Matlab Beginner Training Session Review: Introduction to Matlab for Graduate Research

Matlab Beginner Training Session Review:

Introduction to Matlab for Graduate Research

Non-Accredited Matlab Tutorial Sessions for beginner to intermediate level users

Winter Session Dates:  February 13, 2007 - March 7, 2007Session times: Tuesdays from 8:30am-10:00am, Wednesdays from 8:30pm-10:00amSession Locations: Humphrey Hall - Room 219

Instructors:

Robert Marino rmarino@biomed.queensu.ca

Course Website:

http://www.queensu.ca/neurosci/Matlab Training Sessions.htm

Last Semester

Weeks:

1. Introduction to Matlab and its Interface

2. Fundamentals (Operators)

3. Fundamentals (Flow)

4. Importing Data

5. Functions and M-Files

6. Plotting (2D and 3D)

7. Statistical Tools in Matlab

8. Analysis and Data Structures

Intermediate Sessions

Intermediate Lectures

•Term 1 review•Loading Binary Data•Nonlinear Curve Fitting•Statistical Tools in Matlab II•Creating Graphic User Interfaces (GUIs)

Other possible topics:•Writing ascii text data files•3D plotting and animating•Debugging Tools•Simulink Toolbox

Why Matlab?

Common Uses for Matlab in Research• Data Acquisition• Multi-platform, Multi Format data importing • Analysis Tools (Existing,Custom)• Statistics• Graphing• Modeling

Why Matlab?Multi-platform, Multi Format data importing • Data can be loaded into Matlab from almost any

format and platform• Binary data files (eg. REX, PLEXON etc.)• Ascii Text (eg. Eyelink I, II) • Analog/Digital Data files

PC

UNIX100101010

Subject 1 143

Subject 2 982

Subject 3 87 …

Why Matlab?Analysis Tools • A Considerable library of analysis tools exist for

data analysis• Provides a framework for the design, creation,

and implementation of any custom analysis tool imaginable

Why Matlab?

Graphing• A Comprehensive array of plotting options

available from 2 to 4 dimensions• Full control of formatting, axes, and other visual

representational elements

Why Matlab?

Modeling• Models of complex dynamic system interactions

can be designed to test experimental data

Understanding the Matlab Environment:

Executing CommandsBasic Calculation Operators:

+ Addition

- Subtraction

* Multiplication

/ Division

^ Exponentiation

Solving equations using variables• Matlab is an expression language• Expressions typed by the user are interpreted and evaluated

by the Matlab system• Variables are names used to store values • Variable names allow stored values to be retrieved for

calculations or permanently saved

Variable = Expression

Or

Expression

**Variable Names are Case Sensitive!

Using Matlab

>> x = 6

x = 6

>> y = 2

y = 2

>> x + y

Ans = 8

>> x * y

Ans = 12

>> x / y

Ans = 3

>> x ^ y

Ans = 36

Using Matlab

Working with Matrices

• Matlab works with essentially only one kind of object, a rectangular numerical matrix

• A matrix is a collection of numerical values that are organized into a specific configuration of rows and columns.

• The number of rows and columns can be any number

Example

3 rows and 4 columns define a 3 x 4 matrix having 12 elements

• A scalar is a single number and is represented by a 1 x 1 matrix in matlab.

• A vector is a one dimensional array of numbers and is represented by an n x 1 column vector or a 1 x n row vector of n elements

ExercisesEnter the following Matrices in matlab using spaces,

commas, and semicolons to separate rows and columns:

7231

9175

6211 553878122641

160

16

22

4

A = B =

C =

19246525

12100

2162855

42166418

D =

65D =

E = a 5 x 9 matrix of 1’s

ExercisesChange the following elements in each matrix:

7231

9175

661 553878122641

160

16

22

4

A = B =

C =

19246525

12100

2162855

42166418

D =

65D =

E = a 5 x 9 matrix of 1’s

76

76

76

76

76

0

0

0

Indexing MatricesA = [1 2 4 5

6 3 8 2]

• The colon operator can can be used to remove entire rows or columns

>> A(:,3) = [ ]

A = [1 2 5

6 3 2]

>> A(2,:) = [ ]

A = [1 2 5]

