INTRODUCTORY NOTES ON MATLAB

Transcription

1 INTRODUCTORY NOTES ON MATLAB C Lamas Fernández, S Marelli, B Sudret CHAIR OF RISK, SAFETY AND UNCERTAINTY QUANTIFICATION STEFANO-FRANSCINI-PLATZ 5 CH-8093 ZÜRICH Risk, Safety & Uncertainty Quantification

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3 Contents Introduction 4 2 Vectors and matrices 4 2 Creation and access 4 22 Indexing 6 23 Matrix operations 7 24 Modifying the content and shape 8 3 Functions 9 3 Definition 9 32 Integration 9 4 Statistics 9 4 Main functions 9 42 Sampling 43 Specific functions 2 5 Display 2 5 Command window 2 52 Plots 3 3

4 Introduction This document is a short introduction to the MATLAB programming language, with an emphasis on commands that are useful in the context of uncertainty quantification in engineering More information about MATLAB can be found in the on-line book: Moler, C B (2008) Numerical computing with MATLAB Siam Some of the chapters more related to the concepts presented here are: Introduction to MATLAB Quadrature Random Numbers In general, the built-in MATLAB documentation is comprehensive For a given command, you can type help cmd for the inline help in the workspace, or doc cmd for the HTML documentation 2 Vectors and matrices 2 Creation and access Vectors and matrices are defined using square brackets The comma is used to separate columns and the semicolon to separate rows For instance, to define : ( ) 2 v = (, 2, 3) B = 3 4 the following syntax is used: v = [, 2, 3]; B = [,2 ; 3,4]; There are some functions available to create special types of matrices and vectors, as listed in Table Table : Creation of special types of matrices / vectors Description Operation result Syntax Matrix with zeros z = ( 0 0 ) z = zeros(2, 3); Matrix with ones o = o = ones(3, 2); Elements from to N s = (, 2,, N) s = :N; 4

5 Elements from 0 to N, step k q = (0, k, 2k,, N) q = 0:k:N; 0 0 Diagonal matrix D = D = diag([, 2, 3]) Identity matrix I = ( ) 0 0 I = eye(2) Accessing parts of a matrix A and a vector v a, a,n A = v = (v,, v M ) a M, a M,N may be carried out using the commands in Table 2 Table 2: Accessing parts of matrices / vectors Description Operation result Syntax Vector i th element v i v(i) Last vector element v M v(end) Matrix element a i,j A(i,j) k th row k = (a k,,, a k,n ) k = A(k,:) l th column l = a,l a M,l l = A(:,p) Diagonal d = (a,, a 2,2, ) d = diag(a) 5

6 22 Indexing MATLAB has two powerful ways to access subsets of vectors and matrices, namely, absolute and logical indexing Absolute indexing It consists in referring to elements with their absolute index inside the array Consider the vector v = (0, 2, 4, 8) As you would expect, v(2) = 2 and v(4) = 8 These indices can also be combined in a vector: v([2, 4]) = [2, 8] The colon definition of vectors works also here: v(2:4) = [2, 4, 8] Logical indexing A logical index of a matrix A consists on a matrix L A of the same size as A that contains logical values only These are typically created using logical operators For example, let us define the two matrices: A = ( ) B = ( ) and check the elements that are equal in these two matrices: >> LA = A == B LA = 0 0 One can use this index to retrieve elements from the matrices A and B, that are always stored as a column vector: >> A(LA) ans = 2 >> A( LA) ans = 3 4 In this code, using LA, returns the elements that are not true on the index LA, this means the elements of A that are different from B Example Let us consider now a vector y = sin(x), where x = (,, 00) With logical indexing, we can easily see how many elements of y are greater than zero: >> y = sin(:00); sum(y > 0) ans = 50 If we want to set the vector elements with negative values to 0, we can do it with this notation as well: >> y(y < 0) = 0; 6

7 Note that this would be much faster (and shorter) than writing the equivalent loop version: for ii = :00 if y(ii) < 0 y(ii) = 0; end end 23 Matrix operations Consider the matrices A and B and the vector v: a, a,n b, b,n A = B = a M, a M,N b M, b M,N v = Matrix operations are summarized in Table 3 Operations on matrix elements are summarized in Table 4 Table 3: Matrix level operations Description Operation result Syntax Transpose of A, A T a, a,m a N, a N,M Matrix sum A + B A + B v v s transpose(a) (for real matrices, A' is equivalent) Matrix multiplication AB T A transpose(b) Matrix division A B A/B A squared A 2 = AA Aˆ2 Matrix inversion A inv(a) A\v Table 4: Element level operations Description Operation result Syntax Element multiplication a, b, a,m b,m a N, b N, a N,M b N,M A B 7

