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1Weizmann 2010 ©
Introduction to Matlab & Data Analysis
Lecture 13: Using Matlab for Numerical
Analysis
2
Outline
Data Smoothing Data interpolation Correlation coefficients Curve Fitting Optimization Derivatives and integrals
3
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
Filtering and Smoothing
factor
response
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
6
The Smooth Functionx = 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
Like very low pass
filter Different method
y_smooth = smooth(x,y,11,'rlowess');
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Data Interpolation Definition
“plot” – Performs linear interpolation between the data points
Interpolation - A way of estimating values of a function between those given by some set of data points.
Interpolation
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);
not_nan_i = ~isnan(x_missing); y_i = … interp1(x_missing(not_nan_i),… y_missing(not_nan_i),… x_i);
Data points which we want to interpolate
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 = -2:.5:2;yy = -2:.5:3;[X,Y] = meshgrid(xx,yy);Z = X.*exp(-X.^2-Y.^2); figure;surf(X,Y,Z);hold on;plot3(X,Y,Z+0.01,'ok', 'MarkerFaceColor','r')
Surf uses linear interpolation
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2D Functions Interpolation
interp2 functionxx_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');
figure;surf(X_i,Y_i,Z_i);hold on;plot3(X,Y,Z+0.01,'ok', 'MarkerFaceColor','r')
Data points
Points to interpolate
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Curve Fitting – Assumptions About The Residuals
Residual = Response – fitted response:
Sum square of residuals Two assumptions:
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.
yyr
n
iii
n
ii yyrS
1
2
1
2 )(
This is what we want to minimize
X
Yy yResidu
al
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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 -3.3890
y_fit1 = polyval(poly_y_fit1,x);y_fit1 = 12.6156*x-3.3890poly_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);
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
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))
Polyfit on the log of the data:
poly_exp_y_fit1 = 1.9562 5.0152
19Weizmann 2010 ©
What about fitting a Curve with a linear function of several variables? Can we put constraints on the coefficients values?
332211ˆ xcxcxcy
20Weizmann 2010 ©
For this type of problems (and much more)lets learn the optimization toolbox
http://www.mathworks.com/products/optimization/description1.html
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Optimization Toolbox Can Solve Many Types of Optimization Problems
Optimization Toolbox – Extends the capability of the MATLAB numeric computing
environment. The toolbox includes routines for many types of
optimization including: 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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Solving Constrained Square Linear Problem
lsqlin (Least Square under Linear constraints)
Starting point
[] – if no constraint
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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 … ]
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]);
))((2
1min 2responsecoeff_linvarssumx
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coeff_lin = -0.2361 -0.8379
xx = -2:.1:2;yy = -2:.1:2;[X,Y] = meshgrid(xx,yy);Z = coeff_lin(1)*X+ coeff_lin(2)*Y;
figure;mesh(X,Y,Z,'FaceAlpha',0.75);colorbar;hold on;plot3(vars(:,1),vars(:,2),response,'ok', 'MarkerFaceColor','r')
Simple Use Of Least Sum of Squares Under No Constraints
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We Can Fit Any Function Using Non-Linear Fitting
You Can fit any non linear function using: nlinfit (Non linear fit) lsqnonlin (least squares non-linear fit) lsqcurvefit (least squares curve fit)
Example: Hougen-Watson model
Write an M-file:function yhat = hougen(beta,x) Run:betafit = nlinfit(reactants,rate,@hougen,beta)
470 300 10285 80 10([x1 x2 x3])…
8.553.79(y)…
@func:Function handle – A way to pass a function as an argument!
1.000.05(coefficients)…
Starting point
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Optimization Toolbox – Example Fitting a Curve With a Non Linear Function
Example for using lsqcurvefit, We will fit the data :
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:
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)
m
ic
ydataxdatacF1
2)),((min
)()*2()1()( ixdataceciydata
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Solving Non Linear System of Equations Using fsolve
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))]; Choose initial guess: x0 = [-5; -5]; Run matlab optimizer:options=optimset('Display','iter'); % Option to display output
[x,fval] = fsolve(@fSolveExampleFunc,x0,options)
2
1
21
21
2
2x
x
exx
exx
02
022
1
21
21
x
x
exx
exx
x = [ 0.5671 0.5671]
32Weizmann 2010 ©
Optimization tool box has several features:
Summary:
Minimization Curve fitting Equations solving
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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 expressionf = 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)
Integrate a function:int(f,x_sym)
Symbolic Matlab can do much more…
This is a good place to stop