Matlab Graphics Tutorial

Matlab Graphics
Cheng-An Yang
September 22, 2013

2D Plots The Graphical User Interface Advanced Topics Animation 3D Plots More Plots Extra
Outline
1 2D Plots
2 The Graphical User Interface
3 Advanced Topics
4 Animation
5 3D Plots
6 More Plots
7 Extra
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Section 1
2D Plots
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Figure: create
Syntax
• To create an empty ﬁgure, call
figure();
• You can also number your ﬁgure like
figure(42);
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Figure: close
Syntax
• Close a specific figure
close(42);
• Close all figures
close all;
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Plot
Syntax
• To create a line plot of vector y versus vector t, use
plot(t,y);
• What happen if you don’t provide the x-axis data? Try
plot(y);
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Multiple Lines
Syntax
• You can input more vector pairs like
plot(t1,y1,t2,y2);
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Exercise: Multiple Lines
Visualize three elementary functions on 0 ≤ t ≤ 2π:
y1 = sin(t)
y2 = cos(t)
y3 = e−t
Plot them on the same ﬁgure.
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Adding Labels
Syntax
• Adding labels on axis
xlabel(’x label’);
ylabel(’y label’);
• Adding title
title(’title for the figure’);
• Adding legends
legend(’first’,’second’,...);
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Grid
Syntax
• Turn on/oﬀ the grid lines
grid on;
grid off;
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Changing the Axes Limits
Syntax
• Set the limits of each axis
axis([xmin xmax ymin ymax]);
• If you want to adjust only x-axis or y-axis,
xlim([xmin xmax]);
ylim([ymin ymax]);
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Customize Plots
• Using the string speciﬁer to change the line style.
• The string speciﬁers contains
• Line style: {’-’,’--’,’:’,’-.’,’none’}
• Marker symbol: {’+’,’o’,’*’,’.’,’x’} and more.
• Color: {’r’,’g’,’b’,’w’,’k’}.
• For example,
plot(t,x,’--or’);
plot(t,x,’r’,t,y,’-.xk’);
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Exercise: The Envelop
Please duplicate the ﬁgure shown below:
0 5 10
−1
−0.5
0
0.5
1
The blue line is
x = t2
cos(5t)e−t
. (1)
The envelope of x(t) is
y = ±t2
e−t
. (2)
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Plot Matrix Data
Syntax
• By default plot(Y) will plot each column of Y.
• When specifying the x vector,
plot(x,Y);
will try to match the dimension of x and Y.
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Exercise: Plot Matrix Data
Let
Y = [ y1; y2; y3 ];
• What is the dimension of Y?
• Try plot(t,Y) and plot(Y). Can you anticipate the
outputs?
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Plot Complex Data
Syntax
• To plot complex array x, use
plot(x);
• It is equivalent to
plot(real(x),imag(x));
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Exercise: The Eigenvalues of Random Matrices
Gaussian Random Matrix
A Gaussian Random Matrix H is a matrix with standard normal
components hij
d
= N (0, 1).
h11
h22
h12 h21
TX1
TX2
RX1
RX2
• Arises in many applications.
• Wireless communication.
• Channel gain hij is random.
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Exercise: The Eigenvalues of Random Matrices
Visualizing the distribution of the
eigenvalues of random matrices.
• Generate H = randn(n,n).
• [V,D] = eig(H).
• Plot its eigenvalues as dots on the
complex plane.
• Increase n form 10 to 1, 000. Can
you tell what’s the pattern of the
eigenvalues?
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Plotting Multiple Lines On the Same Axes
• If you call plot twice, the first plot will be erased.
• To retain current graph when adding new graph, tell Matlab to
hold on;
• If you want different lines to have different color, use
hold all;
• The default is
hold off;
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Exercise: Exponent and Convexity
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
x
n
0.1
0.5
1
2
10
• The function xα is convex
when α ≥ 1.
• When 0 < α < 1 is it
concave.
• Use a for loop and hold to
verify this.
• α = {0.1, 0.5, 1, 2, 10}.
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Section 2
The Graphical User Interface
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The GUI
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Saving and Loading
Some tips:
• Save your ﬁle in vector formats, such as .eps and .pdf.
• Use ‘copy ﬁgure’ between applications.
• Keep a .fig copy so you can edit it later.
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Customization Using GUI
• Click the ‘Edit Plot’ icon.
• Double click on the ﬁgure to
enter the editing mode.
• Select an object, and the
property editor will appear.
