Analysis of System Performance IN2072 Chapter M Matlab Tutorial
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1 Chair for Network Architectures and Services Prof. Carle Department of Computer Science TU München Analysis of System Performance IN2072 Chapter M Matlab Tutorial Dr. Alexander Klein Prof. Dr.-Ing. Georg Carle Chair for Network Architectures and Services Department of Computer Science Technische Universität München
2 Matlab - Tutorial Introduction Basics Plots Exercise Block A Fitting Practical Examples Advanced Topics Exercise Block B Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/2012 2
3 Matlab - Tutorial Alternatives: Commercial Mathematica Large library Excel Maple Limited number of mathematical functions Only efficient on small data sets Large library Free Octave Free software Partially compatible to MATLAB (available as plugin) R (Project) Free software Powerful function library Freemat Matlab clone Limited function library Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/2012 3
4 Matlab - Tutorial One of the most popular pograms for numeric calculations Large library of mathematic functions Provides methods for statistical evaluation and visualization Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/2012 4
5 Vectors: Generation: Example 1: Matlab: a = [ ] Output: a = Example 2: Matlab: a = [ ]; Output: none Example 3: Matlab: a = 1:6 Output: a = Example 4: Matlab: a(1:6) = 1 Output: a = Semi-colon prevents output on the console Colon represents the most practical way to generate vectors and matrices Index 1,2,..,6 of vector a are set to 1 Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/2012 5
6 Vectors: Operation: a = [ ]; b = [ ]; Example 1 Addition / Subtraction: Matlab: c = a + b Output: c = Example 2 Addition / Subtraction: Matlab: c = a 1 Output: c = Example 3 Transpose: Matlab: a Output: a = 1 6 Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/2012 6
7 Vectors: Operation: a = [ ]; b = [ ]; Example 4 Multiplication: Matlab: c = 2 * a Output: c = [ ] The scalar operation is performed on every index of the vector Example 5 Multiplication: Matlab: c = a*b Output: c = 36 Example 6 Multiplication: Matlab: c = a.* b Output: c = Dot indicates index-wise multiplication c(1)=a(1)*b(1),. c(n) = a(n)*b(n) Analogue - Division Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/2012 7
8 : - Colon: Usage: Start value : Maximum Value or Start value : Step size : Maximum Value Matlab assumes a step size of one if no step size is given Matlab adds step size to the start value as long as the calculated value is smaller than the maximum value Example 1: Matlab: a = 2:2:10 Output: a = Step size Example 2: Matlab: a = 2:2:9 Output: a = Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/2012 8
9 Matrices: Generation: Example 1: Matlab: m = [ ; ; ] Output: m = Example 2: Matlab: m(1:3,1:3) = 1 Output: m = Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/2012 9
10 Matrices: Access: a = a = [ ] Element: a ( i, j) Row: a ( i, :) Column: a ( :, j) Sub matrix: a (1:2, 2:3) = Note: It is also possible to access matrix elements like vectors Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
11 Basic functions: Sum (vector), mean (vector): a = 1:9 a = [ ] Example 1: Matlab: c = sum(a) Output: c = 45 Example 2: Matlab: c = mean(a) Output: c = 5 Note: Matlab supports a large number of very useful basic functions! Explore the help doc before rewriting a function. Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
12 Basic functions: a = [ ] Find (condition) Returns a vector of indices of elements which hold the given condition Example 1: Matlab: c = find (a > 5) ~= a(a>5) Output: c = 1 5 Example 2: Matlab: c = a(find (a > 5)) Output: c = 9 6 Contains all values of a which are larger than 5 Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
13 Basic functions: a = [ ] Minimum min(vector) / Maximum min(vector) Example 1: Matlab: c = min (a) Output: c = 3 a = [ ] Example 2: Matlab: c = min (a) Output: c = Min and Max return the minimum / maximum element of each column if a matrix is passed as argument to the function Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
14 Basic functions: a = [ ] Sort - Sort(vector) Example 1: Matlab: c = sort (a) Output: c = a = [ ] Example 2: Matlab: c = sort (a) Output: c = Sort returns the sorted column vectors if a matrix is passed as argument to the function Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
15 Basic functions: a = [ ] b = [ ] Matrix concatenation: Example 1: Matlab: c = [ a b] Output: c = Example 2: Matlab: c = [ a ; b ] Output: c = Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
16 Basic functions: a = [ ] b = [ ] Save a variable save( absolute_file_path, name_of_variable, options) Example 1: Matlab: save ( d:\workdir\variable_a.txt, a, -ascii ) Creates a file named variable_a.txt in the workdir folder with the following content:» e e e e e e e+000 (\n) Note that Matlab does not store the name of the variable or any other related information in the file if the ascii option is used! Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
