MATLAB: Linear Regression and Curve Fitting

curve fittinglinear regression

I have a model and some data I'd like to fit to it: X_t = B1*cos(2*pi*omega*t) + B2*sin(2*pi*omega*t) + eta_t

What function would I use to conduct linear regression here, to find B1 and B2?

Best Answer

Fs = 1000;
t = 0:1/Fs:1-1/Fs;
y = 1.5*cos(2*pi*100*t)+0.5*sin(2*pi*100*t)+randn(size(t));
y = y(:);
X = ones(length(y),3);
X(:,2) = cos(2*pi*100*t)';
X(:,3) = sin(2*pi*100*t)';
beta = X\y;

beta(1) is the estimate of the constant term, beta(2) the estimate of B1 and beta(3) the estimate of B2.

If you set the random number generator to its default for reproducible results:

rng default
Fs = 1000;
t = 0:1/Fs:1-1/Fs;
y = 1.5*cos(2*pi*100*t)+0.5*sin(2*pi*100*t)+randn(size(t));
y = y(:);
X = ones(length(y),3);
X(:,2) = cos(2*pi*100*t)';
X(:,3) = sin(2*pi*100*t)';
beta = X\y;

The results are:

beta =

   -0.0326
    1.5284
    0.4643

pretty good.

Related Solutions

MATLAB: Is it so hard for MATLAB to solve the following symbolic equation

This seems to work

syms rp alpha beta theta psi z
A=cos(alpha)-cos(beta);
B=sin(alpha)-sin(beta);
C=(cos(beta)-1)/tan(beta)-(cos(alpha)-1)/tan(alpha);
R=A*(cos(theta)-cos(psi))+B*cos(theta)*sin(psi);
S=A*sin(theta)*sin(psi)-B*cos(theta)+C*cos(psi);
phi=atan(-R/S);
x=-rp*(cos(theta)*cos(phi)+sin(psi)*sin(theta)*sin(phi))*cos(alpha)-rp*(-cos(theta)*sin(phi)+sin(psi)*sin(theta)*cos(phi))*sin(alpha)+(rp/tan(alpha))*(cos(psi)*sin(phi)*(cos(alpha)-1)+cos(psi)*cos(phi)*sin(alpha));
y=-cos(psi)*sin(phi)*rp;
p=[x;y;z];
T=Rot('y',theta)*Rot('x',psi)*Rot('z',phi);
a1=T*[rp;0;0];
a2=T*[rp*cos(alpha);rp*sin(alpha);0];
a3=T*[rp*cos(beta);rp*sin(beta);0];  
Con1=(p+a1)'*[0;1;0];
Con2=(p+a2)'*[-sin(alpha);cos(alpha);0];
Con3=(p+a3)'*[-sin(beta);cos(beta);0];
disp([Con1; Con2; Con3])
function output=Rot(axis,angle)
syms x y z
if axis==x
    output=[1,0,0;0,cos(angle),-sin(angle);0,sin(angle),cos(angle)];
end
if axis==y
    output=[cos(angle),0,sin(angle);0,1,0;-sin(angle),0,cos(angle)];
end
if axis==z
    output=[cos(angle),-sin(angle),0;sin(angle),cos(angle),0;0,0,1];
end
end

MATLAB: How to find annual amplitude and phase of time series

There are a number of ways you can try and estimate it. Do you have a priori knowledge of the frequency? If so you can use linear regression to find the amplitude and phase.

For example, assume I know the frequency is 10 Hz for data sampled at 100 Hz.

    Fs = 100;
    t = 0:1/Fs:2-1/Fs;
    x = 1.5*cos(2*pi*10*t-pi/4)+randn(size(t));
    % create the design matrix
    X = ones(length(x),3);
    X(:,2) = sin(2*pi*10*t);
    X(:,3) = cos(2*pi*10*t);
    x = x(:);
    beta = X\x;

The amplitude estimate is:

   amp_est = norm([beta(2) beta(3)],2)

The phase angle estimate is:

   phase_est = atan2(-beta(2),beta(3))

If you don't know the frequency a priori, then look at the following example:

https://www.mathworks.com/help/signal/ug/frequency-domain-linear-regression.html

Best Answer

Related Solutions

MATLAB: Is it so hard for MATLAB to solve the following symbolic equation

MATLAB: How to find annual amplitude and phase of time series

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