IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

IEEE upgrading knowledgeMatLab workshop (second part)

Georgios Drakopoulos

CEID

December 15, 2013

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Agenda

Topics

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Agenda

Topics

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Overview

Official MatLab users community

Questions can be posted and answered.

Feedback required.

Login credentials required; Free account.

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Agenda

Topics

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

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Final notes

Cipher system

Data

A lowercase letter string.

Task

A string where each original letter is shifted right by three.

Wrap around ’z’.

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GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

First approach

Code

function y = cipher1(x)%CIPHER1 First approach to cipher system.% y = cipher1(x) transforms x into cipher y using% arithmetic operations.

if isa(x, 'char')na = 'a'; nz = 'z';y = na + mod(((x − na) + 3), nz − na + 1);y = char(y);

elseerror('Input is not a string.');

endreturn

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

if conditional

Syntax

if x == 5disp('five!');

elseif x == 6disp('now six!');

elsedisp('something else!');

end

Notes

Equality with ==

Assignment with =

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Class check

Syntax

isa(x, 'char')

Frequent classes

single

double

logical

(u)int8/16/32/64

cell

struct

function handle

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Assessment

Plus

Simple.

Quick.

Minus

Assumes linear character representation.

True for ASCII character set.

Still machine dependent.

Notes

Frequently used by Julius Cæsar.

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GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Second approach

Code

function y = cipher2(x)%CIPHER2 Second approach to cipher system.% y = cipher2(x) transforms x into cipher y using tables.

T = 'a':'z'; %not proper but ok ..

if isa(x, 'char')n = length(x);for k = 1:n

xpos = find(x(k) == T);ypos = circshift(xpos, 3);y = [y T(ypos)];

end;else

error('Input is not a string.');endreturn

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GeorgiosDrakopoulos

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Cipher system

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Final notes

for loop

Syntax

for k = a:s:b% do something with kend

for k = a:b% assumes s = 1end

Notes

a must be lower than b when s is positive.

a must be greater than b when s is negative.

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

while loop

Syntax

while a > b% do somethingend

Notes

for and while are equivalent.

One form may be more convenient.

Be careful when bounds change.

Make sure loop terminates.

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GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

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Final notes

find function

Syntax

% find non−zero entries of X by defultI = find(X);[I, J] = find(X);

% find positive entries of XI = find(X > 0)

% find the first n positive entries of XI = find(X > 0, n);

Notes

Use sub2ind to switch from pairwise to linear.

Use ind2sub to switch from linear to pairwise.

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Help functions

help

Displays text based help.

Preamble comments are important.

doc

Graphical help.

Extended function description.

lookfor

Term based search.

Similar to UNIX apropos command.

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Wrapper function

Code

function y = cipherw(x, method)%CIPHERW Wrapper function to ciphper1 and cipher2.

switch nargincase 1

y = cipher2(x);case 2

switch methodcase 'numeric'

y = cipher1(x);case 'table'

y = cipher2(x);otherwise

error('Incorrect method.');end

endreturn

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MatLab central

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Final notes

switch conditional

Syntax

switch methodcase 'zero'

disp('It''s zero point!');case { 'one', 'two' }

fprintf('%s\n', 'A string');case 4

pause(4)otherwise

warning('Droplets in the ocean!');end

Notes

cell groups strings and matrices.

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

break and continue

Syntax

for k = 1:−0.005:1if k ˜= 0% do somethingelse

continueend

end

Notes

break terminates innermost loop.

continue jumps to next iteration.

Same functionality with their C counterparts.

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Agenda

Topics

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Least squaresThe problem

Data

n observations in (xk , yk) tabular format.

Linear relation between x and y .

yk = α0 xk + β0, 1 ≤ k ≤ n

Result

Compute α0 and β0 which minimize the cost function

J (α0 , β0) =n∑

k=1

(yk − (α0 xk + β0))2

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GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

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Final notes

Least squaresSolution

Setting the derivative to zero yieldsx1 1x2 1...

...xn 1

︸ ︷︷ ︸

A

[α0

β0

]︸ ︷︷ ︸

x

=

y1

y2

...yn

︸ ︷︷ ︸

b

Least squares solution (ATA

)x̂ = AT b

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Final notes

Least squaresThe task

Formulation[∑nk=1 x

2k

∑nk=1 xk∑n

k=1 xk n

] [α0

β0

]=

[∑nk=1 xk yk∑nk=1 yk

]

Data

Create the straight line y = 5x − 1

α0 = 5β0 = −1

Corrupt it with noise.

Record performance vs noise power.