Matrix Operations

Scalar Operations• Scalar (single value) calculations can be can performed on

matrices and arrays

Basic Calculation Operators

+ Addition

- Subtraction

* Multiplication

/ Division

^ Exponentiation

Matrix Operations

Element by Element Multiplication• Element by element multiplication of matrices is performed with

the .* operator• Matrices must have identical dimensions

A = [1 2 B = [1 D = [2 2 E = [2 4 3 6]

6 3 ] 7 2 2 ]

3

3]

>>A .* D

Ans = [ 2 4

12 6]

Matrix Operations

Element by Element Division• Element by element division of matrices is performed with the ./

operator• Matrices must have identical dimensions

A = [1 2 4 5 B = [1 D = [2 2 2 2 E = [2 4 3 6]

6 3 8 2] 7 2 2 2 2]

3

3]

>>A ./ D

Ans = [ 0.5000 1.0000 2.0000 2.5000

3.0000 1.5000 4.0000 1.0000 ]

Matrix Operations

Matrix Exponents

• Built in matrix Exponentiation in Matlab is either:

1. A series of Algebraic dot products

2. Element by element exponentiation

Examples:• A^2 = A * A (Matrix must be square)• A.^2 = A .* A

Matrix Operations

Shortcut: Transposing Matrices• The transpose of a matrix is the matrix formed by interchanging

the rows and columns of a given matrix

A = [1 2 4 5 B = [1

6 3 8 2] 7

3

3]

>> transpose(A) >> B’

A = [1 6 B = [1 7 3 3]

2 3

4 8

5 2]

Matrix Operations

• Relational operators are used to compare two scaler values or matrices of equal dimensions

Relational Operators

< less than

<= less than or equal to

> Greater than

>= Greater than or equal to

== equal

~= not equal

Relational Operators

• Comparison occurs between pairs of corresponding elements • A 1 or 0 is returned for each comparison indicating TRUE or

FALSE• Matrix dimensions must be equal!

>> 5 == 5

Ans 1

>> 20 >= 15

Ans 1

Relational Operators

A = [1 2 4 5 B = 7 C = [2 2 2 2

6 3 8 2] 2 2 2 2]

Try:

>>A > B

>> A < C

Relational Operators

The Find Function

A = [1 2 4 5 B = 7 C = [2 2 2 2 D = [0 2 0 5 0 2]

6 3 8 2] 2 2 2 2]

• The ‘find’ function can also return the row and column indexes of of matching elements by specifying row and column arguments

>> [x,y] = find(A == 5)

• The matching elements will be indexed by (x1,y1), (x2,y2), …

>> A(x,y) = 10

A = [ 1 2 4 10

6 3 8 2 ]

Relational Operators

• Control flow capability enables matlab to function beyond the level of a simple desk calculator

• With control flow statements, matlab can be used as a complete high-level matrix language

• Flow control in matlab is performed with condition statements and loops

Control and Flow

Advantages of M-files

• Easy editing and saving of work• Undo changes• Readability/Portability - non executable comments can be

added using the ‘%’ symbol to make make commands easier to understand

• Saving M-files is far more memory efficient than saving a workspace

Matlab Scripts

• It is often necessary to only perform matlab operations when certain conditions are met

• Relational and Logical operators are used to define specific conditions

• Simple flow control in matlab is performed with the ‘If’, ‘Else’, ‘Elseif’ and ‘Switch’ statements

Condition Statements

If, Else, and Elseif

• An if statement evaluates a logical expression and evaluates a group of commands when the logical expression is true

• The list of conditional commands are terminated by the end statement

• If the logical expression is false, all the conditional commands are skipped

• Execution of the script resumes after the end statement

Basic form:

if logical_expression

commands

end

Condition Statements

Example

A = 6 B = 0

if A > 3

D = [1 2 6]

A = A + 1

elseif A > 2

D = D + 1

A = A + 2

end

What is evaluated in the code above?

