging to each of the bins and displays the results as a box plot. Try plotting the histograms for the sample averages and dispersion using the data with increasing sample size (N=200+). Plot also the histograms of a dispersion. Pay attention to the fact that the range of the averages and dispersion decreases as the sample size increases both the average and dispersion become concentrated more and more near certain values as N becomes larger and larger.! If you type "hist(a(1,:),15) Enter hist(a(2,:),15)" in sequence, the first box plot will be cleared from the figure frame, and the second box plot will appear. If you want to compare the plots, you might want to call a separate figure before plotting the second histogram: "hist(a(1,:),15) Enter figure Enter hist(a(2,:),15)." Each figure will be assigned a number; you can edit the figure n by first pointing to it using the command "figure(n) Enter", and then typing the editing commands of your choice. For example, you want to label the axis and make a figure title (close all figures first): >>figure >>hist(a(1,:),15) >>figure >>hist(a(2,:),15)

6 >>figure(1) >>xlabel('a') >>ylabel('number of observations') >>title('histogram of a sample mean (N=100)') >>figure(2) >>xlabel('a') >>ylabel('number of observations') >>title('histogram of a sample mean (N=200)') >>set(gca,'xlim',[ ]) >>figure(1) >>set(gca,'xlim',[ ]) The last commands above set the x-axis limits to be the same (from 0.4 to 0.6) in both plots so that you could compare the two plots more easily ("gca" means "get current axes" and you can do a number of things using it change fonts and font sizes in the figure, colors, etc.). The histogram is not always convenient, since the total number of events depends on the sample size. A more convenient way to visualize our data would be to scale histograms to cover a unit area using command "histc" (see "help histc"). Here is an example of how to do it: >> mina=min(min(a));! min(x) computes minimum value of a vector or creates a row vector of column mins: min(min(a)) returns the entry of a 2-D matrix with the minimum (of all elements) value (see "help min") >> maxa=max(max(a));! maximum value of a matrix; we need the min and max values to determine the

7 range of our data >> da=(maxa-mina)/15;!divide the range into 15 equal intervals; da is the length of each interval >> edgesa=[mina:da:maxa];!define the vector of interval boundaries >> h1=histc(a(1,:),edgesa);!intrinsic function histc computes the number of observations (a(1,:)) that fall between the elements in the edgesa vector (see "help histc" for more info for example, histc can be applied, as everything in MATLAB, to matrices and tensors) >> whos h1 Name Size Bytes Class h1 1x double array Grand total is 16 elements using 128 bytes >> size(h1,1)!size command ans = 1 >> size(h1,2) ans = 16 >>h1=h1/(size(h1,2)*mean(h1,2));! normalize

8 >> mean(h1)*size(h1,2) ans = 1! Here is our unit area! >>figure >> plot(edgesa,h1)! plot the normalized histogram >> >> h10=histc(a(10,:),edgesa);! do the same for a different sample size (N=1000) >> h10=h10/(size(h10,2)*mean(h10,2)); >> hold on! retain the previous plot in the figure (the inverse command will be "hold off") >> plot(edgesa,h10,'r')! notice that we added color (default is blue). Type "help plot" for available line types and colors. >> legend('n=100','n=1000')! automatic figure legend >>xlabel('a') >>ylabel('p(a)') >>title('estimated PDF of an average of a sample of size N drawn from a uniform distribution on [0,1]') >>grid on! adds the grid (4) At your spare time, you can plot sample distributions of s2 in the same way. Notice that both the distributions of sample average and sample dispersion have a bell shape, with the top of the bell situated at the coordinate

9 corresponding to the expected values of a and s2; these numerically determined values appear to be independent of a sample size (as predicted by the theory). In contrast, the dispersion of the sample mean and sample dispersion (do not get confused: we are talking about "dispersion of a dispersion, " that is the quantity associated with the expected spread of individual samples' dispersions about their ensemble-mean value) both decrease as the sample size increases. Can we see how exactly these two quantities change? >>mean(a')! here is the expected value (ensemble average) of our sample means depending on the sample size (the prime is a transpose operator which changes rows into columns and vice versa) >>var(a')! dispersion of the sample average as a function of a sample size >>N=[100:100:1000]; sample sizes! vector of >>figure >>plot(n,1./var(a'),'x')! inverse of the sample dispersion as a function of N; the syntax 1./matrix means that the inverse of each matrix element is taken >>hold on >>plot(n,12*n,'r')! theoretical prediction

10 >>legend('numerical result','theoretical prediction') >>xlabel('n') >>ylabel('var{n}') >>title('numerical and theoretical estimates of the sample-mean\prime s dispersion as a function of a sample size N') >>grid on (5) You can also compute and compare the observed and theoretical values of the dispersion of s2. In the next lecture, we will derive the theoretical results for the expectation and variance of the sample's average and dispersion for a general case in which each observation is assumed to be drawn from a different (but known) theoretical distribution. It is a good idea to save your MATLAB session if you wish to be able to easily reproduce the results you have obtained previously. MATLAB saves your command history automatically, and it's possible to save it to a file with an extension.m, which you can then edit. For example, most of our present session could be saved in the file (matlab script) session1.m, which could look something like this: Session 1 (script session1.m) clear T=rand(1000,1000); you can put your comments here...

11 for i=1:10...or actually anywhere a(i,:)=mean(t(1:i*100,:)); end for i=1:10 s2(i,:)=var(t(1:i*100,:)); end figure Figure 1 hist(a(1,:),15) figure Figure 2 hist(a(2,:),15) figure(1) xlabel('a') ylabel('number of observations') title('histogram of a sample mean (N=100)') figure(2) xlabel('a') ylabel('number of observations') title('histogram of a sample mean (N=200)') set(gca,'xlim',[ ])

12 figure(1) set(gca,'xlim',[ ]) mina=min(min(a)); maxa=max(max(a)); da=(maxa-mina)/15; edgesa=[mina:da:maxa]; h1=histc(a(1,:),edgesa); whos h1 size(h1,1) size(h1,2) h1=h1/(size(h1,2)*mean(h1,2)); mean(h1)*size(h1,2) figure Figure 3 plot(edgesa,h1) h10=histc(a(10,:),edgesa); h10=h10/(size(h10,2)*mean(h10,2)); hold on plot(edgesa,h10,'r') legend('n=100','n=1000') xlabel('a')

13 ylabel('p(a)') title('estimated PDF of an average of a sample of size N drawn from a uniform distribution on [0,1]') grid on mean(a') var(a') N=[100:100:1000]; figure Figure 4 plot(n,1./var(a'),'x') hold on plot(n,12*n,'r') legend('numerical result','theoretical prediction') xlabel('n') ylabel('var{n}') title('numerical and theoretical estimates of the sample-mean\prime s dispersion as a function of a sample size N') grid on End of script

14 To run the script, change the directory to the one in which the script is stored (you can also add this directory to the MATLAB path see "set path" command of MATLAB menu) and type "session1 Enter"). You can now run the script however many times you want (notice that if you do it in sequence within the same MATLAB session, you will get different realizations of your uniformly-distributed samples and your results will be slightly different from one realization to another. The general result will hold, however, as your sample size increases, the distributions of sample-averaged quantities will become more and more concentrated in the vicinity of expected values, while the distributions of these quantities will be more and more Gaussian (we will discuss this important property shortly).