I have this matlab code for regression with one indepenpent variable, but what if I have two independent variables(x1 and x2)? How should I modify this code of polynomial regression?
x = linspace(0,10,200)'; % independent variable
y = x + 1.5*sin(x) + randn(size(x,1),1); % dependent variable
A = [x.^0, x]; % construct a matrix of permutations
w = (A'*A)\(A'*y); % solve the normal equation
y2 = A*w; % restore the dependent variable
r = y-y1; % find the vector of regression residual
plot(x, [y y2]);
Matlab has facilities for polynomial regression function polyfit. Have you tried that?
http://www.mathworks.com/help/techdoc/data_analysis/f1-8450.html
http://www.mathworks.com/help/toolbox/stats/bq_676m-2.html#bq_676m-3
But if you want to workout your own formulation,you should probably look at textbook or some online resources on regression e.g.
http://www.edwardtufte.com/tufte/dapp/DAPP3a.pdf
Related
I have this hard coded version which fits data to a curve for linear, quadratic and cubic polynomials:
for some data x and a function y
M=[x.^0 x.^1];
L=[x.^0 x.^1 x.^2];
linear = (M'*M)\(M'*y);
plot(x, linear(1)+linear(2)*x, ';linear;r');
deg2 = (L'*L)\(L'*y);
plot(x, deg2(1)+deg2(2)*x+deg2(3)*(x.*x), ';quadratic plot;b');
I am wondering how can I turn this into a for loop to plot curves for degree n polynomials? The part I'm stuck on is the plotting part, how would I be able to translate the increase in the number of coefficients in to the for loop?
what I have:
for i = 1:5 % say we're trying to plot curves up to degree 5 polynomials...
curr=x.^(0:i);
degI = (curr'*curr)\(curr'*y);
plot(x, ???) % what goes in here<-
end
If it is only the plotting, you can use the polyval function to evaluate polynomials of desired grade by supplying a vector of coefficients
% For example, some random coefficients for a 5th order polynomial
% degI = (curr'*curr)\(curr'*y) % Your case
degi = [3.2755 0.8131 0.5950 2.4918 4.7987 1.5464]; % for 5th order polynomial
x = linspace(-2, 2, 10000);
hold on
% Using polyval to loop over the grade of the polynomials
for i = 1:length(degI)
plot(x, polyval(degI(1:i), x))
end
gives the all polynomials in one plot
I believe this should answer your question exactly. You just have to be careful with the matrix dimensions.
for i = 1:5
curr=x.^(0:i);
degI = (curr'*curr)\(curr'*y);
plot(x, x.^(0:i)*degI)
end
I have three vectors, one of X locations, another of Y locations and the third is a f(x, y). I want to find the algebraic expression interpolation polynomial (using matlab) since I will later on use the result in an optimization problem in AMPL. As far as I know, there are not any functions that return the interpolation polynomial.
I have tried https://la.mathworks.com/help/matlab/ref/griddedinterpolant.html, but this function only gives the interpolated values at certain points.
I have also tried https://la.mathworks.com/help/matlab/ref/triscatteredinterp.html as sugested in Functional form of 2D interpolation in Matlab, but the output isn't the coefficents of the polynomial. I cannot see it, it seems to be locked inside of a weird variable.
This is a small program that I have done to test what I am doing:
close all
clear
clc
[X,Y] = ndgrid(1:10,1:10);
V = X.^2 + 3*(Y).^2;
F = griddedInterpolant(X,Y,V,'cubic');
[Xq,Yq] = ndgrid(1:0.5:10,1:0.5:10);
Vq = F(Xq,Yq);
mesh(Xq,Yq,Vq)
figure
mesh(X, Y, V)
I want an output that instead of returning the value at grid points returns whatever it has used to calculate said values. I am aware that it can be done in mathematica with https://reference.wolfram.com/language/ref/InterpolatingPolynomial.html, so I find weird that matlab can't.
You can use fit if you have the curve fitting toolbox.
If it's not the case you can use a simple regression, if I take your example:
% The example data
[X,Y] = ndgrid(1:10,1:10);
V = X.^2 + 3*(Y).^2;
% The size of X
s = size(X(:),1);
% Let's suppose that you want to fit a polynome of degree 2.