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Cipher system

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Final notes

Sneak peek

Code

a0 = 5;b0 = −1;x = 0:15;yc = a0*x + b0;

rng('shuffle');Ey = norm(yc);y = yc + sqrt(0.05*Ey)*randn(size(x));save lsqtest a0 b0 x y

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GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

First approach

Code

function xls = lsq1(x, b)%LSQ1 First approach to least squares.% xls = lsq1(x, b) solves normal equations.

T = [ norm(x)ˆ2 sum(x) ; sum(x) length(x) ];g = [ dot(x,b) ; sum(b) ];xls = inv(T) * g; %not always ok ...return

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GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Second approach

Code

function xls = lsq2(x, b)%LSQ2 Second approach to least squares.% xls = lsq2(x, b) exploits slash operator.

T = [ x(:) ones(length(x), 1) ];xls = T \ b(:);return

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Wrapper functionControl struct

Code

load lsqtests = struct(

'x', x, ...'b', y, ....'method', 'slash' );

Notes

Older functions rely on nargin.

Newer functions rely on struct.

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GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

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Final notes

struct handling

Functions

getfield

setfield

isfield

rmfield

fieldnames

Notes

z = struct('a', 1, 'b', 2);z = setfield(z, 'c', 3);

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Wrapper functionFunction

Code

function xls = lsqw(s)%LSQW Least squares wrapper function.% xls = lsqw(s)

switch s.methodcase 'slash'

xls = lsq2(s.x, s.b);case 'normal'

xls = lsq1(s.x, s.b);otherwise

warning('Incorrect method.');endreturn

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

PlotNoisy data

Code

xi = linspace(min(x), max(x), 10*length(x));yi = interp1(x, y, xi, 'spline');figurehold onplot(x, y, 'k'), plot(xi, yi, 'r'), plot(x, y, 'ro')xlabel('Variable x')ylabel('Variable y ')legend('Linear', 'Cubic', 'Discrete', −1)title('Least squares problem (linear and cubic data fit)')print(gcf, '−depsc2', 'ieee2013 matlab2 lsq00.eps')saveas(gcf, 'lsq00', 'fig')hold off

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

PlotNoisy data (figure)

0 5 10 15−10

0

10

20

30

40

50

60

70

80

Variable x

Variable

y

Least squares problem (linear and cubic data fit)

Linear

Cubic

Discrete

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MatLab central

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Final notes

PlotProjected data

Code

yp = xls2(1) * x + xls2(2);plot(x, yc, 'k', x, y, 'r', x, yp, 'b')xlabel('Variable x')ylabel('Variable y = \alpha 0 x + \beta 0')title('Least squares problem ...

(argmin \{ | | y − A x | | 2ˆ2 \})')saveas(gcf, 'lsq01', 'fig')print(gcf, '−depsc2', 'fig01.eps')

Notes

Limited LATEX support in title and x/ylabel.

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GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

PlotProjected data (figure)

0 5 10 15−10

0

10

20

30

40

50

60

70

80

Variable x

Variable

y =

α0 x

+ β

0

Least squares problem (argmin { || y − A x||2

2 })

Original

Given

Projected

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Graphics

Handle

plot

hold

stem

figure

subplot

xlabel

title

legend

Print

print

saveas

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Agenda

Topics

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Another (hello) worldFunction

Code

function dyn(s)%DYN Displays dynamic hello world.%% dyn(s) displays "s: Hello world!"

z = [ 'disp( [ '' ' ...char(s) ...' '' '': Hello world!''] )' ];

disp(z);eval(z);return

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Another (hello) worldAssessment

Plus

Shorter code.

Flexible code.

Easy to understand.

Minus

Overhead.

Difficult to understand.

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Agenda

Topics

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

IEEE upgradingknowledge

GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Code accelerationOverview

Topics

Critical issue.

Sometimes even for proof of concept.

MatLab JIT compiler slow code compensation.

Vectorization.

Column-based operations.

MatLab uses column-major storage format.

Memory preallocation.

Known operand sizes.

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Fast forward

Topics

mex connects C to MatLab.

matlab compiler compiles MatLab code.

Eclipse plugin for MatLab.

Package mcode for LATEX.

Alternatives

Octave.

ScalaLab.

SciLab.

NumPy.

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GeorgiosDrakopoulos

MatLab central

Cipher system

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Final notes

Final message

In brief

Join MatLab community.

Exploit MatLab flexibility.

Be careful though.

Utilize help, doc, and lookfor.

There is more than one way to do things.

Wrapper functions.

save intermediate data.

Optimize.

Have fun!

Programming is fun!

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GeorgiosDrakopoulos

MatLab central

Cipher system

Least squares

Dynamic code

Final notes

Thank you!Questions?