I'm trying to draw random numbers from a multivariate t-student distribution (with a specified mean, variance and df) in Matlab.
Looking into the matlab documentation, I've found the "mvtrnd" function (http://www.mathworks.it/it/help/stats/mvtrnd.html), but unfortunately, it returns values centered in 0.
Could you suggest me any other function or approach to solve my problem?
EDIT:
I think I have solved the problem writing the function below. I have found in wikipedia that if X is distributed as a multivariate t-student (mu, $$\Sigma$$, df) and Y as a multivariate normal (0, $$\Sigma$$), than $$X = \mu + Y * \sqrt{\frac{df}{\chi^2_{df}}}$$.
function [ output ] = st_rndmvt( mu, sigma, df, cases )
if nargin < 4
cases = 1;
end
mu = mu(:)';
output = zeros(cases, length(mu));
for i=1:cases
output(i, :) = mu + mvnrnd(zeros(length(sigma), 1), sigma) * sqrt(df/chi2rnd(df));
end
end
Related
I want to sample R random numbers from a custom probability density function in Matlab.
This is the expression of the probability density function evaluated at x.
I thought about using slicesample
R=10^6;
f = #(x) 1/(2*pi^(1/2))*(1/(x^(3/2)))*exp(-1/(4*x));
epsilon= slicesample(0.3,R,'pdf',f,'thin',1,'burnin',1000);
However, it does not work because I get the error
Error using slicesample (line 175)
The step-out procedure failed.
I tried to change starting value and values of thin and burning parameters but it does not seem to work. Could you advise, either on how to make slicesample work or on alternative solutions to sample random numbers from a custom probability density function in Matlab?
Let X be a random variable distributed according to your target pdf. Applying the change of variable y = 1/x and using the well-known theorem for a function of a random variable, the distribution of Y = 1/X is recognized to be a Gamma distribution with parameters α = 1/2, β = 1/4.
Therefore, it suffices to generate a Gamma random variable (using gamrnd) with those parameters and take the inverse. Note that Matlab's definition of the Gamma distribution uses parameters A = α, B = 1/β.
R = 1e5; % desired sample size
x = 1./gamrnd(1/2, 4, [1 R]); % result
Check:
histogram(x, 'Normalization', 'pdf', 'BinEdges', 0:.1:10)
hold on
f = #(x) 1/2/sqrt(pi)./x.^(3/2).*exp(-1/4./x); % target pdf
fplot(f, 'linewidth', .75)
I have a set of frequency data with peaks to which I need to fit a Gaussian curve and then get the full width half maximum from. The FWHM part I can do, I already have a code for that but I'm having trouble writing code to fit the Gaussian.
Does anyone know of any functions that'll do this for me or would be able to point me in the right direction? (I can do least squares fitting for lines and polynomials but I can't get it to work for gaussians)
Also it would be helpful if it was compatible with both Octave and Matlab as I have Octave at the moment but don't get access to Matlab until next week.
Any help would be greatly appreciated!
Fitting a single 1D Gaussian directly is a non-linear fitting problem. You'll find ready-made implementations here, or here, or here for 2D, or here (if you have the statistics toolbox) (have you heard of Google? :)
Anyway, there might be a simpler solution. If you know for sure your data y will be well-described by a Gaussian, and is reasonably well-distributed over your entire x-range, you can linearize the problem (these are equations, not statements):
y = 1/(σ·√(2π)) · exp( -½ ( (x-μ)/σ )² )
ln y = ln( 1/(σ·√(2π)) ) - ½ ( (x-μ)/σ )²
= Px² + Qx + R
where the substitutions
P = -1/(2σ²)
Q = +2μ/(2σ²)
R = ln( 1/(σ·√(2π)) ) - ½(μ/σ)²
have been made. Now, solve for the linear system Ax=b with (these are Matlab statements):
% design matrix for least squares fit
xdata = xdata(:);
A = [xdata.^2, xdata, ones(size(xdata))];
% log of your data
b = log(y(:));
% least-squares solution for x
x = A\b;
The vector x you found this way will equal
x == [P Q R]
which you then have to reverse-engineer to find the mean μ and the standard-deviation σ:
mu = -x(2)/x(1)/2;
sigma = sqrt( -1/2/x(1) );
Which you can cross-check with x(3) == R (there should only be small differences).