% Create all the possible combination for a polynome of degree 2
% cst x y x^2 y^2 x*y
A = [ones(s,1), X(:), Y(:), X(:).^2, Y(:).^2, X(:).*Y(:)]
% Then using mldivide
p = A\V(:)
% We obtain:
p =
0 % cst
0 % x
0 % y
1 % x^2
3 % y^2
0 % x*y
Is there any way to fit a function with n variables in Matlab? Any example would be very useful.
Till now I used curve fitting toolbox, which provides solution I need for functions with 2 arguments. But now I need to fit a function with much more variables.
The worst thing is that dependance is non-linear (probably something like a/x+b/y+c/z+…, but it's only a hypothesis). If it was linear, '\' operator would do the trick.
lsqnonlin will do, e.g.
%% generate noisy points for fitting
a = 1; b = 2; c = 3;
x = rand(100,3);
y = a./x(:,1) + b./x(:,2) + c./x(:,3) + 0.1*rand(1,1);
%% fitting
% define residual vector
minRes = #(p) (p(1) ./ x(:,1) + p(2) ./ x(:,2) + p(3) ./ x(:,3) - y);
% start values
par0 = [1,1,1];
% optimize
par = lsqnonlin(minRes, par0);
lsqnonlin
function included in Matlab
Given a M x M desired covariance, R, and a desired number of sample vectors, N calculate a N x M Gaussian random vector, X in vanilla MATLAB (i.e. can't use r = mvnrnd(MU,SIGMA,cases)).
Not really sure how to tackle this, usually you need a covariance AND mean to generate a Gaussian random variable. I think sqrtm and chol could be useful.
If you have access to the MATLAB statistics toolbox you can type edit mvnrnd in MATLAB to see their solution.
[T p] = chol(sigma);
if m1 == c
mu = mu';
end
mu = mu(ones(cases,1),:);
r = randn(cases,c) * T + mu;
It feels almost like cheating to point this out, but editing MATLAB's source is very useful to understand things in general. You can also search for mvnrnd.m on google if you don't have the toolbox.
Example:
% Gaussian mean and covariance
d = 2; % number of dimensions
mu = rand(1,d);
sigma = rand(d,d); sigma = sigma*sigma';
% generate 100 samples from above distribution
num = 100;
X = mvnrnd(mu, sigma, num);
% plot samples (only for 2D case)
scatter(X(:,1), X(:,2), 'filled'), hold on
ezcontour(#(x,y) mvnpdf([x y], mu, sigma), xlim(), ylim())
title('X~N(\mu,\sigma)')
xlabel('X_1'), ylabel('X_2')
The above code uses functions from the Statistics toolbox (mvnrnd and mvnpdf). If you don't have access to it, consider these replacements (using the same concepts mentioned by others):
mvnrnd = #(mu,S,num) bsxfun(#plus, randn(num,numel(mu))*cholcov(S), mu);
mvnpdf = #(x,mu,S) exp(-0.5*(x-mu)*(S\(x-mu)')) / sqrt((2*pi)^d*det(S));
I am new to statistics. I have a discriminant function:
g(x) = ln p(x| w)+ lnP(w)
I know it has a normal distribution. I know mü and sigma variables. How can I plot pdf function of it at Matlab?
Here is a conversation: How to draw probability density function in MatLab? however I don't want to use any toolbax of Matlab.
Use normpdf, or mvnpdf for a multivariate Normal distribution:
mu = 0;
sigma = 1;
xs = [-5:.1:5];
ys = normpdf(xs, mu, sigma);
clf;
plot(xs, ys);
MATLAB plots vectors of data, so you'll need to make an X-vector, and a Y-vector.
If I had a function, say, x^2, i might do:
x = -1:.01:1; %make the x-vector
y = x.^2; %square x
plot(x,y);
You know the function of the PDF (y = exp(-x.^2./sigma^2).*1/sqrt(2*pi*sigma^2) ), so all you have to do is make the x-vector, and plot away!
Based upon a comment from #kamaci, this question gives a complete answer using #Pete's answer.
To avoid using a MATLAB toolbox, just plot the Normal probability density function (PDF) directly.
mu = 12.5; % mean
sigma = 3.75; % standard deviation
fh=#(x) exp(-((x-mu).^2)./(2*(sigma^2)))*(1/sqrt(2*pi*(sigma^2))); % PDF function
X = 0:.01:25;
p = plot(X,fh(X),'b-')