Matlab_Tutorial_PartII_2005 - MATLAB Tutorial Part II...

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MATLAB Tutorial: Part II. Conchi Ausín Olivera Econometrics Ph.D. Program in Business Administration and Quantitative Methods. April 4, 2005 Contents 1 Statistical analysis and simulation using MATLAB 2 1.1 Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Numerical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Descriptive statistics with the Statistics Toolbox of MATLAB . . . . . . . . . 3 1.2 Probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Probability distributions with the Statistics Toolbox of MATLAB. . . . . . . 6 1.3 Multiple linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Programming functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Multiple linear regression with the Statistics Toolbox of MATLAB . . . . . . 9 1.4 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.1 Bernoulli process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.2 Random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.3 Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.4 Autoregressive process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.5 Moving average process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Monte Carlo simulation of known results . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.1 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.2 Some properties of a random sample from a normal population . . . . . . . . 13 1.6 Optimization: Newton-Raphson method to obtain maximum likelihood estimators . 14 1
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1 Statistical analysis and simulation using MATLAB The Statistics Toolbox for MATLAB provides a large number of function for analyzing data. How- ever, in practical situations, we are often faced with the di culty of programming. In these notes, we will consider di ff erent problems that will be solved both by programming and by using the Statistical Toolbox for MATLAB. This toolbox includes functions for Probability distributions, Parameter estimations, Descriptive Statistics, Multivariate statistics, Linear and nonlinear mod- eling, Statistical plotting, Statistical Process Control, Design of Experiments, Hypothesis Tests, Cluster Analysis. See help stats . 1.1 Descriptive statistics 1.1.1 Numerical description We can de fi ne this simple function to compute the mean value of the elements of a vector. function xbar = mean1(x) % For a vector x, mean1(x) returns the mean of x. n = length(x); xbar = x(1); for i = 2:n xbar = xbar + x(i); end xbar = xbar/n; Now, you can type the following statement to see how this function works: >> x = rand(1000,1); >> m = mean1(x) The mean of x given by m should be close to 0.5 . Alternatively, we could have make use of the sum command to avoid including a for loop: function xbar = mean2(x) % For a vector x, mean1(x) returns the mean of x.. n = length(x); xbar = sum(x); xbar = xbar/n; Observe that the function mean2 (unlike mean1 ) can also be applied to matrices where each column is a sample of data from a di ff erent variable. For matrices, mean2(x) is a row vector containing the mean value of each column. We can construct a function to compute simultaneously the sample mean, ¯ x, standard deviation, s, skewness, SK, and kurtosis, K, for each column of a matrix where, ¯ x = n P i =1 x i n , s = v u u u t n P i =1 ( x i ¯ x ) 2 n , SK = n P i =1 ( x i ¯ x ) 3 ns 3 , K = n P i =1 ( x i ¯ x ) 3 ns 4 . function [xbar,s,SK,K] = statistics(X) 2
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% X is a matrix whose columns are sample % of data from a certain variable.
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