Matlab Training Sessions 8: Introduction to Statistics

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Matlab Training Sessions 8Introduction to
Statistics
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• Course Outline
• Weeks
• Introduction to Matlab and its Interface (Jan 13
  2009)
• Fundamentals (Operators)
• Fundamentals (Flow)
• Functions and M-Files
• Importing Data
• Plotting (2D and 3D)
• Plotting (2D and 3D)
• Statistical Tools in Matlab
• Additional classes will begin next week (Feb 10
  2009) and will continue from where the first 8
  sessions left off.
• These sessions will be run by Andrew Pruszynski
  (4jap1_at_qlink.queensu.ca)
• Course Website
• http//www.queensu.ca/neurosci/matlab.php

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• Week 8 Lecture Outline
• Basic Matlab Statistics
• Basic Matlab Statistics
• Mean, Median, Variance
• Correlations
• Statistics Toolbox
• Parametric and Non-parametric statistical tests
• Curve fitting

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Part A Basics

• The Matlab installation contains basic
  statistical tools.
• Including, mean, median, standard deviation,
  error variance, and correlations
• More advanced statistics are available from the
  statistics toolbox and include parametric and
  non-parametric comparisons, analysis of variance
  and curve fitting tools

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Mean and Median
Mean Average or mean value of a
distribution Median Middle value of a sorted
distribution M mean(A), M median(A) M
mean(A,dim), M median(A,dim) M mean(A), M
median(A) Returns the mean or median value of
vector A. If A is a multidimensional mean/median
returns an array of mean values. Example A
0 2 5 7 20 B 1 2 3
3 3 6 4 6 8 4 7 7
mean(A) 6.8 mean(B) 3.0000 4.5000 6.0000
(column-wise mean) mean(B,2) 2.0000 4.0000
6.0000 6.0000 (row-wise mean)
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Mean and Median
Examples A 0 2 5 7 20 B 1 2 3
3 3 6 4 6
8 4 7 7 Mean mean(A) 6.8 mean(B)
3.0 4.5 6.0 (column-wise mean) mean(B,2) 2.0
4.0 6.0 6.0 (row-wise mean) Median median(A)
5 median(B) 3.5 4.5 6.5 (column-wise
median) median(B,2) 2.0
3.0 6.0
7.0 (row-wise median)
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Standard Deviation and Variance

• Standard deviation is calculated using the std()
  function
• std(X) Calcuate the standard deviation of
  vector x
• If x is a matrix, std() will return the standard
  deviation of each column
• Variance (defined as the square of the standard
  deviation) is calculated using the var() function
• var(X) Calcuate the variance of vector x
• If x is a matrix, var() will return the standard
  deviation of each column

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Standard Error of the Mean

• Often the most appropriate measure of
  error/variance is the standard error of the mean
• Matlab does not contain a standard error function
  so it is useful to create your own.
• The standard error of the mean is defined as the
  standard deviation divided by the square root of
  the number of samples

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Standard Error of the Mean

• In Class Exercise 1
• Create a function called se that calculates the
  standard error of some vector supplied to the
  function
• Eg. se(x) should return the standard error of
  matrix x

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Standard Error of the Mean

• In Class Exercise 1 Solution
• function result se(input_vect)
• result STD(input_vect)/sqrt(length(input_vec
  t))
• return

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In Class Exercise 2

• From the class website download the file
  testdata1.txt (http//www.queensu.ca/neurosci/matl
  ab.php)
• This text file contains data from two subjects
  arranged in columns
• Load the text file into matlab using any method
  you like (load, import, textread(), fscanf())
• Calculate the mean and standard error for each
  subject
• In figure 1, plot the data distribution for each
  subject using the hist() plotting function
• In figure 2, plot the mean and standard error of
  each subject using a bar graph (bar() function
  and errorbar() functions).

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In Class Exercise 2Solution
read data subj1, subj2 textread('testdata1.t
xt','ff','headerlines',1) plot distributions
of each subject figure(1) hold on subplot(2,1,1) h
ist(subj1) subplot(2,1,2) hist(subj2) plot mean
and standard error on bar graph figure(2) hold
on bar(1,2,mean(subj1),mean(subj2)) errorbar(
1,2,mean(subj1),mean(subj2),se(subj1),
se(subj2),'r')
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In Class Exercise 2Solution
Subject 1
Subject 2
Subject 1
Subject 2
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Data Correlations

• Matlab can calculate statistical correlations
  using the corrcoef() function
• R,P corrcoef(A,B)
• Calculates a matrix of R correlation
  coefficiencts and P significance values (95
  confidence intervals) for variables A and B
• A B
• R A AcorA BcorA
• B AcorB BcorB

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Data Correlations

• Matlab can calculate statistical correlations
  using the corrcoef() function
• R,P corrcoef(A,B)
• Calculates a matrix of R correlation
  coefficiencts and P significance values (95
  confidence intervals) for variables A and B
• A B
• R A AcorA BcorA 1
  BcorA
• B AcorB BcorB AcorB
  1

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Data Correlations

• Matlab can calculate statistical correlations
  using the corrcoef() function
• R,P corrcoef(A,B)
• Calculates a matrix of R correlation
  coefficiencts and P significance values (95
  confidence intervals) for variables A and B
• A B
• R A AcorA BcorA 1
  BcorA
• B AcorB BcorB AcorB
  1
• A B
• P A sig(AcorA) sig(BcorA)
  1 sig(BcorA)
• B sig(AcorB) sig(BcorB)
  sig(AcorB) 1