8 Element division Square root Standard function, eg sin Sum Product a, b, a N, b N, a,m b,m a N,M b N,M a, an, a,m an,m sin(a, ) sin(a,m ) sin(a N, ) sin(a N,M) M i= v i M i= v i A/B sqrt(a) sin(a) sum(v) prod(v) Most MATLAB functions, like the trigonometric functions, exponential, probability density functions, etc act element-wise on matrices However, you can always check this behaviour on the function s help 24 Modifying the content and shape Scalar value assignment to a matrix, or part of it, sets each element to its value: ( ) ( ) 0 5 A = A = 0 5 >> A = eye(2); A(A == 0) = 5; Setting a row or column to [], deletes it from the matrix A = 0 0 A = >> A = eye(3); A(:,3) = []; The function repmat(a,m,n) creates a matrix that is made of a grid of M N times the A matrix v = 2 A = >> v = transpose(:3); A = repmat(v,,3); 8

9 3 Functions 3 Definition Consider the function f(x, y) = xy In MATLAB it can be defined easily as an anonymous function: f y) x y; However, it is convenient to be able to evaluate more than one point at once We usually adapt to the notation ( ) x x X = N y y N to represent a sample of N points with two coordinates More generally, X has a dimension M N and each column represents a M dimensional point of a sample with N elements Adapting this convention to MATLAB we can define the previous function in a vectorized fashion as: fv X(,:) X(2,:); 32 Integration The integral of an anonymous function f can be easily approximated in the interval [min, max] using the command integral(f, min, max) >> f xˆ2; integral(f, 0, 2) ans = Note that it is also possible to use ± as boundaries For example, we can integrate the standard normal pdf in (, ) using: >> f =@(x) pdf('normal', x, 0, ); integral(f, Inf, Inf) ans = Statistics 4 Main functions Distribution functions MATLAB can handle various probability distributions through its statistics toolbox, in particular it has the functions: pdf(distr,x,par,par2) that computes the probability density function of the distribution distr with parameters par and par2 on the point X cdf(distr,x,par,par2) that works the same but for the cumulative distribution function 9

10 icdf(distr,x,par,par2) for handling the inverse cumulative distribution function Table 5 gathers a list of the most common keywords that can be used for distr For a full list you can type help pdf Table 5: Keywords and expected parameters for various distributions in MATLAB Keywords Distribution Par Par 2 Obs 'beta' Beta, B(a, b) a b Def only for x (0, ) 'ev' or 'Extreme Value' Extreme value µ β The MATLAB default corresponds to Same, but using X Gumbel, G(µ, β) µ β the Gumbel minima distribution 'gam' or 'Gamma' Gamma Γ(λ, k) k λ Inverted order of parameters 'logn' or 'Lognormal' Lognormal, LN (µ, σ) µ σ 'norm' or 'Normal' Normal, N (µ, σ) µ σ 'unif' or 'Uniform' Uniform, U(a, b) a b 'wbl' or 'Weibull' Weibull, W(α, β) α β Data estimates The basic functions for estimating parameters from data are also available on MATLAB Some of them are presented on Table 6 Table 6: Parameter estimation functions in MATLAB Function mean(x) std(x) var(x) skewness(x) kurtosis(x) Description Mean of X Standard deviation of X Variance of X Skewness of X Kurtosis of X Kernel smoothing MATLAB provides a function for kernel smoothing: [fhat, x] = ksdensity(v) It provides a estimate fhat of the value of the probability density function on the points x 0

11 If you wish to specify the abscissae where the function shall be estimated, the syntax fhat = ksdensity(v, points) is also possible, where points contains the vector of abscissae Example Estimate the PDF ˆf(0), for x = 0, from a set made of 000 realizations of a standard normal variable N (0, ) (The actual value is φ(0) = 03989) >> x = randn(,000); fi = ksdensity(x, 0) fi = 0396 For more advanced tuning, ksdensity accepts other input parameters, such as choosing the bandwith (eg [fhat, x] = ksdensity(x,'width',8);) or estimating the cumulative distribution function: [Fhat, x] = ksdensity(x,'function', 'cdf'); Example Calling the function without output arguments, will directly produce the plot, in this case for the estimate of the CDF of a standard normal distribution Sampling v = randn(, e4); ksdensity(v, 'function', 'cdf') Random numbers To generate uniformly distributed random numbers in the interval (0, ), one can simply type: rand The same function with an argument n, will give a random matrix of size n n: >> rand(2) ans = Calling it with two arguments, eg rand(m,n), returns a M N matrix with random numbers Random sampling MATLAB provides the function random(distr, par, par2, M, N), that allows one to generate random numbers from various distributions, specifying them in the distr parameter, as in Table 5 The arguments par and par2 stand for the parameters of the distribution and M and N for the dimensions of the generated random matrix

