Introduction to Matlab & Data analysis

Report
Introduction to Matlab
& Data Analysis
Lecture 13:
Using Matlab for Numerical Analysis
Weizmann 2010 ©
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Outline

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Data Smoothing
Data interpolation
Correlation coefficients
Curve Fitting
Optimization
Derivatives and integrals
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Filtering and Smoothing

Assume we measured the response in time or other input
factor, for example:
 Reaction product as function of substrate
 Cell growth as function of time
response
factor
Our measuring device has some random noise
One way to subtract the noise from the results is to smooth
each data point using its close environment

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Smoothing –
Moving Average
span
Remark: The Span should be odd
span
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Smoothing –
Behavior at the Edges
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The Smooth Function
x = linspace(0, 4 * pi, len_of_vecs);
y = sin(x) + (rand(1,len_of_vecs)-0.5)*error_rat;
Data:
y
Generating Function:
sin(x)
Smoothed data:
smooth(x,y)
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The Smooth Quality Is Affected By
The Smooth Function And The Span
y_smooth = smooth(x,y,11,'rlowess');
Like very low pass filter
Different method
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Data Interpolation Definition
Interpolation A way of estimating values of a function
between those given by some set of data points.
Interpolation
Data points
“plot” – Performs linear interpolation between the data points
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Interpolating Data Using
interp1 Function
x_full = linspace(0, 2.56 * pi, 32);
y_full
= sin(x_full);
x_missing = x_full;
x_missing([14:15,20:23]) = NaN;
y_missing = sin(x_missing);
x_i = linspace(0, 2.56 * pi, 64);
Data points which we want to interpolate
not_nan_i = ~isnan(x_missing);
y_i = …
interp1(x_missing(not_nan_i),…
y_missing(not_nan_i),…
x_i);
Default: Linear interpolation
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interp1 Function Can Use
Different Interpolation Methods
y_i=interp1(x_missing,y_missing,x_i,'cubic');
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2D Functions Interpolation


Also 2D functions can be
interpolated
Assume we have some data
points of a 2D function
xx
yy
[X,Y]
Z
=
=
=
=
-2:.5:2;
-2:.5:3;
meshgrid(xx,yy);
X.*exp(-X.^2-Y.^2);
Surf uses linear
interpolation
figure;
surf(X,Y,Z);
hold on;
plot3(X,Y,Z+0.01,'ok', 'MarkerFaceColor','r')
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2D Functions Interpolation

interp2 function
xx_i = -2:.1:2;
yy_i = -2:.1:3;
[X_i,Y_i] = meshgrid(xx_i,yy_i);
Z_i = interp2(xx,yy,Z,X_i,Y_i,'cubic');
Data points
Points to interpolate
figure;
surf(X_i,Y_i,Z_i);
hold on;
plot3(X,Y,Z+0.01,'ok', 'MarkerFaceColor','r')
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Optimization and Curve Fitting
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Curve Fitting –
Assumptions About The Residuals
Residual
Y
y
yˆ

Residual = Response – fitted response: r  y  y
X

n


n
 2
Sum square of residuals S   ri   ( yi  yi )
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Two assumptions:


1
i 1
This is iwhat
we
want to minimize
The error exists only in the response data,
and not in the predictor data.
The errors are random and follow a normal
(Gaussian) distribution with zero mean and
constant variance.
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corrcoef Computes the
Correlation coefficients

Consider the following data:
x = sort(repmat(linspace(0,10,11),1,20));
y_p = 10 + 3*x + x.^2 + (rand(size(x))-0.5).*x*10;

In many cases we start with
computing the correlation
between the variables:
cor_mat = corrcoef(x , y_p);
cor = cor_mat(1,2);
figure;
plot(x,y_p,'b.');
xlabel('x');ylabel('y');
title(['Correlation: ' num2str(cor)]);
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Curve fitting Using a GUI
Tool (Curve Fitting Tool Box)


cftool – A graphical tool
for curve fitting
Example:

Fitting
x_full =
linspace(0, 2.56 * pi, 32);
y_full = sin(x_full);

With cubic polynomial
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polyfit Fits a Curve By a
Polynomial of the Variable

Find a polynomial fit:
poly_y_fit1 = polyfit(x,y_p,1);
poly_y_fit1 = 12.6156 X + ( -3.3890 )
y_fit1 = polyval(poly_y_fit1,x);
y_fit1 = 12.6156*x-3.3890
poly_y_fit2 = polyfit(x,y_p,2);
y_fit2 = polyval(poly_y_fit2,x);
poly_y_fit3 = polyfit(x,y_p,3);
y_fit3 = polyval(poly_y_fit3,x);

We can estimate
the fit quality by:
mean((y_fit1-y_p).^2)
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We Can Use polyfit to Fit Exponential
Data Using Log Transformation
poly_exp_y_fit1 =
1.9562
5.0152

Polyfit on the log of
the data:
x = sort(repmat(linspace(0,1,11),1,20));
y_exp = exp(5 + 2*x + (rand(size(x))-0.5).*x);
poly_exp_y_fit1 = polyfit(x,log(y_exp),1);
y_exp_fit1 = exp(polyval(poly_exp_y_fit1,x))
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What about fitting a Curve
with a linear function of several
variables?
Can we put constraints on the
coefficients values?
yˆ  c1 x1  c2 x2  c3 x3
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For this type of problems
(and much more)
lets learn the
optimization toolbox
http://www.mathworks.com/products/optimization/description1.html
Weizmann 2010 ©
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Optimization Toolbox Can Solve Many
Types of Optimization Problems