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Exercise: Frequency Response
• The transfer function of a second order system has the form
H(s) =
ω2
n
s2 + 2ζωns + ω2
n
,
where ζ is the damping ratio and ωn is the natural frequency.
• The frequency response of a system is characterized by
• The magnitude |H(ω)|
• The phase ∠H(ω)
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Exercise: Frequency Response
10
−1
10
0
10
1
−100
−50
0
50
Frequency (rad)
Magnitude(dB)
10
−1
10
0
10
1
−π
−π/2
0
Frequency (rad)
Phase(rad)
• Let ωn = 1, ζ = 0.2.
• Frequencies from 10−1 to 10
(rad).
• The unit imaginary number
in Matlab is 1j and 1i.
• Note the magnitude
response is deﬁned as
20 log(|H(ω)|).
• You may want to use
angle and logspace.
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Graded Task 1
The Bessel functions Jα(x) of the ﬁrst kind and order α is the
solution to the
Bessel’s Equation
x2 d2y
dx2
+ x
dy
dx
+ (x2
− α2
)y = 0.
• Separable solution to many important PDE in cylindrical
coordinate.
• Useful in physics, signal processing and statistics.
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Graded Task 1
Bessel function has the following representation
Bessel Functions of the First Kind
Jα(x) =
∞
m=0
(−1)m
m! Γ(m + α + 1)
1
2 x
2m+α
.
Matlab has built-in function for Jα(x)
Syntax
J = besselj(alpha, x);
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Graded Task 1
Visualizing Jα(x) with diﬀerent α. Please create a ﬁgure like this:
0 1 2 3 10 15
0.0
1.0
x
Jα(x)
α = 0
α = 1
α = 2
α = 3
α = 10
• α = {0, 1, 2, 3, 10}.
• 0 ≤ x ≤ 15. Note that some
tick labels are omitted.
• Use alpha for α.
• Insert a text box near the
maximum amplitude of each
function.
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Exploring Data
• Change view point, zoom
in/out, pan.
• Create multiple datatips.
• Select/Brush Data tool.
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Select And Linking Data
• Open the Variable Editor.
• Click the ‘Link Plot’ icon.
• Click the ‘Select/Brush
Data’ icon.
• Select the data of interest.
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Exercise: Wave Editing
• Load the chirp.mat ﬁle.
• Type sound(y,Fs) to
play it.
• Use data selection tool to
select and ‘mute’ the chirps.
• Play the modiﬁed recording.
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Section 3
Advanced Topics
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Why Learning Commands?
• Almost everything can be adjusted using GUI, why learning
commands?
• If you only need to edit a ﬁgure or two, use GUI.
• If you have a dozen of ﬁgures, let computers do the work.
• To create an animation, you have to know the commands.
• Programmer’s pride.
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Hierarchy of The Graphic Objects
A Matlab plot is composed of (at least) three objects:
(a) Figure window. (b) Axis object. (c) Line series object.
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Get a Handle on Objects
• Each object has a unique identiﬁer, called the handle.
• You can ask Matlab to get and set the properties of the object
with its handle.
• To get the handle of the current ﬁgure, type
h_fig = gcf;
• To get the handle of the current axis, type
h_ax = gca;
• To get the handle of the line series, use
h = plot(t,x);
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Get and Set Property
• Get the value of the property:
value = get(h,’PropertyName’);
• Set using property-value pair:
set(h,’PropertyName’,value);
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Exercise: Batch Process
• The command
saveas(h_fig, ’file_name’, ’fig’);
will save the figure h fig as file name.fig.
• You can replace the third argument by other format. Type
help saveas for more information.
• Write a script that does the following:
• For each frequency ω = {1, 2, . . . , 10}, plot sin(ωt) on
0 ≤ t ≤ 2π in 10 separate figures.
• Save and name them as freq1.fig, freq2.fig,...
freq10.fig files.
• Hint: the command sprintf(’freq%d.fig’,I) will
generate the desired file name, where I is an integer.
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Subplot
Syntax
subplot(m,n,1);
plot(t,x);
Will create a m-by-n subplot and place the plot of x versus t at
location 1.
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Exercise: Sine Wave Matrix
Create 25 subplots of sine wave with increasing frequency.
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
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Section 4
Animation
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Making Animated Sequences
• The simplest way: continually erase and redraw your ﬁgure.
• Key ingredients:
• Loop.
• Update data.
• Erase and redraw.
• Pause.
Example
for I = 1:N % Loop.
x = sin(t-I); % Update data.
plot(t,x); % Erase and Redraw.
pause(0.1); % Pause for 0.1 sec.
end
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Example: Vibrating String
Standing Wave
As oppose to a travelling wave, a standing wave oscillates in place
without propagating.