Perhaps this has the thing you are looking for? Not sure about compatability:
http://www.mathworks.com/matlabcentral/fileexchange/11733-gaussian-curve-fit
From its documentation:
[sigma,mu,A]=mygaussfit(x,y)
[sigma,mu,A]=mygaussfit(x,y,h)
this function is doing fit to the function
y=A * exp( -(x-mu)^2 / (2*sigma^2) )
the fitting is been done by a polyfit
the lan of the data.
h is the threshold which is the fraction
from the maximum y height that the data
is been taken from.
h should be a number between 0-1.
if h have not been taken it is set to be 0.2
as default.
i had similar problem.
this was the first result on google, and some of the scripts linked here made my matlab crash.
finally i found here that matlab has built in fit function, that can fit Gaussians too.
it look like that:
>> v=-30:30;
>> fit(v', exp(-v.^2)', 'gauss1')
ans =
General model Gauss1:
ans(x) = a1*exp(-((x-b1)/c1)^2)
Coefficients (with 95% confidence bounds):
a1 = 1 (1, 1)
b1 = -8.489e-17 (-3.638e-12, 3.638e-12)
c1 = 1 (1, 1)
I found that the MATLAB "fit" function was slow, and used "lsqcurvefit" with an inline Gaussian function. This is for fitting a Gaussian FUNCTION, if you just want to fit data to a Normal distribution, use "normfit."
Check it
% % Generate synthetic data (for example) % % %
nPoints = 200; binSize = 1/nPoints ;
fauxMean = 47 ;fauxStd = 8;
faux = fauxStd.*randn(1,nPoints) + fauxMean; % REPLACE WITH YOUR ACTUAL DATA
xaxis = 1:length(faux) ;fauxData = histc(faux,xaxis);
yourData = fauxData; % replace with your actual distribution
xAxis = 1:length(yourData) ;
gausFun = #(hms,x) hms(1) .* exp (-(x-hms(2)).^2 ./ (2*hms(3)^2)) ; % Gaussian FUNCTION
% % Provide estimates for initial conditions (for lsqcurvefit) % %
height_est = max(fauxData)*rand ; mean_est = fauxMean*rand; std_est=fauxStd*rand;
x0 = [height_est;mean_est; std_est]; % parameters need to be in a single variable
options=optimset('Display','off'); % avoid pesky messages from lsqcurvefit (optional)
[params]=lsqcurvefit(gausFun,x0,xAxis,yourData,[],[],options); % meat and potatoes
lsq_mean = params(2); lsq_std = params(3) ; % what you want
% % % Plot data with fit % % %
myFit = gausFun(params,xAxis);
figure;hold on;plot(xAxis,yourData./sum(yourData),'k');
plot(xAxis,myFit./sum(myFit),'r','linewidth',3) % normalization optional
xlabel('Value');ylabel('Probability');legend('Data','Fit')
I have a problem when calculate discrete Fourier transform in MATLAB, apparently get the right result but when plot the amplitude of the frequencies obtained you can see values very close to zero which should be exactly zero. I use my own implementation:
function [y] = Discrete_Fourier_Transform(x)
N=length(x);
y=zeros(1,N);
for k = 1:N
for n = 1:N
y(k) = y(k) + x(n)*exp( -1j*2*pi*(n-1)*(k-1)/N );
end;
end;
end
I know it's better to use fft of MATLAB, but I need to use my own implementation as it is for college.
The code I used to generate the square wave:
x = [ones(1,8), -ones(1,8)];
for i=1:63
x = [x, ones(1,8), -ones(1,8)];
end
MATLAB version: R2013a(8.1.0.604) 64 bits
I have tried everything that has happened to me but I do not have much experience using MATLAB and I have not found information relevant to this issue in forums. I hope someone can help me.
Thanks in advance.
This will be a numerical problem. The values are in the range of 1e-15, while the DFT of your signal has values in the range of 1e+02. Most likely this won't lead to any errors when doing further processing. You can calculate the total squared error between your DFT and the MATLAB fft function by
y = fft(x);
yh = Discrete_Fourier_Transform(x);
sum(abs(yh - y).^2)
ans =
3.1327e-20
which is basically zero. I would therefore conclude: your DFT function works just fine.
