echo on
% ***********************************************************************
% INTRODUCTION TO MATLAB
% ***********************************************************************
% Matlab is a system for doing numerical and graphical computation.
% It was initially written to provide a interface for the LINPACK
% and EISPACK linear algebra packages. It has evolved over the years
% to become one of the premier high level languages for doing
% mathematical computation.
% The goal of this tutorial to describe some basic features of the
% Matlab language with particular emphasis on its abilities in the
% areas of data manipulation, data analysis, and graphing.
%***********************************************************************
% NOTE
%-----------------------------------------------------------------------
% WHENEVER YOU SEE THE WORD PAUSE THE M-FILE WILL PAUSE
% PRESS ANY KEY TO CONTINUE.
%***********************************************************************
pause
%------------------------------------------------------------------------
% MATRIX
%------------------------------------------------------------------------
% The basic structure in Matlab is a matrix. Let's look at a dataset
% that is discussed in Chapters 3 and 4. For 30 students in a statistics
% class, we record the grade (A, B, C, D, F) and the score on the SAT
% exam that was taken during high school.
% We represent this data by a matrix named 'class' that consists of 30 rows
% and 2 columns -- the first column contains the grades (coded 1 = F, 2 = D,
% 3 = C, 4 = B, 5 = A) and the second column contains the SAT scores.
% We use square brackets are used to describe the matrix. Carriage
% returns to describe different rows. Alternatively, a semicolon
% can be used to separate rows. Thee command is concluded with
% a semicolon.
echo off
class=[ 3 525
2 533
3 545
4 582
2 581
1 576
3 572
4 609
2 559
1 543
3 576
4 525
0 574
1 582
2 574
3 471
3 595
2 557
4 557
4 584
3 599
2 517
4 649
2 584
1 463
3 591
2 488
3 563
3 553
4 549 ];
echo on
pause
% We display 'class' by typing it without a semicolon at the end.
class
pause
%------------------------------------------------------------------------
% MATRIX MANIPULATIONS
%------------------------------------------------------------------------
% Let's illustrate some basic manipulations of this data matrix.
% We define a list of numbers by a colon:
1:5
pause
% We look at the data for students 1-5 by use of the command
class(1:5,:)
pause
% Here we are extracting rows 1-5 and all columns (indicated by a :).
% Similarly, suppose we wish to extract the first column of the
% data matrix and assign it a column vector called 'grade'. We extract
% the second column of 'class' and assign it to a variable called 'sat'.
grade=class(:,1); sat=class(:,2);
pause
% Note that we can have multiple statements on a line -- we separate the
% statements by ; (or ,)
% We can perform various computations on vectors and matrices.
% Suppose that we wish to scale the grades by multiplying by 10 and adding 5
% -- we'll call the new grade 'new_grade':
new_grade=10*grade+5
pause
% Suppose that we wish to square each grade -- we do this by the .^ operator.
grade.^2
pause
% grade is currently a column vector, we can convert it to a row vector by
% the transpose (') operator:
grade'
pause
% Suppose that we wish to add 'new_grade' to 'sat'. The addition (+)
% operator works for matrices if the dimensions line up.
new_grade+sat
pause
% Matlab also supports logical operators.
% Suppose that we want to define a passing grade -- if grade is 3 or higher,
% then pass = 1, otherwise pass = 0. We define this using >=
pass=(grade>=3)
pause
% We want to define a variable 'C', which is equal to 1 if the grade is 3
% and 0 otherwise. We use the logical equal (==) operator:
C=(grade==3)
pause
%------------------------------------------------------------------------
% DATA ANALYSIS COMMANDS
%------------------------------------------------------------------------
% Matlab supports many data analysis operations. The ones described below
% are intrinsic to Matlab. Many others are available in the Statistics
% Toolbox.
% The 'length' command gives the number of entries in a vector:
length(grade)
pause
% The commands below find the mean and standard deviation of the column
% vector 'sat':
mean(sat), std(sat)
pause
% Here is an alternative way of computing a standard deviation -- it
% illustrates the 'sum' and 'sqrt' commands:
sqrt(sum((sat-mean(sat)).^2/(length(grade)-1)))
pause
% Suppose we wish to tally the different grades in the vector 'grade'.