17 Basic functions: a = [ ] b = [ ] Save a variable save( absolute_file_path, name_of_variable, options) Example 2: Matlab: save ( d:\workdir\variable_a.mat, a, b, -mat ) Creates a file named variable_a.mat in the workdir folder which contains both variables in a Matlab format The variables can be loaded directly from the file into the workspace Note that Matlab will create or overwrite variables in the workspace which have the same name as those stored in the file! Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
18 Basic functions: Load a variable load( absolute_file_path ) Example 1: Matlab: load ( d:\workdir\variable_a.mat ) Generates variables the stored variables in the workspace. It also possible to only partially load the file See help load for more information Note that Matlab supports a large number of different file types such as xls or other spreadsheet formats Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
19 Basic functions: Random numbers Rand (#rows, #columns) Returns uniform distributed values Randn (#rows, #columns) Returns normal distributed values Example 1: Matlab: c = rand(100,1) Output: c = [ ] RV X ~ U(0,1) RV X ~ N(0,1) ( = 1, = 1) Example 2: Matlab: c = rand(m,n) Output: c = Matrix (m,n) containing uniform distributed random numbers Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
20 Basic functions: sum(vector), prod(vector) s = [ ] Examples: ilength(s) i1 s(i) sum(s) = = 15 ilength(s) i1 s(i) prod(s) = = 120 ceil(value) Rounds the element to the nearest integer greater than or equal to the element. floor(value) Rounds the element to the nearest integer lower than or equal to the element. Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
21 Basic functions: cumsum(vector), cumprod(vector) Cumsum and cumprod return the cumulative sum / product of the elements of the input vector. s = [ ] Example 1: Matlab: result = cumsum(s) Output: result = Example 2: Matlab: result = cumprod(s) Output: result = Both functions are very useful, especially cumsum which is typically used to calculate cumulative density functions. Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
22 Basic functions: Histc( Vector sample, intervals) Histc counts the number of values that are within a certain interval Example: Matlab: sample = [ ]; Matlab: result = histc(sample, 1:10) Intervals [1;2[, [2;3[, [9;10[,[10; [ Output: result = [ ] Matlab: plot(result); Output: Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
23 Plots: Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
24 Plots: Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
25 Matlab Tutorial Plots Plots: Standard 2D-Line Plot plot( Vector x, Vector y, Layout.) Example 1: Matlab: x = 2:2:20 Will be introduced at the end of the section Output: x = Matlab: y = rand (10,1) Output: y = Matlab: plot (x,y) Output: Matlab draws the lines between the points (x(1),y(1)),, (x(n),y(n)) Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
26 Matlab Tutorial Plots Plots: Standard 3D-Line Plot plot3( Vector x, Vector y, Vector z) Example 1: Matlab: x = cos(pi/20:pi/20:8*pi); %Vector with 160 elements Matlab: y = sin(pi/20:pi/20:8*pi); %Vector with 160 elements Matlab: z = 1:160; Matlab: plot3(x,y,z) Matlab: grid; %Adds a nice grid to the figure Output: Matlab draws the lines between the points (x(1),y(1)),, (x(n),y(n)) Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
27 Matlab Tutorial Plots Plots: Scatter plots are similar to standard line plots Example 1: Matlab: x = cos(pi/20:pi/20:8*pi); %Vector with 160 elements Matlab: y = cos(pi/20:pi/20:8*pi); %Vector with 160 elements Matlab: z = 1:160; Matlab: scatter3(x,y,z) Matlab: grid; %Adds a nice grid to the figure Output: Scatter plots are often used to visualize point fields, point densities and autocorrelation plots Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
28 Matlab Tutorial Plots Plots: Stem - stem( Vector x, Vector y) / stem3( Vector x, Vector y, Vector z) Example 1: Matlab: x = 1:20; %Vector with 20 elements Matlab: y = randn(20,1);%vector with 20 normal distributed elements Matlab: stem (x, y) Output: Stem plots are optimal for visualization of errors or discrete random variable Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
29 Matlab Tutorial Plots Plots: Histogram hist (Vector x) / hist (Vector x, #bins) / Example 1: hist(vector x, Vector bins) Matlab: x = randn(1000,1); %Vector with 1000 elements Matlab: hist(x) Output: histogram with 10 bins Example 2: Matlab: hist(randn(10000,1),-5:0.2:5) Output: histogram with 51 bins with bin size 0.2 starting from -5 to +5 Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
30 Matlab Tutorial Plots Plots: Errorbar - errorbar( Vector sample, Vector error) Example: Matlab: sample = 0.5*randn(10,1)+10; Matlab: error = 0.2*rand(10,1)+0.5; Matlab: errorbar (sample, error) Output: The shown error bars reflect twice the error value. Typical values are standard deviation or confidence intervals of the samples. Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
31 Matlab Tutorial Plots Layout - Basics Matlab supports many features to modify an existing plot. This tutorial just gives a brief overview of the most important ones. Line style: - Solid line (default) -- Dashed line Example: Matlab: plot (1:10, LineStyle, - ); hold all; Matlab: plot (1:0.9:9, LineStyle, -- ); : Dotted line -. Dash-dot line plot (1:0.7:8, LineStyle, : ); plot (1:0.6:7, LineStyle, -. ); Use different line styles since many colors cannot be distinguished if they are printed in b/w. Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