Condition Statements

Switch• The switch statement can act as many elseif statements • Only the one case statement who’s value satisfies the original

expression is evaluated

Basic form:

switch expression (scalar or string)

case value 1

commands 1

case value 2

commands 2

case value n

commands n

end

Condition Statements

Example

A = 6 B = 0

switch A

case 4

D = [ 0 0 0]

A = A - 1

case 5

B = 1

case 6

D = [1 2 6]

A = A + 1

end

** Only case 6 is evaluated

Condition Statements

• Loops are an important component of flow control that enables matlab to repeat multiple statements in specific and controllable ways

• Simple repetition in matlab is controlled by two types of loops:

1. For loops

2. While loops

Loops

For Loops

• The for loop executes a statement or group of statements a predetermined number of times

Basic Form:

for index = start:increment:end

statements

end

** If ‘increment’ is not specified, an increment of 1 is assumed by matlab

Loops

For Loops

Examples:

for i = 1:1:100

x(i) = 0

end

• Assigns 0 to the first 100 elements of vector x• If x does not exist or has fewer than 100 elements,

additional space will be automatically allocated

Loops

For Loops

• Loops can be nested in other loops

A = [ ]

for i = 1:m

for j = 1:n

A(i,j) = i + j

end

end• Creates an m by n matrix A whose elements are the

sum of their matrix position

Loops

While Loops

• The while loop executes a statement or group of statements repeatedly as long as the controlling expression is true

Basic Form:

while expression

statements

end

Loops

While Loops

Examples:

A = 6 B = 15

while A > 0 & B < 10

A = A + 1

B = B – 2

end• Iteratively increase A and decrease B until the two

conditions of the while loop are met

** Be very careful to ensure that your while loop will eventually reach its termination condition to prevent an infinite loop

Loops

Breaking out of loops

• The ‘break’ command instantly terminates a for and while loop

• When a break is encountered by matlab, execution of the script continues outside and after the loop

Loops

Breaking out of loopsExample:

A = 6 B = 15

count = 1

while A > 0 & B < 10

A = A + 1

B = B + 2

count = count + 1

if count > 100

break

end

end

• Break out of the loop after 100 repetitions if the while condition has not been met

Loops

Functions in Matlab

• In Matlab, each function is a .m file– It is good protocol to name your .m file the same as

your function name, i.e. funcname.m

• function outargs=funcname(inargs);

Function outputinput

Importing Data

• Basic issue:– How do we get data from other sources into

Matlab so that we can play with it?

• Other Issues:– Where do we get the data?– What types of data can we import

• Easily or Not

Basics• Matlab has a powerful plotting engine that can

generate a wide variety of plots.

Generating Data• Matlab does not understand functions, it can

only use arrays of numbers.– a=t2

– b=sin(2*pi*t)– c=e-10*t note: matlab command is exp()– d=cos(4*pi*t)– e=2t3-4t2+t

• Generate it numerically over specific range• Try and generate a-e over the interval 0:0.01:2

t=0:0.01:10; %make x vector

y=t.^2; %now we have the appropriate y

% but only over the specified range

Quick Assignment 1• Plot a as a thick black line• Plot b as a series of red circles. • Label each axis, add a title and a legend

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (ms)

f(t)

Mini Assignment #1

t2

sin(2*pi*t)

Quick Assignment 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (ms)

f(t)

Mini Assignment #1

t2

sin(2*pi*t)

figureplot(t,a,'k','LineWidth',3); hold on;plot(t,b,'ro')xlabel('Time (ms)');ylabel('f(t)');legend('t^2','sin(2*pi*t)');title('Mini Assignment #1')

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

Mean and MedianMean: Average or mean value of a distributionMedian: 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) = 6.8mean(B) = 3.0000 4.5000 6.0000 (column-wise mean)mean(B,2) = 2.0000 4.0000 6.0000 6.0000 (row-wise mean)

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

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

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 BR = A AcorA BcorA B AcorB BcorB

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

1. Parametric and non-parametric comparisons

2. Curve Fitting

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)

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

Parametric Tests - TTESTExample:For the data from exercise 3

[H,P] = ttest2(var1,var2)

>> [H,P] = ttest2(var1,var2)

H =1

P = 0.00000000000014877

-3 -2 -1 0 1 2 3-4

-3

-2

-1

0

1

2

3

4

5

Var

iabl

e 1

Variable 2

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

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.

1 2 3 4 5 6 7 8 9 10

-2

0

2

4

6

8

10

12

Least squaresRobust