Just one small remark: You can easily vectorize the DFT.
n = 0:1:N-1;
k = 0:1:N-1;
y = exp(-1j*2*pi/N * n'*k) * x(:);
With n'*k you create a matrix with all combinations of n and k. You then take the exp(...) of each of those matrix elements. With x(:) you make sure x is a column vector, so you can do the matrix multiplication (...)*x which automatically sums over all k's. Actually, I just notice, this is exactly the well-known matrix form of the DFT.
I'am new in matlab programming,I should Write a script to generate a random sequence (x1,..., XN) of size N following the normal distribution N (0, 1) and calculate the empirical mean mN and variance σN^2
Then,I should plot them:
this is my essai:
function f = normal_distribution(n)
x =randn(n);
muem = 1./n .* (sum(x));
muem
%mean(muem)
vaem = 1./n .* (sum((x).^2));
vaem
hold on
plot(x,muem,'-')
grid on
plot(x,vaem,'*')
NB:those are the formules that I have used:
I have obtained,a Figure and I don't know if is it correct or not ,thanks for Help
From your question, it seems what you want to do is calculate the mean and variance from a sample of size N (nor an NxN matrix) drawn from a standard normal distribution. So you may want to use randn(n, 1), instead of randn(n). Also as #ThP pointed out, it does not make sense to plot mean and variance vs. x. What you could do is to calculate means and variances for inceasing sample sizes n1, n2, ..., nm, and then plot sample size vs. mean or variance, to see them converge to 0 and 1. See the code below:
function [] = plotMnV(nIter)
means = zeros(nIter, 1);
vars = zeros(nIter, 1);
for pow = 1:nIter
n = 2^pow;
x =randn(n, 1);
means(pow) = 1./n * sum(x);
vars(pow) = 1./n * sum(x.^2);
end
plot(1:nIter, means, 'o-');
hold on;
plot(1:nIter, vars, '*-');
end
For example, plotMnV(20) gave me the plot below.
I have a data-set which is loaded into matlab. I need to do exponential fitting for the plotted curve without using the curve fitting tool cftool.
I want to do this manually through executing a code/function that will output the values of a and b corresponding to the equation:
y = a*exp(b*x)
Then be using those values, I will do error optimization and create the best fit for the data I have.
Any help please?
Thanks in advance.
Try this...
f = fit(x,y,'exp1');
I think the typical objective in this type of assignment is to recognize that by taking the log of both sides, various methods of polynomial fit approaches can be used.
ln(y) = ln(a) + ln( exp(x).^b )
ln(y) = ln(a) + b * ln( exp(x) )
There can be difficulties with this approach when errors such as noise are involved due to the behavior of ln as it approaches zero.
In this exercise I have a set of data that present an exponential curve and I want to fit them exponentially and get the values of a and b. I used the following code and it worked with the data I have.
"trail.m" file:
%defining the data used
load trialforfitting.txt;
xdata= trialforfitting(:,1);
ydata= trialforfitting(:,2);
%calling the "fitcurvedemo" function
[estimates, model] = fitcurvedemo(xdata,ydata)
disp(sse);
plot(xdata, ydata, 'o'); %Data curve
hold on
[sse, FittedCurve] = model(estimates);
plot(xdata, FittedCurve, 'r')%Fitted curve
xlabel('Voltage (V)')
ylabel('Current (A)')
title('Exponential Fitting to IV curves');
legend('data', ['Fitting'])
hold off
"fitcurvedemo.m" file:
function [estimates, model] = fitcurvedemo(xdata, ydata)
%Call fminsearch with a random starting point.
start_point = rand(1, 2);
model = #expfun;
estimates = fminsearch(model, start_point);
%"expfun" accepts curve parameters as inputs, and outputs
%the sum of squares error [sse] expfun is a function handle;
%a value that contains a matlab object methods and the constructor
%"FMINSEARCH" only needs sse
%estimate returns the value of A and lambda
%model computes the exponential function
function [sse, FittedCurve] = expfun(params)
A = params(1);
lambda = params(2);
%exponential function model to fit
FittedCurve = A .* exp(lambda * xdata);
ErrorVector = FittedCurve - ydata;
%output of the expfun function [sum of squares of error]
sse = sum(ErrorVector .^ 2);
end
end
I have a new set of data that doesn't work with this code and give the appropriate exponential fit for the data curve plotted.