32 Matlab Tutorial Plots Layout - Basics Matlab supports many features to modify an existing plot. This tutorial just gives a brief overview of the most important ones. Line style Matlab: plot(x, 'LineStyle','--'); Line width Color Matlab: plot(x, 'LineStyle','--', 'LineWidth',2); Matlab: plot(x, 'LineStyle','--', 'LineWidth',2, 'Color', 'red'); Marker type Matlab: plot(x, 'LineStyle','--', 'LineWidth',2, 'Marker','diamond'); Marker size Always use a line width of 2 since it looks much better in most plots Matlab: plot(x, 'LineStyle','--', 'Marker','diamond', 'MarkerSize', 10); Take a look at Matlab Help linespec for more information Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
33 Matlab Tutorial Plots Layout - Basics Brief command description: Command: axis normal Stretches the current figure such that it fits fills the frame. Command: set(gcf, 'Position', [ ]); Moves the current figure to position [200;200] and sets its size to 800x640 pixels Command: grid A grid is plotted which simplifies reading the graphs that are plotted. The grid is removed it is already present. Command: box A box is plotted around the figure which is strongly recommended for 2D plots. The box is removed by the command if already present. Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
34 Matlab Tutorial Plots Layout - Basics Brief command description: Command: xlabel( This is the label of the x axis') Sets the name of the x axis to the string argument.(ylabel / zlabel) Command: xlim([0 10]); ylim([5 10]); zlim([3 10]); Sets the limit of the axis to the given interval. Command: set(gca, 'FontSize', 14); Sets the font size of the axis label and tick labels to the given value. Command: set(gca, 'XTickLabel', {'0','5','10','15','20','25','30','35'}, 'XTick',[ ]); Sets the labels and the position of the labels to the given values. XTick describes the position of the label whereas XTickLabel holds the string of the label. Here it is used to shift the graph by one unit. Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
35 Matlab Practical Examples Probability Density Function Sample Vector Vector x is the vector which holds all elements of the sample /measurement Example: Matlab: x = *randn(10000,1);%Returns a vector with random values with a mean of 100 Matlab: result = histc(x, 1:200); Intervals [1;2[, [2;3[, [199;200[,[200; [ Matlab: result = result / length(x); %Normalize the result Matlab: plot(result); Output: Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
36 Matlab Tutorial Plots P(robability)P(robability)-Plot Plot two sample vectors against each other to outline their relative difference in a single plot. (Typically only done with distributions) Example 1: Matlab: i = 1:100; % Generate a reference vector with 100 elements Matlab: j = 1:100; % Second vector Matlab: b(1:10) = 0.1; %Generate a filter vector Matlab: k = 20*randn(1,100); Matlab: k = filter(b,1,k);%smoothed random vector Matlab: plot (i, j+k);hold all;plot(1:100);%plot i against j+k. Then hold the figure and plot the reference graph Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
37 Matlab Practical Examples Cumulative Density Function Sample Vector Vector x is the vector which holds all elements of the sample /measurement Example: Matlab: x = *randn(10000,1);%Returns a vector with random values with a mean of 100 Matlab: result = histc(x, 1:200); Intervals [1;2[, [2;3[, [199;200[,[200; [ Matlab: result = result / length(x); %Normalize the result Matlab: plot(cumsum(result)); Output: Calculates the sum of the elements which corresponds to the CDF if the input vector represents the PDF Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
38 Matlab Practical Examples p-quantil The p-quantile for a random variable is the value x such that the probability that the random variable will be less than x is at most p sample = randn(10000,1)+10; Example: Matlab: p = 0.95;%Probability that a randomly chosen sample value is lower than the p-quantile Matlab: size = length(sample);%sometimes the length of the sample is not known in advance. Thus, length should be used. Matlab: result = sort(sample);%sort the samples Matlab: index = ceil(size*p);%calculates the corresponding index Matlab: p_quantile = result (index); The p-quantile can also be directly read from the CDF which should be done if it is already calculated. Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
39 Matlab Practical Examples Moving Average Filter filter (Vector b, Vector a, Vector x) y(n) b(1) x(n) b(2) x(3) b(k) x(n k) Vector x is the sample vector that has to be smoothed Variable a should be set to 1 c.f. Matlab Help Vector b defines the length and the weights for the smoothing vector Example 1: x = rand(10,1);%vector with 10 uniform distributed elements a =1;% Set a to 1 b = [ ];%Will result in a smoothing over the last two values y = filter(b, a, x);%store the smoothed vector in variable y plot(x);hold all; plot(y); %Plots x and y in the same figure Vector x Vector y Output Network IN2045 Security, Discrete Event WS 2008/09, Simulation, Chapter WS 92011/
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