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));
Related
I am varying the signal strength for synthetic images. I need the signal to vary between 0 and 0.1, but I need to do this with a gamma distribution so that more of them fall around the .01/.02 range. The problem is that I am using the 2010 version of Matlab without the Statistics Toolbox that doesn't have the gamrnd function a part of its library.
Any and all help is greatly appreciated.
You can use the Inverse transform sampling method to convert a uniform distribution to any other distribution:
P = rand(1000);
X = gaminv(P(:),2,2); % with k = 2 and theta = 2
Here is a litle demonstration:
for k = [1 3 9]
for theta = [0.5 1 2]
X = gaminv(P(:),k,theta);
histogram(X,50)
hold on
end
end
Which gives:
Edit:
Without the statistics toolbox, you can use the Marsaglia's simple transformation-rejection method to generate random numbers from gamma distribution with rand and randn:
N = 10000; % no. of tries
% distribution parameters:
a = 0.5;
b = 0.1;
% Marsaglia's simple transformation-rejection:
d = a - 1/3;
x = randn(N,1);
U = rand(N,1);
v = (1+x./sqrt(9*d)).^3;
accept = log(U)<(0.5*x.^2+d-d*v+d*log(v));
Y = d*(v(accept)).*b;
Now Y is distributed like gamma(a,b). We can test the result using the gamrnd function:
n = size(Y,1);
X = gamrnd(a,b,n,1);
And the histograms of Y, and X are:
However, keep in mind that gamma distribution might not fit your needs because it has no specific upper bound (i.e. goes to infinity). So you may want to use another (bounded) distribution, like beta divided by 10.
I want generate a number in Gaussian and Uniform distributions in matlab.
I know this function randi and rand() but all of them are in normal (Gaussian) distribution. How can a generate a random number in uniform distribution?
Use rand(dimensions) for a Uniform Distribution between 0 and 1.
Use randn(dimensions) * sqrt(sigma) + mu for a Gaussian Distribution with a mean of mu and standard deviation of sigma.
randn is the function to generate Gaussian distributed variables (randi and rand produce uniformly distributed ones).
You can generate any distribution from rand().
For example , lets say you want to generate 100000 samples for rayleigh dist.The way to do this is that you invert the cdf of that particular function.The basic idea is that since the cdf has to be between 0 and 1 , we can find the value of the random variable by inputting the value of cdf b/w 0 and 1. So for rayleigh, it would be
for i = 1:100000
data(i) = (2*sigma^2 *(-(log(1 - rand(1,1)))))^.5;
end
You can do something similar for gaussian distribution.
Congrulations, you already generating pseudo-random numbers with a gaussian distribution. Normal distribution is a synonym for it.
The only other possible interpretation I can get from your question is that you want something that has mean != 0 and/or variance != 1. To do that, simply perform mean + sqrt(var) * randn(X).
It is true you can generate just about anything from rand but that it isn't always convenient, especially for some complicated distributions.
MATLAB has introduced Probability Distribution Objects which make this a lot easier and allow you to seamlessly access mean, var, truncate, pdf, cdf, icdf (inverse transform), median, and other functions.
You can fit a distribution to data. In this case, we use makedist to define the probability distribution object. Then we can generate using random.
% Parameters
mu = 10;
sigma = 3;
a = 5; b = 15;
N = 5000;
% Older Approaches Still Work
rng(1775)
Z = randn(N,1); % Standard Normal Z~N(0,1)
X = mu + Z*sigma; % X ~ Normal(mu,sigma)
U = rand(N,1); % U ~ Uniform(0,1)
V = a + (b-a)*U; % V ~ Uniform(a,b)
% New Approaches Are Convenient
rng(1775)
pdX = makedist('Normal',mu,sigma);
X2 = random(pdX,N,1);
pdV = makedist('Uniform',a,b);
V2 = random(pdV,N,1);
A reproducible example:
Support = (0:0.01:20)';
figure
s(1) = subplot(2,2,1)
h(1) = histogram(X,'Normalization','pdf')
xlabel('Normal')
s(2) = subplot(2,2,2)
h(2) = histogram(V,'Normalization','pdf')
xlabel('Uniform')
s(3) = subplot(2,2,3), hold on, box on
h(3) = histogram(X2,'Normalization','pdf')
plot(Support,pdf(pdX,Support),'r-','LineWidth',1.2)
xlabel('Normal (new)')
s(4) = subplot(2,2,4), hold on, box on
h(4) = histogram(V2,'Normalization','pdf')
plot(Support,pdf(pdV,Support),'r-','LineWidth',1.2)
xlabel('Uniform (new)')
xlim(s,[0 20])
References:
Uniform Distribution
Normal (Gaussian) Distribution
Following raj's answer: by using the Box-Muller Transform you can generate independent standard Normal/Gaussian random numbers:
N = 1e6; z = sqrt(-2*log(rand(N, 1))) .* cos(2*pi * rand(N, 1)); figure; hist(z, 100)
N = 1e6; z = sqrt(-2*log(rand(N, 1))) .* sin(2*pi * rand(N, 1)); figure; hist(z, 100)
If you want to apply the Inverse Transformation Method, you can use the Inverse Complementary Error Function (erfcinv):
N = 1e6; z = -sqrt(2) * erfcinv(2 * rand(1e6, 1)); figure; hist(z, 100)
But I hope randn works better.
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-')
By using randn function I want to create a Gaussian random variable X such that X ~ N(2,4) and plot this simulated PDF together with theoretic curve.
Matlab randn generates realisations from a normal distribution with zero mean and a standard deviation of 1.
Samples from any other normal distribution can simply be generated via:
numSamples = 1000;
mu = 2;
sigma = 4;
samples = mu + sigma.*randn(numSamples, 1);
You can verify this by plotting the histogram:
figure;hist(samples(:));
See the matlab help.
N = 1000;
x = [-20:20];
samples = 2 + 4*randn(N, 1);
ySamples = histc(samples,x) / N;
yTheoretical = pdf('norm', x, 2, 4);
plot(x, yTheoretical, x, ySamples)
randn(N, 1) creates an N-by-1 vector.
histc is histogram count by bins given in x - you can use hist to plot the result immediately, but here we want to divide it by N.
pdf contains many useful PDFs, normal is just one example.
remember this: X ~ N(mean, variance)
randn in matlab produces normal distributed random variables W with zero mean and unit variance.
To change the mean and variance to be the random variable X (with custom mean and variance), follow this equation:
X = mean + standard_deviation*W
Please be aware of that standard_deviation is square root of variance.
N = 1000;
x = [-20:20];
samples = 2 + sqrt(4)*randn(N, 1);
ySamples = histc(samples,x) / N;
yTheoretical = pdf('norm', x, 2, sqrt(4)); %put std_deviation not variance
plot(x, yTheoretical, x, ySamples)
A quick and easy way to achieve this using one line of code is to use :
mu = 2;
sigma = 2;
samples = normrnd(mu,sigma,M,N);
This will generate an MxN matrix, sampled from N(μ,𝜎), (= N(2,2) in this particular case).
For additional information, see normrnd.
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