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AMS597 Spring 2011 Hao Han April 05,

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Introduction to MATLAB The name MATLAB stands for MATrix LABoratory. Typical uses include: Math and computation – Algorithm development – Data acquisition – Modeling, simulation, and prototyping Data analysis, exploration, and visualization – Scientific and engineering graphics – Application development, including graphical user interface (GUI) building We will focus on the statistical computing in MATLAB. 2

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3 Desktop Tools & Development Environment Workspace Browser – View and make changes to the contents of the workspace. Command Windows – Run MATLAB statements (commands). Hotkey: Ctrl+c -> break while the status is busy M-file Editor – Creating, Editing, Debugging and Running Files.

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MATLAB Variables Variable names are case sensitive. Variable names must start with a letter and can be followed by digits and underscores. MATLAB does not require any type of declarations or dimension statements. When it encounters a new variable name, it automatically creates the variable and allocates the appropriate amount of storage. For example: New_student = 25; To view the matrix assigned to any variable, simply enter the variable name. Special Variables: pivalue of π epssmallest incremental number inf infinity NaNnot a number realminthe smallest usable positive real number realmaxthe largest usable positive real number 4

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MATLAB Matrices MATLAB treats all variables as rectangular matrices. Separate the elements of a row with blanks or commas. Use a semicolon ‘;’ to indicate the end of each row. Surround the entire list of elements with square brackets ‘[ ]’. Claim a matrix: a = [1 2 3; 4 5 6; 7 8 9] a = Subscripts: the element in row i and column j of A is denoted by A(i,j). a(3,2)=8 or a(6)=8 Claim a scalar: x = 2; Claim a row vector: r = [1 2 3] r = [1,2,3] Claim a column vector: c = [1;2;3] c = [1 2 3]’ 5

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Matrix Manipulations The Colon Operator: 1:5 is a row vector containing integers from 1 to 5. To obtain non-unit spacing, specify an increment. For example, 100:-7:50 Extracting a sub-matrix: Sub_matrix = matrix(r1:r2,c1:c2); sub_a = a(2:3,1:2) sub_a = Replication: b = [1 2; 3 4]; b_rep = repmat(b,1,2) b_rep = Concatenation: c = ones(2,2); c_cat = [c 2*c; 3*c 4*c] c_cat = c_cat = cat(DIM,A,B); Deleting rows or columns: c_cat(:,2)=[]; 6

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Structures and Cell Arrays Structure Cell Array Way of organizing related data Create a structure, s, with fields, x, y, and name s.y = 1; s.x = [1 1]; s.name = 'foo'; % or equivalenty s2 = struct('y',1,'x',[1 1],'name','foo'); Test for equality: % works for any s1, s2 isequal(s1,s2); Cell arrays can have entries of arbitrary datatype % create 3 by 2 cell array a = cell(3,2); a{1,1} = 1; a{3,1} = 'hello'; a{2,2} = randn(100,100); Using cell arrays with other datatypes can be tricky % create 2 by 1 cell array a = {[1 2], 3}; y = a{1}; % y is 1 by 2 numeric array ycell =a(1); % is 1 by 1 cell array x = y+1; % allowed xcell = ycell+1; % not allowed onetwothree = [a{1:2}]; % = [1 2 3] 7

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MATLAB Operators Relational operators: Less than< Less than or Equal<= Great than or Equal>= Equal to== Not equal to ~= Logical operators: not ~ % highest precedence and & % equal precedence with or or| % equal precedence with and Matrix computations: + - * / ^ A’; % transpose A \ b; % returns x s.t. A*x=b A / b; % returns x s.t. x*A=b Element wise operators: + Addition - Subtraction.*Element-by-element multiplication./Element-by-element division.\Element-by-element left division.^Element-by-element power.' Unconjugated array transpose 8

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MATLAB Functions MATLAB provides a large number of standard elementary mathematical functions, including abs, sqrt, exp, and sin. For a list of the elementary mathematical functions, type: help elfun For a list of more advanced mathematical and matrix functions: help specfun help elmat Seek help for MATLAB function references, type: help somefun or more detailed doc somefun 9

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Flow Control (‘if’ statement) The general form of the ‘ if ’ statement is if expression … elseif expression … else … end Example 1: if i == j a(i,j) = 2; elseif i >= j a(i,j) = 1; else a(i,j) = 0; end Example 2: if (common>60)&&(area>60) pass = 1; end 10

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Flow Control (‘switch’ statement) switch Switch among several cases based on expression The general form of the switch statement is: switch switch_expr case case_expr1 … case case_expr2 … otherwise … end Example : x = 2, y = 3; switch x case x==y disp('x and y are equal'); case x>y disp('x is greater than y'); otherwise disp('x is less than y'); end % x is less than y 11

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Flow Control (‘for’ loop) for Repeat statements a specific number of times The general form of a for statement is for variable=expression … end Example 1: for x = 0:0.05:1 fprintf('%3.2f\n',x); end Example 2: a = zeros(3,4); for i = 1:3 for j = 1:4 a(i,j) = 1/(i+j); end 12

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Flow Control (‘while’ loop) while Repeat statements an indefinite number of times The general form of a while statement is while expression … end Example 1: n = 1; y = zeros(1,10); while n <= 10 y(n) = 2*n/(n+1); n = n+1; end Example 2: x = 1; while x %execute statements end 13