Optimization Toolbox –
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Extends the capability of the MATLAB numeric computing
environment.
The toolbox includes routines for many types of
optimization including:
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Unconstrained nonlinear minimization
Constrained nonlinear minimization, including goal attainment
problems and minimax problems
Semi-infinite minimization problems
Quadratic and linear programming
Nonlinear least-squares and curve fitting
Nonlinear system of equation solving
Constrained linear least squares
Sparse structured large-scale problems
==
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Optimization Toolbox GUI
Can Generate M-files
The GUI contains
many options.
Everything can be
done using coding.
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Lets learn some of the things
the optimization tool box can do
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Solving Constrained Square
Linear Problem

lsqlin (Least Square under Linear constraints)
[] – if no constraint
Starting point
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Simple Use Of Least Squares
Under No Constrains
Assume a response that is a linear combination of two variables
vars =
[ 1 1
-1 1.5
…
]
response =
[ 0.2
0.4
…
]
1
min sum (( vars  coeff_lin  response ) 2 )
x
2
coeff_lin = lsqlin(vars,response,[],[]);
We can also put constraints on the value of the coefficients:
coeff_lin = lsqlin(vars,response,[],[],[],[],[-1 -1],[1 1]);
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Simple Use Of Least Sum of
Squares Under No Constraints
xx = -2:.1:2;
yy = -2:.1:2;
[X,Y] = meshgrid(xx,yy);
Z = coeff_lin(1)*X+
coeff_lin(2)*Y;
coeff_lin =
-0.2361
-0.8379
figure;
mesh(X,Y,Z,'FaceAlpha',0.75);colorbar;
hold on;
plot3(vars(:,1),vars(:,2),response,
'ok', 'MarkerFaceColor','r')
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What about fitting a Curve
with a non linear function?
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We Can Fit Any Function
Using Non-Linear Fitting

You Can fit any non linear function using:
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nlinfit (Non linear fit)
lsqnonlin (least squares non-linear fit)
lsqcurvefit (least squares curve fit)
@func:
Example:

Hougen-Watson model
Function handle –
A way to pass a function as
an argument!
Write an M-file:
function yhat = hougen(beta,x)
Starting point
 Run:
betafit = nlinfit(reactants,rate,@hougen,beta)

470 300 10
285 80 10
([x1 x2 x3])…
8.55
3.79
(y)…
1.00
0.05
(coefficients)…
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Optimization Toolbox – Example
Fitting a Curve With a Non Linear Function

Example
for using lsqcurvefit, We will fit the data :
m
min  ( F (c, xdata)  ydata) 2
c

i 1
Assume we have the following data:
xdata = [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3];
ydata = [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];

We want to fit the data with our model:
ydata(i)  c(1)  ec ( 2)*xdata(i )

Steps:

Write a function which implements the above model:
function y_hat = lsqcurvefitExampleFunction(c,xdata)

Solve:
c0 = [100; -1] % Starting guess
[c,resnorm] =
lsqcurvefit(@lsqcurvefitExampleFunction,c0,xdata,ydata)
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What about solving non linear
system of equations?
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Solving Non Linear System
of Equations Using fsolve

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Assume we want to solve:
We can express it as:
Solving it:

Write the function M-file:
function f = fSolveExampleFunc(x)
f = [2*x(1) - x(2) - exp(-x(1));
-x(1) + 2*x(2) - exp(-x(2))];

2 x1  x2  e
 x1
 x1  2 x2  e
 x2
2 x1  x2  e  x1  0
 x1  2 x2  e
 x2
0
Choose initial guess: x0 = [-5; -5];
Run matlab optimizer:
options=optimset('Display','iter');
% Option to display output

[x,fval] = fsolve(@fSolveExampleFunc,x0,options)
x = [ 0.5671 0.5671]
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Summary:
Optimization tool box
has several features:
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Minimization
Curve fitting
Equations solving
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A taste of Symbolic matlab:
Derivatives and integrals
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What Is Symbolic Matlab?


“Symbolic Math Toolbox uses symbolic
objects to represent symbolic variables,
expressions, and matrices.”
“Internally, a symbolic object is a data
structure that stores a string
representation of the symbol.”
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Defining Symbolic Variables
and Functions
Define symbolic variables:
a_sym = sym('a')
b_sym = sym('b')
c_sym = sym('c')
x_sym = sym('x')
 Define a symbolic expression
f = sym('a*x^2 + b*x + c')
 Substituting variables:
g = subs(f,x_sym,3)

g = 9*a+3*b+c
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We Can Derive And Integrate
Symbolic Functions

Deriving a function:
diff(f,x_sym)
diff('sin(x)',x_sym)

This is a good
place to stop
Integrate a function:
int(f,x_sym)

Symbolic Matlab can do
much more…
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Summary
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Matlab is not Excel…
If you know what you want to do –
You will find the right tool!
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