% This can be done using the Matlab 'hist' function (the graphical version
% of this command will be illustrated later).
freq=hist(grade,0:4)
% We see that there is 1 F, 4 D's, 8 C's, 10 B's, and 7 A's.
pause
% We can find the number of D's by use of a == logical operator and a 'sum':
num_D=sum(grade==1)
pause
% Likewise, the proportion of A's is given by
prop_A=sum(grade==4)/length(grade)
pause
%------------------------------------------------------------------------
% GRAPHING COMMANDS
%------------------------------------------------------------------------
% Matlab has a large number of graphing commands. Here we illustrate some
% basic Matlab commands that are helpful in data analysis.
% A histogram of the SAT scores is constructed by the 'hist' command.
hist(sat)
pause
% Suppose that we wish to use bin midpoints of 460, 480, ..., 660. We first
% define a variable, called 'bins', which contains 460 to 660 in steps of 20:
bins=460:20:600;
pause
% The histogram using these bin midpoints is given by
hist(sat,bins)
pause
% Suppose that we wish to construct a scatterplot of 'grade' (vertical axis)
% against 'sat' (horizontal axis). We use the 'plot' command. The horizontal
% variable goes first, and we indicate a plotting symbol at the end.
plot(sat,grade,'o')
pause
% We label this plot by the 'xlabel', 'ylabel', 'title' commands:
xlabel('SAT'); ylabel('GRADE'); title('SCATTERPLOT OF GRADE AGAINST SAT')
pause
% We look at this graph -- we see a slight positive relationship between
% SAT and GRADE. This motivates us to fit a line.
% We first form a regression matrix. The matrix X contains two columns - the first
% column is a ones vector (called ones(30,1)) and the second column are the sat scores.
X=[ones(30,1) sat];
pause
% A least-squares fit of SAT on GRADE is found in Matlab using a matrix divide notation --
% the column vector 'b' contains the regression coefficients:
b=X\grade
pause
% Let's put this least-squares fit on the graph. We define points on the line:
xpt=[450 750]; ypt=b(1)+b(2)*xpt;
pause
% The 'line' command will connect the points and put the line on the current plot:
line(xpt,ypt)
pause
%------------------------------------------------------------------------
% WORKING WITH SIMULATED DATA
%------------------------------------------------------------------------
% Let's review some of the above commands and introduce some new ones by
% doing some data analysis on simulated data.
% We simulate numbers from a standard normal distribution. We create a
% matrix of 10 rows and 100 columns and assign it to the matrix 'rand_norm':
rand_norm=randn(10,100);
pause
% Let's construct a histogram of the first row of this matrix -- it should
% look bell shaped and centered about the value 0.
hist(rand_norm(1,:))
pause
% Another property of these simulated values is that they are uncorrelated.
% To show this, we plot the entire stream of simulated values. We collapse
% the matrix 'rand_norm' to a column vector 'rand_c' (using the colon
% operation and then plot the elements of 'rand_c' as a line graph.
rand_c=rand_norm(:);
plot(rand_c)
% Note that there is no pattern or trend in the graph which indicates that
% they are uncorrelated.
pause
% Next, let's explore characteristics of sample means of normal samples.
% The following command finds the mean of each column of 'rand_norm' -- the
% resulting row vector is stored in 'means':
means=mean(rand_norm);
pause
% The values in 'means' are normally distributed with mean 0 and standard
% deviation sqrt(1/10) -- let's display the distribution.
hist(means)
pause
% We can use the random matrix 'rand_norm' to illustrate other functions of
% normals. First we'll square each element of the matrix (by the .^ operator)
% and put the result in 'chisqs' -- the elements are independent
% chi-square(1):
chisqs=rand_norm.^2;
pause
% If we sum across rows, we'll get a random sample from a chi square
% distribution with 10 degrees of freedom.
chisq10=sum(chisqs);
pause
% We'll display a histogram of these values.
hist(chisq10)
pause
% Looking at a chi-square table, we see that the upper 10 percentage point
% of a chi-square (10) is 14.68. So we expect that roughly 10 percent of
% our random chi-square(10)'s are larger than 14.68. Let's check:
sum(chisq10>14.68)
% This is the number of simulated values larger than 14.68 out of 100.
% Hopefully this is about 10 percent.
echo off