12 Example Plot a sample of 00 points for two independent variables following the distributions x N (0, 5) and y N (0, 05) x = random('normal', 0, 5,, 00); y = random('normal', 0, 5,, 00); plot(x, y, ' '); Specific functions Note that apart from this syntax, there are also specific functions in MATLAB for some of the distributions For example, for the normal distribution, you can use directly normpdf, normcdf, norminv and normrnd or randn for the probability density function, the cumulative distribution function, the inverse cumulative distribution function and for generating random numbers, respectively 5 Display 5 Command window disp(var) The function disp allows you to quickly display information of any kind within the command window: >> a = rand(,00); disp('the last element of a is:'); disp(a(end)); The last element of a is: 0622 disp always displays the results in a new line, and it works regardless of the type of variable that you use as an argument In general, it is the same as not typing the semicolon at the end of the sentences, but disp does not show the name of the variable being displayed fprintf(string,) For a more sophisticated and powerful way of displaying information, you can use fprintf It works by printing a string in which some special characters are included, that are to be replaced by the arguments passed to the function For example, the special character %s is used for including strings, %f for double precision numbers and the escape character \n for writing a new line: >> a = rand(,00); name = 'a'; fprintf('the last element of vector %s is: %f\n', name, a(end)); The last element of vector a is:

13 This function can be very powerful, as it allows you to control, for example, the number of decimal digits to be included, to choose or not an exponential notation, etc For a complete list of its capabilities and special characters, you can type doc fprintf or help fprintf 52 Plots There are many functions in MATLAB that can create plots reviewed in the sequel Only a few of them are plot(x,y,) This function plots a vector with x values against a vector of the same size with y values Let us consider two plots representing y = sin(x), z = sin(2x), where x [ π, π] On x, we will consider a discrete set of points with steps of size 0 and boundaries, so as to capture the values in π x = 32::32; y = sin(x); z = sin(2 x); Now, we create a figure and tell MATLAB to keep it open for plotting different curves (ie not opening new graphical windows) figure hold on The command plot with the vector arguments previously defined will create the plots We can also specify a third argument that indicates the color of the line, in this case 'r' that stands for red There are many options that can be specified on the plot command (color, line style, markers, line width, etc ), check help plot for more details The following code will produce this plot: Example 05 plot(x, y); plot(x, z, 'r'); Table 7 there are a few commands that can be used to modify how the plot looks like You can issue them line by line to see the effect they have on the figure: 3

14 Table 7: Commands modifying the layout of a figure Command set(gca,'fontsize', 24); ylim([ 5, 5]) xlim([ pi, pi]) xlabel('x'); ylabel('y'); title('sinusoidal plots'); legend({'sin(x)', 'sin(2x)'}); grid on box on set(gca, 'xtick', pi:pi/2:pi); set(gca, 'xtick', 'ytick', [, 0, ]); Description Make the fonts larger Change the limits on the y axis Change the limits on the x axis Add a label on the x axis Add a label on the y axis Add a title on the plot Add a legend Show a grid on the x and y axis marks Box the figure Modify the place where the grid lines and axis marks are located for x and y respectively After these commands, the previous plot will look like this: Sinusoidal plots sin(x) sin(2x) y x 4

15 Markers Let us now create an example of plot using some of the techniques shown in the previous sections (anonymous functions and logical indexing) Create a figure that allows for several plots: figure hold on Create a function f = x 2, evaluate it in x [0, 6]: f xˆ2; fx = 0::6; fy = f(fx); Plot the function with a discontinuous black line and a line width of 3: plot(fx, fy, ' k', 'linewidth', 3); Generate now 200 random points (x, y), x N (2, ) and y U(0, 4) x = random('normal', 2,,, 200); y = random('uniform', 0, 4,, 200); 2 Use logical indexing now to find those points that are above the line and plot them in different colors, only with markers: idx = y > f(x); plot(x(idx), y(idx), 'b '); plot(x( idx), y( idx), 'r '); Set the limits of the axis according to the random sample and add box and grid: xlim([min(x), max(x)]); ylim([min(y), max(y)]); grid on box on Additional tricks for customizing MATLAB plots are available at: imagesc(a) This function is useful to quickly show the contents of a matrix, scaling the data to use the whole color range Let us create a matrix that contains the basis for a color gradient 5

16 Matrix Code Result A = :00; A = repmat(a, 0, ); imagesc(a); colorbar; hist(v, n) The function hist(v) allows one to plot an histogram of the data contained in a vector v Optionally, you can input a second argument hist(v, N) to specify the number of bins in the plot Example 600 Histogram of the vector v, containing 0 4 random numbers generated from a standard normal distribution: v = randn(,e4); hist(v, 50);