Kernel Density for 2D data

I have two series of data(both of type double). I want to generate a kernel density plot from these. Pls help. My coding is given below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
curs1 = exec(conn, 'select sp_x, sp_y from road_part6_trajectories_oneway2_new_segments_cartesian2');

format long;

curs1 = fetch(curs1);

AA = curs1.Data;

x = [AA{:,1}]';

y = [AA{:,2}]';

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

I want to generate the kernel density or any other suitable density plot for x and y. pls advise.

Answer 1

[the cyclist]

The function ksdensity() will do kernel density estimation. It's not clear to me what your x and y represent, so it's hard to give more specific advice than that.

In response to comments, here is some code with an example of 2-d kernel density estimation, with a plot of the results. Important note: this assumes that x and y are independent from each other.

% Generate some normally distributed data
x = randn(50,1);
y = randn(50,1);

% Estimate a continuous pdf from the discrete data
[pdfx xi]= ksdensity(x);
[pdfy yi]= ksdensity(y);

% Create 2-d grid of coordinates and function values, suitable for 3-d plotting
[xxi,yyi]     = meshgrid(xi,yi);
[pdfxx,pdfyy] = meshgrid(pdfx,pdfy);

% Calculate combined pdf, under assumption of independence
pdfxy = pdfxx.*pdfyy; 

% Plot the results
mesh(xxi,yyi,pdfxy)
set(gca,'XLim',[min(xi) max(xi)])
set(gca,'YLim',[min(yi) max(yi)])

[Devinya Herath]

I have a table that contains two columns called sp_x and sp_y. The values of sp_x and sp_y corresponds to coordinates of positions in cartesian space (hence both columns contain double precision values). I downloaded these two columns to matlab (pls see the code) and assign the values to x and y (x contains all values in sp_x column and y contains of all values of sp_y column). I want to construct kernel density plot using both x and y. I guess this is a bivariate case as coordinates exist as (x,y) pairs. Please advise

[the cyclist]

Can you assume (or test?) that x and y are independent and uncorrelated? If they are, then you could use ksdensity() separately for x and for y, then multiply the two resulting functions.

[Devinya Herath]

no. they are related. for example, (x1, y1) defines one position(location/point) in cartesian plane , (x2, y2) another position likewise. therefore, they are related. if kernel density is not suitable for this kind of density calculation, do you know any other way to do that? please explain.

[Image Analyst]

I'm not sure what you want to get the density of. Can't you just plot them as an image and look for a pattern that would indicate some correlation between different rows or different x and y? There are a variety of ways of determining uniformity of a 2D array.

[Devinya Herath]

yes. exactly that's what I want. These (xi,yi)points are expected to exhibit some patterns. For example, in some locations, the point density is high. I want to see these areas where point density is high.

[Devinya Herath]

I tried to use mvnpdf to find the probability density. Please see the following code. But, I get a series of zeros as the result of p.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

curs1 = exec(conn, 'select sp_x, sp_y from road_part6_trajectories_oneway2_new_segments_cartesian2');

format long;

curs1 = fetch(curs1);

r=rows(curs1);

AA = curs1.Data;

x = [AA{:,1}]';

y = [AA{:,2}]';

for i=1:5

xy(i,1)=x(i);

xy(i,2)=y(i);

end;

p=mvnpdf(xy)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[the cyclist]

From everything you have written so far, I can no longer tell for sure if you are trying to simply plot what the 2-d frequency distribution looks like, or if you are trying to do kernel density estimation (http://en.wikipedia.org/wiki/Kernel_density_estimation). I'll shortly edit my answer to give an example of the latter.

[Devinya Herath]

yes. that's right. I want to plot the 2-d frequency distribution. I have plot the histogram (using hist3) but I want a plot which clearly shows the dense areas. I initially thought that a kernel density would be suitable for this

[Bjorn Gustavsson]

Then you should start with a two-dimensional histogram. There are several contributions with such tools on the file exchange. To use the cyclist's code you have to show that the variables are independent. A quick look with a 2-D histogram will help you get a look of that.

Answer 2

[Tunde]

Very helpful. After changing x and y to variables from my data. I have this diagram

Is there any way to change the z-axis to normalized density estimates or probabilities?

[Joseph Byrnes]

normalize the area under the curve to 1