I have a 2x1800 dimensional data. Both features I approximated separately with the following distributions:
How to combine these two distributions to plot them as a multivariate distrubution's surface graph?
For independent random variables the joint distribution is the product of the marginal distributions. Use meshgrid to generate the appropriate indicies. Here we assume the marginal distributions are stored in arrays px and py
[xidx,yidx] = meshgrid(1:numel(px),1:numel(py));
pxy = px(xidx).*py(yidx);
surf(xidx,yidx,pxy);
shading('interp');
Related
I have created a bivariate gaussian using the scipy.state kde library:
k = kde.gaussian_kde(data.T)
However I don't understand how the contours or z values relate to the probability of a new data point falling within those contours,
ngenerated distribution with contours
I would like to be able to define a contour with the equivalent of p=0.001, or put another way a contour defining 99.9% of expected observations.
I have an image with multivariate Gaussian distribution in histogram. I want to segment the image to two regions so that they both can follow the normal distribution like the red and blue curves shows in histogram. I know Gaussian mixture model potentially works for that. I tried to use fitgmdist function and then clustering the two parts but still not work well. Any suggestion will be appreciated.
Below is the Matlab code for my appraoch.
% Read Image
I = imread('demo.png');
I = rgb2gray(I);
data = I(:);
% Fit a gaussian mixture model
obj = fitgmdist(data,2);
idx = cluster(obj,data);
cluster1 = data(idx == 1,:);
cluster2 = data(idx == 2,:);
% Display Histogram
histogram(cluster1)
histogram(cluster2)
Your solution is correct
The way you are displaying your histogram poorly represents the detected distributions.
Normalize the bin sizes because histogram is a frequency count
Make the axes limits consistent (or plot on same axis)
These two small changes show that you're actually getting a pretty good distribution fit.
histogram(cluster1,0:.01:1); hold on;
histogram(cluster2,0:.01:1);
Re-fit a gaussian-curve to each cluster
Once you have your clusters if you treat them as independent distributions, you can smooth the tails where the two distributions merge.
gcluster1 = fitdist(cluster1,'Normal');
gcluster2 = fitdist(cluster2,'Normal');
x_values = 0:.01:1;
y1 = pdf(gcluster1,x_values);
y2 = pdf(gcluster2,x_values);
plot(x_values,y1);hold on;
plot(x_values,y2);
How are you trying to use this 'model'? If the data is constant, then why dont you measure, the mean/variances for the two gaussians seperately?
And if you are trying to generate new values from this mixed distribution, then you can look into a mixture model with weights given to each of the above distributions.
I know for a random variable x that P(x=i) for each i=1,2,...,100. Then how may I sample x by a multinomial distribution, based on the given P(x=i) in Matlab?
I am allowed to use the Matlab built-in commands rand and randi, but not mnrnd.
In general, you can sample numbers from any 1 dimensional probability distribution X using a uniform random number generator and the inverse cumulative distribution function of X. This is known as inverse transform sampling.
random_x = xcdf_inverse(rand())
How does this apply here? If you have your vector p of probabilities defining your multinomial distribution, F = cumsum(p) gives you a vector that defines the CDF. You can then generate a uniform random number on [0,1] using temp = rand() and then find the first row in F greater than temp. This is basically using the inverse CDF of the multinomial distribution.
Be aware though that for some distributions (eg. gamma distribution), this turns out to be an inefficient way to generate random draws because evaluating the inverse CDF is so slow (if the CDF cannot expressed analytically, slower numerical methods must be used).
I'm trying to fit a multivariate normal distribution to data that I collected, in order to take samples from it.
I know how to fit a (univariate) normal distribution, using the fitdist function (with the 'Normal' option).
How can I do something similar for a multivariate normal distribution?
Doesn't using fitdist on every dimension separately assumes the variables are uncorrelated?
There isn't any need for a specialized fitting function; the maximum likelihood estimates for the mean and variance of the distribution are just the sample mean and sample variance. I.e., compute the sample mean and sample variance and you're done.
Estimate the mean with mean and the variance-covariance matrix with cov.
Then you can generate random numbers with mvnrnd.
It is also possible to use fitmgdist, but for just a multivariate normal distribution mean and cov are enough.
Yes, using fitdist on every dimension separately assumes the variables are uncorrelated and it's not what you want.
You can use [sigma,mu] = robustcov(X) function, where X is your multivariate data, i.e. X = [x1 x2 ... xn] and xi is a column vector data.
Then you can use Y = mvnpdf(X,mu,sigma) to get the values of the estimated normal probability density function.
https://www.mathworks.com/help/stats/normfit.html
https://www.mathworks.com/help/stats/mvnpdf.html
How would you normalize a histogram A so the sum of each bin is 1
Dividing the histogram by the width of the bin, how do you draw it
I have this
dist = rand(50)
average = mean(dist, 1);
[c,x] = hist(average, 15);
normalized = c/sum(c);
bar(x, normalized, 1)
In this case, n = 50,
What is it the formula to get values
of mean and variance^2? We write N(mean, (variance^2) / 50), but how?
How do you plot both uniform distribution and normal distribution?.
The histogram must be close to the normal distribution.
That is a very unusual way of normalizing a probability density function. I assume you want to normalize such that the area under the curve is 1. In that case, this is what you should do.
[c,x]=hist(average,15);
normalized=c/trapz(x,c);
bar(x,normalized)
Either way, to answer your question, you can use randn to generate a normal distribution. You're now generating a 50x50 uniform distribution matrix and summing along one dimension to approximate a normal Gaussian. This is unnecessary. To generate a normal distribution of 1000 points, use randn(1000,1) or if you want a row vector, transpose it or flip the numbers. To generate a Gaussian distribution of mean mu and variance sigma2, and plot its pdf, you can do (an example)
mu=2;
sigma2=3;
dist=sqrt(sigma2)*randn(1000,1)+mu;
[c,x]=hist(dist,50);
bar(x,c/trapz(x,c))
Although these can be done with dedicated functions from the statistics toolbox, this is equally straightforward, simple and requires no additional toolboxes.
EDIT
I missed the part where you wanted to know how to generate a uniform distribution. rand, by default gives you a random variable from a uniform distribution on [0,1]. To get a r.v. from a uniform distribution between [a, b], use a+(b-a)*rand