Mathematically it is described by
y = A cos(ωt) sin(kx),
where ω is the angular frequency
and k is the wave number.
• Let ω = 5, k = 1.5, A = 2.
• Let 0 ≤ x ≤ 2π.
• Make an animation of such
a standing wave.
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Graded Task 2
Standing Wave
A standing wave y can be seen as the result of interference
between two waves y1 and y2 travelling in opposite directions.
Mathematically,
y1 = A sin(kx − ωt)
y2 = A sin(kx + ωt)
and
y = A sin(kx−ωt)+A sin(kx+ωt).
• A: amplitude.
• k: wave number.
• ω: angular velocity.
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Graded Task 2
Create an animation of a standing wave.
0 2 4 6 8 10
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
• A = 1, k = 1, ω = 3.
• Red wave: from left to right.
• Blue wave: from right to
left.
• Black wave: the resulting
standing wave.
• Try update rate dt = 0.05.
• Add a reference line x = 0.
• Bonus problem: labelling the
nodes. (the red circles)
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Section 5
3D Plots
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3D Line Plot
For parametric data (x(t), y(t), z(t)), we can use the 3D version
of the plot function.
Syntax
• Plot lines in 3D
plot3(x,y,z);
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Exercise: Brownian Motion
−50
0
50
−20
0
20
−20
0
20
The position vector xn of a
Brownian particle at time n is
given by
xn = xn−1 + v,
where v is a standard normal
random vector.
• Simulate the path of two
Brownian particles starting
at the origin.
• Take N = 500 steps.
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Representing Matrix Data
−10
0
10
−10
0
10
−0.5
0
0.5
1
• Line series are represented
by vectors.
• Data with 2D indices are
represented by matrices.
• Think of the values of
matrix Z as “height”.
• The indices of x-axis and
y-axis are stored in matrix X
and Y.
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Mesh Grid and Surface Plot
To visualize a function f (x, y) on (x, y) ∈ [a, b] × [c, d],
• First we need to create a rectangular grid.
• Matlab provides a useful function to create such a grid:
[X,Y] = meshgrid(x,y);
where x and y are the sampling points on both axis.
• Then we can use
surf(X,Y,f(X,Y));
to produce a surface plot.
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Exercise: The 3D Sinc Function
3D Sinc Function
sinc(r) :=
sin(r)
r
, where r2 = x2 + y2.
• Visualize the 3D sinc function on (x, y) ∈ [−8, 8] × [−8, 8].
• One way to avoid the divided-by-zero error is to use
sin(r)./(r+esp) instead of sin(r)./r.
• esp is the smallest representable ﬂoating point number in
Matlab.
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Contour Plot
• Surface plot is fancy but not suitable for putting in your paper!
• Contour plot might be more appropriate.
Syntax
• The basic syntax is
contour(Z);
• You can specify the number of contour level
contour(Z, n_level);
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Exercise: Scalar Field
Let
f (x, y) = x2
− 3 sin(xy).
Use contour plot to visualize f (x, y) on [−2, 2] × [−2, 2].
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Vector Field
A 2D vector field is a vector-valued function
z(x, y) =
zx (x)
zy (y)
.
Matlab provides a function to visualize vector files.
Syntax
• Plot the vector field Zx and Zy as a function of X and Y
quiver(X,Y,Zx,Zy);
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Exercise: Scalar Field and its Gradient
• The gradient of a scalar field f (x, y) is defined as
f =
∂
∂x f
∂
∂y f
.
• Numerically, we can approximate the partial derivative by
∂
∂x
f (x, y) ≈
f (x + h, y) − f (x, y)
h
for some small number h.
• In Matlab, we can use gradient(F) to find the gradient of
the scalar field F. See help for more detail.
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Exercise: Scalar Field and its Gradient
−2 −1 0 1
−2
−1
0
1
• Find the gradient of the
scalar ﬁeld
f (x, y) = x2
− 3 sin(xy).
• Plot f (x, y) using
contour().
• Plot its gradient using
gradient() in the same
ﬁgure.
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Section 6
More Plots
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Histogram
Histogram is useful to visualize the distribution of univariate data.
Syntax
• Create a histogram of data vector x
hist(x);
• To specify number of bins, use
hist(x,m);
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Exercise: Random Number Generators
• Two important random number generators in Matlab:
• Uniform between 0 and 1: rand().
• Standard normal: randn().
• Exercise:
• Generate N random samples from rand().
• Use hist to visualize the distribution.