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Flow Control (‘break’ statement) break terminates the execution of for and while loops In nested loops, break terminates from the innermost loop only Example: y = 3; for x = 1:10 fprintf( ' %d\n ',x); if (x>y) break; end % Question: what is the output? 14

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Graphics: 2-D plot Basic commands: Example 1 [plot(vector)]: plot(x, 's') plot(x,y, 's') plot(x1, y1, 's1', x2,y2, 's2', …) title('…') xlabel('…') ylabel('…') legend('…', '…') x=0:pi/10:2*pi; x=[sin(x)' cos(x)']; figure; plot(x) 15

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Graphics: 2-D plot (cont’d) Example 2: Example 3 [plot(vector,matrix)]: t=(0:pi/50:2*pi)'; k=0.4:0.1:1; Y=cos(t)*k; plot(t,Y) x = 0:0.01:2*pi; y = sin(x); z = cos(x); hold on; plot(x,y, 'b'); plot(x,z, 'g'); hold off; 16

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Graphics: 2-D plot (cont’d) plot(x1, y1,’s1’, x2,y2,’s2’, …) t=(0:pi/100:pi)'; y1=sin(t)*[1,-1]; y2=sin(t).*sin(9*t); t3=pi*(0:9)/9; y3=sin(t3).*sin(9*t3); plot(t,y1,'r:',t,y2,'b',t3,y3,'bo') axis([0,pi,-1,1]) Linetype - : Color b g r c m y k w Markertype. + * ^ v d h o p s x plot(t,y1,'.r',t,y2, 'b+',t3,y3,'ob:') 17

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Subplots >> subplot(2,2,1) >> … >> subplot(2,2,2) >> … >> subplot(2,2,3) >> … >> subplot(2,2,4) >> … 18

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Graphics: 3-D plot plot3(x,y,z) t=(0:0.02:2)*pi;x=sin(t);y=cos(t);z=cos(2*t); plot3(x,y,z,'b-',x,y,z,'bd'); view([-82,58]); box on; legend('Chain','Gemstone') 19

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Basic Data Analysis Import/Export data: Use the system import wizard File -> import data -> find and open files -> finish Use commands as follows: 1. help load & help save 2. help xlsread & help xlswrite % Reading into a text file fid = fopen(‘filename.txt’,‘r’); X = fscanf(fid,‘%5d’); % or fread fclose(fid); % Writing onto a text file fid = fopen(‘filename.txt’,‘w’); count = fwrite(fid,x); % or fprintf fclose(fid); Scatter plot Statistics Toolbox: help stats Basic Data Analysis Function (help datafun) FunctionDescription cumprod Cumulative product of elements. cumsum Cumulative sum of elements. cumtrapz Cumulative trapezoidal numerical integration. diff Difference function and approximate derivative. max Largest component. mean Average or mean value. median Median value. min Smallest component. prod Product of elements. sort Sort array elements in ascending or descending order. sortrows Sort rows in ascending order. std Standard deviation. sum Sum of elements. trapz Trapezoidal numerical integration. covCovariance matrix corrcoefCorrelation coefficients 21

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22 Data Preprocessing Missing values: You should remove NaNs from the data before performing statistical computations. Removing outliers: You can remove outliers or misplaced data points from a data set in much the same manner as NaNs. 1. Calculate the mean and standard deviation from the data set. 2. Get the column of points that lies outside the 3*std. (3σ-rule) 3. Remove these points CodeDescription i = find(~isnan(x)); x = x(i) Find indices of elements in vector that are not NaNs, then keep only the non-NaN elements. x = x(find(~isnan(x)))Remove NaNs from vector. x = x(~isnan(x));Remove NaNs from vector (faster). x(isnan(x)) = [];Remove NaNs from vector. X(any(isnan(X)'),:) = [];Remove any rows of matrix X containing NaNs.

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23 Regression and Curve Fitting The easiest way to find estimated regression coefficients efficiently is by using the MATLAB backslash operator. Note that we should avoid matrix inversion (from slow to fast…): % Fit X*b=Y xx = x’*x;xy=x’*y; tic; bhat1 = (xx)ˆ(−1)*xy; toc; tic; bhat2 = inv(xx)*xy; toc; tic; bhat3 = xx \ xy; toc; Other ways use build-in functions: regress() or glmfit() Multiple linear regression model: y = b 0 + b 1 x 1 + b 2 x 2 + … Example: Suppose you measure a quantity y at several values of time t. t=[ ]'; y=[ ]'; plot(t,y,'o') grid on

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Regression Example (cont’d) Polynomial regression: There are six equations in three unknowns, represented by the 6-by-3 matrix X = [ones(size(t)) t t.^2] The solution is found with the backslash operator. a = X\y a = Now evaluate the model at regularly spaced points and overlay the original data in a plot. T=(0:0.1:2.5)'; Y=[ones(size(T)) T T.^2]*a; plot(T,Y,'-',t,y,'o') grid on 24

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Regression Example (cont’d) Linear-in-the-parameters regression, e.g. exponential function: X = [ones(size(t)) exp(-t) t.*exp(-t)]; a = X\y a = T=(0:0.1:2.5)'; Y=[ones(size(T)) exp(-T) T.*exp(-T)]*a; plot(T,Y,'-',t,y,'o') grid on 25

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