How to plot bivariate cumulative probability centred to a point?

Hi,

I am trying to create a probability contour plot from a scatter plot centred in the areas with the largest density.

Basically going from this:

To this:

Where:

Probability for a point to be in the red area is 68% Probability for a point to be in the yellow area is 95% Probability for a point to be in the green area is 99.7% Probability for a point to be in the blue area is 99.99%

(I am not sure about my colour choices, I should probably have drawn it from blue being 68% to red being 99.99%).

I have tried to use histcounts2 using the following code:

x = randn(1000,1) + randn(1000,1);
y= randn(1000,1) + randn(1000,1);
X = [x,y];

N=histcounts2(x,y);
N1 = histcounts2(x,y,'Normalization','countdensity');
N2 = histcounts2(x,y,'Normalization','cumcount');
N3 = histcounts2(x,y,'Normalization','probability');
N4 = histcounts2(x,y,'Normalization','pdf');
N5 = histcounts2(x,y,'Normalization','cdf');

figure

subplot(3,3,1);
scatter(x,y, '.');
title('Scatter plot of the data');

subplot(3,3,2); %to visualize the density for each bin
h = histogram2(x,y,'DisplayStyle','tile','ShowEmptyBins','on');
title('Density plot from the data - using histogram2');

subplot(3,3,3);
contour(N);
colorbar
title('contour plot - using histcounts2 - default normalization');

subplot(3,3,4);
contour(N1);
colorbar
title('contour plot - using histcounts2 - countdensity normalization');

subplot(3,3,5);
contour(N2);
colorbar
title('contour plot - using histcounts2 - cumcount normalization');

subplot(3,3,6);
contour(N3);
colorbar
title('contour plot - using histcounts2 - probability normalization');

subplot(3,3,7);
contour(N4);
colorbar
title('contour plot - using histcounts2 - pdf normalization');

subplot(3,3,8);
contour(N5);
colorbar
title('contour plot - using histcounts2 - cdf normalization');

I get the following outcome from running this code (note that outcomes might vary due to the random nature of the first two lines of code):

N =
     0     0     0     0     0     1     0     0     0     0
     0     0     0     1     0     1     0     0     0     0
     0     0     1     7     2     3     1     1     0     0
     0     1     1    15    15    14    12     0     0     0
     2     3     7    31    43    40    32    10     1     0
     0     2    22    40    65    64    46    22     4     1
     0     3    22    42    71    61    38     8     5     1
     0     3     7    21    46    51    25    11     1     1
     0     0     3     6    15    11    10     6     0     0
     0     0     2     2     5     1     4     2     0     0
     0     0     0     1     2     1     0     1     0     0

The result is as expected, as the 'cumcount' or 'cdf' starts its cumulative count from the origin of the matrix (cf. Matlab documentation - histcounts2 -> Output arguments -> N> ).

Is there any way to 'cumcount' or 'cdf' centered in the most dense area so that I have an output similar to the one hand drawn? Basically, I would like to centre it to (5,7) for N, as it is its largest value (71).

Thank you in advance,

[Benoit Espinola]

Would it make sense if I made:

x = randn(1000,1) + randn(1000,1);
y= randn(1000,1) + randn(1000,1);
X = [x,y];
N = 1-histcounts2(x,y,'Normalization','pdf');
figure();
contour(N);
colorbar

?

I get this:

[Benoit Espinola]

Running the last comment several times, I get this figure:

implying I have two areas with 93% probability... I would think that the two areas combined have a 93% probability to find a point of the scatter plot (and not each area individually).

Am I wrong?