• Experiment with diﬀerent N and diﬀerent number of bins.
• How many samples are required to produce a good
approximation of it distribution?
• Repeat for randn().
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Area, Bar and Pie Chart
Syntax
• Stacks each data series and ﬁll the underlying area with
diﬀerent colors
area(X,Y);
• Create a bar chart
bar(Y);
Each column of Y will have the same color and rows are
grouped together.
• Pie chart
pie(Y);
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Exercise: Operating System War!
• Go to http://www.netmarketshare.com/ and
download the operating system share trend data (Excel ﬁle).
• Import it into workspace.
• Create the following plots:
• Line plot.
• Area chart.
• Bar chart.
• Pie chart on April, 2012.
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Scatter Plot
Scatter plot is used to visualize the distribution of two dimensional
data.
Syntax
• To generate the scatter plot for the data vector X and Y
scatter(X,Y)
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Exercise: Basic Regression
• Suppose we are given a data set x and y.
• It is assumed that y = ax + b + noise.
• Find the best linear ﬁt ˆy = ˆax + ˆb.
0 0.5 1
2
3
4
5
6
x
y
ˆy = 2.0x + 3.0
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Section 7
Extra
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Solving Differential Equations
• Differential equation involves derivatives of a function in its
independent variables.
• Almost everything is described by some differential equation.
• Mostafa will talk about solving differential equations using
powerful tools provided by Matlab.
• We can create a poor man’s differential equation solver.
• The idea of Finite Difference Method (FDM):
∂u
∂t
≈
u(t + h) − u(t)
h
for small h.
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The Finite Difference Method
• Consider the wave equation
utt = c2
∆u − but,
where the Laplacian operator ∆ is defined as
∆u = uxx + uyy .
• Use finite difference to approximate derivatives:
ut ≈
u(t + ∆t) − u(t)
∆t
utt ≈
u(t + ∆t) − 2u(t) + u(t − ∆t)
∆t2
.
Similarly for uxx and uyy .
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The Finite Difference Method
• In Matlab, we can use the discrete Laplacian function del2
to approximate
∆u ≈
4
h2
del2(u),
where h is the spacing in the spatial grid.
• Substitute the derivatives by their finite difference
approximation, we get (verify this!)
u(t+∆t) ≈ 2u(t)−u(t−∆t)+h2
∆u(t)−b∆t(u(t)−u(t−∆t)).
(3)
• Provide the initial data of u and ut at t = 0 and the boundary
condition, we can approximate the solution of the wave
equation by recursively solving (3).
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Demo: 2D Wave Equation
−4
−2
0
2
4
−4
−2
0
2
4
0
2
4
6
8
10
Figure: The simulated solution to the wave equation.
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Exercise: Linear Congruential Generator (LCG)
• LCG is a popular (and old) Pseudo-Random Numbers
Generator.
• Simple and eﬃcient to compute.
• Poor choice of parameters lead to bad performance.
LCG
xn+1 ≡ (axn + b) (mod m), (4)
• m > 0: the modulus
• a > 0: the multiplier
• b ≥ 0: the increment
• x0: the seed
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Histogram
0 10 20 30
0
5
10
15
• Try diﬀerent parameters and use
hist to evaluate its performance.
• If you can’t ﬁgure out what
combination of parameters would
work, try
• a = 3
• b = 0
• m = 31
• x0 = 1
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Polar Plot
• Sometimes it is easier to express coordinate in the polar form.
• Let (x, y) be the coordinates in the Cartesian coordinate
system, its corresponding polar coordinates is given by
r = x2 + y2 (5)
θ = tan−1
(y/x). (6)
Syntax
• Plot r versus theta in the polar coordinate
polar(theta,r);
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Exercise: Butterfly Curve
The butterfly fly curve, discovered by Temple H. Fay, is generated
by the equations
r = esin θ
− 2 cos(4θ) + sin5 2θ − π
24
. (7)
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
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Demo: Lorenz Attractor
Lorenz system is a simpliﬁed model for atmospheric convection, it
is modeled by the ordinary diﬀerential equations
dx
dt
= σ(y − x), (8)
dy
dt
= x(ρ − z) − y, (9)
dz
dt
= xy − βz, (10)
where x, y and z are the coordinate of the state, t represents time,
ρ, σ and β are parameters.
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3D Line Plot
Example Lorenz Attractor
−20 0 20
−50
0
50
0
20
40
60
x
y
z
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Exercise: Visualizing Gibbs’ Phenomena
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Exercise: Complex Data as 2D Representation
Plot the Hypocycloid.
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