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Data Correlations
Variable 1
Variable 2
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Data Correlations
Variable 1
Variable 2
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Data Correlations
Compute sample correlation r, p
corrcoef(var1,var2)
Variable 1
Variable 2
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Data Correlations
Compute sample correlation r, p
corrcoef(var1,var2) r 1.0000 0.7051
0.7051 1.0000 p 1.0000 0.0000
0.0000 1.0000
Variable 1
Variable 2
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In Class Exercise 3

• From the class website download the file
  testdata2.txt (http//www.queensu.ca/neurosci/matl
  ab.php)
• This text file contains data from variables
  arranged in columns
• Load the text file into matlab using any method
  you like (load, import, textread(), fscanf())
• Plot the data points
• Calculate the Correlation

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In Class Exercise 3Solution
read data var1, var2 textread('testdata2.txt'
,'ff','headerlines',1) Compute sample
correlation r corrcoef(var1,var2) Plot
data points figure(1) plot(var1,var2,'ro')
Variable 1
Variable 2
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Part B Statistics Toolbox

• The Statistics tool box contains a large array of
  statistical tools.
• This lecture will concentrate on some of the most
  commonly used statistics for research
• Parametric and non-parametric comparisons
• Curve Fitting

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Comparison of Means

• A wide variety of mathametical methods exist for
  determining whether the means of different groups
  are statistically different
• Methods for comparing means can be either
  parametric (assumes data is normally distributed)
  or non-parametric (does not assume normal
  distribution)

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Parametric Tests - TTEST

• H,P ttest2(X,Y)
• Determines whether the means from matrices X and
  Y are statistically different.
• H return a 0 or 1 indicating accept or reject nul
  hypothesis (that the means are the same)
• P will return the significance level

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Parametric Tests - TTEST

• H,P ttest2(X,Y)
• Determines whether the means from matrices X and
  Y are statistically different.
• H return a 0 or 1 indicating accept or reject nul
  hypothesis (that the means are the same)
• P will return the significance level

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Parametric Tests - TTEST

• Example
• For the data from exercise 3
• H,P ttest2(var1,var2)
• gtgt H,P ttest2(var1,var2)
• H 1
• P 0.00000000000014877

Variable 1
Variable 2

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Non-Parametric Tests Ranksum

• The wilcoxin ranksum test assesses whether the
  means of two groups are statistically different
  from each other.
• This test is non-parametric and should be used
  when data is not normally distributed
• Matlab implements the wilcoxin ranksum test using
  the ranksum() function
• ranksum(X,Y) statistically compares the means of
  two data distributions X and Y

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Non-Parametric Tests - RankSum

• Example
• For the data from exercise 3
• P,H ranksum(var1,var2)
• P 1.1431e-014
• H 1

Variable 1

Variable 2
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Curve Fitting

• Plotting a line of best fit in Matlab can be
  performed using either a traditional least
  squares fit or a robust fitting method.

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10
8
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Least squares

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Robust
2
0
-2
1
2
3
4
5
6
7
8
9
10
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Curve Fitting

• A least squares linear fit minimizes the square
  of the distance between every data point and the
  line of best fit
• polyfit(X,Y,N) finds the coefficients of a
  polynomial P(X) of degree N that fits the data
• Uses least-square minimization
• N 1 (linear fit)
• P polyfit(X,Y,N) returns P, a matrix
  containing the slope and the x intercept for a
  linear fit
• Y polyval(P,X) calculates the Y values for
  every X point on the line of best fit

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Curve Fitting

• Example
• Draw a line of best fit using least squares
  approximation for the data in exercise 2
• var1, var2 textread('testdata2.txt','ff','he
  aderlines',1)
• P polyfit(var1,var2,1)
• Y polyval(P,var1)
• close all
• figure(1)
• hold on
• plot(var1,var2,'ro')
• plot(var1,Y)

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Curve Fitting

• A least squares linear fit minimizes the square
  of the distance between every data point and the
  line of best fit
• P robustfit(X,Y) returns the vector B of the y
  intercept and slope, obtained by performing
  robust linear fit

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Curve Fitting

• Example
• Draw a line of best fit using robust fit
  approximation for the data in exercise 2
• var1, var2 textread('testdata2.txt','ff','he
  aderlines',1)
• P robustfit(var1,var2,1)
• Y polyval(P(2),P(1),var1)
• close all
• figure(1)
• hold on
• plot(var1,var2,'ro')
• plot(var1,Y)

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Ideas for Next Term?

• Additional Statistics, ANOVAs ect..
• Curve fitting with quadratic functions and cubic
  splines
• Algorithms and Data structures
• Improving Program Execution Time
• Assistance Tutorials for individual programming
  problems
• Any Suggestions?

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Getting Help

• Help and Documentation
• Digital
• Accessible Help from the Matlab Start Menu
• Updated online help from the Matlab Mathworks
  website
• http//www.mathworks.com/access/helpdesk/help/tech
  doc/matlab.html
• Matlab command prompt function lookup
• Built in Demos
• Websites
• Hard Copy
• Books, Guides, Reference
• The Student Edition of Matlab pub. Mathworks Inc.